en>MusikAnimal: Reverted edits by 2601:A:F80:3BE:75FD:1A60:99B:3E43 (talk) to last revision by Wcherowi (HG)

2014-02-03T21:30:12Z

Reverted edits by 2601:A:F80:3BE:75FD:1A60:99B:3E43 (talk) to last revision by Wcherowi (HG)

← Older revision		Revision as of 23:30, 3 February 2014
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	~~My name~~ is ~~Elise~~ (~~47 years old~~) and ~~my hobbies are Association football~~ and ~~Element collecting~~.<br><br>~~My blog post :: home makeover ideas (~~[http://www.~~homeimprovementdaily~~.com ~~just click~~ the ~~up coming document~~])		'''Adaptive coordinate descent''' is an extension of [[Coordinate descent]] [[algorithm]] to non-separable [[Mathematical optimization\|optimization]]. Adaptive coordinate descent approach gradually builds a transformation of the coordinate system such that the new coordinates are as decorrelated as possible with respect to the objective function. The adaptive coordinate descent was shown to be competitive to the state-of-the-art [[evolutionary algorithms]] and has the following invariance properties:<ref>{{cite conference
			\| first = I.
			\| last = Loshchilov
			\| coauthors = M. Schoenauer and M. Sebag
			\| title = Adaptive Coordinate Descent
			\| booktitle = Genetic and Evolutionary Computation Conference (GECCO)
			\| pages = 885--892
			\| publisher = ACM Press
			\| date = 2011
			\| location =
			\| url = http://www.loshchilov.com/publications/GECCO2011_AdaptiveCoordinateDescent.pdf
			\| accessdate =
			\| id =
			}}</ref>
			# Invariance with respect to monotonous transformations of the function (scaling)
			# Invariance with respect to [[orthogonal transform]]ations of the search space (rotation).

			[[CMA-ES\|CMA]]-like Adaptive Encoding Update (b) mostly based on [[Principal Component Analysis]] (a) is used to extend Coordinate Descent method (c) to the optimization of non-separable problems (d).

			[[File:Adaptive Coordinate Descent illustration.png\|center\|x400px]]

			The adaptation of an appropriate coordinate system allows Adaptive Coordinate Descent to outperform Coordinate Descent on non-separable functions. The following figure illustrates the convergence of both algorithms on 2-dimensional Rosenbrock function up to a target function value <math>10^{-10}</math>, starting from the initial point <math>x_0=(-3,-4)</math>.

			[[File:Rosenbrock2D.png\|center\|x400px]]

			The Adaptive Coordinate Descent reaches the target value after only 325 function evaluations (about 70 times faster than Coordinate Descent), that is comparable to [[Gradient descent\|gradient-based methods]]. The algorithm has linear time complexity if update coordinate system every D iterations, it is also suitable for large-scale (D>>100) non-linear optimization.

			==Relevant Approaches==

			First approaches to optimization using adaptive coordinate system were proposed already in the 1960s (see, e.g., [[Rosenbrock methods\|Rosenbrock's method]]). PRincipal Axis (PRAXIS) algorithm, also referred to as Brent's algorithm, is an derivative-free algorithm which assumes quadratic form of the optimized function and repeatedly updates a set of conjugate search directions.<ref>
			{{cite conference
			\| first = R.P.
			\| last = Brent
			\| booktitle = Algorithms for minimization without derivatives
			\| publisher = Prentice-Hall
			\| date = 1972
			}}</ref>
			The algorithm, however, is not invariant to scaling of the objective function and may fail under its certain rank-preserving transformations (e.g., will lead to a non-quadratic shape of the objective function). A recent analysis of PRAXIS can be found in
			.<ref>
			{{cite conference
			\| first = U.
			\| last = Ali
			\| coauthors = Kickmeier-Rust, M.D.
			\| title = Implementation and Applications of a Three-Round User Strategy for Improved Principal Axis Minimization
			\| booktitle = Journal of Applied Quantitative Methods
			\| pages = 505--513
			\| date = 2008
			}}</ref>
			For practical applications see,<ref>
			{{cite conference
			\| first = D.
			\| last = Pavlov
			\| title = Manipulator path planning in 3-dimensional space
			\| booktitle = Computer Science--Theory and Applications
			\| pages = 505--513
			\| publisher = Springer
			\| date = 2006
			}}</ref> where an adaptive coordinate descent approach with step-size adaptation and local coordinate system rotation was proposed
			for robot-manipulator path planning in 3D space with static polygonal obstacles.

			==See also==
			* [[Coordinate descent]]
			* [[CMA-ES]]
			* [[Rosenbrock methods]]
			* [[Mathematical optimization]]

			==References==
			{{reflist}}

			*
			*
			*
			*

			== External links ==
			* [http://www.loshchilov.com/acid.html SOURCE CODE ACD] ACD is a MATLAB source code for Adaptive Coordinate Descent

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			[[Category:Optimization algorithms and methods]]

en>David Eppstein: equidissection

2012-08-29T17:10:08Z

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My name is Elise (47 years old) and my hobbies are Association football and Element collecting.<br><br>My blog post :: home makeover ideas ([http://www.homeimprovementdaily.com just click the up coming document])

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en>MusikAnimal: Reverted edits by 2601:A:F80:3BE:75FD:1A60:99B:3E43 (talk) to last revision by Wcherowi (HG)

en>David Eppstein: equidissection