Cited by 127Numerical Methods and AlgorithmsMathematical and Computational MethodsMathematical functions and polynomials

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Accurately computing the log-sum-exp and softmax functions

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Blanchard, Pierre, Higham, Desmond J, Higham, Nicholas J·2021·IMA Journal of Numerical Analysis

DOI 10.1093/imanum/draa038

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Abstract

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 and shifting can introduce subtractive cancellation. 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, so can safely be used, but that a division-free variant of softmax can suffer from loss of accuracy.

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Authors: Blanchard, P., Higham, D. J., & Higham, N. J.
Affiliations: Department of Mathematics, University of Manchester, Manchester, M13 9PL, UK
University of Edinburgh (GB)
University of Manchester (GB)
Corresponding author: Higham, Nicholas J
ORCID: Higham, Desmond J
Higham, Nicholas J
Venue: IMA Journal of Numerical Analysis
Publisher: Oxford University Press (OUP)
Published: 2020-08-19
Volume: 41
Issue: 4
Pages: 2311-2330
ISSN: 0272-4979 (ISSN-L)
1464-3642
DOI: 10.1093/imanum/draa038
OpenAlex: W3004059350
Cited by: 126
References: 31
Citations by year: 2026: 32 · 2025: 30 · 2024: 23 · 2023: 20 · 2022: 12 · 2021: 9
FWCI: 5.10
Type: journal article
Language: en
Open access: hybrid
doi.org
Licence: cc-by
creativecommons.org/licenses/by/4.0/ (vor, 1 d)
Locations: IMA Journal of Numerical Analysis (publishedVersion, cc-by)
Research Explorer (The University of Manchester) (publishedVersion)
MIMS EPrints (University of Southampton) (acceptedVersion)
Funders: Engineering and Physical Sciences Research Council (EP/P020720/1)
MathWorks
Royal Society
Topics: Physical Sciences › Computer Science › Computational Theory and Mathematics › Numerical Methods and Algorithms
Physical Sciences › Mathematics › Modeling and Simulation › Mathematical and Computational Methods
Physical Sciences › Mathematics › Applied Mathematics › Mathematical functions and polynomials
Keywords: softmax, floating-point arithmetic, rounding error analysis, condition number, overflow and underflow, numerical accuracy

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Accurately computing the log-sum-exp and softmax functions

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