Learning Two-Layer Residual Networks with Nonparametric Function Estimation by Convex Programming

Zhunxuan Wang, Linyun He, Chunchuan Lyu, Shay B. Cohen

Abstract: We describe an algorithm that learns two-layer residual units with 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 such a residual unit, given by \[\mathbf{y}= \boldsymbol{B}^\ast\left[\left(\boldsymbol{A}^\ast\mathbf{x}\right)^+ + \mathbf{x}\right]\text{,}\] where ground-truth network parameters $\boldsymbol{A}^\ast \in \mathbb{R}^{d\times d}$ is a nonnegative full-rank matrix and $\boldsymbol{B}^\ast \in \mathbb{R}^{m\times d}$ is full-rank with $m \geq d$ and for $\mathbf{c} \in \mathbb{R}^d$, $[\mathbf{c}^{+}]_i = \max\{0, c_i\}$. 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 statistical strong consistency of our algorithm, and demonstrate the robustness and sample efficiency of our algorithm by experiments.