{"ID":23507279,"CreatedAt":"2026-09-18T02:21:44.056544415Z","UpdatedAt":"2026-09-20T18:11:56.143995915Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20409","arxiv_id":"2609.20409","title":"The Bias of Nonlinear Two-Time-scale Stochastic Approximation under Constant Step-Sizes","abstract":"Two-timescale stochastic approximation (TTSA) is a fundamental tool for analyzing coupled iterative algorithms in reinforcement learning, optimization, and stochastic control. However, finite-time guarantees for nonlinear two-timescale schemes remain difficult to obtain, especially under constant step-sizes. In this paper, we study nonlinear TTSA with step-sizes $α\\ggβ$. Under standard stability, regularity, and Markovian noise assumptions, we upper bound the mean-squared error and the bias of both iterates around their limiting equilibria. Our bounds scale as $O(α+β^2/α^2)$, which we prove to be tight when $β\\leα^{3/2}$. The analysis separates the contributions of initial conditions, fast-timescale tracking error, Markovian dependence, and timescale coupling, thereby clarifying the origin of the $β^2/α^2$ term. Our results reveal qualitative differences from the linear TTSA setting previously studied, showing that nonlinear dynamics introduce additional finite-time effects that are absent in the linear case.","short_abstract":"Two-timescale stochastic approximation (TTSA) is a fundamental tool for analyzing coupled iterative algorithms in reinforcement learning, optimization, and stochastic control. However, finite-time guarantees for nonlinear two-timescale schemes remain difficult to obtain, especially under constant step-sizes. In this pa...","url_abs":"https://arxiv.org/abs/2609.20409","url_pdf":"https://arxiv.org/pdf/2609.20409v1","authors":"[\"Djamel Rassem Lamouri\",\"Dorian Baudry\",\"Nicolas Gast\"]","published":"2026-09-17T13:54:20Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"math.OC\",\"stat.ML\"]","methods":"[\"Reinforcement Learning\"]","has_code":false}