# Deep Neural Network Approximation Theory

## TL;DR

This paper establishes the fundamental theoretical limits of deep neural networks in function approximation, demonstrating their optimality and exponential accuracy across various function classes, including fractal and oscillatory functions.

## Contribution

It develops a theory linking the complexity of functions to the network architecture, proving deep networks' optimality and exponential approximation capabilities for diverse function classes.

## Key findings

- Deep networks are Kolmogorov-optimal for various function classes.
- Exponential approximation accuracy achieved for multiple functions.
- Finite-width networks require less connectivity than wide, shallow networks for smooth functions.

## Abstract

This paper develops fundamental limits of deep neural network learning by characterizing what is possible if no constraints are imposed on the learning algorithm and on the amount of training data. Concretely, we consider Kolmogorov-optimal approximation through deep neural networks with the guiding theme being a relation between the complexity of the function (class) to be approximated and the complexity of the approximating network in terms of connectivity and memory requirements for storing the network topology and the associated quantized weights. The theory we develop establishes that deep networks are Kolmogorov-optimal approximants for markedly different function classes, such as unit balls in Besov spaces and modulation spaces. In addition, deep networks provide exponential approximation accuracy - i.e., the approximation error decays exponentially in the number of nonzero weights in the network - of the multiplication operation, polynomials, sinusoidal functions, and certain smooth functions. Moreover, this holds true even for one-dimensional oscillatory textures and the Weierstrass function - a fractal function, neither of which has previously known methods achieving exponential approximation accuracy. We also show that in the approximation of sufficiently smooth functions finite-width deep networks require strictly smaller connectivity than finite-depth wide networks.