Beneath the valley of the noncommutative arithmetic-geometric mean inequality: conjectures, case-studies, and consequences

TL;DR

This paper investigates the performance difference between sampling with- and without-replacement in randomized algorithms, proposing a noncommutative inequality that could explain faster convergence with without-replacement sampling.

Contribution

It formulates a noncommutative arithmetic-geometric mean inequality and demonstrates its validity for various classes of random matrices, advancing theoretical understanding of sampling methods.

Findings

The inequality holds for many classes of random matrices.

A deterministic worst-case bound on the sampling discrepancy is provided.

Implications for stochastic gradient descent and Kaczmarz algorithm are discussed.

Abstract

Randomized algorithms that base iteration-level decisions on samples from some pool are ubiquitous in machine learning and optimization. Examples include stochastic gradient descent and randomized coordinate descent. This paper makes progress at theoretically evaluating the difference in performance between sampling with- and without-replacement in such algorithms. Focusing on least means squares optimization, we formulate a noncommutative arithmetic-geometric mean inequality that would prove that the expected convergence rate of without-replacement sampling is faster than that of with-replacement sampling. We demonstrate that this inequality holds for many classes of random matrices and for some pathological examples as well. We provide a deterministic worst-case bound on the gap between the discrepancy between the two sampling models, and explore some of the impediments to proving this inequality in full generality. We detail the consequences of this inequality for stochastic gradient descent and the randomized Kaczmarz algorithm for solving linear systems.