Winnow (algorithm): Difference between revisions

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The '''winnow algorithm'''<ref name="littlestone88"> Nick Littlestone (1988). "Learning Quickly When Irrelevant Attributes Abound: A New Linear-threshold Algorithm", ''[httphttps://wwwdoi.springerlinkorg/10.com/content/j0k7t38567325716/1023%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, while WInnowWinnow uses a [[Multiplicative Weight 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, Winnow 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. The algorithm can also be used in the [[Online machine learning|online learning]] setting, where the learning and the classification phase are not clearly separated.

The '''winnow algorithm'''<ref name="littlestone88">

Nick Littlestone (1988). "Learning Quickly When Irrelevant Attributes Abound: A New Linear-threshold Algorithm", ''[http://www.springerlink.com/content/j0k7t38567325716/ Machine Learning 285-318(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, while WInnow uses a multiplicative scheme that allows it to perform much better when many dimensions are irrelevant (hence its name). It is not a sophisticated algorithm but it scales well to high-dimensional spaces. During training, Winnow 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. The algorithm can also be used in the [[Online machine learning|online learning]] setting, where the learning and the classification phase are not clearly separated.

==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>, it applies the typical prediction rule for linear classifiers:

The basic algorithm, Winnow1, is given 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...n\}</math>, which are initially set to 1, one weight for each feature. When the learner is given an example <math>(x_1,...x_n)</math>, it applies the typical prediction rule for linear classifiers:

* '''If''' <math>\sum_{i=1}^n w_i x_i > \Theta </math>, '''then''' predict 1

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Here <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 applesapplies the following update rule:

* 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>, implicatedthe 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 implicated in<math>x_{i}=1</math>, the mistakecorresponding are multiplied byweight <math>\alphaw_{i}</math> multiplied by {{mvar|&alpha;}}(promotion step).

*: <math>\forall x_{i} = 1, w_{i} = \alpha w_{i}</math>

Here, "implicated" means weights on features of the instance that have value 1. A typical value for <math>\{{mvar|&alpha</math>;}} is 2.

There are many variations to this basic approach. ''Winnow2''<ref name="littlestone88"/> is similar 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==

In certain 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 independent 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>.<ref>

Nick Littlestone (1989). "Mistake bounds and logarithmic linear-threshold learning algorithms".

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[[Category:Classification algorithms]]

{{Algorithm-stub}}