论文标题
Recht-ré非交通算术几何平均猜想是错误的
Recht-Ré Noncommutative Arithmetic-Geometric Mean Conjecture is False
论文作者
论文摘要
在现代机器学习中，随机优化算法已成为必不可少的。在该领域的一个尚未解决的基础问题是，用替换抽样和没有替换抽样的差异 - 后者与前者的收敛速率是否优于较高的收敛速率？ Recht和Ré的开创性结果将问题降低到了算术几何平均不平等的非交通类似物，其中$ n $正数被$ n $ n $阳性确定的矩阵取代。如果这种不平等适用于所有$ n $，那么没有替换抽样的情况确实超过了用替换抽样的表现。迄今为止，猜想的Recht-Ré不平等现已以$ n = 2 $和$ n = 3 $的特殊情况建立。我们将证明Recht-Ré的猜想对于一般$ n $是错误的。我们的方法依赖于非交通性的阳性，这使我们能够将猜想的不等式减少到半限定程序的不平等，以及对最佳值的猜想的有效性以及某些界限的有效性，我们表明的是$ n = 5 $。
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 $n = 5$.
