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The '''cross-entropy''' ('''CE''') '''method''' developedis bya [[ReuvenMonte RubinsteinCarlo method|Monte Carlo]] ismethod afor general[[importance sampling]] and [[MonteOptimization Carlo(mathematics)|optimization]]. methodIt is applicable to both [[Combinatorial optimization|Montecombinatorial]] Carloand [[Continuous optimization|continuous]] approachproblems, towith either a static or noisy objective.
[[Combinatorial optimization|combinatorial]] and [[Continuous optimization|continuous]] multi-extremal [[Optimization (mathematics)|optimization]] and [[importance sampling]].
The method originated from the field of ''rare event simulation'', where
very small probabilities need to be accurately estimated, for example in network reliability analysis, queueing models, or performance analysis of telecommunication systems.
The CE method can be applied to static and noisy combinatorial optimization problems such as the [[traveling salesman problem]], the [[quadratic assignment problem]], [[Sequence alignment|DNA sequence alignment]], the [[Maxcut|max-cut]] problem and the buffer allocation problem, as well as continuous [[global optimization]] problems with many local [[extremum|extrema]].
InThe amethod nutshell,approximates approximates the CEoptimal methodimportance sampling estimator consistsby ofrepeating two phases <ref> Rubinstein, R.Y. and Kroese, D.P. (2004), The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation, and Machine Learning, Springer-Verlag, New York {{ISBN|978-0-387-21240-1}}.</ref>:
#Draw a sample from a probability distribution.
#Minimize the [[cross entropy|''cross-entropy'']] between this distribution and a target distribution to produce a better sample in the next iteration.
[[Reuven Rubinstein]] developed the method in the context of ''rare event simulation'', where tiny probabilities must be estimated, for example in network reliability analysis, queueing models, or performance analysis of telecommunication systems. The method has also been applied to the [[traveling salesman problem|traveling salesman]], [[quadratic assignment problem|quadratic assignment]], [[Sequence alignment|DNA sequence alignment]], [[Maxcut|max-cut]] and buffer allocation problems.
#Generate a random data sample (trajectories, vectors, etc.) according to a specified mechanism.
#Update the parameters of the random mechanism based on the data to produce a "better" sample in the next iteration. This step involves minimizing the [[cross entropy|''cross-entropy'']] or [[Kullback–Leibler divergence]].
==Estimation via importance sampling==
Consider the general problem of estimating the quantity
<math>\hat{\ell} = \frac{1}{N} \sum_{i=1}^N H(\mathbf{X}_i) \frac{f(\mathbf{X}_i; \mathbf{u})}{g(\mathbf{X}_i)}</math>,
where <math>\mathbf{X}_1,\dots,\mathbf{X}_N</math> is a random sample from <math>g\,</math>. For positive <math>H</math>, the theoretically ''optimal'' importance sampling [[probability density function|density]] (pdfPDF) is given by
<math> g^*(\mathbf{x}) = H(\mathbf{x}) f(\mathbf{x};\mathbf{u})/\ell</math>.
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