Machine Learning 4771
|
|
- Allen Cameron
- 5 years ago
- Views:
Transcription
1 Machine Learning 4771 Instructor: Tony Jebara
2 Topic 3 Additive Models and Linear Regression Sinusoids and Radial Basis Functions Classification Logistic Regression Gradient Descent
3 Polynomial Basis Functions To fit a P th order polynomial function to multivariate data: concatenate columns of all monomials up to power P E.g. dimensional data and nd order polynomial (quadratic) x 1 ( 1) x 1 ( ) x i 1 ( ) x i ( ) x N 1 ( ) x N ( ) 1 x 1 ( 1) x 1 ( ) x 1 1 ( )x 1 1 ( ) x 1 1 ( )x 1 ( ) x 1 ( )x 1 1 x i ( 1) x i ( ) x i ( 1)x i ( 1) x i ( 1)x i ( ) x i ( )x i ( ) ( ) 1 x N ( 1) x N ( ) x N ( 1)x N ( 1) x N ( 1)x N ( ) x N ( )x N ( )
4 Sinusoidal Basis Functions More generally, we don t just have to deal with polynomials, use any set of basis fn s: ( ) P = θ p φ ( p x) f x;θ + θ 0 p=1 These are generally called Additive Models Regression adds linear combinations of the basis fn s For example: Fourier (sinusoidal) basis φ k ( x ) i = sin kx i ( ) φ ( x ) = cos( kx ) k+1 i i Note, don t have to be a basis per se, usually subset θ 1 + θ + θ 3
5 Radial Basis Functions Can act as prototypes of the data itself f ( x;θ) = θ k exp 1 N x x k=1 σ k Parameter σ = standard deviation σ = covariance controls how wide bumps are what happens if too big/small? Also works in multi-dimensions Called RBF for short
6 Radial Basis Functions Each training point leads to a bump function f ( x;θ) = θ k exp 1 N x x k=1 σ k Reuse solution from linear regression: θ * = X T X Can view the data instead as X, a big matrix of size N x N X = exp 1 σ x 1 x 1 exp 1 σ x x 1 exp 1 σ x 3 x 1 exp 1 σ x 1 x exp 1 σ x x exp 1 σ x 3 x ( ) 1 X T y For RBFs, X is square and symmetric, so solution is just exp 1 σ x 1 x 3 exp 1 σ x x 3 exp 1 σ x 3 x 3 θ R = 0 X T Xθ = X T y Xθ = y θ * = X 1 y
7 Evaluating Our Learned Function We minimized empirical risk to get θ* How well does f(x;θ*) perform on future data? It should Generalize and have low True Risk: R true ( θ) = P ( x,y)l( y, f ( x;θ) )dx dy Can t compute true risk, instead use Testing Empirical Risk We randomly split data into training and testing portions {( x 1,y 1 ),,( x N,y )} ( x,y N N +1 N +1),, x N +M,y N +M { ( )} Find θ* with training data: Evaluate it with testing data: R train R test ( θ) = 1 N ( θ) = 1 M N i=1 N +M i=n +1 L( y i, f ( x i ;θ)) L( y i, f ( x i ;θ))
8 Crossvalidation Try fitting with different sigma radial basis function widths Select sigma which gives lowest R test (θ*) Loss R ( test θ * ) underfitting overfitting R ( train θ * ) 1/σ Best sigma Think of sigma as a measure of the simplicity of the model Thinner RBFs are more flexible and complex
9 Regularized Risk Minimization Empirical Risk Minimization gave overfitting & underfitting We want to add a penalty for using too many theta values This gives us the Regularized Risk R regularized ( θ) = R ( empirical θ) + Penalty ( θ) = 1 N N i=1 L( y i, f ( x i ;θ)) + λ N θ Solution for Regularized Risk with Least Squares Loss: 1 θ R regularized = 0 θ N θ * = ( X T X + λi ) 1 X T y y Xθ + λ N θ = 0
10 Regularized Risk Minimization Have D=16 features (or P=15 throughout) Try minimizing R regularized (θ) to get θ* with different λ Note that λ=0 give back Empirical Risk Minimization
11 Crossvalidation Try fitting with different lambda regularization levels Select lambda which gives lowest R test (θ*) Risk R ( test θ * ) underfitting overfitting R ( train θ * ) 1/λ Best lambda Lambda measures simplicity of the model Models with low lambda are more flexible
12 From Regression To Classification Classification is another important learning problem Regression Classification { ( )} x RD y R 1 X = ( x 1,y 1 ),( x,y ),, x N,y N { ( )} x RD y { 0,1} X = ( x 1,y 1 ),( x,y ),, x N,y N E.g. Given x = [tumor size, tumor density] Predict y in {benign,malignant} Should we solve this as a least squares regression problem? x x x x x x x x x x O O O O O O O O O
13 Classification vs. Regression a) Classification needs binary answers like {0,1} b) Least squares is an unfair measure of risk here e.g. Why penalize a correct but large positive y answer? e.g. Why penalize a correct but large negative y answer? Example: not good to use regression output for a decision f(x)>0.5 Class 1 f(x)<0.5 Class 0 if f(x)=-3.8 & correct class=0, squared error penalizes it f(x) o o x x x x x x x We pay a hefty squared error loss here even if we got the correct classification result. The thick solid line model makes two mistakes while the dashed model is perfect
14 Classification vs. Regression We will consider the following four steps to improve from naïve regression to get better classification learning: 1) Fix functions f(x) to give binary output (logistic neuron) ) Fix our definition of the Risk we will minimize so that we get good classification accuracy (logistic loss) and later on 3) Make an even better fix on f(x) to binarize (perceptron) 4) Make an even better risk (perceptron loss)
15 Logistic Neuron (McCullough-Pitts) To output binary, use squashing function g(). f ( x;θ) = θ T x ( ) = g θ T x ( ) = 1 + exp( z ) f x;θ g z ( ) ( ) 1 Linear neuron Logistic Neuron This squashing is called sigmoid or logistic function
16 Logistic Regression Given a classification problem with binary outputs X = ( x 1,y 1 ),( x,y ),,( x N,y ) N Use this function and output 1 if f(x)>0.5 and 0 otherwise ( ) = 1 + exp θ T x f x;θ { } x R D y 0,1 ( ( )) 1 { }
17 Short hand for Linear Functions What happened to adding the intercept? f x;θ ( ) = θ T x + θ 0 = θ(1) θ() θ(d) T x(1) x() x(d) + θ 0 = θ 0 θ(1) θ() θ(d) T 1 x(1) x() x(d) = θ T x
18 Logistic Regression Given a classification problem with binary outputs X = ( x 1,y 1 ),( x,y ),,( x N,y ) N Fix#1: use f(x) below, output 1 if f(x)>0.5 and 0 otherwise ( ) = 1 + exp θ T x f x;θ { } x R D y 0,1 ( ( )) 1 { }
19 Logistic Regression Given a classification problem with binary outputs X = ( x 1,y 1 ),( x,y ),,( x N,y ) N Fix#1: use f(x) below, output 1 if f(x)>0.5 and 0 otherwise ( ) = 1 + exp θ T x f x;θ ( ( )) 1 Fix#: instead of squared loss, use Logistic Loss L log ( y, f ( x;θ) ) = y i 1 ( )log( 1 f ( x;θ) ) y i log f x;θ ( ( )) This method is called Logistic Regression. But Empirical Risk Minimization has no closed-form sol n: R emp { } x R D y 0,1 ( θ) = 1 N N ( y i 1)log( 1 f ( x i ;θ)) y i log f x i ;θ i=1 { } ( ( ))
20 Logistic Regression With logistic squashing function, minimizing R(θ) is harder R emp ( θ) = 1 N N ( y i 1)log( 1 f ( x i ;θ)) y i log f x i ;θ i=1 1 y θ R = 1 i N 1 f ( x i ;θ) y N i f ( x i ;θ) f ' ( x ;θ ) = 0??? i=1 i Can t minimize risk and find best theta analytically! Let s try finding best theta numerically. Use the following to compute gradient ( ) = 1 + exp θ T x f x;θ ( ( )) 1 = g ( θ T x) Here, g() is the logistic squashing function ( ) = 1 + exp( z ) g z ( ) 1 g ' z ( ) ( ) = g ( z) 1 g ( z) ( ( ))
21 Gradient Descent Useful when we can t get minimum solution in closed form Gradient points in direction of fastest increase Take step in the opposite direction! Gradient Descent Algorithm choose scalar step size η, & tolerance ε initialize θ 0 = small random vector θ 1 = θ 0 η θ R emp θ 0, t = 1 while θ t θ t 1 { θ t+1 = θ t η θ R emp, t = t + 1 θ t } For appropriate η, this will converge to local minimum
22 Logistic Regression Logistic regression gives better classification performance Its empirical risk is N R emp ( y i 1)log( 1 f ( x i ;θ)) y i log( f ( x i ;θ)) This R(θ) is convex so gradient descent always converges to the same solution Make predictions using ( ) = 1 + exp θ T x f x;θ ( θ) = 1 N i=1 ( ( )) 1 Output 1 if f > 0.5 Output 0 otherwise
Machine Learning 4771
Machine Learning 477 Instructor: Tony Jebara Topic Regression Empirical Risk Minimization Least Squares Higher Order Polynomials Under-fitting / Over-fitting Cross-Validation Regression Classification
More informationMachine Learning 4771
Machine Learning 477 Instructor: Tony Jebara Topic 5 Generalization Guarantees VC-Dimension Nearest Neighbor Classification (infinite VC dimension) Structural Risk Minimization Support Vector Machines
More information18.9 SUPPORT VECTOR MACHINES
744 Chapter 8. Learning from Examples is the fact that each regression problem will be easier to solve, because it involves only the examples with nonzero weight the examples whose kernels overlap the
More informationCOMPUTATIONAL INTELLIGENCE (INTRODUCTION TO MACHINE LEARNING) SS16
COMPUTATIONAL INTELLIGENCE (INTRODUCTION TO MACHINE LEARNING) SS6 Lecture 3: Classification with Logistic Regression Advanced optimization techniques Underfitting & Overfitting Model selection (Training-
More information6.036 midterm review. Wednesday, March 18, 15
6.036 midterm review 1 Topics covered supervised learning labels available unsupervised learning no labels available semi-supervised learning some labels available - what algorithms have you learned that
More informationClassification CE-717: Machine Learning Sharif University of Technology. M. Soleymani Fall 2012
Classification CE-717: Machine Learning Sharif University of Technology M. Soleymani Fall 2012 Topics Discriminant functions Logistic regression Perceptron Generative models Generative vs. discriminative
More informationProbabilistic classification CE-717: Machine Learning Sharif University of Technology. M. Soleymani Fall 2016
Probabilistic classification CE-717: Machine Learning Sharif University of Technology M. Soleymani Fall 2016 Topics Probabilistic approach Bayes decision theory Generative models Gaussian Bayes classifier
More informationLogistic Regression Logistic
Case Study 1: Estimating Click Probabilities L2 Regularization for Logistic Regression Machine Learning/Statistics for Big Data CSE599C1/STAT592, University of Washington Carlos Guestrin January 10 th,
More informationMachine Learning 4771
Machine Learning 4771 Instructor: Tony Jebara Topic 7 Unsupervised Learning Statistical Perspective Probability Models Discrete & Continuous: Gaussian, Bernoulli, Multinomial Maimum Likelihood Logistic
More informationReview: Support vector machines. Machine learning techniques and image analysis
Review: Support vector machines Review: Support vector machines Margin optimization min (w,w 0 ) 1 2 w 2 subject to y i (w 0 + w T x i ) 1 0, i = 1,..., n. Review: Support vector machines Margin optimization
More informationClassification Based on Probability
Logistic Regression These slides were assembled by Byron Boots, with only minor modifications from Eric Eaton s slides and grateful acknowledgement to the many others who made their course materials freely
More informationRidge Regression: Regulating overfitting when using many features. Training, true, & test error vs. model complexity. CSE 446: Machine Learning
Ridge Regression: Regulating overfitting when using many features Emily Fox University of Washington January 3, 207 Training, true, & test error vs. model complexity Overfitting if: Error y Model complexity
More informationLogistic Regression. Machine Learning Fall 2018
Logistic Regression Machine Learning Fall 2018 1 Where are e? We have seen the folloing ideas Linear models Learning as loss minimization Bayesian learning criteria (MAP and MLE estimation) The Naïve Bayes
More informationLecture 2 Machine Learning Review
Lecture 2 Machine Learning Review CMSC 35246: Deep Learning Shubhendu Trivedi & Risi Kondor University of Chicago March 29, 2017 Things we will look at today Formal Setup for Supervised Learning Things
More informationMachine Learning and Data Mining. Linear classification. Kalev Kask
Machine Learning and Data Mining Linear classification Kalev Kask Supervised learning Notation Features x Targets y Predictions ŷ = f(x ; q) Parameters q Program ( Learner ) Learning algorithm Change q
More information15-388/688 - Practical Data Science: Nonlinear modeling, cross-validation, regularization, and evaluation
15-388/688 - Practical Data Science: Nonlinear modeling, cross-validation, regularization, and evaluation J. Zico Kolter Carnegie Mellon University Fall 2016 1 Outline Example: return to peak demand prediction
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 informationCS 340 Lec. 16: Logistic Regression
CS 34 Lec. 6: Logistic Regression AD March AD ) March / 6 Introduction Assume you are given some training data { x i, y i } i= where xi R d and y i can take C different values. Given an input test data
More informationLinear Models in Machine Learning
CS540 Intro to AI Linear Models in Machine Learning Lecturer: Xiaojin Zhu jerryzhu@cs.wisc.edu We briefly go over two linear models frequently used in machine learning: linear regression for, well, regression,
More informationToday. Calculus. Linear Regression. Lagrange Multipliers
Today Calculus Lagrange Multipliers Linear Regression 1 Optimization with constraints What if I want to constrain the parameters of the model. The mean is less than 10 Find the best likelihood, subject
More informationLecture 10. Neural networks and optimization. Machine Learning and Data Mining November Nando de Freitas UBC. Nonlinear Supervised Learning
Lecture 0 Neural networks and optimization Machine Learning and Data Mining November 2009 UBC Gradient Searching for a good solution can be interpreted as looking for a minimum of some error (loss) function
More informationLogistic Regression. Robot Image Credit: Viktoriya Sukhanova 123RF.com
Logistic Regression These slides were assembled by Eric Eaton, with grateful acknowledgement of the many others who made their course materials freely available online. Feel free to reuse or adapt these
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 informationMachine Learning - Waseda University Logistic Regression
Machine Learning - Waseda University Logistic Regression AD June AD ) June / 9 Introduction Assume you are given some training data { x i, y i } i= where xi R d and y i can take C different values. Given
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 informationMachine Learning (CSE 446): Neural Networks
Machine Learning (CSE 446): Neural Networks Noah Smith c 2017 University of Washington nasmith@cs.washington.edu November 6, 2017 1 / 22 Admin No Wednesday office hours for Noah; no lecture Friday. 2 /
More informationMachine Learning Lecture 7
Course Outline Machine Learning Lecture 7 Fundamentals (2 weeks) Bayes Decision Theory Probability Density Estimation Statistical Learning Theory 23.05.2016 Discriminative Approaches (5 weeks) Linear Discriminant
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 informationThe Perceptron algorithm
The Perceptron algorithm Tirgul 3 November 2016 Agnostic PAC Learnability A hypothesis class H is agnostic PAC learnable if there exists a function m H : 0,1 2 N and a learning algorithm with the following
More informationMachine Learning Basics Lecture 2: Linear Classification. Princeton University COS 495 Instructor: Yingyu Liang
Machine Learning Basics Lecture 2: Linear Classification Princeton University COS 495 Instructor: Yingyu Liang Review: machine learning basics Math formulation Given training data x i, y i : 1 i n i.i.d.
More informationCPSC 340: Machine Learning and Data Mining. Gradient Descent Fall 2016
CPSC 340: Machine Learning and Data Mining Gradient Descent Fall 2016 Admin Assignment 1: Marks up this weekend on UBC Connect. Assignment 2: 3 late days to hand it in Monday. Assignment 3: Due Wednesday
More informationCOMP 551 Applied Machine Learning Lecture 3: Linear regression (cont d)
COMP 551 Applied Machine Learning Lecture 3: Linear regression (cont d) Instructor: Herke van Hoof (herke.vanhoof@mail.mcgill.ca) Slides mostly by: Class web page: www.cs.mcgill.ca/~hvanho2/comp551 Unless
More informationMachine 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
More informationLinear discriminant functions
Andrea Passerini passerini@disi.unitn.it Machine Learning Discriminative learning Discriminative vs generative Generative learning assumes knowledge of the distribution governing the data Discriminative
More informationLINEAR CLASSIFICATION, PERCEPTRON, LOGISTIC REGRESSION, SVC, NAÏVE BAYES. Supervised Learning
LINEAR CLASSIFICATION, PERCEPTRON, LOGISTIC REGRESSION, SVC, NAÏVE BAYES Supervised Learning Linear vs non linear classifiers In K-NN we saw an example of a non-linear classifier: the decision boundary
More informationMachine Learning. Kernels. Fall (Kernels, Kernelized Perceptron and SVM) Professor Liang Huang. (Chap. 12 of CIML)
Machine Learning Fall 2017 Kernels (Kernels, Kernelized Perceptron and SVM) Professor Liang Huang (Chap. 12 of CIML) Nonlinear Features x4: -1 x1: +1 x3: +1 x2: -1 Concatenated (combined) features XOR:
More informationCPSC 340: Machine Learning and Data Mining
CPSC 340: Machine Learning and Data Mining Linear Classifiers: predictions Original version of these slides by Mark Schmidt, with modifications by Mike Gelbart. 1 Admin Assignment 4: Due Friday of next
More informationLecture 4 Logistic Regression
Lecture 4 Logistic Regression Dr.Ammar Mohammed Normal Equation Hypothesis hθ(x)=θ0 x0+ θ x+ θ2 x2 +... + θd xd Normal Equation is a method to find the values of θ operations x0 x x2.. xd y x x2... xd
More informationLinear classifiers: Logistic regression
Linear classifiers: Logistic regression STAT/CSE 416: Machine Learning Emily Fox University of Washington April 19, 2018 How confident is your prediction? The sushi & everything else were awesome! The
More informationMachine Learning Practice Page 2 of 2 10/28/13
Machine Learning 10-701 Practice Page 2 of 2 10/28/13 1. True or False Please give an explanation for your answer, this is worth 1 pt/question. (a) (2 points) No classifier can do better than a naive Bayes
More informationRirdge Regression. Szymon Bobek. Institute of Applied Computer science AGH University of Science and Technology
Rirdge Regression Szymon Bobek Institute of Applied Computer science AGH University of Science and Technology Based on Carlos Guestrin adn Emily Fox slides from Coursera Specialization on Machine Learnign
More informationNotes on Discriminant Functions and Optimal Classification
Notes on Discriminant Functions and Optimal Classification Padhraic Smyth, Department of Computer Science University of California, Irvine c 2017 1 Discriminant Functions Consider a classification problem
More informationLogistic Regression. COMP 527 Danushka Bollegala
Logistic Regression COMP 527 Danushka Bollegala Binary Classification Given an instance x we must classify it to either positive (1) or negative (0) class We can use {1,-1} instead of {1,0} but we will
More informationRecap from previous lecture
Recap from previous lecture Learning is using past experience to improve future performance. Different types of learning: supervised unsupervised reinforcement active online... For a machine, experience
More information10-701/ Machine Learning - Midterm Exam, Fall 2010
10-701/15-781 Machine Learning - Midterm Exam, Fall 2010 Aarti Singh Carnegie Mellon University 1. Personal info: Name: Andrew account: E-mail address: 2. There should be 15 numbered pages in this exam
More informationSingle layer NN. Neuron Model
Single layer NN We consider the simple architecture consisting of just one neuron. Generalization to a single layer with more neurons as illustrated below is easy because: M M The output units are independent
More informationLogistic Regression Review Fall 2012 Recitation. September 25, 2012 TA: Selen Uguroglu
Logistic Regression Review 10-601 Fall 2012 Recitation September 25, 2012 TA: Selen Uguroglu!1 Outline Decision Theory Logistic regression Goal Loss function Inference Gradient Descent!2 Training Data
More informationWarm up: risk prediction with logistic regression
Warm up: risk prediction with logistic regression Boss gives you a bunch of data on loans defaulting or not: {(x i,y i )} n i= x i 2 R d, y i 2 {, } You model the data as: P (Y = y x, w) = + exp( yw T
More informationCS229 Supplemental Lecture notes
CS229 Supplemental Lecture notes John Duchi Binary classification In binary classification problems, the target y can take on at only two values. In this set of notes, we show how to model this problem
More informationNeural Networks: Introduction
Neural Networks: Introduction Machine Learning Fall 2017 Based on slides and material from Geoffrey Hinton, Richard Socher, Dan Roth, Yoav Goldberg, Shai Shalev-Shwartz and Shai Ben-David, and others 1
More informationClassification Logistic Regression
Classification Logistic Regression Machine Learning CSE546 Kevin Jamieson University of Washington October 16, 2016 1 THUS FAR, REGRESSION: PREDICT A CONTINUOUS VALUE GIVEN SOME INPUTS 2 Weather prediction
More informationDiscriminative Models
No.5 Discriminative Models Hui Jiang Department of Electrical Engineering and Computer Science Lassonde School of Engineering York University, Toronto, Canada Outline Generative vs. Discriminative models
More informationLinear Regression (continued)
Linear Regression (continued) Professor Ameet Talwalkar Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 6, 2017 1 / 39 Outline 1 Administration 2 Review of last lecture 3 Linear regression
More informationUNIVERSITY of PENNSYLVANIA CIS 520: Machine Learning Final, Fall 2013
UNIVERSITY of PENNSYLVANIA CIS 520: Machine Learning Final, Fall 2013 Exam policy: This exam allows two one-page, two-sided cheat sheets; No other materials. Time: 2 hours. Be sure to write your name and
More informationIntroduction to Machine Learning
Introduction to Machine Learning Logistic Regression Varun Chandola Computer Science & Engineering State University of New York at Buffalo Buffalo, NY, USA chandola@buffalo.edu Chandola@UB CSE 474/574
More informationMachine Learning Basics III
Machine Learning Basics III Benjamin Roth CIS LMU München Benjamin Roth (CIS LMU München) Machine Learning Basics III 1 / 62 Outline 1 Classification Logistic Regression 2 Gradient Based Optimization Gradient
More informationLecture 3: Statistical Decision Theory (Part II)
Lecture 3: Statistical Decision Theory (Part II) Hao Helen Zhang Hao Helen Zhang Lecture 3: Statistical Decision Theory (Part II) 1 / 27 Outline of This Note Part I: Statistics Decision Theory (Classical
More informationBinary Classification / Perceptron
Binary Classification / Perceptron Nicholas Ruozzi University of Texas at Dallas Slides adapted from David Sontag and Vibhav Gogate Supervised Learning Input: x 1, y 1,, (x n, y n ) x i is the i th data
More informationBias-Variance Tradeoff
What s learning, revisited Overfitting Generative versus Discriminative Logistic Regression Machine Learning 10701/15781 Carlos Guestrin Carnegie Mellon University September 19 th, 2007 Bias-Variance Tradeoff
More informationIntroduction to Machine Learning
Introduction to Machine Learning Linear Regression Varun Chandola Computer Science & Engineering State University of New York at Buffalo Buffalo, NY, USA chandola@buffalo.edu Chandola@UB CSE 474/574 1
More informationCheng 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 (Many figures from C. M. Bishop, "Pattern Recognition and ") 1of 254 Part V
More informationSVAN 2016 Mini Course: Stochastic Convex Optimization Methods in Machine Learning
SVAN 2016 Mini Course: Stochastic Convex Optimization Methods in Machine Learning Mark Schmidt University of British Columbia, May 2016 www.cs.ubc.ca/~schmidtm/svan16 Some images from this lecture are
More informationComments. x > w = w > x. Clarification: this course is about getting you to be able to think as a machine learning expert
Logistic regression Comments Mini-review and feedback These are equivalent: x > w = w > x Clarification: this course is about getting you to be able to think as a machine learning expert There has to be
More informationMachine Learning, Fall 2009: Midterm
10-601 Machine Learning, Fall 009: Midterm Monday, November nd hours 1. Personal info: Name: Andrew account: E-mail address:. You are permitted two pages of notes and a calculator. Please turn off all
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 informationFA Homework 2 Recitation 2
FA17 10-701 Homework 2 Recitation 2 Logan Brooks,Matthew Oresky,Guoquan Zhao October 2, 2017 Logan Brooks,Matthew Oresky,Guoquan Zhao FA17 10-701 Homework 2 Recitation 2 October 2, 2017 1 / 15 Outline
More informationIntroduction to Machine Learning
Introduction to Machine Learning Vapnik Chervonenkis Theory Barnabás Póczos Empirical Risk and True Risk 2 Empirical Risk Shorthand: True risk of f (deterministic): Bayes risk: Let us use the empirical
More informationNeural networks and support vector machines
Neural netorks and support vector machines Perceptron Input x 1 Weights 1 x 2 x 3... x D 2 3 D Output: sgn( x + b) Can incorporate bias as component of the eight vector by alays including a feature ith
More informationECS171: Machine Learning
ECS171: Machine Learning Lecture 4: Optimization (LFD 3.3, SGD) Cho-Jui Hsieh UC Davis Jan 22, 2018 Gradient descent Optimization Goal: find the minimizer of a function min f (w) w For now we assume f
More informationMachine learning comes from Bayesian decision theory in statistics. There we want to minimize the expected value of the loss function.
Bayesian learning: Machine learning comes from Bayesian decision theory in statistics. There we want to minimize the expected value of the loss function. Let y be the true label and y be the predicted
More informationCOMP 551 Applied Machine Learning Lecture 13: Dimension reduction and feature selection
COMP 551 Applied Machine Learning Lecture 13: Dimension reduction and feature selection Instructor: Herke van Hoof (herke.vanhoof@cs.mcgill.ca) Based on slides by:, Jackie Chi Kit Cheung Class web page:
More informationThe exam is closed book, closed notes except your one-page (two sides) or two-page (one side) crib sheet.
CS 189 Spring 013 Introduction to Machine Learning Final You have 3 hours for the exam. The exam is closed book, closed notes except your one-page (two sides) or two-page (one side) crib sheet. Please
More informationMIDTERM SOLUTIONS: FALL 2012 CS 6375 INSTRUCTOR: VIBHAV GOGATE
MIDTERM SOLUTIONS: FALL 2012 CS 6375 INSTRUCTOR: VIBHAV GOGATE March 28, 2012 The exam is closed book. You are allowed a double sided one page cheat sheet. Answer the questions in the spaces provided on
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 informationECE521 week 3: 23/26 January 2017
ECE521 week 3: 23/26 January 2017 Outline Probabilistic interpretation of linear regression - Maximum likelihood estimation (MLE) - Maximum a posteriori (MAP) estimation Bias-variance trade-off Linear
More informationGaussian and Linear Discriminant Analysis; Multiclass Classification
Gaussian and Linear Discriminant Analysis; Multiclass Classification Professor Ameet Talwalkar Slide Credit: Professor Fei Sha Professor Ameet Talwalkar CS260 Machine Learning Algorithms October 13, 2015
More informationComputational statistics
Computational statistics Lecture 3: Neural networks Thierry Denœux 5 March, 2016 Neural networks A class of learning methods that was developed separately in different fields statistics and artificial
More informationRegression and Classification" with Linear Models" CMPSCI 383 Nov 15, 2011!
Regression and Classification" with Linear Models" CMPSCI 383 Nov 15, 2011! 1 Todayʼs topics" Learning from Examples: brief review! Univariate Linear Regression! Batch gradient descent! Stochastic gradient
More informationSupport 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
More informationLinear classifiers: Overfitting and regularization
Linear classifiers: Overfitting and regularization Emily Fox University of Washington January 25, 2017 Logistic regression recap 1 . Thus far, we focused on decision boundaries Score(x i ) = w 0 h 0 (x
More informationLecture 3 Feedforward Networks and Backpropagation
Lecture 3 Feedforward Networks and Backpropagation CMSC 35246: Deep Learning Shubhendu Trivedi & Risi Kondor University of Chicago April 3, 2017 Things we will look at today Recap of Logistic Regression
More informationNeural Networks Learning the network: Backprop , Fall 2018 Lecture 4
Neural Networks Learning the network: Backprop 11-785, Fall 2018 Lecture 4 1 Recap: The MLP can represent any function The MLP can be constructed to represent anything But how do we construct it? 2 Recap:
More informationCOMP 551 Applied Machine Learning Lecture 2: Linear Regression
COMP 551 Applied Machine Learning Lecture 2: Linear Regression Instructor: Herke van Hoof (herke.vanhoof@mail.mcgill.ca) Slides mostly by: Class web page: www.cs.mcgill.ca/~hvanho2/comp551 Unless otherwise
More informationSTA 414/2104, Spring 2014, Practice Problem Set #1
STA 44/4, Spring 4, Practice Problem Set # Note: these problems are not for credit, and not to be handed in Question : Consider a classification problem in which there are two real-valued inputs, and,
More informationNONLINEAR CLASSIFICATION AND REGRESSION. J. Elder CSE 4404/5327 Introduction to Machine Learning and Pattern Recognition
NONLINEAR CLASSIFICATION AND REGRESSION Nonlinear Classification and Regression: Outline 2 Multi-Layer Perceptrons The Back-Propagation Learning Algorithm Generalized Linear Models Radial Basis Function
More informationSupport 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
More informationJeff 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
More informationStatistical Data Mining and Machine Learning Hilary Term 2016
Statistical Data Mining and Machine Learning Hilary Term 2016 Dino Sejdinovic Department of Statistics Oxford Slides and other materials available at: http://www.stats.ox.ac.uk/~sejdinov/sdmml Naïve Bayes
More informationLinear models: the perceptron and closest centroid algorithms. D = {(x i,y i )} n i=1. x i 2 R d 9/3/13. Preliminaries. Chapter 1, 7.
Preliminaries Linear models: the perceptron and closest centroid algorithms Chapter 1, 7 Definition: The Euclidean dot product beteen to vectors is the expression d T x = i x i The dot product is also
More informationLinear & nonlinear classifiers
Linear & nonlinear classifiers Machine Learning Hamid Beigy Sharif University of Technology Fall 1394 Hamid Beigy (Sharif University of Technology) Linear & nonlinear classifiers Fall 1394 1 / 34 Table
More informationIs the test error unbiased for these programs?
Is the test error unbiased for these programs? Xtrain avg N o Preprocessing by de meaning using whole TEST set 2017 Kevin Jamieson 1 Is the test error unbiased for this program? e Stott see non for f x
More informationArtificial Intelligence
Artificial Intelligence Jeff Clune Assistant Professor Evolving Artificial Intelligence Laboratory Announcements Be making progress on your projects! Three Types of Learning Unsupervised Supervised Reinforcement
More informationDiscriminative Models
No.5 Discriminative Models Hui Jiang Department of Electrical Engineering and Computer Science Lassonde School of Engineering York University, Toronto, Canada Outline Generative vs. Discriminative models
More informationVBM683 Machine Learning
VBM683 Machine Learning Pinar Duygulu Slides are adapted from Dhruv Batra Bias is the algorithm's tendency to consistently learn the wrong thing by not taking into account all the information in the data
More informationLecture 4: Training a Classifier
Lecture 4: Training a Classifier Roger Grosse 1 Introduction Now that we ve defined what binary classification is, let s actually train a classifier. We ll approach this problem in much the same way as
More informationWe choose parameter values that will minimize the difference between the model outputs & the true function values.
CSE 4502/5717 Big Data Analytics Lecture #16, 4/2/2018 with Dr Sanguthevar Rajasekaran Notes from Yenhsiang Lai Machine learning is the task of inferring a function, eg, f : R " R This inference has to
More informationStatistical Machine Learning Hilary Term 2018
Statistical Machine Learning Hilary Term 2018 Pier Francesco Palamara Department of Statistics University of Oxford Slide credits and other course material can be found at: http://www.stats.ox.ac.uk/~palamara/sml18.html
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 informationLogistic Regression Trained with Different Loss Functions. Discussion
Logistic Regression Trained with Different Loss Functions Discussion CS640 Notations We restrict our discussions to the binary case. g(z) = g (z) = g(z) z h w (x) = g(wx) = + e z = g(z)( g(z)) + e wx =
More informationRegression. Machine Learning and Pattern Recognition. Chris Williams. School of Informatics, University of Edinburgh.
Regression Machine Learning and Pattern Recognition Chris Williams School of Informatics, University of Edinburgh September 24 (All of the slides in this course have been adapted from previous versions
More information