Steady-State Convergence of Stochastic Approximation

• URL: http://arxiv.org/abs/2609.14922v1
• Date: Mon, 14 Sep 2026 02:15:03 GMT
• Title: Steady-State Convergence of Stochastic Approximation
• Abstract summary: We develop a unified SSC theory for constant-stepsize contractive SA driven by Markovian, multiplicative noise.<n>A key methodological contribution is a multi-step framework that progressively reduces the original recursion to tractable auxiliary dynamics.<n>We show that the leading-order bias can be universality of order $sqrt$, in contrast to the $$-order bias in the smooth regime.
• Score: 19.40811084751516
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: For constant-stepsize stochastic approximation (SA), the iterates converge in distribution to a stationary law that depends on the stepsize $α.$ Steady-state convergence (SSC) concerns the limit of the scaled stationary distribution as $α\downarrow 0.$ Existing SSC theory requires i.i.d. or additive noise and global differentiability of the mean operator, and yields suboptimal rates. We develop a unified SSC theory for constant-stepsize contractive SA driven by Markovian, multiplicative noise, covering both locally differentiable and locally nondifferentiable mean operators. A key methodological contribution is a multi-step universality framework that progressively reduces the original stochastic recursion to tractable auxiliary dynamics while preserving its steady-state limit. Under local quadratic linearization at the fixed point, we obtain a Gaussian approximation of the scaled steady state at the optimal rate $O(\sqrtα)$ in Wasserstein-2 distance, which further gives finite-time Gaussian approximations for the raw iterates. In the locally nondifferentiable regime, we establish a general SSC result and show that the leading-order asymptotic bias can be of order $\sqrtα$, in contrast to the $α$-order bias in the smooth regime. We apply the theory to Markovian linear SA and asynchronous Q-learning, neither of which is covered by prior results. We further propose a bias-reduction scheme for Q-learning that requires no knowledge of the local smoothness regime, validated by numerical experiments.

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