Accurate Computation of the Log-Sum-Exp and Softmax Functions

TL;DR

This paper analyzes the numerical stability of various algorithms for computing the log-sum-exp and softmax functions, providing error bounds and practical recommendations for accurate implementation in low precision environments.

Contribution

It offers a detailed rounding error analysis of different evaluation algorithms and compares their accuracy, guiding better implementation choices.

Findings

Shifted formulas are similarly accurate to unshifted ones.

Shifted softmax is generally more accurate than division-free variants.

Numerical experiments confirm the theoretical error bounds.

Abstract

Evaluating the log-sum-exp function or the softmax function is a key step in many modern data science algorithms, notably in inference and classification. Because of the exponentials that these functions contain, the evaluation is prone to overflow and underflow, especially in low precision arithmetic. Software implementations commonly use alternative formulas that avoid overflow and reduce the chance of harmful underflow, employing a shift or another rewriting. Although mathematically equivalent, these variants behave differently in floating-point arithmetic. We give rounding error analyses of different evaluation algorithms and interpret the error bounds using condition numbers for the functions. We conclude, based on the analysis and numerical experiments, that the shifted formulas are of similar accuracy to the unshifted ones and that the shifted softmax formula is typically more accurate than a division-free variant.