Abstract: Which concepts can we learn efficiently on average? In this paper, we investigate the capability of a natural average-case learning framework, heuristic PAC (heurPAC) learning to answer this and some other related questions. Roughly speaking, we say that a concept class is heurPAC learnable if there exists a learning algorithm that given $n, s\in\mathbb{N}$ and $\epsilon,\delta,\eta\in(0,1]$ as input, learns all but $\eta$ fraction of $n$-input target functions represented as $s$-bit strings in the class from passively collected examples and then outputs an $\epsilon$-close hypothesis with failure probability at most $\delta$ in polynomial-time in $n$, $s$, $\epsilon^{-1},\delta^{-1}$, and $\eta^{-1}$, where each example is generated according to some example distribution. First, we establish a positive learnability result. Specifically, we show that a simple Fourier-based algorithm heurPAC learns $\Omega(\log n)$-junta functions on the uniform distribution, which is a central open question in the original PAC learning model. Our technical contribution is to introduce the notion of elusive functions that captures hard-to-learn cases and to establish a polynomial relation between the running time and the fraction of such elusive functions. Second, we present clear relations between heurPAC learnability and cryptography. Particularly, we show that for any efficiently evaluated class $\mathscr{C}$, (1) if $\mathscr{C}$ is not heurPAC learnable, then an auxiliary-input one-way function (AIOWF) exists; (2) if $\mathscr{C}$ is not heurPAC learnable on the uniform distribution, then an infinitely-often one-way function (io-OWF) exists. As a corollary, we also present new characterizations for AIOWF and io-OWF based on heurPAC learnability, which is conceptually stronger than the previous ones that are based on average-case learnability for fixed parameters. These results show that our framework might yield heuristic learners with theoretical guarantees for broader classes than the usual PAC learning framework, and any efficiently evaluated class has a potential for such a heuristic learner or a secure cryptographic primitive. Through this paper, we suggest further research toward the win-win ``learning vs. cryptography'' paradigm.

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