The Elliptical Potential Lemma for General Distributions with an Application to Linear Thompson Sampling

Abstract

In this note, we introduce a general version of the well-known elliptical potential lemma that is a widely used technique in the analysis of algorithms in sequential learning and decision-making problems. We consider a stochastic linear bandit setting where a decision-maker sequentially chooses among a set of given actions, observes their noisy rewards, and aims to maximize her cumulative expected reward over a decision-making horizon. The elliptical potential lemma is a key tool for quantifying uncertainty in estimating parameters of the reward function, but it requires the noise and the prior distributions to be Gaussian. Our general elliptical potential lemma relaxes this Gaussian requirement which is a highly non-trivial extension for a number of reasons; unlike the Gaussian case, there is no closed-form solution for the covariance matrix of the posterior distribution, the covariance matrix is not a deterministic function of the actions, and the covariance matrix is not decreasing with respect to the semidefinite inequality. While this result is of broad interest, we showcase an application of it to prove an improved Bayesian regret bound for the well-known Thompson sampling algorithm in stochastic linear bandits with changing action sets where prior and noise distributions are general. This bound is minimax optimal up to constants.