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

Blanchard, Pierre and Higham, Desmond J. and Higham, Nicholas J. (2019) Accurate Computation of the Log-Sum-Exp and Softmax Functions. [MIMS Preprint]

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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.

Item Type: MIMS Preprint
Uncontrolled Keywords: log-sum-exp, softmax, floating-point arithmetic, rounding error analysis, overflow, underflow, condition number
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Nick Higham
Date Deposited: 08 Sep 2019 11:10
Last Modified: 08 Sep 2019 11:10

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