Contextual Bandits with Similarity Information (0907.3986v5)

Abstract: In a multi-armed bandit (MAB) problem, an online algorithm makes a sequence of choices. In each round it chooses from a time-invariant set of alternatives and receives the payoff associated with this alternative. While the case of small strategy sets is by now well-understood, a lot of recent work has focused on MAB problems with exponentially or infinitely large strategy sets, where one needs to assume extra structure in order to make the problem tractable. In particular, recent literature considered information on similarity between arms. We consider similarity information in the setting of "contextual bandits", a natural extension of the basic MAB problem where before each round an algorithm is given the "context" -- a hint about the payoffs in this round. Contextual bandits are directly motivated by placing advertisements on webpages, one of the crucial problems in sponsored search. A particularly simple way to represent similarity information in the contextual bandit setting is via a "similarity distance" between the context-arm pairs which gives an upper bound on the difference between the respective expected payoffs. Prior work on contextual bandits with similarity uses "uniform" partitions of the similarity space, which is potentially wasteful. We design more efficient algorithms that are based on adaptive partitions adjusted to "popular" context and "high-payoff" arms.

• The paper introduces an adaptive partitioning algorithm for contextual bandits that refines exploration based on context popularity and payoff expectations.
• It achieves provable regret bounds by leveraging both context and arm similarity, significantly outperforming traditional uniform partition methods.
• The approach extends to specialized bandit problems, offering versatile solutions for dynamic settings like online advertising and sponsored search.

Contextual Bandits with Similarity Information

The paper examines the multi-armed bandit (MAB) problem with an emphasis on large or infinite strategy sets where contextual information is available. The challenge in such settings is to efficiently maximize the total payoff by balancing exploration and exploitation. This work builds on the concept of contextual bandits, an extension of the basic MAB model where each decision round is informed by a context, effectively providing additional information to aid decision-making. This has clear applications in sponsored search and online advertisements where context, such as user profile data, could inform which ad to display.

Key Contributions

1. Adaptive Partitions: The paper critiques prior approaches that rely on uniform partitions of the similarity space, which do not account for specific structures in payoff functions. Instead, it proposes an algorithm that employs adaptive partitions, refining these partitions based on context popularity and payoff expectations. This methodology ensures improved performance by focusing exploration efforts on regions with high potential payoffs.
2. Provable Guarantees: The proposed algorithm demonstrates improved performance through rigorous regret bounds. Specifically, it achieves regret bounds dependent on both the context and arms space, thus offering refined guarantees over uniform partition methods used in existing literature. The results show optimization by leveraging the benign nature of context arrivals and payoff structures.
3. Applications to Other Bandit Problems: The algorithm is versatile, applicable to constrained and evolving bandit settings such as those with constrained temporal changes and sleeping bandits. Notably, it achieves meaningful bounds within these specialized contexts, demonstrating the utility of contextual zooming in broader MAB settings.
4. Lower Bound Discussions: The work extensively discusses the conditions under which the regret bounds are tight, verifying their efficacy across different scenarios mediated by the variability in context and arm similarity spaces.

Implications and Future Directions

From a practical standpoint, the advancement in adaptive partitioning has significant implications for online advertising and other decision-making systems impacted by dynamic user contexts. The theoretical contribution lies in showing that meaningful improvements in algorithm performance can be achieved without sacrificing robustness in worst-case scenarios.

Theoretically, this paper opens avenues for further research in MAB algorithms by implying that other forms of structure, beyond similarity, could be exploited to achieve faster convergence and lower regret. Future work may extend these ideas to non-Lipschitz settings or assume alternative structures on payoffs, exploring whether similar performance benefits can be achieved under weaker assumptions.

In conclusion, this paper advances our understanding of leveraging context in MAB frameworks, proposing an algorithm that is both theoretically sound and practically applicable to industries reliant on fine-grained decision-making mechanisms.