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Randomized algorithms that base iteration-level decisions on samples fromsome pool are ubiquitous in machine learning and optimization. Examples includestochastic gradient descent and randomized coordinate descent. This paper makesprogress at theoretically evaluating the difference in performance betweensampling with- and without-replacement in such algorithms. Focusing on leastmeans squares optimization, we formulate a noncommutative arithmetic-geometricmean inequality that would prove that the expected convergence rate ofwithout-replacement sampling is faster than that of with-replacement sampling.We demonstrate that this inequality holds for many classes of random matricesand for some pathological examples as well. We provide a deterministicworst-case bound on the gap between the discrepancy between the two samplingmodels, and explore some of the impediments to proving this inequality in fullgenerality. We detail the consequences of this inequality for stochasticgradient descent and the randomized Kaczmarz algorithm for solving linearsystems.

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