arg min 的含义

﹏ヽ暗。殇╰゛Y 2023-08-17 16:54 28阅读 0赞

最通俗的理解：表示使目标函数取最小值时的变量值  
From Wikipedia

In [mathematics][], **arg max** (or **argmax**) stands for the **argument of the maximum**, that is to say, the set of points of the given [argument][] for which the value of the given expression attains its [maximum][] value:[\[note 1\]][note 1]

![\\underset\{x\}\{\\operatorname\{arg\\,max\}\} \\, f(x) := \\\{x\\ |\\ \\forall y : f(y) \\le f(x)\\\}][underset_x_operatorname_arg_max_ _ f_x_ _x_ _ _forall y_ f_y_ _le f_x]

In other words,

![\\underset\{x\}\{\\operatorname\{arg\\,max\}\} \\, f(x)][underset_x_operatorname_arg_max_ _ f_x]

is the set of values of *x* for which *f*(*x*) has the largest value *M.* For example, if *f*(*x*) is 1−|*x*|, then it attains its maximum value of 1 at *x* = 0 and only there, so ![\\underset\{x\}\{\\operatorname\{arg\\,max\}\} \\, (1-|x|) = \\\{0\\\}][underset_x_operatorname_arg_max_ _ _1-_x_ _ _0].

Equivalently, if *M* is the maximum of *f,* then the arg max is the [level set][] of the maximum:

![\\underset\{x\}\{\\operatorname\{arg\\,max\}\} \\, f(x) = f^\{-1\}(M) = \\\{x\\ |\\ f(x) = M \\\}][underset_x_operatorname_arg_max_ _ f_x_ _ f_-1_M_ _ _x_ _ f_x_ _ M]

If the maximum is reached at a single value, then one refers to the point as *the* arg max, meaning we define the arg max as a point, not a set of points. So, for example,

![\\underset\{x\\in \\Bbb\{R\}\}\{\\operatorname\{arg\\,max\}\} (x(10-x)) = 5][underset_x_in _Bbb_R_operatorname_arg_max_ _x_10-x_ _ 5]

(rather than the [singleton][] set \{5\}), since the maximum value of *x*(10 − *x*) is 25, which happens when *x* = 5.[\[note 2\]][note 2]

However, in case the maximum is reached at many values, **arg max** is a *set* of points.

Then, we have for example

![\\underset\{x \\in \[0,4\\pi\]\}\{\\operatorname\{arg\\,max\}\} \\, \\cos(x) = \\\{0,2\\pi,4\\pi\\\}][underset_x _in _0_4_pi_operatorname_arg_max_ _ _cos_x_ _ _0_2_pi_4_pi]

since the maximum value of cos(*x*) is 1, which happens on this interval when *x* = 0, 2π or 4π. On the whole real line, the arg max is ![\\\{0, 2\\pi, -2\\pi, 4\\pi, \\dots \\\}.][0_ 2_pi_ -2_pi_ 4_pi_ _dots _.]

**arg min** (or **argmin**) is defined analogously.

Note also that functions do not in general attain a maximum value, and hence will in general not have an arg max: ![\\underset\{x\\in \\Bbb\{R\}\}\{\\operatorname\{arg\\,max\}\}\\, x][underset_x_in _Bbb_R_operatorname_arg_max_ x] is undefined, as *x*is [unbounded][] on the real line. However, by the [extreme value theorem][] (or the classical [compactness argument][]), a continuous function on a[compact][] [interval][] has a maximum, and thus an arg max.

转载于:https://www.cnblogs.com/stucp/archive/2012/11/16/2773292.html

[mathematics]: http://www.cppblog.com/wiki/Mathematics
[argument]: http://www.cppblog.com/wiki/Parameter
[maximum]: http://www.cppblog.com/wiki/Maximum
[note 1]: http://www.cppblog.com/guijie/archive/2010/12/13/136273.html#cite_note-0
[underset_x_operatorname_arg_max_ _ f_x_ _x_ _ _forall y_ f_y_ _le f_x]: /images/20230809/44aab7bfe08e4ceab29b6159f442c0b5.png
[underset_x_operatorname_arg_max_ _ f_x]: /images/20230809/b2e7352a950e4ec89e6ff3e9ef938396.png
[underset_x_operatorname_arg_max_ _ _1-_x_ _ _0]: /images/20230809/a49ea1ae56db4ccd833be68a22052822.png
[level set]: http://www.cppblog.com/wiki/Level_set
[underset_x_operatorname_arg_max_ _ f_x_ _ f_-1_M_ _ _x_ _ f_x_ _ M]: /images/20230809/5ea4a07dbe8d41b68cb3bfc89cf0c9fb.png
[underset_x_in _Bbb_R_operatorname_arg_max_ _x_10-x_ _ 5]: /images/20230809/22e425ba91cf4b639d777b3697b0eea9.png
[singleton]: http://www.cppblog.com/wiki/Singleton_%28mathematics%29
[note 2]: http://www.cppblog.com/guijie/archive/2010/12/13/136273.html#cite_note-1
[underset_x _in _0_4_pi_operatorname_arg_max_ _ _cos_x_ _ _0_2_pi_4_pi]: /images/20230809/ea36735ba6ca40b7a0c825968f4b9f7e.png
[0_ 2_pi_ -2_pi_ 4_pi_ _dots _.]: /images/20230809/98b6448b8e9c4f2483505054b5792ccc.png
[underset_x_in _Bbb_R_operatorname_arg_max_ x]: http://upload.wikimedia.org/math/2/8/1/2819cfd4c0287ffe40d49f826a8f02f1.png
[unbounded]: http://www.cppblog.com/wiki/Unbounded
[extreme value theorem]: http://www.cppblog.com/wiki/Extreme_value_theorem
[compactness argument]: http://www.cppblog.com/wiki/Compact_set#Theorems
[compact]: http://www.cppblog.com/wiki/Compact_space
[interval]: http://www.cppblog.com/wiki/Interval