Identify the correct logarithmic factor in the expected cumulative regret for 1D stochastic convex bandits using bisection

Ascertain the precise logarithmic dependence in the expected cumulative regret for the one-dimensional stochastic convex bandit setting when using the bisection-based algorithm, determining whether the optimal expectation scales with log log(n), log(n), or another logarithmic factor.

Background

The chapter presents a bisection-based algorithm for one-dimensional stochastic convex bandits and compares cumulative and simple regret. Prior work (Cheshire et al.) established optimal simple regret scaling with sqrt{(log log n)/n}, while the derived bound here for simple regret via cumulative regret is O(log(n)/sqrt(n)).

The author notes that, unlike simple regret where the optimal logarithmic term is known, the exact logarithmic dependence for expected cumulative regret has not been pinned down.