An Introduction to Ensemble Methods Boosting Bagging Random

An Introduction to Ensemble Methods Boosting, Bagging, Random Forests and More Yisong Yue

Supervised Learning • Goal: learn predictor h(x) – High accuracy (low error) – Using training data {(x 1, y 1), …, (xn, yn)}

h(x) = sign(w. Tx-b) Person Age Male? Height > 55” Alice 14 0 1 Bob 10 1 1 Carol 13 0 1 Dave 8 1 0 Erin 11 0 0 Frank 9 1 1 Gena 8 0 0

Male? Yes 1 Person Age Male? Height > 55” Alice 14 0 1 Yes No Bob 10 1 1 Age>9? Age>10? Carol 13 0 1 Dave 8 1 0 Erin 11 0 0 Frank 9 1 1 Gena 8 0 0 No 0 Yes 1 No 0

Splitting Criterion: entropy reduction (info gain) Complexity: number of leaves, size of leaves

Outline • Bias/Variance Tradeoff • Ensemble methods that minimize variance – Bagging – Random Forests • Ensemble methods that minimize bias – Functional Gradient Descent – Boosting – Ensemble Selection

Generalization Error • “True” distribution: P(x, y) – Unknown to us • Train: h(x) = y – Using training data S = {(x 1, y 1), …, (xn, yn)} – Sampled from P(x, y) • Generalization Error: – L(h) = E(x, y)~P(x, y)[ f(h(x), y) ] – E. g. , f(a, b) = (a-b)2

Person Age Male? Height > 55” James 11 1 1 Jessica 14 0 1 Alice 14 0 1 Amy 12 0 1 Bob 10 1 1 Xavier 9 1 0 Cathy 9 0 1 Carol 13 0 1 Eugene 13 1 0 Rafael 12 1 1 Dave 8 1 0 Peter 9 1 0 Henry 13 1 0 Erin 11 0 0 Rose 7 0 0 Iain 8 1 1 Paulo 12 1 0 Margaret 10 0 1 Frank 9 1 1 Jill 13 0 0 Leon 10 1 0 Sarah 12 0 0 Gena 8 0 0 Patrick 5 1 1 Person Age Male? Height > 55” Alice 14 0 1 Bob 10 1 1 Carol 13 0 1 Dave 8 1 0 Erin 11 0 0 Frank 9 1 1 Gena 8 0 0 y … Generalization Error: L(h) = E(x, y)~P(x, y)[ f(h(x), y) ] h(x)

Bias/Variance Tradeoff • Treat h(x|S) has a random function – Depends on training data S • L = ES[ E(x, y)~P(x, y)[ f(h(x|S), y) ] ] – Expected generalization error – Over the randomness of S

Bias/Variance Tradeoff • Squared loss: f(a, b) = (a-b)2 • Consider one data point (x, y) • Notation: – Z = h(x|S) – y – ž = ES[Z] – Z-ž = h(x|S) – ES[h(x|S)] ES[(Z-ž)2] = ES[Z 2 – 2 Zž + ž 2] = ES[Z 2] – 2 ES[Z]ž + ž 2 = ES[Z 2] – ž 2 Expected Error ES[f(h(x|S), y)] = ES[Z 2] = ES[(Z-ž)2] + ž 2 Bias/Variance for all (x, y) is expectation over P(x, y). Can also incorporate measurement noise. (Similar flavor of analysis for other loss functions. ) Variance Bias

Example y x

h(x|S)

h(x|S)

h(x|S)

Bias Variance Bias ES[(h(x|S) - y)2] = ES[(Z-ž)2] + ž 2 Expected Error Variance Bias Variance Z = h(x|S) – y ž = ES[Z]

Outline • Bias/Variance Tradeoff • Ensemble methods that minimize variance – Bagging – Random Forests • Ensemble methods that minimize bias – Functional Gradient Descent – Boosting – Ensemble Selection

Bagging • Goal: reduce variance P(x, y) S’ sampled independently • Ideal setting: many training sets S’ – Train model using each S’ – Average predictions ES[(h(x|S) - y)2] = ES[(Z-ž)2] + ž 2 Expected Error Variance “Bagging Predictors” [Leo Breiman, 1994] Bias http: //statistics. berkeley. edu/sites/default/files/tech-reports/421. pdf Variance reduces linearly Bias unchanged Z = h(x|S) – y ž = ES[Z]

Bagging • Goal: reduce variance S S’ from S • In practice: resample S’ with replacement – Train model using each S’ – Average predictions ES[(h(x|S) - y)2] = ES[(Z-ž)2] + ž 2 Expected Error Variance “Bagging Predictors” [Leo Breiman, 1994] Variance reduces sub-linearly (Because S’ are correlated) Bias often increases slightly Z = h(x|S) – y ž = ES[Z] Bias http: //statistics. berkeley. edu/sites/default/files/tech-reports/421. pdf Bagging = Bootstrap Aggregation

DT Bagged DT Better Variance Bias “An Empirical Comparison of Voting Classification Algorithms: Bagging, Boosting, and Variants” Eric Bauer & Ron Kohavi, Machine Learning 36, 105– 139 (1999)

Random Forests • Goal: reduce variance – Bagging can only do so much – Resampling training data asymptotes • Random Forests: sample data & features! – Sample S’ – Train DT Further de-correlates trees • At each node, sample features (sqrt) – Average predictions “Random Forests – Random Features” [Leo Breiman, 1997] http: //oz. berkeley. edu/~breiman/random-forests. pdf

Better Average performance over many datasets Random Forests perform the best “An Empirical Evaluation of Supervised Learning in High Dimensions” Caruana, Karampatziakis & Yessenalina, ICML 2008

Structured Random Forests • DTs normally train on unary labels y=0/1 • What about structured labels? – Must define information gain of structured labels • Edge detection: – E. g. , structured label is a 16 x 16 image patch – Map structured labels to another space • where entropy is well defined “Structured Random Forests for Fast Edge Detection” Dollár & Zitnick, ICCV 2013

Image Ground Truth RF Prediction Comparable accuracy vs state-of-the-art Orders of magnitude faster “Structured Random Forests for Fast Edge Detection” Dollár & Zitnick, ICCV 2013

Outline • Bias/Variance Tradeoff • Ensemble methods that minimize variance – Bagging – Random Forests • Ensemble methods that minimize bias – Functional Gradient Descent – Boosting – Ensemble Selection

Functional Gradient Descent h(x) = h 1(x) + h 2(x) + … + hn(x) S’ = {(x, y)} S’ = {(x, y-h 1(x) - … - hn-1(x))} … h 1(x) http: //statweb. stanford. edu/~jhf/ftp/trebst. pdf h 2(x) hn(x)

Coordinate Gradient Descent • Learn w so that h(x) = w. Tx • Coordinate descent – Init w = 0 – Choose dimension with highest gain • Set component of w – Repeat

Functional Gradient Descent h(x) = h 1(x) + h 2(x) + … + hn(x) Coordinate descent in function space Restrict weights to be 0, 1, 2, … “Function Space” (All possible DTs)

Boosting (Ada. Boost) h(x) = a 1 h 1(x) + a 2 h 2(x) + … + a 3 hn(x) S’ = {(x, y, u 1)} S’ = {(x, y, u 2)} S’ = {(x, y, u 3))} … h 1(x) h 2(x) u – weighting on data points a – weight of linear combination https: //www. cs. princeton. edu/~schapire/papers/explaining-adaboost. pdf hn(x) Stop when validation performance plateaus (will discuss later)

Initial Distribution of Data Train model Error of model Coefficient of model Update Distribution Final average Theorem: training error drops exponentially fast https: //www. cs. princeton. edu/~schapire/papers/explaining-adaboost. pdf

DT Bagging Ada. Boost Better Variance Boosting often uses weak models E. g, “shallow” decision trees Weak models have lower variance Bias “An Empirical Comparison of Voting Classification Algorithms: Bagging, Boosting, and Variants” Eric Bauer & Ron Kohavi, Machine Learning 36, 105– 139 (1999)

Ensemble Selection Training S’ S H = {2000 models trained using S’} Validation V’ Maintain ensemble model as combination of H: h(x) = h 1(x) + h 2(x) + … + hn(x) + hn+1(x) Denote as hn+1 Add model from H that maximizes performance on V’ Repeat “Ensemble Selection from Libraries of Models” Caruana, Niculescu-Mizil, Crew & Ksikes, ICML 2004 Models are trained on S’ Ensemble built to optimize V’

Ensemble Selection often outperforms a more homogenous sets of models. Reduces overfitting by building model using validation set. Ensemble Selection won KDD Cup 2009 http: //www. niculescu-mizil. org/papers/KDDCup 09. pdf “Ensemble Selection from Libraries of Models” Caruana, Niculescu-Mizil, Crew & Ksikes, ICML 2004

Method Minimize Bias? Minimize Variance? Other Comments Bagging Complex model class. Bootstrap aggregation Does not work for (Deep DTs) (resampling training data) simple models. Random Forests Complex model class. Bootstrap aggregation (Deep DTs) + bootstrapping features Only for decision trees. Gradient Optimize training Boosting performance. (Ada. Boost) Simple model class. (Shallow DTs) Determines which model to add at runtime. Ensemble Selection Optimize validation performance Pre-specified dictionary of models learned on training set. Optimize validation performance …and many other ensemble methods as well. • State-of-the-art prediction performance – Won Netflix Challenge – Won numerous KDD Cups – Industry standard

References & Further Reading “An Empirical Comparison of Voting Classification Algorithms: Bagging, Boosting, and Variants” Bauer & Kohavi, Machine Learning, 36, 105– 139 (1999) “Bagging Predictors” Leo Breiman, Tech Report #421, UC Berkeley, 1994, http: //statistics. berkeley. edu/sites/default/files/tech-reports/421. pdf “An Empirical Comparison of Supervised Learning Algorithms” Caruana & Niculescu-Mizil, ICML 2006 “An Empirical Evaluation of Supervised Learning in High Dimensions” Caruana, Karampatziakis & Yessenalina, ICML 2008 “Ensemble Methods in Machine Learning” Thomas Dietterich, Multiple Classifier Systems, 2000 “Ensemble Selection from Libraries of Models” Caruana, Niculescu-Mizil, Crew & Ksikes, ICML 2004 “Getting the Most Out of Ensemble Selection” Caruana, Munson, & Niculescu-Mizil, ICDM 2006 “Explaining Ada. Boost” Rob Schapire, https: //www. cs. princeton. edu/~schapire/papers/explaining-adaboost. pdf “Greedy Function Approximation: A Gradient Boosting Machine”, Jerome Friedman, 2001, http: //statweb. stanford. edu/~jhf/ftp/trebst. pdf “Random Forests – Random Features” Leo Breiman, Tech Report #567, UC Berkeley, 1999, “Structured Random Forests for Fast Edge Detection” Dollár & Zitnick, ICCV 2013 “ABC-Boost: Adaptive Base Class Boost for Multi-class Classification” Ping Li, ICML 2009 “Additive Groves of Regression Trees” Sorokina, Caruana & Riedewald, ECML 2007, http: //additivegroves. net/ “Winning the KDD Cup Orange Challenge with Ensemble Selection”, Niculescu-Mizil et al. , KDD 2009 “Lessons from the Netﬂix Prize Challenge” Bell & Koren, SIGKDD Exporations 9(2), 75— 79, 2007