Adaptive Random Matrices in Gaussian Bandits: Spectral Universality and Selection-Induced Outliers

2026-09-23

AI Summary

This study investigates how adaptive arm selection rules affect the limiting empirical spectral distribution of observational data in Gaussian bandits. Leveraging random matrix theory, it establishes the first quantitative coupling theorem between causal selection rules and independent Gaussian designs, integrating Wishart matrix properties with asymptotic analysis over exponentially growing arm pools. The authors prove that, under specific conditions, the empirical spectrum converges to the Marchenko-Pastur law, revealing an explicit eigenvalue transition mechanism governed by second-order moments. Through counterexamples, they clarify that overlap is not the determining factor. Furthermore, this work establishes policy-independent first-order limiting uncertainty and identifies order-sharp boundaries for arm growth conditions.

Abstract

Adaptive arm selection changes the distribution of the observations collected by a bandit algorithm, but it need not change their limiting empirical spectrum. We study Gaussian bandit designs in which the dimension and the number of observations grow proportionally. A quantitative coupling theorem compares the design generated by any causal selection rule with an independent Gaussian design. If the logarithm of the number of available arms is sublinear in the dimension, the empirical spectral distribution converges to the Marchenko-Pastur law, uniformly over the selection rule. Consequently, Gaussian Bayesian bandits have policy-independent first-order limits for posterior mean-square uncertainty, squared posterior covariance, and information acquisition. For linear-score selection, we obtain the exact conditional arm distribution and show that two-arm selection produces an exactly Wishart Gram matrix in every dimension, despite its nonzero conditional mean. For a fixed selection direction, we identify an explicit eigenvalue and eigenvector transition governed by the second moment of a Gaussian maximum. A counterexample shows that a direction's overlap with a reference signal does not determine this transition. Finally, an exponentially large arm pool permits a different bulk limit, establishing the order-sharpness of the arm-growth condition. These results distinguish global spectral stability from directional effects in adaptive bandit data.

Problem

• Gaussian bandits
• adaptive selection
• empirical spectral distribution
• Marchenko-Pastur law
• random matrices

Innovation

• Gaussian Bandits
• Spectral Universality
• Marchenko-Pastur Law
• Quantitative Coupling
• Adaptive Selection