A Single Timescale Stochastic Approximation Method for Nested Stochastic Optimization

Abstract

We study constrained nested stochastic optimization problems in which the objective function is a composition of two smooth functions whose exact values and derivatives are not available. We propose a single time-scale stochastic approximation algorithm, which we call the Nested Averaged Stochastic Approximation (NASA), to find an approximate stationary point of the problem. The algorithm has two auxiliary averaged sequences (filters) which estimate the gradient of the composite objective function and the inner function value. By using a special Lyapunov function, we show that NASA achieves the sample complexity of O(1/ε 2 ) for finding an ε-approximate stationary point, thus outperforming all extant methods for nested stochastic approximation. Our method and its analysis are the same for both unconstrained and constrained problems, without any need of batch samples for constrained nonconvex stochastic optimization. We also present a simplified parameter-free variant of the NASA method for solving constrained single level stochastic optimization problems, and we prove the same complexity result for both unconstrained and constrained problems.

Streaming constrained binary logistic regression with online standardized data

“…to ensure the a.s. convergence of the algorithm (X n ) ([23], Theorem 1). It requires that the algorithm (X n ) with step-size (α n ) must be slower than the algorithm (Y n ) with step-size (β n ), which decreases the speed of convergence and creates practical difficulties according to [11]. B.…”

“…B. Consider the algorithm (X n ) defined in equation (11). To estimate E R j and Var R j for j = 1, .…”

“…M p n 1) and Γ n−1 by a (p + 1, p + 1) diagonal matrix with diagonal elements , p, and 1. Then it can be proved that (M n ) converges a.s. to E [g (R)] and (X n ) defined by (11) to θ under the following assumptions on (β n ):…”

“…C. The aim of Stochastic Compositional Optimization (SCO [23], [11], [25] for multilevel compositional optimization, and references therein) is to minimize the composition of two expected-value functions:…”

Convergence of a stochastic subgradient method with averaging for nonsmooth nonconvex constrained optimization

“…We now proceed by induction. Suppose the relations (17)- (19) are true for m + 1, the sequence {u k m+1 } is bounded a.s., and u k m+1 2 is bounded as well. We shall verify these properties for m. From (15) for m and (17) for m + 1 we obtain…”