Convergence and concentration properties of constant step-size SGD through Markov chains
This work provides theoretical guarantees for SGD convergence and concentration, which is incremental but important for machine learning practitioners and theorists seeking robust optimization methods.
The paper tackles the analysis of constant step-size stochastic gradient descent (SGD) for smooth and strongly convex objectives by modeling it as a Markov chain, showing convergence to an invariant distribution in total variation and Wasserstein-2 distances under mild assumptions, and deriving non-asymptotic high-confidence bounds, including a dimension-free deviation bound for Polyak-Ruppert averages in linear cases.
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 under a relaxed assumption on the gradient noise distribution compared to previous work. Our analysis shows that the SGD iterates and their invariant limit distribution \emph{inherit} 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.