通过算法重新审视不稳定公式定理
The unstable formula theorem revisited via algorithms
AI总结：
本文提出PEC统计学习模型，建立等价定理，并基于此给出Shelah不稳定公式定理的完整算法类比，用算法性质刻画Littlestone类。
摘要：
本文探讨了模型论中关于理论稳定性的一个基础性结果与学习中算法稳定性之间令人惊讶的相互作用。首先，针对现有学习模型中的不足，我们引入了一种新的统计学习模型，称为“概率最终正确”（Probably Eventually Correct，简称PEC）。我们利用该模型刻画了Littlestone（稳定）类。作为推论，Littlestone类在自然的统计意义下具有频繁的短定义。为了从频繁定义的角度获得Littlestone类的刻画，我们建立了一个等价定理，突出许多现有近似算法与新PEC模型之间的共同点。这一工作受到模型论中类型可定义性类比的启发，但具有其自身的特点。基于这些定理以及其他近期工作，我们给出了Shelah著名的不稳定公式定理的一个完整算法类比，其中算法性质取代了无限性的地位。
英文摘要：
This paper is about the surprising interaction of a foundational result from model theory, about stability of theories, with algorithmic stability in learning. First, in response to gaps in existing learning models, we introduce a new statistical learning model, called ``Probably Eventually Correct'' or PEC. We characterize Littlestone (stable) classes in terms of this model. As a corollary, Littlestone classes have frequent short definitions in a natural statistical sense. In order to obtain a characterization of Littlestone classes in terms of frequent definitions, we build an equivalence theorem highlighting what is common to many existing approximation algorithms, and to the new PEC. This is guided by an analogy to definability of types in model theory, but has its own character. Drawing on these theorems and on other recent work, we present a complete algorithmic analogue of Shelah's celebrated Unstable Formula Theorem, with algorithmic properties taking the place of the infinite.
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