Recht-Re Noncommutative Arithmetic-Geometric Mean Conjecture is False

Stochastic optimization algorithms have become indispensable in 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 Re reduces the problem to a noncommutative analogue of the arithmetic-geometric mean inequality where 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--Re inequality has so far only been established for n = 2 and a special case of n = 3. We will show that the Recht--Re 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.