Linear Contextual Bandits with Hybrid Payoff: Revisited
(2406.10131)Abstract
We study the Linear Contextual Bandit problem in the hybrid reward setting. In this setting every arm's reward model contains arm specific parameters in addition to parameters shared across the reward models of all the arms. We can reduce this setting to two closely related settings (a) Shared - no arm specific parameters, and (b) Disjoint - only arm specific parameters, enabling the application of two popular state of the art algorithms - $\texttt{LinUCB}$ and $\texttt{DisLinUCB}$ (Algorithm 1 in (Li et al. 2010)). When the arm features are stochastic and satisfy a popular diversity condition, we provide new regret analyses for both algorithms, significantly improving on the known regret guarantees of these algorithms. Our novel analysis critically exploits the hybrid reward structure and the diversity condition. Moreover, we introduce a new algorithm $\texttt{HyLinUCB}$ that crucially modifies $\texttt{LinUCB}$ (using a new exploration coefficient) to account for sparsity in the hybrid setting. Under the same diversity assumptions, we prove that $\texttt{HyLinUCB}$ also incurs only $O(\sqrt{T})$ regret for $T$ rounds. We perform extensive experiments on synthetic and real-world datasets demonstrating strong empirical performance of $\texttt{HyLinUCB}$.For number of arm specific parameters much larger than the number of shared parameters, we observe that $\texttt{DisLinUCB}$ incurs the lowest regret. In this case, regret of $\texttt{HyLinUCB}$ is the second best and extremely competitive to $\texttt{DisLinUCB}$. In all other situations, including our real-world dataset, $\texttt{HyLinUCB}$ has significantly lower regret than $\texttt{LinUCB}$, $\texttt{DisLinUCB}$ and other SOTA baselines we considered. We also empirically observe that the regret of $\texttt{HyLinUCB}$ grows much slower with the number of arms compared to baselines, making it suitable even for very large action spaces.
Overview
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The paper 'Linear Contextual Bandits with Hybrid Payoff: Revisited' explores the Linear Contextual Bandits problem with hybrid rewards, addressing the challenge of combining arm-specific and shared parameters for better decision-making models.
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It introduces a novel regret analysis for existing algorithms like LinUCB and DisLinUCB under hybrid settings, achieving improved regret bounds by leveraging feature diversity and the hybrid model's structure.
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The authors propose a new algorithm, HyLinUCB, which significantly outperforms existing methods in both theoretical and empirical analyses, showcasing its effectiveness through extensive experiments on both synthetic and real-world datasets.
An Analytical Study of Linear Contextual Bandits with Hybrid Payoff
The research paper titled "Linear Contextual Bandits with Hybrid Payoff: Revisited" by Nirjhar Das and Gaurav Sinha addresses significant gaps in the literature regarding the Linear Contextual Bandits (LinearCB) problem with hybrid rewards. This specific setting, where each arm's reward model comprises arm-specific parameters and parameters shared across all arms, presents unique challenges.
Key Contributions
Reduction to Shared and Disjoint Settings: The hybrid reward model combines elements of both shared and disjoint settings. The authors adeptly reduce the hybrid setting into two simplified forms: the fully shared model and the fully disjoint model, thereby enabling the use of well-established LinearCB algorithms such as LinUCB and DisLinUCB. This reduction also appropriately increases the feature space dimensionality, providing theoretical insights into how to optimize regret bounds for these conditions.
Improved Regret Analysis: The paper delivers a novel regret analysis for both LinUCB and DisLinUCB under the hybrid model when arm features satisfy a stochastic diversity condition. By leveraging the hybrid structure and feature diversity, the authors achieve regret bounds that outperform the known guarantees. Specifically, they demonstrate that LinUCB incurs a regret of $\tilde{O}(\sqrt{dKT})$ (improved from $\tilde{O}(d\sqrt{T})$), provided $T = \tilde{\Omega}(K3)$. Similarly, DisLinUCB achieves a regret of $\tilde{O}(\sqrt{(d1 + d2)KT})$ vs. the previously known $\tilde{O}((d1 + d2)\sqrt{KT})$.
Introduction of HyLinUCB: To further refine the exploration-exploitation trade-off in hybrid models, a new algorithm named HyLinUCB is introduced. This algorithm modifies LinUCB with a new exploration coefficient to account for sparsity in the hybrid setting. HyLinUCB achieves a regret bound of $\tilde{O}(\sqrt{K3 T} + \sqrt{dKT})$ under the same diversity assumptions. Extensive experiments show that HyLinUCB significantly outperforms LinUCB, DisLinUCB, and other state-of-the-art baselines, particularly when the number of shared parameters is substantial.
Theoretical Underpinnings
The paper's theoretical contributions lie in the detailed and rigorous analysis of regret bounds for the LinearCB algorithms in hybrid settings. The hybrid nature introduces sparsity that influences the effective dimensionality of feature vectors, which is meticulously leveraged in the algorithm design and analysis.
Assumptions and Bounds
Diversity Assumption (Assumption 1): The assumption stipulates that the feature vectors of pulled arms exhibit a diversity condition, ensuring that their covariance matrices have eigenvalues bounded away from zero. This assumption is critical for enabling the improved regret bounds.
Boundedness Assumption (Assumption 2): Both shared and arm-specific parameters, as well as feature vectors, are bounded. This is common in bandit literature to facilitate meaningful regret analysis.
Practical Implications and Experiments
Empirical results validate the theoretical findings. Experiments are carried out on synthetic data under varying settings of parameters $d1$, $d2$, and $K$, as well as on real-world datasets like the Yahoo! Front Page Dataset. These experiments illustrate that HyLinUCB consistently demonstrates low regret in numerous scenarios, evidencing its robustness and applicability in large action spaces.
Speculative Future Directions
The promising results and methodologies introduced in this paper open multiple avenues for future research. One notable direction would be to formalize tighter regret bounds for HyLinUCB, potentially reducing its dependence on the number of arms, $K$. Furthermore, exploring the application of these methodologies to non-linear contextual bandits and other more complex reward models could be valuable. Additionally, the framework established here could extend to online learning scenarios with non-stationary environments.
In conclusion, "Linear Contextual Bandits with Hybrid Payoff: Revisited" provides significant advancements in understanding and efficiently tackling the LinearCB problem in hybrid settings. The theoretical improvements and practical efficacy of the proposed methods like HyLinUCB promise substantial contributions to the field of sequential decision-making and reinforce the potential for further explorations and enhancements.
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