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Almost Sure Convergence of Nonlinear Stochastic Approximation: An Interplay of Noise and Step Size

Quang Dinh Thien Nguyen, Duc Anh Nguyen, Hoang Huy Nguyen, Siva Theja Maguluri

TL;DR

This work establishes almost sure convergence of nonlinear stochastic approximation under a general negative drift and a noise process with finite $p$-th moment ($p>1$). It proves that any non-summable but $p$-th power summable step size sequence suffices, with explicit $\alpha_k=\alpha(k+K)^{-\xi}$ requirements $\xi\in(1/p,1]$, and extends to multiplicative noise. A key innovation is a universal Lyapunov drift argument and, for $p>2$, a novel iterate projection technique that handles nonlinear and high-moment terms. The results generalize classical $p=2$ theory, connect to SLLN, and apply to contractive, linear, SGD, and non-expansive settings, offering a refined noise-step-size trade-off with potential finite-time implications.

Abstract

We study the almost sure convergence of the Stochastic Approximation algorithm to the fixed point $x^\star$ of a nonlinear operator under a negative drift condition and a general noise sequence with finite $p$-th moment for some $p > 1$. Classical almost sure convergence results of Stochastic Approximation are mostly analyzed for the square-integrable noise setting, and it is shown that any non-summable but square-summable step size sequence is sufficient to obtain almost sure convergence. However, such a limitation prevents wider algorithmic application. In particular, many applications in Machine Learning and Operations Research admit heavy-tailed noise with infinite variance, rendering such guarantees inapplicable. On the other hand, when a stronger condition on the noise is available, such guarantees on the step size would be too conservative, as practitioners would like to pick a larger step size for a more preferable convergence behavior. To this end, we show that any non-summable but $p$-th power summable step size sequence is sufficient to guarantee almost sure convergence, covering the gap in the literature. Our guarantees are obtained using a universal Lyapunov drift argument. For the regime $p \in (1, 2)$, we show that using the Lyapunov function $\norm{x-x^\star}^p$ and applying a Taylor-like bound suffice. For $p > 2$, such an approach is no longer applicable, and therefore, we introduce a novel iterate projection technique to control the nonlinear terms produced by high-moment bounds and multiplicative noise. We believe our proof techniques and their implications could be of independent interest and pave the way for finite-time analysis of Stochastic Approximation under a general noise condition.

Almost Sure Convergence of Nonlinear Stochastic Approximation: An Interplay of Noise and Step Size

TL;DR

This work establishes almost sure convergence of nonlinear stochastic approximation under a general negative drift and a noise process with finite -th moment (). It proves that any non-summable but -th power summable step size sequence suffices, with explicit requirements , and extends to multiplicative noise. A key innovation is a universal Lyapunov drift argument and, for , a novel iterate projection technique that handles nonlinear and high-moment terms. The results generalize classical theory, connect to SLLN, and apply to contractive, linear, SGD, and non-expansive settings, offering a refined noise-step-size trade-off with potential finite-time implications.

Abstract

We study the almost sure convergence of the Stochastic Approximation algorithm to the fixed point of a nonlinear operator under a negative drift condition and a general noise sequence with finite -th moment for some . Classical almost sure convergence results of Stochastic Approximation are mostly analyzed for the square-integrable noise setting, and it is shown that any non-summable but square-summable step size sequence is sufficient to obtain almost sure convergence. However, such a limitation prevents wider algorithmic application. In particular, many applications in Machine Learning and Operations Research admit heavy-tailed noise with infinite variance, rendering such guarantees inapplicable. On the other hand, when a stronger condition on the noise is available, such guarantees on the step size would be too conservative, as practitioners would like to pick a larger step size for a more preferable convergence behavior. To this end, we show that any non-summable but -th power summable step size sequence is sufficient to guarantee almost sure convergence, covering the gap in the literature. Our guarantees are obtained using a universal Lyapunov drift argument. For the regime , we show that using the Lyapunov function and applying a Taylor-like bound suffice. For , such an approach is no longer applicable, and therefore, we introduce a novel iterate projection technique to control the nonlinear terms produced by high-moment bounds and multiplicative noise. We believe our proof techniques and their implications could be of independent interest and pave the way for finite-time analysis of Stochastic Approximation under a general noise condition.
Paper Structure (27 sections, 15 theorems, 94 equations, 2 figures)

This paper contains 27 sections, 15 theorems, 94 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Literature overview
  4. Problem setting and Main Results
  5. Problem setting
  6. Main results
  7. Example: Contractive operators
  8. Example: Linear operators
  9. Example: Stochastic Gradient Descent (SGD)
  10. Example: Strong Law of Large Numbers (SLLN) with step-sizes
  11. Generalization to non-expansive operators
  12. Proof outline
  13. $p = 2$
  14. $p \in (1,2)$
  15. $p > 2$
  16. ...and 12 more sections

Key Result

Theorem 2.1

Suppose Assumption assumption: general-drift-condition, assumption: noise-unbiasedness, assumption: noise-condition-p for some $p > 1$ and assumption: Lipschitz hold, then for any step size sequence $\{\alpha_k\}_{k \geq 0}$ of the update equation: main_iterate to be a non-increasing sequence that s then the iteration $x_k$ converges almost surely to the unique solution of problem: fixed_point_equ

Figures (2)

  • Figure 1: Visual summary of the Lyapunov drift argument with the iterate projection trick.
  • Figure 2: Selector control with theoretical noise and multiple step sizes, here $p^{-1} = 0.625$ is the divergence threshold for $p = 1.6$. Our experimental results confirm the tightness of the condition $\xi > p^{-1}$ for $p = 1.6$.

Theorems & Definitions (31)

  • Theorem 2.1
  • Corollary 2.1
  • proof
  • Corollary 2.2
  • proof
  • Theorem 2.2
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • proof
  • ...and 21 more