Practical Quasi-Newton Methods for Training Deep Neural Networks

TL;DR

This paper introduces practical stochastic quasi-Newton methods, specifically Kronecker-factored block-diagonal BFGS and L-BFGS, for efficient training of deep neural networks by approximating the Hessian with structured, layer-wise Kronecker products.

Contribution

The paper proposes novel Kronecker-factored block-diagonal quasi-Newton methods with a damping strategy, improving training efficiency and performance over existing methods like KFAC and first-order approaches.

Findings

Outperformed or matched KFAC and first-order methods on multiple datasets

Efficiently approximated the Hessian for large-scale neural networks

Demonstrated effectiveness on multi-layer autoencoder models

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.