Abstract
We study the stochastic linear bandits with parameter noise model, in which the reward of action a is a θ where θ is sampled i.i.d. We show a regret upper bound of O (√d T log (K/δ) σ²_max) for a horizon T, general action set of size K of dimension d, and where σ²_max is the maximal variance of the reward for any action. We further provide a lower bound of Ω (d √T σ²_max) which is tight (up to logarithmic factors) whenever log (K) ≈ d. For more specific action sets, ℓ_p unit balls with p ≤ 2 and dual norm q, we show that the minimax regret is Θ (√dT σ²_q), where σ²_q is a variance-dependent quantity that is always at most 4. This is in contrast to the minimax regret attainable for such sets in the classic additive noise model, where the regret is of order d √T. Surprisingly, we show that this optimal (up to logarithmic factors) regret bound is attainable using a very simple explore-exploit algorithm.