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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク simultaneous perturbation stochastic approximation ： ウィキペディア英語版 simultaneous perturbation stochastic approximation Simultaneous perturbation stochastic approximation (SPSA) is an algorithmic method for optimizing systems with multiple unknown parameters. It is a type of stochastic approximation algorithm. As an optimization method, it is appropriately suited to large-scale population models, adaptive modeling, simulation optimization, and atmospheric modeling. Many examples are presented at the SPSA website http://www.jhuapl.edu/SPSA. A comprehensive recent book on the subject is Bhatnagar et al. (2013). An early paper on the subject is Spall (1987) and the foundational paper providing the key theory and justification is Spall (1992). SPSA is a descent method capable of finding global minima, sharing this property with other methods as simulated annealing. Its main feature is the gradient approximation that requires only two measurements of the objective function, regardless of the dimension of the optimization problem. Recall that we want to find the optimal control $u^$ * with loss function $J(u)$: :$u^$ * = \arg \min_ J(u). Both Finite Differences Stochastic Approximation (FDSA) and SPSA use the same iterative process: :$u\_\; =\; u\_n\; -\; a\_n\backslash hat\_n(u\_n),$ where $u\_n=((u\_n)\_1,(u\_n)\_2,\backslash ldots,(u\_n)\_p)^T$ represents the $n^$ iterate, $\backslash hat\_n(u\_n)$ is the estimate of the gradient of the objective function $g(u)=\; \backslash fracJ(u)$ evaluated at $$, and $\backslash$ is a positive number sequence converging to 0. If $u\_n$ is a ''p''-dimensional vector, the $i^$ component of the symmetric finite difference gradient estimator is: :FD: $(\backslash hat(u\_n))\_i\; =\; \backslash frac,$ ''1 ≤i ≤p'', where $e\_i$ is the unit vector with a 1 in the $i^$ place, and $c\_n$is a small positive number that decreases with ''n''. With this method, ''2p'' evaluations of ''J'' for each $g\_n$ are needed. Clearly, when ''p'' is large, this estimator loses efficiency. Let now $\backslash Delta\_n$ be a random perturbation vector. The $i^$ component of the stochastic perturbation gradient estimator is: :SP: $(\backslash hat(u\_n))\_i\; =\; \backslash frac.$ Remark that FD perturbs only one direction at a time, while the SP estimator disturbs all directions at the same time (the numerator is identical in all ''p'' components). The number of loss function measurements needed in the SPSA method for each $g\_n$ is always 2, independent of the dimension ''p''. Thus, SPSA uses ''p'' times fewer function evaluations than FDSA, which makes it a lot more efficient. Simple experiments with ''p=2'' showed that SPSA converges in the same number of iterations as FDSA. The latter follows approximately the steepest descent direction, behaving like the gradient method. On the other hand, SPSA, with the random search direction, does not follow exactly the gradient path. In average though, it tracks it nearly because the gradient approximation is an almost unbiased estimator of the gradient, as shown in the following lemma. == Convergence lemma == Denote by :$b\_n\; =\; E()\; -\backslash nabla\; J(u\_n)$ the bias in the estimator $\backslash hat\_n$. Assume that $\backslash$ are all mutually independent with zero-mean, bounded second moments, and $E(|(\backslash Delta\_n)\_i|^)$ uniformly bounded. Then $b\_n$→0 w.p. 1. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「simultaneous perturbation stochastic approximation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.