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What is The Optimal Number of Features? A learning theoretic Perspective

What is The Optimal Number of Features? A learning theoretic Perspective. Amir Navot Joint work with Ran Gilad-Bachrach, Yiftah Navot and Naftali Tishby. What is Feature Selection?. Feature selection: select a “good” small subset of features (out of the given set)

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What is The Optimal Number of Features? A learning theoretic Perspective

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  1. What is The Optimal Number of Features?A learning theoretic Perspective Amir Navot Joint work with Ran Gilad-Bachrach, Yiftah Navot and Naftali Tishby

  2. What is Feature Selection? • Feature selection: select a “good” small subset of features (out of the given set) • “Good” subset: enables to build good classifiers • Feature selection is a special form of dimensionality reduction

  3. Reasons to do Feature Selection • Reduces computational complexity • Saves the cost of measuring extra features • The selected features can provide insights about the nature of the problem • Improves accuracy • Improves accuracy

  4. The Questions • Under which conditions can feature selection improve classification accuracy? • What is the optimal number of features? • How does this number depend on the training set size? We discuss these questions by analyzing one simple setting

  5. Two Gaussians - Problem Setting • Binary classification task • The coordinates are the features • Optimal classifier: • given features subset F : • If is known: using all the features is optimal

  6. Problem Setting – Cont. • Assume that is estimated from a sample of size m, • Given an estimator and a features subset F , we consider the classifier • We want to find a subset of features that minimizes the average generalization error: • We assume ) we only need to find the optimal number of features • We consider the optimal estimator:

  7. Illustration  -

  8. Result • The number of features that minimizes the average error for a training set of size m is: • Observations: • If , there is a non trivial optimal n • When then is optimal choice • Decision on adding depends on other features

  9. Solving for Specific 

  10. Solving for Specific  -Cont. x-axis: number of features

  11. Proof • For a given estimator for , the generalization error of the classifier is: ( is the CDF function of a standard Gaussian )

  12. Proof – Cont. • We want to find the number of features nthat minimizes the average error: • Lemma: is a good approximation for when m is large enough.

  13. Proof – Cont. • Therefore, minimizing is equivalent to maximizing • For the optimal estimator , and Therefore,

  14. “Empirical Proof” of the Lemma x-axis: number of features

  15. Linear SVM Error(averaged on 200 repeats, c=0.01, using Gavin Cawley’s tool box) x-axis: number of features

  16. Conclusions • Even when all the features carry information and are independent, • Using all the features may be suboptimal • The decision to add a feature depends on the others • The optimal number of features depends critically on the sample size ( the quality of model estimation)

  17. What is The Optimal Number of Features?A learning theoretic Perspective Amir Navot Joint work with Ran Gilad-Bachrach, Yiftah Navot and Naftali Tishby

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