Nonparametric Learning of Two-Layer ReLU Residual Units

TL;DR

This paper introduces a nonparametric, convex optimization-based algorithm for learning two-layer residual neural networks with ReLU activations, demonstrating strong theoretical guarantees and empirical performance.

Contribution

It develops a layer-wise convex optimization framework for learning residual units, providing statistical consistency and efficient solution methods.

Findings

Algorithm is statistically consistent.

Solutions can be efficiently obtained via linear programming.

Effective on synthetic and benchmark datasets.

Abstract

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.