When the noncommutative AM-GM inequality holds

Abstract:In this note, we prove that the noncommutative AM-GM inequality holds if $n\ge 2\lceil m/2 \rceil^2$. The motivation comes from counterexamples constructed in [De Sa, Random reshuffling is not always better, NeurIPS2020]. The proof constructs a vertex measure based on the Chebyshev nodes on the Boolean cube to extract the distinct indices. The main difficulty is that the measure is not positive on non-integer nodes. The key technique comes from [Grigoriev, Complexity of Positivstellensatz proofs for the knapsack, Computational Complexity (2001)] and eventually transforms the problem into a quadrature estimate.

Probability (math.PR); Machine Learning (cs.LG); Optimization and Control (math.OC)

arXiv:2610.04874 [math.PR]

(or arXiv:2610.04874v1 [math.PR] for this version)

https://doi.org/10.48550/arXiv.2610.04874 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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