stoppkriterium

Stoppkriterium, or stopping criterion, is a condition used to terminate an iterative algorithm when a solution is deemed sufficiently accurate or when further computation is not justified. It is common in numerical optimization, root finding, linear algebra and machine learning. The criterion helps balance accuracy against computational cost and numerical stability.

Common forms include: residual-based criteria such as the norm of the residual ||f(x_k)|| being less than a tolerance; gradient-based criteria such as ||∇f(x_k)|| ≤ tol; change-based criteria such as ||x_{k+1}-x_k|| ≤ tol; and improvement-based criteria such as |f(x_k) - f(x_{k-1})| ≤ tol. Resource-based criteria, like reaching a maximum number of iterations (k_max) or a time limit, are also widely used. In stochastic settings, early stopping may be based on validation performance rather than the current objective value.

Practical use often combines criteria, stopping when any one is satisfied or when multiple are satisfied. The choice depends on the problem, the desired accuracy, and the available computational budget. A criterion that is too loose may yield imprecise results, while one that is too strict can cause excessive computation or instability, especially for poorly conditioned problems. Tolerances are typically tuned to the scale of the objective and the expected rate of convergence.

Notes: a stopping criterion is not a guarantee of global optimality; it signals convergence to a solution under the algorithm’s model. The term is used across languages, with variations such as stoppkriterium or Stopkriterium in Nordic and Germanic literature, and stopping criterion in English.