Boltzmann operator edit

For large positive values of the parameter ${\displaystyle \alpha >0}$ , the following formulation is a smooth, differentiable approximation of the maximum function. For negative values of the parameter that are large in absolute value, it approximates the minimum.

${\displaystyle {\mathcal {S}}_{\alpha }(x_{1},\ldots ,x_{n})={\frac {\sum _{i=1}^{n}x_{i}e^{\alpha x_{i}}}{\sum _{i=1}^{n}e^{\alpha x_{i}}}}}$ 

${\displaystyle {\mathcal {S}}_{\alpha }}$  has the following properties:

1. ${\displaystyle {\mathcal {S}}_{\alpha }\to \max }$  as ${\displaystyle \alpha \to \infty }$ 
2. ${\displaystyle {\mathcal {S}}_{0}}$  is the arithmetic mean of its inputs
3. ${\displaystyle {\mathcal {S}}_{\alpha }\to \min }$  as ${\displaystyle \alpha \to -\infty }$ 

The gradient of ${\displaystyle {\mathcal {S}}_{\alpha }}$  is closely related to softmax and is given by

${\displaystyle \nabla _{x_{i}}{\mathcal {S}}_{\alpha }(x_{1},\ldots ,x_{n})={\frac {e^{\alpha x_{i}}}{\sum _{j=1}^{n}e^{\alpha x_{j}}}}[1+\alpha (x_{i}-{\mathcal {S}}_{\alpha }(x_{1},\ldots ,x_{n}))].}$ 

This makes the softmax function useful for optimization techniques that use gradient descent.

This operator is sometimes called the Boltzmann operator,^[1] after the Boltzmann distribution.

LogSumExp edit

Another smooth maximum is LogSumExp:

${\displaystyle \mathrm {LSE} _{\alpha }(x_{1},\ldots ,x_{n})={\frac {1}{\alpha }}\log \sum _{i=1}^{n}\exp \alpha x_{i}}$ 

This can also be normalized if the ${\displaystyle x_{i}}$  are all non-negative, yielding a function with domain ${\displaystyle [0,\infty )^{n}}$  and range ${\displaystyle [0,\infty )}$ :

${\displaystyle g(x_{1},\ldots ,x_{n})=\log \left(\sum _{i=1}^{n}\exp x_{i}-(n-1)\right)}$ 

The ${\displaystyle (n-1)}$  term corrects for the fact that ${\displaystyle \exp(0)=1}$  by canceling out all but one zero exponential, and ${\displaystyle \log 1=0}$  if all ${\displaystyle x_{i}}$  are zero.

Mellowmax edit

The mellowmax operator^[1] is defined as follows:

${\displaystyle \mathrm {mm} _{\alpha }(x)={\frac {1}{\alpha }}\log {\frac {1}{n}}\sum _{i=1}^{n}\exp \alpha x_{i}}$ 

It is a non-expansive operator. As ${\displaystyle \alpha \to \infty }$ , it acts like a maximum. As ${\displaystyle \alpha \to 0}$ , it acts like an arithmetic mean. As ${\displaystyle \alpha \to -\infty }$ , it acts like a minimum. This operator can be viewed as a particular instantiation of the quasi-arithmetic mean. It can also be derived from information theoretical principles as a way of regularizing policies with a cost function defined by KL divergence. The operator has previously been utilized in other areas, such as power engineering.^[2]

p-Norm edit

Another smooth maximum is the p-norm:

${\displaystyle \|(x_{1},\ldots ,x_{n})\|_{p}=\left(\sum _{i=1}^{n}|x_{i}|^{p}\right)^{\frac {1}{p}}}$ 

which converges to ${\displaystyle \|(x_{1},\ldots ,x_{n})\|_{\infty }=\max _{1\leq i\leq n}|x_{i}|}$  as ${\displaystyle p\to \infty }$ .

An advantage of the p-norm is that it is a norm. As such it is scale invariant (homogeneous): ${\displaystyle \|(\lambda x_{1},\ldots ,\lambda x_{n})\|_{p}=|\lambda |\cdot \|(x_{1},\ldots ,x_{n})\|_{p}}$ , and it satisfies the triangle inequality.

Smooth maximum unit edit

The following binary operator is called the Smooth Maximum Unit (SMU):^[3]

${\displaystyle {\begin{aligned}\textstyle \max _{\varepsilon }(a,b)&={\frac {a+b+|a-b|_{\varepsilon }}{2}}\\&={\frac {a+b+{\sqrt {(a-b)^{2}+\varepsilon }}}{2}}\end{aligned}}}$ 

where ${\displaystyle \varepsilon \geq 0}$  is a parameter. As ${\displaystyle \varepsilon \to 0}$ , ${\displaystyle |\cdot |_{\varepsilon }\to |\cdot |}$  and thus ${\displaystyle \textstyle \max _{\varepsilon }\to \max }$ .

• LogSumExp
• Softmax function
• Generalized mean

https://www.johndcook.com/soft_maximum.pdf

M. Lange, D. Zühlke, O. Holz, and T. Villmann, "Applications of lp-norms and their smooth approximations for gradient based learning vector quantization," in Proc. ESANN, Apr. 2014, pp. 271-276. (https://www.elen.ucl.ac.be/Proceedings/esann/esannpdf/es2014-153.pdf)