Cryptographic hardness for learning intersections of halfspaces

We give the first representation-independent hardness results for PAC learning intersections of halfspaces, a central concept class in computational learning theory. Our hardness results are derived from two public-key cryptosystems due to Regev, which are based on the worst-case hardness of well-studied lattice problems. Specifically, we prove that a polynomial-time algorithm for PAC learning intersections of nϵ halfspaces (for a constant ϵ>0) in n dimensions would yield a polynomial-time solution to -uSVP (unique shortest vector problem). We also prove that PAC learning intersections of nϵ low-weight halfspaces would yield a polynomial-time quantum solution to -SVP and -SIVP (shortest vector problem and shortest independent vector problem, respectively). Our approach also yields the first representation-independent hardness results for learning polynomial-size depth-2 neural networks and polynomial-size depth-3 arithmetic circuits.