In mathematics, a smooth maximum of an indexed family x1, ..., xn of numbers is a smooth approximation to the maximum function meaning a parametric family of functions such that for every α, the function is smooth, and the family converges to the maximum function as . 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, as and as . 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

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Boltzmann operator

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

For large positive values of the parameter  , 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.

 

  has the following properties:

  1.   as  
  2.   is the arithmetic mean of its inputs
  3.   as  

The gradient of   is closely related to softmax and is given by

 

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

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Another smooth maximum is LogSumExp:

 

This can also be normalized if the   are all non-negative, yielding a function with domain   and range  :

 

The   term corrects for the fact that   by canceling out all but one zero exponential, and   if all   are zero.

Mellowmax

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The mellowmax operator[1] is defined as follows:

 

It is a non-expansive operator. As  , it acts like a maximum. As  , it acts like an arithmetic mean. As  , 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

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Another smooth maximum is the p-norm:

 

which converges to   as  .

An advantage of the p-norm is that it is a norm. As such it is scale invariant (homogeneous):  , and it satisfies the triangle inequality.

Smooth maximum unit

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

 

where   is a parameter. As  ,   and thus  .

See also

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References

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  1. ^ a b Asadi, Kavosh; Littman, Michael L. (2017). "An Alternative Softmax Operator for Reinforcement Learning". PMLR. 70: 243–252. arXiv:1612.05628. Retrieved January 6, 2023.
  2. ^ Safak, Aysel (February 1993). "Statistical analysis of the power sum of multiple correlated log-normal components". IEEE Transactions on Vehicular Technology. 42 (1): {58–61. doi:10.1109/25.192387. Retrieved January 6, 2023.
  3. ^ Biswas, Koushik; Kumar, Sandeep; Banerjee, Shilpak; Ashish Kumar Pandey (2021). "SMU: Smooth activation function for deep networks using smoothing maximum technique". arXiv:2111.04682 [cs.LG].

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)