Heavy-Tailed and Long-Range Dependent Noise in Stochastic Approximation: A Finite-Time Analysis
| Authors | Siddharth Chandak et al. |
| Year | 2026 |
| Field | Machine Learning |
| arXiv | 2603.19648 |
| Download | |
| Categories | cs.LG, stat.ML |
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.
Engineering Breakdown
Plain English
This paper extends stochastic approximation (SA)—a foundational optimization technique used throughout reinforcement learning and control systems—to handle heavy-tailed and long-range dependent noise, which are common in real-world settings like finance and wireless communications but poorly understood theoretically. The authors prove the first finite-time convergence guarantees for SA under these non-classical noise models, providing explicit rates that show how heavy tails and temporal dependence degrade convergence speed. Their key insight is a noise-averaging argument that dampens noise effects without changing the actual algorithm being run, making it broadly applicable. This bridges a critical gap between classical theory (which assumes bounded second moments and independence) and practical environments where noise violates these assumptions.
Core Technical Contribution
The core novelty is establishing finite-time moment bounds for stochastic approximation under heavy-tailed and long-range dependent noise—two noise models that prior analyses could not handle rigorously. Previous work relied on martingale difference or Markov noise assumptions with bounded second moments, which fails when noise has infinite variance or temporal structure spanning many iterations. The authors introduce a noise-averaging regularization technique that operates purely at the analysis level without modifying the algorithm itself, allowing them to derive convergence rates that explicitly quantify the cost of heavy tails (expressed through moment bounds higher than second moment) and temporal dependence (through autocorrelation metrics). This is the first work to provide computable convergence guarantees under both noise pathologies simultaneously in the strongly monotone operator setting.
How It Works
The algorithm operates on the problem of finding the root of a strongly monotone operator—a generalization of solving equations and convex optimization—using iterative stochastic approximation updates. At each iteration t, the algorithm computes a noisy gradient or operator evaluation and moves in the negative direction with some stepsize, similar to stochastic gradient descent but on operator roots rather than function minimization. The heavy-tailed noise (extreme outliers with probability decaying slowly) and long-range dependence (current noise correlated with distant past noise) would normally cause divergence or very slow convergence. The authors' noise-averaging argument groups consecutive iterates and analyzes the averaged error, which mathematically suppresses the impact of heavy tails through a regularization effect and decouples the long-range dependence structure by examining blocks of iterations rather than individual steps. This transforms an intractable problem into one where standard concentration inequalities apply, yielding explicit convergence rates that depend on the tail index (how heavy the tails are) and the temporal dependence parameter (how far correlations extend).
Production Impact
For engineers building reinforcement learning systems in finance or communications, this work validates using standard SA algorithms in environments with extreme noise and temporal structure without custom modifications—you get theoretical guarantees that your algorithm will converge despite noise pathologies, with understood convergence rates. In production financial data pipelines, where price changes exhibit heavy tails and autocorrelation over hours or days, this means you can use classical stochastic optimization (like for portfolio optimization or option pricing) without developing specialized heavy-tailed algorithms, reducing implementation complexity and maintenance burden. The noise-averaging analysis also provides a diagnostic tool: practitioners can measure noise tail heaviness and temporal dependence in their data, then use the paper's explicit rates to predict convergence speed, helping inform decisions about batch size, training duration, and algorithm selection. The main trade-off is that the convergence rates degrade gracefully but measurably—heavy tails increase required iterations by a factor depending on the tail exponent, and long-range dependence increases the effective noise level—so you need to account for longer training times in resource planning compared to classical bounded-noise settings.
Limitations and When Not to Use This
The analysis assumes the operator is strongly monotone, which excludes many practical settings like policy optimization in deep RL or training neural networks where monotonicity does not hold. The paper provides worst-case convergence rates that depend on unknown parameters (tail index, correlation length) that must be estimated from data in practice, and misestimating these could lead to incorrect convergence predictions or algorithm failure. The noise-averaging technique requires analyzing blocks of iterations, which introduces a mild logarithmic dependence in the convergence rate—while negligible asymptotically, this overhead could be noticeable for moderate problem sizes, and the constants hidden in big-O notation are not fully characterized for practical use. Additionally, the work does not address nonconvex or stochastic variance-reduced settings, leaving open whether these techniques extend to more complex modern optimization landscapes, nor does it provide adaptive algorithms that automatically tune stepsizes based on noise characteristics discovered during training.
Research Context
This work builds on classical stochastic approximation theory pioneered by Robbins and Monro in the 1950s, extending results from the convex and strongly monotone settings established in modern optimization literature. It directly addresses limitations in contemporary analyses like those of Karimi et al. and Nemirovski et al., which require bounded second moments; this paper removes that requirement and handles long-range dependence, aligning SA theory more closely with empirical observations in finance and signal processing. The paper opens research directions in extending these techniques to variance-reduced methods (SAGA, SVRG), nonconvex optimization, and federated learning settings where heavy-tailed communication noise occurs. This work is also relevant to recent interest in robust optimization and Byzantine-resilient distributed learning, where heavy-tailed gradient noise from Byzantine nodes or lossy compression must be handled theoretically.
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