PAPER: INTRODUCTIONDistributed machine learning, a.k.a. federated learning, has emerged as a dominant paradigm to cope with the increasing computational cost of learning tasks, mainly due to growing model sizes and datasets (Kairouz et al., 2021). Worker machines, holding each a fraction of the training dataset, collaborate over a network to learn an optimal common model over the collection of their datasets. Workers typically collaborate with the help of a central coordinator, that we call server (McMahan et al., 2017). Besides scalability, distributed learning is also helpful in preserving data ownership and sovereignty, since the workers do not have to share their local datasets during the learning.Conventional distributed learning algorithms are known to be vulnerable to misbehaving workers that could behave unpredictably (Blanchard et al., 2017;Kairouz et al., 2021;Guerraoui et al., 2023). Misbehavior may result from software and hardware bugs, data poisoning, or malicious players controlling part of the network. In the parlance of distributed computing, misbehaving workers are referred to as Byzantine (Lamport et al., 1982). Due to the growing influence of distributed learning in critical public-domain applications such as healthcare (Nguyen et al., 2022) and finance (Long et al., 2020), the problem of robustness to misbehaving workers, a.k.a. robust distributed learning, has received significant attention (Yin et al., 2018;Farhadkhani et al., 2022;Karimireddy et al., 2022;Gorbunov et al., 2023;Allouah et al., 2023a;Farhadkhani et al., 2023;El-Mhamdi et al., 2023).Robust distributed learning algorithms primarily rely on robust aggregation, such as coordinatewise trimmed mean (CWTM) (Yin et al., 2018), geometric median (GM) (Chen et al., 2017) and Multi-Krum (MK) (Blanchard et al., 2017). Specifically, in robust distributed gradient descent (Robust-DGD), the server aggregates the workers' local gradients using a robust aggregation method, instead of simply averaging them. This protects the learning from erroneous gradients sent by misbehaving workers. Recent work has made significant improvements over these aggregation techniques by incorporating a pre-aggregation step such as bucketing (Karimireddy et al., 2022;Gorbunov et al., 2023) and nearest-neighbor mixing (NNM) (Allouah et al., 2023a), to tackle gradient dissimilarity resulting from data heterogeneity. The learning guarantee of the resulting Robust-DGD has been proven optimal (Allouah et al., 2023b), i.e., it cannot be improved without additional assumptions under the standard heterogeneity model of (G, B)-gradient dissimilarity (Karimireddy et al., 2020).Despite its theoretical tightness, the empirical success of Robust-DGD has unknowingly relied on pre-aggregation gradient clipping (Mhamdi et al., 2021;Farhadkhani et al., 2022;Allouah et al., 2023a). Specifically, clipping the gradients of the workers prior to aggregation has been observed to sometimes enhance the algorithm's empirical performance in the presence of adversarial workers (as evidenced in Figure 7a). Yet, this improvement lacks a concrete explanation, raising the question of whether the observed benefits of clipping are merely anecdotal. This leads to the natural inquiry: Why, and when, does pre-aggregation clipping improve robustness?In particular, using a constant clipping threshold, referred to as static clipping, has exhibited mixed results. Figure 7 shows that while static clipping effectively mitigates sign-flipping (SF) attacks, it completely fails under label-flipping (LF). This indicates an inherent fragility of static clipping. Indeed, we prove in our work that static clipping breaks the standard (f, κ)-robustness property of an aggregation method (Allouah et al., 2023a). This highlights a key shortcoming of existing empirical results that rely on static clipping (Mhamdi et al., 2021;Farhadkhani et al., 2022;Allouah et al., 2023a). To overcome the limitations of static clipping but preserve its empirical benefits at the same time, we introduce a novel adaptive clipping scheme, termed Adaptive Robust Clipping (ARC).ARC dynamically adjusts the clipping threshold as per the gradients sent by the workers and the fraction of adversarial workers to be tolerated. We demonstrate that integrating ARC into Robust-DGD consistently improves its empirical performance (see Figures 7 and1a), while also preserving the convergence guarantee of the original Robust-DGD algorithm. Moreover, we show that when the model initialization is good, ARC provably improves the robustness of Robust-DGD. The benefits of ARC are more pronounced as the fraction of misbehaving workers approaches the system's breakdown point1 and when the data across the workers is highly heterogeneous. Our key results are summarized below. Critical comparisons to prior work are deferred to Section 6. On the left, we vary the number of adversarial workers. On the right, we vary the initialization conditions by scaling a well-chosen set of initial parameters (CWTM • NNM is used, and f = 1). More details on the experimental setup can be found in Sections 4 and 5.2, and Appendix D. Main results & contributions. We consider a system comprising n workers and a server. The goal is to tolerate up to f adversarial workers.(1) Adaptive robust clipping (ARC). We propose ARC, wherein prior to aggregating the gradients, the server clips the largest k := ⌊2(f /n)(n -f )⌋ gradients using a clipping parameter given by the (Euclidean) norm of the (k + 1)-th largest gradient. It is important to note that in contrast to existing adaptive clipping schemes (Diakonikolas et al., 2020;Abdalla & Zhivotovskiy, 2024), ARC does not require additional a priori information on honest workers' gradients. We prove that ARC preserves the robustness guarantee of the original robust aggregation method.(2) Improved empirical robustness. We conduct experiments on MNIST (Deng, 2012), Fashion-MNIST (Xiao et al., 2017), and CIFAR-10 ( Krizhevsky et al., 2014), across various data heterogeneity settings and adversarial regimes. Our results demonstrate that ARC significantly enhances the performance of state-of-the-art Robust-DGD methods, particularly in scenarios with high data heterogeneity (Figure 1a) and a large number of adversarial workers (Figure 4b).(3) Improved learning guarantee. We demonstrate that ARC possesses an additional property that is not satisfied by classical robust aggregation methods. Specifically, ARC constrains the norm of an adversarial gradient by that of an honest (non-adversarial) gradient. Leveraging this property, we show that ARC circumvents the lower bound established under data heterogeneity in Allouah et al. (2023b), provided the honest gradients are bounded at model initialization. An empirical validation of this insight is shown in Figure 1b. Such model initialization is often satisfiable in practice (Glorot & Bengio, 2010), highlighting the practical relevance of ARC. When the model is arbitrarily initialized, ARC recovers the original convergence guarantee of Robust-DGD in the worst case.
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REVIEW
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# Summary Of The Paper

The paper introduces **Adaptive Robust Clipping (ARC)**, a dynamic gradient clipping technique aimed at enhancing the robustness of distributed learning algorithms in the presence of Byzantine adversaries. Static clipping techniques, commonly used in robust distributed learning, are shown to be fragile under certain adversarial attacks (such as label flipping) and may compromise the robustness guarantees of aggregation methods. The proposed ARC method dynamically adjusts the clipping threshold based on the norms of the gradients from honest workers and the number of adversarial workers $ f $. Theoretical analysis demonstrates that ARC preserves the robustness guarantees of $(f, \kappa)$-robust aggregation methods, and extensive empirical evaluations on datasets like MNIST, Fashion-MNIST, and CIFAR-10 confirm that ARC improves the performance of Robust-DGD, especially in scenarios with high data heterogeneity and a large number of adversarial workers.

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# Strengths

1. **Novelty and Motivation**: The paper identifies a critical flaw in the use of static clipping in robust distributed learning—its inability to handle certain adversarial attacks—and proposes a principled adaptive clipping strategy, ARC, which is motivated by theoretical considerations.

2. **Theoretical Guarantees**: The authors rigorously prove that **ARC preserves the robustness guarantees** of $(f, \kappa)$-robust aggregation methods (Theorem 3.2). This is a strong theoretical contribution, as it ensures that the robustness of the aggregation method is maintained even after clipping.

3. **Empirical Validation Across Scenarios**: The paper evaluates ARC on multiple datasets (MNIST, Fashion-MNIST, CIFAR-10) and considers various levels of data heterogeneity and numbers of adversarial workers. The results show consistent performance gains of ARC over static clipping and standard aggregation methods.

4. **Analysis of Breakdown Point Improvement**: The paper investigates how **ARC improves the breakdown point** of robust aggregation methods under certain initialization conditions. This contributes to understanding the limits of robustness in distributed learning.

5. **Comprehensive Comparison with Prior Work**: The paper discusses the differences between ARC and prior adaptive clipping techniques, emphasizing that ARC does not require a priori knowledge of honest gradients, distinguishing it from earlier approaches.

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# Weaknesses

1. **Limited Breadth of Empirical Evaluation** (High Severity):
   - The empirical evaluation is largely focused on **smaller datasets (MNIST, CIFAR-10)** and relatively **few adversarial workers (up to f = 4)**. There is insufficient exploration of higher values of $ f/n $, such as $ f/n \approx 0.4 $, which are closer to the theoretical limit of robustness.
   - The paper does not evaluate **larger neural networks** or **more complex model architectures**, limiting the generalizability of the results.

2. **Insufficient Statistical Analysis** (Medium Severity):
   - Many of the empirical results (e.g., "ARC consistently improves") are presented without **confidence intervals or p-values**, making it hard to judge the statistical significance of the improvements.
   - The variation in performance across different hyperparameters (e.g., clipping thresholds, initialization scales) is not clearly quantified or discussed in detail.

3. **Ambiguity in Definitions and Criteria** (Medium Severity):
   - The term "good initialization" is referenced frequently, but **no clear definition or quantitative criterion** is provided. This makes it difficult to interpret the results and reproduce the experiments.
   - The phrase "well-chosen set of initial parameters" is used without explanation, leaving ambiguity about how these parameters were selected.

4. **Comparison with Prior Methods** (Low Severity):
   - While the paper contrasts ARC with prior adaptive clipping techniques, it does not clearly explain **how ARC differs fundamentally** from these methods in implementation or performance. This weakens the novelty claim slightly.
   - The comparison with static clipping is based on anecdotal observations rather than systematic ablation studies.

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# Questions For The Authors

1. **Clarify the derivation of Lemma 3.1**: Could you elaborate on the steps taken to conclude that static clipping breaks $(f, \kappa')$-robustness? What assumptions are necessary for this conclusion?

2. **Justify the derivation in Theorem 3.2**: How exactly is the expression $\kappa + 2f/(n-2f)$ derived? Does this imply that the robustness coefficient increases by a multiplicative factor of 3?

3. **Mathematical justification for equation (19)**: In Lemma 5.1, the claim that “at least one of the vectors has its norm equal to the clipping threshold” is based on the clipping threshold being the norm of the $(k+1)$-th largest gradient. Could you provide a more formal mathematical justification for this assertion?

4. **Formal proof of Lemma 5.1**: The argument that the norm of the aggregated output is bounded by the maximum norm of the honest gradients is stated informally. Could you provide a more rigorous derivation?

5. **Address the violation of bounded output property in Lemma C.1**: The counterexamples involving CWTM and GM show that these aggregation methods violate the bounded output property. However, the paper claims that **ARC ensures this property**. Why aren’t CWTM • NNM or GM • NNM affected by this issue?

6. **Report statistical significance**: Could the authors provide **standard deviations or confidence intervals** for the reported results (e.g., in Table 1 and Figures 1–4)?

7. **Clarify the meaning of “good initialization”**: What specific criteria or procedures were used to define “good initialization”? Are these based on prior knowledge or experimental calibration?

8. **Define “well-chosen set of initial parameters”**: How were the initial parameters determined to be “well-chosen” in the experiments?

9. **Specify the values of μ in Section 5.2**: What exact values of $ \mu $ were used in the experiments to scale the initial weights?

10. **Compare with prior adaptive clipping methods**: Could the authors clarify the **key differences** between ARC and prior adaptive clipping methods (e.g., Diakonikolas et al., 2020; Abdalla & Zhivotovskiy, 2024) in terms of assumptions, implementation, and performance?

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# Limitations Not Addressed By The Authors

- **Scalability and Generalization**: The paper does not discuss how ARC might perform in **high-dimensional spaces** or with **very large datasets**, nor does it explore its behavior under **non-iid data distributions** outside the $(G, B)$-gradient dissimilarity framework.

- **Robustness under Arbitrary Initialization**: While the paper shows that ARC improves robustness under good initialization, it does not evaluate its performance under **arbitrarily bad initialization**. This leaves open the question of whether the benefits of ARC are universally applicable.

- **Ethical and Practical Implications**: The paper does not include an **ethics section**, which is becoming standard in top-tier conferences. This omission may reflect a lack of consideration for the ethical implications of deploying robust learning systems in sensitive domains such as healthcare or finance.

- **Reproducibility Details**: Although the paper mentions that experiments are repeated with seeds 1 to 5, it does not provide **full details of the hyperparameter settings** or **exact implementations** of the aggregation methods used (e.g., NNM, CWTM, etc.), which could hinder reproducibility.

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# Soundness

**Score: 3 (Good)** – The paper presents a sound theoretical foundation and conducts reasonable empirical validations. However, the lack of statistical significance testing and limited breadth of experiments reduce the overall confidence in the robustness and generalizability of the results.

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# Contribution

**Score: 3 (Good)** – The paper makes a meaningful contribution by introducing a new adaptive clipping technique, ARC, that preserves the robustness guarantees of aggregation methods. Its empirical results support the theoretical claims, and the discussion of breakdown point improvements adds value. However, the novelty is somewhat diluted by similarities to prior work, and the lack of broad empirical coverage limits the perceived impact.

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# Confidence

**Score: 3 (Moderately Confident)** – The technical arguments appear correct, and the empirical results are plausible. However, the lack of statistical rigor and limited experimental scope reduce my confidence in the robustness and general applicability of the proposed method.

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# Rating

**Score: 7 (Accept)** – The paper presents a solid and well-motivated approach to improving the robustness of distributed learning via adaptive clipping. The theoretical analysis is rigorous, and the empirical results are encouraging. However, the limited experimental breadth and lack of statistical depth prevent it from receiving a higher score.

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# Brief Justification For Rating

The paper introduces a novel and theoretically grounded method, ARC, for improving the robustness of distributed learning. The theoretical proofs are sound, and the empirical results are compelling, especially in high-heterogeneity and high-adversarial-worker settings. However, the lack of statistical rigor, limited experimental breadth, and incomplete discussion of the method's limitations prevent it from being a stronger contribution. Overall, the paper deserves acceptance as it advances the field with a principled and well-analyzed approach.

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