DOCSLIB.ORG

Recent Methods from Statistics and Machine Learning for Credit Scoring

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Recent Methods from Statistics and Machine Learning for Credit Scoring Recent Methods from Statistics and Machine Learning for Credit Scoring Anne Kraus M¨unchen2014 Recent Methods from Statistics and Machine Learning for Credit Scoring Anne Kraus Dissertation an der Fakult¨atf¨urMathematik, Informatik und Statistik der Ludwig{Maximilians{Universit¨at M¨unchen vorgelegt von Anne Kraus aus Schweinfurt M¨unchen, den 10. M¨arz2014 Erstgutachter: Prof. Dr. Helmut K¨uchenhoff Zweitgutachter: Prof. Dr. Martin Missong Tag der Disputation: 22. Mai 2014 Mein besonderer Dank gilt • Prof. Dr. Helmut K¨uchenhoff, f¨urdie M¨oglichkeit bei ihm zu promovieren, die Begeisterung f¨urdas Thema und die hervorragende Betreuung in den letzten Jahren • Prof. Stefan Mittnik, Ph.D., f¨urdie Zweitbetreuung im Rahmen des Promotionsprogramms • Prof. Dr. Martin Missong, f¨urdie Bereitschaft, das Zweitgutachten zu ¨ubernehmen • allen Doktoranden und Mitarbeitern am Institut f¨urStatistik, f¨urdie freundliche Aufnahme und Unterst¨utzung,ganz besonders Monia Mahling • meinem Arbeitgeber, f¨urdie M¨oglichkeit der Freistellung • meinen Kollegen, f¨urihr Interesse und viele gemeinsame Mittagspausen • meinen Freunden, f¨urviel Verst¨andnisund Motivation, allen voran Karin Schr¨oter • meinen Geschwistern Eva Schmitt und Wolfgang Kraus samt Familien, f¨ur Aufmunterung und Ablenkung • meinen Eltern, f¨urihre grenzenlose Unterst¨utzungin jeglicher Hinsicht • Martin Tusch, f¨urunendlichen R¨uckhalt Abstract Credit scoring models are the basis for financial institutions like retail and consumer credit banks. The purpose of the models is to evaluate the likelihood of credit applicants defaulting in order to decide whether to grant them credit. The area under the receiver operating characteristic (ROC) curve (AUC) is one of the most commonly used measures to evaluate predictive performance in credit scoring. The aim of this thesis is to benchmark different methods for building scoring models in order to maximize the AUC. While this measure is used to evaluate the predictive accuracy of the presented algorithms, the AUC is especially introduced as direct optimization criterion. The logistic regression model is the most widely used method for creating credit scorecards and classifying applicants into risk classes. Since this development process, based on the logit model, is standard in the retail banking practice, the predictive accuracy of this proceeding is used for benchmark reasons throughout this thesis. The AUC approach is a main task introduced within this work. Instead of using the maximum likelihood estimation, the AUC is considered as objective function to optimize it directly. The coefficients are estimated by calculating the AUC measure with Wilcoxon{ Mann{Whitney and by using the Nelder{Mead algorithm for the optimization. The AUC optimization denotes a distribution-free approach, which is analyzed within a simulation study for investigating the theoretical considerations. It can be shown that the approach still works even if the underlying distribution is not logistic. In addition to the AUC approach and classical well-known methods like generalized additive models, new methods from statistics and machine learning are evaluated for the credit scoring case. Conditional inference trees, model-based recursive partitioning methods and random forests are presented as recursive partitioning algorithms. Boosting algorithms are also explored by additionally using the AUC as a loss function. The empirical evaluation is based on data from a German bank. From the application scoring, 26 attributes are included in the analysis. Besides the AUC, different performance measures are used for evaluating the predictive performance of scoring models. While classification trees cannot improve predictive accuracy for the current credit scoring case, the AUC approach and special boosting methods provide outperforming results compared to the robust classical scoring models regarding the predictive performance with the AUC measure. Zusammenfassung Scoringmodelle dienen Finanzinstituten als Grundlage daf¨ur,die Ausfallwahrscheinlichkeit von Kreditantragstellern zu berechnen und zu entscheiden ob ein Kredit gew¨ahrtwird oder nicht. Das AUC (area under the receiver operating characteristic curve) ist eines der am h¨aufigsten verwendeten Maße, um die Vorhersagekraft im Kreditscoring zu bewerten. Demzufolge besteht das Ziel dieser Arbeit darin, verschiedene Methoden zur Scoremodell- Bildung hinsichtlich eines optimierten AUC Maßes zu benchmarken\. W¨ahrenddas " genannte Maß dazu dient die vorgestellten Algorithmen hinsichtlich ihrer Trennsch¨arfezu bewerten, wird das AUC insbesondere als direktes Optimierungskriterium eingef¨uhrt. Die logistische Regression ist das am h¨aufigstenverwendete Verfahren zur Entwicklung von Scorekarten und die Einteilung der Antragsteller in Risikoklassen. Da der Entwick- lungsprozess mittels logistischer Regression im Retail-Bankenbereich stark etabliert ist, wird die Trennsch¨arfedieses Verfahrens in der vorliegenden Arbeit als Benchmark verwendet. Der AUC Ansatz wird als entscheidender Teil dieser Arbeit vorgestellt. Anstatt die Maximum Likelihood Sch¨atzungzu verwenden, wird das AUC als direkte Zielfunktion zur Optimierung verwendet. Die Koeffizienten werden gesch¨atzt,indem f¨urdie Berechnung des AUC die Wilcoxon Statistik und f¨urdie Optimierung der Nelder-Mead Algorithmus verwendet wird. Die AUC Optimierung stellt einen verteilungsfreien Ansatz dar, der im Rahmen einer Simulationsstudie untersucht wird, um die theoretischen Uberlegungen¨ zu analysieren. Es kann gezeigt werden, dass der Ansatz auch dann funktioniert, wenn in den Daten kein logistischer Zusammenhang vorliegt. Zus¨atzlich zum AUC Ansatz und bekannten Methoden wie Generalisierten Additiven Modellen, werden neue Methoden aus der Statistik und dem Machine Learning f¨urdas Kreditscoring evaluiert. Klassifikationsb¨aume,Modell-basierte Recursive Partitioning Methoden und Random Forests werden als Recursive Paritioning Methoden vorgestellt. Dar¨uberhinaus werden Boosting Algorithmen untersucht, die auch das AUC Maß als Verlustfunktion verwenden. Die empirische Analyse basiert auf Daten einer deutschen Kreditbank. 26 Variablen werden im Rahmen der Analyse untersucht. Neben dem AUC Maß werden verschieden Per- formancemaße verwendet, um die Trennsch¨arfevon Scoringmodellen zu bewerten. W¨ahrend Klassifikationsb¨aume im vorliegenden Kreditscoring Fall keine Verbesserungen erzielen, weisen der AUC Ansatz und einige Boosting Verfahren gute Ergebnisse im Vergleich zum robusten klassischen Scoringmodell hinsichtlich des AUC Maßes auf. Contents Abstract vi Zusammenfassung viii 1 Introduction1 1.1 Credit Scoring..................................1 1.2 Scope of the Work...............................3 2 Measures of Performance and Data Description5 2.1 Receiver Operating Characteristic and Area under the Curve........5 2.2 Data Description................................9 3 Logistic Regression 11 3.1 A Short Introduction to Logistic Regression................. 11 3.2 The Development of Scorecards........................ 12 3.3 Discussion of the Logistic Regression Model in Credit Scoring....... 16 4 Optimization AUC 19 4.1 A New Approach for AUC Optimization................... 19 4.2 Optimality Properties of the AUC Approach................. 21 4.2.1 Theoretical Considerations....................... 21 4.2.2 Simulation Study............................ 23 4.3 The AUC Approach in Credit Scoring..................... 32 4.4 Discussion of the AUC Approach in Credit Scoring............. 35 5 Generalized Additive Model 37 5.1 A Short Overview of Generalized Additive Model.............. 37 5.2 Generalized Additive Model for Credit Scoring................ 38 5.3 Discussion of a Generalized Additive Model in Credit Scoring....... 42 6 Recursive Partitioning 45 6.1 Classification and Regression Trees...................... 45 6.1.1 CART Algorithm............................ 46 6.1.2 Conditional Inference Trees...................... 49 xii Contents 6.2 Model-Based Recursive Partitioning...................... 52 6.3 Random Forests................................. 56 6.3.1 Random Forest Algorithm....................... 56 6.3.2 Importance Measures.......................... 57 6.4 Recursive Partitioning Methods for Credit Scoring.............. 60 6.4.1 Classification Trees for Credit Scoring................. 60 6.4.2 Model-Based Recursive Partitioning in Credit Scoring........ 69 6.4.3 Random Forests in Credit Scoring................... 74 6.5 Discussion of Recursive Partitioning Methods for Credit Scoring...... 85 7 Boosting 87 7.1 Boosting Algorithms.............................. 87 7.1.1 Rationale of Gradient Boosting.................... 87 7.1.2 Component-Wise Gradient Boosting................. 88 7.1.3 Base Learners.............................. 90 7.2 Boosting for Credit Scoring.......................... 91 7.2.1 Discrete, Real and Gentle AdaBoost................. 91 7.2.2 Boosting Logistic Regression...................... 94 7.2.3 Boosting Generalized Additive Model................. 97 7.3 Boosting and the Optimization concerning AUC............... 101 7.3.1 AUC as a Loss Function in Boosting Algorithms........... 101 7.3.2 Boosting AUC in Credit Scoring.................... 102 7.4 Discussion of Boosting Methods in Credit Scoring.............. 107 8 Summary and Outlook 109 A Supplementary Material 115 A.1 Optimization AUC - Chapter4........................ 115 A.2 Generalized Additive Model - Chapter5................... 118 A.3 Recursive Partitioning - Chapter6...................... 119 A.4
Load more

Recommended publications
  • Machine Learning Or Econometrics for Credit Scoring: Let’S Get the Best of Both Worlds Elena Dumitrescu, Sullivan Hué, Christophe Hurlin, Sessi Tokpavi
    Machine Learning or Econometrics for Credit Scoring: Let’s Get the Best of Both Worlds Elena Dumitrescu, Sullivan Hué, Christophe Hurlin, Sessi Tokpavi To cite this version: Elena Dumitrescu, Sullivan Hué, Christophe Hurlin, Sessi Tokpavi. Machine Learning or Econometrics for Credit Scoring: Let’s Get the Best of Both Worlds. 2020. hal-02507499v2 HAL Id: hal-02507499 https://hal.archives-ouvertes.fr/hal-02507499v2 Preprint submitted on 2 Nov 2020 (v2), last revised 15 Jan 2021 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Machine Learning or Econometrics for Credit Scoring: Let's Get the Best of Both Worlds∗ Elena, Dumitrescu,y Sullivan, Hu´e,z Christophe, Hurlin,x Sessi, Tokpavi{ October 18, 2020 Abstract In the context of credit scoring, ensemble methods based on decision trees, such as the random forest method, provide better classification performance than standard logistic regression models. However, logistic regression remains the benchmark in the credit risk industry mainly because the lack of interpretability of ensemble methods is incompatible with the requirements of financial regulators. In this paper, we pro- pose to obtain the best of both worlds by introducing a high-performance and inter- pretable credit scoring method called penalised logistic tree regression (PLTR), which uses information from decision trees to improve the performance of logistic regression.
    [Show full text]
  • 18.650 (F16) Lecture 10: Generalized Linear Models (Glms)
    Statistics for Applications Chapter 10: Generalized Linear Models (GLMs) 1/52 Linear model A linear model assumes Y X (µ(X), σ2I), | ∼ N And ⊤ IE(Y X) = µ(X) = X β, | 2/52 Components of a linear model The two components (that we are going to relax) are 1. Random component: the response variable Y X is continuous | and normally distributed with mean µ = µ(X) = IE(Y X). | 2. Link: between the random and covariates X = (X(1),X(2), ,X(p))⊤: µ(X) = X⊤β. · · · 3/52 Generalization A generalized linear model (GLM) generalizes normal linear regression models in the following directions. 1. Random component: Y some exponential family distribution ∼ 2. Link: between the random and covariates: ⊤ g µ(X) = X β where g called link function� and� µ = IE(Y X). | 4/52 Example 1: Disease Occuring Rate In the early stages of a disease epidemic, the rate at which new cases occur can often increase exponentially through time. Hence, if µi is the expected number of new cases on day ti, a model of the form µi = γ exp(δti) seems appropriate. ◮ Such a model can be turned into GLM form, by using a log link so that log(µi) = log(γ) + δti = β0 + β1ti. ◮ Since this is a count, the Poisson distribution (with expected value µi) is probably a reasonable distribution to try. 5/52 Example 2: Prey Capture Rate(1) The rate of capture of preys, yi, by a hunting animal, tends to increase with increasing density of prey, xi, but to eventually level off, when the predator is catching as much as it can cope with.
    [Show full text]
  • Maximum Likelihood from Incomplete Data Via the EM Algorithm A. P. Dempster; N. M. Laird; D. B. Rubin Journal of the Royal Stati
    Maximum Likelihood from Incomplete Data via the EM Algorithm A. P. Dempster; N. M. Laird; D. B. Rubin Journal of the Royal Statistical Society. Series B (Methodological), Vol. 39, No. 1. (1977), pp. 1-38. Stable URL: http://links.jstor.org/sici?sici=0035-9246%281977%2939%3A1%3C1%3AMLFIDV%3E2.0.CO%3B2-Z Journal of the Royal Statistical Society. Series B (Methodological) is currently published by Royal Statistical Society. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/rss.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact [email protected]. http://www.jstor.org Fri Apr 6 01:07:17 2007 Maximum Likelihood from Incomplete Data via the EM Algorithm By A. P. DEMPSTER,N. M. LAIRDand D. B. RDIN Harvard University and Educational Testing Service [Read before the ROYALSTATISTICAL SOCIETY at a meeting organized by the RESEARCH SECTIONon Wednesday, December 8th, 1976, Professor S.
    [Show full text]
  • Analysis of Computer Experiments Using Penalized Likelihood in Gaussian Kriging Models
    Editor’s Note: This article will be presented at the Technometrics invited session at the 2005 Joint Statistical Meetings in Minneapolis, Minnesota, August 2005. Analysis of Computer Experiments Using Penalized Likelihood in Gaussian Kriging Models Runze LI Agus SUDJIANTO Department of Statistics Risk Management Quality & Productivity Pennsylvania State University Bank of America University Park, PA 16802 Charlotte, NC 28255 ([email protected]) ([email protected]) Kriging is a popular analysis approach for computer experiments for the purpose of creating a cheap- to-compute “meta-model” as a surrogate to a computationally expensive engineering simulation model. The maximum likelihood approach is used to estimate the parameters in the kriging model. However, the likelihood function near the optimum may be flat in some situations, which leads to maximum likelihood estimates for the parameters in the covariance matrix that have very large variance. To overcome this difficulty, a penalized likelihood approach is proposed for the kriging model. Both theoretical analysis and empirical experience using real world data suggest that the proposed method is particularly important in the context of a computationally intensive simulation model where the number of simulation runs must be kept small because collection of a large sample set is prohibitive. The proposed approach is applied to the reduction of piston slap, an unwanted engine noise due to piston secondary motion. Issues related to practical implementation of the proposed approach are discussed. KEY WORDS: Computer experiment; Fisher scoring algorithm; Kriging; Meta-model; Penalized like- lihood; Smoothly clipped absolute deviation. 1. INTRODUCTION e.g., Ye, Li, and Sudjianto 2000) and by the meta-modeling ap- proach itself (Jin, Chen, and Simpson 2000).
    [Show full text]
  • Properties of the Zero-And-One Inflated Poisson Distribution and Likelihood
    Statistics and Its Interface Volume 9 (2016) 11–32 Properties of the zero-and-one inflated Poisson distribution and likelihood-based inference methods Chi Zhang, Guo-Liang Tian∗, and Kai-Wang Ng The over-dispersion may result from other phenomena such as a high proportion of both zeros and ones in the To model count data with excess zeros and excess ones, count data. For example, Eriksson and Aberg˚ [8] reported a in their unpublished manuscript, Melkersson and Olsson two-year panel data from Swedish Level of Living Surveys (1999) extended the zero-inflated Poisson distribution to in 1974 and 1991, where the visits to a dentist have higher a zero-and-one-inflated Poisson (ZOIP) distribution. How- proportions of zeros and ones and one-visit observations are ever, the distributional theory and corresponding properties even much more frequent than zero-visits. It is common to of the ZOIP have not yet been explored, and likelihood- visit a dentist for a routine control, e.g., school-children go based inference methods for parameters of interest were not to a dentist once a year almost as a rule. As the second ex- well developed. In this paper, we extensively study the ZOIP ample, Carrivick et al.[3] reported that much of the data distribution by first constructing five equivalent stochastic collected on occupational safety involves accident or injury representations for the ZOIP random variable and then de- counts with many zeros and ones. In general, the safety reg- riving other important distributional properties. Maximum ulations protect most workers from injury so that many ze- likelihood estimates of parameters are obtained by both the ros are observed.
    [Show full text]
  • A Brief Survey of Modern Optimization for Statisticians1
    International Statistical Review (2014), 82, 1, 46–70 doi:10.1111/insr.12022 A Brief Survey of Modern Optimization for Statisticians1 Kenneth Lange1,2,EricC.Chi2 and Hua Zhou3 1Departments of Biomathematics and Statistics, University of California, Los Angeles, CA 90095-1766, USA E-mail: [email protected] 2Department of Human Genetics, University of California, Los Angeles, CA 90095-1766, USA E-mail: [email protected] 3Department of Statistics, North Carolina State University, Raleigh, NC 27695-8203, USA E-mail: [email protected] Summary Modern computational statistics is turning more and more to high-dimensional optimization to handle the deluge of big data. Once a model is formulated, its parameters can be estimated by optimization. Because model parsimony is important, models routinely include non-differentiable penalty terms such as the lasso. This sober reality complicates minimization and maximization. Our broad survey stresses a few important principles in algorithm design. Rather than view these principles in isolation, it is more productive to mix and match them. A few well-chosen examples illustrate this point. Algorithm derivation is also emphasized, and theory is downplayed, particularly the abstractions of the convex calculus. Thus, our survey should be useful and accessible to a broad audience. Key words: Block relaxation; Newton’s method; MM algorithm; penalization; augmented Lagrangian; acceleration. 1 Introduction Modern statistics represents a confluence of data, algorithms, practical inference, and subject area knowledge. As data mining expands, computational statistics is assuming greater promi- nence. Surprisingly, the confident prediction of the previous generation that Bayesian methods would ultimately supplant frequentist methods has given way to a realization that Markov chain Monte Carlo may be too slow to handle modern data sets.
    [Show full text]
  • On the Maximization of Likelihoods Belonging to the Exponential Family
    arXiv.org manuscript No. (will be inserted by the editor) On the maximization of likelihoods belonging to the exponential family using ideas related to the Levenberg-Marquardt approach Marco Giordan · Federico Vaggi · Ron Wehrens Received: date / Revised: date Abstract The Levenberg-Marquardt algorithm is a flexible iterative procedure used to solve non-linear least squares problems. In this work we study how a class of possible adaptations of this procedure can be used to solve maximum likelihood problems when the underlying distributions are in the exponential family. We formally demonstrate a local convergence property and we discuss a possible implementation of the penalization involved in this class of algorithms. Applications to real and simulated compositional data show the stability and efficiency of this approach. Keywords Aitchison distribution · compositional data · Dirichlet distribution · generalized linear models · natural link · optimization. arXiv:1410.0793v1 [stat.CO] 3 Oct 2014 M. Giordan IASMA Research and Innovation Centre, Biostatistics and Data Managment E-mail: [email protected] F. Vaggi IASMA Research and Innovation Centre, Integrative Genomic Research E-mail: [email protected] Ron Wehrens IASMA Research and Innovation Centre, Biostatistics and Data Managment E-mail: [email protected] 2 Marco Giordan et al. 1 Introduction The exponential family is a large class of probability distributions widely used in statistics. Distributions in this class possess many useful properties that make them useful for inferential and algorithmic purposes, especially with reference to maximum likelihood estimation. Despite this, when closed form solutions are not available, computational problems can arise in the fitting of a particular distribution in this family to real data.
    [Show full text]
  • Gaussian Process Learning Via Fisher Scoring of Vecchia's Approximation
    Gaussian Process Learning via Fisher Scoring of Vecchia's Approximation Joseph Guinness Cornell University, Department of Statistics and Data Science Abstract We derive a single pass algorithm for computing the gradient and Fisher information of Vecchia's Gaussian process loglikelihood approximation, which provides a computationally efficient means for applying the Fisher scoring algorithm for maximizing the loglikelihood. The advantages of the optimization techniques are demonstrated in numerical examples and in an application to Argo ocean temperature data. The new methods are more accurate and much faster than an optimization method that uses only function evaluations, especially when the covariance function has many parameters. This allows practitioners to fit nonstationary models to large spatial and spatial-temporal datasets. 1 Introduction The Gaussian process model is an indispensible tool for the analysis of spatial and spatial-temporal datasets and has become increasingly popular as a general-purpose model for functions. Because of its high computational burden, researchers have devoted substantial effort to developing nu- merical approximations for Gaussian process computations. Much of the work focuses on efficient approximation of the likelihood function. Fast likelihood evaluations are crucial for optimization procedures that require many evaluations of the likelihood, such as the default Nelder-Mead al- gorithm (Nelder and Mead, 1965) in the R optim function. The likelihood must be repeatedly evaluated in MCMC algorithms as well. Compared to the amount of literature on efficient likelihood approximations, there has been considerably less development of techniques for numerically maximizing the likelihood (see Geoga et al. (2018) for one recent example). This article aims to address the disparity by providing: 1.
    [Show full text]
  • Stochastic Gradient Descent As Approximate Bayesian Inference
    Journal of Machine Learning Research 18 (2017) 1-35 Submitted 4/17; Revised 10/17; Published 12/17 Stochastic Gradient Descent as Approximate Bayesian Inference Stephan Mandt [email protected] Data Science Institute Department of Computer Science Columbia University New York, NY 10025, USA Matthew D. Hoffman [email protected] Adobe Research Adobe Systems Incorporated 601 Townsend Street San Francisco, CA 94103, USA David M. Blei [email protected] Department of Statistics Department of Computer Science Columbia University New York, NY 10025, USA Editor: Manfred Opper Abstract Stochastic Gradient Descent with a constant learning rate (constant SGD) simulates a Markov chain with a stationary distribution. With this perspective, we derive several new results. (1) We show that constant SGD can be used as an approximate Bayesian posterior inference algorithm. Specif- ically, we show how to adjust the tuning parameters of constant SGD to best match the stationary distribution to a posterior, minimizing the Kullback-Leibler divergence between these two distri- butions. (2) We demonstrate that constant SGD gives rise to a new variational EM algorithm that optimizes hyperparameters in complex probabilistic models. (3) We also show how to tune SGD with momentum for approximate sampling. (4) We analyze stochastic-gradient MCMC algorithms. For Stochastic-Gradient Langevin Dynamics and Stochastic-Gradient Fisher Scoring, we quantify the approximation errors due to finite learning rates. Finally (5), we use the stochastic process per- spective to give a short proof of why Polyak averaging is optimal. Based on this idea, we propose a scalable approximate MCMC algorithm, the Averaged Stochastic Gradient Sampler.
    [Show full text]
  • Stochastic Gradient MCMC: Algorithms and Applications
    UNIVERSITY OF CALIFORNIA, IRVINE Stochastic Gradient MCMC: Algorithms and Applications DISSERTATION submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Computer Science by Sungjin Ahn Dissertation Committee: Professor Max Welling, Chair Professor Babak Shahbaba Professor Charless Fowlkes 2015 c 2015 Sungjin Ahn TABLE OF CONTENTS Page LIST OF FIGURES v LIST OF TABLES vii LIST OF ALGORITHMS viii ACKNOWLEDGMENTS ix CURRICULUM VITAE x ABSTRACT OF THE DISSERTATION xii 1 Introduction 1 1.1 Bayesian Inference for Machine Learning . 3 1.2 Advantages of Bayesian Inference . 6 1.3 Large-Scale Machine Learning . 7 1.4 Research Challenges . 10 1.5 Outline of the Thesis . 11 2 Markov Chain Monte Carlo 13 2.1 Monte Carlo Method . 13 2.2 Markov Chains . 15 2.3 MCMC Algorithms . 17 2.3.1 The Metropolis-Hastings Algorithm . 17 2.3.2 Gibbs Sampling . 18 2.3.3 Langevin Monte Carlo . 20 2.4 Scalability of Traditional MCMC . 21 3 Stochastic Gradient Langevin Dynamics 24 3.1 Stochastic Gradient Langevin Dynamics . 25 3.1.1 SGLD with a constant step-size . 27 3.1.2 SGLD during burn-in . 27 4 Stochastic Gradient Fisher Scoring 29 4.1 Motivation . 29 4.2 Preliminaries and Notation . 30 ii 4.3 Stochastic Gradient Fisher Scoring . 32 4.3.1 Sampling from the Gaussian Approximate Posterior . 32 4.3.2 Stochastic Gradient Fisher Scoring . 33 4.4 Computational Efficiency . 38 4.5 Experiments . 39 4.5.1 Logistic Regression . 39 4.5.2 SGFS on Neural Networks . 42 4.5.3 Discriminative Restricted Boltzmann Machine (DRBM) .
    [Show full text]
  • Recursive Importance Sketching for Rank Constrained Least Squares: Algorithms and High-Order Convergence
    Recursive Importance Sketching for Rank Constrained Least Squares: Algorithms and High-order Convergence Yuetian Luo1, Wen Huang2, Xudong Li3, and Anru R. Zhang1;4 March 16, 2021 Abstract In this paper, we propose a new Recursive Importance Sketching algorithm for Rank con- strained least squares Optimization (RISRO). As its name suggests, the algorithm is based on a new sketching framework, recursive importance sketching. Several existing algorithms in the literature can be reinterpreted under the new sketching framework and RISRO offers clear ad- vantages over them. RISRO is easy to implement and computationally efficient, where the core procedure in each iteration is only solving a dimension reduced least squares problem. Different from numerous existing algorithms with locally geometric convergence rate, we establish the local quadratic-linear and quadratic rate of convergence for RISRO under some mild conditions. In addition, we discover a deep connection of RISRO to Riemannian manifold optimization on fixed rank matrices. The effectiveness of RISRO is demonstrated in two applications in machine learning and statistics: low-rank matrix trace regression and phase retrieval. Simulation studies demonstrate the superior numerical performance of RISRO. Keywords: Rank constrained least squares, Sketching, Quadratic convergence, Riemannian man- ifold optimization, Low-rank matrix recovery, Non-convex optimization 1 Introduction The focus of this paper is on the rank constrained least squares: 1 2 min fpXq :“ }y ´ ApXq}2 ; subject to rankpXq “ r: (1) XPRp1ˆp2 2 n p p n Here, y P R is the given data and A P R 1ˆ 2 Ñ R is a known linear map that can be explicitly arXiv:2011.08360v2 [math.OC] 14 Mar 2021 represented as J ApXq “ rxA1; Xy;:::; xAn; Xys ; xAi; Xy “ pAiqrj;ksXrj;ks (2) 1¤j¤p ;1¤k¤p ¸1 2 p ˆp with given measurement matrices Ai P R 1 2 , i “ 1; : : : ; n.
    [Show full text]
  • An Approximate Fisher Scoring Algorithm for Finite Mixtures of Multinomials Andrew M
    An Approximate Fisher Scoring Algorithm for Finite Mixtures of Multinomials Andrew M. Raim, Minglei Liu, Nagaraj K. Neerchal and Jorge G. Morel Abstract Finite mixture distributions arise naturally in many applications including clus- tering and classification. Since they usually do not yield closed forms for maximum likelihood estimates (MLEs), numerical methods using the well known Fisher Scoring or Expectation-Maximization algorithms are considered. In this work, an approximation to the Fisher Information Matrix of an arbitrary mixture of multinomial distributions is introduced. This leads to an Approximate Fisher Scoring algorithm (AFSA), which turns out to be closely related to Expectation-Maximization, and is more robust to the choice of initial value than Fisher Scoring iterations. A combination of AFSA and the classical Fisher Scoring iterations provides the best of both computational efficiency and stable convergence properties. Key Words: Multinomial; Finite mixture, Maximum likelihood, Fisher Information Matrix, Fisher Scoring. 1 Introduction A finite mixture model arises when observations are believed to originate from one of several populations, but it is unknown to which population each observation belongs. Model iden- tifiability of mixtures is an important issue with a considerable literature; see, for example, Robbins (1948) and Teicher (1960). The book by McLachlan and Peel (2000) is a good entry point to the literature on mixtures. A majority of the mixture literature deals with mix- tures of normal distributions; however, Blischke (1962; 1964), Krolikowska (1976), and Kabir (1968) are a few early works which address mixtures of discrete distributions. The present work focuses on mixtures of multinomials, which have wide applications in clustering and classification problems, as well as modeling overdispersion data (Morel and Nagaraj, 2012).
    [Show full text]