2511.0006 Multi-Agent Adaptive Variance Reduction Technique for Decentralized Nonsmooth Nonconvex Stochastic Optimization v1

🎯 ICAIS2025 Accepted Paper
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📋 Summary

This paper introduces MAAVRT, a decentralized zeroth-order algorithm designed for nonsmooth nonconvex optimization. The core contribution lies in the integration of three key components: randomized smoothing, adaptive variance reduction, and topology-aware consensus. The algorithm aims to address the challenges of high variance inherent in zeroth-order methods, particularly within a decentralized setting. The authors propose a modular convergence analysis that decomposes the convergence error into four explicit components: optimization error, smoothing bias, variance error, and consensus disagreement. This decomposition allows for a detailed understanding of how each component contributes to the overall convergence rate. The theoretical analysis demonstrates that MAAVRT achieves a sample complexity of $O(d ext{δ}^{-1} e^{-3})$, which matches known lower bounds up to network factors, indicating near-optimal performance. The algorithm's adaptive variance reduction mechanism is designed to mitigate the high variance associated with zeroth-order gradient estimates. This is achieved by maintaining an exponential moving average of recent gradient estimates, where the weight of the moving average adapts based on the local gradient magnitude. The topology-aware consensus component leverages the network's spectral gap to facilitate efficient information aggregation across the decentralized system. The algorithm's performance is evaluated on three standard benchmark datasets: IJCNN, COVTYPE, and A9A. The empirical results demonstrate that MAAVRT achieves substantially lower gradient norms and higher test accuracy compared to baseline methods, including a decentralized zeroth-order method without variance reduction (DGFM) and a decentralized first-order method (DPSGD). These findings suggest that the proposed adaptive variance reduction mechanism and topology-aware consensus approach are effective in improving the performance of decentralized zeroth-order optimization. The paper's significance lies in its theoretical analysis, which provides a modular framework for understanding the convergence of decentralized zeroth-order algorithms. The proposed algorithm, MAAVRT, offers a practical approach to addressing the challenges of nonsmooth nonconvex optimization in decentralized settings, achieving near-optimal sample complexity and demonstrating strong empirical performance on standard benchmarks. The paper also highlights the importance of considering network topology in decentralized optimization and provides a rigorous analysis of the impact of the spectral gap on convergence. Overall, the paper makes a valuable contribution to the field of decentralized optimization by providing a novel algorithm with strong theoretical guarantees and empirical validation.

✅ Strengths

The paper presents a well-structured and clearly written account of a novel decentralized zeroth-order optimization algorithm, MAAVRT. The core strength of this work lies in its theoretical analysis, which provides a modular decomposition of the convergence error into four distinct components: optimization error, smoothing bias, variance error, and consensus disagreement. This decomposition offers a clear and intuitive understanding of the factors influencing the algorithm's performance. The authors' ability to achieve a sample complexity of $O(d ext{δ}^{-1} e^{-3})$ that matches known lower bounds up to network factors is a significant achievement, demonstrating the near-optimal nature of the proposed algorithm. The integration of randomized smoothing, adaptive variance reduction, and topology-aware consensus is another notable strength. The adaptive variance reduction mechanism effectively addresses the high variance inherent in zeroth-order gradient estimates, while the topology-aware consensus component leverages the network's spectral gap to facilitate efficient information aggregation. The empirical results presented in the paper further support the theoretical findings. The experiments on standard benchmark datasets (IJCNN, COVTYPE, and A9A) demonstrate that MAAVRT achieves substantially lower gradient norms and higher test accuracy compared to baseline methods. This provides strong evidence for the practical effectiveness of the proposed algorithm. The paper's clear presentation of the algorithm's components and the detailed theoretical analysis make it accessible and easy to follow. The authors have successfully combined several existing techniques in a novel way to address the challenging problem of decentralized nonsmooth nonconvex optimization. The modular nature of the convergence analysis is particularly noteworthy, as it provides a framework for analyzing other decentralized zeroth-order algorithms and suggests clear directions for further improvements. The paper also highlights the importance of considering network topology in decentralized optimization and provides a rigorous analysis of the impact of the spectral gap on convergence. The connection between randomized smoothing and the Goldstein subdifferential, as discussed in the theoretical implications section, is another valuable contribution, providing a rigorous foundation for applying zeroth-order methods to nonsmooth objectives. Overall, the paper's strengths lie in its theoretical rigor, its practical effectiveness, and its clear and well-organized presentation.

❌ Weaknesses

While the paper presents a novel algorithm with promising theoretical and empirical results, several limitations warrant careful consideration. A primary concern, consistently raised by all reviewers, is the assumption of synchronous communication. The algorithm's design relies on simultaneous updates across all agents, which is unrealistic in many real-world decentralized systems. The paper's analysis does not account for the potential impact of stragglers or intermittent communication failures, which are common in practical decentralized settings. This reliance on synchronous updates makes the algorithm vulnerable to delays caused by slower agents, potentially leading to stale gradient information and hindered convergence. The algorithm description, particularly in Algorithm 1, line 18, explicitly shows that each agent updates its state by communicating with its neighbors simultaneously, which implies a synchronous communication model. The paper acknowledges this limitation in the discussion section, stating, 'our experiments assume synchronous communication, but many decentralized systems operate asynchronously with communication delays and agent failures. Extending MAAVRT to handle asynchrony and Byzantine faults is an important direction for practical applications.' However, the paper does not provide any analysis or experimental results to address this issue. This limitation significantly restricts the practical applicability of the proposed algorithm in realistic decentralized environments. Another significant weakness, also highlighted by multiple reviewers, is the memory overhead associated with the moving average buffer used for variance reduction. The paper states that 'the variance reduction mechanism requires maintaining moving-average buffers, which adds O(d) memory per agent.' This O(d) memory requirement per agent can be a substantial limitation for large-scale problems or resource-constrained agents. The paper does not provide a detailed analysis of the memory requirements as a function of the buffer size and the dimensionality of the problem. Furthermore, it does not explore alternative variance reduction techniques that might be more memory-efficient, such as stochastic averaging or adaptive learning rates. The paper also lacks a detailed analysis of how the buffer size affects the convergence rate and overall performance of the algorithm. This lack of analysis makes it difficult to assess the practical implications of the memory overhead and to determine the optimal buffer size for different problem settings. The experimental evaluation, while demonstrating the effectiveness of MAAVRT on three standard datasets, is limited in scope. The experiments use a fixed ring topology and do not explore the algorithm's performance under different network topologies, such as sparse networks or hierarchical networks. The paper also does not investigate the algorithm's robustness to noise or communication delays. The lack of experiments under asynchronous conditions makes it difficult to assess the algorithm's robustness in realistic decentralized environments. The paper's comparison with baseline methods is also limited, as it only compares against a decentralized zeroth-order method without variance reduction (DGFM) and a decentralized first-order method (DPSGD). A more comprehensive comparison with other state-of-the-art decentralized optimization algorithms, particularly those designed for asynchronous communication or zeroth-order settings, would be beneficial. The paper also lacks a detailed discussion of the algorithm's limitations in terms of scalability, robustness to noisy gradients, and sensitivity to hyperparameter choices. While the paper mentions the matching of lower bounds, it does not discuss the practical implications of these bounds or how they translate to real-world performance. The paper also does not provide a sensitivity analysis of the algorithm's performance with respect to the choice of parameters, such as the smoothing parameter and the step size. This lack of a detailed discussion of limitations and sensitivity analysis makes it difficult to assess the practical applicability of the proposed algorithm in different settings. Finally, the paper does not provide a detailed explanation of the specific implementation differences between MAAVRT and existing methods, particularly those employing variance reduction techniques. While the authors mention adaptive variance reduction and topology-aware consensus as key components, a deeper dive into how these are implemented and how they differ from prior approaches would be valuable. The paper also lacks a detailed explanation of the technical contributions that enable the tighter sample complexity bounds. The authors mention a modular error decomposition and a refined analysis of the smoothing operator, but they do not elaborate on the specific challenges in adapting existing error decomposition techniques to the decentralized setting or provide concrete examples of their application in the analysis. This lack of detailed explanation makes it difficult to fully appreciate the novelty and significance of the proposed algorithm and its theoretical contributions. The confidence level for all these identified weaknesses is high, as they are consistently supported by the paper's content and the reviewers' analyses.

💡 Suggestions

To address the identified weaknesses, several concrete improvements can be made. First and foremost, the authors should explore extensions of MAAVRT that can handle asynchronous communication and communication delays. This could involve incorporating techniques such as gradient tracking or asynchronous variance reduction methods. Specifically, the authors could investigate how to adapt the variance reduction mechanism to account for delayed gradient information, possibly by using time-weighted averaging or by introducing correction terms based on the age of the gradient. Furthermore, the authors should analyze the convergence properties of the algorithm under asynchronous conditions, providing theoretical guarantees for its performance in the presence of communication delays and agent failures. This would involve modifying the convergence analysis to account for the stochastic nature of asynchronous updates and the potential for stragglers. It would also be beneficial to explore the use of fault-tolerant mechanisms to mitigate the impact of agent failures on the overall performance of the algorithm. This would significantly enhance the practical applicability of the proposed algorithm in realistic decentralized environments. Second, the authors should investigate alternative variance reduction techniques that are more memory-efficient. This could involve exploring methods such as stochastic averaging, where only a subset of past gradients are used to estimate the variance, or adaptive learning rate methods that adjust the step size based on the observed variance. The authors should also analyze the trade-off between memory usage and convergence rate for different variance reduction techniques, providing guidance on how to choose the most appropriate method for a given problem and resource constraint. Additionally, the authors could explore techniques for compressing or sparsifying the gradient information to reduce the memory footprint of the algorithm. A detailed analysis of the memory requirements and how they scale with the problem dimension and the number of agents would also be beneficial. This would help to mitigate the memory overhead associated with the moving average buffer and make the algorithm more suitable for large-scale problems and resource-constrained agents. Third, the authors should significantly expand the experimental evaluation to include a wider range of datasets and problem settings. They should consider including experiments on datasets with different characteristics, such as high dimensionality, non-linear relationships, and imbalanced classes. They should also explore the algorithm's performance under different network topologies, such as sparse networks and hierarchical networks. Furthermore, the authors should investigate the algorithm's robustness to noise and communication delays. The experiments should also include a comparison with other state-of-the-art decentralized optimization algorithms, providing a more comprehensive evaluation of the algorithm's performance. The authors should also provide a more detailed analysis of the experimental results, including a discussion of the statistical significance of the observed differences. This would provide a more robust and comprehensive evaluation of the algorithm's performance and its applicability to different problem settings. Fourth, the authors should provide a more detailed discussion of the algorithm's limitations, including scalability, robustness to noisy gradients, and sensitivity to hyperparameter choices. They should also discuss the practical implications of the theoretical bounds and how they translate to real-world performance. A sensitivity analysis of the algorithm's performance with respect to the choice of parameters, such as the smoothing parameter and the step size, would also be beneficial. This would provide a more balanced view of the proposed algorithm and its applicability. Finally, the authors should provide a more detailed explanation of the specific implementation differences between MAAVRT and existing methods, particularly those employing variance reduction techniques. They should also elaborate on the technical contributions that enable the tighter sample complexity bounds, providing concrete examples of how these techniques are applied in the analysis. This would significantly enhance the reader's understanding of the algorithm's novelty and practical advantages.

❓ Questions

Several key questions arise from my analysis of this paper. First, how can the MAAVRT algorithm be modified to effectively handle asynchronous communication and communication delays in real-world decentralized systems? Specifically, what are the challenges in adapting the variance reduction mechanism to account for delayed gradient information, and what are the potential trade-offs between convergence rate and robustness to asynchrony? Second, what is the precise impact of the moving average buffer on the memory usage of the algorithm, especially for large-scale problems or resource-constrained agents? How does the buffer size affect the convergence rate and overall performance of the algorithm, and what are the optimal strategies for choosing the buffer size in different problem settings? Are there alternative variance reduction techniques that can achieve comparable performance with lower memory overhead? Third, how does the algorithm perform in asynchronous communication settings, and what are the potential modifications required to extend it to such settings? What are the specific challenges in adapting the consensus mechanism to asynchronous updates, and how can the algorithm be made robust to stragglers and intermittent communication failures? Fourth, what are the specific algorithmic differences between MAAVRT and existing methods, particularly those employing variance reduction techniques? How does the specific adaptation mechanism in MAAVRT differ from other adaptive gradient methods, and what are the precise modifications made to standard variance reduction techniques to accommodate the decentralized setting? Finally, what are the key technical contributions that lead to the tighter bounds achieved by MAAVRT? How does the modular error decomposition and the refined analysis of the smoothing operator differ from existing approaches, and how do these techniques enable a more efficient control of smoothing bias without sacrificing convergence rate? These questions target core methodological choices and seek clarification of critical assumptions, aiming to further understand the algorithm's practical applicability and limitations.

📊 Scores

Soundness:3.0
Presentation:3.0
Contribution:3.0
Rating: 6.0

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📋 Summary

The paper proposes MAAVRT, a decentralized zeroth-order algorithm for nonsmooth nonconvex stochastic optimization. MAAVRT combines (i) randomized smoothing via a two-point estimator (Equation (2)), (ii) an adaptive exponential moving average with an anchor-based correction for variance reduction (Equations (3)–(4)), and (iii) topology-aware consensus using a symmetric, doubly-stochastic mixing matrix W (Equation (5)). The analysis decomposes error into optimization, smoothing bias, estimator variance, and consensus disagreement, and claims sample complexity O(d δ^{-1} ε^{-3}) for achieving E[||g_μ(x)||] ≤ ε and consensus gap ≤ δ (Theorem 3.1; Equations (6)–(7)). Empirically, on IJCNN, COVTYPE, and A9A with 20 agents on a ring, MAAVRT is reported to reduce gradient norms and improve test accuracy versus DGFM (a zeroth-order baseline) and DPSGD (first-order, used as an upper bound), with fixed σ=0.1 and anchor period M=10 (Section 4). Practical considerations and limitations involve parameter tuning for σ, synchronous communication, and memory overhead (Section 5.2).

✅ Strengths

  • Addresses an important and challenging regime: decentralized, nonsmooth, nonconvex optimization with only function-value queries.
  • Clear algorithmic structure combining smoothing, adaptive variance reduction, and consensus; parallelizable and fully decentralized (Algorithm 1).
  • Theoretical goal is ambitious: near-optimal sample complexity O(d δ^{-1} ε^{-3}), with modular error decomposition (Section 3.4).
  • Empirical results show substantial improvements in reported gradient norms and test accuracy on multiple datasets (Figure 1; Table 1).
  • Limitations and practical considerations are discussed, including tuning of σ and synchronous assumptions (Section 5.2).

❌ Weaknesses

  • Potentially incorrect zeroth-order estimator scaling in Equation (2): for Gaussian smoothing with u ~ N(0, σ^2 I), the unbiased gradient estimator typically uses 1/σ^2 (not d/σ^2), while the d factor is characteristic of uniform-sphere estimators; this mismatch threatens the correctness of the analysis.
  • The variance-reduction mechanism in Equation (4) (EMA with an anchor) is nonstandard and asserted to yield Var = O(1/t) without a clear derivation; it is unclear whether the estimator remains unbiased (or with controlled bias) and how the variance decays under decentralization.
  • Consensus analysis appears optimistic: claiming purely exponential decay O(exp(-c t κ)) overlooks steady-state error due to stochastic noise and gradient steps, unless stepsizes vanish or gradient tracking is used. Theorem 3.1 and the proof sketch do not specify network-dependent constants or steady-state terms.
  • Notation and assumptions are inconsistent: σ denotes both the smoothing parameter and in (A2) the oracle noise variance bound; Theorem 3.1 uses μ for smoothing with μ ≤ δ (A5) while the algorithm uses σ. The paper claims a refined link to Goldstein-subdifferential stationarity (Section 2.2), but no formal statement/lemma is provided.
  • The claimed near-optimality (removing a sqrt(d) penalty) is not rigorously substantiated in the current text; explicit constants and precise lower-bound comparisons (in the decentralized setting with δ) are missing.
  • Experimental clarity and fairness: unclear how gradient norm ||∇f(x)|| is computed under a zeroth-order (black-box) assumption; tuning protocols for σ, M, and step sizes across methods are not fully detailed; limited baselines (missing recent ZO VR methods), no multi-seed variance, and possible artifacts in Table 1 (learning rate columns shown as '-' multiple times).
  • Limited analysis of communication-computation trade-offs; no ablation on consensus frequency, network topology spectrum (κ), or anchor period M.

❓ Questions

  • Zeroth-order estimator (Eq. (2)): With u_i^(t) ~ N(0, σ^2 I), why is the scaling d/σ^2 used? Standard Gaussian smoothing yields ∇f_σ(x) = (1/σ^2) E[(f(x+z) - f(x)) z] with z ~ N(0, σ^2 I). Please clarify the distribution and scaling to ensure unbiasedness.
  • Variance reduction (Eq. (4)): Is \tilde{g}_i^(t) an unbiased estimator of ∇f_i^σ(x_i^(t))? Please provide a detailed variance analysis showing the claimed Var(\tilde{g}_i^(t)) = O(1/t) in the decentralized setting, including dependence on σ, d, and κ.
  • Consensus error: Under persistent stochastic noise and nonvanishing stepsizes, how do you avoid a nonzero steady-state disagreement term? Can you provide explicit bounds that separate the exponential mixing component and the noise-induced floor, with their dependence on κ and stepsizes?
  • Lower bound matching: Which precise lower bound (problem class, oracle model, stationarity notion) does O(d δ^{-1} ε^{-3}) match? Is it centralized or decentralized? Please include a theorem-level statement and a direct comparison including network-dependent factors.
  • Smoothing-to-subdifferential: The paper claims a refined conversion to Goldstein-subdifferential stationarity (Section 2.2). Could you provide the formal lemma/theorem, explicit constants, and how this avoids a sqrt(d) penalty?
  • Evaluation of gradient norm: How is ||∇f(x)|| computed in experiments under black-box access? If approximated, what estimator and budget are used? Alternatively, if exact gradients are computed for evaluation, please discuss the implications for fairness.
  • Tuning protocol: What is the hyperparameter search space for each method (σ, M, stepsize schedules)? Were σ and M tuned for baselines that also require smoothing? Please report sensitivity analyses and ensure comparable tuning budgets.
  • Reproducibility: Could you provide details on random seeds, number of independent runs, confidence intervals, and computational budget? Are code and scripts available?
  • Network dependence: Can you include experiments varying the spectral gap (e.g., ring vs. expander) and report communication rounds and wall-clock time? How does performance scale with κ?
  • Ablations: Please include ablations on the EMA parameter α_t, anchor period M, and the VR coefficient η_vr to substantiate the contribution of each component.

⚠️ Limitations

  • Dependence on problem-specific constants for tuning (σ, stepsizes), acknowledged in Section 5.2; this may hinder deployment without adaptive, constant-free schemes.
  • Synchronous communication assumption; no treatment of delays, asynchrony, or Byzantine robustness (Section 5.2).
  • Memory overhead O(d) per agent to maintain moving-average buffers can be prohibitive in very high dimensions (Section 5.2).
  • Experimental scope: limited baselines, lack of multi-seed statistics, and no communication metrics make it hard to assess robustness and practicality.
  • Potential mis-specification of the estimator scaling (Eq. (2)) and incomplete theoretical details (variance analysis, consensus bounds) threaten the soundness of the claimed guarantees.
  • Societal implications are limited here; however, federated/decentralized optimization has privacy/security considerations (e.g., information leakage via messages) not discussed.

🖼️ Image Evaluation

Cross‑Modal Consistency: 28/50

Textual Logical Soundness: 18/30

Visual Aesthetics & Clarity: 10/20

Overall Score: 56/100

Detailed Evaluation (≤500 words):

1. Cross‑Modal Consistency

• Major 1: Iteration budget mismatch. Text states 30,000 iterations, while all shown plots have Iteration axis 0–100, and claims like “<0.01 within 10,000 iters” cannot be verified. Evidence: “All methods … evaluated over 30,000 iterations” (Sec. 4.1 Baselines) vs. Fig. 1 panels with x‑axis 0–100.

• Major 2: Table‑learning‑rate mapping missing; the “Learning Rate” column shows “–” so rows cannot be associated with η∈{1e‑2,1e‑3,1e‑4}. Evidence: Table 1 first column entries are “-” while Sec. 4.1 says “We sweep learning rates over {10^{-2},10^{-3},10^{-4}}”.

• Minor 1: Extra “Final Performance Comparison” bar chart appears without a figure number or in‑text reference, creating ambiguity whether it is part of Fig. 1. Evidence: image titled “Final Performance Comparison” not cited in prose.

• Minor 2: Table 1 contains repeated values (e.g., 0.3197 across methods), suggesting copy/paste or aggregation ambiguity. Evidence: Table 1 last block shows identical “0.3197”.

2. Text Logic

• Major 1: Notation collision and inconsistency: σ denotes both smoothing radius and oracle noise variance; stationarity and theorem use μ, creating confusion about which parameter is tuned. Evidence: (A2) “noise variance … σ^2”; (A5) defines μ‑smoothing; Theorem 3.1 uses g_μ.

• Major 2: Misstatement about Eq. (2): “scaling factor d/σ^2 compensates for the smoothing‑induced bias” — scaling ensures unbiasedness for the smoothed gradient; it does not remove smoothing bias. Evidence: Sec. 3.2 sentence after Eq. (2).

• Major 3: Evaluation uses ∥∇f(x)∥ while the setting is zeroth‑order; the paper does not describe how this gradient norm is computed/estimated, so claims on stationarity cannot be audited. Evidence: Sec. 4.1 “Gradient norm ∥∇f(x)∥ …” with no estimator protocol.

• Minor 1: Variance‑reduction claim Var(ĝ)=O(1/t) stated without derivation or citation tailored to this construction. Evidence: After Eq. (4): “yielding variance decay O(1/t)”.

3. Figure Quality

• Major 1: Final 8‑panel composite (covtype/a9a) is illegible at print size (axes/legends unreadable). Evidence: last image resolution ≈203×512 px.

• Minor 1: Panels lack (a)/(b)/(c) labels; captions don’t enumerate sub‑figures, hindering cross‑reference. Evidence: Fig. 1 shows three panels with no labels.

• Minor 2: Table 1 promises “highlighted in green,” but no highlighting appears. Evidence: Table 1 rendering.

Key strengths:

  • Clear high‑level algorithm structure and modular error decomposition.
  • Emphasis on topology via spectral gap is well‑motivated.

Key weaknesses:

  • Critical figure‑text mismatches (iteration counts; table labeling).
  • Notation inconsistencies (μ vs. σ; σ reused for noise).
  • Empirical methodology under‑specified (gradient‑norm computation).
  • One core figure illegible; sub‑figure labeling absent.

Recommendations:

  • Fix iteration scales in plots or text; report the actual budget used.
  • Populate Table 1 with explicit η values; add uncertainty (mean±std, seeds).
  • Unify notation (use μ for smoothing; different symbol for noise).
  • Correct Eq. (2) discussion; detail how ∥∇f(x)∥ is estimated.
  • Relabel panels (a–d), increase font sizes, and provide high‑resolution composites.

📊 Scores

Originality:3
Quality:2
Clarity:2
Significance:2
Soundness:2
Presentation:2
Contribution:2
Rating: 3

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📋 Summary

The paper introduces MAAVRT, a decentralized zeroth-order algorithm designed to address the challenges of nonsmooth and nonconvex optimization in a distributed setting. The key methodological approaches include the use of randomized smoothing to approximate gradients, adaptive variance reduction to improve the stability of gradient estimates, and a topology-aware consensus mechanism to ensure agreement among agents in the network. The theoretical analysis is structured around a modular error decomposition, which allows for a clear understanding of the contributions of each component to the overall convergence rate. The authors claim that MAAVRT achieves a sample complexity of $O(d

u^{-1}

g^{-3})$, matching the known lower bounds for this class of problems. Empirically, the paper demonstrates the effectiveness of MAAVRT on three standard machine learning datasets: IJCNN, COVTYPE, and A9A. The results show that MAAVRT outperforms a basic decentralized zeroth-order method (DGFM) and a decentralized first-order method (DPSGD) in terms of gradient norm, training loss, and test accuracy. However, the paper's significance is somewhat undermined by the limited scope of the experimental validation and the lack of comparison with more recent and relevant baselines. Despite these limitations, the paper makes a valuable contribution to the field of decentralized optimization, particularly in the context of zeroth-order methods and nonsmooth, nonconvex objectives.

✅ Strengths

One of the paper's core strengths lies in its theoretical analysis, which is both novel and insightful. The modular error decomposition into optimization, smoothing, variance, and consensus components provides a clear and structured understanding of the algorithm's performance. This decomposition is particularly useful for identifying the key factors that influence convergence and for designing more efficient algorithms in the future. The authors' ability to achieve a near-optimal sample complexity that matches known lower bounds is a significant theoretical achievement, especially given the challenging nature of decentralized nonsmooth nonconvex optimization with zeroth-order queries. The paper also demonstrates the practical effectiveness of MAAVRT through empirical validation on standard benchmarks. The results show that MAAVRT outperforms DGFM and DPSGD in terms of gradient norm, training loss, and test accuracy, which is a promising indication of the algorithm's potential in real-world applications. The inclusion of a topology-aware consensus mechanism is another strong point, as it addresses the practical challenges of communication in decentralized networks. The adaptive variance reduction technique, which uses moving average buffers to reduce estimator variance online, is a technical innovation that enhances the stability and efficiency of the algorithm. Overall, the paper makes a valuable contribution to the field by integrating these techniques and providing a comprehensive theoretical and empirical analysis.

❌ Weaknesses

Despite the paper's strengths, several limitations and weaknesses are evident. One of the most significant concerns is the limited novelty of the proposed algorithm. The core components of MAAVRT—randomized smoothing, adaptive variance reduction, and topology-aware consensus—are well-established techniques in the literature. The paper does not provide a detailed comparison with existing methods that combine similar techniques, such as those found in recent works on decentralized and zeroth-order optimization. This lack of comparison makes it difficult to assess the unique contributions of MAAVRT and its advantages over existing approaches. For instance, the paper cites works like Lin et al. (2024) and Kornowski & Shamir (2024) but does not explicitly discuss how MAAVRT's integration of these techniques differs from or improves upon these prior works. The theoretical analysis, while sound, is also somewhat limited in its scope. The paper focuses on the (C̃-stationarity) measure, which is a common choice in the literature, but does not provide a clear justification for why this measure is more meaningful than other stationarity measures, such as the Goldstein subdifferential. This omission leaves the reader questioning the practical relevance of the theoretical results. Additionally, the paper's claim of achieving near-optimal sample complexity is not fully supported by a detailed comparison with the most relevant lower bounds. The authors state that the complexity $O(d

u^{-1}

g^{-3})$ matches known lower bounds, but they do not provide a rigorous analysis of the constants and assumptions underlying these bounds. Another critical weakness is the experimental setup. The experiments are conducted on relatively simple machine learning tasks with low-dimensional datasets (IJCNN: 22 features, COVTYPE: 54 features, A9A: 123 features). These datasets do not fully capture the challenges of high-dimensional and complex optimization problems, which are more common in real-world applications. The choice of baselines is also problematic. The paper compares MAAVRT with DGFM, a basic decentralized zeroth-order method, and DPSGD, a decentralized first-order method. However, DGFM is not a state-of-the-art algorithm, and DPSGD assumes the availability of gradients, which is not a fair comparison in the context of zeroth-order methods. The paper would benefit from including more recent and relevant baselines, such as those from Lin et al. (2024) and Kornowski & Shamir (2024), to provide a more comprehensive evaluation of MAAVRT's performance. Furthermore, the paper lacks a detailed discussion of the practical implications of the algorithm, particularly in terms of computational and communication costs. While the authors mention the per-iteration computational cost of $O(d)$, they do not provide a thorough analysis of how the variance reduction window $M$ and the smoothing parameter $ au$ impact these costs. This omission is significant, as the choice of these parameters can greatly affect the algorithm's efficiency and scalability. The paper also does not address the practical challenges of deploying MAAVRT in asynchronous communication settings, which are common in real-world decentralized systems. The assumption of synchronous communication is a limitation that could be mitigated by extending the analysis to asynchronous settings. Finally, the paper's presentation could be improved. The introduction is somewhat vague and does not clearly articulate the specific limitations of existing methods that MAAVRT aims to address. The related work section is overly technical and does not provide a clear overview of the state-of-the-art. The paper would benefit from a more intuitive explanation of the algorithm's components and a clearer connection between the theoretical results and the empirical findings.

💡 Suggestions

To address the identified weaknesses, I recommend several concrete and actionable improvements. First, the authors should provide a more detailed comparison with existing methods that combine similar techniques, such as those found in recent works on decentralized and zeroth-order optimization. This comparison should highlight the unique contributions of MAAVRT and explain how it differs from or improves upon these prior works. For example, the authors could discuss the specific innovations in the adaptive variance reduction technique and how it addresses the limitations of existing methods. Second, the theoretical analysis should be expanded to include a more comprehensive comparison with lower bounds. The authors should explicitly state the assumptions underlying the lower bounds and provide a detailed analysis of how MAAVRT's complexity matches these bounds. This would strengthen the claim of near-optimality and provide a clearer understanding of the algorithm's theoretical performance. Third, the experimental validation should be significantly enhanced. The authors should include experiments on more complex and high-dimensional tasks, such as non-convex bandit optimization or black-box optimization problems. These experiments would better demonstrate the algorithm's practical effectiveness and robustness. Additionally, the authors should compare MAAVRT with more recent and relevant baselines, such as those from Lin et al. (2024) and Kornowski & Shamir (2024). This would provide a more comprehensive evaluation of MAAVRT's performance and highlight its advantages over state-of-the-art methods. The authors should also conduct an ablation study to isolate the impact of each component of MAAVRT, such as the adaptive variance reduction and topology-aware consensus. This would help to understand the specific contributions of each technique and identify areas for further improvement. Fourth, the paper should include a detailed discussion of the practical implications of the algorithm, particularly in terms of computational and communication costs. The authors should analyze how the variance reduction window $M$ and the smoothing parameter $ au$ affect these costs and provide guidelines for choosing these parameters in practice. This analysis should consider the trade-offs between computational efficiency, communication overhead, and convergence rate. Fifth, the authors should extend the analysis to asynchronous communication settings, which are more common in real-world decentralized systems. This extension would make the algorithm more practical and applicable to a wider range of problems. Finally, the paper's presentation should be improved. The introduction should clearly articulate the specific limitations of existing methods that MAAVRT aims to address. The related work section should be more accessible and provide a clear overview of the state-of-the-art. The authors should also include more intuitive explanations of the algorithm's components and a clearer connection between the theoretical results and the empirical findings. These improvements would make the paper more accessible and impactful.

❓ Questions

1. Could the authors provide a more detailed comparison of MAAVRT with existing methods that combine similar techniques, such as those from Lin et al. (2024) and Kornowski & Shamir (2024)? Specifically, how does the adaptive variance reduction technique in MAAVRT differ from or improve upon the variance reduction methods used in these prior works?

2. What is the rationale behind the choice of the (C̃-stationarity) measure for theoretical analysis? Could the authors provide a clear justification for why this measure is more meaningful than other stationarity measures, such as the Goldstein subdifferential, in the context of decentralized nonsmooth nonconvex optimization?

3. How does the variance reduction window $M$ and the smoothing parameter $ au$ impact the computational and communication costs of MAAVRT? Could the authors provide a detailed analysis of these trade-offs and guidelines for choosing these parameters in practice?

4. Could the authors extend the analysis to asynchronous communication settings? How would the convergence guarantees and sample complexity be affected in such settings, and what modifications to the algorithm would be necessary?

5. What are the specific challenges and limitations of applying MAAVRT to high-dimensional and complex optimization problems? Could the authors provide experimental results on such problems to better demonstrate the algorithm's practical effectiveness and robustness?

📊 Scores

Soundness:2.75
Presentation:2.0
Contribution:2.5
Rating: 4.5

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