The Price of Hidden Curvature: An $\widetildeΩ (d^{5/4} \sqrt{T})$ Lower Bound for Bandit Convex Optimization

We establish a $\widetildeΩ(d^{5/4}\sqrt T)$ lower bound on the minimax expected regret of stochastic bandit convex optimization of $1$-Lipschitz functions on the Euclidean ball. This presents the first nontrivial regret lower bound that grows faster than $d\sqrt{T}$ for this problem, establishing that stochastic bandit convex optimization is fundamentally harder than linear bandits.

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• reinforcement-learning