Linear Contextual Bandits with Hybrid Payoff: Revisited

TL;DR

This paper revisits the linear contextual bandit problem with hybrid rewards, introducing a new algorithm HyLinUCB that improves regret bounds and performs well empirically across various settings, especially with many arm-specific parameters.

Contribution

The paper provides new regret analyses for existing algorithms under hybrid rewards, and introduces HyLinUCB, a modified algorithm with improved theoretical guarantees and empirical performance.

Findings

HyLinUCB achieves $O( oot T)$ regret under certain conditions.

DisLinUCB performs best when many arm-specific parameters are present.

HyLinUCB's regret grows slower with the number of arms compared to baselines.

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.