RegML 2018 Class 2 Tikhonov regularization and kernels
|
|
- Emery Rice
- 5 years ago
- Views:
Transcription
1 RegML 2018 Class 2 Tikhonov regularization and kernels Lorenzo Rosasco UNIGE-MIT-IIT June 17, 2018
2 Learning problem Problem For H {f f : X Y }, solve min E(f), f H dρ(x, y)l(f(x), y) given S n = (x i, y i ) n (ρ, fixed, unknown).
3 Empirical Risk Minimization (ERM) min f H E(f) min Ê(f) f H proxy to E Ê(f) = 1 n L(f(x i ), y i )
4 From ERM to regularization ERM can be a bad idea if n is small and H is big Regularization min Ê(f) min f H f H Ê(f) + λ R(f) }{{} regularization λ regularization parameter
5 Examples of regularizers Let p f(x) = φ j (x)w j j=1 l 2 p R(f) = w 2 = w j 2, j=1 l 1 p R(f) = w 1 = w j, j=1 Differential operators R(f) = f(x) 2 dρ(x),... X
6 From statistics to optimization Problem Solve min Ê(w) + λ w 2 w Rp with Ê(w) = 1 n L(w x i, y i ).
7 Minimization min Ê(w) + λ w 2 w Strongly convex functional Computations depends on the considered function
8 Logistic regression Ê λ (w) = 1 log(1 + e yiw x i ) + λ w 2. n }{{} smooth and strongly convex Êλ(w) = 1 n x i y i 1 + e yiw x i + 2λw
9 Gradient descent Let F : R d R differentiable, (strictly) convex and such that F (w) F (w ) L w w (e.g. sup w H(w) L) }{{} hessian Then w 0 = 0, w t+1 = w t 1 L F (w t), converges to the minimizer of F.
10 Gradient descent for LR 1 min w R p n log(1 + e yiw x i ) + λ w 2 Consider w t+1 = w t 1 L [ 1 n x i y i 1 + e yiw t xi + 2λw t ]
11 Complexity Logistic: O(ndT ) n number of examples, d dimensionality, T number of steps What if n d? Can we get better complexities?
12 Representer theorems Idea Show that f(x) = w x = x i xc i, c i R. i.e. w = n x ic i. Compute c = (c 1,..., c n ) R n rather than w R d.
13 Representer theorem for GD & LR By induction c t+1 = c t 1 L [ 1 n e i y i 1 + e yift(xi) + 2λc t ] with e i the i-th element of the canonical basis and f t (x) = x x i (c t ) i
14 Non-linear features d p f(x) = w i x i f(x) = w i φ i (x i ). Model Φ(x) = (φ 1 (x),..., φ p (x)),
15 Non-linear features d p f(x) = w i x i f(x) = w i φ i (x i ). Computations Same up-to the change Φ(x) = (φ 1 (x),..., φ p (x)), x Φ(x)
16 Representer theorem with non-linear features f(x) = x i xc i f(x) = Φ(x i ) Φ(x)c i
17 Rewriting logistic regression and gradient descent By induction c t+1 = c t 1 L [ 1 n e i y i 1 + e yift(xi) + 2λc t ] with e i the i-th element of the canonical basis and f t (x) = Φ(x) Φ(x) i (c t ) i
18 Hinge loss and support vector machines Ê λ (w) = 1 1 y i w x i + + λ w 2 n }{{} non-smooth & strongly-convex Consider left derivative S i (w) = w t+1 = w t 1 L t ( 1 n ) S i (w t ) + 2λw t { y i x i if y i w x i 1, L = sup x + 2λ. 0 otherwise x X
19 Subgradient Let F : R p R convex, Subgradient F (w 0 ) set of vectors v R p such that, for every w R p F (w) F (w 0 ) (w w 0 ) v In one dimension F (w 0 ) = [F (w 0 ), F +(w 0 )].
20 Subgradient method Let F : R p R strictly convex, with bounded subdifferential, and γ t = 1/t then, w t+1 = w t γ t v t with v t F (w t ) converges to the minimizer of F.
21 Subgradient method for SVM 1 min w R p n 1 y i w x i + + λ w 2 Consider w t+1 = w t 1 t ( 1 n ) S i (w t ) + 2λw t S i (w t ) = { y i x i if y i w x i 1 0 otherwise
22 Representer theorem of SVM By induction c t+1 = c t 1 t ( 1 n ) S i (c t ) + 2λc t with e i the i-th element of the canonical basis, f t (x) = x x i (c t ) i and S i (c t ) = { y i e i if y i f t (x i ) < 1. 0 otherwise
23 Rewriting SVM by subgradient By induction c t+1 = c t 1 t ( 1 n ) S i (c t ) + 2λc t with e i the i-th element of the canonical basis, f t (x) = Φ(x) Φ(x) i (c t ) i and S i (c t ) = { y i e i if y i f t (x i ) < 1. 0 otherwise
24 Optimality condition for SVM Smooth Convex F (w ) = 0 Non-smooth Convex 0 F (w) 0 F (w ) 0 1 y i w x i + + λ2w w 1 2λ 1 y iw x i +.
25 Optimality condition for SVM (cont.) The optimality condition can be rewritten as 0 = 1 n ( y i x i c i ) + 2λw w = where c i = c i (w) [V ( y i w x i ), V + ( y i w x i )]. x i ( y ic i 2λn ). A direct computation gives c i = 1 if yf(x i ) < 1 0 c i 1 if yf(x i ) = 1 c i = 0 if yf(x i ) > 1
26 Support vectors c i = 1 if yf(x i ) < 1 0 c i 1 if yf(x i ) = 1 c i = 0 if yf(x i ) > 1
27 Complexity Without representer Logistic: O(ndT ) SVM: O(ndT ) With representer Logistic: O(n 2 (d + T )) SVM: O(n 2 (d + T )) n number of example, d dimensionality, T number of steps
28 Are loss functions all the same? min Ê(w) + λ w 2 w each loss has a different target function and different computations The choice of the loss is problem dependent
29 So far regularization by penalization iterative optimization linear/non-linear parametric models What about nonparametric models?
30 From features to kernels f(x) = x i xc i f(x) = Φ(x i ) Φ(x)c i Kernels Φ(x) Φ(x ) K(x, x ) f(x) = K(x i, x)c i
31 LR and SVM with kernels As before: LR SVM But now c t+1 = c t 1 L [ c t+1 = c t 1 t f t (x) = 1 n ( 1 n e i y i 1 + e yift(xi) + 2λc t ) S i (c t ) + 2λc t K(x, x i )(c t ) i ]
32 Examples of kernels Linear K(x, x ) = x x Polynomial K(x, x ) = (1 + x x) p, with p N Gaussian K(x, x ) = e γ x x 2, with γ > 0 f(x) = c i K(x i, x)
33 Kernel engineering Kernels for Strings, Graphs, Histograms, Sets,...
34 What is a kernel? K(x, x ) Similarity measure Inner product Positive definite function
35 Positive definite function K : X X R is positive definite, when for any n N, x 1,..., x n X, let K n such that K n R n n, (K n ) ij = K(x i, x j ), then K n is positive semidefinite, (eigenvalues 0)
36 PD functions and RKHS Each PD Kernel defines a function space called Reproducing kernel Hilbert space (RKHS) H = span {K(, x) x X}.
37 Nonparametrics and kernels Number of parameters automatically determined by number of points f(x) = K(x i, x)c i Compare to p f(x) = φ j (x)w j j=1
38 This class Learning and Regularization: logistic regression and SVM Optimization with first order methods Linear and Non-linear parametric models Non-parametric models and kernels
39 Next class Beyond penalization Regularization by subsampling stochastic projection
Oslo Class 2 Tikhonov regularization and kernels
RegML2017@SIMULA Oslo Class 2 Tikhonov regularization and kernels Lorenzo Rosasco UNIGE-MIT-IIT May 3, 2017 Learning problem Problem For H {f f : X Y }, solve min E(f), f H dρ(x, y)l(f(x), y) given S n
More informationOslo Class 4 Early Stopping and Spectral Regularization
RegML2017@SIMULA Oslo Class 4 Early Stopping and Spectral Regularization Lorenzo Rosasco UNIGE-MIT-IIT June 28, 2016 Learning problem Solve min w E(w), E(w) = dρ(x, y)l(w x, y) given (x 1, y 1 ),..., (x
More informationMLCC 2017 Regularization Networks I: Linear Models
MLCC 2017 Regularization Networks I: Linear Models Lorenzo Rosasco UNIGE-MIT-IIT June 27, 2017 About this class We introduce a class of learning algorithms based on Tikhonov regularization We study computational
More informationReproducing Kernel Hilbert Spaces
Reproducing Kernel Hilbert Spaces Lorenzo Rosasco 9.520 Class 03 February 12, 2007 About this class Goal To introduce a particularly useful family of hypothesis spaces called Reproducing Kernel Hilbert
More informationTUM 2016 Class 3 Large scale learning by regularization
TUM 2016 Class 3 Large scale learning by regularization Lorenzo Rosasco UNIGE-MIT-IIT July 25, 2016 Learning problem Solve min w E(w), E(w) = dρ(x, y)l(w x, y) given (x 1, y 1 ),..., (x n, y n ) Beyond
More informationReproducing Kernel Hilbert Spaces
Reproducing Kernel Hilbert Spaces Lorenzo Rosasco 9.520 Class 03 February 9, 2011 About this class Goal To introduce a particularly useful family of hypothesis spaces called Reproducing Kernel Hilbert
More informationReproducing Kernel Hilbert Spaces Class 03, 15 February 2006 Andrea Caponnetto
Reproducing Kernel Hilbert Spaces 9.520 Class 03, 15 February 2006 Andrea Caponnetto About this class Goal To introduce a particularly useful family of hypothesis spaces called Reproducing Kernel Hilbert
More informationReproducing Kernel Hilbert Spaces
Reproducing Kernel Hilbert Spaces Lorenzo Rosasco 9.520 Class 03 February 11, 2009 About this class Goal To introduce a particularly useful family of hypothesis spaces called Reproducing Kernel Hilbert
More informationReproducing Kernel Hilbert Spaces
Reproducing Kernel Hilbert Spaces Lorenzo Rosasco 9.520 Class 03 February 9, 2011 About this class Goal In this class we continue our journey in the world of RKHS. We discuss the Mercer theorem which gives
More informationRegularization via Spectral Filtering
Regularization via Spectral Filtering Lorenzo Rosasco MIT, 9.520 Class 7 About this class Goal To discuss how a class of regularization methods originally designed for solving ill-posed inverse problems,
More informationSpectral Regularization
Spectral Regularization Lorenzo Rosasco 9.520 Class 07 February 27, 2008 About this class Goal To discuss how a class of regularization methods originally designed for solving ill-posed inverse problems,
More informationTUM 2016 Class 1 Statistical learning theory
TUM 2016 Class 1 Statistical learning theory Lorenzo Rosasco UNIGE-MIT-IIT July 25, 2016 Machine learning applications Texts Images Data: (x 1, y 1 ),..., (x n, y n ) Note: x i s huge dimensional! All
More informationSupport Vector Machine
Support Vector Machine Fabrice Rossi SAMM Université Paris 1 Panthéon Sorbonne 2018 Outline Linear Support Vector Machine Kernelized SVM Kernels 2 From ERM to RLM Empirical Risk Minimization in the binary
More informationOslo Class 6 Sparsity based regularization
RegML2017@SIMULA Oslo Class 6 Sparsity based regularization Lorenzo Rosasco UNIGE-MIT-IIT May 4, 2017 Learning from data Possible only under assumptions regularization min Ê(w) + λr(w) w Smoothness Sparsity
More informationThe Learning Problem and Regularization Class 03, 11 February 2004 Tomaso Poggio and Sayan Mukherjee
The Learning Problem and Regularization 9.520 Class 03, 11 February 2004 Tomaso Poggio and Sayan Mukherjee About this class Goal To introduce a particularly useful family of hypothesis spaces called Reproducing
More informationMIT 9.520/6.860, Fall 2018 Statistical Learning Theory and Applications. Class 04: Features and Kernels. Lorenzo Rosasco
MIT 9.520/6.860, Fall 2018 Statistical Learning Theory and Applications Class 04: Features and Kernels Lorenzo Rosasco Linear functions Let H lin be the space of linear functions f(x) = w x. f w is one
More informationLecture 14 : Online Learning, Stochastic Gradient Descent, Perceptron
CS446: Machine Learning, Fall 2017 Lecture 14 : Online Learning, Stochastic Gradient Descent, Perceptron Lecturer: Sanmi Koyejo Scribe: Ke Wang, Oct. 24th, 2017 Agenda Recap: SVM and Hinge loss, Representer
More informationReproducing Kernel Hilbert Spaces
9.520: Statistical Learning Theory and Applications February 10th, 2010 Reproducing Kernel Hilbert Spaces Lecturer: Lorenzo Rosasco Scribe: Greg Durrett 1 Introduction In the previous two lectures, we
More informationRegML 2018 Class 8 Deep learning
RegML 2018 Class 8 Deep learning Lorenzo Rosasco UNIGE-MIT-IIT June 18, 2018 Supervised vs unsupervised learning? So far we have been thinking of learning schemes made in two steps f(x) = w, Φ(x) F, x
More informationBasis Expansion and Nonlinear SVM. Kai Yu
Basis Expansion and Nonlinear SVM Kai Yu Linear Classifiers f(x) =w > x + b z(x) = sign(f(x)) Help to learn more general cases, e.g., nonlinear models 8/7/12 2 Nonlinear Classifiers via Basis Expansion
More informationStochastic optimization in Hilbert spaces
Stochastic optimization in Hilbert spaces Aymeric Dieuleveut Aymeric Dieuleveut Stochastic optimization Hilbert spaces 1 / 48 Outline Learning vs Statistics Aymeric Dieuleveut Stochastic optimization Hilbert
More informationHilbert Space Methods in Learning
Hilbert Space Methods in Learning guest lecturer: Risi Kondor 6772 Advanced Machine Learning and Perception (Jebara), Columbia University, October 15, 2003. 1 1. A general formulation of the learning problem
More informationIntroductory Machine Learning Notes 1. Lorenzo Rosasco
Introductory Machine Learning Notes 1 Lorenzo Rosasco DIBRIS, Universita degli Studi di Genova LCSL, Massachusetts Institute of Technology and Istituto Italiano di Tecnologia lrosasco@mit.edu October 10,
More informationThe Representor Theorem, Kernels, and Hilbert Spaces
The Representor Theorem, Kernels, and Hilbert Spaces We will now work with infinite dimensional feature vectors and parameter vectors. The space l is defined to be the set of sequences f 1, f, f 3,...
More informationIntroductory Machine Learning Notes 1
Introductory Machine Learning Notes 1 Lorenzo Rosasco DIBRIS, Universita degli Studi di Genova LCSL, Massachusetts Institute of Technology and Istituto Italiano di Tecnologia lrosasco@mit.edu December
More informationMIT 9.520/6.860, Fall 2018 Statistical Learning Theory and Applications. Class 08: Sparsity Based Regularization. Lorenzo Rosasco
MIT 9.520/6.860, Fall 2018 Statistical Learning Theory and Applications Class 08: Sparsity Based Regularization Lorenzo Rosasco Learning algorithms so far ERM + explicit l 2 penalty 1 min w R d n n l(y
More informationMachine Learning. Support Vector Machines. Fabio Vandin November 20, 2017
Machine Learning Support Vector Machines Fabio Vandin November 20, 2017 1 Classification and Margin Consider a classification problem with two classes: instance set X = R d label set Y = { 1, 1}. Training
More informationOnline Gradient Descent Learning Algorithms
DISI, Genova, December 2006 Online Gradient Descent Learning Algorithms Yiming Ying (joint work with Massimiliano Pontil) Department of Computer Science, University College London Introduction Outline
More informationGeneralization theory
Generalization theory Daniel Hsu Columbia TRIPODS Bootcamp 1 Motivation 2 Support vector machines X = R d, Y = { 1, +1}. Return solution ŵ R d to following optimization problem: λ min w R d 2 w 2 2 + 1
More informationLess is More: Computational Regularization by Subsampling
Less is More: Computational Regularization by Subsampling Lorenzo Rosasco University of Genova - Istituto Italiano di Tecnologia Massachusetts Institute of Technology lcsl.mit.edu joint work with Alessandro
More informationLecture 18: Kernels Risk and Loss Support Vector Regression. Aykut Erdem December 2016 Hacettepe University
Lecture 18: Kernels Risk and Loss Support Vector Regression Aykut Erdem December 2016 Hacettepe University Administrative We will have a make-up lecture on next Saturday December 24, 2016 Presentations
More informationCIS 520: Machine Learning Oct 09, Kernel Methods
CIS 520: Machine Learning Oct 09, 207 Kernel Methods Lecturer: Shivani Agarwal Disclaimer: These notes are designed to be a supplement to the lecture They may or may not cover all the material discussed
More information2 Tikhonov Regularization and ERM
Introduction Here we discusses how a class of regularization methods originally designed to solve ill-posed inverse problems give rise to regularized learning algorithms. These algorithms are kernel methods
More informationLinear classifiers selecting hyperplane maximizing separation margin between classes (large margin classifiers)
Support vector machines In a nutshell Linear classifiers selecting hyperplane maximizing separation margin between classes (large margin classifiers) Solution only depends on a small subset of training
More informationFunctional Gradient Descent
Statistical Techniques in Robotics (16-831, F12) Lecture #21 (Nov 14, 2012) Functional Gradient Descent Lecturer: Drew Bagnell Scribe: Daniel Carlton Smith 1 1 Goal of Functional Gradient Descent We have
More informationMidterm exam CS 189/289, Fall 2015
Midterm exam CS 189/289, Fall 2015 You have 80 minutes for the exam. Total 100 points: 1. True/False: 36 points (18 questions, 2 points each). 2. Multiple-choice questions: 24 points (8 questions, 3 points
More informationSupport Vector Machines for Classification: A Statistical Portrait
Support Vector Machines for Classification: A Statistical Portrait Yoonkyung Lee Department of Statistics The Ohio State University May 27, 2011 The Spring Conference of Korean Statistical Society KAIST,
More informationStat542 (F11) Statistical Learning. First consider the scenario where the two classes of points are separable.
Linear SVM (separable case) First consider the scenario where the two classes of points are separable. It s desirable to have the width (called margin) between the two dashed lines to be large, i.e., have
More informationLess is More: Computational Regularization by Subsampling
Less is More: Computational Regularization by Subsampling Lorenzo Rosasco University of Genova - Istituto Italiano di Tecnologia Massachusetts Institute of Technology lcsl.mit.edu joint work with Alessandro
More informationSVMs: Non-Separable Data, Convex Surrogate Loss, Multi-Class Classification, Kernels
SVMs: Non-Separable Data, Convex Surrogate Loss, Multi-Class Classification, Kernels Karl Stratos June 21, 2018 1 / 33 Tangent: Some Loose Ends in Logistic Regression Polynomial feature expansion in logistic
More informationIndirect Rule Learning: Support Vector Machines. Donglin Zeng, Department of Biostatistics, University of North Carolina
Indirect Rule Learning: Support Vector Machines Indirect learning: loss optimization It doesn t estimate the prediction rule f (x) directly, since most loss functions do not have explicit optimizers. Indirection
More informationAbout this class. Maximizing the Margin. Maximum margin classifiers. Picture of large and small margin hyperplanes
About this class Maximum margin classifiers SVMs: geometric derivation of the primal problem Statement of the dual problem The kernel trick SVMs as the solution to a regularization problem Maximizing the
More informationMLCC 2018 Variable Selection and Sparsity. Lorenzo Rosasco UNIGE-MIT-IIT
MLCC 2018 Variable Selection and Sparsity Lorenzo Rosasco UNIGE-MIT-IIT Outline Variable Selection Subset Selection Greedy Methods: (Orthogonal) Matching Pursuit Convex Relaxation: LASSO & Elastic Net
More informationChapter 9. Support Vector Machine. Yongdai Kim Seoul National University
Chapter 9. Support Vector Machine Yongdai Kim Seoul National University 1. Introduction Support Vector Machine (SVM) is a classification method developed by Vapnik (1996). It is thought that SVM improved
More informationMachine Learning. Classification, Discriminative learning. Marc Toussaint University of Stuttgart Summer 2015
Machine Learning Classification, Discriminative learning Structured output, structured input, discriminative function, joint input-output features, Likelihood Maximization, Logistic regression, binary
More informationDiffeomorphic Warping. Ben Recht August 17, 2006 Joint work with Ali Rahimi (Intel)
Diffeomorphic Warping Ben Recht August 17, 2006 Joint work with Ali Rahimi (Intel) What Manifold Learning Isn t Common features of Manifold Learning Algorithms: 1-1 charting Dense sampling Geometric Assumptions
More informationBits of Machine Learning Part 1: Supervised Learning
Bits of Machine Learning Part 1: Supervised Learning Alexandre Proutiere and Vahan Petrosyan KTH (The Royal Institute of Technology) Outline of the Course 1. Supervised Learning Regression and Classification
More informationIntroductory Machine Learning Notes 1. Lorenzo Rosasco
Introductory Machine Learning Notes Lorenzo Rosasco lrosasco@mit.edu April 4, 204 These are the notes for the course master level course Intelligent Systems and Machine Learning (ISML)- Module II for the
More informationOverfitting, Bias / Variance Analysis
Overfitting, Bias / Variance Analysis Professor Ameet Talwalkar Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 8, 207 / 40 Outline Administration 2 Review of last lecture 3 Basic
More informationECS289: Scalable Machine Learning
ECS289: Scalable Machine Learning Cho-Jui Hsieh UC Davis Sept 29, 2016 Outline Convex vs Nonconvex Functions Coordinate Descent Gradient Descent Newton s method Stochastic Gradient Descent Numerical Optimization
More informationIntroduction to Logistic Regression and Support Vector Machine
Introduction to Logistic Regression and Support Vector Machine guest lecturer: Ming-Wei Chang CS 446 Fall, 2009 () / 25 Fall, 2009 / 25 Before we start () 2 / 25 Fall, 2009 2 / 25 Before we start Feel
More information9.520 Problem Set 2. Due April 25, 2011
9.50 Problem Set Due April 5, 011 Note: there are five problems in total in this set. Problem 1 In classification problems where the data are unbalanced (there are many more examples of one class than
More informationMachine Learning. Support Vector Machines. Manfred Huber
Machine Learning Support Vector Machines Manfred Huber 2015 1 Support Vector Machines Both logistic regression and linear discriminant analysis learn a linear discriminant function to separate the data
More informationEE613 Machine Learning for Engineers. Kernel methods Support Vector Machines. jean-marc odobez 2015
EE613 Machine Learning for Engineers Kernel methods Support Vector Machines jean-marc odobez 2015 overview Kernel methods introductions and main elements defining kernels Kernelization of k-nn, K-Means,
More informationKernel Methods. Outline
Kernel Methods Quang Nguyen University of Pittsburgh CS 3750, Fall 2011 Outline Motivation Examples Kernels Definitions Kernel trick Basic properties Mercer condition Constructing feature space Hilbert
More informationClass 2 & 3 Overfitting & Regularization
Class 2 & 3 Overfitting & Regularization Carlo Ciliberto Department of Computer Science, UCL October 18, 2017 Last Class The goal of Statistical Learning Theory is to find a good estimator f n : X Y, approximating
More informationKernel Methods. Jean-Philippe Vert Last update: Jan Jean-Philippe Vert (Mines ParisTech) 1 / 444
Kernel Methods Jean-Philippe Vert Jean-Philippe.Vert@mines.org Last update: Jan 2015 Jean-Philippe Vert (Mines ParisTech) 1 / 444 What we know how to solve Jean-Philippe Vert (Mines ParisTech) 2 / 444
More informationMIT 9.520/6.860, Fall 2017 Statistical Learning Theory and Applications. Class 19: Data Representation by Design
MIT 9.520/6.860, Fall 2017 Statistical Learning Theory and Applications Class 19: Data Representation by Design What is data representation? Let X be a data-space X M (M) F (M) X A data representation
More informationLecture 10 February 23
EECS 281B / STAT 241B: Advanced Topics in Statistical LearningSpring 2009 Lecture 10 February 23 Lecturer: Martin Wainwright Scribe: Dave Golland Note: These lecture notes are still rough, and have only
More informationKernel Learning via Random Fourier Representations
Kernel Learning via Random Fourier Representations L. Law, M. Mider, X. Miscouridou, S. Ip, A. Wang Module 5: Machine Learning L. Law, M. Mider, X. Miscouridou, S. Ip, A. Wang Kernel Learning via Random
More informationSVMs, Duality and the Kernel Trick
SVMs, Duality and the Kernel Trick Machine Learning 10701/15781 Carlos Guestrin Carnegie Mellon University February 26 th, 2007 2005-2007 Carlos Guestrin 1 SVMs reminder 2005-2007 Carlos Guestrin 2 Today
More informationSTA141C: Big Data & High Performance Statistical Computing
STA141C: Big Data & High Performance Statistical Computing Lecture 8: Optimization Cho-Jui Hsieh UC Davis May 9, 2017 Optimization Numerical Optimization Numerical Optimization: min X f (X ) Can be applied
More informationNotes on Regularized Least Squares Ryan M. Rifkin and Ross A. Lippert
Computer Science and Artificial Intelligence Laboratory Technical Report MIT-CSAIL-TR-2007-025 CBCL-268 May 1, 2007 Notes on Regularized Least Squares Ryan M. Rifkin and Ross A. Lippert massachusetts institute
More informationA summary of Deep Learning without Poor Local Minima
A summary of Deep Learning without Poor Local Minima by Kenji Kawaguchi MIT oral presentation at NIPS 2016 Learning Supervised (or Predictive) learning Learn a mapping from inputs x to outputs y, given
More informationConvex Optimization Lecture 16
Convex Optimization Lecture 16 Today: Projected Gradient Descent Conditional Gradient Descent Stochastic Gradient Descent Random Coordinate Descent Recall: Gradient Descent (Steepest Descent w.r.t Euclidean
More informationDeviations from linear separability. Kernel methods. Basis expansion for quadratic boundaries. Adding new features Systematic deviation
Deviations from linear separability Kernel methods CSE 250B Noise Find a separator that minimizes a convex loss function related to the number of mistakes. e.g. SVM, logistic regression. Systematic deviation
More informationl 1 and l 2 Regularization
David Rosenberg New York University February 5, 2015 David Rosenberg (New York University) DS-GA 1003 February 5, 2015 1 / 32 Tikhonov and Ivanov Regularization Hypothesis Spaces We ve spoken vaguely about
More informationBig Data Analytics: Optimization and Randomization
Big Data Analytics: Optimization and Randomization Tianbao Yang Tutorial@ACML 2015 Hong Kong Department of Computer Science, The University of Iowa, IA, USA Nov. 20, 2015 Yang Tutorial for ACML 15 Nov.
More informationMark your answers ON THE EXAM ITSELF. If you are not sure of your answer you may wish to provide a brief explanation.
CS 189 Spring 2015 Introduction to Machine Learning Midterm You have 80 minutes for the exam. The exam is closed book, closed notes except your one-page crib sheet. No calculators or electronic items.
More informationKernel methods CSE 250B
Kernel methods CSE 250B Deviations from linear separability Noise Find a separator that minimizes a convex loss function related to the number of mistakes. e.g. SVM, logistic regression. Deviations from
More informationSVMC An introduction to Support Vector Machines Classification
SVMC An introduction to Support Vector Machines Classification 6.783, Biomedical Decision Support Lorenzo Rosasco (lrosasco@mit.edu) Department of Brain and Cognitive Science MIT A typical problem We have
More informationKernelized Perceptron Support Vector Machines
Kernelized Perceptron Support Vector Machines Emily Fox University of Washington February 13, 2017 What is the perceptron optimizing? 1 The perceptron algorithm [Rosenblatt 58, 62] Classification setting:
More informationRKHS, Mercer s theorem, Unbounded domains, Frames and Wavelets Class 22, 2004 Tomaso Poggio and Sayan Mukherjee
RKHS, Mercer s theorem, Unbounded domains, Frames and Wavelets 9.520 Class 22, 2004 Tomaso Poggio and Sayan Mukherjee About this class Goal To introduce an alternate perspective of RKHS via integral operators
More informationCase Study 1: Estimating Click Probabilities. Kakade Announcements: Project Proposals: due this Friday!
Case Study 1: Estimating Click Probabilities Intro Logistic Regression Gradient Descent + SGD Machine Learning for Big Data CSE547/STAT548, University of Washington Sham Kakade April 4, 017 1 Announcements:
More informationStatistical Optimality of Stochastic Gradient Descent through Multiple Passes
Statistical Optimality of Stochastic Gradient Descent through Multiple Passes Francis Bach INRIA - Ecole Normale Supérieure, Paris, France ÉCOLE NORMALE SUPÉRIEURE Joint work with Loucas Pillaud-Vivien
More informationStatistical learning on graphs
Statistical learning on graphs Jean-Philippe Vert Jean-Philippe.Vert@ensmp.fr ParisTech, Ecole des Mines de Paris Institut Curie INSERM U900 Seminar of probabilities, Institut Joseph Fourier, Grenoble,
More informationMATH 829: Introduction to Data Mining and Analysis Support vector machines and kernels
1/12 MATH 829: Introduction to Data Mining and Analysis Support vector machines and kernels Dominique Guillot Departments of Mathematical Sciences University of Delaware March 14, 2016 Separating sets:
More informationSupport Vector Machine
Andrea Passerini passerini@disi.unitn.it Machine Learning Support vector machines In a nutshell Linear classifiers selecting hyperplane maximizing separation margin between classes (large margin classifiers)
More informationKernel Methods for Computational Biology and Chemistry
Kernel Methods for Computational Biology and Chemistry Jean-Philippe Vert Jean-Philippe.Vert@ensmp.fr Centre for Computational Biology Ecole des Mines de Paris, ParisTech Mathematical Statistics and Applications,
More informationRepresenter theorem and kernel examples
CS81B/Stat41B Spring 008) Statistical Learning Theory Lecture: 8 Representer theorem and kernel examples Lecturer: Peter Bartlett Scribe: Howard Lei 1 Representer Theorem Recall that the SVM optimization
More informationMLCC 2015 Dimensionality Reduction and PCA
MLCC 2015 Dimensionality Reduction and PCA Lorenzo Rosasco UNIGE-MIT-IIT June 25, 2015 Outline PCA & Reconstruction PCA and Maximum Variance PCA and Associated Eigenproblem Beyond the First Principal Component
More information(Kernels +) Support Vector Machines
(Kernels +) Support Vector Machines Machine Learning Torsten Möller Reading Chapter 5 of Machine Learning An Algorithmic Perspective by Marsland Chapter 6+7 of Pattern Recognition and Machine Learning
More informationUnsupervised Learning Techniques Class 07, 1 March 2006 Andrea Caponnetto
Unsupervised Learning Techniques 9.520 Class 07, 1 March 2006 Andrea Caponnetto About this class Goal To introduce some methods for unsupervised learning: Gaussian Mixtures, K-Means, ISOMAP, HLLE, Laplacian
More informationNesterov s Optimal Gradient Methods
Yurii Nesterov http://www.core.ucl.ac.be/~nesterov Nesterov s Optimal Gradient Methods Xinhua Zhang Australian National University NICTA 1 Outline The problem from machine learning perspective Preliminaries
More informationConvex optimization COMS 4771
Convex optimization COMS 4771 1. Recap: learning via optimization Soft-margin SVMs Soft-margin SVM optimization problem defined by training data: w R d λ 2 w 2 2 + 1 n n [ ] 1 y ix T i w. + 1 / 15 Soft-margin
More informationMachine Learning Support Vector Machines. Prof. Matteo Matteucci
Machine Learning Support Vector Machines Prof. Matteo Matteucci Discriminative vs. Generative Approaches 2 o Generative approach: we derived the classifier from some generative hypothesis about the way
More informationStatistical learning theory, Support vector machines, and Bioinformatics
1 Statistical learning theory, Support vector machines, and Bioinformatics Jean-Philippe.Vert@mines.org Ecole des Mines de Paris Computational Biology group ENS Paris, november 25, 2003. 2 Overview 1.
More informationA GENERAL FORMULATION FOR SUPPORT VECTOR MACHINES. Wei Chu, S. Sathiya Keerthi, Chong Jin Ong
A GENERAL FORMULATION FOR SUPPORT VECTOR MACHINES Wei Chu, S. Sathiya Keerthi, Chong Jin Ong Control Division, Department of Mechanical Engineering, National University of Singapore 0 Kent Ridge Crescent,
More informationThe Learning Problem and Regularization
9.520 Class 02 February 2011 Computational Learning Statistical Learning Theory Learning is viewed as a generalization/inference problem from usually small sets of high dimensional, noisy data. Learning
More informationRobust Support Vector Machines for Probability Distributions
Robust Support Vector Machines for Probability Distributions Andreas Christmann joint work with Ingo Steinwart (Los Alamos National Lab) ICORS 2008, Antalya, Turkey, September 8-12, 2008 Andreas Christmann,
More informationOslo Class 7 Structured sparsity
RegML2017@SIMULA Oslo Class 7 Structured sparsity Lorenzo Rosasco UNIGE-MIT-IIT May 4, 2017 Exploiting structure Building blocks of a function can be more structured than single variables Sparsity Variables
More informationSupport Vector Machines
Two SVM tutorials linked in class website (please, read both): High-level presentation with applications (Hearst 1998) Detailed tutorial (Burges 1998) Support Vector Machines Machine Learning 10701/15781
More informationKernel Methods. Barnabás Póczos
Kernel Methods Barnabás Póczos Outline Quick Introduction Feature space Perceptron in the feature space Kernels Mercer s theorem Finite domain Arbitrary domain Kernel families Constructing new kernels
More information5.6 Nonparametric Logistic Regression
5.6 onparametric Logistic Regression Dmitri Dranishnikov University of Florida Statistical Learning onparametric Logistic Regression onparametric? Doesnt mean that there are no parameters. Just means that
More informationMax Margin-Classifier
Max Margin-Classifier Oliver Schulte - CMPT 726 Bishop PRML Ch. 7 Outline Maximum Margin Criterion Math Maximizing the Margin Non-Separable Data Kernels and Non-linear Mappings Where does the maximization
More informationStatistical Methods for Data Mining
Statistical Methods for Data Mining Kuangnan Fang Xiamen University Email: xmufkn@xmu.edu.cn Support Vector Machines Here we approach the two-class classification problem in a direct way: We try and find
More informationDecentralized Detection and Classification using Kernel Methods
Decentralized Detection and Classification using Kernel Methods XuanLong Nguyen Computer Science Division University of California, Berkeley xuanlong@cs.berkeley.edu Martin J. Wainwright Electrical Engineering
More informationBayesian Support Vector Machines for Feature Ranking and Selection
Bayesian Support Vector Machines for Feature Ranking and Selection written by Chu, Keerthi, Ong, Ghahramani Patrick Pletscher pat@student.ethz.ch ETH Zurich, Switzerland 12th January 2006 Overview 1 Introduction
More informationConvex Optimization Algorithms for Machine Learning in 10 Slides
Convex Optimization Algorithms for Machine Learning in 10 Slides Presenter: Jul. 15. 2015 Outline 1 Quadratic Problem Linear System 2 Smooth Problem Newton-CG 3 Composite Problem Proximal-Newton-CD 4 Non-smooth,
More informationApproximation Theoretical Questions for SVMs
Ingo Steinwart LA-UR 07-7056 October 20, 2007 Statistical Learning Theory: an Overview Support Vector Machines Informal Description of the Learning Goal X space of input samples Y space of labels, usually
More information