A Direct Approach for Handling Contextual Bandits with Latent State Dynamics

We revisit the finite-armed linear bandit model by Nelson et al. (2022), where contexts and rewards are governed by a finite hidden Markov chain. Nelson et al. (2022) approach this model by a reduction to linear contextual bandits; but to do so, they actually introduce a simplification in which rewards are linear functions of the posterior probabilities over the hidden states given the observed contexts, rather than functions of the hidden states themselves. Their analysis (but not their algorithm) also does not take into account the estimation of the HMM parameters, and only tackles expected, not high-probability, bounds, which suffer in addition from unnecessary complex dependencies on the model (like reward gaps). We instead study the more natural model incorporating direct dependencies in the hidden states (on top of dependencies on the observed contexts, as is natural for contextual bandits) and also obtain stronger, high-probability, regret bounds for a fully adaptive strategy that estimates HMM parameters online. These bounds do not depend on the reward functions and only depend on the model through the estimation of the HMM parameters.