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Infinite Ensemble Learning with Support Vector Machines

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Infinite Ensemble Learning with Support Vector Machines Infinite Ensemble Learning with Support Vector Machines Thesis by Hsuan-Tien Lin In Partial Fulfillment of the Requirements for the Degree of Master of Science California Institute of Technology Pasadena, California 2005 (Submitted May 18, 2005) ii c 2005 Hsuan-Tien Lin All Rights Reserved iii Acknowledgements I am very grateful to my advisor, Professor Yaser S. Abu-Mostafa, for his help, sup- port, and guidance throughout my research and studies. It has been a privilege to work in the free and easy atmosphere that he provides for the Learning Systems Group. I have enjoyed numerous discussions with Ling Li and Amrit Pratap, my fellow members of the group. I thank them for their valuable input and feedback. I am particularly indebted to Ling, who not only collaborates with me in some parts of this work, but also gives me lots of valuable suggestions in research and in life. I also appreciate Lucinda Acosta for her help in obtaining necessary resources for research. I thank Kai-Min Chung for providing useful comments on an early publication of this work. I also want to address a special gratitude to Professor Chih-Jen Lin, who brought me into the fascinating area of Support Vector Machine five years ago. Most importantly, I thank my family and friends for their endless love and support. I am thankful to my parents, who have always encouraged me and have believed in me. In addition, I want to convey my personal thanks to Yung-Han Yang, my girlfriend, who has shared every moment with me during days of our lives in the U.S. This work is supported primarily by the Engineering Research Centers Program of the National Science Foundation under award EEC-9402726. iv Abstract Ensemble learning algorithms achieve better out-of-sample performance by averaging over the predictions of some base learners. Theoretically, the ensemble could include an infinite number of base learners. However, the construction of an infinite ensemble is a challenging task. Thus, traditional ensemble learning algorithms usually have to rely on a finite approximation or a sparse representation of the infinite ensemble. It is not clear whether the performance could be further improved by a learning algorithm that actually outputs an infinite ensemble. In this thesis, we exploit the Support Vector Machine (SVM) to develop such learning algorithm. SVM is capable of handling infinite number of features through the use of kernels, and has a close connection to ensemble learning. These properties allow us to formulate an infinite ensemble learning framework by embedding the base learners into an SVM kernel. With the framework, we could derive several new kernels for SVM, and give novel interpretations to some existing kernels from an ensemble point-of-view. We would construct several useful and concrete instances of the framework. Exper- imental results show that SVM could perform well even with kernels that embed very simple base learners. In addition, the framework provides a platform to fairly compare SVM with traditional ensemble learning algorithms. It is observed that SVM could usually achieve better performance than traditional ensemble learning algorithms, which provides us further understanding of both SVM and ensemble learning. v Contents Acknowledgements iii Abstract iv 1 Introduction 1 1.1 The Learning Problem . 1 1.1.1 Formulation . 1 1.1.2 Capacity of the Learning Model . 3 1.2 Ensemble Learning . 6 1.2.1 Formulation . 6 1.2.2 Why Ensemble Learning? . 7 1.3 Infinite Ensemble Learning . 9 1.3.1 Why Infinite Ensemble Learning? . 9 1.3.2 Dealing with Infinity . 10 1.4 Overview . 12 2 Connection between SVM and Ensemble Learning 13 2.1 Support Vector Machine . 13 2.1.1 Basic SVM Formulation . 14 2.1.2 Nonlinear Soft-Margin SVM Formulation . 16 2.2 Ensemble Learning and Large-Margin Concept . 20 2.2.1 Adaptive Boosting . 20 2.2.2 Linear Programming Boosting . 22 2.3 Connecting SVM to Ensemble Learning . 24 vi 3 SVM-based Framework for Infinite Ensemble Learning 29 3.1 Embedding Learning Model into the Kernel . 29 3.2 Assuming Negation Completeness . 32 3.3 Properties of the Framework . 34 3.3.1 Embedding Multiple Learning Models . 34 3.3.2 Averaging Ambiguous Hypotheses . 35 4 Concrete Instances of the Framework 38 4.1 Stump Kernel . 38 4.1.1 Formulation . 38 4.1.2 Power of the Stump Ensemble . 40 4.1.3 Stump Kernel and Radial Basis Function . 43 4.1.4 Averaging Ambiguous Stumps . 44 4.2 Perceptron Kernel . 45 4.2.1 Formulation . 45 4.2.2 Properties of the Perceptron Kernel . 48 4.3 Kernels that Embed Combined Hypotheses . 50 4.3.1 Logic Kernels . 50 4.3.2 Multiplication of Logic Kernels . 51 4.3.3 Laplacian Kernel and Exponential Kernel . 55 4.3.4 Discussion on RBF Kernels . 58 5 Experiments 60 5.1 Setup . 60 5.2 Comparison of Ensemble Learning Algorithms . 62 5.2.1 Experimental Results . 62 5.2.2 Regularization and Sparsity . 63 5.3 Comparison of RBF Kernels . 65 6 Conclusion 67 vii List of Figures 1.1 Illustration of the learning scenario . 3 1.2 Overfitting and underfitting . 5 2.1 Illustration of the margin, where yi = 2 · I[circle is empty] − 1 . 15 2.2 The power of the feature mapping in (2.1) . 17 2.3 LPBoost can only choose one between h1 and h2 . 24 4.1 Illustration of the decision stump s+1;2,α(x) . 39 4.2 The XOR training set. 43 4.3 Illustration of the perceptron p+1,θ,α(x) . 46 4.4 Illustration of AND combination . 52 4.5 Combining s 1;1;2(x) and s+1;2;1(x) . 55 − 5.1 Decision boundaries of SVM-Stump (left) and AdaBoost-Stump (right) on a 2-D twonorm dataset . 64 viii List of Tables 5.1 Details of the datasets . 61 5.2 Test error (%) comparison of ensemble learning algorithms . 63 5.3 Test error (%) comparison on sparsity and regularization . 65 5.4 Test error (%) comparison of RBF kernels . 66 5.5 Parameter selection time (sec.) comparison of RBF kernels . 66 ix List of Algorithms 1 AdaBoost . 27 2 LPBoost . 28 3 SVM-based framework for infinite ensemble learning . 33 1 Chapter 1 Introduction This thesis is about infinite ensemble learning, a promising paradigm in machine learning. In this chapter, we would introduce the learning problem, the ensemble learning paradigm, and the infinite ensemble learning paradigm. We would build up our scheme of notations, and show our motivations for this work. 1.1 The Learning Problem 1.1.1 Formulation In this thesis, we would study the problem of learning from examples (Abu-Mostafa N 1989). For such a learning problem, we are given a training set Z = fzi : zi = (xi; yi)gi=1 which consists of the training examples zi. We assume that the training vectors xi are drawn independently from an unknown probability measure P (x) on X ⊆ RD, X and their labels yi are computed from yi = f(xi). Here f : X ! Y is called the target function, and is also assumed to be unknown. With the given training set, we want to obtain a function g∗ : X ! Y as our inference of the target function. The function g∗ is usually chosen from a collection G of candidate functions, called the learning model. Briefly speaking, the task of the learning problem is to use the information in the training set Z to find some g∗ 2 G that approximates f well. For example, we may want to build a recognition system that transforms an image of a written digit to its intended meaning. This goal is usually called an inverse 2 problem, but we can also formulate it as a learning problem. We first ask someone to write down N digits, and represent their images by the training vectors xi. We then label the digits by yi 2 f0; 1; · · · ; 9g according to their meanings. The target function f here encodes the process of our human-based recognition system. The task of this learning problem would be setting up an automatic recognition system (function) g∗ that is almost as good as our own recognition system, even on the yet unseen images of written digits in the future. Throughout this thesis, we would work on binary classification problems, in which Y = {−1; +1g. We call a function of the form X ! {−1; +1g a classifier, and define the deviation between two classifiers g and f on a point x 2 X to be I[g(x) =6 f(x)].1 The overall deviation, called the out-of-sample error, would be π(g) = I[f(x) =6 g(x)] dP (x): X ZX We say that g approximates f well, or generalizes well, if π(g) is small. We design the learning algorithm A to solve the learning problem. Generally, the algorithm takes the training set Z and the learning model G, and outputs a decision function g∗ 2 G by minimizing a predefined error cost eZ (g). The full scenario of learning is illustrated in Figure 1.1. To obtain a decision function that generalizes well, we ideally desires eZ (g) = π(g) for all g 2 G. However, both f and p are assumed to be unknown, and it is hence X impossible to compute and minimize π(g) directly. A substitute quantity that depends only on Z is called the in-sample error N 1 ν(g) = I[y =6 g(x )] · : i i N i=1 X Many learning algorithms consider ν(g) to be an important part of the error cost, because ν(g) is an unbiased estimate of π(g).
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