Bandit Convex Optimization: sqrt{T} Regret in One Dimension
This work addresses a fundamental theoretical problem in online optimization for researchers, though it is incremental as it focuses on the one-dimensional case and does not provide a constructive algorithm.
The paper tackled the adversarial bandit convex optimization problem in one dimension, proving that the minimax regret is $\widetilde{\Theta}(\sqrt{T})$, partially resolving a long-standing open problem.
We analyze the minimax regret of the adversarial bandit convex optimization problem. Focusing on the one-dimensional case, we prove that the minimax regret is $\widetildeΘ(\sqrt{T})$ and partially resolve a decade-old open problem. Our analysis is non-constructive, as we do not present a concrete algorithm that attains this regret rate. Instead, we use minimax duality to reduce the problem to a Bayesian setting, where the convex loss functions are drawn from a worst-case distribution, and then we solve the Bayesian version of the problem with a variant of Thompson Sampling. Our analysis features a novel use of convexity, formalized as a "local-to-global" property of convex functions, that may be of independent interest.