  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 $\omega(1)$ halfspaces, DNF formulas with $\omega(1)$ terms,
and ReLU networks with $\omega(1)$ 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.
