From Local Pseudorandom Generators to Hardness of Learning
This work addresses foundational challenges in computational learning theory by establishing hardness results for key learning problems, which is incremental as it builds on existing assumptions to extend the scope of proven hardness.
The paper tackles the problem of proving hardness-of-learning for various basic problems, such as learning shallow ReLU neural networks and intersections of halfspaces, by assuming the existence of local pseudorandom generators, surpassing current state-of-the-art results with no prior hardness results for these tasks.
We prove hardness-of-learning results under a well-studied assumption on the existence of local pseudorandom generators. As we show, this assumption allows us to surpass the current state of the art, and prove hardness of various basic problems, with no hardness results to date. Our results include: hardness of learning shallow ReLU neural networks under the Gaussian distribution and other distributions; hardness of learning intersections of $ω(1)$ halfspaces, DNF formulas with $ω(1)$ terms, and ReLU networks with $ω(1)$ hidden neurons; hardness of weakly learning deterministic finite automata under the uniform distribution; hardness of weakly learning depth-$3$ Boolean circuits under the uniform distribution, as well as distribution-specific hardness results for learning DNF formulas and intersections of halfspaces. We also establish lower bounds on the complexity of learning intersections of a constant number of halfspaces, and ReLU networks with a constant number of hidden neurons. Moreover, our results imply the hardness of virtually all improper PAC-learning problems (both distribution-free and distribution-specific) that were previously shown hard under other assumptions.