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 $1 - \beta$, a convergence rate of order $\frac{\log \log k + \log (1/\beta)}{k}$ simultaneously holds for all $k \in \mathbb{N}_+$, 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-{\L}ojasiewicz functions and indicate an $\mathcal{O}(k^{-1} \log \log k)$ convergence rate for contractive stochastic approximation with additive noise.