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

arXiv:2608.06656v1 Announce Type: new Abstract: 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 K^{5/4}sqrt{T} worst-case regret and incurs an additional etasqrt{K}log K for eta-smooth losses. We close both gaps with a one-line forecaster. After observing class counts c_{t-1}, draw the next prediction from operatorname{Dir}(c_{t-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 operatorname{Dir}(alpha) 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 sup_{ell}Eoperatorname{Reg}{ell}leq 4sqrt{S_T T}leq 4sqrt{K T} and Eoperatorname{Reg}{ell}leq frac{5}{2}eta(1+log T) for every eta-smooth proper loss. Here S_T 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.

Source: arXiv cs.LG | 2026-08-10