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Register a new userUpper Counterfactual Confidence Bounds: a New Optimism Principle for Contextual Bandits
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Yunbei Xu
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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The principle of optimism in the face of uncertainty is one of the most widely used and successful ideas in multi-armed bandits and reinforcement learning. However, existing optimistic algorithms (primarily UCB and its variants) are often unable to deal with large context spaces. Essentially all existing well performing algorithms for general contextual bandit problems rely on weighted action allocation schemes; and theoretical guarantees for optimism-based algorithms are only known for restricted formulations. In this paper we study general contextual bandits under the realizability condition, and propose a simple generic principle to design optimistic algorithms, dubbed Upper Counterfactual Confidence Bounds (UCCB). We show that these algorithms are provably optimal and efficient in the presence of large context spaces. Key components of UCCB include: 1) a systematic analysis of confidence bounds in policy space rather than in action space; and 2) the potential function perspective that is used to express the power of optimism in the contextual setting. We further show how the UCCB principle can be extended to infinite action spaces, by constructing confidence bounds via the newly introduced notion of counterfactual action divergence.
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Bandit learning algorithms typically involve the balance of exploration and exploitation. However, in many practical applications, worst-case scenarios needing systematic exploration are seldom encountered. In this work, we consider a smoothed setting for structured linear contextual bandits where the adversarial contexts are perturbed by Gaussian noise and the unknown parameter $theta^*$ has structure, e.g., sparsity, group sparsity, low rank, etc. We propose simple greedy algorithms for both the single- and multi-parameter (i.e., different parameter for each context) settings and provide a unified regret analysis for $theta^*$ with any assumed structure. The regret bounds are expressed in terms of geometric quantities such as Gaussian widths associated with the structure of $theta^*$. We also obtain sharper regret bounds compared to earlier work for the unstructured $theta^*$ setting as a consequence of our improved analysis. We show there is implicit exploration in the smoothed setting where a simple greedy algorithm works.
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