Recht-Re Noncommutative Arithmetic-Geometric Mean Conjecture is False

Zehua Lai, Lek-Heng Lim
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:5608-5617, 2020.

Abstract

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 (also known as random reshuffling) indeed outperforms with-replacement sampling in some important optimization problems. 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 $n = 5$.

Cite this Paper

BibTeX
@InProceedings{pmlr-v119-lai20a, title = {Recht-Re Noncommutative Arithmetic-Geometric Mean Conjecture is False}, author = {Lai, Zehua and Lim, Lek-Heng}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {5608--5617}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/lai20a/lai20a.pdf}, url = {https://proceedings.mlr.press/v119/lai20a.html}, abstract = {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 (also known as random reshuffling) indeed outperforms with-replacement sampling in some important optimization problems. 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 $n = 5$.} }
Endnote
%0 Conference Paper %T Recht-Re Noncommutative Arithmetic-Geometric Mean Conjecture is False %A Zehua Lai %A Lek-Heng Lim %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-lai20a %I PMLR %P 5608--5617 %U https://proceedings.mlr.press/v119/lai20a.html %V 119 %X 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 (also known as random reshuffling) indeed outperforms with-replacement sampling in some important optimization problems. 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 $n = 5$.
APA
Lai, Z. & Lim, L.. (2020). Recht-Re Noncommutative Arithmetic-Geometric Mean Conjecture is False. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:5608-5617 Available from https://proceedings.mlr.press/v119/lai20a.html.

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