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Stochastic Contextual Bandits with Known Reward Functions

Many sequential decision-making problems in communication networks can be modeled as contextual bandit problems, which are natural extensions of the well-known multi-armed bandit problem. In contextual bandit problems, at each time, an agent observes some side information or context, pulls one arm and receives the reward for that arm. We consider a stochastic formulation where the context-reward tuples are independently drawn from an unknown distribution in each trial. Motivated by networking applications, we analyze a setting where the reward is a known non-linear function of the context and the chosen arm's current state. We first consider the case of discrete and finite context-spaces and propose DCB($ε$), an algorithm that we prove, through a careful analysis, yields regret (cumulative reward gap compared to a distribution-aware genie) scaling logarithmically in time and linearly in the number of arms that are not optimal for any context, improving over existing algorithms where the regret scales linearly in the total number of arms. We then study continuous context-spaces with Lipschitz reward functions and propose CCB($ε, δ$), an algorithm that uses DCB($ε$) as a subroutine. CCB($ε, δ$) reveals a novel regret-storage trade-off that is parametrized by $δ$. Tuning $δ$ to the time horizon allows us to obtain sub-linear regret bounds, while requiring sub-linear storage. By exploiting joint learning for all contexts we get regret bounds for CCB($ε, δ$) that are unachievable by any existing contextual bandit algorithm for continuous context-spaces. We also show similar performance bounds for the unknown horizon case.