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Stochastic Compositional Gradient Descent: Algorithms for Minimizing Compositions of Expected-Value Functions

Mengdi Wang (mengdiw***at***princeton.edu)
Ethan Fang (xingyuan***at***princeton.edu)
Han Liu (hanliu***at***princeton.edu)

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., problems of the form $\min_x \E_v\[f_v\big(\E_w [g_w(x)]\big) \]$. 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 $f_v,g_{w}$ and use an auxiliary variable to track the unknown quantity $\E_w\[g_w(x)\]$. 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 achieve a convergence rate of $\O(k^{-1/4})$ in the general case and $\O(k^{-2/3})$ in the strongly convex case, after taking $k$ samples. For smooth convex problems, the SCGD can be accelerated to converge at a rate of $\O(k^{-2/7})$ in the general case and $\O(k^{-4/5})$ 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.

Keywords: stochastic optimization, stochastic gradient, sample complexity, statistical learning, composition

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Stochastic Programming

Category 3: Applications -- Science and Engineering (Statistics )

Citation:

Download: [PDF]

Entry Submitted: 11/13/2014
Entry Accepted: 11/14/2014
Entry Last Modified: 09/01/2015

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