The LogSumExp (LSE) (also called RealSoftMax[1] or multivariable softplus) function is a smooth maximum – a smooth approximation to the maximum function, mainly used by machine learning algorithms.[2] It is defined as the logarithm of the sum of the exponentials of the arguments:

Properties

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The LogSumExp function domain is  , the real coordinate space, and its codomain is  , the real line. It is an approximation to the maximum   with the following bounds   The first inequality is strict unless  . The second inequality is strict unless all arguments are equal. (Proof: Let  . Then  . Applying the logarithm to the inequality gives the result.)

In addition, we can scale the function to make the bounds tighter. Consider the function  . Then   (Proof: Replace each   with   for some   in the inequalities above, to give   and, since     finally, dividing by   gives the result.)

Also, if we multiply by a negative number instead, we of course find a comparison to the   function:  

The LogSumExp function is convex, and is strictly increasing everywhere in its domain.[3] It is not strictly convex, since it is affine (linear plus a constant) on the diagonal and parallel lines:[4]

 

Other than this direction, it is strictly convex (the Hessian has rank  ), so for example restricting to a hyperplane that is transverse to the diagonal results in a strictly convex function. See  , below.

Writing   the partial derivatives are:   which means the gradient of LogSumExp is the softmax function.

The convex conjugate of LogSumExp is the negative entropy.

log-sum-exp trick for log-domain calculations

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The LSE function is often encountered when the usual arithmetic computations are performed on a logarithmic scale, as in log probability.[5]

Similar to multiplication operations in linear-scale becoming simple additions in log-scale, an addition operation in linear-scale becomes the LSE in log-scale:

  A common purpose of using log-domain computations is to increase accuracy and avoid underflow and overflow problems when very small or very large numbers are represented directly (i.e. in a linear domain) using limited-precision floating point numbers.[6]

Unfortunately, the use of LSE directly in this case can again cause overflow/underflow problems. Therefore, the following equivalent must be used instead (especially when the accuracy of the above 'max' approximation is not sufficient).

  where  

Many math libraries such as IT++ provide a default routine of LSE and use this formula internally.

A strictly convex log-sum-exp type function

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LSE is convex but not strictly convex. We can define a strictly convex log-sum-exp type function[7] by adding an extra argument set to zero:

  This function is a proper Bregman generator (strictly convex and differentiable). It is encountered in machine learning, for example, as the cumulant of the multinomial/binomial family.

In tropical analysis, this is the sum in the log semiring.

See also

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References

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  1. ^ Zhang, Aston; Lipton, Zack; Li, Mu; Smola, Alex. "Dive into Deep Learning, Chapter 3 Exercises". www.d2l.ai. Retrieved 27 June 2020.
  2. ^ Nielsen, Frank; Sun, Ke (2016). "Guaranteed bounds on the Kullback-Leibler divergence of univariate mixtures using piecewise log-sum-exp inequalities". Entropy. 18 (12): 442. arXiv:1606.05850. Bibcode:2016Entrp..18..442N. doi:10.3390/e18120442. S2CID 17259055.
  3. ^ El Ghaoui, Laurent (2017). Optimization Models and Applications.
  4. ^ "convex analysis - About the strictly convexity of log-sum-exp function - Mathematics Stack Exchange". stackexchange.com.
  5. ^ McElreath, Richard. Statistical Rethinking. OCLC 1107423386.
  6. ^ "Practical issues: Numeric stability". CS231n Convolutional Neural Networks for Visual Recognition.
  7. ^ Nielsen, Frank; Hadjeres, Gaetan (2018). "Monte Carlo Information Geometry: The dually flat case". arXiv:1803.07225 [cs.LG].