Practical Quasi-Newton Methods for Training Deep Neural Networks

• URL: http://arxiv.org/abs/2006.08877v3
• Date: Thu, 7 Jan 2021 19:36:34 GMT
• Title: Practical Quasi-Newton Methods for Training Deep Neural Networks
• Authors: Donald Goldfarb, Yi Ren, Achraf Bahamou
• Abstract summary: In training, the number of variables and components of the gradient $n$ is often of the order of tens of millions and the Hessian has $n2$ elements. We approximate the Hessian by a block-diagonal matrix and use the structure of the gradient and Hessian to further approximate these blocks. Because the indefinite and highly variable nature of the Hessian in a DNN, we also propose a new damping approach to keep the upper as well as the lower bounds of the BFGS and L-BFGS approximations bounded.
• Score: 12.48022619079224
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider the development of practical stochastic quasi-Newton, and in particular Kronecker-factored block-diagonal BFGS and L-BFGS methods, for training deep neural networks (DNNs). In DNN training, the number of variables and components of the gradient $n$ is often of the order of tens of millions and the Hessian has $n^2$ elements. Consequently, computing and storing a full $n \times n$ BFGS approximation or storing a modest number of (step, change in gradient) vector pairs for use in an L-BFGS implementation is out of the question. In our proposed methods, we approximate the Hessian by a block-diagonal matrix and use the structure of the gradient and Hessian to further approximate these blocks, each of which corresponds to a layer, as the Kronecker product of two much smaller matrices. This is analogous to the approach in KFAC, which computes a Kronecker-factored block-diagonal approximation to the Fisher matrix in a stochastic natural gradient method. Because the indefinite and highly variable nature of the Hessian in a DNN, we also propose a new damping approach to keep the upper as well as the lower bounds of the BFGS and L-BFGS approximations bounded. In tests on autoencoder feed-forward neural network models with either nine or thirteen layers applied to three datasets, our methods outperformed or performed comparably to KFAC and state-of-the-art first-order stochastic methods.

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