Concentration of General Stochastic Approximation Under Heavy-Tailed Markovian Noise

2026-05-21 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, medium

Summary

This paper establishes maximal concentration bounds for stochastic approximation (SA) algorithms, which are fundamental to modern optimization and machine learning, including SGD and Q-learning. The analysis covers algorithms with general step sizes α₊=α/(k+h)⁺ and noise comprising a finite-state Markovian component plus a Martingale-difference component. For bounded Martingale-difference noise, the error tail can be sub-Gaussian (when z=1 and operator is non-expansive, or z ∈ (0,1) and operator is contractive), sub-Weibull (when z=1 generally, or z ∈ (0,1) and operator is non-expansive), or heavier than Weibull but lighter than Pareto (when z ∈ (0,1) and operator is expansive). For unbounded Martingale-difference noise with ℴ(1/k) step sizes and contractive operators, the error tail is at most three times heavier than the noise tail if the operator is almost surely non-expansive, but can be substantially heavier if expansive. The methodology introduces a novel Lyapunov function using a Poisson equation's moment-generating function and a black-box truncation argument.

Key takeaway

For research scientists designing or analyzing stochastic approximation algorithms, understanding the interplay between noise characteristics, step sizes, and operator properties is crucial. Your choice of step size exponent z and the operator's contractiveness directly dictate the error tail distribution, ranging from sub-Gaussian to substantially heavier. You should account for these factors to accurately predict algorithm stability and confidence levels, especially when dealing with heavy-tailed or Markovian noise.

Key insights

The concentration of SA iterates under heavy-tailed Markovian noise depends critically on step sizes and operator properties.

Principles

• SA error tails vary from sub-Gaussian to Pareto-like.
• Operator contractiveness impacts tail heaviness.
• Step size exponent z dictates error distribution.

Method

The analysis uses a novel Lyapunov function based on the moment-generating function of a Poisson equation solution, an auxiliary projected algorithm, and a black-box truncation for unbounded noise.

In practice

• Tail bounds inform algorithm stability and confidence.
• Consider operator type for noise sensitivity.
• Step size choice impacts error distribution.

Topics

• Stochastic Approximation
• Markovian Noise
• Heavy-Tailed Noise
• Concentration Bounds
• Lyapunov Functions
• Stochastic Gradient Descent

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.