In this paper, we address the stochastic contextual linear bandit problem, where a decision maker is provided a context (a random set of actions drawn from a distribution). The expected reward of each action is specified by the inner product of the action and an unknown parameter. The goal is to design an algorithm that learns to play as close as possible to the unknown optimal policy after a number of action plays. This problem is considered more challenging than the linear bandit problem, which can be viewed as a contextual bandit problem with a \emph{fixed} context. Surprisingly, in this paper, we show that the stochastic contextual problem can be solved as if it is a linear bandit problem. In particular, we establish a novel reduction framework that converts every stochastic contextual linear bandit instance to a linear bandit instance, when the context distribution is known. When the context distribution is unknown, we establish an algorithm that reduces the stochastic contextual instance to a sequence of linear bandit instances with small misspecifications and achieves nearly the same worst-case regret bound as the algorithm that solves the misspecified linear bandit instances. As a consequence, our results imply a $O(d\sqrt{T\log T})$ high-probability regret bound for contextual linear bandits, making progress in resolving an open problem in (Li et al., 2019), (Li et al., 2021). Our reduction framework opens up a new way to approach stochastic contextual linear bandit problems, and enables improved regret bounds in a number of instances including the batch setting, contextual bandits with misspecifications, contextual bandits with sparse unknown parameters, and contextual bandits with adversarial corruption.
翻译:在本文中, 我们处理的是背景背景的线性土匪问题, 在这种问题上, 决策者可以提供上下文( 随机的一组从分布中抽取的动作 ) 。 每个动作的预期回报由动作的内部产物和一个未知参数来指定。 目标是设计一个算法, 在一系列动作播放后, 学会尽可能接近未知的最佳政策。 这个问题比线性土匪问题更具有挑战性。 这个问题可以被看成是一个背景的土匪问题。 令人惊讶的是, 在本文中, 我们显示, 每个动作的随机背景问题可以像线性土匪问题一样得到解决。 特别是, 我们建立一个新的递减框架, 将每个随机背景线性线性线性线性土匪事件转换成一个线性土匪事件。 当背景分布不明时, 我们建立一个算法, 将线性背景背景问题降低到直线性土匪事件序列的顺序( 细略微的直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直的直径直径直径直径直径直径直径直径直,, 直直直直直直直直直直直直至直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直的直的直的直直直直的直的直的直的直的直的直的直的直的直的直的直的直的直直直的直的直的直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直