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:

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

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

(Proof: Replace each ${\displaystyle x_{i}}$ with ${\displaystyle tx_{i}}$ for some ${\displaystyle t>0}$ in the inequalities above, to give and, since ${\displaystyle t>0}$ 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:

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:

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:

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

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:

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