Efficient Nonconvex Empirical Risk Minimization via Adaptive Sample Size Methods

摘要

In this paper, we are interested in finding a local minimizer of an empirical risk minimization (ERM) problem where the loss associated with each sample is possibly a nonconvex function. Unlike traditional deterministic and stochastic algorithms that attempt to solve the ERM problem for the full training set, we propose an adaptive sample size scheme to reduce the overall computational complexity of finding a local minimum. To be more precise, we first find an approximate local minimum of the ERM problem corresponding to a small number of samples and use the uniform convergence theory to show that if the population risk is a Morse function, by properly increasing the size of training set the iterates generated by the proposed procedure always stay close to a local minimum of the corresponding ERM problem. Therefore, eventually, the proposed procedure finds a local minimum of the ERM corresponding to the full training set which happens to also be close to a local minimum of the expected risk minimization problem with high probability. We formally state the conditions on the size of the initial sample set and characterize the required accuracy for obtaining an approximate local minimum to ensure that the iterates always stay in a neighborhood of a local minimum and do not get attracted to saddle points.