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In this note, we introduce a general version of the well-known ellipticalpotential lemma that is a widely used technique in the analysis of algorithmsin sequential learning and decision-making problems. We consider a stochasticlinear bandit setting where a decision-maker sequentially chooses among a setof given actions, observes their noisy rewards, and aims to maximize hercumulative expected reward over a decision-making horizon. The ellipticalpotential lemma is a key tool for quantifying uncertainty in estimatingparameters of the reward function, but it requires the noise and the priordistributions to be Gaussian. Our general elliptical potential lemma relaxesthis Gaussian requirement which is a highly non-trivial extension for a numberof reasons; unlike the Gaussian case, there is no closed-form solution for thecovariance matrix of the posterior distribution, the covariance matrix is not adeterministic function of the actions, and the covariance matrix is notdecreasing with respect to the semidefinite inequality. While this result is ofbroad interest, we showcase an application of it to prove an improved Bayesianregret bound for the well-known Thompson sampling algorithm in stochasticlinear bandits with changing action sets where prior and noise distributionsare general. This bound is minimax optimal up to constants.

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