一般分布的椭圆位引理及其在线性Thompson抽样中的应用
The Elliptical Potential Lemma for General Distributions with an Application to Linear Thompson Sampling
摘要：
在这篇文章中，我们介绍了一个通用版本的著名的椭圆势引理，这是一个广泛使用的技术在顺序学习和决策问题的算法分析。我们考虑一个随机线性的强盗设置，决策者顺序选择一组给定的行动，观察他们的噪音奖励，并旨在最大化她的累积预期奖励在决策的地平线。椭圆势引理是一种量化奖励函数参数估计中不确定性的关键工具，但它要求噪声和先验分布为高斯分布。我们的一般椭圆势引理放松了这个高斯要求，这是一个高度非平凡的扩展，原因有很多;不像高斯情况下，没有封闭形式的解决方案的协方差矩阵的后验分布，协方差阵不是一个确定性的功能的行动，和协方差阵是不是减少相对于半定不等式。虽然这一结果是广泛的兴趣，我们展示了它的应用，以证明一个改进的贝叶斯遗憾界的著名的汤普森采样算法在随机线性土匪不断变化的行动集，先验和噪声分布是一般的。这个界是极大极小最优的常数。
英文摘要：
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.
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