argmaxa

Argmaxa is not a standard mathematical term and may be a misspelling or a context-specific variant of argmax. In common usage, argmax denotes the argument that maximizes a function.

For a function f: X → R, the argmax of f is the set of points x in X at which f attains its maximum value. It is written as argmax_{x ∈ X} f(x). The argmax returns the argument (the input), not the maximum value itself. The set can be a single point or multiple points if several inputs yield the same maximum.

Existence and uniqueness depend on the domain and the function. If X is compact and f is continuous, a maximum exists and the argmax is nonempty. If the maximum is achieved at a unique point, argmax is a singleton; if multiple maximizers exist, argmax is a larger set. In finite discrete domains, argmax corresponds to the indices with the highest score; when ties occur, a convention may be chosen (for example, selecting the smallest index) though mathematically the argmax remains the set of all maximizers.

In applications, argmax is widely used in optimization and machine learning to choose the most likely or best-scoring option. It is often nondifferentiable, which motivates differentiable surrogates such as softmax or Gumbel-softmax in training procedures.

Related concepts include argmin (the input(s) that minimize a function) and maximizers in optimization theory.