An Asymptotically Optimal Primal-Dual Incremental Algorithm for Contextual Linear Bandits
This work addresses the problem of inefficient exploration in contextual linear bandits for reinforcement learning and decision-making systems, offering a novel and scalable solution with proven optimality.
The paper tackles the asymptotic suboptimality of optimistic algorithms in contextual linear bandits by introducing a primal-dual incremental algorithm that is asymptotically optimal, robust to unbalanced context distributions, and scalable, with regret bounds scaling logarithmically with the number of arms and demonstrating better empirical performance than state-of-the-art baselines.
In the contextual linear bandit setting, algorithms built on the optimism principle fail to exploit the structure of the problem and have been shown to be asymptotically suboptimal. In this paper, we follow recent approaches of deriving asymptotically optimal algorithms from problem-dependent regret lower bounds and we introduce a novel algorithm improving over the state-of-the-art along multiple dimensions. We build on a reformulation of the lower bound, where context distribution and exploration policy are decoupled, and we obtain an algorithm robust to unbalanced context distributions. Then, using an incremental primal-dual approach to solve the Lagrangian relaxation of the lower bound, we obtain a scalable and computationally efficient algorithm. Finally, we remove forced exploration and build on confidence intervals of the optimization problem to encourage a minimum level of exploration that is better adapted to the problem structure. We demonstrate the asymptotic optimality of our algorithm, while providing both problem-dependent and worst-case finite-time regret guarantees. Our bounds scale with the logarithm of the number of arms, thus avoiding the linear dependence common in all related prior works. Notably, we establish minimax optimality for any learning horizon in the special case of non-contextual linear bandits. Finally, we verify that our algorithm obtains better empirical performance than state-of-the-art baselines.