# LogSumExp

The LogSumExp (LSE) (also called RealSoftMax or multivariable softplus) function is a smooth maximum – a smooth approximation to the maximum function, mainly used by machine learning algorithms. It is defined as the logarithm of the sum of the exponentials of the arguments:
$\mathrm {LSE} (x_{1},\dots ,x_{n})=\log \left(\exp(x_{1})+\cdots +\exp(x_{n})\right).$

## Properties

The LogSumExp function domain is $\mathbb {R} ^{n}$, the real coordinate space, and its codomain is $\mathbb {R}$, the real line. It is an approximation to the maximum $\max _{i}x_{i}$ with the following bounds $\max {\{x_{1},\dots ,x_{n}\}}\leq \mathrm {LSE} (x_{1},\dots ,x_{n})\leq \max {\{x_{1},\dots ,x_{n}\}}+\log(n).$ The first inequality is strict unless $n=1$. The second inequality is strict unless all arguments are equal. (Proof: Let $m=\max _{i}x_{i}$. Then $\exp(m)\leq \sum _{i=1}^{n}\exp(x_{i})\leq n\exp(m)$. Applying the logarithm to the inequality gives the result.)
In addition, we can scale the function to make the bounds tighter. Consider the function ${\frac {1}{t}}\mathrm {LSE} (tx_{1},\dots ,tx_{n})$. Then $\max {\{x_{1},\dots ,x_{n}\}}<{\frac {1}{t}}\mathrm {LSE} (tx_{1},\dots ,tx_{n})\leq \max {\{x_{1},\dots ,x_{n}\}}+{\frac {\log(n)}{t}}.$ (Proof: Replace each $x_{i}$ with $tx_{i}$ for some $t>0$ in the inequalities above, to give $\max {\{tx_{1},\dots ,tx_{n}\}}<\mathrm {LSE} (tx_{1},\dots ,tx_{n})\leq \max {\{tx_{1},\dots ,tx_{n}\}}+\log(n).$ and, since $t>0$ $t\max {\{x_{1},\dots ,x_{n}\}}<\mathrm {LSE} (tx_{1},\dots ,tx_{n})\leq t\max {\{x_{1},\dots ,x_{n}\}}+\log(n).$ finally, dividing by $t$ gives the result.)
Also, if we multiply by a negative number instead, we of course find a comparison to the $\min$ function: $\min {\{x_{1},\dots ,x_{n}\}}-{\frac {\log(n)}{t}}\leq {\frac {1}{-t}}\mathrm {LSE} (-tx)<\min {\{x_{1},\dots ,x_{n}\}}.$
The LogSumExp function is convex, and is strictly increasing everywhere in its domain. It is not strictly convex, since it is affine (linear plus a constant) on the diagonal and parallel lines:
    $\mathrm {LSE} (x_{1}+c,\dots ,x_{n}+c)=\mathrm {LSE} (x_{1},\dots ,x_{n})+c.$
Other than this direction, it is strictly convex (the Hessian has rank ⁠$n-1$⁠), so for example restricting to a hyperplane that is transverse to the diagonal results in a strictly convex function. See $\mathrm {LSE} _{0}^{+}$, below.
Writing $\mathbf {x} =(x_{1},\dots ,x_{n}),$ the partial derivatives are: ${\frac {\partial }{\partial x_{i}}}{\mathrm {LSE} (\mathbf {x} )}={\frac {\exp x_{i}}{\sum _{j}\exp {x_{j}}}},$ 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.
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:
$\mathrm {LSE} (\log(x_{1}),...,\log(x_{n}))=\log(x_{1}+\dots +x_{n})$ 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.
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).
$\mathrm {LSE} (x_{1},\dots ,x_{n})=x^{*}+\log \left(\exp(x_{1}-x^{*})+\cdots +\exp(x_{n}-x^{*})\right)$ where $x^{*}=\max {\{x_{1},\dots ,x_{n}\}}$
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 by adding an extra argument set to zero:
$\mathrm {LSE} _{0}^{+}(x_{1},...,x_{n})=\mathrm {LSE} (0,x_{1},...,x_{n})$ 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.