Competing with the Empirical Risk Minimizer in a Single Pass

TL;DR

This paper introduces a simple, single-pass streaming algorithm that matches the statistical performance of the empirical risk minimizer while significantly reducing computational resources, making it efficient and scalable for large data problems.

Contribution

The authors present a novel streaming algorithm that achieves ERM-like statistical convergence in linear time and space, with super-polynomial error decay and easy parallelization.

Findings

Algorithm runs in linear time with a single data pass.

Achieves the same statistical rate as ERM on all problems.

Performance improves super-polynomially with initial error.

Abstract

In many estimation problems, e.g. linear and logistic regression, we wish to minimize an unknown objective given only unbiased samples of the objective function. Furthermore, we aim to achieve this using as few samples as possible. In the absence of computational constraints, the minimizer of a sample average of observed data -- commonly referred to as either the empirical risk minimizer (ERM) or the $M$-estimator -- is widely regarded as the estimation strategy of choice due to its desirable statistical convergence properties. Our goal in this work is to perform as well as the ERM, on every problem, while minimizing the use of computational resources such as running time and space usage. We provide a simple streaming algorithm which, under standard regularity assumptions on the underlying problem, enjoys the following properties: * The algorithm can be implemented in linear time with a single pass of the observed data, using space linear in the size of a single sample. * The algorithm achieves the same statistical rate of convergence as the empirical risk minimizer on every problem, even considering constant factors. * The algorithm's performance depends on the initial error at a rate that decreases super-polynomially. * The algorithm is easily parallelizable. Moreover, we quantify the (finite-sample) rate at which the algorithm becomes competitive with the ERM.