High-dimensional Nonparametric Contextual Bandit Problem
This addresses a bottleneck in decision-making systems like online advertising, offering a theoretical advance for high-dimensional nonparametric models, though it is incremental relative to prior kernelized bandit work.
The paper tackles the kernelized contextual bandit problem in high-dimensional settings, where existing methods fail with trivial bounds, and shows that no-regret learning is achievable even when dimensions grow with samples, deriving specific regret rates.
We consider the kernelized contextual bandit problem with a large feature space. This problem involves $K$ arms, and the goal of the forecaster is to maximize the cumulative rewards through learning the relationship between the contexts and the rewards. It serves as a general framework for various decision-making scenarios, such as personalized online advertising and recommendation systems. Kernelized contextual bandits generalize the linear contextual bandit problem and offers a greater modeling flexibility. Existing methods, when applied to Gaussian kernels, yield a trivial bound of $O(T)$ when we consider $Ω(\log T)$ feature dimensions. To address this, we introduce stochastic assumptions on the context distribution and show that no-regret learning is achievable even when the number of dimensions grows up to the number of samples. Furthermore, we analyze lenient regret, which allows a per-round regret of at most $Δ> 0$. We derive the rate of lenient regret in terms of $Δ$.