Stochastic approximation methods are a family of iterative methods typically used for root-finding problems or for optimization problems. The recursive update rules of stochastic approximation methods can be used, among other things, for solving linear systems when the collected data is corrupted by noise, or for approximating extreme values of functions which cannot be computed directly, but only estimated via noisy observations.

In a nutshell, stochastic approximation algorithms deal with a function of the form ${\textstyle f(\theta )=\operatorname {E} _{\xi }}$ which is the expected value of a function depending on a random variable ${\textstyle \xi }$. The goal is to recover properties of such a function ${\textstyle f}$ without evaluating it directly. Instead, stochastic approximation algorithms use random samples of ${\textstyle F(\theta ,\xi )}$ to efficiently approximate properties of ${\textstyle f}$ such as zeros or extrema.

Recently, stochastic approximations have found extensive applications in the fields of statistics and machine learning, especially in settings with big data. These applications range from stochastic optimization methods and algorithms, to online forms of the EM algorithm, reinforcement learning via temporal differences, and deep learning, and others.^[1] Stochastic approximation algorithms have also been used in the social sciences to describe collective dynamics: fictitious play in learning theory and consensus algorithms can be studied using their theory.^[2]

The earliest, and prototypical, algorithms of this kind are the Robbins–Monro and Kiefer–Wolfowitz algorithms introduced respectively in 1951 and 1952.

Robbins–Monro algorithm

The Robbins–Monro algorithm, introduced in 1951 by Herbert Robbins and Sutton Monro,^[3] presented a methodology for solving a root finding problem, where the function is represented as an expected value. Assume that we have a function ${\textstyle M(\theta )}$, and a constant ${\textstyle \alpha }$, such that the equation ${\textstyle M(\theta )=\alpha }$ has a unique root at ${\textstyle \theta ^{*}}$. It is assumed that while we cannot directly observe the function ${\textstyle M(\theta )}$, we can instead obtain measurements of the random variable ${\textstyle N(\theta )}$ where ${\textstyle \operatorname {E} =M(\theta )}$. The structure of the algorithm is to then generate iterates of the form:

${\displaystyle \theta _{n+1}=\theta _{n}-a_{n}(N(\theta _{n})-\alpha )}$

Here, ${\displaystyle a_{1},a_{2},\dots }$ is a sequence of positive step sizes. Robbins and Monro proved^{[3], Theorem 2} that ${\displaystyle \theta _{n}}$ converges in ${\displaystyle L^{2}}$ (and hence also in probability) to ${\displaystyle \theta ^{*}}$, and Blum^[4] later proved the convergence is actually with probability one, provided that:

• ${\textstyle N(\theta )}$ is uniformly bounded,
• ${\textstyle M(\theta )}$ is nondecreasing,
• ${\textstyle M'(\theta ^{*})}$ exists and is positive, and
• The sequence ${\textstyle a_{n}}$ satisfies the following requirements:

${\displaystyle \qquad \sum _{n=0}^{\infty }a_{n}=\infty \quad {\mbox{ and }}\quad \sum _{n=0}^{\infty }a_{n}^{2}<\infty \quad }$

A particular sequence of steps which satisfy these conditions, and was suggested by Robbins–Monro, have the form: ${\textstyle a_{n}=a/n}$, for ${\textstyle a>0}$. Other series are possible but in order to average out the noise in ${\textstyle N(\theta )}$, the above condition must be met.

Complexity results

1. If ${\textstyle f(\theta )}$ is twice continuously differentiable, and strongly convex, and the minimizer of ${\textstyle f(\theta )}$ belongs to the interior of ${\textstyle \Theta }$, then the Robbins–Monro algorithm will achieve the asymptotically optimal convergence rate, with respect to the objective function, being ${\textstyle \operatorname {E} =O(1/n)}$, where ${\textstyle f^{*}}$ is the minimal value of ${\textstyle f(\theta )}$ over ${\textstyle \theta \in \Theta }$.^[5][6]
2. Conversely, in the general convex case, where we lack both the assumption of smoothness and strong convexity, Nemirovski and Yudin^[7] have shown that the asymptotically optimal convergence rate, with respect to the objective function values, is ${\textstyle O(1/{\sqrt {n}})}$. They have also proven that this rate cannot be improved.

Subsequent developments and Polyak–Ruppert averaging

While the Robbins–Monro algorithm is theoretically able to achieve ${\textstyle O(1/n)}$ under the assumption of twice continuous differentiability and strong convexity, it can perform quite poorly upon implementation. This is primarily due to the fact that the algorithm is very sensitive to the choice of the step size sequence, and the supposed asymptotically optimal step size policy can be quite harmful in the beginning.^[6][8]

Chung (1954)^[9] and Fabian (1968)^[10] showed that we would achieve optimal convergence rate ${\textstyle O(1/{\sqrt {n}})}$ with ${\textstyle a_{n}=\bigtriangledown ^{2}f(\theta ^{*})^{-1}/n}$ (or ${\textstyle a_{n}={\frac {1}{(nM'(\theta ^{*}))}}}$). Lai and Robbins^[11][12] designed adaptive procedures to estimate ${\textstyle M'(\theta ^{*})}$ such that ${\textstyle \theta _{n}}$ has minimal asymptotic variance. However the application of such optimal methods requires much a priori information which is hard to obtain in most situations. To overcome this shortfall, Polyak (1991)^[13] and Ruppert (1988)^[14] independently developed a new optimal algorithm based on the idea of averaging the trajectories. Polyak and Juditsky^[15] also presented a method of accelerating Robbins–Monro for linear and non-linear root-searching problems through the use of longer steps, and averaging of the iterates. The algorithm would have the following structure: