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Stochastic approximation (SA) and stochastic gradient descent (SGD)algorithms are work-horses for modern machine learning algorithms. Theirconstant stepsize variants are preferred in practice due to fast convergencebehavior. However, constant step stochastic iterative algorithms do notconverge asymptotically to the optimal solution, but instead have a stationarydistribution, which in general cannot be analytically characterized. In thiswork, we study the asymptotic behavior of the appropriately scaled stationarydistribution, in the limit when the constant stepsize goes to zero.Specifically, we consider the following three settings: (1) SGD algorithms withsmooth and strongly convex objective, (2) linear SA algorithms involving aHurwitz matrix, and (3) nonlinear SA algorithms involving a contractiveoperator. When the iterate is scaled by $1/\sqrt{\alpha}$, where $\alpha$ isthe constant stepsize, we show that the limiting scaled stationary distributionis a solution of an integral equation. Under a uniqueness assumption (which canbe removed in certain settings) on this equation, we further characterize thelimiting distribution as a Gaussian distribution whose covariance matrix is theunique solution of a suitable Lyapunov equation. For SA algorithms beyond thesecases, our numerical experiments suggest that unlike central limit theorem typeresults: (1) the scaling factor need not be $1/\sqrt{\alpha}$, and (2) thelimiting distribution need not be Gaussian. Based on the numerical study, wecome up with a formula to determine the right scaling factor, and makeinsightful connection to the Euler-Maruyama discretization scheme forapproximating stochastic differential equations.

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