Variance-Reduced Stochastic Quasi-Newton Methods for Decentralized Learning: Part II

TL;DR

This paper introduces two fully decentralized stochastic quasi-Newton methods, DFP and BFGS, that efficiently construct Hessian inverse approximations for decentralized learning, achieving faster convergence than first-order methods.

Contribution

The paper specifies two new decentralized stochastic quasi-Newton algorithms that adaptively build Hessian inverse approximations without extra sampling or communication, ensuring linear convergence.

Findings

Methods outperform existing decentralized stochastic first-order algorithms.

Damped regularized DFP improves performance with regularization.

Limited-memory BFGS reduces storage and computation complexity.

Abstract

In Part I of this work, we have proposed a general framework of decentralized stochastic quasi-Newton methods, which converge linearly to the optimal solution under the assumption that the local Hessian inverse approximations have bounded positive eigenvalues. In Part II, we specify two fully decentralized stochastic quasi-Newton methods, damped regularized limited-memory DFP (Davidon-Fletcher-Powell) and damped limited-memory BFGS (Broyden-Fletcher-Goldfarb-Shanno), to locally construct such Hessian inverse approximations without extra sampling or communication. Both of the methods use a fixed moving window of $M$ past local gradient approximations and local decision variables to adaptively construct positive definite Hessian inverse approximations with bounded eigenvalues, satisfying the assumption in Part I for the linear convergence. For the proposed damped regularized limited-memory DFP, a regularization term is added to improve the performance. For the proposed damped limited-memory BFGS, a two-loop recursion is applied, leading to low storage and computation complexity. Numerical experiments demonstrate that the proposed quasi-Newton methods are much faster than the existing decentralized stochastic first-order algorithms.