Stochastic compositional gradient descent: algorithms for minimizing compositions of expected-value functions

Abstract

Classical stochastic gradient methods are well suited for minimizing expected-value objective functions. However, they do not apply to the minimization of a nonlinear function involving expected values or a composition of two expected-value functions, i.e., the problem min𝑥𝐄𝑣[𝑓𝑣(𝐄𝑤[𝑔𝑤(𝑥)])]. In order to solve this stochastic composition problem, we propose a class of stochastic compositional gradient descent (SCGD) algorithms that can be viewed as stochastic versions of quasi-gradient method. SCGD update the solutions based on noisy sample gradients of 𝑓𝑣,𝑔𝑤 and use an auxiliary variable to track the unknown quantity 𝐄𝑤[𝑔𝑤(𝑥)]. We prove that the SCGD converge almost surely to an optimal solution for convex optimization problems, as long as such a solution exists. The convergence involves the interplay of two iterations with different time scales. For nonsmooth convex problems, the SCGD achieves a convergence rate of (𝑘−1/4) in the general case and (𝑘−2/3) in the strongly convex case, after taking k samples. For smooth convex problems, the SCGD can be accelerated to converge at a rate of (𝑘−2/7) in the general case and (𝑘−4/5) in the strongly convex case. For nonconvex problems, we prove that any limit point generated by SCGD is a stationary point, for which we also provide the convergence rate analysis. Indeed, the stochastic setting where one wants to optimize compositions of expected-value functions is very common in practice. The proposed SCGD methods find wide applications in learning, estimation, dynamic programming, etc.