Title: 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 $x\in \mathbb{R}^d$ is from Gaussian distribution and the
label $y = a^\top \sigma(Bx) + \xi$, where $a$ is a nonnegative vector in
$\mathbb{R}^m$ with $m\le d$, $B\in \mathbb{R}^{m\times d}$ 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 $G(\cdot)$
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.