Finite-Time Analysis of Kernelised Contextual Bandits (1309.6869v1)

Abstract: We tackle the problem of online reward maximisation over a large finite set of actions described by their contexts. We focus on the case when the number of actions is too big to sample all of them even once. However we assume that we have access to the similarities between actions' contexts and that the expected reward is an arbitrary linear function of the contexts' images in the related reproducing kernel Hilbert space (RKHS). We propose KernelUCB, a kernelised UCB algorithm, and give a cumulative regret bound through a frequentist analysis. For contextual bandits, the related algorithm GP-UCB turns out to be a special case of our algorithm, and our finite-time analysis improves the regret bound of GP-UCB for the agnostic case, both in the terms of the kernel-dependent quantity and the RKHS norm of the reward function. Moreover, for the linear kernel, our regret bound matches the lower bound for contextual linear bandits.

• The paper presents the KernelUCB algorithm, extending UCB methods into RKHS to model complex, non-linear reward functions.
• It derives a novel cumulative regret bound that scales with the effective dimension, offering a more tailored analysis than traditional approaches.
• The study shows practical improvements in online decision-making, notably in recommendation systems and advertising, by leveraging kernel methods.

An Expert Overview of "Finite-Time Analysis of Kernelised Contextual Bandits"

The paper "Finite-Time Analysis of Kernelised Contextual Bandits" authored by Michal Valko, Nathan Korda, Rémi Munos, Ilias Flaounas, and Nello Cristianini provides an in-depth exploration into the enhancement of contextual bandit algorithms through kernelisation. This paper proposes the KernelUCB algorithm, offering a fresh perspective by extending the classical Upper Confidence Bound (UCB) methods into the field of Reproducing Kernel Hilbert Spaces (RKHS). Here, we will discuss the primary contributions, theoretical insights, and implications stemming from this work.

Main Contributions

The central thesis of the paper is to address the challenge in contextual bandit settings where the number of actions is so large that exploring each action becomes impractical. Instead of relying on all actions being explicitly sampled, the paper leverages similarity information between actions' contexts, which can be efficiently encoded in the RKHS framework using kernels.

The noteworthy contributions of this paper include:

1. KernelUCB Algorithm: The authors introduce KernelUCB, a kernelised version of the LinUCB algorithm, designed for scenarios where action contexts can be projected into an infinite-dimensional RKHS. This formulation allows capturing complex, non-linear relationships between contexts and rewards beyond the traditional linear contextual models.
2. Cumulative Regret Analysis: A novel aspect of the paper is its finite-time analysis, yielding a cumulative regret bound that scales with the effective dimension $d$ of the data in the RKHS. This is a significant innovation as it adapts the regret analysis uniquely to data-dependent characteristics via the effective dimension rather than relying on a fixed-dimensional space.
3. Connections and Improvements Over GP-UCB: The analysis shows that GP-UCB can be seen as a special case of KernelUCB when regularisation aligns with model noise. More importantly, the KernelUCB algorithm offers improved regret bounds over GP-UCB in the agnostic setting, where the reward function is not sampled from a Gaussian Process (GP).

Theoretical Insights

The work derives cumulative regret bounds for KernelUCB under the assumption that expected rewards are arbitrary linear functions of the contexts in the RKHS. The theoretical underpinnings rely heavily on the kernel trick, transforming the problem into the dual space, which leads to efficient computations via kernel matrices.

The regret bound's dependence on the effective dimension $d$ provides a unique advantage by encapsulating the complexity of underlying data distributions. It reflects the principal directions in the RKHS where the data predominantly lies, thus achieving a more tailored and potentially tighter analysis than previous approaches.

Practical Implications and Future Directions

The implications of this paper are far-reaching in applications like online advertising and recommendation systems, where actions—such as which ad to display—must be chosen from a vast set of possibilities. The incorporation of similarity measures through kernelisation allows for more sophisticated modelling of user preferences and content relevance at reduced computational costs.

Furthermore, the paper lays a strong foundation for future exploration in adaptive regularisation based on data distribution properties, which can enhance the versatility and performance of contextual bandit frameworks. This approach could also be extended to other forms of non-linear bandits or settings with more complex reward structures.

In conclusion, the paper provides a significant step forward in contextual bandit research by empowering algorithms with kernel methods. The theoretical developments and connections to existing approaches like GP-UCB advance both the understanding and application of bandit algorithms in high-dimensional spaces. Going forward, researchers might build upon this work to explore alternative kernel functions or integrate this methodology with deep learning-based contextual representations to tackle even more complex decision-making environments.