Daily Papers

We introduce CamMimic, an innovative algorithm tailored for dynamic video editing needs. It is designed to seamlessly transfer the camera motion observed in a given reference video onto any scene of the user's choice in a zero-shot manner without requiring any additional data. Our algorithm achieves this using a two-phase strategy by leveraging a text-to-video diffusion model. In the first phase, we develop a multi-concept learning method using a combination of LoRA layers and an orthogonality loss to capture and understand the underlying spatial-temporal characteristics of the reference video as well as the spatial features of the user's desired scene. The second phase proposes a unique homography-based refinement strategy to enhance the temporal and spatial alignment of the generated video. We demonstrate the efficacy of our method through experiments conducted on a dataset containing combinations of diverse scenes and reference videos containing a variety of camera motions. In the absence of an established metric for assessing camera motion transfer between unrelated scenes, we propose CameraScore, a novel metric that utilizes homography representations to measure camera motion similarity between the reference and generated videos. Extensive quantitative and qualitative evaluations demonstrate that our approach generates high-quality, motion-enhanced videos. Additionally, a user study reveals that 70.31% of participants preferred our method for scene preservation, while 90.45% favored it for motion transfer. We hope this work lays the foundation for future advancements in camera motion transfer across different scenes.

Medical vision-language models enable co-learning and integrating features from medical imaging and clinical text. However, these models are not easy to train and the latent representation space can be complex. Here we propose a novel way for pre-training and regularising medical vision-language models. The proposed method, named Medical vision-language pre-training with Frozen language models and Latent spAce Geometry optimization (M-FLAG), leverages a frozen language model for training stability and efficiency and introduces a novel orthogonality loss to harmonize the latent space geometry. We demonstrate the potential of the pre-trained model on three downstream tasks: medical image classification, segmentation, and object detection. Extensive experiments across five public datasets demonstrate that M-FLAG significantly outperforms existing medical vision-language pre-training approaches and reduces the number of parameters by 78\%. Notably, M-FLAG achieves outstanding performance on the segmentation task while using only 1\% of the RSNA dataset, even outperforming ImageNet pre-trained models that have been fine-tuned using 100\% of the data.

Deep neural networks have achieved remarkable performance on a range of classification tasks, with softmax cross-entropy (CE) loss emerging as the de-facto objective function. The CE loss encourages features of a class to have a higher projection score on the true class-vector compared to the negative classes. However, this is a relative constraint and does not explicitly force different class features to be well-separated. Motivated by the observation that ground-truth class representations in CE loss are orthogonal (one-hot encoded vectors), we develop a novel loss function termed `Orthogonal Projection Loss' (OPL) which imposes orthogonality in the feature space. OPL augments the properties of CE loss and directly enforces inter-class separation alongside intra-class clustering in the feature space through orthogonality constraints on the mini-batch level. As compared to other alternatives of CE, OPL offers unique advantages e.g., no additional learnable parameters, does not require careful negative mining and is not sensitive to the batch size. Given the plug-and-play nature of OPL, we evaluate it on a diverse range of tasks including image recognition (CIFAR-100), large-scale classification (ImageNet), domain generalization (PACS) and few-shot learning (miniImageNet, CIFAR-FS, tiered-ImageNet and Meta-dataset) and demonstrate its effectiveness across the board. Furthermore, OPL offers better robustness against practical nuisances such as adversarial attacks and label noise. Code is available at: https://github.com/kahnchana/opl.

A deep hashing model typically has two main learning objectives: to make the learned binary hash codes discriminative and to minimize a quantization error. With further constraints such as bit balance and code orthogonality, it is not uncommon for existing models to employ a large number (>4) of losses. This leads to difficulties in model training and subsequently impedes their effectiveness. In this work, we propose a novel deep hashing model with only a single learning objective. Specifically, we show that maximizing the cosine similarity between the continuous codes and their corresponding binary orthogonal codes can ensure both hash code discriminativeness and quantization error minimization. Further, with this learning objective, code balancing can be achieved by simply using a Batch Normalization (BN) layer and multi-label classification is also straightforward with label smoothing. The result is an one-loss deep hashing model that removes all the hassles of tuning the weights of various losses. Importantly, extensive experiments show that our model is highly effective, outperforming the state-of-the-art multi-loss hashing models on three large-scale instance retrieval benchmarks, often by significant margins. Code is available at https://github.com/kamwoh/orthohash

Learning discriminative shape representations is a crucial issue for large-scale 3D shape retrieval. In this paper, we propose the Collaborative Inner Product Loss (CIP Loss) to obtain ideal shape embedding that discriminative among different categories and clustered within the same class. Utilizing simple inner product operation, CIP loss explicitly enforces the features of the same class to be clustered in a linear subspace, while inter-class subspaces are constrained to be at least orthogonal. Compared to previous metric loss functions, CIP loss could provide more clear geometric interpretation for the embedding than Euclidean margin, and is easy to implement without normalization operation referring to cosine margin. Moreover, our proposed loss term can combine with other commonly used loss functions and can be easily plugged into existing off-the-shelf architectures. Extensive experiments conducted on the two public 3D object retrieval datasets, ModelNet and ShapeNetCore 55, demonstrate the effectiveness of our proposal, and our method has achieved state-of-the-art results on both datasets.

Sparse autoencoders (SAEs) are widely used in mechanistic interpretability research for large language models; however, the state-of-the-art method of using k-sparse autoencoders lacks a theoretical grounding for selecting the hyperparameter k that represents the number of nonzero activations, often denoted by ell_0. In this paper, we reveal a theoretical link that the ell_2-norm of the sparse feature vector can be approximated with the ell_2-norm of the dense vector with a closed-form error, which allows sparse autoencoders to be trained without the need to manually determine ell_0. Specifically, we validate two applications of our theoretical findings. First, we introduce a new methodology that can assess the feature activations of pre-trained SAEs by computing the theoretically expected value from the input embedding, which has been overlooked by existing SAE evaluation methods and loss functions. Second, we introduce a novel activation function, top-AFA, which builds upon our formulation of approximate feature activation (AFA). This function enables top-k style activation without requiring a constant hyperparameter k to be tuned, dynamically determining the number of activated features for each input. By training SAEs on three intermediate layers to reconstruct GPT2 hidden embeddings for over 80 million tokens from the OpenWebText dataset, we demonstrate the empirical merits of this approach and compare it with current state-of-the-art k-sparse autoencoders. Our code is available at: https://github.com/SewoongLee/top-afa-sae.

Retrieval systems rely on representations learned by increasingly powerful models. However, due to the high training cost and inconsistencies in learned representations, there is significant interest in facilitating communication between representations and ensuring compatibility across independently trained neural networks. In the literature, two primary approaches are commonly used to adapt different learned representations: affine transformations, which adapt well to specific distributions but can significantly alter the original representation, and orthogonal transformations, which preserve the original structure with strict geometric constraints but limit adaptability. A key challenge is adapting the latent spaces of updated models to align with those of previous models on downstream distributions while preserving the newly learned representation spaces. In this paper, we impose a relaxed orthogonality constraint, namely λ-Orthogonality regularization, while learning an affine transformation, to obtain distribution-specific adaptation while retaining the original learned representations. Extensive experiments across various architectures and datasets validate our approach, demonstrating that it preserves the model's zero-shot performance and ensures compatibility across model updates. Code available at: https://github.com/miccunifi/lambda_orthogonality.git{https://github.com/miccunifi/lambda\_orthogonality}.

Imposing orthogonality on the layers of neural networks is known to facilitate the learning by limiting the exploding/vanishing of the gradient; decorrelate the features; improve the robustness. This paper studies the theoretical properties of orthogonal convolutional layers.We establish necessary and sufficient conditions on the layer architecture guaranteeing the existence of an orthogonal convolutional transform. The conditions prove that orthogonal convolutional transforms exist for almost all architectures used in practice for 'circular' padding.We also exhibit limitations with 'valid' boundary conditions and 'same' boundary conditions with zero-padding.Recently, a regularization term imposing the orthogonality of convolutional layers has been proposed, and impressive empirical results have been obtained in different applications (Wang et al. 2020).The second motivation of the present paper is to specify the theory behind this.We make the link between this regularization term and orthogonality measures. In doing so, we show that this regularization strategy is stable with respect to numerical and optimization errors and that, in the presence of small errors and when the size of the signal/image is large, the convolutional layers remain close to isometric.The theoretical results are confirmed with experiments and the landscape of the regularization term is studied. Experiments on real data sets show that when orthogonality is used to enforce robustness, the parameter multiplying the regularization termcan be used to tune a tradeoff between accuracy and orthogonality, for the benefit of both accuracy and robustness.Altogether, the study guarantees that the regularization proposed in Wang et al. (2020) is an efficient, flexible and stable numerical strategy to learn orthogonal convolutional layers.

We propose an image-to-image translation framework for facial attribute editing with disentangled interpretable latent directions. Facial attribute editing task faces the challenges of targeted attribute editing with controllable strength and disentanglement in the representations of attributes to preserve the other attributes during edits. For this goal, inspired by the latent space factorization works of fixed pretrained GANs, we design the attribute editing by latent space factorization, and for each attribute, we learn a linear direction that is orthogonal to the others. We train these directions with orthogonality constraints and disentanglement losses. To project images to semantically organized latent spaces, we set an encoder-decoder architecture with attention-based skip connections. We extensively compare with previous image translation algorithms and editing with pretrained GAN works. Our extensive experiments show that our method significantly improves over the state-of-the-arts.

Model merging has recently emerged as a cost-efficient paradigm for multi-task learning. Among current approaches, task arithmetic stands out for its simplicity and effectiveness. In this paper, we motivate the effectiveness of task vectors by linking them to multi-task gradients. We show that in a single-epoch scenario, task vectors are mathematically equivalent to the gradients obtained via gradient descent in a multi-task setting, and still approximate these gradients in subsequent epochs. Furthermore, we show that task vectors perform optimally when equality is maintained, and their effectiveness is largely driven by the first epoch's gradient. Building on this insight, we propose viewing model merging as a single step in an iterative process that Alternates between Tuning and Merging (ATM). This method acts as a bridge between model merging and multi-task gradient descent, achieving state-of-the-art results with the same data and computational requirements. We extensively evaluate ATM across diverse settings, achieving up to 20% higher accuracy in computer vision and NLP tasks, compared to the best baselines. Finally, we provide both empirical and theoretical support for its effectiveness, demonstrating increased orthogonality between task vectors and proving that ATM minimizes an upper bound on the loss obtained by jointly finetuning all tasks.

We propose a unified framework to study policy evaluation (PE) and the associated temporal difference (TD) methods for reinforcement learning in continuous time and space. We show that PE is equivalent to maintaining the martingale condition of a process. From this perspective, we find that the mean--square TD error approximates the quadratic variation of the martingale and thus is not a suitable objective for PE. We present two methods to use the martingale characterization for designing PE algorithms. The first one minimizes a "martingale loss function", whose solution is proved to be the best approximation of the true value function in the mean--square sense. This method interprets the classical gradient Monte-Carlo algorithm. The second method is based on a system of equations called the "martingale orthogonality conditions" with test functions. Solving these equations in different ways recovers various classical TD algorithms, such as TD(lambda), LSTD, and GTD. Different choices of test functions determine in what sense the resulting solutions approximate the true value function. Moreover, we prove that any convergent time-discretized algorithm converges to its continuous-time counterpart as the mesh size goes to zero, and we provide the convergence rate. We demonstrate the theoretical results and corresponding algorithms with numerical experiments and applications.

Beyond minimizing a single training loss, many deep learning estimation pipelines rely on an auxiliary objective to quantify and encourage desirable properties of the model (e.g. performance on another dataset, robustness, agreement with a prior). Although the simplest approach to incorporating an auxiliary loss is to sum it with the training loss as a regularizer, recent works have shown that one can improve performance by blending the gradients beyond a simple sum; this is known as gradient surgery. We cast the problem as a constrained minimization problem where the auxiliary objective is minimized among the set of minimizers of the training loss. To solve this bilevel problem, we follow a parameter update direction that combines the training loss gradient and the orthogonal projection of the auxiliary gradient to the training gradient. In a setting where gradients come from mini-batches, we explain how, using a moving average of the training loss gradients, we can carefully maintain this critical orthogonality property. We demonstrate that our method, Bloop, can lead to much better performances on NLP and vision experiments than other gradient surgery methods without EMA.

To bridge the ever increasing gap between deep neural networks' complexity and hardware capability, network quantization has attracted more and more research attention. The latest trend of mixed precision quantization takes advantage of hardware's multiple bit-width arithmetic operations to unleash the full potential of network quantization. However, this also results in a difficult integer programming formulation, and forces most existing approaches to use an extremely time-consuming search process even with various relaxations. Instead of solving a problem of the original integer programming, we propose to optimize a proxy metric, the concept of network orthogonality, which is highly correlated with the loss of the integer programming but also easy to optimize with linear programming. This approach reduces the search time and required data amount by orders of magnitude, with little compromise on quantization accuracy. Specifically, we achieve 72.08% Top-1 accuracy on ResNet-18 with 6.7Mb, which does not require any searching iterations. Given the high efficiency and low data dependency of our algorithm, we used it for the post-training quantization, which achieve 71.27% Top-1 accuracy on MobileNetV2 with only 1.5Mb. Our code is available at https://github.com/MAC-AutoML/OMPQ.

Dictionary learning is a widely used unsupervised learning method in signal processing and machine learning. Most existing works of dictionary learning are in an offline manner. There are mainly two offline ways for dictionary learning. One is to do an alternative optimization of both the dictionary and the sparse code; the other way is to optimize the dictionary by restricting it over the orthogonal group. The latter one is called orthogonal dictionary learning which has a lower complexity implementation, hence, it is more favorable for lowcost devices. However, existing schemes on orthogonal dictionary learning only work with batch data and can not be implemented online, which is not applicable for real-time applications. This paper proposes a novel online orthogonal dictionary scheme to dynamically learn the dictionary from streaming data without storing the historical data. The proposed scheme includes a novel problem formulation and an efficient online algorithm design with convergence analysis. In the problem formulation, we relax the orthogonal constraint to enable an efficient online algorithm. In the algorithm design, we propose a new Frank-Wolfe-based online algorithm with a convergence rate of O(ln t/t^(1/4)). The convergence rate in terms of key system parameters is also derived. Experiments with synthetic data and real-world sensor readings demonstrate the effectiveness and efficiency of the proposed online orthogonal dictionary learning scheme.

As neural networks become deeper, the redundancy within their parameters increases. This phenomenon has led to several methods that attempt to reduce the correlation between convolutional filters. We propose a computationally efficient regularization technique that encourages orthonormality between groups of filters within the same layer. Our experiments show that when incorporated into recent adaptation methods for diffusion models and vision transformers (ViTs), this regularization improves performance on downstream tasks. We further show improved robustness when group orthogonality is enforced during adversarial training. Our code is available at https://github.com/YoavKurtz/GOR.

We present Neural Spectral Methods, a technique to solve parametric Partial Differential Equations (PDEs), grounded in classical spectral methods. Our method uses orthogonal bases to learn PDE solutions as mappings between spectral coefficients. In contrast to current machine learning approaches which enforce PDE constraints by minimizing the numerical quadrature of the residuals in the spatiotemporal domain, we leverage Parseval's identity and introduce a new training strategy through a spectral loss. Our spectral loss enables more efficient differentiation through the neural network, and substantially reduces training complexity. At inference time, the computational cost of our method remains constant, regardless of the spatiotemporal resolution of the domain. Our experimental results demonstrate that our method significantly outperforms previous machine learning approaches in terms of speed and accuracy by one to two orders of magnitude on multiple different problems. When compared to numerical solvers of the same accuracy, our method demonstrates a 10times increase in performance speed.

Orthogonalized-update optimizers such as Muon improve training of matrix-valued parameters, but existing extensions typically either rescale updates after orthogonalization or use heavier whitening-based preconditioners before it. We introduce {\method}, a lightweight family of pre-orthogonalization equilibration schemes for Muon with three forms: two-sided row/column normalization (RC), row normalization (R), and column normalization (C). By rebalancing the momentum matrix before finite-step Newton--Schulz orthogonalization, {\method} improves the geometry seen by orthogonalization. We show that finite-step orthogonalization is governed by the input spectrum, especially stable rank and condition number, and that row/column normalization acts as a zeroth-order surrogate for whitening. For hidden matrix weights, R is the default variant. Theoretically, {\method} (R) retains the standard mathcal O(T^{-1/4}) Muon-type nonconvex stationarity guarantee with decoupled weight decay and a horizon-free diminishing learning-rate schedule, and extends it to finite-step NS5 up to an explicit inexactness constant. In LLaMA2 pretraining on C4, {\method} (R) consistently outperforms Muon on 130M, 350M, and 1B models, with faster convergence and lower validation perplexity. The code is available at the https://github.com/MaeChd/muon-eq{MuonEq codebase}.

As generative models become increasingly powerful and pervasive, the ability to unlearn specific data, whether due to privacy concerns, legal requirements, or the correction of harmful content, has become increasingly important. Unlike in conventional training, where data are accumulated and knowledge is reinforced, unlearning aims to selectively remove the influence of particular data points without costly retraining from scratch. To be effective and reliable, such algorithms need to achieve (i) forgetting of the undesired data, (ii) preservation of the quality of the generation, (iii) preservation of the influence of the desired training data on the model parameters, and (iv) small number of training steps. We propose fast unlearning algorithms based on loss gradient orthogonalization. We show that our algorithms are able to forget data while maintaining the fidelity of the original model. Using MNIST and CelebA data, we demonstrate that our algorithms achieve orders of magnitude faster unlearning times than their predecessors, such as gradient surgery.

Optimization with matrix gradient orthogonalization has recently demonstrated impressive results in the training of deep neural networks (Jordan et al., 2024; Liu et al., 2025). In this paper, we provide a theoretical analysis of this approach. In particular, we show that the orthogonalized gradient method can be seen as a first-order trust-region optimization method, where the trust-region is defined in terms of the matrix spectral norm. Motivated by this observation, we develop the stochastic non-Euclidean trust-region gradient method with momentum, which recovers the Muon optimizer (Jordan et al., 2024) as a special case, along with normalized SGD and signSGD with momentum (Cutkosky and Mehta, 2020; Sun et al., 2023). In addition, we prove state-of-the-art convergence results for the proposed algorithm in a range of scenarios, which involve arbitrary non-Euclidean norms, constrained and composite problems, and non-convex, star-convex, first- and second-order smooth functions. Finally, our theoretical findings provide an explanation for several practical observations, including the practical superiority of Muon compared to the Orthogonal-SGDM algorithm of Tuddenham et al. (2022) and the importance of weight decay in the training of large-scale language models.

The optimization of large language models (LLMs) remains a critical challenge, particularly as model scaling exacerbates sensitivity to algorithmic imprecision and training instability. Recent advances in optimizers have improved convergence efficiency through momentum orthogonalization, but suffer from two key robustness limitations: dimensional fragility in orthogonalization precision and vulnerability to outlier-induced noise. To address these robustness challenges, we introduce ROOT, a Robust Orthogonalized Optimizer that enhances training stability through dual robustness mechanisms. First, we develop a dimension-robust orthogonalization scheme using adaptive Newton iterations with fine-grained coefficients tailored to specific matrix sizes, ensuring consistent precision across diverse architectural configurations. Second, we introduce an optimization-robust framework via proximal optimization that suppresses outlier noise while preserving meaningful gradient directions. Extensive experiments demonstrate that ROOT achieves significantly improved robustness, with faster convergence and superior final performance compared to both Muon and Adam-based optimizers, particularly in noisy and non-convex scenarios. Our work establishes a new paradigm for developing robust and precise optimizers capable of handling the complexities of modern large-scale model training. The code will be available at https://github.com/huawei-noah/noah-research/tree/master/ROOT.

While Large Language Models (LLMs) have achieved remarkable capabilities, they unintentionally memorize sensitive data, posing critical privacy and security risks. Machine unlearning is pivotal for mitigating these risks, yet existing paradigms face a fundamental dilemma: aggressive unlearning often induces catastrophic forgetting that degrades model utility, whereas conservative strategies risk superficial forgetting, leaving models vulnerable to adversarial recovery. To address this trade-off, we propose AGT^{AO} (Adversarial Gating Training with Adaptive Orthogonality), a unified framework designed to reconcile robust erasure with utility preservation. Specifically, our approach introduces Adaptive Orthogonality (AO) to dynamically mitigate geometric gradient conflicts between forgetting and retention objectives, thereby minimizing unintended knowledge degradation. Concurrently, Adversarial Gating Training (AGT) formulates unlearning as a latent-space min-max game, employing a curriculum-based gating mechanism to simulate and counter internal recovery attempts. Extensive experiments demonstrate that AGT^{AO} achieves a superior trade-off between unlearning efficacy (KUR approx 0.01) and model utility (MMLU 58.30). Code is available at https://github.com/TiezMind/AGT-unlearning.

Initialization of parameters in deep neural networks has been shown to have a big impact on the performance of the networks (Mishkin & Matas, 2015). The initialization scheme devised by He et al, allowed convolution activations to carry a constrained mean which allowed deep networks to be trained effectively (He et al., 2015a). Orthogonal initializations and more generally orthogonal matrices in standard recurrent networks have been proved to eradicate the vanishing and exploding gradient problem (Pascanu et al., 2012). Majority of current initialization schemes do not take fully into account the intrinsic structure of the convolution operator. Using the duality of the Fourier transform and the convolution operator, Convolution Aware Initialization builds orthogonal filters in the Fourier space, and using the inverse Fourier transform represents them in the standard space. With Convolution Aware Initialization we noticed not only higher accuracy and lower loss, but faster convergence. We achieve new state of the art on the CIFAR10 dataset, and achieve close to state of the art on various other tasks.

Model dimension (d_{model}) is a fundamental hyperparameter in transformer language models, yet its role in setting the geometric limits of feature representation remains under-explored. Grounded in the Linear Representation and Superposition Hypotheses - which propose that models encode features as near-orthogonal directions in latent space - we develop a framework for estimating how many such directions a model can support. We first establish the embedding matrix as a measurable proxy for near-orthogonality constraints across the latent space: the boundary between meaningful token relationships and incidental similarity in the pairwise cosine similarity distribution gives a concrete estimate of the model's accepted deviation varepsilon from perfect orthogonality. Applying this metric across dozens of open-source models reveals two classes: models with high varepsilon whose embeddings lack near-orthogonal structure, and models with low varepsilon that maintain it. We then show that the standard Johnson-Lindenstrauss lemma greatly underestimates the packing efficiency of trained representations, and derive an adjusted capacity formula in which the number of near-orthogonal directions depends on the ratio of vectors to dimensions (k/d) rather than the raw count - a single modification that cuts prediction error by two orders of magnitude with no extra parameters. Combining these results, we define representational capacity as an upper bound on the number of distinguishable directions available for features and embeddings in a model's latent space. Capacity is exponentially sensitive to varepsilon, and larger models favor tighter orthogonality constraints over maximizing raw capacity - a pattern compatible with several explanations (a stability-capacity trade-off, a ceiling on usable concepts, or confounds with model scale) that we leave to future work.

We propose Orthogonal Monte Carlo Dropout, a mechanism that enforces strict orthogonality when combining sparse semantic vectors without extra time complexity. Low-Rank Adaptation (LoRA), a popular fine-tuning method for large models, typically trains a module to represent a specific concept such as an object or a style. When multiple LoRA modules are merged, for example to generate an object in a particular style, their outputs (semantic vectors) may interfere with each other. Our method guarantees that merged LoRA modules remain orthogonal and thus free from direct interference. However, empirical analysis reveals that such orthogonality does not lead to the semantic disentanglement highlighted in prior work on compositional adaptation. This finding suggests that inter-LoRA orthogonality alone may be insufficient for achieving true semantic compositionality, prompting a re-examination of its role in adapter merging.

With the increasingly powerful performances and enormous scales of Pretrained Language Models (PLMs), promoting parameter efficiency in fine-tuning has become a crucial need for effective and efficient adaptation to various downstream tasks. One representative line of fine-tuning methods is Orthogonal Fine-tuning (OFT), which rigorously preserves the angular distances within the parameter space to preserve the pretrained knowledge. Despite the empirical effectiveness, OFT still suffers low parameter efficiency at O(d^2) and limited capability of downstream adaptation. Inspired by Givens rotation, in this paper, we proposed quasi-Givens Orthogonal Fine-Tuning (qGOFT) to address the problems. We first use O(d) Givens rotations to accomplish arbitrary orthogonal transformation in SO(d) with provable equivalence, reducing parameter complexity from O(d^2) to O(d). Then we introduce flexible norm and relative angular adjustments under soft orthogonality regularization to enhance the adaptation capability of downstream semantic deviations. Extensive experiments on various tasks and PLMs validate the effectiveness of our methods.

Orthogonality-based optimizers, such as Muon, have recently shown strong performance across large-scale training and community-driven efficiency challenges. However, these methods rely on a costly gradient orthogonalization step. Even efficient iterative approximations such as Newton-Schulz remain expensive, typically requiring dozens of matrix multiplications to converge. We introduce a preconditioning procedure that accelerates Newton-Schulz convergence and reduces its computational cost. We evaluate its impact and show that the overhead of our preconditioning can be made negligible. Furthermore, the faster convergence it enables allows us to remove one iteration out of the usual five without degrading approximation quality. Our publicly available implementation achieves up to a 2.8x speedup in the Newton-Schulz approximation. We also show that this has a direct impact on end-to-end training runtime with 5-10% improvement in realistic training scenarios across two efficiency-focused tasks. On challenging language or vision tasks, we validate that our method maintains equal or superior model performance while improving runtime. Crucially, these improvements require no hyperparameter tuning and can be adopted as a simple drop-in replacement. Our code is publicly available on github.

The problem of computing optimal orthogonal approximation to a given matrix has attracted growing interest in machine learning. Notable applications include the recent Muon optimizer or Riemannian optimization on the Stiefel manifold. Among existing approaches, the Newton-Schulz iteration has emerged as a particularly effective solution, as it relies solely on matrix multiplications and thus achieves high computational efficiency on GPU hardware. Despite its efficiency, the method has inherent limitations - its coefficients are fixed and thus not optimized for a given matrix. In this paper we address this issue by proposing a Chebyshev-optimized version of Newton-Schulz (CANS). Based on the Chebyshev's alternance theorem, we theoretically derive optimal coefficients for the 3-rd order Newton-Schulz iteration and apply a Remez algorithm to compute optimal higher-degree polynomials. We leverage these polynomials to construct controlled approximate orthogonalization schemes, which is of interest in deep learning applications. Practically, we demonstrate the method's effectiveness in two key applications: orthogonalization in the Muon optimizer, and providing an efficient retraction alternative for Riemannian optimization on the Stiefel manifold.

Neural operators, as an efficient surrogate model for learning the solutions of PDEs, have received extensive attention in the field of scientific machine learning. Among them, attention-based neural operators have become one of the mainstreams in related research. However, existing approaches overfit the limited training data due to the considerable number of parameters in the attention mechanism. To address this, we develop an orthogonal attention based on the eigendecomposition of the kernel integral operator and the neural approximation of eigenfunctions. The orthogonalization naturally poses a proper regularization effect on the resulting neural operator, which aids in resisting overfitting and boosting generalization. Experiments on six standard neural operator benchmark datasets comprising both regular and irregular geometries show that our method can outperform competing baselines with decent margins.

The trade-off between regret and computational cost is a fundamental problem for online kernel regression, and previous algorithms worked on the trade-off can not keep optimal regret bounds at a sublinear computational complexity. In this paper, we propose two new algorithms, AOGD-ALD and NONS-ALD, which can keep nearly optimal regret bounds at a sublinear computational complexity, and give sufficient conditions under which our algorithms work. Both algorithms dynamically maintain a group of nearly orthogonal basis used to approximate the kernel mapping, and keep nearly optimal regret bounds by controlling the approximate error. The number of basis depends on the approximate error and the decay rate of eigenvalues of the kernel matrix. If the eigenvalues decay exponentially, then AOGD-ALD and NONS-ALD separately achieves a regret of O(L(f)) and O(d_{eff}(mu)T) at a computational complexity in O(ln^2{T}). If the eigenvalues decay polynomially with degree pgeq 1, then our algorithms keep the same regret bounds at a computational complexity in o(T) in the case of p>4 and pgeq 10, respectively. L(f) is the cumulative losses of f and d_{eff}(mu) is the effective dimension of the problem. The two regret bounds are nearly optimal and are not comparable.

Orthogonal gradient updates have emerged as a promising direction in optimization for machine learning. However, traditional approaches such as SVD/QR decomposition incur prohibitive computational costs of O(n^3) and underperform compared to well-tuned SGD with momentum, since momentum is applied only after strict orthogonalization. Recent advances, such as Muon, improve efficiency by applying momentum before orthogonalization and producing semi-orthogonal matrices via Newton-Schulz iterations, reducing complexity to O(n^2). Nevertheless, quadratic costs remain a bottleneck. In this work, we study the semi-orthogonal properties of momentum-based updates and develop a method to bound momentum updates under a spectral-norm trust region, preserving directional information without requiring explicit semi-orthogonalization. We propose AuON (Alternative Unit-norm momentum updates by Normalized nonlinear scaling), a linear-time optimizer that achieves strong performance without constructing semi-orthogonal matrices, while preserving structural alignment and reconditioning ill-posed updates. Our approach combines hyperbolic-cosine RMS scaling transformations with normalization, demonstrating both effectiveness and computational efficiency compared to Newton-Schulz methods. We further introduce a hybrid variant (Hybrid-AuON) that applies a single Newton-Schulz iteration. Experiments across vision and language benchmarks show that AuON and its hybrid variant achieve performance comparable to strong baselines such as AdamW and Muon. Code is available at: https://github.com/ryyzn9/AuON

While following different technical routes, both low-rank and orthogonal adaptation techniques can efficiently adapt large-scale pre-training models in specific tasks or domains based on a small piece of trainable parameters. In this study, we bridge the gap between these two techniques, proposing a simple but effective adaptation method based on Householder reflections. Given a pre-trained model, our method fine-tunes its layers by multiplying each frozen weight matrix with an orthogonal matrix constructed by a chain of learnable Householder reflections (HRs). This HR-based orthogonal fine-tuning is equivalent to an adaptive low-rank adaptation. Moreover, we show that the orthogonality of the reflection planes corresponding to the HRs impacts the model capacity and regularity. The analysis motivates us to regularize the orthogonality of the HRs, leading to different implementations of the proposed Householder reflection adaptation (HRA) method. Compared with state-of-the-art methods, HRA achieves superior performance with fewer learnable parameters when adapting large language models and conditional image generators. The code is available at https://github.com/DaShenZi721/HRA

We initiate the study of proper losses for evaluating generative models in the discrete setting. Unlike traditional proper losses, we treat both the generative model and the target distribution as black-boxes, only assuming ability to draw i.i.d. samples. We define a loss to be black-box proper if the generative distribution that minimizes expected loss is equal to the target distribution. Using techniques from statistical estimation theory, we give a general construction and characterization of black-box proper losses: they must take a polynomial form, and the number of draws from the model and target distribution must exceed the degree of the polynomial. The characterization rules out a loss whose expectation is the cross-entropy between the target distribution and the model. By extending the construction to arbitrary sampling schemes such as Poisson sampling, however, we show that one can construct such a loss.

Supervised contrastive loss (SCL) is a competitive and often superior alternative to the cross-entropy loss for classification. While prior studies have demonstrated that both losses yield symmetric training representations under balanced data, this symmetry breaks under class imbalances. This paper presents an intriguing discovery: the introduction of a ReLU activation at the final layer effectively restores the symmetry in SCL-learned representations. We arrive at this finding analytically, by establishing that the global minimizers of an unconstrained features model with SCL loss and entry-wise non-negativity constraints form an orthogonal frame. Extensive experiments conducted across various datasets, architectures, and imbalance scenarios corroborate our finding. Importantly, our experiments reveal that the inclusion of the ReLU activation restores symmetry without compromising test accuracy. This constitutes the first geometry characterization of SCL under imbalances. Additionally, our analysis and experiments underscore the pivotal role of batch selection strategies in representation geometry. By proving necessary and sufficient conditions for mini-batch choices that ensure invariant symmetric representations, we introduce batch-binding as an efficient strategy that guarantees these conditions hold.

Orthogonal finetuning (OFT) offers highly parameter-efficient adaptation while preventing catastrophic forgetting, but its high runtime and memory demands limit practical deployment. We identify the core computational bottleneck in OFT as its weight-centric implementation, which relies on costly matrix-matrix multiplications with cubic complexity. To overcome this, we propose OFTv2, an input-centric reformulation that instead uses matrix-vector multiplications (i.e., matrix-free computation), reducing the computational cost to quadratic. We further introduce the Cayley-Neumann parameterization, an efficient orthogonal parameterization that approximates the matrix inversion in Cayley transform via a truncated Neumann series. These modifications allow OFTv2 to achieve up to 10x faster training and 3x lower GPU memory usage without compromising performance. In addition, we extend OFTv2 to support finetuning quantized foundation models and show that it outperforms the popular QLoRA in training stability, efficiency, and memory usage.

Matrix-based optimizers have attracted growing interest for improving LLM training efficiency, with significant progress centered on orthogonalization/whitening based methods. While yielding substantial performance gains, a fundamental question arises: can we develop new paradigms beyond orthogonalization, pushing the efficiency frontier further? We present Adaptively Rotated Optimization (ARO, a new matrix optimization framework that treats gradient rotation as a first class design principle. ARO accelerates LLM training by performing normed steepest descent in a rotated coordinate system, where the rotation is determined by a novel norm-informed policy. This perspective yields update rules that go beyond existing orthogonalization and whitening optimizers, improving sample efficiency in practice. To make comparisons reliable, we propose a rigorously controlled benchmarking protocol that reduces confounding and bias. Under this protocol, ARO consistently outperforms AdamW (by 1.3 sim1.35times) and orthogonalization methods (by 1.1sim1.15times) in LLM pretraining at up to 8B activated parameters, and up to 8times overtrain budget, without evidence of diminishing returns. Finally, we discuss how ARO can be reformulated as a symmetry-aware optimizer grounded in rotational symmetries of residual streams, motivating advanced designs that enable computationally efficient exploitation of cross-layer/cross module couplings.

Kernel online convex optimization (KOCO) is a framework combining the expressiveness of non-parametric kernel models with the regret guarantees of online learning. First-order KOCO methods such as functional gradient descent require only O(t) time and space per iteration, and, when the only information on the losses is their convexity, achieve a minimax optimal O(T) regret. Nonetheless, many common losses in kernel problems, such as squared loss, logistic loss, and squared hinge loss posses stronger curvature that can be exploited. In this case, second-order KOCO methods achieve O(log(Det(K))) regret, which we show scales as O(d_{eff}log T), where d_{eff} is the effective dimension of the problem and is usually much smaller than O(T). The main drawback of second-order methods is their much higher O(t^2) space and time complexity. In this paper, we introduce kernel online Newton step (KONS), a new second-order KOCO method that also achieves O(d_{eff}log T) regret. To address the computational complexity of second-order methods, we introduce a new matrix sketching algorithm for the kernel matrix K_t, and show that for a chosen parameter γleq 1 our Sketched-KONS reduces the space and time complexity by a factor of γ^2 to O(t^2γ^2) space and time per iteration, while incurring only 1/γ times more regret.

Lasso and Ridge are important minimization problems in machine learning and statistics. They are versions of linear regression with squared loss where the vector θinR^d of coefficients is constrained in either ell_1-norm (for Lasso) or in ell_2-norm (for Ridge). We study the complexity of quantum algorithms for finding varepsilon-minimizers for these minimization problems. We show that for Lasso we can get a quadratic quantum speedup in terms of d by speeding up the cost-per-iteration of the Frank-Wolfe algorithm, while for Ridge the best quantum algorithms are linear in d, as are the best classical algorithms. As a byproduct of our quantum lower bound for Lasso, we also prove the first classical lower bound for Lasso that is tight up to polylog-factors.

While orthogonal subspace methods try to mitigate task interference in Continual Learning (CL), they often suffer from energy diffusion across the basis, hindering knowledge compaction and exhausting capacity for future tasks. We observe that output feature drift induced by parameter updates is inherently low-rank, and theoretically prove that preserving parameters along the principal directions of this drift minimizes the output reconstruction error. Motivated by this, we propose Energy-Concentrated and Energy-Ordered Low-Rank Adaptation (E^2-LoRA). By explicitly ordering and concentrating knowledge into leading ranks, E^2-LoRA frees capacity for subsequent tasks. Furthermore, we design a dynamic rank allocation strategy to balance stability and plasticity by jointly optimizing energy retention and model plasticity. Extensive experiments across multiple benchmarks demonstrate that E^2-LoRA achieves state-of-the-art performance.

We present AERO, a audio super-resolution model that processes speech and music signals in the spectral domain. AERO is based on an encoder-decoder architecture with U-Net like skip connections. We optimize the model using both time and frequency domain loss functions. Specifically, we consider a set of reconstruction losses together with perceptual ones in the form of adversarial and feature discriminator loss functions. To better handle phase information the proposed method operates over the complex-valued spectrogram using two separate channels. Unlike prior work which mainly considers low and high frequency concatenation for audio super-resolution, the proposed method directly predicts the full frequency range. We demonstrate high performance across a wide range of sample rates considering both speech and music. AERO outperforms the evaluated baselines considering Log-Spectral Distance, ViSQOL, and the subjective MUSHRA test. Audio samples and code are available at https://pages.cs.huji.ac.il/adiyoss-lab/aero

The choice of optimizer significantly impacts the training efficiency and computational costs of large language models (LLMs). Recently, the Muon optimizer has demonstrated promising results by orthogonalizing parameter updates, improving optimization geometry through better conditioning. Despite Muon's emergence as a candidate successor to Adam, the potential for jointly leveraging their strengths has not been systematically explored. In this work, we bridge this gap by proposing NorMuon (Neuron-wise Normalized Muon), an optimizer that synergistically combines orthogonalization with neuron-level adaptive learning rates. Our analysis reveals that while Muon effectively reduces condition numbers, the resulting updates exhibit highly non-uniform neuron norms, causing certain neurons to dominate the optimization process. NorMuon addresses this imbalance by maintaining second-order momentum statistics for each neuron and applying row-wise normalization after orthogonalization, ensuring balanced parameter utilization while preserving Muon's conditioning benefits. To enable practical deployment at scale, we develop an efficient distributed implementation under the FSDP2 framework that strategically distributes orthogonalization computations across devices. Experiments across multiple model scales demonstrate that NorMuon consistently outperforms both Adam and Muon, achieving 21.74% better training efficiency than Adam and 11.31% improvement over Muon on 1.1 B pretraining setting, while maintaining a comparable memory footprint to Muon. Our findings suggest that orthogonalization and adaptive learning rates are complementary rather than competing approaches, opening new avenues for optimizer design in large-scale deep learning.

Nearly all practical neural models for classification are trained using cross-entropy loss. Yet this ubiquitous choice is supported by little theoretical or empirical evidence. Recent work (Hui & Belkin, 2020) suggests that training using the (rescaled) square loss is often superior in terms of the classification accuracy. In this paper we propose the "squentropy" loss, which is the sum of two terms: the cross-entropy loss and the average square loss over the incorrect classes. We provide an extensive set of experiments on multi-class classification problems showing that the squentropy loss outperforms both the pure cross entropy and rescaled square losses in terms of the classification accuracy. We also demonstrate that it provides significantly better model calibration than either of these alternative losses and, furthermore, has less variance with respect to the random initialization. Additionally, in contrast to the square loss, squentropy loss can typically be trained using exactly the same optimization parameters, including the learning rate, as the standard cross-entropy loss, making it a true "plug-and-play" replacement. Finally, unlike the rescaled square loss, multiclass squentropy contains no parameters that need to be adjusted.

Offline optimization has recently emerged as an increasingly popular approach to mitigate the prohibitively expensive cost of online experimentation. The key idea is to learn a surrogate of the black-box function that underlines the target experiment using a static (offline) dataset of its previous input-output queries. Such an approach is, however, fraught with an out-of-distribution issue where the learned surrogate becomes inaccurate outside the offline data regimes. To mitigate this, existing offline optimizers have proposed numerous conditioning techniques to prevent the learned surrogate from being too erratic. Nonetheless, such conditioning strategies are often specific to particular surrogate or search models, which might not generalize to a different model choice. This motivates us to develop a model-agnostic approach instead, which incorporates a notion of model sharpness into the training loss of the surrogate as a regularizer. Our approach is supported by a new theoretical analysis demonstrating that reducing surrogate sharpness on the offline dataset provably reduces its generalized sharpness on unseen data. Our analysis extends existing theories from bounding generalized prediction loss (on unseen data) with loss sharpness to bounding the worst-case generalized surrogate sharpness with its empirical estimate on training data, providing a new perspective on sharpness regularization. Our extensive experimentation on a diverse range of optimization tasks also shows that reducing surrogate sharpness often leads to significant improvement, marking (up to) a noticeable 9.6% performance boost. Our code is publicly available at https://github.com/cuong-dm/IGNITE

Cross-entropy is a widely used loss function in applications. It coincides with the logistic loss applied to the outputs of a neural network, when the softmax is used. But, what guarantees can we rely on when using cross-entropy as a surrogate loss? We present a theoretical analysis of a broad family of loss functions, comp-sum losses, that includes cross-entropy (or logistic loss), generalized cross-entropy, the mean absolute error and other cross-entropy-like loss functions. We give the first H-consistency bounds for these loss functions. These are non-asymptotic guarantees that upper bound the zero-one loss estimation error in terms of the estimation error of a surrogate loss, for the specific hypothesis set H used. We further show that our bounds are tight. These bounds depend on quantities called minimizability gaps. To make them more explicit, we give a specific analysis of these gaps for comp-sum losses. We also introduce a new family of loss functions, smooth adversarial comp-sum losses, that are derived from their comp-sum counterparts by adding in a related smooth term. We show that these loss functions are beneficial in the adversarial setting by proving that they admit H-consistency bounds. This leads to new adversarial robustness algorithms that consist of minimizing a regularized smooth adversarial comp-sum loss. While our main purpose is a theoretical analysis, we also present an extensive empirical analysis comparing comp-sum losses. We further report the results of a series of experiments demonstrating that our adversarial robustness algorithms outperform the current state-of-the-art, while also achieving a superior non-adversarial accuracy.

As the field of representation learning grows, there has been a proliferation of different loss functions to solve different classes of problems. We introduce a single information-theoretic equation that generalizes a large collection of modern loss functions in machine learning. In particular, we introduce a framework that shows that several broad classes of machine learning methods are precisely minimizing an integrated KL divergence between two conditional distributions: the supervisory and learned representations. This viewpoint exposes a hidden information geometry underlying clustering, spectral methods, dimensionality reduction, contrastive learning, and supervised learning. This framework enables the development of new loss functions by combining successful techniques from across the literature. We not only present a wide array of proofs, connecting over 23 different approaches, but we also leverage these theoretical results to create state-of-the-art unsupervised image classifiers that achieve a +8% improvement over the prior state-of-the-art on unsupervised classification on ImageNet-1K. We also demonstrate that I-Con can be used to derive principled debiasing methods which improve contrastive representation learners.

To adapt a well-trained large model to downstream tasks, we propose constraining learning within its original latent space by leveraging linear combinations of its basis vectors. This approach ensures stable training without compromising the model's capabilities. Traditionally, constructing orthonormal bases from a matrix requires a transfer matrix, which significantly increases storage and computational overhead for parameters and feature maps. In this paper, we introduce Absorb and Decompose for Q, K, V, and O matrices, enabling their orthogonalization without the need for transfer matrices. Furthermore, the Absorb-Decompose operation eliminates redundant vectors, reducing the encoder attention parameters of Whisper-large-v3 by 46.42% without requiring additional training. For parameter-efficient and stable fine-tuning, we orthonormalized Q, K, V, and O and fine-tuned only the singular values, allowing efficient adaptation while constraining changes to the original latent space. When fine-tuning LLaMA-2-7B on eight commonsense reasoning datasets, our method outperforms LoRA by 5.4% and DoRA by 4.4%.

Decentralized learning often involves a weighted global loss with heterogeneous node weights λ. We revisit two natural strategies for incorporating these weights: (i) embedding them into the local losses to retain a uniform weight (and thus a doubly stochastic matrix), and (ii) keeping the original losses while employing a λ-induced row-stochastic matrix. Although prior work shows that both strategies target the same λ-weighted global loss, it remains unclear whether the Euclidean-space guarantees are tight and what fundamentally differentiates their behaviors. To clarify this, we develop a weighted Hilbert-space framework L^2(λ;R^d) and obtain convergence rates that are strictly tighter than those from standard Euclidean analysis. In this geometry, the row-stochastic matrix becomes self-adjoint whereas the doubly stochastic one does not, creating additional penalty terms that amplify consensus error, thereby slowing convergence. Consequently, the difference in convergence arises not only from spectral gaps but also from these penalty terms. We then derive sufficient conditions under which the row-stochastic design converges faster even with a smaller spectral gap. Finally, by using a Rayleigh-quotient and Loewner-order eigenvalue comparison, we further obtain topology conditions that guarantee this advantage and yield practical topology-design guidelines.

Preconditioned adaptive methods have gained significant attention for training deep neural networks, as they capture rich curvature information of the loss landscape. The central challenge in this field lies in balancing preconditioning effectiveness with computational efficiency of implementing the preconditioner. Among recent advances, Muon stands out by using Newton-Schulz iteration to obtain preconditioned updates without explicitly constructing the preconditioning matrix. Despite its advantages, the efficiency of Muon still leaves room for further improvement. In this paper, we introduce RMNP (Row Momentum Normalized Preconditioning), an optimizer that replaces Newton-Schulz iteration with a simple row-wise (d_{in}) ell_2 normalization operation, motivated by the empirically observed diagonal block structure of the Transformer layerwise Hessian. We empirically verified that orthogonalization and row-wise (on input dim) ell_2 normalization are asymptotically equivalent in the case of the transformer. This substitution reduces the per-iteration computational complexity from {O}(mncdotmin(m,n)) to {O}(mn) for an mtimes n weight matrix while maintaining comparable optimization performance. Theoretically, we establish convergence guarantees for RMNP in the non-convex setting that match recent results for Muon optimizers, achieving the minimax optimal complexity. Extensive experiments on large language model pretraining show that RMNP delivers competitive optimization performance compared with Muon while substantially reducing preconditioning wall-clock time. Our code is available at https://github.com/Dominator-Index/RMNP.

In this work, we present a variety of novel information-theoretic generalization bounds for learning algorithms, from the supersample setting of Steinke & Zakynthinou (2020)-the setting of the "conditional mutual information" framework. Our development exploits projecting the loss pair (obtained from a training instance and a testing instance) down to a single number and correlating loss values with a Rademacher sequence (and its shifted variants). The presented bounds include square-root bounds, fast-rate bounds, including those based on variance and sharpness, and bounds for interpolating algorithms etc. We show theoretically or empirically that these bounds are tighter than all information-theoretic bounds known to date on the same supersample setting.

The asymptotically precise estimation of the generalization of kernel methods has recently received attention due to the parallels between neural networks and their associated kernels. However, prior works derive such estimates for training by kernel ridge regression (KRR), whereas neural networks are typically trained with gradient descent (GD). In the present work, we consider the training of kernels with a family of spectral algorithms specified by profile h(lambda), and including KRR and GD as special cases. Then, we derive the generalization error as a functional of learning profile h(lambda) for two data models: high-dimensional Gaussian and low-dimensional translation-invariant model. Under power-law assumptions on the spectrum of the kernel and target, we use our framework to (i) give full loss asymptotics for both noisy and noiseless observations (ii) show that the loss localizes on certain spectral scales, giving a new perspective on the KRR saturation phenomenon (iii) conjecture, and demonstrate for the considered data models, the universality of the loss w.r.t. non-spectral details of the problem, but only in case of noisy observation.

Residual connections are pivotal for deep neural networks, enabling greater depth by mitigating vanishing gradients. However, in standard residual updates, the module's output is directly added to the input stream. This can lead to updates that predominantly reinforce or modulate the existing stream direction, potentially underutilizing the module's capacity for learning entirely novel features. In this work, we introduce Orthogonal Residual Update: we decompose the module's output relative to the input stream and add only the component orthogonal to this stream. This design aims to guide modules to contribute primarily new representational directions, fostering richer feature learning while promoting more efficient training. We demonstrate that our orthogonal update strategy improves generalization accuracy and training stability across diverse architectures (ResNetV2, Vision Transformers) and datasets (CIFARs, TinyImageNet, ImageNet-1k), achieving, for instance, a +4.3\%p top-1 accuracy gain for ViT-B on ImageNet-1k.

Cross-entropy loss and focal loss are the most common choices when training deep neural networks for classification problems. Generally speaking, however, a good loss function can take on much more flexible forms, and should be tailored for different tasks and datasets. Motivated by how functions can be approximated via Taylor expansion, we propose a simple framework, named PolyLoss, to view and design loss functions as a linear combination of polynomial functions. Our PolyLoss allows the importance of different polynomial bases to be easily adjusted depending on the targeting tasks and datasets, while naturally subsuming the aforementioned cross-entropy loss and focal loss as special cases. Extensive experimental results show that the optimal choice within the PolyLoss is indeed dependent on the task and dataset. Simply by introducing one extra hyperparameter and adding one line of code, our Poly-1 formulation outperforms the cross-entropy loss and focal loss on 2D image classification, instance segmentation, object detection, and 3D object detection tasks, sometimes by a large margin.

Higher-order optimizers such as Muon and SOAP offer faster convergence than AdamW, but their computational cost and numerical stability challenges have limited adoption at scale. In this work, we adapt and enhance preconditioned gradient methods to overcome the practical challenges of large-scale LLM pretraining. We first identify instabilities in SOAP at large batch sizes and propose algorithmic modifications including per-step QR orthogonalization and improved preconditioning strategies that eliminate loss spikes and enable stable training in these regimes. We then present a unified empirical study of SOAP, Muon, and AdamW using update-RMS matching to ensure fair learning rate transfer across optimizers. As part of this analysis, we empirically evaluate the orthogonalization quality of Muon. Our experiments on multi-billion-parameter models trained on trillions of tokens reveal that SOAP and Muon consistently outperform AdamW at the scales we tested. Notably, at batch sizes of up to 100M tokens for next-token prediction, these optimizers maintain training stability and quality while AdamW degrades. To enable efficient training at large scale, we introduce a layer-wise distributed optimizer compatible with Megatron-LM. Our implementation balances memory and hides communication while avoiding approximations to the optimizer computations, thus retaining their convergence benefits. Additionally, we identify and build specific system-level improvements to further accelerate our layer-wise implementation. To support the research community, we release a codebase that contains emerging algorithms for optimization: https://github.com/NVIDIA-NeMo/Emerging-Optimizers

Cross-entropy loss has long been the standard choice for training deep neural networks, yet it suffers from interpretability limitations, unbounded weight growth, and inefficiencies that can contribute to costly training dynamics. The harmonic loss is a distance-based alternative grounded in Euclidean geometry that improves interpretability and mitigates phenomena such as grokking, or delayed generalization on the test set. However, the study of harmonic loss remains narrow: only Euclidean distance is explored, and no systematic evaluation of computational efficiency or sustainability was conducted. We extend harmonic loss by systematically investigating a broad spectrum of distance metrics as replacements for the Euclidean distance. We comprehensively evaluate distance-tailored harmonic losses on both vision backbones and large language models. Our analysis is framed around a three-way evaluation of model performance, interpretability, and sustainability. On vision tasks, cosine distances provide the most favorable trade-off, consistently improving accuracy while lowering carbon emissions, whereas Bray-Curtis and Mahalanobis further enhance interpretability at varying efficiency costs. On language models, cosine-based harmonic losses improve gradient and learning stability, strengthen representation structure, and reduce emissions relative to cross-entropy and Euclidean heads. Our code is available at: https://anonymous.4open.science/r/rethinking-harmonic-loss-5BAB/.

Representing a manifold of very high-dimensional data with generative models has been shown to be computationally efficient in practice. However, this requires that the data manifold admits a global parameterization. In order to represent manifolds of arbitrary topology, we propose to learn a mixture model of variational autoencoders. Here, every encoder-decoder pair represents one chart of a manifold. We propose a loss function for maximum likelihood estimation of the model weights and choose an architecture that provides us the analytical expression of the charts and of their inverses. Once the manifold is learned, we use it for solving inverse problems by minimizing a data fidelity term restricted to the learned manifold. To solve the arising minimization problem we propose a Riemannian gradient descent algorithm on the learned manifold. We demonstrate the performance of our method for low-dimensional toy examples as well as for deblurring and electrical impedance tomography on certain image manifolds.

Fine-tuning large language models (LLMs) with high parameter efficiency for downstream tasks has become a new paradigm. Low-Rank Adaptation (LoRA) significantly reduces the number of trainable parameters for fine-tuning. Although it has demonstrated commendable performance, updating parameters within a single scale may not be the optimal choice for complex downstream tasks.In this paper, we extend the LoRA to multiple scales, dubbed as LoRA^2. We first combine orthogonal projection theory to train a set of LoRAs in two mutually orthogonal planes. Then, we improve the importance score algorithm, which reduce parameter sensitivity score calculations by approximately 98.5\%. By pruning singular values with lower importance scores, thereby enhancing adaptability to various downstream tasks. Extensive experiments are conducted on two widely used pre-trained models to validate the effectiveness of LoRA^2. Results show that it significantly reduces the number of trainable parameters to just 0.72\% compared to full fine-tuning, while still delivering highly impressive performance. Even when the parameters are further reduced to 0.17M, it still achieves comparable results to the baseline with 8 times more parameters. Our code is available here: https://anonymous.4open.science/r/LoRA-2-5B4C

We propose a new bound for generalization of neural networks using Koopman operators. Whereas most of existing works focus on low-rank weight matrices, we focus on full-rank weight matrices. Our bound is tighter than existing norm-based bounds when the condition numbers of weight matrices are small. Especially, it is completely independent of the width of the network if the weight matrices are orthogonal. Our bound does not contradict to the existing bounds but is a complement to the existing bounds. As supported by several existing empirical results, low-rankness is not the only reason for generalization. Furthermore, our bound can be combined with the existing bounds to obtain a tighter bound. Our result sheds new light on understanding generalization of neural networks with full-rank weight matrices, and it provides a connection between operator-theoretic analysis and generalization of neural networks.

We propose a class of greedy algorithms for weighted sparse recovery by considering new loss function-based generalizations of Orthogonal Matching Pursuit (OMP). Given a (regularized) loss function, the proposed algorithms alternate the iterative construction of the signal support via greedy index selection and a signal update based on solving a local data-fitting problem restricted to the current support. We show that greedy selection rules associated with popular weighted sparsity-promoting loss functions admit explicitly computable and simple formulas. Specifically, we consider ell^0 - and ell^1 -based versions of the weighted LASSO (Least Absolute Shrinkage and Selection Operator), the Square-Root LASSO (SR-LASSO) and the Least Absolute Deviations LASSO (LAD-LASSO). Through numerical experiments on Gaussian compressive sensing and high-dimensional function approximation, we demonstrate the effectiveness of the proposed algorithms and empirically show that they inherit desirable characteristics from the corresponding loss functions, such as SR-LASSO's noise-blind optimal parameter tuning and LAD-LASSO's fault tolerance. In doing so, our study sheds new light on the connection between greedy sparse recovery and convex relaxation.

The current best bounds on the matrix multiplication exponent ω are obtained through a refinement of the laser method called combination loss analysis (Duan et al., 2022; Williams et al., 2024; Alman et al., 2025). In this note, we address the optimization problem at the core of this approach and propose several improvements. First, we reformulate the optimization problem allowing us to solve it in a larger setting than was previously possible. Second, we leverage recent advances in machine learning to design a new optimization algorithm for this problem. Finally, we refine the resulting optimization algorithm with AlphaEvolve. Our combined approach yields an upper bound of ω < 2.371177, improving the previous best bound of 2.371339.

We consider a deep matrix factorization model of covariance matrices trained with the Bures-Wasserstein distance. While recent works have made important advances in the study of the optimization problem for overparametrized low-rank matrix approximation, much emphasis has been placed on discriminative settings and the square loss. In contrast, our model considers another interesting type of loss and connects with the generative setting. We characterize the critical points and minimizers of the Bures-Wasserstein distance over the space of rank-bounded matrices. For low-rank matrices the Hessian of this loss can theoretically blow up, which creates challenges to analyze convergence of optimizaton methods. We establish convergence results for gradient flow using a smooth perturbative version of the loss and convergence results for finite step size gradient descent under certain assumptions on the initial weights.

Neural operators, as a powerful approximation to the non-linear operators between infinite-dimensional function spaces, have proved to be promising in accelerating the solution of partial differential equations (PDE). However, it requires a large amount of simulated data, which can be costly to collect. This can be avoided by learning physics from the physics-constrained loss, which we refer to it as mean squared residual (MSR) loss constructed by the discretized PDE. We investigate the physical information in the MSR loss, which we called long-range entanglements, and identify the challenge that the neural network requires the capacity to model the long-range entanglements in the spatial domain of the PDE, whose patterns vary in different PDEs. To tackle the challenge, we propose LordNet, a tunable and efficient neural network for modeling various entanglements. Inspired by the traditional solvers, LordNet models the long-range entanglements with a series of matrix multiplications, which can be seen as the low-rank approximation to the general fully-connected layers and extracts the dominant pattern with reduced computational cost. The experiments on solving Poisson's equation and (2D and 3D) Navier-Stokes equation demonstrate that the long-range entanglements from the MSR loss can be well modeled by the LordNet, yielding better accuracy and generalization ability than other neural networks. The results show that the Lordnet can be 40times faster than traditional PDE solvers. In addition, LordNet outperforms other modern neural network architectures in accuracy and efficiency with the smallest parameter size.

Recent works have shown a reduction from contextual bandits to online regression under a realizability assumption [Foster and Rakhlin, 2020, Foster and Krishnamurthy, 2021]. In this work, we investigate the use of neural networks for such online regression and associated Neural Contextual Bandits (NeuCBs). Using existing results for wide networks, one can readily show a {O}(T) regret for online regression with square loss, which via the reduction implies a {O}(K T^{3/4}) regret for NeuCBs. Departing from this standard approach, we first show a O(log T) regret for online regression with almost convex losses that satisfy QG (Quadratic Growth) condition, a generalization of the PL (Polyak-\L ojasiewicz) condition, and that have a unique minima. Although not directly applicable to wide networks since they do not have unique minima, we show that adding a suitable small random perturbation to the network predictions surprisingly makes the loss satisfy QG with unique minima. Based on such a perturbed prediction, we show a {O}(log T) regret for online regression with both squared loss and KL loss, and subsequently convert these respectively to mathcal{O}(KT) and mathcal{O}(KL^* + K) regret for NeuCB, where L^* is the loss of the best policy. Separately, we also show that existing regret bounds for NeuCBs are Omega(T) or assume i.i.d. contexts, unlike this work. Finally, our experimental results on various datasets demonstrate that our algorithms, especially the one based on KL loss, persistently outperform existing algorithms.

Low-rank optimization has emerged as a promising direction in training large language models (LLMs) to reduce the memory usage of adaptive optimizers by constraining learning to a lower-dimensional space. Prior work typically projects gradients of linear layers using approaches based on Singular Value Decomposition (SVD). However, applying SVD-based procedures individually to each layer in large models is computationally expensive and incurs additional memory costs due to storing the projection matrices. In this work, we propose a computationally efficient and conceptually simple two-step procedure to approximate SVD-based gradient projections into lower-dimensional spaces. First, we construct a complete orthogonal basis using predefined orthogonal matrices of the Discrete Cosine Transform (DCT). Second, we adaptively select basis columns based on their alignment with the gradient of each layer. Each projection matrix in our method is obtained via a single matrix multiplication followed by a lightweight sorting step to identify the most relevant basis vectors. Due to the predefined nature of the orthogonal bases, they are computed once at the start of training. During training, we store only the indices of the selected columns, avoiding the need to store full projection matrices for each layer. Our numerical experiments on both pre-training and fine-tuning tasks demonstrate the effectiveness of our dual strategy in approximating optimal low-rank projections, matching the performance of costly SVD-based methods while achieving faster runtime and reduced memory usage.

Loss functions serve as the foundation of supervised learning and are often chosen prior to model development. To avoid potentially ad hoc choices of losses, statistical decision theory describes a desirable property for losses known as properness, which asserts that Bayes' rule is optimal. Recent works have sought to learn losses and models jointly. Existing methods do this by fitting an inverse canonical link function which monotonically maps R to [0,1] to estimate probabilities for binary problems. In this paper, we extend monotonicity to maps between R^{C-1} and the projected probability simplex Delta^{C-1} by using monotonicity of gradients of convex functions. We present {\sc LegendreTron} as a novel and practical method that jointly learns proper canonical losses and probabilities for multiclass problems. Tested on a benchmark of domains with up to 1,000 classes, our experimental results show that our method consistently outperforms the natural multiclass baseline under a t-test at 99% significance on all datasets with greater than 10 classes.

The Muon optimizer has rapidly emerged as a powerful, geometry-aware alternative to AdamW, demonstrating strong performance in large-scale training of neural networks. However, a critical theory-practice disconnect exists: Muon's efficiency relies on fast, approximate orthogonalization, yet all prior theoretical work analyzes an idealized, computationally intractable version assuming exact SVD-based updates. This work moves beyond the ideal by providing the first analysis of the inexact orthogonalized update at Muon's core. We develop our analysis within the general framework of Linear Minimization Oracle (LMO)-based optimization, introducing a realistic additive error model to capture the inexactness of practical approximation schemes. Our analysis yields explicit bounds that quantify performance degradation as a function of the LMO inexactness/error. We reveal a fundamental coupling between this inexactness and the optimal step size and momentum: lower oracle precision requires a smaller step size but larger momentum parameter. These findings elevate the approximation procedure (e.g., the number of Newton-Schulz steps) from an implementation detail to a critical parameter that must be co-tuned with the learning schedule. NanoGPT experiments directly confirm the predicted coupling, with optimal learning rates clearly shifting as approximation precision changes.

Unified fashion generation integrates tasks like virtual try-on and garment reconstruction into a single model to reduce task-specific adaptation costs. However, naive parameter sharing across semantically distinct tasks induces negative transfer through severe inter-task gradient conflict. We propose OrthoTryOn, a unified framework mitigating this interference within a shared Low-Rank Adaptation (LoRA) module. Its Orthogonal Subspace Projection (OSP) applies task-specific orthogonal rotations to bottleneck features, mapping them into decorrelated coordinate frames. To address residual semantic coupling at inference time, we further propose Fisher-guided Negative Guidance (FNG), a parameter-free strategy that utilizes diagonal Fisher information to quantify inter-task sensitivity overlap and explicitly repels generation trajectories from the most confusable task via Classifier-Free Guidance. Extensive experiments demonstrate that OrthoTryOn avoids the severe performance degradation typical of naive unified training and even surpasses independently trained task-specific models, achieving state-of-the-art results across multiple benchmarks while generalizing robustly across diverse diffusion backbones. Code is available at https://github.com/NJU-PCALab/OrthoTryOn.

In this work, we study the performance of sub-gradient method (SubGM) on a natural nonconvex and nonsmooth formulation of low-rank matrix recovery with ell_1-loss, where the goal is to recover a low-rank matrix from a limited number of measurements, a subset of which may be grossly corrupted with noise. We study a scenario where the rank of the true solution is unknown and over-estimated instead. The over-estimation of the rank gives rise to an over-parameterized model in which there are more degrees of freedom than needed. Such over-parameterization may lead to overfitting, or adversely affect the performance of the algorithm. We prove that a simple SubGM with small initialization is agnostic to both over-parameterization and noise in the measurements. In particular, we show that small initialization nullifies the effect of over-parameterization on the performance of SubGM, leading to an exponential improvement in its convergence rate. Moreover, we provide the first unifying framework for analyzing the behavior of SubGM under both outlier and Gaussian noise models, showing that SubGM converges to the true solution, even under arbitrarily large and arbitrarily dense noise values, and--perhaps surprisingly--even if the globally optimal solutions do not correspond to the ground truth. At the core of our results is a robust variant of restricted isometry property, called Sign-RIP, which controls the deviation of the sub-differential of the ell_1-loss from that of an ideal, expected loss. As a byproduct of our results, we consider a subclass of robust low-rank matrix recovery with Gaussian measurements, and show that the number of required samples to guarantee the global convergence of SubGM is independent of the over-parameterized rank.

Modern deep neural networks have achieved impressive performance on tasks from image classification to natural language processing. Surprisingly, these complex systems with massive amounts of parameters exhibit the same structural properties in their last-layer features and classifiers across canonical datasets when training until convergence. In particular, it has been observed that the last-layer features collapse to their class-means, and those class-means are the vertices of a simplex Equiangular Tight Frame (ETF). This phenomenon is known as Neural Collapse (NC). Recent papers have theoretically shown that NC emerges in the global minimizers of training problems with the simplified "unconstrained feature model". In this context, we take a step further and prove the NC occurrences in deep linear networks for the popular mean squared error (MSE) and cross entropy (CE) losses, showing that global solutions exhibit NC properties across the linear layers. Furthermore, we extend our study to imbalanced data for MSE loss and present the first geometric analysis of NC under bias-free setting. Our results demonstrate the convergence of the last-layer features and classifiers to a geometry consisting of orthogonal vectors, whose lengths depend on the amount of data in their corresponding classes. Finally, we empirically validate our theoretical analyses on synthetic and practical network architectures with both balanced and imbalanced scenarios.

Large Language Models (LLMs) often memorize sensitive or harmful information, necessitating effective machine unlearning techniques. While existing parameter-efficient unlearning methods have shown promise, they still struggle with the forget-retain trade-off. This can be attributed to their reliance on parameter importance metrics to identify parameters that are important exclusively for the forget set, which is fundamentally limited by the superposition phenomenon. Due to the polysemantic nature of LLM parameters, such an importance metric may struggle to disentangle parameters associated with the forget and retain sets. In this work, we propose Representation-Guided Low-rank Unlearning (REGLU), a novel approach that leverages the geometric properties of representation spaces to achieve robust and precise unlearning. First, we develop a representation-guided initialization for LoRA that identifies the optimal subspace for selective forgetting. Second, we introduce a regularization loss that constrains the outputs of the LoRA update to lie in the orthogonal complement of the retain set's representation subspace, thereby minimizing interference with the model's performance on the retain set. We evaluate REGLU on the TOFU and WMDP benchmarks across multiple models. Our results demonstrate that REGLU consistently outperforms state-of-the-art baselines, achieving superior unlearning quality while maintaining higher model utility.

Updating a truncated Singular Value Decomposition (SVD) is crucial in representation learning, especially when dealing with large-scale data matrices that continuously evolve in practical scenarios. Aligning SVD-based models with fast-paced updates becomes increasingly important. Existing methods for updating truncated SVDs employ Rayleigh-Ritz projection procedures, where projection matrices are augmented based on original singular vectors. However, these methods suffer from inefficiency due to the densification of the update matrix and the application of the projection to all singular vectors. To address these limitations, we introduce a novel method for dynamically approximating the truncated SVD of a sparse and temporally evolving matrix. Our approach leverages sparsity in the orthogonalization process of augmented matrices and utilizes an extended decomposition to independently store projections in the column space of singular vectors. Numerical experiments demonstrate a remarkable efficiency improvement of an order of magnitude compared to previous methods. Remarkably, this improvement is achieved while maintaining a comparable precision to existing approaches.

The capabilities and adoption of deep neural networks (DNNs) grow at an exhilarating pace: Vision models accurately classify human actions in videos and identify cancerous tissue in medical scans as precisely than human experts; large language models answer wide-ranging questions, generate code, and write prose, becoming the topic of everyday dinner-table conversations. Even though their uses are exhilarating, the continually increasing model sizes and computational complexities have a dark side. The economic cost and negative environmental externalities of training and serving models is in evident disharmony with financial viability and climate action goals. Instead of pursuing yet another increase in predictive performance, this dissertation is dedicated to the improvement of neural network efficiency. Specifically, a core contribution addresses the efficiency aspects during online inference. Here, the concept of Continual Inference Networks (CINs) is proposed and explored across four publications. CINs extend prior state-of-the-art methods developed for offline processing of spatio-temporal data and reuse their pre-trained weights, improving their online processing efficiency by an order of magnitude. These advances are attained through a bottom-up computational reorganization and judicious architectural modifications. The benefit to online inference is demonstrated by reformulating several widely used network architectures into CINs, including 3D CNNs, ST-GCNs, and Transformer Encoders. An orthogonal contribution tackles the concurrent adaptation and computational acceleration of a large source model into multiple lightweight derived models. Drawing on fusible adapter networks and structured pruning, Structured Pruning Adapters achieve superior predictive accuracy under aggressive pruning using significantly fewer learned weights compared to fine-tuning with pruning.

We study the task of agnostically learning halfspaces under the Gaussian distribution. Specifically, given labeled examples (x,y) from an unknown distribution on R^n times { pm 1}, whose marginal distribution on x is the standard Gaussian and the labels y can be arbitrary, the goal is to output a hypothesis with 0-1 loss OPT+epsilon, where OPT is the 0-1 loss of the best-fitting halfspace. We prove a near-optimal computational hardness result for this task, under the widely believed sub-exponential time hardness of the Learning with Errors (LWE) problem. Prior hardness results are either qualitatively suboptimal or apply to restricted families of algorithms. Our techniques extend to yield near-optimal lower bounds for related problems, including ReLU regression.

In our era of enormous neural networks, empirical progress has been driven by the philosophy that more is better. Recent deep learning practice has found repeatedly that larger model size, more data, and more computation (resulting in lower training loss) improves performance. In this paper, we give theoretical backing to these empirical observations by showing that these three properties hold in random feature (RF) regression, a class of models equivalent to shallow networks with only the last layer trained. Concretely, we first show that the test risk of RF regression decreases monotonically with both the number of features and the number of samples, provided the ridge penalty is tuned optimally. In particular, this implies that infinite width RF architectures are preferable to those of any finite width. We then proceed to demonstrate that, for a large class of tasks characterized by powerlaw eigenstructure, training to near-zero training loss is obligatory: near-optimal performance can only be achieved when the training error is much smaller than the test error. Grounding our theory in real-world data, we find empirically that standard computer vision tasks with convolutional neural tangent kernels clearly fall into this class. Taken together, our results tell a simple, testable story of the benefits of overparameterization, overfitting, and more data in random feature models.

Merging finetuned Large Language Models (LLMs) has become increasingly important for integrating diverse capabilities into a single unified model. However, prevailing model merging methods rely on linear arithmetic in Euclidean space, which often destroys the intrinsic geometric properties of pretrained weights, such as hyperspherical energy. To address this, we propose Orthogonal Model Merging (OrthoMerge), a method that performs merging operations on the Riemannian manifold formed by the orthogonal group to preserve the geometric structure of the model's weights. By mapping task-specific orthogonal matrices learned by Orthogonal Finetuning (OFT) to the Lie algebra, OrthoMerge enables a principled yet efficient integration that takes into account both the direction and intensity of adaptations. In addition to directly leveraging orthogonal matrices obtained by OFT, we further extend this approach to general models finetuned with non-OFT methods (i.e., low-rank finetuning, full finetuning) via an Orthogonal-Residual Decoupling strategy. This technique extracts the orthogonal components of expert models by solving the orthogonal Procrustes problem, which are then merged on the manifold of the orthogonal group, while the remaining linear residuals are processed through standard additive merging. Extensive empirical results demonstrate the effectiveness of OrthoMerge in mitigating catastrophic forgetting and maintaining model performance across diverse tasks.

Fully-connected deep neural networks with weights initialized from independent Gaussian distributions can be tuned to criticality, which prevents the exponential growth or decay of signals propagating through the network. However, such networks still exhibit fluctuations that grow linearly with the depth of the network, which may impair the training of networks with width comparable to depth. We show analytically that rectangular networks with tanh activations and weights initialized from the ensemble of orthogonal matrices have corresponding preactivation fluctuations which are independent of depth, to leading order in inverse width. Moreover, we demonstrate numerically that, at initialization, all correlators involving the neural tangent kernel (NTK) and its descendants at leading order in inverse width -- which govern the evolution of observables during training -- saturate at a depth of sim 20, rather than growing without bound as in the case of Gaussian initializations. We speculate that this structure preserves finite-width feature learning while reducing overall noise, thus improving both generalization and training speed. We provide some experimental justification by relating empirical measurements of the NTK to the superior performance of deep nonlinear orthogonal networks trained under full-batch gradient descent on the MNIST and CIFAR-10 classification tasks.

Adapting large-scale pre-trained generative models in a parameter-efficient manner is gaining traction. Traditional methods like low rank adaptation achieve parameter efficiency by imposing constraints but may not be optimal for tasks requiring high representation capacity. We propose a novel spectrum-aware adaptation framework for generative models. Our method adjusts both singular values and their basis vectors of pretrained weights. Using the Kronecker product and efficient Stiefel optimizers, we achieve parameter-efficient adaptation of orthogonal matrices. We introduce Spectral Orthogonal Decomposition Adaptation (SODA), which balances computational efficiency and representation capacity. Extensive evaluations on text-to-image diffusion models demonstrate SODA's effectiveness, offering a spectrum-aware alternative to existing fine-tuning methods.

Whitening loss offers a theoretical guarantee against feature collapse in self-supervised learning (SSL) with joint embedding architectures. Typically, it involves a hard whitening approach, transforming the embedding and applying loss to the whitened output. In this work, we introduce Spectral Transformation (ST), a framework to modulate the spectrum of embedding and to seek for functions beyond whitening that can avoid dimensional collapse. We show that whitening is a special instance of ST by definition, and our empirical investigations unveil other ST instances capable of preventing collapse. Additionally, we propose a novel ST instance named IterNorm with trace loss (INTL). Theoretical analysis confirms INTL's efficacy in preventing collapse and modulating the spectrum of embedding toward equal-eigenvalues during optimization. Our experiments on ImageNet classification and COCO object detection demonstrate INTL's potential in learning superior representations. The code is available at https://github.com/winci-ai/INTL.

Neural scaling laws approximate a language model's loss as a power-law function of parameter count N and token count D. Following Chinchilla-style compute-optimal training, many studies fit scaling laws from runs performed under a fixed tokens-per-parameter (TPP) ratio k and set D = kN. We show that this collinear design, combined with the empirically common near-equality of the exponents governing N and D, induces an inherent ill-conditioning in the Gauss-Newton least-squares problem: the condition number of the design grows as the inverse square of the gap between the N and D-exponents. The scale coefficients become practically unidentifiable, with confidence intervals inflating by an order of magnitude or more, yielding a ``sloppy'' model whose extrapolations degrade sharply off the training ray. We prove this for four scaling-law formalisms and derive a closed-form TPP-diversity threshold that is necessary and sufficient for well-conditioned estimation. Empirically, non-collinear designs outperform collinear ones on held-out splits with a 97.3\% win rate across four laws, five corpora, multiple floating point precision modes. We further show the degeneracy is rooted in Jacobian geometry and is not an artifact of the loss function: any smooth estimation objective whose curvature involves the Jacobian inherits the same ill-conditioning.

Machine learning weather prediction (MLWP) models have demonstrated remarkable potential in delivering accurate forecasts at significantly reduced computational cost compared to traditional numerical weather prediction (NWP) systems. However, challenges remain in ensuring the physical consistency of MLWP outputs, particularly in deterministic settings. This study presents FastNet, a graph neural network (GNN)-based global prediction model, and investigates the impact of alternative loss function designs on improving the physical realism of its forecasts. We explore three key modifications to the standard mean squared error (MSE) loss: (1) a modified spherical harmonic (MSH) loss that penalises spectral amplitude errors to reduce blurring and enhance small-scale structure retention; (2) inclusion of horizontal gradient terms in the loss to suppress non-physical artefacts; and (3) an alternative wind representation that decouples speed and direction to better capture extreme wind events. Results show that while the MSH and gradient-based losses alone may slightly degrade RMSE scores, when trained in combination the model exhibits very similar MSE performance to an MSE-trained model while at the same time significantly improving spectral fidelity and physical consistency. The alternative wind representation further improves wind speed accuracy and reduces directional bias. Collectively, these findings highlight the importance of loss function design as a mechanism for embedding domain knowledge into MLWP models and advancing their operational readiness.

We study the problem of classification with a reject option for a fixed predictor, applicable in natural language processing. We introduce a new problem formulation for this scenario, and an algorithm minimizing a new surrogate loss function. We provide a complete theoretical analysis of the surrogate loss function with a strong H-consistency guarantee. For evaluation, we choose the decontextualization task, and provide a manually-labelled dataset of 2mathord,000 examples. Our algorithm significantly outperforms the baselines considered, with a sim!!25% improvement in coverage when halving the error rate, which is only sim!! 3 % away from the theoretical limit.

We show that deep neural networks (DNNs) can efficiently learn any composition of functions with bounded F_{1}-norm, which allows DNNs to break the curse of dimensionality in ways that shallow networks cannot. More specifically, we derive a generalization bound that combines a covering number argument for compositionality, and the F_{1}-norm (or the related Barron norm) for large width adaptivity. We show that the global minimizer of the regularized loss of DNNs can fit for example the composition of two functions f^{*}=hcirc g from a small number of observations, assuming g is smooth/regular and reduces the dimensionality (e.g. g could be the modulo map of the symmetries of f^{*}), so that h can be learned in spite of its low regularity. The measures of regularity we consider is the Sobolev norm with different levels of differentiability, which is well adapted to the F_{1} norm. We compute scaling laws empirically and observe phase transitions depending on whether g or h is harder to learn, as predicted by our theory.

Generalization bounds which assess the difference between the true risk and the empirical risk, have been studied extensively. However, to obtain bounds, current techniques use strict assumptions such as a uniformly bounded or a Lipschitz loss function. To avoid these assumptions, in this paper, we follow an alternative approach: we relax uniform bounds assumptions by using on-average bounded loss and on-average bounded gradient norm assumptions. Following this relaxation, we propose a new generalization bound that exploits the contractivity of the log-Sobolev inequalities. These inequalities add an additional loss-gradient norm term to the generalization bound, which is intuitively a surrogate of the model complexity. We apply the proposed bound on Bayesian deep nets and empirically analyze the effect of this new loss-gradient norm term on different neural architectures.

We consider the problem of learning a target function corresponding to a single hidden layer neural network, with a quadratic activation function after the first layer, and random weights. We consider the asymptotic limit where the input dimension and the network width are proportionally large. Recent work [Cui & al '23] established that linear regression provides Bayes-optimal test error to learn such a function when the number of available samples is only linear in the dimension. That work stressed the open challenge of theoretically analyzing the optimal test error in the more interesting regime where the number of samples is quadratic in the dimension. In this paper, we solve this challenge for quadratic activations and derive a closed-form expression for the Bayes-optimal test error. We also provide an algorithm, that we call GAMP-RIE, which combines approximate message passing with rotationally invariant matrix denoising, and that asymptotically achieves the optimal performance. Technically, our result is enabled by establishing a link with recent works on optimal denoising of extensive-rank matrices and on the ellipsoid fitting problem. We further show empirically that, in the absence of noise, randomly-initialized gradient descent seems to sample the space of weights, leading to zero training loss, and averaging over initialization leads to a test error equal to the Bayes-optimal one.

In this paper, we propose new randomized algorithms for estimating the two-to-infinity and one-to-two norms in a matrix-free setting, using only matrix-vector multiplications. Our methods are based on appropriate modifications of Hutchinson's diagonal estimator and its Hutch++ version. We provide oracle complexity bounds for both modifications. We further illustrate the practical utility of our algorithms for Jacobian-based regularization in deep neural network training on image classification tasks. We also demonstrate that our methodology can be applied to mitigate the effect of adversarial attacks in the domain of recommender systems.

In this work, we revisit Transformer optimization through the lens of second-order geometry and establish a direct connection between architectural design, activation scale, the Hessian matrix, and the maximum tolerable learning rate. We introduce a simple normalization strategy, termed SimpleNorm, which stabilizes intermediate activation scales by construction. Then, by analyzing the Hessian of the loss with respect to network activations, we theoretically show that SimpleNorm significantly reduces the spectral norm of the Hessian, thereby permitting larger stable learning rates. We validate our theoretical findings through extensive experiments on large GPT models at parameter scales 1B, 1.4B, 7B and 8B. Empirically, SimpleGPT, our SimpleNorm-based network, tolerates learning rates 3times-10times larger than standard convention, consistently demonstrates strong optimization stability, and achieves substantially better performance than well-established baselines. Specifically, when training 7B-scale models for 60K steps, SimpleGPT achieves a training loss that is 0.08 lower than that of LLaMA2 with QKNorm, reducing the loss from 2.290 to 2.208. Our source code will be released at https://github.com/Ocram7/SimpleGPT.

Task arithmetic provides an efficient, training-free way to edit pre-trained models, yet lacks a fundamental theoretical explanation for its success. The existing concept of ``weight disentanglement" describes the ideal outcome of non-interfering task composition but does not reveal its underlying cause. Crucially, what intrinsic properties of the pre-trained model (θ_0) or the task vectors (τ_t) enable this disentanglement remains underexplored. In this paper, we introduce Task-Feature Specialization (TFS), a model's ability to allocate distinct internal features to different tasks, as the fundamental principle. We first prove that TFS is a sufficient condition for weight disentanglement. More importantly, we find that TFS also gives rise to an observable geometric consequence: weight vector orthogonality. This positions TFS as the common cause for both the desired functional outcome (disentanglement) and a measurable geometric property (orthogonality). This relationship provides the key insight for our method: since the abstract TFS property is intractable to enforce directly, we can instead promote weight disentanglement by shaping its concrete geometric consequence, orthogonality. Therefore, we propose OrthoReg, a simple and effective regularization method that actively enforces an internal orthogonal structure on weight updates (ΔW) that constitute τ_t during fine-tuning. And we theoretically prove that OrthoReg promotes disentanglement. Extensive experiments demonstrate that OrthoReg consistently and significantly enhances the performance of various task arithmetic methods. Code is available at https://github.com/RL-MIND/OrthoReg{https://github.com/RL-MIND/OrthoReg}.

Dual-encoder retrievers depend on the principle that relevant documents should score higher than irrelevant ones for a given query. Yet the dominant Noise Contrastive Estimation (NCE) objective, which underpins Contrastive Loss, optimizes a softened ranking surrogate that we rigorously prove is fundamentally oblivious to score separation quality and unrelated to AUC. This mismatch leads to poor calibration and suboptimal performance in downstream tasks like retrieval-augmented generation (RAG). To address this fundamental limitation, we introduce the MW loss, a new training objective that maximizes the Mann-Whitney U statistic, which is mathematically equivalent to the Area under the ROC Curve (AUC). MW loss encourages each positive-negative pair to be correctly ranked by minimizing binary cross entropy over score differences. We provide theoretical guarantees that MW loss directly upper-bounds the AoC, better aligning optimization with retrieval goals. We further promote ROC curves and AUC as natural threshold free diagnostics for evaluating retriever calibration and ranking quality. Empirically, retrievers trained with MW loss consistently outperform contrastive counterparts in AUC and standard retrieval metrics. Our experiments show that MW loss is an empirically superior alternative to Contrastive Loss, yielding better-calibrated and more discriminative retrievers for high-stakes applications like RAG.

We address the computational challenges of solving parametric PDEs with non parametrized geometric variations and non-reducible problems, such as those involving shocks and discontinuities of variable positions. Traditional dimensionality reduction methods like POD struggle with these scenarios due to slowly decaying Kolmogorov widths. To overcome this, we propose a novel non-linear dimensionality reduction technique to reduce the required modes for representation. The non-linear reduction is obtained through a POD after applying a transformation on the fields, which we call optimal mappings, and is a solution to an optimization problem in infinite dimension. The proposed learning framework combines morphing techniques, non-linear dimensionality reduction, and Gaussian Process Regression (GPR). The problem is reformulated on a reference geometry before applying the dimensionality reduction. Our method learns both the optimal mapping, and the solution fields, using a series of GPR models, enabling efficient and accurate modeling of complex parametric PDEs with geometrical variability. The results obtained concur with current state-of-the-art models. We mainly compare our method with the winning solution of the ML4CFD NeurIPS 2024 competition.

We study the sparse recovery problem with an underdetermined linear system characterized by a Kronecker-structured dictionary and a Kronecker-supported sparse vector. We cast this problem into the sparse Bayesian learning (SBL) framework and rely on the expectation-maximization method for a solution. To this end, we model the Kronecker-structured support with a hierarchical Gaussian prior distribution parameterized by a Kronecker-structured hyperparameter, leading to a non-convex optimization problem. The optimization problem is solved using the alternating minimization (AM) method and a singular value decomposition (SVD)-based method, resulting in two algorithms. Further, we analytically guarantee that the AM-based method converges to the stationary point of the SBL cost function. The SVD-based method, though it adopts approximations, is empirically shown to be more efficient and accurate. We then apply our algorithm to estimate the uplink wireless channel in an intelligent reflecting surface-aided MIMO system and extend the AM-based algorithm to address block sparsity in the channel. We also study the SBL cost to show that the minima of the cost function are achieved at sparse solutions and that incorporating the Kronecker structure reduces the number of local minima of the SBL cost function. Our numerical results demonstrate the effectiveness of our algorithms compared to the state-of-the-art.

This paper presents a novel information-theoretic perspective on generalization in machine learning by framing the learning problem within the context of lossy compression and applying finite blocklength analysis. In our approach, the sampling of training data formally corresponds to an encoding process, and the model construction to a decoding process. By leveraging finite blocklength analysis, we derive lower bounds on sample complexity and generalization error for a fixed randomized learning algorithm and its associated optimal sampling strategy. Our bounds explicitly characterize the degree of overfitting of the learning algorithm and the mismatch between its inductive bias and the task as distinct terms. This separation provides a significant advantage over existing frameworks. Additionally, we decompose the overfitting term to show its theoretical connection to existing metrics found in information-theoretic bounds and stability theory, unifying these perspectives under our proposed framework.

There is a growing gap between the impressive results of deep image generative models and classical algorithms that offer theoretical guarantees. The former suffer from mode collapse or memorization issues, limiting their application to scientific data. The latter require restrictive assumptions such as log-concavity to escape the curse of dimensionality. We partially bridge this gap by introducing conditionally strongly log-concave (CSLC) models, which factorize the data distribution into a product of conditional probability distributions that are strongly log-concave. This factorization is obtained with orthogonal projectors adapted to the data distribution. It leads to efficient parameter estimation and sampling algorithms, with theoretical guarantees, although the data distribution is not globally log-concave. We show that several challenging multiscale processes are conditionally log-concave using wavelet packet orthogonal projectors. Numerical results are shown for physical fields such as the varphi^4 model and weak lensing convergence maps with higher resolution than in previous works.

The Fisher information is a fundamental concept for characterizing the sensitivity of parameters in neural networks. However, leveraging the full observed Fisher information is too expensive for large models, so most methods rely on simple diagonal approximations. While efficient, this approach ignores parameter correlations, often resulting in reduced performance on downstream tasks. In this work, we mitigate these limitations and propose Generalized Fisher-Weighted SVD (GFWSVD), a post-training LLM compression technique that accounts for both diagonal and off-diagonal elements of the Fisher information matrix, providing a more accurate reflection of parameter importance. To make the method tractable, we introduce a scalable adaptation of the Kronecker-factored approximation algorithm for the observed Fisher information. We demonstrate the effectiveness of our method on LLM compression, showing improvements over existing compression baselines. For example, at a 20 compression rate on the MMLU benchmark, our method outperforms FWSVD, which is based on a diagonal approximation of the Fisher information, by 5 percent, SVD-LLM by 3 percent, and ASVD by 6 percent compression rate.

Large foundation models are becoming ubiquitous, but training them from scratch is prohibitively expensive. Thus, efficiently adapting these powerful models to downstream tasks is increasingly important. In this paper, we study a principled finetuning paradigm -- Orthogonal Finetuning (OFT) -- for downstream task adaptation. Despite demonstrating good generalizability, OFT still uses a fairly large number of trainable parameters due to the high dimensionality of orthogonal matrices. To address this, we start by examining OFT from an information transmission perspective, and then identify a few key desiderata that enable better parameter-efficiency. Inspired by how the Cooley-Tukey fast Fourier transform algorithm enables efficient information transmission, we propose an efficient orthogonal parameterization using butterfly structures. We apply this parameterization to OFT, creating a novel parameter-efficient finetuning method, called Orthogonal Butterfly (BOFT). By subsuming OFT as a special case, BOFT introduces a generalized orthogonal finetuning framework. Finally, we conduct an extensive empirical study of adapting large vision transformers, large language models, and text-to-image diffusion models to various downstream tasks in vision and language.

Finding the mode of a high dimensional probability distribution D is a fundamental algorithmic problem in statistics and data analysis. There has been particular interest in efficient methods for solving the problem when D is represented as a mixture model or kernel density estimate, although few algorithmic results with worst-case approximation and runtime guarantees are known. In this work, we significantly generalize a result of (LeeLiMusco:2021) on mode approximation for Gaussian mixture models. We develop randomized dimensionality reduction methods for mixtures involving a broader class of kernels, including the popular logistic, sigmoid, and generalized Gaussian kernels. As in Lee et al.'s work, our dimensionality reduction results yield quasi-polynomial algorithms for mode finding with multiplicative accuracy (1-epsilon) for any epsilon > 0. Moreover, when combined with gradient descent, they yield efficient practical heuristics for the problem. In addition to our positive results, we prove a hardness result for box kernels, showing that there is no polynomial time algorithm for finding the mode of a kernel density estimate, unless P = NP. Obtaining similar hardness results for kernels used in practice (like Gaussian or logistic kernels) is an interesting future direction.

In representation learning, uniformity refers to the uniform feature distribution in the latent space (i.e., unit hypersphere). Previous work has shown that improving uniformity contributes to the learning of under-represented classes. However, most of the previous work focused on classification; the representation space of imbalanced regression remains unexplored. Classification-based methods are not suitable for regression tasks because they cluster features into distinct groups without considering the continuous and ordered nature essential for regression. In a geometric aspect, we uniquely focus on ensuring uniformity in the latent space for imbalanced regression through two key losses: enveloping and homogeneity. The enveloping loss encourages the induced trace to uniformly occupy the surface of a hypersphere, while the homogeneity loss ensures smoothness, with representations evenly spaced at consistent intervals. Our method integrates these geometric principles into the data representations via a Surrogate-driven Representation Learning (SRL) framework. Experiments with real-world regression and operator learning tasks highlight the importance of uniformity in imbalanced regression and validate the efficacy of our geometry-based loss functions.

Self-predictive unsupervised learning methods such as BYOL or SimSiam have shown impressive results, and counter-intuitively, do not collapse to trivial representations. In this work, we aim at exploring the simplest possible mathematical arguments towards explaining the underlying mechanisms behind self-predictive unsupervised learning. We start with the observation that those methods crucially rely on the presence of a predictor network (and stop-gradient). With simple linear algebra, we show that when using a linear predictor, the optimal predictor is close to an orthogonal projection, and propose a general framework based on orthonormalization that enables to interpret and give intuition on why BYOL works. In addition, this framework demonstrates the crucial role of the exponential moving average and stop-gradient operator in BYOL as an efficient orthonormalization mechanism. We use these insights to propose four new closed-form predictor variants of BYOL to support our analysis. Our closed-form predictors outperform standard linear trainable predictor BYOL at 100 and 300 epochs (top-1 linear accuracy on ImageNet).

Recent advancements in operator-type neural networks have shown promising results in approximating the solutions of spatiotemporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training expenses, and may not always achieve the desired accuracy required in many scientific and engineering disciplines. In this paper, we propose a new Spatiotemporal Fourier Neural Operator (SFNO) that learns maps between Bochner spaces, and a new learning framework to address these issues. This new paradigm leverages wisdom from traditional numerical PDE theory and techniques to refine the pipeline of commonly adopted end-to-end neural operator training and evaluations. Specifically, in the learning problems for the turbulent flow modeling by the Navier-Stokes Equations (NSE), the proposed architecture initiates the training with a few epochs for SFNO, concluding with the freezing of most model parameters. Then, the last linear spectral convolution layer is fine-tuned without the frequency truncation. The optimization uses a negative Sobolev norm for the first time as the loss in operator learning, defined through a reliable functional-type a posteriori error estimator whose evaluation is almost exact thanks to the Parseval identity. This design allows the neural operators to effectively tackle low-frequency errors while the relief of the de-aliasing filter addresses high-frequency errors. Numerical experiments on commonly used benchmarks for the 2D NSE demonstrate significant improvements in both computational efficiency and accuracy, compared to end-to-end evaluation and traditional numerical PDE solvers.

We present a probabilistic data-driven weather model capable of providing an ensemble of high spatial resolution realizations of 87 variables at arbitrary forecast length and ensemble size. The model uses a stretched grid, dedicating 2.5 km resolution to a region of interest, and 31 km resolution elsewhere. Based on a stochastic encoder-decoder architecture, the model is trained using a loss function based on the Continuous Ranked Probability Score (CRPS) evaluated point-wise in real and spectral space. The spectral loss components is shown to be necessary to create fields that are spatially coherent. The model is compared to high-resolution operational numerical weather prediction forecasts from the MetCoOp Ensemble Prediction System (MEPS), showing competitive forecasts when evaluated against observations from surface weather stations. The model produced fields that are more spatially coherent than mean squared error based models and CRPS based models without the spectral component in the loss.

Catastrophic overfitting (CO) in single-step adversarial training (AT) results in abrupt drops in the adversarial test accuracy (even down to 0%). For models trained with multi-step AT, it has been observed that the loss function behaves locally linearly with respect to the input, this is however lost in single-step AT. To address CO in single-step AT, several methods have been proposed to enforce local linearity of the loss via regularization. However, these regularization terms considerably slow down training due to Double Backpropagation. Instead, in this work, we introduce a regularization term, called ELLE, to mitigate CO effectively and efficiently in classical AT evaluations, as well as some more difficult regimes, e.g., large adversarial perturbations and long training schedules. Our regularization term can be theoretically linked to curvature of the loss function and is computationally cheaper than previous methods by avoiding Double Backpropagation. Our thorough experimental validation demonstrates that our work does not suffer from CO, even in challenging settings where previous works suffer from it. We also notice that adapting our regularization parameter during training (ELLE-A) greatly improves the performance, specially in large epsilon setups. Our implementation is available in https://github.com/LIONS-EPFL/ELLE .

With more open-source models available for diverse tasks, model merging has gained attention by combining models into one, reducing training, storage, and inference costs. Current research mainly focuses on model merging for full fine-tuning, overlooking the popular LoRA. However, our empirical analysis reveals that: a) existing merging methods designed for full fine-tuning perform poorly on LoRA; b) LoRA modules show much larger parameter magnitude variance than full fine-tuned weights; c) greater parameter magnitude variance correlates with worse merging performance. Considering that large magnitude variances cause deviations in the distribution of the merged parameters, resulting in information loss and performance degradation, we propose a Decoupled and Orthogonal merging approach(DO-Merging). By separating parameters into magnitude and direction components and merging them independently, we reduce the impact of magnitude differences on the directional alignment of the merged models, thereby preserving task information. Furthermore, we introduce a data-free, layer-wise gradient descent method with orthogonal constraints to mitigate interference during the merging of direction components. We provide theoretical guarantees for both the decoupling and orthogonal components. And we validate through extensive experiments across vision, language, and multi-modal domains that our proposed DO-Merging can achieve significantly higher performance than existing merging methods at a minimal cost. Notably, each component can be flexibly integrated with existing methods, offering near free-lunch improvements across tasks.

As Large Language Models (LLMs) become increasingly prevalent, their security vulnerabilities have already drawn attention. Machine unlearning is introduced to seek to mitigate these risks by removing the influence of undesirable data. However, existing methods not only rely on the retained dataset to preserve model utility, but also suffer from cumulative catastrophic utility loss under continuous unlearning requests. To solve this dilemma, we propose a novel method, called Rotation Control Unlearning (RCU), which leverages the rotational salience weight of RCU to quantify and control the unlearning degree in the continuous unlearning process. The skew symmetric loss is designed to construct the existence of the cognitive rotation space, where the changes of rotational angle can simulate the continuous unlearning process. Furthermore, we design an orthogonal rotation axes regularization to enforce mutually perpendicular rotation directions for continuous unlearning requests, effectively minimizing interference and addressing cumulative catastrophic utility loss. Experiments on multiple datasets confirm that our method without retained dataset achieves SOTA performance.

Offline optimization is an important task in numerous material engineering domains where online experimentation to collect data is too expensive and needs to be replaced by an in silico maximization of a surrogate of the black-box function. Although such a surrogate can be learned from offline data, its prediction might not be reliable outside the offline data regime, which happens when the surrogate has narrow prediction margin and is (therefore) sensitive to small perturbations of its parameterization. This raises the following questions: (1) how to regulate the sensitivity of a surrogate model; and (2) whether conditioning an offline optimizer with such less sensitive surrogate will lead to better optimization performance. To address these questions, we develop an optimizable sensitivity measurement for the surrogate model, which then inspires a sensitivity-informed regularizer that is applicable to a wide range of offline optimizers. This development is both orthogonal and synergistic to prior research on offline optimization, which is demonstrated in our extensive experiment benchmark.

Autoencoders are a popular model in many branches of machine learning and lossy data compression. However, their fundamental limits, the performance of gradient methods and the features learnt during optimization remain poorly understood, even in the two-layer setting. In fact, earlier work has considered either linear autoencoders or specific training regimes (leading to vanishing or diverging compression rates). Our paper addresses this gap by focusing on non-linear two-layer autoencoders trained in the challenging proportional regime in which the input dimension scales linearly with the size of the representation. Our results characterize the minimizers of the population risk, and show that such minimizers are achieved by gradient methods; their structure is also unveiled, thus leading to a concise description of the features obtained via training. For the special case of a sign activation function, our analysis establishes the fundamental limits for the lossy compression of Gaussian sources via (shallow) autoencoders. Finally, while the results are proved for Gaussian data, numerical simulations on standard datasets display the universality of the theoretical predictions.

In this manuscript we consider the problem of generalized linear estimation on Gaussian mixture data with labels given by a single-index model. Our first result is a sharp asymptotic expression for the test and training errors in the high-dimensional regime. Motivated by the recent stream of results on the Gaussian universality of the test and training errors in generalized linear estimation, we ask ourselves the question: "when is a single Gaussian enough to characterize the error?". Our formula allow us to give sharp answers to this question, both in the positive and negative directions. More precisely, we show that the sufficient conditions for Gaussian universality (or lack of thereof) crucially depend on the alignment between the target weights and the means and covariances of the mixture clusters, which we precisely quantify. In the particular case of least-squares interpolation, we prove a strong universality property of the training error, and show it follows a simple, closed-form expression. Finally, we apply our results to real datasets, clarifying some recent discussion in the literature about Gaussian universality of the errors in this context.

Generative Adversarial Networks (GANs), especially the recent style-based generators (StyleGANs), have versatile semantics in the structured latent space. Latent semantics discovery methods emerge to move around the latent code such that only one factor varies during the traversal. Recently, an unsupervised method proposed a promising direction to directly use the eigenvectors of the projection matrix that maps latent codes to features as the interpretable directions. However, one overlooked fact is that the projection matrix is non-orthogonal and the number of eigenvectors is too large. The non-orthogonality would entangle semantic attributes in the top few eigenvectors, and the large dimensionality might result in meaningless variations among the directions even if the matrix is orthogonal. To avoid these issues, we propose Householder Projector, a flexible and general low-rank orthogonal matrix representation based on Householder transformations, to parameterize the projection matrix. The orthogonality guarantees that the eigenvectors correspond to disentangled interpretable semantics, while the low-rank property encourages that each identified direction has meaningful variations. We integrate our projector into pre-trained StyleGAN2/StyleGAN3 and evaluate the models on several benchmarks. Within only 1% of the original training steps for fine-tuning, our projector helps StyleGANs to discover more disentangled and precise semantic attributes without sacrificing image fidelity.

The weights of neural networks (NNs) have recently gained prominence as a new data modality in machine learning, with applications ranging from accuracy and hyperparameter prediction to representation learning or weight generation. One approach to leverage NN weights involves training autoencoders (AEs), using contrastive and reconstruction losses. This allows such models to be applied to a wide variety of downstream tasks, and they demonstrate strong predictive performance and low reconstruction error. However, despite the low reconstruction error, these AEs reconstruct NN models with deteriorated performance compared to the original ones, limiting their usability with regard to model weight generation. In this paper, we identify a limitation of weight-space AEs, specifically highlighting that a structural loss, that uses the Euclidean distance between original and reconstructed weights, fails to capture some features critical for reconstructing high-performing models. We analyze the addition of a behavioral loss for training AEs in weight space, where we compare the output of the reconstructed model with that of the original one, given some common input. We show a strong synergy between structural and behavioral signals, leading to increased performance in all downstream tasks evaluated, in particular NN weights reconstruction and generation.

Lipschitz neural networks are well-known for providing certified robustness in deep learning. In this paper, we present a novel, efficient Block Reflector Orthogonal (BRO) layer that enhances the capability of orthogonal layers on constructing more expressive Lipschitz neural architectures. In addition, by theoretically analyzing the nature of Lipschitz neural networks, we introduce a new loss function that employs an annealing mechanism to increase margin for most data points. This enables Lipschitz models to provide better certified robustness. By employing our BRO layer and loss function, we design BRONet - a simple yet effective Lipschitz neural network that achieves state-of-the-art certified robustness. Extensive experiments and empirical analysis on CIFAR-10/100, Tiny-ImageNet, and ImageNet validate that our method outperforms existing baselines. The implementation is available at https://github.com/ntuaislab/BRONet.

We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for partial differential equations. Algorithms for approximating the solution to these systems are often the bottleneck in problems that require their solution, particularly for modern applications that require many millions of unknowns. Indeed, numerical linear algebra techniques have been investigated for many decades to alleviate this computational burden. Recently, data-driven techniques have also shown promise for these problems. Motivated by the conjugate gradients algorithm that iteratively selects search directions for minimizing the matrix norm of the approximation error, we design an approach that utilizes a deep neural network to accelerate convergence via data-driven improvement of the search directions. Our method leverages a carefully chosen convolutional network to approximate the action of the inverse of the linear operator up to an arbitrary constant. We train the network using unsupervised learning with a loss function equal to the L^2 difference between an input and the system matrix times the network evaluation, where the unspecified constant in the approximate inverse is accounted for. We demonstrate the efficacy of our approach on spatially discretized Poisson equations with millions of degrees of freedom arising in computational fluid dynamics applications. Unlike state-of-the-art learning approaches, our algorithm is capable of reducing the linear system residual to a given tolerance in a small number of iterations, independent of the problem size. Moreover, our method generalizes effectively to various systems beyond those encountered during training.

Stochastic Gradient Descent (SGD) stands as a cornerstone optimization algorithm with proven real-world empirical successes but relatively limited theoretical understanding. Recent research has illuminated a key factor contributing to its practical efficacy: the implicit regularization it instigates. Several studies have investigated the linear stability property of SGD in the vicinity of a stationary point as a predictive proxy for sharpness and generalization error in overparameterized neural networks (Wu et al., 2022; Jastrzebski et al., 2019; Cohen et al., 2021). In this paper, we delve deeper into the relationship between linear stability and sharpness. More specifically, we meticulously delineate the necessary and sufficient conditions for linear stability, contingent on hyperparameters of SGD and the sharpness at the optimum. Towards this end, we introduce a novel coherence measure of the loss Hessian that encapsulates pertinent geometric properties of the loss function that are relevant to the linear stability of SGD. It enables us to provide a simplified sufficient condition for identifying linear instability at an optimum. Notably, compared to previous works, our analysis relies on significantly milder assumptions and is applicable for a broader class of loss functions than known before, encompassing not only mean-squared error but also cross-entropy loss.

While LLMs have demonstrated impressive performance across various domains and tasks, their security issues have become increasingly severe. Machine unlearning (MU) has emerged as a promising solution to address these issues by removing the influence of undesired data on the target model without compromising its utility in other aspects. MU typically assumes full access to the original training data to preserve utility, which is difficult to achieve in LLM unlearning. Existing LLM unlearning methods often assume access to data most affected by undesired data unlearning. However, this assumption underestimates the entanglement among various LLM capabilities and ignores data access limitations due to various issues. Moreover, these LLM unlearning methods do not sufficiently consider that unlearning requests in real-world scenarios are continuously emerging. To overcome these challenges and achieve practical LLM unlearning, we propose the O3 framework. The O3 framework includes an Out-Of-Distribution (OOD) detector to measure the similarity between input and unlearning data, and an Orthogonal low-rank adapter (LoRA) for continuously unlearning requested data. The OOD detector is trained with a novel contrastive entropy loss and utilizes a local-global layer-aggregated scoring mechanism. The orthogonal LoRA achieves parameter disentanglement among continual unlearning requests. During inference, our O3 framework can smartly decide whether and to what extent to load the unlearning LoRA based on the OOD detector's predictions. Notably, O3's effectiveness does not rely on any retained data. We conducted extensive experiments on O3 and state-of-the-art LLM unlearning methods across three tasks and seven datasets. The results indicate that O3 consistently achieves the best trade-off between unlearning effectiveness and utility preservation, especially when facing continuous unlearning requests.

Artificial Intelligence and Machine learning have been widely used in various fields of mathematical computing, physical modeling, computational science, communication science, and stochastic analysis. Approaches based on Deep Artificial Neural Networks (DANN) are very popular in our days. Depending on the learning task, the exact form of DANNs is determined via their multi-layer architecture, activation functions and the so-called loss function. However, for a majority of deep learning approaches based on DANNs, the kernel structure of neural signal processing remains the same, where the node response is encoded as a linear superposition of neural activity, while the non-linearity is triggered by the activation functions. In the current paper, we suggest to analyze the neural signal processing in DANNs from the point of view of homogeneous chaos theory as known from polynomial chaos expansion (PCE). From the PCE perspective, the (linear) response on each node of a DANN could be seen as a 1^{st} degree multi-variate polynomial of single neurons from the previous layer, i.e. linear weighted sum of monomials. From this point of view, the conventional DANN structure relies implicitly (but erroneously) on a Gaussian distribution of neural signals. Additionally, this view revels that by design DANNs do not necessarily fulfill any orthogonality or orthonormality condition for a majority of data-driven applications. Therefore, the prevailing handling of neural signals in DANNs could lead to redundant representation as any neural signal could contain some partial information from other neural signals. To tackle that challenge, we suggest to employ the data-driven generalization of PCE theory known as arbitrary polynomial chaos (aPC) to construct a corresponding multi-variate orthonormal representations on each node of a DANN to obtain Deep arbitrary polynomial chaos neural networks.

We study the problem of learning hierarchical polynomials over the standard Gaussian distribution with three-layer neural networks. We specifically consider target functions of the form h = g circ p where p : R^d rightarrow R is a degree k polynomial and g: R rightarrow R is a degree q polynomial. This function class generalizes the single-index model, which corresponds to k=1, and is a natural class of functions possessing an underlying hierarchical structure. Our main result shows that for a large subclass of degree k polynomials p, a three-layer neural network trained via layerwise gradient descent on the square loss learns the target h up to vanishing test error in mathcal{O}(d^k) samples and polynomial time. This is a strict improvement over kernel methods, which require widetilde Theta(d^{kq}) samples, as well as existing guarantees for two-layer networks, which require the target function to be low-rank. Our result also generalizes prior works on three-layer neural networks, which were restricted to the case of p being a quadratic. When p is indeed a quadratic, we achieve the information-theoretically optimal sample complexity mathcal{O}(d^2), which is an improvement over prior work~nichani2023provable requiring a sample size of widetildeTheta(d^4). Our proof proceeds by showing that during the initial stage of training the network performs feature learning to recover the feature p with mathcal{O}(d^k) samples. This work demonstrates the ability of three-layer neural networks to learn complex features and as a result, learn a broad class of hierarchical functions.

In this paper we introduce the SCoRe (Submodular Combinatorial Representation Learning) framework, a novel approach in representation learning that addresses inter-class bias and intra-class variance. SCoRe provides a new combinatorial viewpoint to representation learning, by introducing a family of loss functions based on set-based submodular information measures. We develop two novel combinatorial formulations for loss functions, using the Total Information and Total Correlation, that naturally minimize intra-class variance and inter-class bias. Several commonly used metric/contrastive learning loss functions like supervised contrastive loss, orthogonal projection loss, and N-pairs loss, are all instances of SCoRe, thereby underlining the versatility and applicability of SCoRe in a broad spectrum of learning scenarios. Novel objectives in SCoRe naturally model class-imbalance with up to 7.6\% improvement in classification on CIFAR-10-LT, CIFAR-100-LT, MedMNIST, 2.1% on ImageNet-LT, and 19.4% in object detection on IDD and LVIS (v1.0), demonstrating its effectiveness over existing approaches.

This manuscript considers the problem of learning a random Gaussian network function using a fully connected network with frozen intermediate layers and trainable readout layer. This problem can be seen as a natural generalization of the widely studied random features model to deeper architectures. First, we prove Gaussian universality of the test error in a ridge regression setting where the learner and target networks share the same intermediate layers, and provide a sharp asymptotic formula for it. Establishing this result requires proving a deterministic equivalent for traces of the deep random features sample covariance matrices which can be of independent interest. Second, we conjecture the asymptotic Gaussian universality of the test error in the more general setting of arbitrary convex losses and generic learner/target architectures. We provide extensive numerical evidence for this conjecture, which requires the derivation of closed-form expressions for the layer-wise post-activation population covariances. In light of our results, we investigate the interplay between architecture design and implicit regularization.

Invariance learning algorithms that conditionally filter out domain-specific random variables as distractors, do so based only on the data semantics, and not the target domain under evaluation. We show that a provably optimal and sample-efficient way of learning conditional invariances is by relaxing the invariance criterion to be non-commutatively directed towards the target domain. Under domain asymmetry, i.e., when the target domain contains semantically relevant information absent in the source, the risk of the encoder varphi^* that is optimal on average across domains is strictly lower-bounded by the risk of the target-specific optimal encoder Phi^*_tau. We prove that non-commutativity steers the optimization towards Phi^*_tau instead of varphi^*, bringing the H-divergence between domains down to zero, leading to a stricter bound on the target risk. Both our theory and experiments demonstrate that non-commutative invariance (NCI) can leverage source domain samples to meet the sample complexity needs of learning Phi^*_tau, surpassing SOTA invariance learning algorithms for domain adaptation, at times by over 2%, approaching the performance of an oracle. Implementation is available at https://github.com/abhrac/nci.

We propose a simple pairwise sigmoid loss for image-text pre-training. Unlike standard contrastive learning with softmax normalization, the sigmoid loss operates solely on image-text pairs and does not require a global view of the pairwise similarities for normalization. The sigmoid loss simultaneously allows further scaling up the batch size, while also performing better at smaller batch sizes. With only four TPUv4 chips, we can train a Base CLIP model at 4k batch size and a Large LiT model at 20k batch size, the latter achieves 84.5% ImageNet zero-shot accuracy in two days. This disentanglement of the batch size from the loss further allows us to study the impact of examples vs pairs and negative to positive ratio. Finally, we push the batch size to the extreme, up to one million, and find that the benefits of growing batch size quickly diminish, with a more reasonable batch size of 32k being sufficient. We hope our research motivates further explorations in improving the quality and efficiency of language-image pre-training.

This paper proposes a novel approach to address the challenge that pretrained VLA models often fail to effectively improve performance and reduce adaptation costs during standard supervised finetuning (SFT). Some advanced finetuning methods with auxiliary training objectives can improve performance and reduce the number of convergence steps. However, they typically incur significant computational overhead due to the additional losses from auxiliary objectives. To simultaneously achieve the enhanced capabilities of auxiliary training with the simplicity of standard SFT, we decouple the two objectives of auxiliary-objective SFT within the parameter space, namely, enhancing general capabilities and fitting task-specific action distributions. To deliver the goal, we only need to train the model to converge on a small-scale task set using two distinct training strategies, resulting in two finetuned models. The parameters' difference between the two models can then be interpreted as capability vectors provided by auxiliary objectives. These vectors are then merged with pretrained parameters to form a capability-enhanced meta model. Moreover, when standard SFT is augmented with a lightweight orthogonal regularization loss, the merged model attains performance comparable to auxiliary finetuned baselines with reduced computational overhead. Internal and external experiments demonstrate that our capability vectors (1) are effective and versatile across diverse models, (2) can generalize to novel environments and embodiments out of the box.

Machine unlearning, the removal of a training subset's influence from a deployed model, is critical for privacy preservation and model reliability, yet gradient ascent on forget samples often harms retained knowledge. Existing approaches face a persistent tradeoff between effective forgetting and preservation on the retain set. While previous methods provide useful heuristics, they often lack a formal analysis on how exactly forgetting updates harm retained knowledge, and whether the side effects can be removed with theoretical guarantees. To explore a theoretically sound and simple solution, we start from the first principle on how performance on the retain set is actually affected: a first-order analysis of the local change of the retain loss under small parameter updates during model training. We start from a crisp equivalence: the retain loss is unchanged to first order iff the update direction is orthogonal to the subspace spanned by retain gradients ("retain-invariant"). This identifies the entangled component as the tangential part of forget update within the retain-gradient subspace, and characterizes disentanglement as orthogonality. Guided by this, we propose the Geometric-disentanglement Unlearning (GU) that decomposes any candidate forget gradient update into tangential and normal components to retain space and executes only the normal component. Under a standard trust-region budget, the projected direction aligned with the raw forget gradient is optimal among all first-order retain-invariant moves, and we also derive the optimal projected direction for joint forget-retain updating objectives. Our method is plug-and-play and can be attached to existing gradient-based unlearning procedures to mitigate side effects. GU achieves consistent improvement on various methods across three benchmarks TOFU, MUSE, and WMDP.

This paper studies the properties of solutions to multi-task shallow ReLU neural network learning problems, wherein the network is trained to fit a dataset with minimal sum of squared weights. Remarkably, the solutions learned for each individual task resemble those obtained by solving a kernel regression problem, revealing a novel connection between neural networks and kernel methods. It is known that single-task neural network learning problems are equivalent to a minimum norm interpolation problem in a non-Hilbertian Banach space, and that the solutions of such problems are generally non-unique. In contrast, we prove that the solutions to univariate-input, multi-task neural network interpolation problems are almost always unique, and coincide with the solution to a minimum-norm interpolation problem in a Sobolev (Reproducing Kernel) Hilbert Space. We also demonstrate a similar phenomenon in the multivariate-input case; specifically, we show that neural network learning problems with large numbers of tasks are approximately equivalent to an ell^2 (Hilbert space) minimization problem over a fixed kernel determined by the optimal neurons.

As models and data scale, independently trained networks often induce analogous notions of similarity. But, matching similarities is weaker than establishing an explicit correspondence between the representation spaces, especially for multimodal models, where consistency must hold not only within each modality, but also for the learned image-text coupling. We therefore ask: given two independently trained multimodal contrastive models (with encoders (f, g) and (f,g)) -- trained on different distributions and with different architectures -- does a systematic geometric relationship exist between their embedding spaces? If so, what form does it take, and does it hold uniformly across modalities? In this work, we show that across model families such as CLIP, SigLIP, and FLAVA, this geometric relationship is well approximated by an orthogonal map (up to a global mean shift), i.e., there exists an orthogonal map Q where Q^top Q = I such that f(x)approx Q f(x) for paired images x. Strikingly, the same Q simultaneously aligns the text encoders i.e., g(y)approx Q g(y) for texts y. Theoretically, we prove that if the multimodal kernel agrees across models on a small anchor set i.e. langle f(x), g(y)rangle approx langle f(x), g(y)rangle, then the two models must be related by a single orthogonal map Q and the same Q maps images and text across models. More broadly, this finding enables backward-compatible model upgrades, avoiding costly re-embedding, and has implications for the privacy of learned representations. Our project page: https://canonical-multimodal.github.io/

Efficient stochastic optimization typically integrates an update direction that performs well in the deterministic regime with a mechanism adapting to stochastic perturbations. While Adam uses adaptive moment estimates to promote stability, Muon utilizes the weight layers' matrix structure via orthogonalized momentum, showing superior performance in large language model training. We propose a new optimizer and a diagonal extension, NAMO and NAMO-D, providing the first principled integration of orthogonalized momentum with norm-based Adam-type noise adaptation. NAMO scales orthogonalized momentum using a single adaptive stepsize, preserving orthogonality while improving upon Muon at negligible additional cost. NAMO-D instead right-multiplies orthogonalized momentum by a diagonal matrix with clamped entries. This design enables neuron-wise noise adaptation and aligns with the common near block-diagonal Hessian structure. Under standard assumptions, we establish optimal convergence rates for both algorithms in the deterministic setting and show that, in the stochastic setting, their convergence guarantees adapt to the noise level of stochastic gradients. Experiments on pretraining GPT-2 models demonstrate improved performance of both NAMO and NAMO-D compared to the AdamW and Muon baselines, with NAMO-D achieving further gains over NAMO via an additional clamping hyperparameter that balances the competing goals of maintaining a well-conditioned update direction and leveraging fine-grained noise adaptation.