Module approximation 

Classical stochastic approximation algorithms.

This submodule implements the three canonical stochastic approximation (SA) methods from the 1950s–1960s together with a modern SPSA variant:

Algorithm	Reference	Use case
Robbins-Monro	Robbins & Monro (1951)	Root-finding under noise
Kiefer-Wolfowitz	Kiefer & Wolfowitz (1952)	Gradient-free stochastic minimisation
SPSA	Spall (1992)	High-dimensional gradient-free SA

Notation

• xₖ : current iterate
• aₖ : gain sequence for update step (must satisfy Σ aₖ = ∞, Σ aₖ² < ∞)
• cₖ : gain sequence for finite-difference width (must → 0)

Structs

KieferWolfowitzOptions

Options for the Kiefer-Wolfowitz algorithm.

KieferWolfowitzResult

Result from the Kiefer-Wolfowitz algorithm.

RobbinsMonroOptions

Options for the Robbins-Monro algorithm.

RobbinsMonroResult

Result from Robbins-Monro root finding.

SpsaOptions

Options for the SPSA optimizer.

SpsaResult

Result from the SPSA algorithm.

Functions

kiefer_wolfowitz

Kiefer-Wolfowitz gradient-free stochastic approximation.

robbins_monro

Robbins-Monro stochastic root-finding algorithm.

spsa_minimize

Simultaneous Perturbation Stochastic Approximation (SPSA) optimizer.

spsa_step

Compute one SPSA gradient-estimate step.