stepdependent

Stepdependent is an adjective used to describe a quantity whose value changes as a function of the step or iteration in a discrete process. The term is commonly applied in contexts where parameters are adjusted over time rather than kept fixed throughout the run.

In numerical optimization and machine learning, stepdependent parameters include learning rate, momentum, and regularization strength, which may follow predefined schedules or adaptive rules. A familiar example is step decay, where the learning rate is reduced by a factor at fixed step counts. Other variants use piecewise-constant or staircase adjustments, while some methods adjust parameters based on elapsed steps or observed performance. Stepdependent settings are often used to improve convergence, stability, or generalization.

In numerical integration and simulations, step-dependent step sizes or tolerances are used to balance stability and accuracy. Parameters may be tuned depending on the current step to control error, stiffness, or computational load. In these contexts, the step index k is used to index the parameter, often written as theta_k with a scheduler function S such that theta_k = S(k) or theta_k = theta_0 · S(k).

Stepdependent schedules can be deterministic, applying predefined changes at specific steps, or adaptive, responding to metrics such as error, gradient norms, or validation performance. The idea behind stepdependence is to allow a process to adjust its behavior as it progresses, rather than relying on a fixed parameter throughout. It is related to broader concepts of time-varying parameters and piecewise-constant functions.