Stochastic Recursive Inclusions under Biased Perturbations: An Input-to-State Stability Perspective
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| Format: | Preprint |
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2026
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| _version_ | 1866917206644752384 |
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| author | Paul, Anik Kumar Shenoy, Karthik Mahindrakar, Arun D. |
| author_facet | Paul, Anik Kumar Shenoy, Karthik Mahindrakar, Arun D. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_11462 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stochastic Recursive Inclusions under Biased Perturbations: An Input-to-State Stability Perspective Paul, Anik Kumar Shenoy, Karthik Mahindrakar, Arun D. Optimization and Control 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. |
| title | Stochastic Recursive Inclusions under Biased Perturbations: An Input-to-State Stability Perspective |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2601.11462 |