Fast large-scale optimization by unifying stochastic gradient and quasi-Newton methods

TL;DR

This paper introduces a scalable optimization algorithm that combines stochastic gradient descent with quasi-Newton methods by maintaining individual Hessian approximations, leading to faster convergence in large-scale problems.

Contribution

It unifies stochastic gradient and quasi-Newton methods through per-function Hessian approximations in a low-dimensional subspace, enhancing efficiency and convergence.

Findings

Improved convergence on seven diverse problems.

Maintains computational efficiency with minimal hyperparameter tuning.

Open source implementation available.

Abstract

We present an algorithm for minimizing a sum of functions that combines the computational efficiency of stochastic gradient descent (SGD) with the second order curvature information leveraged by quasi-Newton methods. We unify these disparate approaches by maintaining an independent Hessian approximation for each contributing function in the sum. We maintain computational tractability and limit memory requirements even for high dimensional optimization problems by storing and manipulating these quadratic approximations in a shared, time evolving, low dimensional subspace. Each update step requires only a single contributing function or minibatch evaluation (as in SGD), and each step is scaled using an approximate inverse Hessian and little to no adjustment of hyperparameters is required (as is typical for quasi-Newton methods). This algorithm contrasts with earlier stochastic second order techniques that treat the Hessian of each contributing function as a noisy approximation to the full Hessian, rather than as a target for direct estimation. We experimentally demonstrate improved convergence on seven diverse optimization problems. The algorithm is released as open source Python and MATLAB packages.