The authors derive a variance-aware regret bound of order d H^2 bar sigma_T sqrt(T) for multinomial logistic MDPs together with a matching lower bound, establishing minimax optimality.
Consequently, by substituting into Equation (12), we can upper bound the problem-dependent constant σT by: σT ≤ vuut(T H)−1 TX t=1 HX h=1 2 H = √ 2H −1/2
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Minimax Optimal Variance-Aware Regret Bounds for Multinomial Logistic MDPs
The authors derive a variance-aware regret bound of order d H^2 bar sigma_T sqrt(T) for multinomial logistic MDPs together with a matching lower bound, establishing minimax optimality.