LogSumExp

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.

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.

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.

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