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

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 $$\min _x \mathbf{E}_v\left[ f_v\big (\mathbf{E}_w [g_w(x)]\big ) \right] .$$minxEvfv(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 $$f_v,g_{w}$$fv,gw and use an auxiliary variable to track the unknown quantity $$\mathbf{E}_w\left[ g_w(x)\right] $$Ewgw(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 $$\mathcal {O}(k^{-1/4})$$O(k-1/4) in the general case and $$\mathcal {O}(k^{-2/3})$$O(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 $$\mathcal {O}(k^{-2/7})$$O(k-2/7) in the general case and $$\mathcal {O}(k^{-4/5})$$O(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.