# The Lagrangian L : R d R m R r R is an (easier to optimize) lower bound on the original problem:

Size: px
Start display at page:

Download "The Lagrangian L : R d R m R r R is an (easier to optimize) lower bound on the original problem:"

## Transcription

1 HT05: SC4 Statistical Data Mining and Machine Learning Dino Sejdinovic Department of Statistics Oxford Convex Optimization and slides based on Arthur Gretton s Advanced Topics in Machine Learning course Optimization and the Lagrangian Optimization problem on x R d / primal, imize f 0 x) subject to f i x) 0 i =,..., m h j x) = 0 j =,... r. Lower bound interpretation of Lagrangian The Lagrangian L : R d R m R r R is an easier to optimize) lower bound on the original problem: Lx, λ, ν) := f 0 x) + λ i f i x) + }} I f i x)) ν j h j x), }} I 0 h j x)) j= domain D := m i=0 domf i r j= domh j nonempty). p : the primal) optimal value Idealy we would want an unconstrained problem where I u) = imize f 0 x) + 0, u 0, u > 0 I f i x)) + and I 0 u) = I 0 h j x)), j= 0, u = 0, u 0. and has domain doml := D R m R r. The vectors λ and ν are called Lagrange multipliers or dual variables. To ensure a lower bound, we require λ 0. I ) f i x) I 0 ) h i x)

2 Lagrange dual: lower bound on optimum p Lagrange dual: lower bound on optimum p The Lagrange dual function: imize Lagrangian When λ 0 and f i x) 0, Lagrange dual function is Simplest example: imize over x the function Lx, λ) = f 0 x) + λf x) f 0 +λf gλ, ν) := inf Lx, λ, ν). A dual feasible pair λ, ν) is a pair for which λ 0 and λ, ν) domg). We will show: next slide) for any λ 0 and ν, wherever gλ, ν) f 0 x) f i x) 0 h j x) = 0 p f 0 f Reders: f 0 is function to be imized. f 0 is inequality constraint λ 0 is Lagrange multiplier p is imum f 0 in constraint set including at optimal point f 0 x ) = p ). f 0 Figure from Boyd and Vandenberghe Lagrange dual: lower bound on optimum p Lagrange dual: lower bound on optimum p Simplest example: imize over x the function Lx, λ) = f 0 x) + λf x)!"#\$%&'") f 0 +λf Simplest example: imize over x the function Lx, λ) = f 0 x) + λf x)!"#\$!%"%& f 0 +λf Reders: Reders: p f 0 is function to be imized. p f 0 is function to be imized. f 0 f 0 is inequality constraint f 0 f 0 is inequality constraint f λ 0 is Lagrange multiplier f λ 0 is Lagrange multiplier p is imum f 0 in p is imum f 0 in constraint set constraint set f 0 f 0 Figure from Boyd and Vandenberghe Figure from Boyd and Vandenberghe

3 Lagrange dual is a lower bound on p Assume x is feasible, i.e. f i x) 0, h i x) = 0, x D, λ 0. Then Thus λ i f i x) + gλ, ν) := inf f 0 x) + f 0 x). f 0 x) + ν i h i x) 0 λ i f i x) + λ i f i x) + ) ν i h i x) ν i h i x) This holds for every feasible x, hence lower bound holds. Best lower bound: maximize the dual Best lower bound gλ, ν) on the optimal solution p of original problem: Lagrange dual problem maximize gλ, ν) subject to λ 0. Dual feasible: λ, ν) with λ 0 and gλ, ν) >. Dual optimal: solutions λ, ν ) to the dual problem, d is optimal value dual always easy to maximize: next slide). Weak duality always holds: max λ 0,ν Lx, λ, ν) }} =gλ,ν) = d p = = max Lx, λ, ν) λ 0,ν }} f 0 x) if constraints satisfied, otherwise. Strong duality: does not always hold, conditions given later): d = p. If strong duality holds: solve the easy concave) dual problem to find p. Maximizing the dual is always easy The Lagrange dual function: imize Lagrangian lower bound) gλ, ν) = inf Lx, λ, ν). Dual function is a pointwise infimum of affine functions of λ, ν), hence concave in λ, ν) with convex constraint set λ 0. How do we know if strong duality holds? Conditions under which strong duality holds are called constraint qualifications they are sufficient, but not necessary) Probably) best known sufficient condition: Strong duality holds if Primal problem is convex, i.e. of the form gλ) λ Example: One inequality constraint, Lx, λ) = f 0 x) + λf x), and assume there are only four possible values for x. Each line represents a different x. imize for convex f 0,..., f m, and f 0 x) subject to f i x) 0 i =,..., n Ax = b Slater s condition: there exists a strictly feasible point x, such that f i x) < 0, i =,..., n reduces to the existence of any feasible point when inequality constraints are affine, i.e., Cx d).

4 A consequence of strong duality......is complementary slackness Assume primal is equal to the dual. What are the consequences? x solution of original problem imum of f 0 under constraints), λ, ν ) solutions to dual f 0 x ) = assumed) = g definition) inf definition) 4) gλ, ν ) inf f 0 x ) + f 0 x ), f 0 x) + λ i f i x) + λ i f i x ) + ) p νi h i x) p νi h i x ) From previous slide, λ i f i x ) = 0, ) which is the condition of complementary slackness. This means λ i > 0 = f i x ) = 0, f i x ) < 0 = λ i = 0. From λ i, read off which inequality constraints are strict. 4): x, λ, ν ) satisfies λ 0, f i x ) 0, and h i x ) = 0. KKT conditions for global optimum KKT conditions for global optimum Assume functions f i, h i are differentiable and strong duality. Since x imizes Lx, λ, ν ), derivative at x is zero, f 0 x ) + λ i f i x ) + νi h i x ) = 0. KKT conditions definition: we are at global optimum, x, λ, ν ) when a) strong duality holds, and b): f 0 x ) + λ i f i x ) + f i x ) 0, i =,..., m h i x ) = 0, i =,..., r λ i 0, i =,..., m λ i f i x ) = 0, i =,..., m νi h i x ) = 0 In summary: if primal problem convex and inequality constraints affine then strong duality holds. If in addition functions f i, h i differentiable then KKT conditions are necessary and sufficient for optimality.

5 Linearly separable points Linearly separable points Classify two clouds of points, where there exists a hyperplane which linearly separates one cloud from the other without error. Classify two clouds of points, where there exists a hyperplane which linearly separates one cloud from the other without error. y i = + y i = + y i = Data given by x i, y i } n, x i R p, y i, +} y i = Hyperplane equation w x + b = 0. Linear discriant given by f x) = signw x + b) Linearly separable points Linearly separable points Classify two clouds of points, where there exists a hyperplane which linearly separates one cloud from the other without error. Classify two clouds of points, where there exists a hyperplane which linearly separates one cloud from the other without error. y i = + w y i = + y i = y i = / w For a datapoint close to the decision boundary, a small change leads to a change in classification. Can we make the classifier more robust? Smallest distance from each class to the separating hyperplane w x + b is called the margin.

6 Maximum margin classifier, linearly separable case This problem can be expressed as follows: ) max margin) = max w,b w,b w subject to w x i + b i : y i = +, w x i + b i : y i =. The resulting classifier is f x) = signw x + b), We can rewrite to obtain a quadratic program: subject to w,b w y i w x i + b). Maximum margin classifier: with errors allowed Allow errors : points within the margin, or even on the wrong side of the decision boudary. Ideally: w,b w + C I[y i w x i + b ) ) < 0], where C controls the tradeoff between maximum margin and loss. Replace with convex upper bound: w,b w + C h y i w x i + b ))). with hinge loss, hα) = α) + = α, α > 0 0, otherwise. Hinge loss Support vector classification Hinge loss: α) + Iα < 0) hα) = α) + = α, α > 0 0, otherwise. α Substituting in the hinge loss, we get w,b w + C h y i w x i + b ))). To simplify, use substitution ξ i = h y i w x i + b )) : ) w,b,ξ w + C ξ i subject to ξ i 0 y i w x i + b ) ξ i

7 Support vector classification Does strong duality hold? ξ/ w y i = w y i = + / w Is the optimization problem convex wrt the variables w, b, ξ? imize subject to f 0 w, b, ξ) := w + C f i w, b, ξ) := ξ i y i w x i + b ) 0, i =,..., n f i w, b, ξ) := ξ i 0, i = n +,..., n Each of f 0, f,..., f n are convex. No equality constraints. Does Slater s condition hold? Yes trivially) since inequality constraints affine. ξ i Thus strong duality holds, the problem is differentiable, hence the KKT conditions hold at the global optimum. Support vector classification: Lagrangian Support vector classification: Lagrangian The Lagrangian: Lw, b, ξ, α, λ) = w + C ξ i + with dual variable constraints α i ξi y i w x i + b )) + λ i ξ i ) Derivative wrt ξ i : Since λ i 0, L ξ i = C α i λ i = 0 α i = C λ i. α i C. α i 0, λ i 0. Minimize wrt the primal variables w, b, and ξ. Derivative wrt w: Derivative wrt b: L w = w α i y i x i = 0 w = L b = i y i α i = 0. α i y i x i. Now use complementary slackness: Non-margin SVs margin errors): α i = C > 0: We immediately have y i w x i + b ) = ξ i. Also, from condition α i = C λ i, we have λ i = 0, so ξ i 0 Margin SVs: 0 < α i < C: We again have y i w x i + b ) = ξ i. This time, from α i = C λ i, we have λ i > 0, hence ξ i = 0. Non-SVs on the correct side of the margin): α i = 0: From α i = C λ i, we have λ i > 0, hence ξ i = 0. Thus, y i w x i + b )

8 The support vectors Support vector classification: dual function Thus, our goal is to maximize the dual, We observe: The solution is sparse: points which are neither on the margin nor margin errors have α i = 0 The support vectors: only those points on the decision boundary, or which are margin errors, contribute. 3 Influence of the non-margin SVs is bounded, since their weight cannot exceed C. gα, λ) = w + C ξ i + α i yi w x i + b ) ) ξ i = = + λ i ξ i ) α i α j y i y j xi x j + C j= b α i y i + α i }} 0 α i ξ i α i α j y i y j xi x j α i ξ i j= α i α j y i y j xi x j. j= C α i )ξ i Support vector classification: dual problem Maximize the dual, subject to the constraints gα) = α i α i α j y i y j xi x j, j= 0 α i C, y i α i = 0 This is a quadratic program. From α, obtain the hyperplane with w = n α iy i x i Offset b can be obtained from any of the margin SVs: = y i w x i + b ).

### ICS-E4030 Kernel Methods in Machine Learning

ICS-E4030 Kernel Methods in Machine Learning Lecture 3: Convex optimization and duality Juho Rousu 28. September, 2016 Juho Rousu 28. September, 2016 1 / 38 Convex optimization Convex optimisation This

### Support Vector Machines for Classification and Regression

CIS 520: Machine Learning Oct 04, 207 Support Vector Machines for Classification and Regression Lecturer: Shivani Agarwal Disclaimer: These notes are designed to be a supplement to the lecture. They may

### CS-E4830 Kernel Methods in Machine Learning

CS-E4830 Kernel Methods in Machine Learning Lecture 3: Convex optimization and duality Juho Rousu 27. September, 2017 Juho Rousu 27. September, 2017 1 / 45 Convex optimization Convex optimisation This

### Support Vector Machines for Classification and Regression. 1 Linearly Separable Data: Hard Margin SVMs

E0 270 Machine Learning Lecture 5 (Jan 22, 203) Support Vector Machines for Classification and Regression Lecturer: Shivani Agarwal Disclaimer: These notes are a brief summary of the topics covered in

### On the Method of Lagrange Multipliers

On the Method of Lagrange Multipliers Reza Nasiri Mahalati November 6, 2016 Most of what is in this note is taken from the Convex Optimization book by Stephen Boyd and Lieven Vandenberghe. This should

### Lecture Notes on Support Vector Machine

Lecture Notes on Support Vector Machine Feng Li fli@sdu.edu.cn Shandong University, China 1 Hyperplane and Margin In a n-dimensional space, a hyper plane is defined by ω T x + b = 0 (1) where ω R n is

### Convex Optimization & Lagrange Duality

Convex Optimization & Lagrange Duality Chee Wei Tan CS 8292 : Advanced Topics in Convex Optimization and its Applications Fall 2010 Outline Convex optimization Optimality condition Lagrange duality KKT

### 14. Duality. ˆ Upper and lower bounds. ˆ General duality. ˆ Constraint qualifications. ˆ Counterexample. ˆ Complementary slackness.

CS/ECE/ISyE 524 Introduction to Optimization Spring 2016 17 14. Duality ˆ Upper and lower bounds ˆ General duality ˆ Constraint qualifications ˆ Counterexample ˆ Complementary slackness ˆ Examples ˆ Sensitivity

### Extreme Abridgment of Boyd and Vandenberghe s Convex Optimization

Extreme Abridgment of Boyd and Vandenberghe s Convex Optimization Compiled by David Rosenberg Abstract Boyd and Vandenberghe s Convex Optimization book is very well-written and a pleasure to read. The

### Duality. Lagrange dual problem weak and strong duality optimality conditions perturbation and sensitivity analysis generalized inequalities

Duality Lagrange dual problem weak and strong duality optimality conditions perturbation and sensitivity analysis generalized inequalities Lagrangian Consider the optimization problem in standard form

### Constrained Optimization and Lagrangian Duality

CIS 520: Machine Learning Oct 02, 2017 Constrained Optimization and Lagrangian Duality Lecturer: Shivani Agarwal Disclaimer: These notes are designed to be a supplement to the lecture. They may or may

### Machine Learning. Lecture 6: Support Vector Machine. Feng Li.

Machine Learning Lecture 6: Support Vector Machine Feng Li fli@sdu.edu.cn https://funglee.github.io School of Computer Science and Technology Shandong University Fall 2018 Warm Up 2 / 80 Warm Up (Contd.)

### 5. Duality. Lagrangian

5. Duality Convex Optimization Boyd & Vandenberghe Lagrange dual problem weak and strong duality geometric interpretation optimality conditions perturbation and sensitivity analysis examples generalized

### Lecture: Duality of LP, SOCP and SDP

1/33 Lecture: Duality of LP, SOCP and SDP Zaiwen Wen Beijing International Center For Mathematical Research Peking University http://bicmr.pku.edu.cn/~wenzw/bigdata2017.html wenzw@pku.edu.cn Acknowledgement:

### Convex Optimization. Dani Yogatama. School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA. February 12, 2014

Convex Optimization Dani Yogatama School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA February 12, 2014 Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12,

### Lagrangian Duality and Convex Optimization

Lagrangian Duality and Convex Optimization David Rosenberg New York University February 11, 2015 David Rosenberg (New York University) DS-GA 1003 February 11, 2015 1 / 24 Introduction Why Convex Optimization?

### Convex Optimization Boyd & Vandenberghe. 5. Duality

5. Duality Convex Optimization Boyd & Vandenberghe Lagrange dual problem weak and strong duality geometric interpretation optimality conditions perturbation and sensitivity analysis examples generalized

### Support vector machines

Support vector machines Guillaume Obozinski Ecole des Ponts - ParisTech SOCN course 2014 SVM, kernel methods and multiclass 1/23 Outline 1 Constrained optimization, Lagrangian duality and KKT 2 Support

### Linear 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

### Introduction to Machine Learning Spring 2018 Note Duality. 1.1 Primal and Dual Problem

CS 189 Introduction to Machine Learning Spring 2018 Note 22 1 Duality As we have seen in our discussion of kernels, ridge regression can be viewed in two ways: (1) an optimization problem over the weights

### Linear 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

### Lecture Support Vector Machine (SVM) Classifiers

Introduction to Machine Learning Lecturer: Amir Globerson Lecture 6 Fall Semester Scribe: Yishay Mansour 6.1 Support Vector Machine (SVM) Classifiers Classification is one of the most important tasks in

### Linear Support Vector Machine. Classification. Linear SVM. Huiping Cao. Huiping Cao, Slide 1/26

Huiping Cao, Slide 1/26 Classification Linear SVM Huiping Cao linear hyperplane (decision boundary) that will separate the data Huiping Cao, Slide 2/26 Support Vector Machines rt Vector Find a linear Machines

### Support Vector Machines

EE 17/7AT: Optimization Models in Engineering Section 11/1 - April 014 Support Vector Machines Lecturer: Arturo Fernandez Scribe: Arturo Fernandez 1 Support Vector Machines Revisited 1.1 Strictly) Separable

### Introduction to Machine Learning Lecture 7. Mehryar Mohri Courant Institute and Google Research

Introduction to Machine Learning Lecture 7 Mehryar Mohri Courant Institute and Google Research mohri@cims.nyu.edu Convex Optimization Differentiation Definition: let f : X R N R be a differentiable function,

### Lecture: Duality.

Lecture: Duality http://bicmr.pku.edu.cn/~wenzw/opt-2016-fall.html Acknowledgement: this slides is based on Prof. Lieven Vandenberghe s lecture notes Introduction 2/35 Lagrange dual problem weak and strong

### Convex Optimization and Support Vector Machine

Convex Optimization and Support Vector Machine Problem 0. Consider a two-class classification problem. The training data is L n = {(x 1, t 1 ),..., (x n, t n )}, where each t i { 1, 1} and x i R p. We

### Lagrange duality. The Lagrangian. We consider an optimization program of the form

Lagrange duality Another way to arrive at the KKT conditions, and one which gives us some insight on solving constrained optimization problems, is through the Lagrange dual. The dual is a maximization

### Support Vector Machine (SVM) and Kernel Methods

Support Vector Machine (SVM) and Kernel Methods CE-717: Machine Learning Sharif University of Technology Fall 2015 Soleymani Outline Margin concept Hard-Margin SVM Soft-Margin SVM Dual Problems of Hard-Margin

### EE364a Review Session 5

EE364a Review Session 5 EE364a Review announcements: homeworks 1 and 2 graded homework 4 solutions (check solution to additional problem 1) scpd phone-in office hours: tuesdays 6-7pm (650-723-1156) 1 Complementary

### Karush-Kuhn-Tucker Conditions. Lecturer: Ryan Tibshirani Convex Optimization /36-725

Karush-Kuhn-Tucker Conditions Lecturer: Ryan Tibshirani Convex Optimization 10-725/36-725 1 Given a minimization problem Last time: duality min x subject to f(x) h i (x) 0, i = 1,... m l j (x) = 0, j =

### Support Vector Machines

Support Vector Machines Support vector machines (SVMs) are one of the central concepts in all of machine learning. They are simply a combination of two ideas: linear classification via maximum (or optimal

### Convex Optimization M2

Convex Optimization M2 Lecture 3 A. d Aspremont. Convex Optimization M2. 1/49 Duality A. d Aspremont. Convex Optimization M2. 2/49 DMs DM par email: dm.daspremont@gmail.com A. d Aspremont. Convex Optimization

### Convex Optimization and SVM

Convex Optimization and SVM Problem 0. Cf lecture notes pages 12 to 18. Problem 1. (i) A slab is an intersection of two half spaces, hence convex. (ii) A wedge is an intersection of two half spaces, hence

### I.3. LMI DUALITY. Didier HENRION EECI Graduate School on Control Supélec - Spring 2010

I.3. LMI DUALITY Didier HENRION henrion@laas.fr EECI Graduate School on Control Supélec - Spring 2010 Primal and dual For primal problem p = inf x g 0 (x) s.t. g i (x) 0 define Lagrangian L(x, z) = g 0

### Support Vector Machines, Kernel SVM

Support Vector Machines, Kernel SVM Professor Ameet Talwalkar Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 27, 2017 1 / 40 Outline 1 Administration 2 Review of last lecture 3 SVM

### Jeff Howbert Introduction to Machine Learning Winter

Classification / Regression Support Vector Machines Jeff Howbert Introduction to Machine Learning Winter 2012 1 Topics SVM classifiers for linearly separable classes SVM classifiers for non-linearly separable

### CSC 411 Lecture 17: Support Vector Machine

CSC 411 Lecture 17: Support Vector Machine Ethan Fetaya, James Lucas and Emad Andrews University of Toronto CSC411 Lec17 1 / 1 Today Max-margin classification SVM Hard SVM Duality Soft SVM CSC411 Lec17

### Outline. Basic concepts: SVM and kernels SVM primal/dual problems. Chih-Jen Lin (National Taiwan Univ.) 1 / 22

Outline Basic concepts: SVM and kernels SVM primal/dual problems Chih-Jen Lin (National Taiwan Univ.) 1 / 22 Outline Basic concepts: SVM and kernels Basic concepts: SVM and kernels SVM primal/dual problems

### Support Vector Machine (SVM) and Kernel Methods

Support Vector Machine (SVM) and Kernel Methods CE-717: Machine Learning Sharif University of Technology Fall 2014 Soleymani Outline Margin concept Hard-Margin SVM Soft-Margin SVM Dual Problems of Hard-Margin

### Machine 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

### Linear 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

### Support 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)

### Max 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

### Lecture 3: Lagrangian duality and algorithms for the Lagrangian dual problem

Lecture 3: Lagrangian duality and algorithms for the Lagrangian dual problem Michael Patriksson 0-0 The Relaxation Theorem 1 Problem: find f := infimum f(x), x subject to x S, (1a) (1b) where f : R n R

### Support Vector Machines: Maximum Margin Classifiers

Support Vector Machines: Maximum Margin Classifiers Machine Learning and Pattern Recognition: September 16, 2008 Piotr Mirowski Based on slides by Sumit Chopra and Fu-Jie Huang 1 Outline What is behind

### Support Vector Machines and Kernel Methods

2018 CS420 Machine Learning, Lecture 3 Hangout from Prof. Andrew Ng. http://cs229.stanford.edu/notes/cs229-notes3.pdf Support Vector Machines and Kernel Methods Weinan Zhang Shanghai Jiao Tong University

### Lecture 3 January 28

EECS 28B / STAT 24B: Advanced Topics in Statistical LearningSpring 2009 Lecture 3 January 28 Lecturer: Pradeep Ravikumar Scribe: Timothy J. Wheeler Note: These lecture notes are still rough, and have only

### A Revisit to Support Vector Data Description (SVDD)

A Revisit to Support Vector Data Description (SVDD Wei-Cheng Chang b99902019@csie.ntu.edu.tw Ching-Pei Lee r00922098@csie.ntu.edu.tw Chih-Jen Lin cjlin@csie.ntu.edu.tw Department of Computer Science, National

### Lagrange Duality. Daniel P. Palomar. Hong Kong University of Science and Technology (HKUST)

Lagrange Duality Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470 - Convex Optimization Fall 2017-18, HKUST, Hong Kong Outline of Lecture Lagrangian Dual function Dual

### subject to (x 2)(x 4) u,

Exercises Basic definitions 5.1 A simple example. Consider the optimization problem with variable x R. minimize x 2 + 1 subject to (x 2)(x 4) 0, (a) Analysis of primal problem. Give the feasible set, the

### EE/AA 578, Univ of Washington, Fall Duality

7. Duality EE/AA 578, Univ of Washington, Fall 2016 Lagrange dual problem weak and strong duality geometric interpretation optimality conditions perturbation and sensitivity analysis examples generalized

### A Brief Review on Convex Optimization

A Brief Review on Convex Optimization 1 Convex set S R n is convex if x,y S, λ,µ 0, λ+µ = 1 λx+µy S geometrically: x,y S line segment through x,y S examples (one convex, two nonconvex sets): A Brief Review

### Support Vector Machines

Support Vector Machines Ryan M. Rifkin Google, Inc. 2008 Plan Regularization derivation of SVMs Geometric derivation of SVMs Optimality, Duality and Large Scale SVMs The Regularization Setting (Again)

### Homework Set #6 - Solutions

EE 15 - Applications of Convex Optimization in Signal Processing and Communications Dr Andre Tkacenko JPL Third Term 11-1 Homework Set #6 - Solutions 1 a The feasible set is the interval [ 4] The unique

### Support Vector Machines. Introduction to Data Mining, 2 nd Edition by Tan, Steinbach, Karpatne, Kumar

Data Mining Support Vector Machines Introduction to Data Mining, 2 nd Edition by Tan, Steinbach, Karpatne, Kumar 02/03/2018 Introduction to Data Mining 1 Support Vector Machines Find a linear hyperplane

### Lecture 18: Optimization Programming

Fall, 2016 Outline Unconstrained Optimization 1 Unconstrained Optimization 2 Equality-constrained Optimization Inequality-constrained Optimization Mixture-constrained Optimization 3 Quadratic Programming

### Support Vector Machine (continued)

Support Vector Machine continued) Overlapping class distribution: In practice the class-conditional distributions may overlap, so that the training data points are no longer linearly separable. We need

### Support Vector Machines

Support Vector Machines Statistical Inference with Reproducing Kernel Hilbert Space Kenji Fukumizu Institute of Statistical Mathematics, ROIS Department of Statistical Science, Graduate University for

### SVM and Kernel machine

SVM and Kernel machine Stéphane Canu stephane.canu@litislab.eu Escuela de Ciencias Informáticas 20 July 3, 20 Road map Linear SVM Linear classification The margin Linear SVM: the problem Optimization in

### Support Vector Machines

Support Vector Machines Le Song Machine Learning I CSE 6740, Fall 2013 Naïve Bayes classifier Still use Bayes decision rule for classification P y x = P x y P y P x But assume p x y = 1 is fully factorized

### CSCI : Optimization and Control of Networks. Review on Convex Optimization

CSCI7000-016: Optimization and Control of Networks Review on Convex Optimization 1 Convex set S R n is convex if x,y S, λ,µ 0, λ+µ = 1 λx+µy S geometrically: x,y S line segment through x,y S examples (one

### Machine Learning And Applications: Supervised Learning-SVM

Machine Learning And Applications: Supervised Learning-SVM Raphaël Bournhonesque École Normale Supérieure de Lyon, Lyon, France raphael.bournhonesque@ens-lyon.fr 1 Supervised vs unsupervised learning Machine

### Optimization for Machine Learning

Optimization for Machine Learning (Problems; Algorithms - A) SUVRIT SRA Massachusetts Institute of Technology PKU Summer School on Data Science (July 2017) Course materials http://suvrit.de/teaching.html

### Support Vector Machine (SVM) and Kernel Methods

Support Vector Machine (SVM) and Kernel Methods CE-717: Machine Learning Sharif University of Technology Fall 2016 Soleymani Outline Margin concept Hard-Margin SVM Soft-Margin SVM Dual Problems of Hard-Margin

### Lecture 9: Large Margin Classifiers. Linear Support Vector Machines

Lecture 9: Large Margin Classifiers. Linear Support Vector Machines Perceptrons Definition Perceptron learning rule Convergence Margin & max margin classifiers (Linear) support vector machines Formulation

### Lecture 2: Linear SVM in the Dual

Lecture 2: Linear SVM in the Dual Stéphane Canu stephane.canu@litislab.eu São Paulo 2015 July 22, 2015 Road map 1 Linear SVM Optimization in 10 slides Equality constraints Inequality constraints Dual formulation

### Lagrangian Duality Theory

Lagrangian Duality Theory Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye Chapter 14.1-4 1 Recall Primal and Dual

### Announcements - Homework

Announcements - Homework Homework 1 is graded, please collect at end of lecture Homework 2 due today Homework 3 out soon (watch email) Ques 1 midterm review HW1 score distribution 40 HW1 total score 35

### Applied inductive learning - Lecture 7

Applied inductive learning - Lecture 7 Louis Wehenkel & Pierre Geurts Department of Electrical Engineering and Computer Science University of Liège Montefiore - Liège - November 5, 2012 Find slides: http://montefiore.ulg.ac.be/

### Support Vector Machines

Support Vector Machines Sridhar Mahadevan mahadeva@cs.umass.edu University of Massachusetts Sridhar Mahadevan: CMPSCI 689 p. 1/32 Margin Classifiers margin b = 0 Sridhar Mahadevan: CMPSCI 689 p.

### Duality Uses and Correspondences. Ryan Tibshirani Convex Optimization

Duality Uses and Correspondences Ryan Tibshirani Conve Optimization 10-725 Recall that for the problem Last time: KKT conditions subject to f() h i () 0, i = 1,... m l j () = 0, j = 1,... r the KKT conditions

### Kernel Machines. Pradeep Ravikumar Co-instructor: Manuela Veloso. Machine Learning

Kernel Machines Pradeep Ravikumar Co-instructor: Manuela Veloso Machine Learning 10-701 SVM linearly separable case n training points (x 1,, x n ) d features x j is a d-dimensional vector Primal problem:

### Machine Learning and Data Mining. Support Vector Machines. Kalev Kask

Machine Learning and Data Mining Support Vector Machines Kalev Kask Linear classifiers Which decision boundary is better? Both have zero training error (perfect training accuracy) But, one of them seems

### 8. Geometric problems

8. Geometric problems Convex Optimization Boyd & Vandenberghe extremal volume ellipsoids centering classification placement and facility location 8 Minimum volume ellipsoid around a set Löwner-John ellipsoid

### Motivation. Lecture 2 Topics from Optimization and Duality. network utility maximization (NUM) problem:

CDS270 Maryam Fazel Lecture 2 Topics from Optimization and Duality Motivation network utility maximization (NUM) problem: consider a network with S sources (users), each sending one flow at rate x s, through

### Linear & nonlinear classifiers

Linear & nonlinear classifiers Machine Learning Hamid Beigy Sharif University of Technology Fall 1396 Hamid Beigy (Sharif University of Technology) Linear & nonlinear classifiers Fall 1396 1 / 44 Table

### LECTURE 7 Support vector machines

LECTURE 7 Support vector machines SVMs have been used in a multitude of applications and are one of the most popular machine learning algorithms. We will derive the SVM algorithm from two perspectives:

### Polyhedral Computation. Linear Classifiers & the SVM

Polyhedral Computation Linear Classifiers & the SVM mcuturi@i.kyoto-u.ac.jp Nov 26 2010 1 Statistical Inference Statistical: useful to study random systems... Mutations, environmental changes etc. life

### Optimization for Communications and Networks. Poompat Saengudomlert. Session 4 Duality and Lagrange Multipliers

Optimization for Communications and Networks Poompat Saengudomlert Session 4 Duality and Lagrange Multipliers P Saengudomlert (2015) Optimization Session 4 1 / 14 24 Dual Problems Consider a primal convex

### Primal/Dual Decomposition Methods

Primal/Dual Decomposition Methods Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470 - Convex Optimization Fall 2018-19, HKUST, Hong Kong Outline of Lecture Subgradients

### Machine 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

### Support Vector Machine (SVM) & Kernel CE-717: Machine Learning Sharif University of Technology. M. Soleymani Fall 2012

Support Vector Machine (SVM) & Kernel CE-717: Machine Learning Sharif University of Technology M. Soleymani Fall 2012 Linear classifier Which classifier? x 2 x 1 2 Linear classifier Margin concept x 2

### COMP 875 Announcements

Announcements Tentative presentation order is out Announcements Tentative presentation order is out Remember: Monday before the week of the presentation you must send me the final paper list (for posting

### Support Vector Machine

Support Vector Machine Kernel: Kernel is defined as a function returning the inner product between the images of the two arguments k(x 1, x 2 ) = ϕ(x 1 ), ϕ(x 2 ) k(x 1, x 2 ) = k(x 2, x 1 ) modularity-

### Introduction to Support Vector Machines

Introduction to Support Vector Machines Shivani Agarwal Support Vector Machines (SVMs) Algorithm for learning linear classifiers Motivated by idea of maximizing margin Efficient extension to non-linear

### Statistical Pattern Recognition

Statistical Pattern Recognition Support Vector Machine (SVM) Hamid R. Rabiee Hadi Asheri, Jafar Muhammadi, Nima Pourdamghani Spring 2013 http://ce.sharif.edu/courses/91-92/2/ce725-1/ Agenda Introduction

### Convex Optimization and Modeling

Convex Optimization and Modeling Duality Theory and Optimality Conditions 5th lecture, 12.05.2010 Jun.-Prof. Matthias Hein Program of today/next lecture Lagrangian and duality: the Lagrangian the dual

### 10701 Recitation 5 Duality and SVM. Ahmed Hefny

10701 Recitation 5 Duality and SVM Ahmed Hefny Outline Langrangian and Duality The Lagrangian Duality Eamples Support Vector Machines Primal Formulation Dual Formulation Soft Margin and Hinge Loss Lagrangian

### Lecture Note 5: Semidefinite Programming for Stability Analysis

ECE7850: Hybrid Systems:Theory and Applications Lecture Note 5: Semidefinite Programming for Stability Analysis Wei Zhang Assistant Professor Department of Electrical and Computer Engineering Ohio State

### Linear and Combinatorial Optimization

Linear and Combinatorial Optimization The dual of an LP-problem. Connections between primal and dual. Duality theorems and complementary slack. Philipp Birken (Ctr. for the Math. Sc.) Lecture 3: Duality

### Support Vector Machine I

Support Vector Machine I Statistical Data Analysis with Positive Definite Kernels Kenji Fukumizu Institute of Statistical Mathematics, ROIS Department of Statistical Science, Graduate University for Advanced

### Introduction to Machine Learning Prof. Sudeshna Sarkar Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur

Introduction to Machine Learning Prof. Sudeshna Sarkar Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Module - 5 Lecture - 22 SVM: The Dual Formulation Good morning.

### Cheng Soon Ong & Christian Walder. Canberra February June 2018

Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 Outlines Overview Introduction Linear Algebra Probability Linear Regression

### Introduction to Mathematical Programming IE406. Lecture 10. Dr. Ted Ralphs

Introduction to Mathematical Programming IE406 Lecture 10 Dr. Ted Ralphs IE406 Lecture 10 1 Reading for This Lecture Bertsimas 4.1-4.3 IE406 Lecture 10 2 Duality Theory: Motivation Consider the following

### 8. Geometric problems

8. Geometric problems Convex Optimization Boyd & Vandenberghe extremal volume ellipsoids centering classification placement and facility location 8 1 Minimum volume ellipsoid around a set Löwner-John ellipsoid

### UVA CS 4501: Machine Learning

UVA CS 4501: Machine Learning Lecture 16 Extra: Support Vector Machine Optimization with Dual Dr. Yanjun Qi University of Virginia Department of Computer Science Today Extra q Optimization of SVM ü SVM

### Tutorial on Convex Optimization for Engineers Part II

Tutorial on Convex Optimization for Engineers Part II M.Sc. Jens Steinwandt Communications Research Laboratory Ilmenau University of Technology PO Box 100565 D-98684 Ilmenau, Germany jens.steinwandt@tu-ilmenau.de