Revisiting Stochastic Gradient Descent for Strongly Convex Objectives: Tight Uniform-in-Time Bounds

Stochastic optimization via Stochastic Gradient Descent (SGD) is a fundamental problem in statistics and optimization. This paper revisits Stochastic Gradient Descent (SGD) for strongly convex objectives, establishing tight, uniform-in-time convergence bounds. We prove that, with probability at least 1 - β, a convergence rate of order fraclog log k + log (1/β)k simultaneously holds for all k ∈ mathbbN_+ , and demonstrate this bound is tight up to constant factors. We also provide an improved last-iterate convergence rate for such objectives. While focused on strongly convex objectives, our results generalize to the Polyak-Lojasiewicz functions and indicate an O(k-1 log log k) convergence rate for contractive stochastic approximation with additive noise.