The Median is Easier than it Looks: Approximation with a Constant-Depth, Linear-Width ReLU Network

TL;DR

This paper demonstrates that ReLU neural networks with constant depth and linear width can approximate the median of high-dimensional inputs with exponentially small error, surpassing previous limitations suggested by maximum function approximations.

Contribution

It introduces a novel constant-depth, linear-width neural network construction for median approximation, breaking prior depth requirements and establishing a reduction from maximum to median.

Findings

Achieves exponentially small approximation error with constant depth and linear width.

Provides a reduction from maximum to median approximation, enabling stronger results.

Surpasses previous depth limitations for median approximation in neural networks.

Abstract

We study the approximation of the median of $d$ inputs using ReLU neural networks. We present depth-width tradeoffs under several settings, culminating in a constant-depth, linear-width construction that achieves exponentially small approximation error with respect to the uniform distribution over the unit hypercube. By further establishing a general reduction from the maximum to the median, our results break a barrier suggested by prior work on the maximum function, which indicated that linear width should require depth growing at least as $\log\log d$ to achieve comparable accuracy. Our construction relies on a multi-stage procedure that iteratively eliminates non-central elements while preserving a candidate set around the median. We overcome obstacles that do not arise for the maximum to yield approximation results that are strictly stronger than those previously known for the maximum itself.