PAC Learning

We study computable probably approximately correct (CPAC) learning, where learners are required to be computable functions. It had been previously observed that the Fundamental Theorem of Statistical Learning, which characterizes PAC learnability by finiteness of the Vapnik-Chervonenkis (VC-)dimension, no longer holds in this framework. Recent works recovered analogs of the Fundamental Theorem in the computable setting, for instance by introducing an effective VC-dimension. Guided by this, we investigate the connection between CPAC learning and recursively enumerable representable (RER) classes, whose members can be algorithmically listed. Our results show that the effective VC-dimensions can take arbitrary values above the traditional one, even for RER classes, which creates a whole family of (non-)examples for various notions of CPAC learning. Yet the two dimensions coincide for classes satisfying sufficiently strong notions of CPAC learning. We then observe that CPAC learnability can also be characterized via containment of RER classes that realize the same samples. Furthermore, it is shown that CPAC learnable classes satisfying a unique identification property are necessarily RER. Finally, we establish that agnostic learnability can be guaranteed for RER classes, by considering the relaxed notion of nonuniform CPAC learning.

In the classic point location problem, one is given an arbitrary dataset $X \subset \mathbb{R}^d$ of $n$ points with query access to an unknown halfspace $f : \mathbb{R}^d \to \{0,1\}$, and the goal is to learn the label of every point in $X$. This problem is extremely well-studied and a nearly-optimal $\widetilde{O}(d \log n)$ query algorithm is known due to Hopkins-Kane-Lovett-Mahajan (FOCS 2020). However, their algorithm is granted the power to query arbitrary points outside of $X$ (point synthesis), and in fact without this power there is an $Ω(n)$ query lower bound due to Dasgupta (NeurIPS 2004). In this work our goal is to design efficient algorithms for learning halfspaces without point synthesis. To circumvent the $Ω(n)$ lower bound, we consider learning halfspaces whose normal vectors come from a set of size $D$, and show tight bounds of $Θ(D + \log n)$. As a corollary, we obtain an optimal $O(d + \log n)$ query deterministic learner for axis-aligned halfspaces, closing a previous gap of $O(d \log n)$ vs. $Ω(d + \log n)$. In fact, our algorithm solves the more general problem of learning a Boolean function $f$ over $n$ elements which is monotone under at least one of $D$ provided orderings. Our technical insight is to exploit the structure in these orderings to perform a binary search in parallel rather than considering each ordering sequentially, and we believe our approach may be of broader interest. Furthermore, we use our exact learning algorithm to obtain nearly optimal algorithms for PAC-learning. We show that $O(\min(D + \log(1/\varepsilon), 1/\varepsilon) \cdot \log D)$ queries suffice to learn $f$ within error $\varepsilon$, even in a setting when $f$ can be adversarially corrupted on a $c\varepsilon$-fraction of points, for a sufficiently small constant $c$. This bound is optimal up to a $\log D$ factor, including in the realizable setting.