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The '''winnow algorithm'''<ref name="littlestone88">Littlestone, N.Nick Littlestone (1988). "Learning Quickly When Irrelevant Attributes Abound: A New Linear-threshold Algorithm", [https://doi.org/10.1023%2FA%3A1022869011914 ''Machine Learning'' 285-318285–318(2)].</ref> is a technique from [[machine learning]] for learning a [[linear classifier]] from labeled examples. It is very similar to the [[perceptron|perceptron algorithm]]. However, the perceptron algorithm uses an additive weight-update scheme, butwhile winnowWinnow uses a multiplicative[[Multiplicative weight-updateWeight Update Method|multiplicative scheme]] that allows it to perform much better when many dimensions are irrelevant (hence its name [[winnowing|winnow]]). It is not a sophisticatedsimple algorithm but itthat scales well to high-dimensional spacesdata. During training, winnowWinnow is shown a sequence of positive and negative examples. From these it learns a decision [[hyperplane]] that can then be used to label novel examples as positive or negative. It The algorithm can also be used in the [[Online machine learning|online learning]] setting, where the learning and the classification phase are not clearly seperatedseparated.
==The algorithmAlgorithm==
The basic algorithm, Winnow1, is as follows. The instance space is <math>X=\{0,1\}^n</math> , that is, each instance is described as a set of [[Boolean-valued]] [[features (pattern recognition)|features]]. The algorithm maintains non-negative weights <math>w_i</math> for <math>i\in \{1 ...,\ldots,n\}</math> , which are initially set to 1 , one weight for each feature. When the learner is given an example <math>(x_1, ...\ldots,x_n)</math>, theit learner followsapplies the followingtypical prediction rule for linear classifiers: ▼
The basic algorithm, Winnow1, is given as follows.
▲The instance space is <math>X=\{0,1\}^n</math>. The algorithm maintains non-negative weights <math>w_i</math> for <math>i\in \{1...n\}</math> which are initially set to 1. When the learner is given an example <math>(x_1,...x_n)</math>, the learner follows the following prediction rule:
* '''If''' <math>\sum_{i=1}^n w_i x_i > \Theta </math>, '''then''' it predictspredict 1
* '''Otherwise''' it predictspredict 0
WhereHere <math>\Theta</math> is a real number that is called the ''threshold''. Together with the weights, the threshold defines a dividing hyperplane in the instance space. Good bounds are obtained if <math>\Theta=n/2</math> (see below).
For each example with which it is presented, the learner applies the following update rule:
The update rule is (loosely):
* If an example is correctly classified, do nothing.
* If an example is predicted to be 1incorrectly butand the correct result was 0, allfor ofeach thefeature weights<math>x_{i}=1</math>, involvedthe incorresponding theweight mistake<math>w_{i}</math> areis set to zero0 (demotion step).
*: <math>\forall x_{i} = 1, w_{i} = 0</math>
* If an example is predicted toincorrectly be 0 butand the correct result was 1, allfor ofeach thefeature weights involved in<math>x_{i}=1</math>, the mistakecorresponding are multiplied byweight <math>\alphaw_{i}</math> multiplied by {{mvar|α}}(promotion step).
*: <math>\forall x_{i} = 1, w_{i} = \alpha w_{i}</math>
A goodtypical value for <math>\{{mvar|&alpha</math>;}} is 2.
VariationsThere are alsomany used.variations to Forthis example,basic Winnow2approach. is the''Winnow2''<ref samename="littlestone88"/> asis Winnow1similar except that in the demotion step the weights are divided by <math>\{{mvar|&alpha</math>;}} instead of being set to 0. ''Balanced Winnow'' maintains two sets of weights, and thus two hyperplanes. This can then be generalized for [[multi-label classification]].
==Mistake bounds==
IfIn Winnow1certain circumstances, it can be shown that the number of mistakes Winnow makes as it learns has an [[Upper and lower bounds|upper bound]] that is runindependent of the number of instances with which it is presented. If the Winnow1 algorithm uses <math>\alpha > 1</math> and <math>\Theta \geq 1/\alpha</math> on a target function that is a <math>k</math>-literal monotone disjunction given by <math>f(x_1,...\ldots,x_n)=x_{i_1}\cup ...\cdots \cup x_{i_k}</math>, then for any sequence of instances the total number of mistakes is bounded by <math>\alpha k ( \log_\alpha \Theta+1)+\frac{n}{\Theta}</math>.:
<math>\alpha k ( \log_\alpha \Theta+1)+\frac{n}{\Theta}</math>.<ref>
Nick Littlestone (1989). "Mistake bounds and logarithmic linear-threshold learning algorithms".
Technical report UCSC-CRL-89-11, University of California, Santa Cruz.</ref>
==References==
===Citations and notes===
<references/>
[[Category:Classification algorithms]]
[[Category:Machine learning]]
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