Constant-Stepsize Stochastic Approximation: Finite-Time Convergence, Gaussian Approximation, and Tail Bounds

Abstract

Constant-stepsize stochastic approximation (SA) is widely used in learning for computational efficiency, yet the distribution of the iterates is typically intractable. Classical asymptotics results give $X_k^{(\alpha)} \approx X^{(\alpha)} \approx x^\star+\sqrt{\alpha}Y$, where $X^{(\alpha)}$ is the steady state and $Y$ is an appropriate Gaussian limit, by progressively taking the time $k\uparrow\infty$ and stepsize $\alpha\downarrow0$. Such limit results, however, do not quantify finite-time, finite-stepsize errors. We develop an explicit pre-limit characterization for SA with i.i.d.\ and Markovian noise. We establish existence and uniqueness of the stationary law, a geometric Wasserstein convergence to stationarity, and almost-sure and $L^3$ convergence of the steady state to the root $x^\star$, identifying the scale $\sqrt{\alpha}$ as first-order fluctuation. At this scale, we derive a higher-order quantitative Gaussian approximation with a Wasserstein error, using Stein's method and Poisson equation techniques. We further obtain non-uniform Berry--Esseen-type tail bounds, incorporating both steady-state approximation and finite-time convergence errors. We instantiate the theory for strongly convex smooth SGD, linear SA, and nonlinear contractive SA. Beyond strong convexity, for general convex SGD, we identify a Gibbs limiting law and prove a pre-limit Wasserstein approximation error under stability and Stein-equation hypothesis, which are validated numerically.