A Subsampling Line-Search Method with Second-Order Results

In many contemporary optimization problems such as those arising in machine learning, it can be computationally challenging or even infeasible to evaluate an entire function or its derivatives. This motivates the use of stochastic algorithms that sample problem data, which can jeopardize the guarantees obtained through classical globalization techniques in optimization, such as a line search. Using subsampled function values is particularly challenging for the latter strategy, which relies upon multiple evaluations. For nonconvex data-related problems, such as training deep learning models, one aims at developing methods that converge to second-order stationary points quickly, that is, escape saddle points efficiently. This is particularly difficult to ensure when one only accesses subsampled approximations of the objective and its derivatives. In this paper, we describe a stochastic algorithm based on negative curvature and Newton-type directions that are computed for a subsampling model of the objective. A line-search technique is used to enforce suitable decrease for this model; for a sufficiently large sample, a similar amount of reduction holds for the true objective. We then present worst-case complexity guarantees for a notion of stationarity tailored to the subsampling context. Our analysis encompasses the deterministic regime and allows us to identify sampling requirements for second-order line-search paradigms. As we illustrate through real data experiments, these worst-case estimates need not be satisfied for our method to be competitive with first-order strategies in practice.