Title: Eigenvalue Decay Implies Polynomial-Time Learnability for Neural Networks,
Abstract: We consider the problem of learning function classes computed by neural
networks with various activations (e.g. ReLU or Sigmoid), a task believed to be
computationally intractable in the worst-case. A major open problem is to
understand the minimal assumptions under which these classes admit provably
efficient algorithms. In this work we show that a natural distributional
assumption corresponding to {\em eigenvalue decay} of the Gram matrix yields
polynomial-time algorithms in the non-realizable setting for expressive classes
of networks (e.g. feed-forward networks of ReLUs). We make no assumptions on
the structure of the network or the labels. Given sufficiently-strong
polynomial eigenvalue decay, we obtain {\em fully}-polynomial time algorithms
in {\em all} the relevant parameters with respect to square-loss. Milder decay
assumptions also lead to improved algorithms. This is the first purely
distributional assumption that leads to polynomial-time algorithms for networks
of ReLUs, even with one hidden layer. Further, unlike prior distributional
assumptions (e.g., the marginal distribution is Gaussian), eigenvalue decay has
been observed in practice on common data sets.