On Kernelized Multi-armed Bandits (1704.00445v2)

Abstract: We consider the stochastic bandit problem with a continuous set of arms, with the expected reward function over the arms assumed to be fixed but unknown. We provide two new Gaussian process-based algorithms for continuous bandit optimization-Improved GP-UCB (IGP-UCB) and GP-Thomson sampling (GP-TS), and derive corresponding regret bounds. Specifically, the bounds hold when the expected reward function belongs to the reproducing kernel Hilbert space (RKHS) that naturally corresponds to a Gaussian process kernel used as input by the algorithms. Along the way, we derive a new self-normalized concentration inequality for vector- valued martingales of arbitrary, possibly infinite, dimension. Finally, experimental evaluation and comparisons to existing algorithms on synthetic and real-world environments are carried out that highlight the favorable gains of the proposed strategies in many cases.

• The paper introduces two novel GP-based algorithms, IGP-UCB and GP-TS, that enhance continuous bandit optimization with improved regret bounds.
• The paper derives a self-normalized concentration inequality for infinite-dimensional vector-valued martingales, providing a robust theoretical foundation.
• Empirical results demonstrate that the proposed methods outperform traditional approaches in both synthetic and real-world scenarios.

An Analytical Approach to Kernelized Multi-Armed Bandits

The paper, "On Kernelized Multi-armed Bandits," by Sayak Ray Chowdhury and Aditya Gopalan, presents a detailed paper of continuous stochastic bandit problems using Gaussian Processes (GP) to model uncertainty. The authors introduce two novel algorithms: Improved GP-UCB (IGP-UCB) and GP-Thomson Sampling (GP-TS). These algorithms are designed to optimize continuous bandit problems where the expected reward function is fixed but unknown, belonging to a reproducing kernel Hilbert space (RKHS) associated with a GP kernel. The paper makes significant technical contributions, including deriving new regret bounds and establishing a self-normalized concentration inequality for vector-valued martingales in potentially infinite dimensions.

Overview of Contributions

1. Algorithmic Development: The paper introduces two algorithms for continuous bandit optimization. IGP-UCB provides an improvement over the existing GP-UCB method by refining the confidence interval used in the upper confidence bound approach, leading to better regret performance. GP-TS extends Thompson Sampling into the nonparametric field by employing Gaussian Processes, achieving a new regret bound for this setting.
2. New Theoretical Tools: A critical contribution is the derivation of a self-normalized concentration inequality for infinite-dimensional vector-valued martingales. This result is pivotal for the analysis of the proposed algorithms and might have implications beyond the scope of this paper, potentially influencing future research in infinite-dimensional statistical learning and decision-making processes.
3. Empirical Validation: Empirical results demonstrate the practical effectiveness of the proposed algorithms in synthetic as well as real-world scenarios. The authors compare the performance of IGP-UCB and GP-TS against existing methods like GP-EI, GP-PI, and the original GP-UCB, highlighting the enhancements provided by their approaches.
4. Analysis of Regret Bounds: The paper provides rigorous analysis with bounds on regret for both algorithms. The regret for IGP-UCB is shown to scale as $O(\sqrt{T}(B\sqrt{\gamma_T}+\gamma_T))$, a notable improvement over previous work by reducing a multiplicative $O(\ln^{3/2}T)$ factor. For GP-TS, the bounds obtained are $\tilde{O}(\gamma_T\sqrt{dT})$, providing insights into nonparametric Thompson Sampling's efficacy.

Implications and Future Directions

The proposed algorithms and theoretical approaches have several implications for the field of AI, particularly in sequential decision-making and reinforcement learning with continuous action spaces. The improved regret bounds indicate more efficient balance between exploration and exploitation, which is critical for applications in dynamic pricing, continuous state-action reinforcement learning, and adaptive communication systems.

Future work might focus on extending these methods to scenarios where the kernel itself is not known and must be learned concurrently with the decision problem. Additionally, exploring computationally efficient implementations for high-dimensional problems remains an open area. Another potential direction lies in integrating the GP-based nonparametric models with other machine learning paradigms, such as deep learning, to handle scalable and complex systems more effectively.

This paper sets a foundation that bridges the gap between theoretical advances and practical implementations, enabling more robust and efficient solutions for real-world problems characterized by uncertainty and complexity.