Apollo: An Adaptive Parameter-wised Diagonal Quasi-Newton Method for Nonconvex Stochastic Optimization

Xuezhe Ma

Keywords: Optimization for Neural Networks, Optimization for Representation Learning, Stochastic Optimization, Nonconvex, Quasi-Newton, Optimization for Deep Learning

Abstract: In this paper, we introduce Apollo, a quasi-Newton method for nonconvex stochastic optimization, which dynamically incorporates the curvature of the loss function by approximating the Hessian via a diagonal matrix. Importantly, the update and storage of the diagonal approximation of Hessian is as efficient as adaptive first-order optimization methods with linear complexity for both time and memory. To handle nonconvexity, we replace the Hessian with its rectified absolute value, which is guaranteed to be positive-definite. Experiments on three tasks of vision and language show that Apollo achieves significant improvements over other stochastic optimization methods, including SGD and variants of Adam, in term of both convergence speed and generalization performance.

One-sentence Summary: An Adaptive Parameter-wised Diagonal Quasi-Newton Method for Nonconvex Stochastic Optimization