Argminm

Argminm is a mathematical notation used to denote the value of a variable m that minimizes a real-valued function f. It is commonly written as argmin_m f(m) and read as “the argument of the minimum with respect to m.” The result is the minimizer m* such that f(m*) ≤ f(m) for all m in the domain of f. If multiple minimizers exist, the argmin is the set of all minimizers.

Examples illustrate its use: for f(m) = (m-3)^2, the argmin m* is 3, with f(3) = 0. For f(m) = |m|, the argmin is m* = 0. In general, the corresponding minimum value is f(m*).

Notes and properties: Argmin focuses on the input that achieves the smallest value of f, not on the minimum value itself. It is related to argmax via argmin_m f(m) = -argmax_m (-f(m)) when considering real-valued functions. If f is convex, the argmin is either unique or forms a convex set; if f is differentiable, a necessary condition for a minimizer is the vanishing of the gradient, ∇f(m*) = 0, with the Hessian positive semidefinite. In non-differentiable cases, subgradients are used.

Computational aspects: Argmin problems are solved by optimization algorithms such as gradient descent for differentiable objectives, subgradient methods for non-differentiable cases, and specialized solvers for convex or discrete problems. Argmin is widely used in statistics and machine learning to denote parameter estimates that minimize a loss function, such as least squares or cross-entropy losses.