We investigate the learning task of language generation in the limit, but shift focus from the traditional time-of-last-mistake metric of a generator's success to a new notion of "mistake-bounded generation." While existing results for language generation in the limit focus on guaranteeing eventual consistency, they are blind to the cumulative error incurred during the learning process. We address this by shifting the goal to minimizing the total number of invalid elements output by a generation algorithm. We establish a formal reduction to the Learning from Correct Demonstrations framework of Joshi et al. (2025), enabling a general recipe for deriving mistake bounds via weighted update rules. For finite classes, we provide an algorithm that simultaneously achieves an optimal last-mistake time of $\mathsf{Cdim}(L)$ and a mistake bound of $\lfloor \log_2 |L| \rfloor$, whereas for the non-uniform setting of countably infinite streams of languages, we prove a fundamental trade-off: achieving logarithmic mistakes $O(\log i)$ necessarily precludes convergence guarantees established in prior work. Finally, we show that our framework can be extended to accommodate noisy adversaries and guarantee mistake bounds that scale with the adversary's suboptimality.

翻译：我们研究极限学习任务中的语言生成，但将焦点从传统上衡量生成器成功的“最后错误时间”指标，转向一种新的“有界错误生成”概念。尽管现有关于极限语言生成的结果侧重于保证最终一致性，但它们忽略了学习过程中累积的误差。我们通过将目标转变为最小化生成算法输出的无效元素总数来解决这一问题。我们建立了一个形式化归约，将其转化为Joshi等人（2025）的“从正确演示中学习”框架，从而通过加权更新规则导出通用错误界方法。对于有限类别，我们提出了一种算法，该算法同时实现了最优的最后错误时间$\mathsf{Cdim}(L)$和错误界$\lfloor \log_2 |L| \rfloor$；而对于语言流可数无穷的非均匀设置，我们证明了一个基本权衡：实现对数级错误$O(\log i)$必然会排除先前工作中建立的收敛保证。最后，我们展示该框架可扩展至对抗噪声对手，并保证错误界随对手次优性缩放。