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

arXiv: https://arxiv.org/abs/2609.32149 · HTML: https://mlsanity.com/paper/2609.32149/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 14 references that are in the map.
- First posted on arXiv 26 Sep 2026.

## Builds on (most cited first, 6 of 6)

- [Learning to Optimize Via Posterior Sampling](https://mlsanity.com/paper/1301.2609/) (2013)
- [On Explore-Then-Commit Strategies](https://mlsanity.com/paper/1605.08988/) (2016)
- [Linear bandits with polylogarithmic minimax regret](https://mlsanity.com/paper/2402.12042/) (2024)
- [When and why randomised exploration works (in linear bandits)](https://mlsanity.com/paper/2502.08870/) (2025)
- [Geometry-Aware Approaches for Balancing Performance and Theoretical Guarantees in Linear Bandits](https://mlsanity.com/paper/2306.14872/) (2023)
- [Tangential Randomization in Linear Bandits (TRAiL): Guaranteed Inference and Regret Bounds](https://mlsanity.com/paper/2411.12154/) (2024)

## Abstract (arXiv)

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 √t with time t, while actions remain sufficiently concentrated for exploitation, then an algorithm produces optimal high-probability O(√Tlog 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.