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A stochastic second-order trust region method is proposed, which can beviewed as a second-order extension of the trust-region-ish (TRish) algorithmproposed by Curtis et al. (INFORMS J. Optim. 1(3) 200-220, 2019). In eachiteration, a search direction is computed by (approximately) solving a trustregion subproblem defined by stochastic gradient and Hessian estimates. Thealgorithm has convergence guarantees for stochastic minimization in the fullystochastic regime, meaning that guarantees hold when each stochastic gradientis required merely to be an unbiased estimate of the true gradient with boundedvariance and when the stochastic Hessian estimates are bounded uniformly innorm. The algorithm is also equipped with a worst-case complexity guarantee inthe nearly deterministic regime, i.e., when the stochastic gradient and Hessianestimates are very close in expectation to the true gradients and Hessians. Theresults of numerical experiments for training convolutional neural networks forimage classification and training a recurrent neural network for time seriesforecasting are presented. These results show that the algorithm can outperforma stochastic gradient approach and the first-order TRish algorithm in practice.

A Fully Stochastic Second-Order Trust Region Method