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

Can a single proper multiclass forecaster attain the minimax-optimal worst-case pseudo-U-calibration rate over all bounded proper losses? Can it do so while retaining the optimal logarithmic dependence on $T$ for every bounded Euclidean-smooth proper loss? Recent work gave the first simultaneous guarantee, but its bounds are roughly $$\begin{gathered} K^{5/4} \\ sqrtT \end{gathered}$$ in the worst case and include an additional $$\begin{gathered} beta \\ sqrtK \\ logK \end{gathered}$$ term for smooth losses. We show that both dimension losses are unnecessary. A one-line, horizon-free Dirichlet forecaster achieves the optimal rates. After observing class counts $c_{t-1}$, draw the next prediction from $operatornameDir(c_{t-1})$, 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 $$\begin{gathered} operatornameDir( \\ alpha) \end{gathered}$$ 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 $$\begin{gathered} su{p}_{ \\ ell} \\ operatorname{Reg}_{ \\ ell} \\ leq4 \\ sqrt{S}_{T}T \\ leq4 \\ sqrt \\ min \\ K,T \\ T. \end{gathered}$$ For every bounded $beta$-smooth proper loss $ell$, $$\begin{gathered} operatorname{Reg}_{ \\ ell} \\ leq \\ frac52 \\ beta(1+ \\ logT). \end{gathered}$$ Here $S_T$ is the number of observed classes. A class-aggregation reduction shows that the dimension-capped rate $$\begin{gathered} sqrt \\ min \\ K,T \\ T \end{gathered}$$ is minimax-optimal up to constants in every regime with $$\begin{gathered} T \\ geq24 \end{gathered}$$, and known smooth-loss lower bounds match the logarithmic dependence on $T$ of the second bound. The proof covers nondifferentiable losses and changes of the active simplex face.