In this paper, we study both multi-armed and contextual bandit problems in censored environments. Our goal is to estimate the performance loss due to censorship in the context of classical algorithms designed for uncensored environments. Our main contributions include the introduction of a broad class of censorship models and their analysis in terms of the effective dimension of the problem -- a natural measure of its underlying statistical complexity and main driver of the regret bound. In particular, the effective dimension allows us to maintain the structure of the original problem at first order, while embedding it in a bigger space, and thus naturally leads to results analogous to uncensored settings. Our analysis involves a continuous generalization of the Elliptical Potential Inequality, which we believe is of independent interest. We also discover an interesting property of decision-making under censorship: a transient phase during which initial misspecification of censorship is self-corrected at an extra cost, followed by a stationary phase that reflects the inherent slowdown of learning governed by the effective dimension. Our results are useful for applications of sequential decision-making models where the feedback received depends on strategic uncertainty (e.g., agents' willingness to follow a recommendation) and/or random uncertainty (e.g., loss or delay in arrival of information).

翻译：本文研究审查环境下多臂赌博机与情境赌博机问题。我们的目标是在针对非审查环境设计的经典算法背景下，评估审查导致的性能损失。主要贡献包括引入一类广泛的审查模型，并从问题的有效维数角度进行分析——有效维数是衡量问题潜在统计复杂性的自然度量，也是遗憾界的主要驱动因素。特别地，有效维数允许我们在保持原问题一阶结构的同时将其嵌入更大空间，从而自然得到与非审查环境类似的结果。我们的分析涉及椭圆势不等式的一种连续推广，我们认为这一推广具有独立研究价值。我们还发现审查环境下决策制定的一项有趣特性：存在一个瞬态阶段，期间审查的初始错误指定会以额外代价自我修正，随后进入反映由有效维数支配的学习固有减速的稳态阶段。我们的研究结果对反馈取决于策略不确定性（例如智能体遵循建议的意愿）和/或随机不确定性（例如信息到达的缺失或延迟）的序列决策模型应用具有重要价值。