Mini-batch stochastic approximation methods for nonconvex stochastic composite optimization

• Published in: Mathematical programming Vol. 155; no. 1-2; pp. 267 - 305
• Main Authors: Ghadimi, Saeed, Lan, Guanghui, Zhang, Hongchao
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2016; Springer Nature B.V
• Subjects: Algorithms; Approximation; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Complexity; Full Length Paper; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical models; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Methods; Numerical Analysis; Optimization; Programming; Random variables; Stochastic models; Stochasticity; Studies; Theoretical; Nonconvex optimization; 90C25; Mini-batch of samples; 90C22; 90C06; Stochastic approximation; 49M37; Zeroth-order method; Constrained stochastic programming; First-order method; Stochastic programming
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-014-0846-1

This paper considers a class of constrained stochastic composite optimization problems whose objective function is given by the summation of a differentiable (possibly nonconvex) component, together with a certain non-differentiable (but convex) component. In order to solve these problems, we propose a randomized stochastic projected gradient (RSPG) algorithm, in which proper mini-batch of samples are taken at each iteration depending on the total budget of stochastic samples allowed. The RSPG algorithm also employs a general distance function to allow taking advantage of the geometry of the feasible region. Complexity of this algorithm is established in a unified setting, which shows nearly optimal complexity of the algorithm for convex stochastic programming. A post-optimization phase is also proposed to significantly reduce the variance of the solutions returned by the algorithm. In addition, based on the RSPG algorithm, a stochastic gradient free algorithm, which only uses the stochastic zeroth-order information, has been also discussed. Some preliminary numerical results are also provided.