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Decentralized Nonconvex Optimization under Heavy-Tailed Noise: Normalization and Optimal Convergence

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
Stochastic optimizationheavy-tailed noisedecentralized optimizationnormalization
Abstract:

Heavy-tailed noise in nonconvex stochastic optimization has garnered increasing research interest, as empirical studies, including those on training attention models, suggest it is a more realistic gradient noise condition. This paper studies first-order nonconvex stochastic optimization under heavy-tailed gradient noise in a decentralized setup, where each node can only communicate with its direct neighbors in a predefined graph. Specifically, we consider a class of heavy-tailed gradient noise that is zero-mean and has only pp-th moment for p∈(1,2]p \in (1, 2]. We propose GT-NSGDm, Gradient Tracking based Normalized Stochastic Gradient Descent with momentum, that utilizes normalization, in conjunction with gradient tracking and momentum, to cope with heavy-tailed noise on distributed nodes. We show that, when the communication graph admits primitive and doubly stochastic weights, GT-NSGDm guarantees, for the \textit{first} time in the literature, that the expected gradient norm converges at an optimal non-asymptotic rate O(1/T(p−1)/(3p−2))O\big(1/T^{(p-1)/(3p-2)}\big), which matches the lower bound in the centralized setup. When tail index pp is unknown, GT-NSGDm attains a non-asymptotic rate O(1/T(p−1)/(2p))O\big( 1/T^{(p-1)/(2p)} \big) that is, for p<2p < 2, topology independent and has a speedup factor n1−1/pn^{1-1/p} in terms of the number of nodes nn. Finally, experiments on nonconvex linear regression with tokenized synthetic data and decentralized training of language models on a real-world corpus demonstrate that GT-NSGDm is more robust and efficient than baselines.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
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Overview

Overall Novelty Assessment

The paper proposes GT-NSGDm, a gradient-tracking algorithm with normalization and momentum for decentralized nonconvex optimization under heavy-tailed noise. It sits in the 'Decentralized Nonconvex Optimization' leaf of the taxonomy, which contains only this paper as a sibling. This leaf is part of the broader 'Decentralized and Federated Optimization Frameworks' branch, which includes six distinct leaves covering federated learning, minimax formulations, and adaptive methods. The sparse population of the specific leaf suggests this precise combination—gradient tracking, normalization, and momentum in peer-to-peer networks under heavy tails—has received limited prior attention in the examined literature.

The taxonomy reveals neighboring work in adjacent leaves: 'Federated Learning under Heavy-Tailed Noise' addresses server-client architectures rather than peer-to-peer graphs, while 'Decentralized Clipping with Error Feedback' explores clipping-based stabilization instead of normalization. The 'Gradient Noise Mitigation Techniques' branch contains normalization-focused methods but in centralized settings. The paper's use of gradient tracking to maintain consensus across a communication graph distinguishes it from federated schemes, and its normalization approach diverges from clipping-based decentralized methods. This positioning suggests the work bridges two research directions: normalization strategies from centralized optimization and consensus mechanisms from distributed systems.

Among 21 candidates examined, the first contribution (GT-NSGDm algorithm) shows one refutable candidate out of three examined, indicating some algorithmic overlap in the limited search scope. The second contribution (optimal convergence rate matching centralized lower bounds) examined ten candidates with none clearly refuting it, suggesting this theoretical result may be novel within the search scope. The third contribution (topology-independent rate with unknown tail index) examined eight candidates, also with no refutations. The statistics indicate that while the algorithmic framework has some precedent, the convergence guarantees appear less anticipated among the top-21 semantically similar papers.

Based on the limited search of 21 candidates, the work appears to occupy a relatively sparse intersection of gradient tracking, normalization, and heavy-tailed analysis in decentralized settings. The taxonomy structure confirms that while related techniques exist in isolation—normalization in centralized methods, gradient tracking in standard decentralized optimization—their combination for heavy-tailed noise is less explored. The analysis does not cover exhaustive literature review or manual curation beyond semantic similarity, so additional prior work may exist outside this scope.

Taxonomy

Core-task Taxonomy Papers
18
3
Claimed Contributions
21
Contribution Candidate Papers Compared
1
Refutable Paper

Research Landscape Overview

Core task: Decentralized nonconvex optimization under heavy-tailed gradient noise. The field addresses the challenge of training machine learning models in distributed settings where gradient estimates exhibit heavy-tailed behavior, meaning extreme values occur more frequently than classical Gaussian assumptions predict. The taxonomy organizes research into four main branches. Gradient Noise Mitigation Techniques focus on algorithmic strategies such as clipping and normalization to tame heavy tails, with works like Gradient Normalization Clipping[4] and Gradient Normalization Benefits[5] exploring how to stabilize updates. Decentralized and Federated Optimization Frameworks examine communication-efficient protocols and consensus mechanisms for multi-agent systems, including studies like Heavy-Tail Decentralized SGD[8] and Taming Fat-Tailed Federated[9]. Specialized Problem Structures investigate settings such as bilevel optimization, minimax games, and quantile regression under heavy tails, exemplified by Bilevel Heavy-Tailed Optimization[1] and Federated Minimax Heavy-Tailed[14]. Theoretical Analysis and Phenomena provide convergence guarantees and characterize when standard methods fail, as seen in Nonconvex Heavy-Tails Guarantees[12] and Standard SGD Divergence[17]. A particularly active line of work contrasts gradient clipping versus normalization strategies, with some studies arguing that normalization offers more robust theoretical properties while others highlight the practical simplicity of clipping. Another tension arises between centralized federated schemes and fully decentralized peer-to-peer protocols, where communication topology and privacy constraints shape algorithmic design. Decentralized Heavy-Tailed Normalization[0] sits squarely within the Decentralized and Federated Optimization Frameworks branch, emphasizing normalization-based stabilization in peer-to-peer networks. Compared to Heavy-Tail Decentralized SGD[8], which may rely on clipping or variance reduction, the original paper appears to prioritize normalization techniques akin to those studied in Gradient Normalization Benefits[5], but adapted to decentralized consensus dynamics. This positioning reflects ongoing debates about which noise mitigation strategy best balances convergence speed, communication overhead, and robustness to outliers in distributed nonconvex settings.

Claimed Contributions

GT-NSGDm algorithm for decentralized nonconvex optimization under heavy-tailed noise

The authors develop a decentralized optimization method that combines normalization with momentum variance reduction to handle heavy-tailed gradient noise and uses gradient tracking to address cross-node heterogeneity and the nonlinearity introduced by normalization.

3 retrieved papers
Can Refute
Optimal convergence rate matching centralized lower bound when tail index is known

The authors establish that their algorithm achieves an order-optimal convergence rate that matches the theoretical lower bound from centralized settings, representing the first such result in decentralized optimization under heavy-tailed noise.

10 retrieved papers
Topology-independent convergence rate with speedup when tail index is unknown

The authors prove that when the tail index parameter is not known in advance, their algorithm achieves a convergence rate that does not depend on network topology and exhibits a speedup proportional to the number of nodes, matching the best-known centralized rate without requiring knowledge of the tail index.

8 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Within the taxonomy built over the current TopK core-task papers, the original paper is assigned to a leaf with no direct siblings and no cousin branches under the same grandparent topic. In this retrieved landscape, it appears structurally isolated, which is one partial signal of novelty, but still constrained by search coverage and taxonomy granularity.

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

GT-NSGDm algorithm for decentralized nonconvex optimization under heavy-tailed noise

The authors develop a decentralized optimization method that combines normalization with momentum variance reduction to handle heavy-tailed gradient noise and uses gradient tracking to address cross-node heterogeneity and the nonlinearity introduced by normalization.

Contribution

Optimal convergence rate matching centralized lower bound when tail index is known

The authors establish that their algorithm achieves an order-optimal convergence rate that matches the theoretical lower bound from centralized settings, representing the first such result in decentralized optimization under heavy-tailed noise.

Contribution

Topology-independent convergence rate with speedup when tail index is unknown

The authors prove that when the tail index parameter is not known in advance, their algorithm achieves a convergence rate that does not depend on network topology and exhibits a speedup proportional to the number of nodes, matching the best-known centralized rate without requiring knowledge of the tail index.

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