When the noncommutative AM-GM inequality holds

2026-10-04

AI Summary

This study addresses the failure of the non-commutative AM-GM inequality under specific conditions and the challenge of non-positivity in non-integer node measures. To overcome these issues, this work proposes a novel framework that reformulates the original problem as one of quadrature estimation. Specifically, it constructs vertex measures using Chebyshev nodes on the Boolean hypercube and leverages Grigoriev’s Positivstellensatz technique to resolve the underlying theoretical difficulties. The primary contribution is a rigorous proof establishing that the inequality holds whenever n ≥ 2⌈m/2⌉², thereby determining the precise quantitative condition for its validity and refining the relevant theoretical bounds.

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.

Problem

• noncommutative AM-GM inequality
• Boolean cube
• Chebyshev nodes
• random reshuffling

Innovation

• Noncommutative AM-GM inequality
• Chebyshev nodes
• Boolean cube
• Vertex measure
• Quadrature estimate