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Entropy Estimation and Applications to Decision Trees

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Estimation Distribution over K=8 classes Repeat 50,000 times: 1.Generate N samples 2.Estimate entropy from samples N=10 N=100 N=50000 H=1.289

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Estimation Estimating the true entropy Goals: 1. Consistency: large N guarantees correct result 2. Low variance: variation of estimates small 3. Low bias: expected estimate should be correct

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Discrete Entropy Estimators

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UCI classification data sets Accuracy on test set Plugin vs. Grassberger Better trees Experimental Results Source: [Nowozin, “Improved Information Gain Estimates for Decision Tree Induction”, ICML 2012]

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In regression, differential entropy – measures remaining uncertainty about y – is a function of a distribution Differential Entropy Estimation Problem – q is not from a parametric family Solution 1: project onto a parametric family Solution 2: non-parametric entropy estimation

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Solution 1: parametric family – Uniform minimum variance unbiased estimator (UMVUE) [Ahmed, Gokhale, “Entropy expressions and their estimators for multivariate distributions”, IEEE Trans. Inf. Theory, 1989]

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Solution 1: parametric family

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Solution 2: Non-parametric entropy estimation [Kozachenko, Leonenko, “Sample estimate of the entropy of a random vector”, Probl. Peredachi Inf., 1987] [Beirlant, Dudewicz, Győrfi, van der Meulen, “Nonparametric entropy estimation: An overview”, 2001] [Wang, Kulkarni, Verdú, “Universal estimation of information measures for analog sources”, FnT Comm. Inf. Th., 2009]

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Solution 2: Non-parametric estimation

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Experimental Results [Nowozin, “Improved Information Gain Estimates for Decision Tree Induction”, ICML 2012]

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Streaming Decision Trees

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Streaming Data “Infinite data” setting 10 possible splits and their scores When to stop and make a decision?

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Streaming Decision Trees [Domingos, Hulten, “Mining High-Speed Data Streams”, KDD 2000] [Jin, Agralwal, “Efficient Decision Tree Construction on Streaming Data”, KDD 2003] [Loh, Nowozin, “Faster Hoeffding racing: Bernstein races via jackknife estimates”, ALT 2013] Score splits on a subset of samples only Domingos/Hulten (Hoeffding Trees), 2000: – Compute sample count n for given precision – Streaming decision tree induction – Incorrect confidence intervals, but work well in practice Jin/Agralwal, 2003: – Tighter confidence interval, asymptotic derivation using delta method Loh/Nowozin, 2013: – Racing algorithm (bad splits are removed early) – Finite sample confidence intervals for entropy and gini

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Multivariate Delta Method [DasGupta, “Asymptotic Theory of Statistics and Probability”, Springer, 2008]

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Delta Method for the Information Gain [Small, “Expansions and Asymptotics for Statistics”, CRC, 2010] [DasGupta, “Asymptotic Theory of Statistics and Probability”, Springer, 2008]

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Delta Method Example

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Statistical problem Large body of literature exists on entropy estimation Better estimators yield better decision trees Distribution of estimate relevant in the streaming setting Conclusion on Entropy Estimation

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