Stochastic Quasi-Newton Methods for Nonconvex Stochastic Optimization

Abstract

In this paper we study stochastic quasi-Newton methods for nonconvex stochastic optimization, where we assume that noisy information about the gradients of the objective function is available via a stochastic first-order oracle (SFO). We propose a general framework for such methods, for which we prove almost sure convergence to stationary points and analyze its worst-case iteration complexity. When a randomly chosen iterate is returned as the output of such an algorithm, we prove that in the worst-case, the SFO-calls complexity is O(^-2) to ensure that the expectation of the squared norm of the gradient is smaller than the given accuracy tolerance . We also propose a specific algorithm, namely a stochastic damped L-BFGS (SdLBFGS) method, that falls under the proposed framework. Moreover, we incorporate the SVRG variance reduction technique into the proposed SdLBFGS method, and analyze its SFO-calls complexity. Numerical results on a nonconvex binary classification problem using SVM, and a multiclass classification problem using neural networks are reported.

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Reproductions