LogSumExp

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:

$${\displaystyle \mathrm {LSE} (x_{1},\dots ,x_{n})=\log \left(\exp(x_{1})+\cdots +\exp(x_{n})\right).}$$

Properties

The LogSumExp function domain is ${\displaystyle \mathbb {R} ^{n}}$, the real coordinate space, and its codomain is ${\displaystyle \mathbb {R} }$, the real line. It is an approximation to the maximum ${\displaystyle \max _{i}x_{i}}$ with the following bounds $${\displaystyle \max {\{x_{1},\dots ,x_{n}\}}\leq \mathrm {LSE} (x_{1},\dots ,x_{n})\leq \max {\{x_{1},\dots ,x_{n}\}}+\log(n).}$$ The first inequality is strict unless ${\displaystyle n=1}$. The second inequality is strict unless all arguments are equal. (Proof: Let ${\displaystyle m=\max _{i}x_{i}}$. Then ${\displaystyle \exp(m)\leq \sum _{i=1}^{n}\exp(x_{i})\leq n\exp(m)}$. Applying the logarithm to the inequality gives the result.)

In addition, we can scale the function to make the bounds tighter. Consider the function ${\displaystyle {\frac {1}{t}}\mathrm {LSE} (tx_{1},\dots ,tx_{n})}$. Then $${\displaystyle \max {\{x_{1},\dots ,x_{n}\}}<{\frac {1}{t}}\mathrm {LSE} (tx_{1},\dots ,tx_{n})\leq \max {\{x_{1},\dots ,x_{n}\}}+{\frac {\log(n)}{t}}.}$$ (Proof: Replace each ${\displaystyle x_{i}}$ with ${\displaystyle tx_{i}}$ for some ${\displaystyle t>0}$ in the inequalities above, to give $${\displaystyle \max {\{tx_{1},\dots ,tx_{n}\}}<\mathrm {LSE} (tx_{1},\dots ,tx_{n})\leq \max {\{tx_{1},\dots ,tx_{n}\}}+\log(n).}$$ and, since ${\displaystyle t>0}$ $${\displaystyle t\max {\{x_{1},\dots ,x_{n}\}}<\mathrm {LSE} (tx_{1},\dots ,tx_{n})\leq t\max {\{x_{1},\dots ,x_{n}\}}+\log(n).}$$ finally, dividing by ${\displaystyle t}$ gives the result.)

Also, if we multiply by a negative number instead, we of course find a comparison to the ${\displaystyle \min }$ function: $${\displaystyle \min {\{x_{1},\dots ,x_{n}\}}-{\frac {\log(n)}{t}}\leq {\frac {1}{-t}}\mathrm {LSE} (-tx)<\min {\{x_{1},\dots ,x_{n}\}}.}$$

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]

${\displaystyle \mathrm {LSE} (x_{1}+c,\dots ,x_{n}+c)=\mathrm {LSE} (x_{1},\dots ,x_{n})+c.}$

Other than this direction, it is strictly convex (the Hessian has rank ⁠${\displaystyle n-1}$⁠), so for example restricting to a hyperplane that is transverse to the diagonal results in a strictly convex function. See ${\displaystyle \mathrm {LSE} _{0}^{+}}$, below.

Writing ${\displaystyle \mathbf {x} =(x_{1},\dots ,x_{n}),}$ the partial derivatives are: $${\displaystyle {\frac {\partial }{\partial x_{i}}}{\mathrm {LSE} (\mathbf {x} )}={\frac {\exp x_{i}}{\sum _{j}\exp {x_{j}}}},}$$ 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

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:

$${\displaystyle \mathrm {LSE} (\log(x_{1}),...,\log(x_{n}))=\log(x_{1}+\dots +x_{n})}$$ 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).

$${\displaystyle \mathrm {LSE} (x_{1},\dots ,x_{n})=x^{*}+\log \left(\exp(x_{1}-x^{*})+\cdots +\exp(x_{n}-x^{*})\right)}$$ where ${\displaystyle x^{*}=\max {\{x_{1},\dots ,x_{n}\}}}$

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

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:

$${\displaystyle \mathrm {LSE} _{0}^{+}(x_{1},...,x_{n})=\mathrm {LSE} (0,x_{1},...,x_{n})}$$ 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

• Logarithmic mean
• Log semiring
• Smooth maximum
• Softmax function

References

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].