Bias-Variance Analysis of Ensemble Learning - Oregon State 偏差，方差分析的集成学习-俄勒冈州.pptVIP

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Bias-Variance Analysis of Ensemble Learning - Oregon State 偏差，方差分析的集成学习-俄勒冈州.ppt

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Bias-Variance Analysis of Ensemble Learning - Oregon State 偏差，方差分析的集成学习-俄勒冈州

Bias-Variance Analysis of Ensemble Learning Thomas G. Dietterich Department of Computer Science Oregon State University Corvallis, Oregon 97331 /~tgd Outline Bias-Variance Decomposition for Regression Bias-Variance Decomposition for Classification Bias-Variance Analysis of Learning Algorithms Effect of Bagging on Bias and Variance Effect of Boosting on Bias and Variance Summary and Conclusion Bias-Variance Analysis in Regression True function is y = f(x) + e where e is normally distributed with zero mean and standard deviation s. Given a set of training examples, {(xi, yi)}, we fit an hypothesis h(x) = w ￠ x + b to the data to minimize the squared error Si [yi – h(xi)]2 Example: 20 pointsy = x + 2 sin(1.5x) + N(0,0.2) 50 fits (20 examples each) Bias-Variance Analysis Now, given a new data point x* (with observed value y* = f(x*) + e), we would like to understand the expected prediction error E[ (y* – h(x*))2 ] Classical Statistical Analysis Imagine that our particular training sample S is drawn from some population of possible training samples according to P(S). Compute EP [ (y* – h(x*))2 ] Decompose this into “bias”, “variance”, and “noise” Lemma Let Z be a random variable with probability distribution P(Z) Let Z = EP[ Z ] be the average value of Z. Lemma: E[ (Z – Z)2 ] = E[Z2] – Z2 E[ (Z – Z)2 ] = E[ Z2 – 2 Z Z + Z2 ] = E[Z2] – 2 E[Z] Z + Z2 = E[Z2] – 2 Z2 + Z2 = E[Z2] – Z2 Corollary: E[Z2] = E[ (Z – Z)2 ] + Z2 Bias-Variance-Noise Decomposition E[ (h(x*) – y*)2 ] = E[ h(x*)2 – 2 h(x*) y* + y*2 ] = E[ h(x*)2 ] – 2 E[ h(x*) ] E[y*] + E[y*2] = E[ (h(x*) – h(x*))2 ] + h(x*)2 (lemma) – 2 h(x*) f(x*) + E[ (y* – f(x*))2 ] + f(x*)2 (lemma) = E[ (h(x*) – h(x*))2 ] + [variance] (h(x*) – f(x*))2 + [bias2] E[ (y* – f(x*))2 ] [noise] D