Stochastic Intermediate Gradient Method for Convex Problems with Stochastic Inexact Oracle

In this paper, we introduce new methods for convex optimization problems with stochastic inexact oracle. Our first method is an extension of the Intermediate Gradient Method proposed by Devolder, Glineur and Nesterov for problems with deterministic inexact oracle. Our method can be applied to problems with composite objective function, both deterministic and stochastic inexactness of the oracle, and allows using a non-Euclidean setup. We estimate the rate of convergence in terms of the expectation of the non-optimality gap and provide a way to control the probability of large deviations from this rate. Also we introduce two modifications of this method for strongly convex problems. For the first modification, we estimate the rate of convergence for the non-optimality gap expectation and, for the second, we provide a bound for the probability of large deviations from the rate of convergence in terms of the expectation of the non-optimality gap. All the rates lead to the complexity estimates for the proposed methods, which up to a multiplicative constant coincide with the lower complexity bound for the considered class of convex composite optimization problems with stochastic inexact oracle.

[1]  Yurii Nesterov,et al.  Gradient methods for minimizing composite functions , 2012, Mathematical Programming.

[2]  Olivier Devolder,et al.  Exactness, inexactness and stochasticity in first-order methods for large-scale convex optimization , 2013 .

[3]  John Darzentas,et al.  Problem Complexity and Method Efficiency in Optimization , 1983 .

[4]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[5]  Yurii Nesterov,et al.  Subgradient methods for huge-scale optimization problems , 2013, Mathematical Programming.

[6]  Saeed Ghadimi,et al.  Optimal Stochastic Approximation Algorithms for Strongly Convex Stochastic Composite Optimization I: A Generic Algorithmic Framework , 2012, SIAM J. Optim..

[7]  Y. Nesterov,et al.  Primal-dual subgradient methods for minimizing uniformly convex functions , 2010, 1401.1792.

[8]  O. Nelles,et al.  An Introduction to Optimization , 1996, IEEE Antennas and Propagation Magazine.

[9]  Alexander Shapiro,et al.  Validation analysis of mirror descent stochastic approximation method , 2012, Math. Program..

[10]  Yurii Nesterov,et al.  First-order methods of smooth convex optimization with inexact oracle , 2013, Mathematical Programming.

[11]  Y. Nesterov,et al.  Intermediate gradient methods for smooth convex problems with inexact oracle , 2013 .

[12]  Saeed Ghadimi,et al.  Optimal Stochastic Approximation Algorithms for Strongly Convex Stochastic Composite Optimization, II: Shrinking Procedures and Optimal Algorithms , 2013, SIAM J. Optim..

[13]  Alexander Shapiro,et al.  Stochastic Approximation approach to Stochastic Programming , 2013 .

[14]  Alexander Shapiro,et al.  Stochastic Approximation approach to Stochastic Programming , 2013 .

[15]  Yurii Nesterov,et al.  Introductory Lectures on Convex Optimization - A Basic Course , 2014, Applied Optimization.