  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 R\'e 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-R\'e inequality has so far only been
established for $n = 2$ and a special case of $n = 3$. We will show that the
Recht-R\'e 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 $n =
5$.
