  We describe an algorithm that learns two-layer residual units using rectified
linear unit (ReLU) activation: suppose the input $\mathbf{x}$ is from a
distribution with support space $\mathbb{R}^d$ and the ground-truth generative
model is a residual unit of this type, given by $\mathbf{y} =
\boldsymbol{B}^\ast\left[\left(\boldsymbol{A}^\ast\mathbf{x}\right)^+ +
\mathbf{x}\right]$, where ground-truth network parameters $\boldsymbol{A}^\ast
\in \mathbb{R}^{d\times d}$ represent a full-rank matrix with nonnegative
entries and $\boldsymbol{B}^\ast \in \mathbb{R}^{m\times d}$ is full-rank with
$m \geq d$ and for $\boldsymbol{c} \in \mathbb{R}^d$, $[\boldsymbol{c}^{+}]_i =
\max\{0, c_i\}$. We design layer-wise objectives as functionals whose analytic
minimizers express the exact ground-truth network in terms of its parameters
and nonlinearities. Following this objective landscape, learning residual units
from finite samples can be formulated using convex optimization of a
nonparametric function: for each layer, we first formulate the corresponding
empirical risk minimization (ERM) as a positive semi-definite quadratic program
(QP), then we show the solution space of the QP can be equivalently determined
by a set of linear inequalities, which can then be efficiently solved by linear
programming (LP). We further prove the strong statistical consistency of our
algorithm, and demonstrate its robustness and sample efficiency through
experimental results on synthetic data and a set of benchmark regression
datasets.
