Smooth maximum

In mathematics, a smooth maximum of an indexed family x₁, ..., x_n of numbers is a smooth approximation to the maximum function $\max(x_{1},\ldots ,x_{n}),$ meaning a parametric family of functions $m_{\alpha }(x_{1},\ldots ,x_{n})$ such that for every α, the function ⁠$m_{\alpha }$⁠ is smooth, and the family converges to the maximum function ⁠$m_{\alpha }\to \max$⁠ as ⁠$\alpha \to \infty$⁠. The concept of smooth minimum is similarly defined. In many cases, a single family approximates both: maximum as the parameter goes to positive infinity, minimum as the parameter goes to negative infinity; in symbols, ⁠$m_{\alpha }\to \max$⁠ as ⁠$\alpha \to \infty$⁠ and ⁠$m_{\alpha }\to \min$⁠ as ⁠$\alpha \to -\infty$⁠. The term can also be used loosely for a specific smooth function that behaves similarly to a maximum, without necessarily being part of a parametrized family.

Examples

Boltzmann operator

Smoothmax of (−x, x) versus x for various parameter values. Very smooth for $\alpha$=0.5, and more sharp for $\alpha$=8.

For large positive values of the parameter $\alpha >0$, 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.

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

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

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

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

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

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

This operator is sometimes called the Boltzmann operator, after the Boltzmann distribution.

LogSumExp

Main article: LogSumExp

Another smooth maximum is LogSumExp:

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

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

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

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

Mellowmax

The mellowmax operator is defined as follows:

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

It is a non-expansive operator. As $\alpha \to \infty$, it acts like a maximum. As $\alpha \to 0$, it acts like an arithmetic mean. As $\alpha \to -\infty$, 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.

p-Norm

Main article: P-norm

Another smooth maximum is the p-norm:

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

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

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

Smooth maximum unit

The following binary operator is called the Smooth Maximum Unit (SMU):

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

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

See also

• LogSumExp
• Softmax function
• Generalized mean

References

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)