发表机构
Johns Hopkins University（约翰斯·霍普金斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出狄利克雷逐次学习器算法，解决了多类别U校准中预测器的悔策率差距问题，得到的算法对有界和光滑真损失均达到最优悔策率。
AI 中文摘要
能否让一个预测器实现每一个有界真损失的最优悔策率，同时适配每一个光滑真损失？近期研究在维度差距问题上给出了答案，其自相容扰动给出的最坏情况悔策约为$K^{5/4}\sqrt{T}$，并对$\beta$-光滑损失额外产生$\beta\sqrt{K}\log K$项。我们用一行代码实现的预测器缩小了这两个差距：在观测到类别计数$c_{t-1}$后，从当前已观测类别面上的$\operatorname{Dir}(c_{t-1})$中抽取下一个预测，这是对结果的全新贝叶斯自助法。分析基于一个精确恒等式：在$\operatorname{Dir}(\alpha)$下对任意有界真损失取平均，等于其狄利克雷平均贝叶斯风险的离散导数。该恒等式使“成为被扰动学习器”项收缩为非正的Jensen间隙，而一个单计数似然比将稳定性限制在该类别计数的平方根倒数范围内。由此得到的单一无 horizon 算法满足：对所有$\ell$，$\mathbb{E}\operatorname{Reg}_{\ell}\leq 4\sqrt{S_T T}\leq 4\sqrt{K T}$；对所有$\beta$-光滑真损失，$\mathbb{E}\operatorname{Reg}_{\ell}\leq \frac{5}{2}\beta(1+\log T)$，其中$S_T$是已观测类别的数量。已知下界表明，这两个速率在其非平凡区域中是最优的，证明涵盖了不可微损失和活动单纯形面的变化。
英文摘要
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 $β\sqrt{K}\log K$ for $β$-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 $\operatorname{Dir}(α)$ 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 $\sup_{\ell}\mathbb{E}\operatorname{Reg}_{\ell}\leq 4\sqrt{S_T T}\leq 4\sqrt{K T}$ and $\mathbb{E}\operatorname{Reg}_{\ell}\leq \frac{5}{2}β(1+\log T)$ for every $β$-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.
