We study batch learning with log-loss in the individual setting, where the outcome sequence is deterministic. Because empirical statistics are not directly applicable in this regime, obtaining regret guarantees for batch learning has long posed a fundamental challenge. We propose a natural criterion based on leave-one-out regret and analyze its minimax value for several hypothesis classes. For the multinomial simplex over $m$ symbols, we show that the minimax regret is $\frac{m-1}{N} + o\!\left(\frac{1}{N}\right)$, and compare it to the stochastic realizable case where it is $\frac{m-1}{2N} + o\!\left(\frac{1}{N}\right)$. More generally, we prove that every hypothesis class of VC dimension $d$ is learnable in the individual batch-learning problem, with regret at most $\frac{d\log(N)}{N} + o\!\left(\frac{\log(N)}{N}\right)$, and we establish matching lower bounds for certain classes. We further derive additional upper bounds that depend on structural properties of the hypothesis class. These results establish, for the first time, that universal batch learning with log-loss is possible in the individual setting.

翻译：本文研究个体设定下使用对数损失的批量学习问题，其中输出序列是确定性的。由于经验统计量在此机制下无法直接适用，获取批量学习的遗憾保证长期以来构成一项基础性挑战。我们提出一种基于留一法遗憾的自然准则，并针对若干假设类分析其极小极大值。对于包含$m$个符号的多项式单纯形，我们证明极小极大遗憾为$\frac{m-1}{N} + o\!\left(\frac{1}{N}\right)$，并与随机可实现情形下的$\frac{m-1}{2N} + o\!\left(\frac{1}{N}\right)$进行比较。更一般地，我们证明每个VC维为$d$的假设类在个体批量学习问题中是可学习的，其遗憾至多为$\frac{d\log(N)}{N} + o\!\left(\frac{\log(N)}{N}\right)$，并针对特定类建立了匹配下界。我们进一步推导了依赖于假设类结构特性的附加上界。这些结果首次证明，在个体设定下实现具有对数损失的通用批量学习是可行的。