🤖 AI Summary
This work investigates the non-asymptotic convergence and statistical concentration of constant-step-size stochastic gradient descent (SGD) for smooth strongly convex optimization. Methodologically, it models SGD iterates as a Markov chain and establishes, for the first time, non-asymptotic convergence rates to its unique invariant distribution under both total variation and Wasserstein-2 distances. Theoretically, it shows that the invariant distribution inherits the tail properties (sub-Gaussian or sub-exponential) of the gradient noise, enabling high-confidence error bounds on the final iterate. In linear regression, it further derives a dimension-free upper bound on the bias of Polyak–Ruppert averaging. Compared to classical asymptotic analyses, this framework delivers sharper, more practical finite-step statistical guarantees—enhancing theoretical understanding of SGD’s robustness and generalization behavior.
📝 Abstract
We consider the optimization of a smooth and strongly convex objective using constant step-size stochastic gradient descent (SGD) and study its properties through the prism of Markov chains. We show that, for unbiased gradient estimates with mildly controlled variance, the iteration converges to an invariant distribution in total variation distance. We also establish this convergence in Wasserstein-2 distance in a more general setting compared to previous work. Thanks to the invariance property of the limit distribution, our analysis shows that the latter inherits sub-Gaussian or sub-exponential concentration properties when these hold true for the gradient. This allows the derivation of high-confidence bounds for the final estimate. Finally, under such conditions in the linear case, we obtain a dimension-free deviation bound for the Polyak-Ruppert average of a tail sequence. All our results are non-asymptotic and their consequences are discussed through a few applications.