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# Exploration in Linear Bandits with Rich Action Sets and its Implications for Inference

*Proceedings of The 26th International Conference on Artificial Intelligence and Statistics*, PMLR 206:8233-8262, 2023.

#### Abstract

We present a non-asymptotic lower bound on the spectrum of the design matrix generated by any linear bandit algorithm with sub-linear regret when the action set has well-behaved curvature. Specifically, we show that the minimum eigenvalue of the expected design matrix grows as $\Omega(\sqrt{n})$ whenever the expected cumulative regret of the algorithm is $O(\sqrt{n})$, where $n$ is the learning horizon, and the action-space has a constant Hessian around the optimal arm. This shows that such action-spaces force a polynomial lower bound on the least eigenvalue, rather than a logarithmic lower bound as shown by Lattimore et al. (2017) for discrete (i.e., well-separated) action spaces. Furthermore, while the latter holds only in the asymptotic regime ($n \to \infty$), our result for these “locally rich” action spaces is any-time. Additionally, under a mild technical assumption, we obtain a similar lower bound on the minimum eigen value holding with high probability. We apply our result to two practical scenarios – model selection and clustering in linear bandits. For model selection, we show that an epoch-based linear bandit algorithm adapts to the true model complexity at a rate exponential in the number of epochs, by virtue of our novel spectral bound. For clustering, we consider a multi agent framework where we show, by leveraging the spectral result, that no forced exploration is necessary—the agents can run a linear bandit algorithm and estimate their underlying parameters at once, and hence incur a low regret.