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Smooth maximum

In mathematics, a smooth maximum of an indexed family x1, ..., xn of numbers is a differentiable approximation to the maximum function

and the concept of smooth minimum is similarly defined.

For large positive values of the parameter , the following formulation is one 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 average 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.

Another formulation is:

The term corrects for the fact that by canceling out all but one zero exponential


Use in numerical methodsEdit

Other choices of smoothing functionEdit

See alsoEdit


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. (