Stochastic approximation methods are a family ofiterative methods typically used forroot-finding problems or foroptimization 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 approximatingextreme 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$f(\theta )={\mathrm{E}}_{\xi }[F(\theta ,\xi )]$which is theexpected value of a function depending on arandom variable$\xi$. The goal is to recover properties of such a function$f$ without evaluating it directly. Instead, stochastic approximation algorithms use random samples of$F(\theta ,\xi )$ to efficiently approximate properties of$f$ such as zeros or extrema.

Recently, stochastic approximations have found extensive applications in the fields of statistics and machine learning, especially in settings withbig data. These applications range fromstochastic optimization methods and algorithms, to online forms of the EM algorithm, reinforcement learning viatemporal differences, anddeep 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 theRobbins–Monro andKiefer–Wolfowitz algorithms introduced respectively in 1951 and 1952.

The Robbins–Monro algorithm, introduced in 1951 byHerbert Robbins andSutton 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$M(\theta )$, and a constant$\alpha$, such that the equation$M(\theta )=\alpha$ has a unique root at${\theta }^{\ast }.$ It is assumed that while we cannot directly observe the function$M(\theta ),$ we can instead obtain measurements of the random variable$N(\theta )$ where$\mathrm{E}[N(\theta )]=M(\theta )$. 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${\displaystyle {\theta }^{\ast }}$, and Blum^[4] later proved the convergence is actually with probability one, provided that:

• $N(\theta )$ is uniformly bounded,
• $M(\theta )$ is nondecreasing,
• $M^{\prime }({\theta }^{\ast })$ exists and is positive, and
• The sequence$a_n$ satisfies the following requirements:

A particular sequence of steps which satisfy these conditions, and was suggested by Robbins–Monro, have the form:$a_n=a/n$, for$a>0$. Other series, such as${\displaystyle a_n=\frac{1}{n\ln n},\frac{1}{n\ln n\ln \ln n},\dots }$ are possible but in order to average out the noise in$N(\theta )$, the above condition must be met.

Consider the problem of estimating the mean${\displaystyle {\theta }^{\ast }}$ of a probability distribution from a stream of independent samples${\displaystyle X_1,X_2,\dots }$.

Let${\displaystyle N(\theta ):=\theta -X}$, then the unique solution to$\mathrm{E}[N(\theta )]=0$ is the desired mean${\displaystyle {\theta }^{\ast }}$. The RM algorithm gives us$${\displaystyle {\theta }_{n+1}={\theta }_n-a_n({\theta }_n-X_n)}$$This is equivalent tostochastic gradient descent with loss function${\displaystyle L(\theta )=\frac{1}{2}\Vert X-\theta {\Vert }^2}$. It is also equivalent to a weighted average:$${\displaystyle {\theta }_{n+1}=(1-a_n){\theta }_n+a_n{X}_n}$$In general, if there exists some function${\displaystyle L}$ such that${\displaystyle \mathrm{\nabla }L(\theta )=N(\theta )-\alpha }$, then the Robbins–Monro algorithm is equivalent to stochastic gradient descent with loss function${\displaystyle L(\theta )}$. However, the RM algorithm does not require${\displaystyle L}$ to exist in order to converge.

1. If$f(\theta )$ is twice continuously differentiable, and strongly convex, and the minimizer of$f(\theta )$ belongs to the interior of$\mathrm{\Theta }$, then the Robbins–Monro algorithm will achieve the asymptotically optimal convergence rate, with respect to the objective function, being$\mathrm{E}[f({\theta }_n)-f^{\ast }]=O(1/n)$, where$f^{\ast }$ is the minimal value of$f(\theta )$ over$\theta \in \mathrm{\Theta }$.^[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$O(1/\sqrt{n})$. They have also proven that this rate cannot be improved.

While the Robbins–Monro algorithm is theoretically able to achieve$O(1/n)$ 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$O(1/\sqrt{n})$ with$a_n={▽}^{2}f({\theta }^{\ast }{)}^{-1}/n$ (or$a_n=\frac{1}{(n{M}^{\prime }({\theta }^{\ast }))}$). Lai and Robbins^[11][12] designed adaptive procedures to estimate$M^{\prime }({\theta }^{\ast })$ such that${\theta }_n$ 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:$${\displaystyle {\theta }_{n+1}-{\theta }_n=a_n(\alpha -N({\theta }_n)),{\overline{\theta }}_n=\frac{1}{n}\sum _{i=0}^{n-1}{\theta }_i}$$The convergence of${\displaystyle {\overline{\theta }}_n}$ to the unique root${\displaystyle {\theta }^{\ast }}$ relies on the condition that the step sequence${\displaystyle \{a_n\}}$ decreases sufficiently slowly. That is

A1)$${\displaystyle a_n\to 0,\frac{a_n-a_{n+1}}{a_n}=o(a_n)}$$

Therefore, the sequence$a_n=n^{-\alpha }$ with satisfies this restriction, but$\alpha =1$ does not, hence the longer steps. Under the assumptions outlined in the Robbins–Monro algorithm, the resulting modification will result in the same asymptotically optimal convergence rate$O(1/\sqrt{n})$ yet with a more robust step size policy.^[15] Prior to this, the idea of using longer steps and averaging the iterates had already been proposed by Nemirovski and Yudin^[16] for the cases of solving the stochastic optimization problem with continuous convex objectives and for convex-concave saddle point problems. These algorithms were observed to attain the nonasymptotic rate$O(1/\sqrt{n})$.

A more general result is given in Chapter 11 of Kushner and Yin^[17] by defining interpolated time$t_n=\sum _{i=0}^{n-1}{a}_i$, interpolated process${\theta }^n(\cdot )$ and interpolated normalized process$U^n(\cdot )$ as

$${\displaystyle {\theta }^n(t)={\theta }_{n+i},U^n(t)=({\theta }_{n+i}-{\theta }^{\ast })/\sqrt{a_{n+i}}\text{for}t\in [t_{n+i}-t_n,t_{n+i+1}-t_n),i\ge 0}$$Let the iterate average be${\displaystyle {\mathrm{\Theta }}_n=\frac{a_n}{t}\sum _{i=n}^{n+t/a_n-1}{\theta }_i}$ and the associate normalized error to be${\displaystyle {\hat{U}}^n(t)=\frac{\sqrt{a_n}}{t}\sum _{i=n}^{n+t/a_n-1}({\theta }_i-{\theta }^{\ast })}$.

With assumptionA1) and the followingA2)

A2)There is a Hurwitz matrix$A$ and a symmetric and positive-definite matrix$\mathrm{\Sigma }$ such that$\{U^n(\cdot )\}$ converges weakly to$U(\cdot )$, where$U(\cdot )$ is the statisolution to$${\displaystyle dU=AUdt+{\mathrm{\Sigma }}^{1/2}dw}$$where$w(\cdot )$ is a standard Wiener process.

satisfied, and define$\overline{V}=(A^{-1}{)}^{\prime }\mathrm{\Sigma }(A^{\prime }{)}^{-1}$. Then for each$t$,

$${\displaystyle {\hat{U}}^n(t)\stackrel{\mathcal{D}}{⟶}\mathcal{N}(0,V_t),\text{where}{V}_t=\overline{V}/t+O(1/t^2).}$$

The success of the averaging idea is because of the time scale separation of the original sequence$\{{\theta }_n\}$ and the averaged sequence$\{{\mathrm{\Theta }}_n\}$, with the time scale of the former one being faster.

Suppose we want to solve the following stochastic optimization problem$${\displaystyle g({\theta }^{\ast })=\underset{\theta \in \mathrm{\Theta }}{min}\mathrm{E}[Q(\theta ,X)],}$$where$g(\theta )=\mathrm{E}[Q(\theta ,X)]$ is differentiable and convex, then this problem is equivalent to find the root${\displaystyle {\theta }^{\ast }}$ of${\displaystyle \mathrm{\nabla }g(\theta )=0}$. Here${\displaystyle Q(\theta ,X)}$ can be interpreted as some "observed" cost as a function of the chosen${\displaystyle \theta }$ and random effects${\displaystyle X}$. In practice, it might be hard to get an analytical form of${\displaystyle \mathrm{\nabla }g(\theta )}$, Robbins–Monro method manages to generate a sequence${\displaystyle ({\theta }_n{)}_{n\ge 0}}$ to approximate${\displaystyle {\theta }^{\ast }}$ if one can generate${\displaystyle (X_n{)}_{n\ge 0}}$ , in which the conditional expectation of${\displaystyle X_n}$ given${\displaystyle {\theta }_n}$ is exactly${\displaystyle \mathrm{\nabla }g({\theta }_n)}$, i.e.${\displaystyle X_n}$ is simulated from a conditional distribution defined by

$${\displaystyle \mathrm{E}[H(\theta ,X)|\theta ={\theta }_n]=\mathrm{\nabla }g({\theta }_n).}$$

Here${\displaystyle H(\theta ,X)}$ is an unbiased estimator of${\displaystyle \mathrm{\nabla }g(\theta )}$. If${\displaystyle X}$ depends on${\displaystyle \theta }$, there is in general no natural way of generating a random outcome${\displaystyle H(\theta ,X)}$ that is an unbiased estimator of the gradient. In some special cases when either IPA or likelihood ratio methods are applicable, then one is able to obtain an unbiased gradient estimator${\displaystyle H(\theta ,X)}$. If${\displaystyle X}$ is viewed as some "fundamental" underlying random process that is generatedindependently of${\displaystyle \theta }$, and under some regularization conditions for derivative-integral interchange operations so that${\displaystyle \mathrm{E}[\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }Q(\theta ,X)]=\mathrm{\nabla }g(\theta )}$, then${\displaystyle H(\theta ,X)=\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }Q(\theta ,X)}$ gives the fundamental gradient unbiased estimate. However, for some applications we have to use finite-difference methods in which${\displaystyle H(\theta ,X)}$ has a conditional expectation close to${\displaystyle \mathrm{\nabla }g(\theta )}$ but not exactly equal to it.

We then define a recursion analogously toNewton's Method in the deterministic algorithm:

$${\displaystyle {\theta }_{n+1}={\theta }_n-{\epsilon }_{n}H({\theta }_n,X_{n+1}).}$$

The following result gives sufficient conditions on${\displaystyle {\theta }_n}$ for the algorithm to converge:^[18]

C1)${\displaystyle {\epsilon }_n\ge 0,\mathrm{\forall }n\ge 0.}$

C2)${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{\epsilon }_n=\mathrm{\infty }}$

C3)

C4)${\displaystyle |X_n|\le B,\text{ for a fixed bound }B.}$

C5)${\displaystyle g(\theta )\text{ is strictly convex, i.e.}}$

Then${\displaystyle {\theta }_n}$ converges to${\displaystyle {\theta }^{\ast }}$ almost surely.

Here are some intuitive explanations about these conditions. Suppose${\displaystyle H({\theta }_n,X_{n+1})}$ is a uniformly bounded random variables. If C2) is not satisfied, i.e. , then$${\displaystyle {\theta }_n-{\theta }_0=-\sum _{i=0}^{n-1}{\epsilon }_{i}H({\theta }_i,X_{i+1})}$$is a bounded sequence, so the iteration cannot converge to${\displaystyle {\theta }^{\ast }}$ if the initial guess${\displaystyle {\theta }_0}$ is too far away from${\displaystyle {\theta }^{\ast }}$. As for C3) note that if${\displaystyle {\theta }_n}$ converges to${\displaystyle {\theta }^{\ast }}$ then

$${\displaystyle {\theta }_{n+1}-{\theta }_n=-{\epsilon }_{n}H({\theta }_n,X_{n+1})\to 0,\text{ as }n\to \mathrm{\infty }.}$$ so we must have${\displaystyle {\epsilon }_n\downarrow 0}$ ，and the condition C3) ensures it. A natural choice would be${\displaystyle {\epsilon }_n=1/n}$. Condition C5) is a fairly stringent condition on the shape of${\displaystyle g(\theta )}$; it gives the search direction of the algorithm.

Suppose${\displaystyle Q(\theta ,X)=f(\theta )+{\theta }^{T}X}$, where${\displaystyle f}$ is differentiable and${\displaystyle X\in {\mathbb{R}}^p}$ is a random variable independent of${\displaystyle \theta }$. Then${\displaystyle g(\theta )=\mathrm{E}[Q(\theta ,X)]=f(\theta )+{\theta }^T\mathrm{E}X}$ depends on the mean of${\displaystyle X}$, and the stochastic gradient method would be appropriate in this problem. We can choose${\displaystyle H(\theta ,X)=\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }Q(\theta ,X)=\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }f(\theta )+X.}$

The Kiefer–Wolfowitz algorithm was introduced in 1952 byJacob Wolfowitz andJack Kiefer,^[19] and was motivated by the publication of the Robbins–Monro algorithm. However, the algorithm was presented as a method which would stochastically estimate the maximum of a function.

Let${\displaystyle M(x)}$ be a function which has a maximum at the point${\displaystyle \theta }$. It is assumed that${\displaystyle M(x)}$ is unknown; however, certain observations${\displaystyle N(x)}$, where${\displaystyle \mathrm{E}[N(x)]=M(x)}$, can be made at any point${\displaystyle x}$. The structure of the algorithm follows a gradient-like method, with the iterates being generated as

${\displaystyle x_{n+1}=x_n+a_n\cdot \left(\frac{N(x_n+c_n)-N(x_n-c_n)}{2c_n}\right)}$

where${\displaystyle N(x_n+c_n)}$ and${\displaystyle N(x_n-c_n)}$ are independent. At every step, the gradient of${\displaystyle M(x)}$ is approximated akin to acentral difference method with${\displaystyle h=2c_n}$. So the sequence${\displaystyle \{c_n\}}$ specifies the sequence of finite difference widths used for the gradient approximation, while the sequence${\displaystyle \{a_n\}}$ specifies a sequence of positive step sizes taken along that direction.

Kiefer and Wolfowitz proved that, if${\displaystyle M(x)}$ satisfied certain regularity conditions, then${\displaystyle x_n}$ will converge to${\displaystyle \theta }$ in probability as${\displaystyle n\to \mathrm{\infty }}$, and later Blum^[4] in 1954 showed${\displaystyle x_n}$ converges to${\displaystyle \theta }$ almost surely, provided that:

• for all${\displaystyle x}$.
• The function${\displaystyle M(x)}$ has a unique point of maximum (minimum) and is strong concave (convex)
  • The algorithm was first presented with the requirement that the function${\displaystyle M(\cdot )}$ maintains strong global convexity (concavity) over the entire feasible space. Given this condition is too restrictive to impose over the entire domain, Kiefer and Wolfowitz proposed that it is sufficient to impose the condition over a compact set${\displaystyle C_0\subset {\mathbb{R}}^d}$ which is known to include the optimal solution.
• The function${\displaystyle M(x)}$ satisfies the regularity conditions as follows:
  • There exists${\displaystyle \beta >0}$ and${\displaystyle B>0}$ such that
  • There exists${\displaystyle \rho >0}$ and${\displaystyle R>0}$ such that
  • For every${\displaystyle \delta >0}$, there exists some${\displaystyle \pi (\delta )>0}$ such that$${\displaystyle |z-\theta |>\delta ⟹\underset{\delta /2>\epsilon >0}{inf}\frac{|M(z+\epsilon )-M(z-\epsilon )|}{\epsilon }>\pi (\delta )}$$
• The selected sequences${\displaystyle \{a_n\}}$ and${\displaystyle \{c_n\}}$ must be infinite sequences of positive numbers such that
  • ${\displaystyle c_n\to 0\text{as}n\to \mathrm{\infty }}$
  • ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{a}_n=\mathrm{\infty }}$
  • 
  • 

A suitable choice of sequences, as recommended by Kiefer and Wolfowitz, would be${\displaystyle a_n=1/n}$ and${\displaystyle c_n=n^{-1/3}}$.

1. The Kiefer Wolfowitz algorithm requires that for each gradient computation, at least${\displaystyle d+1}$ different parameter values must be simulated for every iteration of the algorithm, where${\displaystyle d}$ is the dimension of the search space. This means that when${\displaystyle d}$ is large, the Kiefer–Wolfowitz algorithm will require substantial computational effort per iteration, leading to slow convergence.
   1. To address this problem, Spall proposed the use ofsimultaneous perturbations to estimate the gradient. This method would require only two simulations per iteration, regardless of the dimension${\displaystyle d}$.^[20]
2. In the conditions required for convergence, the ability to specify a predetermined compact set that fulfills strong convexity (or concavity) and contains the unique solution can be difficult to find. With respect to real world applications, if the domain is quite large, these assumptions can be fairly restrictive and highly unrealistic.

An extensive theoretical literature has grown up around these algorithms, concerning conditions for convergence, rates of convergence, multivariate and other generalizations, proper choice of step size, possible noise models, and so on.^[21][22] These methods are also applied incontrol theory, in which case the unknown function which we wish to optimize or find the zero of may vary in time. In this case, the step size${\displaystyle a_n}$ should not converge to zero but should be chosen so as to track the function.^{[21], 2nd ed., chapter 3}

C. Johan Masreliez andR. Douglas Martin were the first to apply stochastic approximation torobustestimation.^[23]

The main tool for analyzing stochastic approximations algorithms (including the Robbins–Monro and the Kiefer–Wolfowitz algorithms) is a theorem byAryeh Dvoretzky published in 1956.^[24]

• Stochastic gradient descent
• Stochastic variance reduction

• Stochastic optimization
• Statistical approximations

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