Non-stationary Contextual Bandits and Universal Learning

We study the fundamental limits of learning in contextual bandits, where a learner's rewards depend on their actions and a known context, which extends the canonical multi-armed bandit to the case where side-information is available. We are interested in universally consistent algorithms, which achieve sublinear regret compared to any measurable fixed policy, without any function class restriction. For stationary contextual bandits, when the underlying reward mechanism is time-invariant, [Blanchard et al.] characterized learnable context processes for which universal consistency is achievable; and further gave algorithms ensuring universal consistency whenever this is achievable, a property known as optimistic universal consistency. It is well understood, however, that reward mechanisms can evolve over time, possibly depending on the learner's actions. We show that optimistic universal learning for non-stationary contextual bandits is impossible in general, contrary to all previously studied settings in online learning -- including standard supervised learning. We also give necessary and sufficient conditions for universal learning under various non-stationarity models, including online and adversarial reward mechanisms. In particular, the set of learnable processes for non-stationary rewards is still extremely general -- larger than i.i.d., stationary or ergodic -- but in general strictly smaller than that for supervised learning or stationary contextual bandits, shedding light on new non-stationary phenomena.