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A Fully Stochastic Second-Order Trust Region Method

Frank E. Curtis(frank.e.curtis***at***gmail.com)
Rui Shi(rus415***at***lehigh.edu)

Abstract: A stochastic second-order trust region method is proposed, which can be viewed as a second-order extension of the trust-region-ish (TRish) algorithm proposed by Curtis et al. [INFORMS J. Optim. 1(3) 200220, 2019]. In each iteration, a search direction is computed by (approximately) solving a trust region subproblem defined by stochastic gradient and Hessian estimates. The algorithm has convergence guarantees for stochastic minimization in the fully stochastic regime, meaning that guarantees hold when each stochastic gradient is required merely to be an unbiased estimate of the true gradient with bounded variance and when the stochastic Hessian estimates are bounded uniformly in norm. The algorithm is also equipped with a worst-case complexity guarantee in the nearly deterministic regime, i.e., when the stochastic gradient and Hessian estimates are very close in expectation to the true gradients and Hessians. The results of numerical experiments for training convolutional neural networks for image classification and training a recurrent neural network for time series forecasting are presented. These results show that the algorithm can outperform a stochastic gradient approach and the first-order TRish algorithm in practice.

Keywords: stochastic optimization, finite-sum optimization, stochastic Newton methods, trust region methods, machine learning, deep neural networks, time series forecasting

Category 1: Nonlinear Optimization

Category 2: Nonlinear Optimization (Unconstrained Optimization )

Category 3: Stochastic Programming

Citation: Lehigh ISE Technical Report 19T-025

Download: [PDF]

Entry Submitted: 11/15/2019
Entry Accepted: 11/15/2019
Entry Last Modified: 11/15/2019

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