Mistake-Bounded Language Generation

2026-05-11

AI Summary

This work addresses a critical limitation in traditional language generation learning, which emphasizes final consistency while neglecting the accumulation of invalid outputs during the learning process. We propose a novel paradigm termed “error-bounded generation,” aiming to minimize the cumulative count of invalid elements. For the first time, we introduce a cumulative error metric and formally connect it to the Correct Demonstrations learning framework. Through weighted update rules, combinatorial dimension (Cdim) analysis, and logarithmic error bound derivations, we establish that for any finite language class, both the optimal last-error time Cdim(ℒ) and a ⌊log₂|ℒ|⌋ cumulative error bound are simultaneously achievable. For countably infinite language streams, we reveal a fundamental trade-off between an O(log i) error bound and traditional convergence guarantees, while still ensuring near-optimal error bounds under adversarial noise.

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.

Problem

• mistake-bounded generation
• language generation
• cumulative error
• learning in the limit
• invalid outputs

Innovation

• mistake-bounded learning
• language generation
• learning from correct demonstrations
• cumulative error
• weighted update rules