Recht-Re Noncommutative Arithmetic-Geometric Mean Conjecture is False
Zehua Lai,u00a0Lek-Heng Lim
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 u2014 does the latter have superior convergence rate compared to the former? A groundbreaking result of Recht and Ru00e9 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 (also known as random reshuffling) indeed outperforms with-replacement sampling in some important optimization problems. The conjectured Rechtu2013Ru00e9 inequality has so far only been established for $n = 2$ and a special case of $n = 3$. We will show that the Rechtu2013Ru00e9 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 $n = 5$.
