Probably Approximately Correct (i.e., PAC) learning is a core concept of sample complexity theory, and efficient PAC learnability is often seen as a natural counterpart to the class P in classical computational complexity. But while the nascent theory of parameterized complexity has allowed us to push beyond the P-NP ``dichotomy'' in classical computational complexity and identify the exact boundaries of tractability for numerous problems, there is no analogue in the domain of sample complexity that could push beyond efficient PAC learnability. As our core contribution, we fill this gap by developing a theory of parameterized PAC learning which allows us to shed new light on several recent PAC learning results that incorporated elements of parameterized complexity. Within the theory, we identify not one but two notions of fixed-parameter learnability that both form distinct counterparts to the class FPT -- the core concept at the center of the parameterized complexity paradigm -- and develop the machinery required to exclude fixed-parameter learnability. We then showcase the applications of this theory to identify refined boundaries of tractability for CNF and DNF learning as well as for a range of learning problems on graphs.

翻译：概率近似正确（即PAC）学习是样本复杂度理论的核心概念，而高效的PAC可学性常被视为经典计算复杂度中P类的自然对应。然而，当参数化复杂度新兴理论使我们能够突破经典计算复杂度中的P-NP“二分法”，并为众多问题精确刻画可处理性边界时，样本复杂度领域却缺乏超越高效PAC可学性的对应物。作为核心贡献，我们通过发展一套参数化PAC学习理论填补了这一空白，该理论为近期若干融入参数化复杂度元素的PAC学习结果提供了全新视角。在该理论中，我们识别出并非一个而是两个固定参数可学性概念，它们分别构成参数化复杂度范式核心概念FPT类的对应物，并开发了排除固定参数可学性所需的工具。随后，我们展示了该理论在刻画CNF与DNF学习以及一系列图学习问题的精细可处理性边界中的应用。