Machine Learning - MT Linear Regression
|
|
- Amberlynn Shepherd
- 5 years ago
- Views:
Transcription
1 Machine Learning - MT Linear Regression Varun Kanade University of Oxford October 12, 2016
2 Announcements All students eligible to take the course for credit can sign-up for classes and practicals Attempt Problem Sheet 0 (contact your class tutor if you intend to attend class in Week 2) Problem Sheet 1 is posted (submit by noon 21 Oct at CS reception) 1
3 Announcement : Strachey Lecture Will finish min early on Monday, October 31 May run over by 5 minutes or so a few other days 2
4 Outline Goals Review the supervised learning setting Describe the linear regression framework Apply the linear model to make predictions Derive the least squares estimate Supervised Learning Setting Data consists of input and output pairs Inputs (also covariates, independent variables, predictors, features) Output (also variates, dependent variable, targets, labels) 3
5 Why study linear regression? Least squares is at least 200 years old going back to Legendre and Gauss Francis Galton (1886): Regression to the mean Often real processes can be approximated by linear models More complex models require understanding linear regression Closed form analytic solutions can be obtained Many key notions of machine learning can be introduced 4
6 A toy example : Commute Times Want to predict commute time into city centre What variables would be useful? Distance to city centre Day of the week Data dist (km) day commute time (min) 2.7 fri mon sun tue sat 22 5
7 Linear Models Suppose the input is a vector x R D and the output is y R. We have data x i, y i N i=1 Notation: data dimension D, size of dataset N, column vectors Linear Model y = w 0 + x 1w x Dw D + ɛ Bias/intercept Noise/uncertainty 6
8 Linear Models : Commute Time Linear Model y = w 0 + x 1w x Dw D + ɛ Bias/intercept Noise/uncertainty Input encoding: mon-sun has to be converted to a number monday: 0, tuesday: 1,..., sunday: 6 0 if weekend, 1 if weekday Say x 1 R (distance) and x 2 {0, 1} (weekend/weekday) Linear model for commute time y = w 0 + w 1x 1 + w 2x 2 + ɛ Using 0-6 is a bad encoding. Use seven 0-1 features instead called one-hot encoding 7
9 Linear Model : Adding a feature for bias term dist day commute time x 1 x 2 y 2.7 fri mon sun tue sat 22 one dist day commute time x 0 x 1 x 2 y fri mon sun tue sat 22 Model y = w 0 + w 1x 1 + w 2x 2 + ɛ Model y = w 0x 0 + w 1x 1 + w 2x 2 + ɛ = w x + ɛ 8
10 Learning Linear Models Data: (x i, y i) N i=1, where x i R D and y i R Model parameter w, where w R D Training phase: (learning/estimation w from data) (x i, y i) N i=1 data Learning Algorithm w (estimate) Testing/Deployment phase: (predict ŷ new = x new w) How different is ŷ new from y new (actual observation)? We should keep some data aside for testing before deploying a model 9
11 (x i, y i) N i=1, where x i R and y i R ŷ(x) = w 0 + x w 1, (no noise term in ŷ) L(w) = L(w 0, w 1) = 1 2N N (ŷ i y i) 2 = 1 N (w 0 + x i w 1 y i) 2 2N i=1 i=1 Loss function Cost function Objective Function Energy Function Notation - L, J, E, R This objective is known as the residual sum of squares or (RSS) The estimate (w 0, w 1) is known as the least squares estimate 10
12 (x i, y i) N i=1, where x i R and y i R ŷ(x) = w 0 + x w 1, (no noise term in ŷ) L(w) = L(w 0, w 1) = 1 2N N (ŷ i y i) 2 = 1 N (w 0 + x i w 1 y i) 2 2N i=1 i=1 L = 1 w 0 N L = 1 w 1 N N (w 0 + w 1 x i y i) i=1 N (w 0 + w 1 x i y i)x i i=1 We obtain the solution for (w 0, w 1) by setting the partial derivatives to 0 and solving the resulting system. (Normal Equations) w 0 w 0 + w 1 i xi N + w1 i xi N = i x2 i N = i yi N i xiyi N (1) (2) x = ȳ = var(x) = ĉov(x, y) = w 1 = i xi N i yi N i x2 i N x2 i xiyi x ȳ N ĉov(x, y) var(x) w 0 = ȳ w 1 x 11
13 Linear Regression : General Case Recall that the linear model is ŷ i = D j=0 x ijw j where we assume that x i0 = 1 for all x i, so that the bias term w 0 does not need to be treated separately. Expressing everything in matrix notation ŷ = Xw Here we have ŷ R N 1, X R N (D+1) and w R (D+1) 1 ŷ N 1 ŷ 1 ŷ 2. ŷ N = X N (D+1) w (D+1) 1 w 0 x T 1 x T 2.. x T N. w D = X N (D+1) x 10 x 1D x 20 x 2D x N0 x ND w (D+1) 1 w 0. w D 12
14 Back to toy example one dist (km) weekday? commute time (min) (fri) (mon) (sun) (tue) (sat) 22 We have N = 5, D + 1 = 3 and so we get y = 15, X = , w = Suppose we get w = [6.09, 6.53, 2.11] T. Then our predictions would be ŷ = w 0 w 1 w 2 13
15 Least Squares Estimate : Minimise the Squared Error L(w) = 1 2N N (x T i w y i) 2 = (Xw y) T (Xw y) i=1 14
16 Finding Optimal Solutions using Calculus L(w) = 1 2N = 1 2N = 1 2N = N i=1 (x T i w y i) 2 = 1 2N (Xw y)t (Xw y) ( w T ( X T X ) ) w w T X T y y T Xw + y T y ( ( ) ) w T X T X w 2 y T Xw + y T y Then, write out all partial derivatives to form the gradient wl L w 0 = L w 1 =. L w D = Instead, we will develop tricks to differentiate using matrix notation directly 15
17 Differentiating Matrix Expressions Rules (Tricks) ) (i) Linear Form Expressions: w (c T w = c c T w = D c jw j j=0 ) (c T w) = c w j j, and so w (c T w (ii) Quadratic Form Expressions: ) w (w T Aw = Aw + A T w ( = 2Aw for symmetric A) w T Aw = (w T Aw) w k = D i=0 j=0 i=0 D w iw ja ij D D w ia ik + A kj w j = A T [:,k]w + A [k,:] w j=0 = c (3) ) w (w T Aw = A T w + Aw (4) 16
18 Deriving the Least Squares Estimate L(w) = 1 2N N i=1 (x T i w y i) 2 = 1 2N ( ( ) ) w T X T X w 2 y T Xw + y T y We compute the gradient wl = 0 using the matrix differentiation rules, wl = 1 ( ( ) X X) T w X T y N By setting wl = 0 and solving we get, ( ) X T X w = X T y ( 1 w = X X) T X T y (Assuming inverse exists) The predictions made by the model on the data X are given by ( 1 ŷ = Xw = X X X) T X T y ( 1 For this reason the matrix X X X) T X T is called the hat matrix 17
19 Least Squares Estimate w = ( X T X) 1 X T y When do we expect X T X to be invertible? rank(x T X) = rank(x) min{d + 1, N} As X T X is D + 1 D + 1, invertible is rank(x) = D + 1 What if we use one-hot encoding for a feature like day? Suppose x mon,..., x sun stand for 0-1 valued variables in the one-hot encoding We always have x mon + + x sun = 1 This introduces a linear dependence in the columns of X reducing the rank In this case, we can drop some features to adjust rank. We ll see alternative approaches later in the course. What is the computational complexity of computing w? Relatively easy to get O(D 2 N) bound 18
20 19
21 Recap : Predicting Commute Time Goal Predict the time taken for commute given distance and day of week Do we only wish to make predictions or also suggestions? Model and Choice of Loss Function Use a linear model y = w 0 + w 1x w Dx D + ɛ = ŷ + ɛ Minimise average squared error 1 (yi ŷ 2N i) 2 Algorithm to Fit Model Simple matrix operations using closed-form solution 20
22 Model and Loss Function Choice Optimisation View of Machine Learning Pick model that you expect may fit the data well enough Pick a measure of performance that makes sense and can be optimised Run optimisation algorithm to obtain model parameters Probabilistic View of Machine Learning Pick a model for data and explicitly formulate the deviation (or uncertainty) from the model using the language of probability Use notions from probability to define suitability of various models Find the parameters or make predictions on unseen data using these suitability criteria (Frequentist vs Bayesian viewpoints) 21
23 Next Time Probabilistic View of Machine Learning (Maximum Likelihood) Non-linearity using basis expansion What to do when you have more features than data? Make sure you re familiar with the the multi-variate Gaussian distribution 22
Machine Learning - MT & 5. Basis Expansion, Regularization, Validation
Machine Learning - MT 2016 4 & 5. Basis Expansion, Regularization, Validation Varun Kanade University of Oxford October 19 & 24, 2016 Outline Basis function expansion to capture non-linear relationships
More informationMachine learning - HT Maximum Likelihood
Machine learning - HT 2016 3. Maximum Likelihood Varun Kanade University of Oxford January 27, 2016 Outline Probabilistic Framework Formulate linear regression in the language of probability Introduce
More informationMachine Learning - MT Classification: Generative Models
Machine Learning - MT 2016 7. Classification: Generative Models Varun Kanade University of Oxford October 31, 2016 Announcements Practical 1 Submission Try to get signed off during session itself Otherwise,
More informationMachine Learning - MT & 14. PCA and MDS
Machine Learning - MT 2016 13 & 14. PCA and MDS Varun Kanade University of Oxford November 21 & 23, 2016 Announcements Sheet 4 due this Friday by noon Practical 3 this week (continue next week if necessary)
More informationModeling Data with Linear Combinations of Basis Functions. Read Chapter 3 in the text by Bishop
Modeling Data with Linear Combinations of Basis Functions Read Chapter 3 in the text by Bishop A Type of Supervised Learning Problem We want to model data (x 1, t 1 ),..., (x N, t N ), where x i is a vector
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 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 informationCPSC 540: Machine Learning
CPSC 540: Machine Learning Matrix Notation Mark Schmidt University of British Columbia Winter 2017 Admin Auditting/registration forms: Submit them at end of class, pick them up end of next class. I need
More informationCOS513: FOUNDATIONS OF PROBABILISTIC MODELS LECTURE 9: LINEAR REGRESSION
COS513: FOUNDATIONS OF PROBABILISTIC MODELS LECTURE 9: LINEAR REGRESSION SEAN GERRISH AND CHONG WANG 1. WAYS OF ORGANIZING MODELS In probabilistic modeling, there are several ways of organizing models:
More informationAssociation studies and regression
Association studies and regression CM226: Machine Learning for Bioinformatics. Fall 2016 Sriram Sankararaman Acknowledgments: Fei Sha, Ameet Talwalkar Association studies and regression 1 / 104 Administration
More informationLecture 4: Types of errors. Bayesian regression models. Logistic regression
Lecture 4: Types of errors. Bayesian regression models. Logistic regression A Bayesian interpretation of regularization Bayesian vs maximum likelihood fitting more generally COMP-652 and ECSE-68, Lecture
More informationCOMP 551 Applied Machine Learning Lecture 20: Gaussian processes
COMP 55 Applied Machine Learning Lecture 2: Gaussian processes Instructor: Ryan Lowe (ryan.lowe@cs.mcgill.ca) Slides mostly by: (herke.vanhoof@mcgill.ca) Class web page: www.cs.mcgill.ca/~hvanho2/comp55
More informationCSC2515 Winter 2015 Introduction to Machine Learning. Lecture 2: Linear regression
CSC2515 Winter 2015 Introduction to Machine Learning Lecture 2: Linear regression All lecture slides will be available as.pdf on the course website: http://www.cs.toronto.edu/~urtasun/courses/csc2515/csc2515_winter15.html
More informationMachine learning - HT Basis Expansion, Regularization, Validation
Machine learning - HT 016 4. Basis Expansion, Regularization, Validation Varun Kanade University of Oxford Feburary 03, 016 Outline Introduce basis function to go beyond linear regression Understanding
More informationMidterm. Introduction to Machine Learning. CS 189 Spring Please do not open the exam before you are instructed to do so.
CS 89 Spring 07 Introduction to Machine Learning Midterm Please do not open the exam before you are instructed to do so. The exam is closed book, closed notes except your one-page cheat sheet. Electronic
More informationBayesian Linear Regression [DRAFT - In Progress]
Bayesian Linear Regression [DRAFT - In Progress] David S. Rosenberg Abstract Here we develop some basics of Bayesian linear regression. Most of the calculations for this document come from the basic theory
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 informationLeast Squares Regression
E0 70 Machine Learning Lecture 4 Jan 7, 03) Least Squares Regression Lecturer: Shivani Agarwal Disclaimer: These notes are a brief summary of the topics covered in the lecture. They are not a substitute
More informationLinear Regression. In this problem sheet, we consider the problem of linear regression with p predictors and one intercept,
Linear Regression In this problem sheet, we consider the problem of linear regression with p predictors and one intercept, y = Xβ + ɛ, where y t = (y 1,..., y n ) is the column vector of target values,
More informationRegression with Numerical Optimization. Logistic
CSG220 Machine Learning Fall 2008 Regression with Numerical Optimization. Logistic regression Regression with Numerical Optimization. Logistic regression based on a document by Andrew Ng October 3, 204
More informationLeast Squares Regression
CIS 50: Machine Learning Spring 08: Lecture 4 Least Squares Regression Lecturer: Shivani Agarwal Disclaimer: These notes are designed to be a supplement to the lecture. They may or may not cover all the
More informationLecture 5: Linear models for classification. Logistic regression. Gradient Descent. Second-order methods.
Lecture 5: Linear models for classification. Logistic regression. Gradient Descent. Second-order methods. Linear models for classification Logistic regression Gradient descent and second-order methods
More informationDD Advanced Machine Learning
Modelling Carl Henrik {chek}@csc.kth.se Royal Institute of Technology November 4, 2015 Who do I think you are? Mathematically competent linear algebra multivariate calculus Ok programmers Able to extend
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 informationOverview. Probabilistic Interpretation of Linear Regression Maximum Likelihood Estimation Bayesian Estimation MAP Estimation
Overview Probabilistic Interpretation of Linear Regression Maximum Likelihood Estimation Bayesian Estimation MAP Estimation Probabilistic Interpretation: Linear Regression Assume output y is generated
More informationMachine Learning Linear Models
Machine Learning Linear Models Outline II - Linear Models 1. Linear Regression (a) Linear regression: History (b) Linear regression with Least Squares (c) Matrix representation and Normal Equation Method
More informationLoss Functions and Optimization. Lecture 3-1
Lecture 3: Loss Functions and Optimization Lecture 3-1 Administrative: Live Questions We ll use Zoom to take questions from remote students live-streaming the lecture Check Piazza for instructions and
More information4 Bias-Variance for Ridge Regression (24 points)
Implement Ridge Regression with λ = 0.00001. Plot the Squared Euclidean test error for the following values of k (the dimensions you reduce to): k = {0, 50, 100, 150, 200, 250, 300, 350, 400, 450, 500,
More informationGI07/COMPM012: Mathematical Programming and Research Methods (Part 2) 2. Least Squares and Principal Components Analysis. Massimiliano Pontil
GI07/COMPM012: Mathematical Programming and Research Methods (Part 2) 2. Least Squares and Principal Components Analysis Massimiliano Pontil 1 Today s plan SVD and principal component analysis (PCA) Connection
More informationLecture 2: Linear regression
Lecture 2: Linear regression Roger Grosse 1 Introduction Let s ump right in and look at our first machine learning algorithm, linear regression. In regression, we are interested in predicting a scalar-valued
More informationComputer Vision Group Prof. Daniel Cremers. 2. Regression (cont.)
Prof. Daniel Cremers 2. Regression (cont.) Regression with MLE (Rep.) Assume that y is affected by Gaussian noise : t = f(x, w)+ where Thus, we have p(t x, w, )=N (t; f(x, w), 2 ) 2 Maximum A-Posteriori
More informationLinear regression COMS 4771
Linear regression COMS 4771 1. Old Faithful and prediction functions Prediction problem: Old Faithful geyser (Yellowstone) Task: Predict time of next eruption. 1 / 40 Statistical model for time between
More informationLecture 13: Simple Linear Regression in Matrix Format. 1 Expectations and Variances with Vectors and Matrices
Lecture 3: Simple Linear Regression in Matrix Format To move beyond simple regression we need to use matrix algebra We ll start by re-expressing simple linear regression in matrix form Linear algebra is
More informationCOMP90051 Statistical Machine Learning
COMP90051 Statistical Machine Learning Semester 2, 2017 Lecturer: Trevor Cohn 17. Bayesian inference; Bayesian regression Training == optimisation (?) Stages of learning & inference: Formulate model Regression
More informationMachine Learning Lecture 5
Machine Learning Lecture 5 Linear Discriminant Functions 26.10.2017 Bastian Leibe RWTH Aachen http://www.vision.rwth-aachen.de leibe@vision.rwth-aachen.de Course Outline Fundamentals Bayes Decision Theory
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 informationJ. Sadeghi E. Patelli M. de Angelis
J. Sadeghi E. Patelli Institute for Risk and, Department of Engineering, University of Liverpool, United Kingdom 8th International Workshop on Reliable Computing, Computing with Confidence University of
More informationWill it rain tomorrow?
Will it rain tomorrow? Bilal Ahmed - 561539 Department of Computing and Information Systems, The University of Melbourne, Victoria, Australia bahmad@student.unimelb.edu.au Abstract With the availability
More informationMultivariate Bayesian Linear Regression MLAI Lecture 11
Multivariate Bayesian Linear Regression MLAI Lecture 11 Neil D. Lawrence Department of Computer Science Sheffield University 21st October 2012 Outline Univariate Bayesian Linear Regression Multivariate
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 informationAdvanced Introduction to Machine Learning
10-715 Advanced Introduction to Machine Learning Homework Due Oct 15, 10.30 am Rules Please follow these guidelines. Failure to do so, will result in loss of credit. 1. Homework is due on the due date
More informationLogistic Regression Introduction to Machine Learning. Matt Gormley Lecture 9 Sep. 26, 2018
10-601 Introduction to Machine Learning Machine Learning Department School of Computer Science Carnegie Mellon University Logistic Regression Matt Gormley Lecture 9 Sep. 26, 2018 1 Reminders Homework 3:
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 89 Part II
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 informationRegression Models - Introduction
Regression Models - Introduction In regression models, two types of variables that are studied: A dependent variable, Y, also called response variable. It is modeled as random. An independent variable,
More informationIntroduction to Gaussian Processes
Introduction to Gaussian Processes Neil D. Lawrence GPSS 10th June 2013 Book Rasmussen and Williams (2006) Outline The Gaussian Density Covariance from Basis Functions Basis Function Representations Constructing
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 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 informationCOMP 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
More informationMachine Learning (COMP-652 and ECSE-608)
Machine Learning (COMP-652 and ECSE-608) Lecturers: Guillaume Rabusseau and Riashat Islam Email: guillaume.rabusseau@mail.mcgill.ca - riashat.islam@mail.mcgill.ca Teaching assistant: Ethan Macdonald Email:
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 informationLast updated: Oct 22, 2012 LINEAR CLASSIFIERS. J. Elder CSE 4404/5327 Introduction to Machine Learning and Pattern Recognition
Last updated: Oct 22, 2012 LINEAR CLASSIFIERS Problems 2 Please do Problem 8.3 in the textbook. We will discuss this in class. Classification: Problem Statement 3 In regression, we are modeling the relationship
More informationWeek 3: Linear Regression
Week 3: Linear Regression Instructor: Sergey Levine Recap In the previous lecture we saw how linear regression can solve the following problem: given a dataset D = {(x, y ),..., (x N, y N )}, learn to
More informationLinear vs Non-linear classifier. CS789: Machine Learning and Neural Network. Introduction
Linear vs Non-linear classifier CS789: Machine Learning and Neural Network Support Vector Machine Jakramate Bootkrajang Department of Computer Science Chiang Mai University Linear classifier is in the
More informationIntroduction to Simple Linear Regression
Introduction to Simple Linear Regression Yang Feng http://www.stat.columbia.edu/~yangfeng Yang Feng (Columbia University) Introduction to Simple Linear Regression 1 / 68 About me Faculty in the Department
More informationCS540 Machine learning Lecture 5
CS540 Machine learning Lecture 5 1 Last time Basis functions for linear regression Normal equations QR SVD - briefly 2 This time Geometry of least squares (again) SVD more slowly LMS Ridge regression 3
More informationCS798: Selected topics in Machine Learning
CS798: Selected topics in Machine Learning Support Vector Machine Jakramate Bootkrajang Department of Computer Science Chiang Mai University Jakramate Bootkrajang CS798: Selected topics in Machine Learning
More informationMachine Learning (CSE 446): Multi-Class Classification; Kernel Methods
Machine Learning (CSE 446): Multi-Class Classification; Kernel Methods Sham M Kakade c 2018 University of Washington cse446-staff@cs.washington.edu 1 / 12 Announcements HW3 due date as posted. make sure
More informationBayesian Machine Learning
Bayesian Machine Learning Andrew Gordon Wilson ORIE 6741 Lecture 2: Bayesian Basics https://people.orie.cornell.edu/andrew/orie6741 Cornell University August 25, 2016 1 / 17 Canonical Machine Learning
More informationLinear Regression 1 / 25. Karl Stratos. June 18, 2018
Linear Regression Karl Stratos June 18, 2018 1 / 25 The Regression Problem Problem. Find a desired input-output mapping f : X R where the output is a real value. x = = y = 0.1 How much should I turn my
More informationLinear Regression. CSL603 - Fall 2017 Narayanan C Krishnan
Linear Regression CSL603 - Fall 2017 Narayanan C Krishnan ckn@iitrpr.ac.in Outline Univariate regression Multivariate regression Probabilistic view of regression Loss functions Bias-Variance analysis Regularization
More informationLinear Regression. CSL465/603 - Fall 2016 Narayanan C Krishnan
Linear Regression CSL465/603 - Fall 2016 Narayanan C Krishnan ckn@iitrpr.ac.in Outline Univariate regression Multivariate regression Probabilistic view of regression Loss functions Bias-Variance analysis
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 informationDetermine the trend for time series data
Extra Online Questions Determine the trend for time series data Covers AS 90641 (Statistics and Modelling 3.1) Scholarship Statistics and Modelling Chapter 1 Essent ial exam notes Time series 1. The value
More informationMachine Learning and Computational Statistics, Spring 2017 Homework 2: Lasso Regression
Machine Learning and Computational Statistics, Spring 2017 Homework 2: Lasso Regression Due: Monday, February 13, 2017, at 10pm (Submit via Gradescope) Instructions: Your answers to the questions below,
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 informationCSCI567 Machine Learning (Fall 2014)
CSCI567 Machine Learning (Fall 24) Drs. Sha & Liu {feisha,yanliu.cs}@usc.edu October 2, 24 Drs. Sha & Liu ({feisha,yanliu.cs}@usc.edu) CSCI567 Machine Learning (Fall 24) October 2, 24 / 24 Outline Review
More informationMIDTERM: CS 6375 INSTRUCTOR: VIBHAV GOGATE October,
MIDTERM: CS 6375 INSTRUCTOR: VIBHAV GOGATE October, 23 2013 The exam is closed book. You are allowed a one-page cheat sheet. Answer the questions in the spaces provided on the question sheets. If you run
More informationClassification with Perceptrons. Reading:
Classification with Perceptrons Reading: Chapters 1-3 of Michael Nielsen's online book on neural networks covers the basics of perceptrons and multilayer neural networks We will cover material in Chapters
More informationRegression, Ridge Regression, Lasso
Regression, Ridge Regression, Lasso Fabio G. Cozman - fgcozman@usp.br October 2, 2018 A general definition Regression studies the relationship between a response variable Y and covariates X 1,..., X n.
More informationReminders. Thought questions should be submitted on eclass. Please list the section related to the thought question
Linear regression Reminders Thought questions should be submitted on eclass Please list the section related to the thought question If it is a more general, open-ended question not exactly related to a
More informationCS168: The Modern Algorithmic Toolbox Lecture #6: Regularization
CS168: The Modern Algorithmic Toolbox Lecture #6: Regularization Tim Roughgarden & Gregory Valiant April 18, 2018 1 The Context and Intuition behind Regularization Given a dataset, and some class of models
More informationCSC321 Lecture 2: Linear Regression
CSC32 Lecture 2: Linear Regression Roger Grosse Roger Grosse CSC32 Lecture 2: Linear Regression / 26 Overview First learning algorithm of the course: linear regression Task: predict scalar-valued targets,
More informationMachine Learning! in just a few minutes. Jan Peters Gerhard Neumann
Machine Learning! in just a few minutes Jan Peters Gerhard Neumann 1 Purpose of this Lecture Foundations of machine learning tools for robotics We focus on regression methods and general principles Often
More informationLearning from Data: Regression
November 3, 2005 http://www.anc.ed.ac.uk/ amos/lfd/ Classification or Regression? Classification: want to learn a discrete target variable. Regression: want to learn a continuous target variable. Linear
More informationMachine Learning Lecture 2
Machine Perceptual Learning and Sensory Summer Augmented 15 Computing Many slides adapted from B. Schiele Machine Learning Lecture 2 Probability Density Estimation 16.04.2015 Bastian Leibe RWTH Aachen
More informationLecture 3: More on regularization. Bayesian vs maximum likelihood learning
Lecture 3: More on regularization. Bayesian vs maximum likelihood learning L2 and L1 regularization for linear estimators A Bayesian interpretation of regularization Bayesian vs maximum likelihood fitting
More informationSparse Linear Models (10/7/13)
STA56: Probabilistic machine learning Sparse Linear Models (0/7/) Lecturer: Barbara Engelhardt Scribes: Jiaji Huang, Xin Jiang, Albert Oh Sparsity Sparsity has been a hot topic in statistics and machine
More informationCS 188: Artificial Intelligence Spring Announcements
CS 188: Artificial Intelligence Spring 2010 Lecture 24: Perceptrons and More! 4/22/2010 Pieter Abbeel UC Berkeley Slides adapted from Dan Klein Announcements W7 due tonight [this is your last written for
More informationLecture 3. Linear Regression II Bastian Leibe RWTH Aachen
Advanced Machine Learning Lecture 3 Linear Regression II 02.11.2015 Bastian Leibe RWTH Aachen http://www.vision.rwth-aachen.de/ leibe@vision.rwth-aachen.de This Lecture: Advanced Machine Learning Regression
More informationLoss Functions and Optimization. Lecture 3-1
Lecture 3: Loss Functions and Optimization Lecture 3-1 Administrative Assignment 1 is released: http://cs231n.github.io/assignments2017/assignment1/ Due Thursday April 20, 11:59pm on Canvas (Extending
More informationMachine Learning. Linear Models. Fabio Vandin October 10, 2017
Machine Learning Linear Models Fabio Vandin October 10, 2017 1 Linear Predictors and Affine Functions Consider X = R d Affine functions: L d = {h w,b : w R d, b R} where ( d ) h w,b (x) = w, x + b = w
More informationIntelligent Systems (AI-2)
Intelligent Systems (AI-2) Computer Science cpsc422, Lecture 19 Oct, 23, 2015 Slide Sources Raymond J. Mooney University of Texas at Austin D. Koller, Stanford CS - Probabilistic Graphical Models D. Page,
More informationLinear Regression. Machine Learning CSE546 Kevin Jamieson University of Washington. Oct 5, Kevin Jamieson 1
Linear Regression Machine Learning CSE546 Kevin Jamieson University of Washington Oct 5, 2017 1 The regression problem Given past sales data on zillow.com, predict: y = House sale price from x = {# sq.
More informationValue Function Methods. CS : Deep Reinforcement Learning Sergey Levine
Value Function Methods CS 294-112: Deep Reinforcement Learning Sergey Levine Class Notes 1. Homework 2 is due in one week 2. Remember to start forming final project groups and writing your proposal! Proposal
More informationSample questions for Fundamentals of Machine Learning 2018
Sample questions for Fundamentals of Machine Learning 2018 Teacher: Mohammad Emtiyaz Khan A few important informations: In the final exam, no electronic devices are allowed except a calculator. Make sure
More informationLinear Models for Classification
Linear Models for Classification Oliver Schulte - CMPT 726 Bishop PRML Ch. 4 Classification: Hand-written Digit Recognition CHINE INTELLIGENCE, VOL. 24, NO. 24, APRIL 2002 x i = t i = (0, 0, 0, 1, 0, 0,
More informationLecture 7: Kernels for Classification and Regression
Lecture 7: Kernels for Classification and Regression CS 194-10, Fall 2011 Laurent El Ghaoui EECS Department UC Berkeley September 15, 2011 Outline Outline A linear regression problem Linear auto-regressive
More informationMachine Learning (CS 567) Lecture 5
Machine Learning (CS 567) Lecture 5 Time: T-Th 5:00pm - 6:20pm Location: GFS 118 Instructor: Sofus A. Macskassy (macskass@usc.edu) Office: SAL 216 Office hours: by appointment Teaching assistant: Cheol
More informationData Analysis and Machine Learning Lecture 12: Multicollinearity, Bias-Variance Trade-off, Cross-validation and Shrinkage Methods.
TheThalesians Itiseasyforphilosopherstoberichiftheychoose Data Analysis and Machine Learning Lecture 12: Multicollinearity, Bias-Variance Trade-off, Cross-validation and Shrinkage Methods Ivan Zhdankin
More informationIntroduction to Machine Learning (67577) Lecture 3
Introduction to Machine Learning (67577) Lecture 3 Shai Shalev-Shwartz School of CS and Engineering, The Hebrew University of Jerusalem General Learning Model and Bias-Complexity tradeoff Shai Shalev-Shwartz
More informationFinal Overview. Introduction to ML. Marek Petrik 4/25/2017
Final Overview Introduction to ML Marek Petrik 4/25/2017 This Course: Introduction to Machine Learning Build a foundation for practice and research in ML Basic machine learning concepts: max likelihood,
More informationGI01/M055: Supervised Learning
GI01/M055: Supervised Learning 1. Introduction to Supervised Learning October 5, 2009 John Shawe-Taylor 1 Course information 1. When: Mondays, 14:00 17:00 Where: Room 1.20, Engineering Building, Malet
More informationCS-E3210 Machine Learning: Basic Principles
CS-E3210 Machine Learning: Basic Principles Lecture 4: Regression II slides by Markus Heinonen Department of Computer Science Aalto University, School of Science Autumn (Period I) 2017 1 / 61 Today s introduction
More informationMachine Learning (CSE 446): Probabilistic Machine Learning
Machine Learning (CSE 446): Probabilistic Machine Learning oah Smith c 2017 University of Washington nasmith@cs.washington.edu ovember 1, 2017 1 / 24 Understanding MLE y 1 MLE π^ You can think of MLE as
More informationPart 6: Multivariate Normal and Linear Models
Part 6: Multivariate Normal and Linear Models 1 Multiple measurements Up until now all of our statistical models have been univariate models models for a single measurement on each member of a sample of
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 informationMachine Learning. Lecture 4: Regularization and Bayesian Statistics. Feng Li. https://funglee.github.io
Machine Learning Lecture 4: Regularization and Bayesian Statistics Feng Li fli@sdu.edu.cn https://funglee.github.io School of Computer Science and Technology Shandong University Fall 207 Overfitting Problem
More informationApprentissage, réseaux de neurones et modèles graphiques (RCP209) Neural Networks and Deep Learning
Apprentissage, réseaux de neurones et modèles graphiques (RCP209) Neural Networks and Deep Learning Nicolas Thome Prenom.Nom@cnam.fr http://cedric.cnam.fr/vertigo/cours/ml2/ Département Informatique Conservatoire
More information