Learning One-hidden-layer Neural Networks with Landscape Design
Abstract¶
We consider the problem of learning a one-hidden-layer neural network: we assume the input xin R^(d) is from Gaussian distribution and the label y = a^(top) sigma(Bx) + xi, where a is a nonnegative vector in R^(m) with m <= d, Bin R^(mx) d is a full-rank weight matrix, and xi 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 result and validate the results by simulations.