Heavy-Tailed and Long-Range Dependent Noise in Stochastic Approximation: A Finite-Time Analysis
arXiv:2603.19648v1 Announce Type: new Abstract: Stochastic approximation (SA) is a fundamental iterative framework with broad applications in reinforcement learning and optimization. Classical analyses typically rely on martingale difference or Markov noise with bounded second moments, but many practical settings, including finance and communications, frequently encounter heavy-tailed and long-range dependent (LRD) noise. In this work, we study SA for finding the root of a strongly monotone operator under these non-classical noise models. We establish the first finite-time moment bounds in both settings, providing explicit convergence rates that quantify the impact of heavy tails and temporal dependence. Our analysis employs a noise-averaging argument that regularizes the impact of noise without modifying the iteration. Finally, we apply our general framework to stochastic gradient descent (SGD) and gradient play, and corroborate our finite-time analysis through numerical experiments.
arXiv:2603.19648v1 Announce Type: new Abstract: Stochastic approximation (SA) is a fundamental iterative framework with broad applications in reinforcement learning and optimization. Classical analyses typically rely on martingale difference or Markov noise with bounded second moments, but many practical settings, including finance and communications, frequently encounter heavy-tailed and long-range dependent (LRD) noise. In this work, we study SA for finding the root of a strongly monotone operator under these non-classical noise models. We establish the first finite-time moment bounds in both settings, providing explicit convergence rates that quantify the impact of heavy tails and temporal dependence. Our analysis employs a noise-averaging argument that regularizes the impact of noise without modifying the iteration. Finally, we apply our general framework to stochastic gradient descent (SGD) and gradient play, and corroborate our finite-time analysis through numerical experiments.
Executive Summary
This article provides a significant contribution to the field of stochastic approximation by analyzing heavy-tailed and long-range dependent noise in the context of finding the root of a strongly monotone operator. Building on the classical martingale difference or Markov noise models, the authors establish the first finite-time moment bounds for these non-classical noise models. The analysis employs a novel noise-averaging argument that regularizes the impact of noise without modifying the iteration. The findings are illustrated through numerical experiments on stochastic gradient descent (SGD) and gradient play, highlighting the importance of considering heavy-tailed and long-range dependent noise in practical applications. The work has far-reaching implications for reinforcement learning and optimization, particularly in finance and communications.
Key Points
- ▸ Establishes finite-time moment bounds for heavy-tailed and long-range dependent noise
- ▸ Introduces a novel noise-averaging argument for regularizing noise impact
- ▸ Applies the framework to stochastic gradient descent (SGD) and gradient play
- ▸ Provides explicit convergence rates for non-classical noise models
Merits
Strength in mathematical rigor
The article demonstrates a high level of mathematical rigor, with a thorough analysis of heavy-tailed and long-range dependent noise. The use of noise-averaging argument is particularly noteworthy, as it provides a nuanced understanding of the impact of non-classical noise models on stochastic approximation.
Demerits
Limited applicability to specific domains
While the article provides a general framework for analyzing heavy-tailed and long-range dependent noise, its application to specific domains such as finance and communications may require further adaptation and refinement.
Expert Commentary
This article is a significant contribution to the field of stochastic approximation, with far-reaching implications for reinforcement learning and optimization. The use of noise-averaging argument is a novel and interesting approach, providing a nuanced understanding of the impact of non-classical noise models. While the article's focus on heavy-tailed and long-range dependent noise is welcome, its limited applicability to specific domains may require further adaptation and refinement. Nonetheless, the work has the potential to reshape our understanding of stochastic approximation and its applications in finance and communications.
Recommendations
- ✓ Future research should focus on adapting the noise-averaging argument to specific domains such as finance and communications.
- ✓ The article's framework should be further refined to account for more complex noise models and their interactions.
Sources
Original: arXiv - cs.LG
