Determine the minimax adversarial regret for bandit convex optimization with bounded convex losses

Determine the exact minimax adversarial regret rate for bandit convex optimization with bounded convex loss functions over a convex action set in R^d, resolving whether the optimal dependence on dimension is Theta(d^{1.5} sqrt(n)) or Theta(d sqrt(n)) and precisely characterizing the correct exponent of d in the leading term.

Background

The book surveys upper and lower bounds for adversarial bandit convex optimization with bounded convex losses. The best known upper bound is d^{2.5} sqrt(n), obtained by non-constructive information-theoretic methods, while the strongest lower bound is d sqrt(n), achieved by linear losses. This gap leaves the true minimax rate unresolved.

The author articulates a conjecture that the true minimax rate scales as d^{1.5} sqrt(n), motivated by positive results for more structured subclasses, but notes that the absence of a matching lower bound leaves open the possibility that d sqrt(n) is minimax.