No-Regret Linear Bandits beyond Realizability

We study linear bandits when the underlying reward function is not linear. Existing work relies on a uniform misspecification parameter ε that measures the sup-norm error of the best linear approximation. This results in an unavoidable linear regret whenever ε > 0. We describe a more natural model of misspecification which only requires the approximation error at each input x to be proportional to the suboptimality gap at x. It captures the intuition that, for optimization problems, near-optimal regions should matter more and we can tolerate larger approximation errors in suboptimal regions. Quite surprisingly, we show that the classical LinUCB algorithm -- designed for the realizable case -- is automatically robust against such gap-adjusted misspecification. It achieves a near-optimal √T regret for problems that the best-known regret is almost linear in time horizon T. Technically, our proof relies on a novel self-bounding argument that bounds the part of the regret due to misspecification by the regret itself.