Risk-Sensitive Non-Parametric and Kernel Bandits

Devdan Dey, Sujoy Bhore, Avishek Ghosh

We study non-parametric bandits in which arms are evaluated using Cumulative Prospect Theory (CPT) rather than expected reward. We consider continuous action spaces with arm-dependent noise and propose Non-Parametric CPT Bandits ($\mathtt{NPCB}$), an optimistic algorithm that decouples mean estimation from CPT-based decision-making. We prove CPT regret of order $\widetilde{\mathcal O}\bigl(\sqrt{T\,\dim_E(\mathcal F,T^{-2})\,\log N(\mathcal F,T^{-2},\|\cdot\|_\infty)}\bigr)$, where $\dim_E$ is the Eluder dimension of the function class $\mathcal F$, $T$ is the time horizon, and $N$ is its covering number in the sup norm. This matches the order-wise regret of classical non-parametric bandits, showing that CPT risk sensitivity incurs no additional order-wise regret cost. We extend $\mathtt{NPCB}$ to unknown noise CDFs, retaining the known-CDF mean-learning term and adding a CDF-learning cost governed by a distributional Eluder dimension. We also study Kernel-CPT bandits and obtain information-gain-based regret bounds for a broad range of kernels, matching the corresponding risk-neutral kernel rates up to logarithmic factors.