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{{Short description|Family of iterative methods}}
'''Stochastic approximation''' methods are a family of [[iterative methods]] typically used for [[root-finding]] problems or for [[optimization problem|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 [[Extremum|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 <math display="inline"> f(\theta) = \operatorname E_{\xi} [F(\theta,\xi)] </math>
which is the [[expected value]] of a function depending on a [[random variable]] <math display="inline">\xi </math>. The goal is to recover properties of such a function <math display="inline">f</math> without evaluating it directly. Instead, stochastic approximation algorithms use random samples of <math display="inline">F(\theta,\xi)</math> to efficiently approximate properties of <math display="inline">f</math> 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 [[Expectation–maximization algorithm| EM algorithm]], reinforcement learning via [[Temporal difference learning|temporal differences]], and [[deep learning]], and others.<ref name=":1">{{cite journal |last1=Toulis |first1=Panos |first2=Edoardo |last2=Airoldi|title=Scalable estimation strategies based on stochastic approximations: classical results and new insights |journal=Statistics and Computing |volume=25 |issue=4 |year=2015 |pages=781–795|doi=10.1007/s11222-015-9560-y|pmid=26139959 |pmc=4484776 }}</ref>
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==Robbins–Monro algorithm==
The Robbins–Monro algorithm, introduced in 1951 by [[Herbert Robbins]] and [[John U. Monro#Personal life|Sutton Monro]],<ref name="rm">{{Cite journal | last1 = Robbins | first1 = H. | author-link = Herbert Robbins| last2 = Monro | first2 = S. | doi = 10.1214/aoms/1177729586 | title = A Stochastic Approximation Method | journal = The Annals of Mathematical Statistics | volume = 22 | issue = 3 | pages = 400 | year = 1951 | doi-access = free }}</ref> presented a methodology for solving a root finding problem, where the function is represented as an expected value. Assume that we have a function <math display="inline">M(\theta)</math>, and a constant <math display="inline">\alpha</math>, such that the equation <math display="inline">M(\theta) = \alpha</math> has a unique root at <math display="inline">\theta^*.</math>
Here, <math>a_1, a_2, \dots</math> is a sequence of positive step sizes. [[Herbert Robbins|Robbins]] and Monro proved<ref name="rm" /><sup>, Theorem 2</sup> that <math>\theta_n</math> [[convergence of random variables|converges]] in <math>L^2</math> (and hence also in probability) to <math>\theta^*</math>, and Blum<ref name=":0">{{Cite journal|last=Blum|first=Julius R.|date=1954-06-01|title=Approximation Methods which Converge with Probability one|journal=The Annals of Mathematical Statistics|language=EN|volume=25|issue=2|pages=382–386|doi=10.1214/aoms/1177728794|issn=0003-4851|doi-access=free}}</ref> later proved the convergence is actually with probability one, provided that:
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* <math display="inline">M'(\theta^*)</math> exists and is positive, and
* The sequence <math display="inline">a_n</math> satisfies the following requirements:
<math display="block">\qquad \sum^{\infty}_{n=0}a_n = \infty \quad \mbox{ and } \quad \sum^{\infty}_{n=0}a^2_n < \infty \quad </math>
A particular sequence of steps which satisfy these conditions, and was suggested by Robbins–Monro, have the form: <math display="inline">a_n=a/n</math>, for <math display="inline"> a > 0 </math>. Other series, such as <math>a_n = \frac{1}{n \ln n}, \frac{1}{n \ln n \ln\ln n}, \dots</math> are possible but in order to average out the noise in <math display="inline">N(\theta)</math>, the above condition must be met. === Example ===
Consider the problem of estimating the mean <math>\theta^*</math> of a probability distribution from a stream of independent samples <math>X_1, X_2, \dots</math>.
Let <math>N(\theta) := \theta - X</math>, then the unique solution to <math display="inline">\operatorname E[N(\theta)] = 0</math> is the desired mean <math>\theta^*</math>. The RM algorithm gives us<math display="block">\theta_{n+1}=\theta_n - a_n(\theta_n - X_n) </math>This is equivalent to [[stochastic gradient descent]] with loss function <math>L(\theta) = \frac 12 \|X - \theta\|^2 </math>. It is also equivalent to a weighted average:<math display="block">\theta_{n+1}=(1-a_n)\theta_n + a_n X_n </math>In general, if there exists some function <math>L</math> such that <math>\nabla L(\theta) = N(\theta) - \alpha </math>, then the Robbins–Monro algorithm is equivalent to stochastic gradient descent with loss function <math>L(\theta)</math>. However, the RM algorithm does not require <math>L</math> to exist in order to converge.
===Complexity results===
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===Subsequent developments and Polyak–Ruppert averaging===
While the Robbins–Monro algorithm is theoretically able to achieve <math display="inline"> O(1/n)</math> 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.<ref name="NJLS" /><ref name="jcsbook">[https://books.google.com/books?id=f66OIvvkKnAC
Chung (1954)<ref>{{Cite journal|last=Chung|first=K. L.|date=1954-09-01|title=On a Stochastic Approximation Method|journal=The Annals of Mathematical Statistics|language=EN|volume=25|issue=3|pages=463–483|doi=10.1214/aoms/1177728716|issn=0003-4851|doi-access=free}}</ref>
'''''A1)'''''
Therefore, the sequence <math display="inline">a_n = n^{-\alpha}</math> with <math display="inline">0 < \alpha < 1</math> satisfies this restriction, but <math display="inline">\alpha = 1</math> 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 <math display="inline">O(1/\sqrt{n})</math> yet with a more robust step size policy.<ref name="pj" /> Prior to this, the idea of using longer steps and averaging the iterates had already been proposed by Nemirovski and Yudin<ref name="NY">On Cezari's convergence of the steepest descent method for approximating saddle points of convex-concave functions, A. Nemirovski and D. Yudin, ''Dokl. Akad. Nauk SSR'' '''2939''', (1978 (Russian)), Soviet Math. Dokl. '''19''' (1978 (English)).</ref> 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 <math display="inline">O(1/\sqrt{n})</math>.
A more general result is given in Chapter 11 of Kushner and Yin<ref>{{Cite book|url=https://www.springer.com/us/book/9780387008943|title=Stochastic Approximation and Recursive Algorithms and {{!}} Harold Kushner {{!}} Springer|
<math display="block">\theta^n(t)=\theta_{n+i},\quad U^n(t)=(\theta_{n+i}-\theta^*)/\sqrt{a_{n+i}}\quad\mbox{for}\quad t\in[t_{n+i}-t_n,t_{n+i+1}-t_n),i\ge0</math>Let the iterate average be <math>\Theta_n=\frac{a_n}{t}\sum_{i=n}^{n+t/a_n-1}\theta_i</math> and the associate normalized error to be <math>\hat{U}^n(t)=\frac{\sqrt{a_n}}{t}\sum_{i=n}^{n+t/a_n-1}(\theta_i-\theta^*)</math>.
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With assumption '''A1)''' and the following '''A2)'''
'''''A2)''''' ''There is a Hurwitz matrix <math display="inline">A</math> and a symmetric and positive-definite matrix <math display="inline">\Sigma</math> such that <math display="inline">\{U^n(\cdot)\}</math> converges weakly to <math display="inline">U(\cdot)</math>, where <math display="inline">U(\cdot)</math> is the statisolution to
satisfied, and define ''<math display="inline">\bar{V}=(A^{-1})'\Sigma(A')^{-1}</math>''. Then for each ''<math display="inline">t</math>'',
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=== Application in stochastic optimization ===
Suppose we want to solve the following stochastic optimization problem<math display="block">g(\theta^*) = \min_{\theta\in\Theta}\operatorname{E}[Q(\theta,X)],</math>where <math display="inline">g(\theta) = \operatorname{E}[Q(\theta,X)]</math> is differentiable and convex, then this problem is equivalent to find the root <math>\theta^*</math> of <math>\nabla g(\theta) = 0</math>. Here <math>Q(\theta,X)</math> can be interpreted as some "observed" cost as a function of the chosen <math>\theta</math> and random effects <math>X</math>. In practice, it might be hard to get an analytical form of <math>\nabla g(\theta)</math>, Robbins–Monro method manages to generate a sequence <math>(\theta_n)_{n\geq 0}</math> to approximate <math>\theta^*</math> if one can generate <math>(X_n)_{n\geq 0}▼
▲<math display="block">g(\theta^*) = \min_{\theta\in\Theta}\operatorname{E}[Q(\theta,X)],</math>where <math display="inline">g(\theta) = \operatorname{E}[Q(\theta,X)]</math> is differentiable and convex, then this problem is equivalent to find the root <math>\theta^*</math> of <math>\nabla g(\theta) = 0</math>. Here <math>Q(\theta,X)</math> can be interpreted as some "observed" cost as a function of the chosen <math>\theta</math> and random effects <math>X</math>. In practice, it might be hard to get an analytical form of <math>\nabla g(\theta)</math>, Robbins–Monro method manages to generate a sequence <math>(\theta_n)_{n\geq 0}</math> to approximate <math>\theta^*</math> if one can generate <math>(X_n)_{n\geq 0}
</math> , in which the conditional expectation of <math>X_n
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Here <math>H(\theta, X)</math> is an unbiased estimator of <math>\nabla g(\theta)</math>. If <math>X</math> depends on <math>\theta</math>, there is in general no natural way of generating a random outcome <math>H(\theta, X)</math> 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 <math>H(\theta, X)</math>. If <math>X</math> is viewed as some "fundamental" underlying random process that is generated ''independently'' of <math>\theta</math>, and under some regularization conditions for derivative-integral interchange operations so that <math>\operatorname{E}\Big[\frac{\partial}{\partial\theta}Q(\theta,X)\Big] = \nabla g(\theta)</math>, then <math>H(\theta, X) = \frac{\partial}{\partial \theta}Q(\theta, X)</math> gives the fundamental gradient unbiased estimate. However, for some applications we have to use finite-difference methods in which <math>H(\theta, X)</math> has a conditional expectation close to <math>\nabla g(\theta)</math> but not exactly equal to it.
We then define a recursion analogously to [[Newton's Method]] in the deterministic algorithm:
: <math display="block">\theta_{n+1} = \theta_n - \varepsilon_n H(\theta_n,X_{n+1}).</math>
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The following result gives sufficient conditions on <math>\theta_n
</math> for the algorithm to converge:<ref>{{Cite book|title=Numerical Methods for stochastic Processes|last1=Bouleau|first1=N.|author-link=Nicolas Bouleau|last2=Lepingle|first2=D.|publisher=John Wiley|year=1994|isbn=9780471546412|___location=New York|url=https://books.google.com/books?id=9MLL2RN40asC}}</ref>
C1) <math>\varepsilon_n \geq 0, \forall\; n\geq 0. </math>
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==Kiefer–Wolfowitz algorithm==
The Kiefer–Wolfowitz algorithm was introduced in 1952 by [[Jacob Wolfowitz]] and [[Jack_Kiefer_(statistician)|Jack Kiefer]],<ref name = "KW">{{Cite journal | last1 = Kiefer | first1 = J. | last2 = Wolfowitz | first2 = J. | doi = 10.1214/aoms/1177729392 | title = Stochastic Estimation of the Maximum of a Regression Function | journal = The Annals of Mathematical Statistics | volume = 23 | issue = 3 | pages = 462 | year = 1952 | doi-access = free }}</ref> 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 <math>M(x) </math> be a function which has a maximum at the point <math>\theta </math>. It is assumed that <math>M(x)</math> is unknown; however, certain observations <math>N(x)</math>, where <math>\operatorname E[N(x)] = M(x)</math>, can be made at any point <math>x</math>. The structure of the algorithm follows a gradient-like method, with the iterates being generated as ::<math> x_{n+1} = x_n + a_n
where <math>N(x_n+c_n)</math> and <math>N(x_n-c_n)</math> are independent
Kiefer and Wolfowitz proved that, if <math>M(x)</math> satisfied certain regularity conditions, then <math>x_n</math> will converge to <math>\theta</math> in probability as <math>n\to\infty </math>, and later Blum<ref name=":0" /> in 1954 showed <math>x_n</math> converges to <math>\theta</math> almost surely, provided that:
* <math>\operatorname{Var}(N(x))\le S<\infty</math> for all <math>x</math>.
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stochastic approximation to [[Robust statistics|robust]] [[estimation]].<ref>{{Cite journal | last1 = Martin | first1 = R. | last2 = Masreliez | first2 = C. | doi = 10.1109/TIT.1975.1055386 | title = Robust estimation via stochastic approximation | journal = IEEE Transactions on Information Theory | volume = 21 | issue = 3 | pages = 263 | year = 1975 }}</ref>
The main tool for analyzing stochastic approximations algorithms (including the Robbins–Monro and the Kiefer–Wolfowitz algorithms) is a theorem by [[Aryeh Dvoretzky]] published in
| last = Dvoretzky | first = Aryeh | author-link = Aryeh Dvoretzky
| editor-last = Neyman | editor-first = Jerzy | editor-link = Jerzy Neyman
| contribution = On stochastic approximation
| contribution-url = https://projecteuclid.org/euclid.bsmsp/1200501645
| mr = 84911
| pages = 39–55
| publisher = University of California Press
| title = Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. I
| year = 1956}}</ref>
==See also==
*[[Stochastic gradient descent]]
*[[Stochastic variance reduction]]
==References==
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