Mistake-Bounded Language Generation

TL;DR

This paper introduces a new mistake-bounded framework for language generation, focusing on minimizing total errors during learning rather than just eventual correctness, with algorithms and bounds for finite and infinite classes.

Contribution

It develops a formal reduction to a known learning framework, providing mistake bounds and trade-offs for finite and infinite language classes, including extensions to noisy adversaries.

Findings

Algorithm achieves optimal last-mistake time for finite classes.

Mistake bound of logarithmic order for finite classes.

Trade-off between mistake bounds and convergence in infinite classes.

Abstract

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.