X-Armed Bandits (1001.4475v2)

Abstract: We consider a generalization of stochastic bandits where the set of arms, $\cX$, is allowed to be a generic measurable space and the mean-payoff function is "locally Lipschitz" with respect to a dissimilarity function that is known to the decision maker. Under this condition we construct an arm selection policy, called HOO (hierarchical optimistic optimization), with improved regret bounds compared to previous results for a large class of problems. In particular, our results imply that if $\cX$ is the unit hypercube in a Euclidean space and the mean-payoff function has a finite number of global maxima around which the behavior of the function is locally continuous with a known smoothness degree, then the expected regret of HOO is bounded up to a logarithmic factor by $\sqrt{n}$, i.e., the rate of growth of the regret is independent of the dimension of the space. We also prove the minimax optimality of our algorithm when the dissimilarity is a metric. Our basic strategy has quadratic computational complexity as a function of the number of time steps and does not rely on the doubling trick. We also introduce a modified strategy, which relies on the doubling trick but runs in linearithmic time. Both results are improvements with respect to previous approaches.

• The paper generalizes traditional bandit problems by extending the arm set to measurable spaces and leveraging local Lipschitz continuity for mean-payoff functions.
• The paper introduces the Hierarchical Optimistic Optimization (HOO) strategy, which smartly balances exploration and exploitation to minimize regret.
• The paper demonstrates that HOO achieves near-minimax optimal regret rates with computational efficiency, even in high-dimensional and complex environments.

Analysis of "$X$–Armed Bandits"

The paper presents a comprehensive exploration of a generalization in the field of stochastic bandit problems, notably enlarging the scope to $X$--armed bandits where the set of arms, $X$, is a generic measurable space. The authors aim to improve the existing regret bounds by introducing a novel arm selection strategy termed Hierarchical Optimistic Optimization (HOO), which effectively utilizes known conditions about the local Lipschitz continuity of the mean-payoff function.

Major Contributions

1. Generalization to Measurable Spaces: The extension of traditional bandit settings to measurable spaces significantly broadens the applicability of stochastic bandit frameworks. The paper harnesses the concept of local Lipschitz continuity in defining the behavior of the mean-payoff function relative to a dissimilarity function, providing a structured method to navigate infinitely many arms.
2. Introduced Algorithm - HOO: The HOO strategy is an innovative approach that incrementally builds hierarchical estimates of the mean-payoff function. The exploration-exploitation balance is maintained by focusing estimation efforts around the function's maxima, leading to enhanced regret minimization.
3. Enhanced Regret Bounds: The authors demonstrate that, under their assumptions, the HOO algorithm achieves superior regret bounds, notably when $X$ is a unit hypercube in Euclidean space. Particularly striking is that the regret is shown to grow at a rate independent of the dimensionality of $X$, merely scaling with $\sqrt{n}$ for specific configurations.
4. Minimax Optimality and Computational Efficiency: Establishing the minimax optimality of HOO under metric dissimilarities, the paper attests to the robustness of the algorithm across diverse practical settings. Furthermore, a computationally efficient variant of HOO is introduced, achieving linearithmic complexity by employing a modified strategy that incorporates the doubling trick.

Implications and Future Directions

From a theoretical standpoint, this work establishes a foundational base for further explorations into environments with complex, high-dimensional parameter spaces. Practically, the implications stretch across scenarios requiring decision-making with infinitely many choices, such as hyperparameter tuning in machine learning.

The strategy elucidated could be extended for various scalable applications, suggesting a potential intersection with reinforcement learning domains and other sequential decision-making frameworks. Future research might focus on adaptive versions of the parameters guiding the HOO strategy, or on employing the algorithm in dynamic environments where the characteristics of the arms or the nature of the dissimilarities evolve over time.

By addressing these aspects, this paper not only opens new avenues in the theoretical landscape of stochastic optimization but also paves the way for its pragmatic implications in real-world scenarios. Such contributions underscore the utility of theoretical advancements when melded with applicable algorithmic strategies.