Paper ID: 2401.15121

Expressive Power of ReLU and Step Networks under Floating-Point Operations

Yeachan Park, Geonho Hwang, Wonyeol Lee, Sejun Park

The study of the expressive power of neural networks has investigated the fundamental limits of neural networks. Most existing results assume real-valued inputs and parameters as well as exact operations during the evaluation of neural networks. However, neural networks are typically executed on computers that can only represent a tiny subset of the reals and apply inexact operations. In this work, we analyze the expressive power of neural networks under a more realistic setup: when we use floating-point numbers and operations. Our first set of results assumes floating-point operations where the significand of a float is represented by finite bits but its exponent can take any integer value. Under this setup, we show that neural networks using a binary threshold unit or ReLU can memorize any finite input/output pairs and can approximate any continuous function within a small error. We also show similar results on memorization and universal approximation when floating-point operations use finite bits for both significand and exponent; these results are applicable to many popular floating-point formats such as those defined in the IEEE 754 standard (e.g., 32-bit single-precision format) and bfloat16.

Submitted: Jan 26, 2024