SAN: Stochastic Average Newton Algorithm for Minimizing Finite Sums

TL;DR

The paper introduces SAN, a stochastic Newton method for finite sum optimization that is incremental, easy to implement, and competitive with existing gradient and Newton methods, without requiring parameter tuning.

Contribution

It proposes a novel stochastic Newton approach based on reformulating stationarity as nonlinear equations and solving them with a subsampled Newton-Raphson method.

Findings

SAN requires only one data point per iteration.

SAN is parameter-free and easy to implement.

SAN performs competitively with existing methods in experiments.

Abstract

We present a principled approach for designing stochastic Newton methods for solving finite sum optimization problems. Our approach has two steps. First, we re-write the stationarity conditions as a system of nonlinear equations that associates each data point to a new row. Second, we apply a Subsampled Newton Raphson method to solve this system of nonlinear equations. Using our approach, we develop a new Stochastic Average Newton (SAN) method, which is incremental by design, in that it requires only a single data point per iteration. It is also cheap to implement when solving regularized generalized linear models, with a cost per iteration of the order of the number of the parameters. We show through extensive numerical experiments that SAN requires no knowledge about the problem, neither parameter tuning, while remaining competitive as compared to classical variance reduced gradient methods (e.g. SAG and SVRG), incremental Newton and quasi-Newton methods (e.g. SNM, IQN).