TeXipedia

argmax

Returns the argument (input value) that maximizes a given function or expression.

Overview

Essential in optimization problems, mathematical analysis, and machine learning applications where finding the input that produces the maximum output is crucial.

  • Commonly used in statistical estimation to find most likely parameters
  • Appears frequently in computer science and artificial intelligence algorithms
  • Key notation in mathematical optimization and decision theory
  • Often paired with objective functions in optimization problems

Examples

Finding the value of x that maximizes a quadratic function.

arg max⁡x∈R(−(x−2)2+4)=2\argmax_{x \in \mathbb{R}} (-(x-2)^2 + 4) = 2 
\argmax_{x \in \mathbb{R}} (-(x-2)^2 + 4) = 2

Maximum likelihood estimation in statistics.

θ^=arg max⁡θL(θ∣x)\hat{\theta} = \argmax_{\theta} \mathcal{L}(\theta|x) 
\hat{\theta} = \argmax_{\theta} \mathcal{L}(\theta|x)

Optimal parameter selection in machine learning.

w∗=arg max⁡w∑i=1nlog⁡P(yi∣xi,w)w^* = \argmax_{w} \sum_{i=1}^n \log P(y_i|x_i,w) 
w^* = \argmax_{w} \sum_{i=1}^n \log P(y_i|x_i,w)