Learning one-hidden-layer neural networks with landscape design

We consider the problem of learning a one-hidden-layer neural network: we assume the input x ∈ R^d is from Gaussian distribution and the label y = a^>σ(Bx) + ξ, where a is a nonnegative vector in R^m with m ≤ d, B ∈ R^m×d is a full-rank weight matrix, and ξ is a noise vector. We first give an analytic formula for the population risk of the standard squared loss and demonstrate that it implicitly attempts to decompose a sequence of low-rank tensors simultaneously. Inspired by the formula, we design a non-convex objective function G(•) whose landscape is guaranteed to have the following properties: 1. All local minima of G are also global minima. 2. All global minima of G correspond to the ground truth parameters. 3. The value and gradient of G can be estimated using samples. With these properties, stochastic gradient descent on G provably converges to the global minimum and learn the ground-truth parameters. We also prove finite sample complexity results and validate the results by simulations.