Stochastic Recursive Inclusions under Biased Perturbations: An Input-to-State Stability Perspective

TL;DR

This paper develops a unified theoretical framework using input-to-state stability of differential inclusions to analyze the almost sure convergence of biased stochastic recursive algorithms, relevant in zeroth-order optimization and distributed learning.

Contribution

It introduces a novel input-to-state stability perspective for biased stochastic recursive inclusions, providing verifiable conditions for convergence and boundedness.

Findings

Iterates converge to a neighborhood of equilibrium under input-to-state stability.

Sufficient conditions for boundedness are established for Lipschitz operators.

Zeroth-order stochastic gradient methods are shown to be input-to-state stable.

Abstract

This paper investigates the asymptotic behavior of stochastic recursive inclusions in the presence of non-zero, non-diminishing bias, a setting that frequently arises in zeroth-order optimization, stochastic approximation with iterate-dependent noise, and distributed learning with adversarial agents. The analysis is conducted through the lens of input-to-state stability of an associated differential inclusion, which serves as the continuous-time limit of the discrete recursion. We first establish that if the limiting differential inclusion is input-to-state stable and the iterates remain almost surely bounded, then the iterates converge almost surely to the neighborhood of desired equilibrium. We then provide a verifiable sufficient condition for almost sure boundedness by assuming that the underlying operator is single-valued and globally Lipschitz. Finally, we show that several zeroth-order variants of stochastic gradient naturally fit within this framework, and we demonstrate their input-to-state stability under standard conditions. Overall, the results provide a unified theoretical foundation for studying almost sure convergence of biased stochastic approximation schemes through the Input to State stability theory of differential inclusions.