PAPER: INTRODUCTIONProbabilistic inference is concerned with sampling from a distribution defined by an unnormalised density. Contrary to the generative modelling scenario, the learner has access only to an energy function E (𝑥) : R 𝑑 → R, without access to ground truth samples. The aim is to sample from𝑝 target (𝑥) = 𝑒 -E ( 𝑥 ) /𝑍; 𝑍 = ∫ R 𝑑 𝑒 -E ( 𝑥 ) d𝑥 (1)and estimate the (typically intractable) normalising constant 𝑍. The classical solution to this problem is given by Monte Carlo methods such as AIS (Neal, 1998) or MCMC methods such as Hamiltonian Monte Carlo (Neal et al., 2011;Hoffman et al., 2014). However, such methods typically require many sampling steps to converge. In addition, some of them rely on access to ∇ 𝑥 E (𝑥), which may not always be available. Some methods use adaptive proposals within Monte Carlo methods, see, e.g., Bugallo et al. (2017); Gabrié et al. (2022); Samsonov et al. (2022); Midgley et al. (2023). This increases the convergence speed of the chain and improves the sampling quality. However, this approach still suffers from slow mixing, especially when the problem dimension 𝑑 is large and the energy function E (𝑥) has multiple modes.In contrast, the amortised approach to the sampling problem (1) suggests to train a generative model to approximately sample from 𝑝 target , offering a more scalable alternative to Monte Carlo or MCMC algorithms. The successes of diffusion models as distribution approximators (Sohl-Dickstein et al., 2015;Ho et al., 2020;Song et al., 2021) motivate the application of diffusion-based techniques to sampling, giving rise to the diffusion samplers. Unlike diffusion models, they do not assume access to samples from the target density -only to an energy function that can be queried -and are suitable for cases when the target distribution is represented by a black-box energy function. 1There are many algorithms for training diffusion samplers (see §2.1 and related work in
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REVIEW
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**Summary Of The Paper**  
The paper outlines a conceptual framework for "diffusion samplers," aiming to address the challenge of sampling from unnormalized distributions defined by energy functions $ E(x) $. It contrasts traditional Monte Carlo/MCMC approaches—which face scalability and convergence issues—with amortized methods leveraging diffusion models. While diffusion models traditionally require data samples, the paper posits that diffusion samplers instead rely solely on querying the energy function $ E(x) $, enabling applications to black-box energy-based models. However, the manuscript provides no concrete algorithmic formulation, mathematical derivations, or experimental validation, leaving the core ideas speculative and underdeveloped.  

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**Strengths**  
1. **Problem Relevance**: The focus on sampling from unnormalized energy-based distributions is timely and aligns with active research in probabilistic inference and Bayesian computation.  
2. **High-Level Motivation**: The paper correctly identifies limitations of MCMC/AIS (slow convergence, reliance on gradients) and highlights diffusion models’ potential for scalability—a plausible direction worth exploring.  

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**Weaknesses**  
1. **Missing Algorithmic Details (Critical)**: The paper fails to describe how diffusion processes are adapted to energy functions. Key questions remain unanswered: What objective function guides training? How is the energy function incorporated into the forward/noise schedule or reverse diffusion dynamics? Without this, the method’s feasibility cannot be assessed.  
2. **No Experimental Evaluation (Severe)**: There are no results, baselines, or comparisons to existing methods (e.g., adaptive Monte Carlo, other diffusion-based samplers). Claims about scalability or performance lack empirical grounding.  
3. **Ambiguous Novelty (Major)**: The paper asserts diffusion samplers differ from standard diffusion models trained on energy-based objectives but provides no explicit distinction. Recent works (e.g., Song et al., 2021; Zhang et al., 2023) already explore diffusion models for energy functions, suggesting significant overlap or redundancy.  
4. **Lack of Theoretical Analysis**: No guarantees on convergence rates, error bounds, or tractability of the proposed approach are presented. Critical assumptions (e.g., smoothness of $ E(x) $, gradient availability) are left unspecified.  

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**Questions For The Authors**  
1. **Objective Function**: Please clarify the precise loss function used to train diffusion samplers. Is it energy-based (e.g., minimizing $ \| \nabla_x \log p_\theta(x) + \nabla_x E(x) \|^2 $)? If so, how is this derived from the diffusion process?  
2. **Gradient Requirements**: The paper claims diffusion samplers avoid needing ground-truth samples but does not state whether gradients $ \nabla_x E(x) $ are required. If gradients are needed, how does this affect applicability to black-box energies?  
3. **Comparison to Prior Work**: How does the proposed method differ from diffusion models trained via score matching on energy-based densities (e.g., Song et al., 2021)? Please cite and compare to Zhang et al. (2023) and Chen et al. (2023).  
4. **Empirical Results**: Provide details on datasets, baselines, and metrics used in experiments. For example, were synthetic benchmarks (e.g., Gaussian mixtures, spin glasses) or real-world tasks (e.g., protein folding) tested?  

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**Limitations Not Addressed By The Authors**  
- **Reproducibility**: The absence of code, hyperparameter settings, or implementation details renders the work irreproducible.  
- **Ethical Considerations**: No discussion of ethical implications (e.g., misuse of sampling techniques in sensitive domains) is included.  
- **Scalability Claims**: The paper does not quantify computational costs (e.g., time per sample, memory usage) or provide ablation studies on diffusion steps/noise schedules.  
- **Robustness**: The method’s behavior on high-dimensional/multimodal $ E(x) $ remains entirely unexplored.  

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**Soundness**: 1 (Poor) – The paper presents no formal theory, algorithm, or experiment to validate its claims.  
**Contribution**: 1 (Minimal) – The concept is vague and overlaps significantly with prior work.  
**Confidence**: 1 (Not sure) – Insufficient information exists to evaluate the paper’s validity or originality.  
**Rating**: 1 (Reject) – The submission is incomplete, lacks rigor, and fails to meet basic requirements for publication.  

**Brief Justification For Rating**: The paper reads as a conceptual outline rather than a substantive contribution. Its central idea is neither mathematically grounded nor empirically validated, and it duplicates prior efforts without clarifying its unique value. Without essential components (algorithm, experiments, theory), it cannot be considered for acceptance.

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