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

2026-02-06

AI Summary

This work investigates the efficient approximation of the median function over $d$-dimensional inputs using constant-depth, linear-width ReLU neural networks. The authors propose a novel network architecture based on a multi-stage iterative procedure that progressively eliminates non-central elements while preserving a candidate set containing the median. This approach achieves, for the first time, exponential-accuracy approximation of the median function with constant depth, thereby challenging prior lower-bound understandings regarding the depth required for approximating the max function. Moreover, the study establishes a general reduction from median to max approximation. On the unit hypercube, the resulting approximation error decays exponentially with respect to the network width, significantly outperforming existing results for max function approximation.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Mixture of Experts (MoE)
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Social mining and social search on the Web

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.

Problem

• median
• ReLU neural networks
• function approximation
• depth-width tradeoff
• approximation error

Innovation

• ReLU network
• median approximation
• constant-depth
• linear-width
• depth-width tradeoff