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.

翻译：随机逼近（SA）是一种基础迭代框架，在强化学习和优化领域具有广泛应用。经典分析通常依赖于鞅差或具有有限二阶矩的马尔可夫噪声，但金融、通信等许多实际场景中常遇到重尾和长程依赖（LRD）噪声。本文针对这些非经典噪声模型，研究了求解强单调算子根的随机逼近问题。我们首次建立了两种场景下的有限时间矩界，给出了量化重尾性和时间依赖性影响的显式收敛速度。分析采用噪声平均化论证，在不修改迭代过程的前提下正则化噪声影响。最后，我们将该通用框架应用于随机梯度下降（SGD）和梯度博弈，通过数值实验验证了有限时间分析的有效性。