The paper studies single-index bandits, where rewards depend on a one-dimensional projection of high-dimensional context vectors through an unknown (possibly non-monotone) reward function. This generalizes linear bandits to settings where you don't know the reward function's form. Previous work achieved Õ(T^(3/4)) regret for non-monotone functions; this paper closes the gap by proving Õ(T^(2/3)) regret is achievable and optimal. Their algorithm first estimates the projection direction using a normalized Stein estimator, then reduces to a one-dimensional bandit problem via discretization and UCB. They also prove a matching lower bound, showing T^(2/3) is the right rate.
Main takeaways:
- Single-index bandits: rewards depend on an unknown one-dimensional projection of high-dimensional contexts via an unknown function
- Prior best regret for non-monotone functions was Õ(T^(3/4)); this paper achieves Õ(T^(2/3))
- Two-phase algorithm: estimate the projection direction, then discretize and use UCB in one dimension
- Prove a matching Ω̃(T^(2/3)) lower bound, establishing optimality
- No additional assumptions (like monotonicity) needed; empirical results confirm effectiveness