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# Extending Learnability to Auxiliary-Input Cryptographic Primitives and Meta-PAC Learning

*Proceedings of Thirty Third Conference on Learning Theory*, PMLR 125:2998-3029, 2020.

#### Abstract

We investigate the meaning of efficient learnability from several different perspectives. The purpose is to give new insights into central problems in computational learning theory (CoLT). Specifically, we discuss the following two questions related to efficient PAC learnability. First, we investigate the gap between PAC learnability for polynomial-size circuits and weak cryptographic primitives taking auxiliary-input. Applebaum et al. observed that such a weak primitive is enough to show the hardness of PAC learning. However, the opposite direction is still unknown. In this paper, we introduce the following two notions: (1) a variant model of PAC learning whose hardness corresponds to auxiliary-input one-way functions; (2) a variant of a hitting set generator corresponding to the hardness of PAC learning. The equivalence gives a clearer insight into the gap between the hardness of learning and weak cryptographic primitives. Second, we discuss why proving efficient learnability is difficult. This question is natural because few classes are known to be polynomially learnable at present. In this paper, we formulate a task of determining efficient learnability as a meta-PAC learning problem and show that our meta-PAC learning is exactly as hard as PAC learning. Our result insists on one possibility: a hard-to-learn instance itself yields the hardness of proving efficient learnability. Our technical contribution is to give (1) a general framework for translating the hardness of PAC learning into auxiliary-input primitives, and (2) a new formulation to discuss the hardness of determining efficient learnability. Our work yields new important frontiers related to CoLT, including investigation of the learning hierarchy.