Stochastic Recursive Inclusions With Biased Perturbations: An ISS Perspective

This letter investigates the asymptotic behavior of stochastic recursive inclusions (SRI) 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. Using input-to-state stability (ISS) of the differential inclusion that arises as the continuous-time limit of the SRI, we show that if the limiting differential inclusion is ISS and the iterates are almost surely bounded, then the iterates converge a.s. to a neighborhood of the equilibrium. We provide a simple, verifiable sufficient condition for almost sure boundedness when the operator is single-valued and globally Lipschitz, and show that several zeroth-order stochastic-gradient variants fit into this SRI/ISS framework under standard assumptions. The results offer a unified theoretical foundation for proving almost sure convergence of biased stochastic approximation schemes via ISS of differential inclusions.