SAN: Stochastic Average Newton Algorithm for Minimizing Finite Sums
Abstract
We present a principled approach for designing stochastic Newton methods for solving finite sum optimization problems. Our approach has two steps. First, we rewrite 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 quasiNewton methods (e.g. SNM, IQN).
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.10520
 Bibcode:
 2021arXiv210610520C
 Keywords:

 Mathematics  Optimization and Control;
 Mathematics  Numerical Analysis
 EPrint:
 Accepted at AISTATS 2022