Dirichlet Follow-the-Leader Closes the Gap in Simultaneous Multiclass U-Calibration

Can one forecaster attain the optimal regret rate for every bounded proper loss and also adapt to every smooth proper loss? Recent work answered this up to a dimension gap. Its self-concordant perturbation gives roughly K5/4√T worst-case regret and incurs an additional β√Klog K for β-smooth losses. We close both gaps with a one-line forecaster. After observing class counts ct-1, draw the next prediction from operatornameDir(ct-1), on the face of classes seen so far. This is a fresh Bayesian bootstrap of the outcomes. The analysis rests on an exact identity: averaging any bounded proper loss under operatornameDir(α) equals a discrete derivative of its Dirichlet-averaged Bayes risk. The identity makes the be-the-perturbed-leader term telescope to a nonpositive Jensen gap. A one-count likelihood ratio then bounds stability by the inverse square root of that class's count. The resulting single, horizon-free algorithm satisfies supℓmathbbEoperatornameRegℓ≤ 4√ST T≤ 4√K T and mathbbEoperatornameRegℓ≤ frac52β(1+log T) for every β-smooth proper loss. Here ST is the number of observed classes. Known lower bounds show that both rates are optimal in their nontrivial regimes. The proof covers nondifferentiable losses and changes of the active simplex face.