A Refined Analysis of UCBVI

In this work, we provide a refined analysis of the UCBVI algorithm (Azar et al., 2017), improving both the bonus terms and the regret analysis. Additionally, we compare our version of UCBVI with both its original version and the state-of-the-art MVP algorithm. Our empirical validation demonstrates that improving the multiplicative constants in the bounds has significant positive effects on the empirical performance of the algorithms.

Open Problem: Tight Bounds for Kernelized Multi-Armed Bandits with Bernoulli Rewards

We consider Kernelized Bandits (KBs) to optimize a function $f : \mathcal{X} \rightarrow [0,1]$ belonging to the Reproducing Kernel Hilbert Space (RKHS) $\mathcal{H}_k$. Mainstream works on kernelized bandits focus on a subgaussian noise model in which observations of the form $f(\mathbf{x}_t)+\epsilon_t$, being $\epsilon_t$ a subgaussian noise, are available (Chowdhury and Gopalan, 2017). Differently, we focus on the case in which we observe realizations $y_t \sim \text{Ber}(f(\mathbf{x}_t))$ sampled from a Bernoulli distribution with parameter $f(\mathbf{x}_t)$. While the Bernoulli model has been investigated successfully in multi-armed bandits (Garivier and Capp\'e, 2011), logistic bandits (Faury et al., 2022), bandits in metric spaces (Magureanu et al., 2014), it remains an open question whether tight results can be obtained for KBs. This paper aims to draw the attention of the online learning community to this open problem.