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argmin

Denotes the argument that minimizes a given function or expression in mathematical optimization.

Overview

Essential in optimization theory, machine learning, and computational mathematics for finding input values that yield the minimum output of an objective function.

  • Commonly used in statistical estimation and model parameter optimization
  • Appears frequently in research papers and technical documentation describing minimization problems
  • Often paired with subscripts to specify the variable or domain over which minimization occurs
  • Distinguished from min by returning the argument (input value) rather than the minimum value itself

Examples

Finding the value of x that minimizes a quadratic function.

x∗=arg min⁡x∈R(x2+2x+1)x^* = \argmin_{x \in \mathbb{R}} (x^2 + 2x + 1) 
x^* = \argmin_{x \in \mathbb{R}} (x^2 + 2x + 1)

Expressing the least squares solution in linear regression.

β=arg min⁡β∑i=1n(yi−Xiβ)2\beta = \argmin_{\beta} \sum_{i=1}^n (y_i - X_i\beta)^2 
\beta = \argmin_{\beta} \sum_{i=1}^n (y_i - X_i\beta)^2

Optimization problem in machine learning with regularization term.

w=arg min⁡w(1n∑i=1nL(w;xi)+λ∥w∥2)w = \argmin_{w} \left(\frac{1}{n}\sum_{i=1}^n L(w;x_i) + \lambda\|w\|^2\right) 
w = \argmin_{w} \left(\frac{1}{n}\sum_{i=1}^n L(w;x_i) + \lambda\|w\|^2\right)