Recht-Ré Noncommutative Arithmetic-Geometric Mean Conjecture is False

Abstract

Stochastic optimization algorithms have become indispensable in modern machine learning. An unresolved foundational question in this area is the difference between with-replacement sampling and without-replacement sampling -- does the latter have superior convergence rate compared to the former? A groundbreaking result of Recht and Ré reduces the problem to a noncommutative analogue of the arithmetic-geometric mean inequality where $n$ positive numbers are replaced by $n$ positive definite matrices. If this inequality holds for all $n$, then without-replacement sampling indeed outperforms with-replacement sampling. The conjectured Recht-Ré inequality has so far only been established for $n=2$ and a special case of $n=3$. We will show that the Recht-Ré conjecture is false for general $n$. Our approach relies on the noncommutative Positivstellensatz, which allows us to reduce the conjectured inequality to a semidefinite program and the validity of the conjecture to certain bounds for the optimum values, which we show are false as soon as $n=5$.

10 pages

References in corpus (4)

• Symmetry groups, semidefinite programs, and sums of squares
• The convex Positivstellensatz in a free algebra
• Sharp Analysis for Nonconvex SGD Escaping from Saddle Points
• SGD without Replacement: Sharper Rates for General Smooth Convex Functions

Cited by in corpus (5)

• Random Reshuffling: Simple Analysis with Vast Improvements
• SGD with shuffling: optimal rates without component convexity and large epoch requirements
• Can Single-Shuffle SGD be Better than Reshuffling SGD and GD?
• Convergence of Random Reshuffling Under The Kurdyka-Łojasiewicz Inequality
• Permutation-Based SGD: Is Random Optimal?