arXiv subjects

Thomas O'Leary-Roseberry

Publications and source records attributed to Thomas O'Leary-Roseberry.

21 records · Page 2Linked to original sources

arXiv2020

Projected Stein Variational Newton: A Fast and Scalable Bayesian Inference Method in High Dimensions

We propose a fast and scalable variational method for Bayesian inference in high-dimensional parameter space, which we call projected Stein variational Newton (pSVN) method. We exploit the intrinsic low-dimensional geometric structure of the posterior distribution in the high-dimensional parameter space via its Hessian (of the log posterior) operator and perform a parallel update of the parameter samples projected into a low-dimensional subspace by an SVN method. The subspace is adaptively constructed using the eigenvectors of the averaged Hessian at the current samples. We demonstrate fast convergence of the proposed method and its scalability with respect to the number of parameters, samples, and processor cores.

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arXiv2020

Ill-Posedness and Optimization Geometry for Nonlinear Neural Network Training

In this work we analyze the role nonlinear activation functions play at stationary points of dense neural network training problems. We consider a generic least squares loss function training formulation. We show that the nonlinear activation functions used in the network construction play a critical role in classifying stationary points of the loss landscape. We show that for shallow dense networks, the nonlinear activation function determines the Hessian nullspace in the vicinity of global minima (if they exist), and therefore determines the ill-posedness of the training problem. Furthermore, for shallow nonlinear networks we show that the zeros of the activation function and its derivatives can lead to spurious local minima, and discuss conditions for strict saddle points. We extend these results to deep dense neural networks, showing that the last activation function plays an important role in classifying stationary points, due to how it shows up in the gradient from the chain rule.

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arXiv2019

Inexact Newton Methods for Stochastic Nonconvex Optimization with Applications to Neural Network Training

We study stochastic inexact Newton methods and consider their application in nonconvex settings. Building on the work of [R. Bollapragada, R. H. Byrd, and J. Nocedal, IMA Journal of Numerical Analysis, 39 (2018), pp. 545--578] we derive bounds for convergence rates in expected value for stochastic low rank Newton methods, and stochastic inexact Newton Krylov methods. These bounds quantify the errors incurred in subsampling the Hessian and gradient, as well as in approximating the Newton linear solve, and in choosing regularization and step length parameters. We deploy these methods in training convolutional autoencoders for the MNIST and CIFAR10 data sets. Numerical results demonstrate that, relative to first order methods, these stochastic inexact Newton methods often converge faster, are more cost-effective, and generalize better.

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