It's Not What Machines Can Learn, It's What We Cannot Teach

Can deep neural networks learn to solve any task, and in particular problems of high complexity? This question attracts a lot of interest, with recent works tackling computationally hard tasks such as the traveling salesman problem and satisfiability. In this work we offer a different perspective on this question. Given the common assumption that $\textit{NP} \neq \textit{coNP}$ we prove that any polynomial-time sample generator for an $\textit{NP}$-hard problem samples, in fact, from an easier sub-problem. We empirically explore a case study, Conjunctive Query Containment, and show how common data generation techniques generate biased datasets that lead practitioners to over-estimate model accuracy. Our results suggest that machine learning approaches that require training on a dense uniform sampling from the target distribution cannot be used to solve computationally hard problems, the reason being the difficulty of generating sufficiently large and unbiased training sets.