LogSumExp

The LogSumExp function domain is ${\displaystyle \mathbb {R} ^{n}}$, the real coordinate space, and its range is ${\displaystyle \mathbb {R} }$, the real line. It is an approximation to the maximum ${\displaystyle \max _{i}x_{i}}$ with the following bounds

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

The first inequality is strict unless ${\displaystyle n=1}$. The second inequality becomes an equality exact when all the 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 ${\displaystyle {\frac {1}{t}}\mathrm {LSE} (tx)}$. Then

${\displaystyle \max {\{x_{1},\dots ,x_{n}\}}<{\frac {1}{t}}\mathrm {LSE} (tx)\leq \max {\{x_{1},\dots ,x_{n}\}}+{\frac {\log(n)}{t}}.}$

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.

The LogSumExp function is convex, and is strictly monotonically increasing everywhere in its domain[3] (but not strictly convex everywhere[4]).

Writing ${\displaystyle \mathbf {x} =(x_{1},\dots ,x_{n}),}$ the partial derivatives are:

${\displaystyle {\frac {\partial }{\partial x_{i}}}{LSE(\mathbf {x} )}={\frac {\exp x_{i}}{\sum _{j}\exp {x_{j}}}}.}$

Which means the gradient of LogSumExp is the softmax function

The convex conjugate of LogSumExp is the negative entropy.

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

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.

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). Therefore, many math libraries such as IT++ provide a default routine of LSE and use this formula internally.

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

where ${\displaystyle x^{*}=\max {\{x_{1},\dots ,x_{n}\}}}$

LSE is convex but not strictly convex. We can define a strictly convex log-sum-exp type function[5] by adding an extra argument set to zero:

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

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.

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