Algorithmic and Minimax Complexities in Kernel Bandits

2026-06-09

AI Summary

This work clarifies the theoretical distinctions between Gaussian Process Upper Confidence Bound (GP-UCB) and the Decision-Estimation Coefficient (DEC) approaches under kernel bandwidth specification, revealing that algorithmic complexity and minimax complexity address fundamentally different questions in overparameterized settings, thereby leading to performance gaps. To reconcile these perspectives, the authors propose a unified MAIR framework that integrates heterogeneous positive semidefinite algorithmic priors, casting both GP-UCB and MAMS within a common algorithmic information-theoretic language and designing a safety-guaranteed master algorithm. Through RKHS bandwidth modeling, construction of algorithmic priors, and trajectory complexity analysis, they establish—for the first time—that algorithmic complexity provides a more informative characterization than class-level minimax or DEC bounds in kernel bandwidth regimes, successfully extending GP-UCB theory and demonstrating its superiority in overparameterized scenarios.

Abstract

Gaussian-process upper confidence bound (GP-UCB) and decision-estimation-coefficient (DEC) methods may appear, at first sight, to belong to different theories. This paper places the two viewpoints in a common algorithmic-information language for frequentist RKHS bandits. GP-UCB fixes an algorithmic, rather than true, Gaussian-process prior and exploits realized-trajectory complexity together with computational tractability, whereas MAMS optimizes a robust class-wide MAIR/DEC envelope. Through the unified MAIR framework and heterogeneous positive-semidefinite algorithmic priors, we generalize both the GP-UCB analysis and the MAMS algorithm, propose a safeguarded master that combines their advantages, and provide a kernel-bandit construction showing that algorithmic complexity can be more informative than class-wide minimax or DEC certificates in overparameterized models. The resulting message is that algorithmic information and class-wide minimax coefficients answer different questions and can lead to different gaps; kernel bandits provide a clean setting in which this distinction becomes mathematically visible.

Problem

• kernel bandits
• algorithmic complexity
• minimax complexity
• decision-estimation coefficient
• RKHS

Innovation

• kernel bandits
• algorithmic information
• minimax complexity
• GP-UCB
• DEC