Stationary Behavior of Constant Stepsize SGD Type Algorithms: An Asymptotic Characterization

Stochastic approximation (SA) and stochastic gradient descent (SGD) algorithms are work-horses for modern machine learning algorithms. Their constant stepsize variants are preferred in practice due to fast convergence behavior. However, constant stepsize SA algorithms do not converge to the optimal solution, but instead have a stationary distribution, which in general cannot be analytically characterized. In this work, we study the asymptotic behavior of the appropriately scaled stationary distribution, in the limit when the constant stepsize goes to zero. Specifically, we consider the following three settings: (1) SGD algorithm with a smooth and strongly convex objective, (2) linear SA algorithm involving a Hurwitz matrix, and (3) nonlinear SA algorithm involving a contractive operator. When the iterate is scaled by 1/α, where α is the constant stepsize, we show that the limiting scaled stationary distribution is a solution of an implicit equation. Under a uniqueness assumption (which can be removed in certain settings) on this equation, we further characterize the limiting distribution as a Gaussian distribution whose covariance matrix is the unique solution of a suitable Lyapunov equation. For SA algorithms beyond these cases, our numerical experiments suggest that unlike central limit theorem type results: (1) the scaling factor need not be 1/α, and (2) the limiting distribution need not be Gaussian. Based on the numerical study, we come up with a heuristic formula to determine the right scaling factor, and make insightful connection to the Euler-Maruyama discretization scheme for approximating stochastic differential equations.