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arXiv2026

A Unified Optimism-Agnostic Framework for Linear Bandits over Spherical Action Sets

Linear bandits model sequential decision-making problems with noisy rewards that are linear in the decision variable, where an agent must simultaneously learn about an unknown parameter that governs the mean rewards, while maximizing (expected) rewards over time. Two prominent algorithmic families--upper confidence bound (UCB) and Thompson sampling (TS)--achieve a balance of exploration (to estimate said parameter) and exploitation (utilization of knowledge about it) across time. The quality of estimation of that parameter depends on the eigenvalues of a design matrix. In this paper, we begin by showing that if the inference quality obtained from exploration, encoded in the minimum eigenvalue of the design matrix, grows $\gtrsim \sqrt{t}$ with time $t$, while actions remain sufficiently concentrated for exploitation, then an algorithm produces optimal high-probability $\mathcal{O}(\sqrt{T}\log T)$-regret rate over a time-horizon $T$ for spherical action sets. This analysis is algorithm-agnostic and follows an alternative route to the classical optimism-based elliptical-potential argument for regret analysis. Then, we illustrate that variants of UCB and TS satisfy the inference and concentration properties and in turn, enjoy optimal regret rate. In effect, our results provide a modular framework that can be used to analyze linear bandit algorithms and explicitly connect quality of parameter estimation to optimal regret accumulation.

stat.ML↗

arXiv2024

Tangential Randomization in Linear Bandits (TRAiL): Guaranteed Inference and Regret Bounds

We propose and analyze TRAiL (Tangential Randomization in Linear Bandits), a computationally efficient regret-optimal forced exploration algorithm for linear bandits on action sets that are sublevel sets of strongly convex functions. TRAiL estimates the governing parameter of the linear bandit problem through a standard regularized least squares and perturbs the reward-maximizing action corresponding to said point estimate along the tangent plane of the convex compact action set before projecting back to it. Exploiting concentration results for matrix martingales, we prove that TRAiL ensures a $Ω(\sqrt{T})$ growth in the inference quality, measured via the minimum eigenvalue of the design (regressor) matrix with high-probability over a $T$-length period. We build on this result to obtain an $\mathcal{O}(\sqrt{T} \log(T))$ upper bound on cumulative regret with probability at least $ 1 - 1/T$ over $T$ periods, and compare TRAiL to other popular algorithms for linear bandits. Then, we characterize an $Ω(\sqrt{T})$ minimax lower bound for any algorithm on the expected regret that covers a wide variety of action/parameter sets and noise processes. Our analysis not only expands the realm of lower-bounds in linear bandits significantly, but as a byproduct, yields a trade-off between regret and inference quality. Specifically, we prove that any algorithm with an $\mathcal{O}(T^α)$ expected regret growth must have an $Ω(T^{1-α})$ asymptotic growth in expected inference quality. Our experiments on the $L^p$ unit ball as action sets reveal how this relation can be violated, but only in the short-run, before returning to respect the bound asymptotically. In effect, regret-minimizing algorithms must have just the right rate of inference -- too fast or too slow inference will incur sub-optimal regret growth.

stat.ML↗