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A stochastic semismooth Newton method for nonsmooth nonconvex optimization

2018-11-25
 

Academic Report

Title: A stochastic semismooth Newton method for nonsmooth nonconvex optimization

Reporter: Andre Milzarek (Beijing International Mathematics Research Center, Peking University)

Time: November 26, 2018 (Monday) AM 10:00-11:00

Location: A1101# room, Innovation Park Building

Contact: A.prof. XIAO Xiantao (tel:84708351-8307)

 

Abstract: In this talk, we present a globalized semismooth Newton method for solving stochastic optimization problems involving smooth nonconvex and nonsmooth convex terms in the objective function. The class of problems that can be solved within our algorithmic framework comprises a large variety of applications such as l1-logistic regression, structured dictionary learning, and other minimization problems arising in machine learning, statistics, or image processing. We assume that only noisy gradient and Hessian information of the smooth part of the objective function is available via calling stochastic first- and second-order oracles. Our approach utilizes approximate second order information and stochastic semismooth Newton steps for a prox-type fixed-point equation, representing the associated optimality conditions, to accelerate the basic stochastic proximal gradient method for convex composite programming. Inexact growth conditions are introduced to monitor the quality and acceptance of the Newton steps and to combine the two different methods. We prove that the proposed algorithm converges globally to stationary points in expectation and almost surely. Moreover, under standard assumptions, the method can be shown to locally turn into a pure semismooth Newton method and fast local convergence can be established with high probability. Finally, we provide numerical experiments illustrating the efficiency of the stochastic semismooth Newton method.

 

The brief introduction to the reporter: Andre Milzarek received his doctoral degree in mathematics from the Technical University of Munich in Germany under the supervision of Michael Ulbrichin2016. Currently, he is a postdoctoral researcher at the Beijing International Center for Mathematical Research at the Peking University. His main research directions and interests cover nonsmooth optimization, large-scale and stochastic optimization, second order methods and theory. From 2010 to 2012 he was supported by the Max-Weber program of the state of Bavaria and in 2017 he received the Boya Postdoctoral Fellowship at Peking University. He published papers in SIAM Journal on Optimization, SIAM Journal on Scientific Computing and SIAM Journal onMatrix Analysis and Applications.