Machine Learning Basics Lecture 2: Linear Classification. Princeton University COS 495 Instructor: Yingyu Liang
|
|
- Garey Rich
- 6 years ago
- Views:
Transcription
1 Machine Learning Basics Lecture 2: Linear Classification Princeton University COS 495 Instructor: Yingyu Liang
2 Review: machine learning basics
3 Math formulation Given training data x i, y i : 1 i n i.i.d. from distribution D Find y = f(x) H that minimizes L f = 1 σ n i=1 n l(f, x i, y i ) s.t. the expected loss is small L f = E x,y ~D [l(f, x, y)]
4 Machine learning Collect data and extract features Build model: choose hypothesis class H and loss function l Optimization: minimize the empirical loss
5 Machine learning Experience Collect data and extract features Build model: choose hypothesis class H and loss function l Optimization: minimize the empirical loss Prior knowledge
6 Example: Linear regression Given training data Find f w x = w T x that minimizes L f w x i, y i : 1 i n i.i.d. from distribution D = 1 σ n n i=1 w T x i y i 2 l 2 loss Linear model H
7 Why l 2 loss Why not choose another loss l 1 loss, hinge loss, exponential loss, Empirical: easy to optimize For linear case: w = X T X 1 X T y Theoretical: a way to encode prior knowledge Questions: What kind of prior knowledge? Principal way to derive loss?
8 Maximum likelihood Estimation
9 Maximum likelihood Estimation (MLE) Given training data x i, y i : 1 i n i.i.d. from distribution D Let {P θ x, y : θ Θ} be a family of distributions indexed by θ Would like to pick θ so that P θ (x, y) fits the data well
10 Maximum likelihood Estimation (MLE) Given training data x i, y i : 1 i n i.i.d. from distribution D Let {P θ x, y : θ Θ} be a family of distributions indexed by θ fitness of θ to one data point x i, y i likelihood θ; x i, y i P θ (x i, y i )
11 Maximum likelihood Estimation (MLE) Given training data x i, y i : 1 i n i.i.d. from distribution D Let {P θ x, y : θ Θ} be a family of distributions indexed by θ fitness of θ to i.i.d. data points { x i, y i } likelihood θ; {x i, y i } P θ {x i, y i } = ς i P θ (x i, y i )
12 Maximum likelihood Estimation (MLE) Given training data x i, y i : 1 i n i.i.d. from distribution D Let {P θ x, y : θ Θ} be a family of distributions indexed by θ MLE: maximize fitness of θ to i.i.d. data points { x i, y i } θ ML = argmax θ Θ ς i P θ (x i, y i )
13 Maximum likelihood Estimation (MLE) Given training data x i, y i : 1 i n i.i.d. from distribution D Let {P θ x, y : θ Θ} be a family of distributions indexed by θ MLE: maximize fitness of θ to i.i.d. data points { x i, y i } θ ML = argmax θ Θ log[ς i P θ x i, y i ] θ ML = argmax θ Θ σ i log[p θ x i, y i ]
14 Maximum likelihood Estimation (MLE) Given training data x i, y i : 1 i n i.i.d. from distribution D Let {P θ x, y : θ Θ} be a family of distributions indexed by θ MLE: negative log-likelihood loss θ ML = argmax θ Θ σ i log(p θ x i, y i ) l P θ, x i, y i = log(p θ x i, y i ) L P θ = σ i log(p θ x i, y i )
15 MLE: conditional log-likelihood Given training data x i, y i : 1 i n i.i.d. from distribution D Let {P θ y x : θ Θ} be a family of distributions indexed by θ MLE: negative conditional log-likelihood loss θ ML = argmax θ Θ σ i log(p θ y i x i ) Only care about predicting y from x; do not care about p(x) l P θ, x i, y i = log(p θ y i x i ) L P θ = σ i log(p θ y i x i )
16 MLE: conditional log-likelihood Given training data x i, y i : 1 i n i.i.d. from distribution D Let {P θ y x : θ Θ} be a family of distributions indexed by θ MLE: negative conditional log-likelihood loss θ ML = argmax θ Θ σ i log(p θ y i x i ) P(y x): discriminative; P(x,y): generative l P θ, x i, y i = log(p θ y i x i ) L P θ = σ i log(p θ y i x i )
17 Example: l 2 loss Given training data Find f θ x that minimizes L f θ x i, y i : 1 i n i.i.d. from distribution D = 1 σ n n i=1 f θ (x i ) y i 2
18 Example: l 2 loss Given training data Find f θ x that minimizes L f θ x i, y i : 1 i n i.i.d. from distribution D = 1 σ n n i=1 Define P θ y x = Normal y; f θ x, σ 2 f θ (x i ) y i 2 log(p θ y i x i ) = 1 (f 2σ 2 θ x i y i ) 2 log(σ) 1 log(2π) 2 1 n θ ML = argmin θ Θ f θ (x i ) y 2 i n σ i=1 l 2 loss: Normal + MLE
19 Linear classification
20 Example 1: image classification indoor Indoor outdoor
21 Example 2: Spam detection # $ # Mr. # sale Spam? Yes No Yes n No New ??
22 Why classification Classification: a kind of summary Easy to interpret Easy for making decisions
23 Linear classification w T x = 0 w T x > 0 Class 1 w w T x < 0 Class 0
24 Linear classification: natural attempt Given training data Hypothesis f w x = w T x y = 1 if w T x > 0 y = 0 if w T x < 0 Prediction: y = step(f w x ) = step(w T x) x i, y i : 1 i n i.i.d. from distribution D Linear model H
25 Linear classification: natural attempt Given training data Find f w x = w T x to minimize L f w x i, y i : 1 i n i.i.d. from distribution D = 1 σ n i=1 n I[step(w T x i ) y i ] Drawback: difficult to optimize NP-hard in the worst case 0-1 loss
26 Linear classification: simple approach Given training data Find f w x = w T x that minimizes L f w x i, y i : 1 i n i.i.d. from distribution D = 1 σ n n i=1 w T x i y i 2 Reduce to linear regression; ignore the fact y {0,1}
27 Linear classification: simple approach Drawback: not robust to outliers Figure borrowed from Pattern Recognition and Machine Learning, Bishop
28 Compare the two y y = w T x y = step(w T x) w T x
29 Between the two Prediction bounded in [0,1] Smooth Sigmoid: σ a = 1 1+exp( a) Figure borrowed from Pattern Recognition and Machine Learning, Bishop
30 Linear classification: sigmoid prediction Squash the output of the linear function Sigmoid w T x = σ w T x = exp( w T x) Find w that minimizes L f w = 1 σ n n i=1 σ(w T x i ) y i 2
31 Linear classification: logistic regression Squash the output of the linear function Sigmoid w T x = σ w T 1 x = 1 + exp( w T x) A better approach: Interpret as a probability P w (y = 1 x) = σ w T 1 x = 1 + exp( w T x) P w y = 0 x = 1 P w y = 1 x = 1 σ w T x
32 Linear classification: logistic regression Given training data x i, y i : 1 i n i.i.d. from distribution D Find w that minimizes n L w = 1 n log P w y x i=1 L w = 1 n yi=1 logσ(w T x i ) 1 n y i =0 log[1 σ w T x i ] Logistic regression: MLE with sigmoid
33 Linear classification: logistic regression Given training data Find w that minimizes x i, y i : 1 i n i.i.d. from distribution D L w = 1 n yi=1 logσ(w T x i ) 1 n y i =0 log[1 σ w T x i ] No close form solution; Need to use gradient descent
34 Properties of sigmoid function Bounded Symmetric σ a = exp( a) (0,1) Gradient 1 σ a = σ (a) = exp a 1 + exp a = 1 exp a + 1 = σ( a) exp a 1 + exp a 2 = σ(a)(1 σ a )
Machine Learning Basics Lecture 7: Multiclass Classification. Princeton University COS 495 Instructor: Yingyu Liang
Machine Learning Basics Lecture 7: Multiclass Classification Princeton University COS 495 Instructor: Yingyu Liang Example: image classification indoor Indoor outdoor Example: image classification (multiclass)
More informationMachine Learning Basics Lecture 4: SVM I. Princeton University COS 495 Instructor: Yingyu Liang
Machine Learning Basics Lecture 4: SVM I 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. from distribution
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. 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 informationLogic and machine learning review. CS 540 Yingyu Liang
Logic and machine learning review CS 540 Yingyu Liang Propositional logic Logic If the rules of the world are presented formally, then a decision maker can use logical reasoning to make rational decisions.
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 informationLinear and Logistic Regression. Dr. Xiaowei Huang
Linear and Logistic Regression Dr. Xiaowei Huang https://cgi.csc.liv.ac.uk/~xiaowei/ Up to now, Two Classical Machine Learning Algorithms Decision tree learning K-nearest neighbor Model Evaluation Metrics
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 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 informationMachine Learning Basics Lecture 3: Perceptron. Princeton University COS 495 Instructor: Yingyu Liang
Machine Learning Basics Lecture 3: Perceptron Princeton University COS 495 Instructor: Yingyu Liang Perceptron Overview Previous lectures: (Principle for loss function) MLE to derive loss Example: linear
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 informationLecture 2: Logistic Regression and Neural Networks
1/23 Lecture 2: and Neural Networks Pedro Savarese TTI 2018 2/23 Table of Contents 1 2 3 4 3/23 Naive Bayes Learn p(x, y) = p(y)p(x y) Training: Maximum Likelihood Estimation Issues? Why learn p(x, y)
More informationNaïve Bayes classification
Naïve Bayes classification 1 Probability theory Random variable: a variable whose possible values are numerical outcomes of a random phenomenon. Examples: A person s height, the outcome of a coin toss
More informationDeep Learning Basics Lecture 8: Autoencoder & DBM. Princeton University COS 495 Instructor: Yingyu Liang
Deep Learning Basics Lecture 8: Autoencoder & DBM Princeton University COS 495 Instructor: Yingyu Liang Autoencoder Autoencoder Neural networks trained to attempt to copy its input to its output Contain
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 informationMachine Learning. Regression-Based Classification & Gaussian Discriminant Analysis. Manfred Huber
Machine Learning Regression-Based Classification & Gaussian Discriminant Analysis Manfred Huber 2015 1 Logistic Regression Linear regression provides a nice representation and an efficient solution to
More informationLogistic Regression. Jia-Bin Huang. Virginia Tech Spring 2019 ECE-5424G / CS-5824
Logistic Regression Jia-Bin Huang ECE-5424G / CS-5824 Virginia Tech Spring 2019 Administrative Please start HW 1 early! Questions are welcome! Two principles for estimating parameters Maximum Likelihood
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 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 informationCSC 411: Lecture 04: Logistic Regression
CSC 411: Lecture 04: Logistic Regression Raquel Urtasun & Rich Zemel University of Toronto Sep 23, 2015 Urtasun & Zemel (UofT) CSC 411: 04-Prob Classif Sep 23, 2015 1 / 16 Today Key Concepts: Logistic
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 informationNaïve Bayes classification. p ij 11/15/16. Probability theory. Probability theory. Probability theory. X P (X = x i )=1 i. Marginal Probability
Probability theory Naïve Bayes classification Random variable: a variable whose possible values are numerical outcomes of a random phenomenon. s: A person s height, the outcome of a coin toss Distinguish
More informationOutline. Supervised Learning. Hong Chang. Institute of Computing Technology, Chinese Academy of Sciences. Machine Learning Methods (Fall 2012)
Outline Hong Chang Institute of Computing Technology, Chinese Academy of Sciences Machine Learning Methods (Fall 2012) Outline Outline I 1 Linear Models for Regression Linear Regression Probabilistic Interpretation
More informationGaussian discriminant analysis Naive Bayes
DM825 Introduction to Machine Learning Lecture 7 Gaussian discriminant analysis Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1. is 2. Multi-variate
More informationStochastic Gradient Descent
Stochastic Gradient Descent Machine Learning CSE546 Carlos Guestrin University of Washington October 9, 2013 1 Logistic Regression Logistic function (or Sigmoid): Learn P(Y X) directly Assume a particular
More informationCPSC 340: Machine Learning and Data Mining. MLE and MAP Fall 2017
CPSC 340: Machine Learning and Data Mining MLE and MAP Fall 2017 Assignment 3: Admin 1 late day to hand in tonight, 2 late days for Wednesday. Assignment 4: Due Friday of next week. Last Time: Multi-Class
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 and logistic regression
Linear and logistic regression Guillaume Obozinski Ecole des Ponts - ParisTech Master MVA Linear and logistic regression 1/22 Outline 1 Linear regression 2 Logistic regression 3 Fisher discriminant analysis
More informationLogistic Regression. Stochastic Gradient Descent
Tutorial 8 CPSC 340 Logistic Regression Stochastic Gradient Descent Logistic Regression Model A discriminative probabilistic model for classification e.g. spam filtering Let x R d be input and y { 1, 1}
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 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 informationAn Introduction to Statistical and Probabilistic Linear Models
An Introduction to Statistical and Probabilistic Linear Models Maximilian Mozes Proseminar Data Mining Fakultät für Informatik Technische Universität München June 07, 2017 Introduction In statistical learning
More informationMachine Learning, Fall 2012 Homework 2
0-60 Machine Learning, Fall 202 Homework 2 Instructors: Tom Mitchell, Ziv Bar-Joseph TA in charge: Selen Uguroglu email: sugurogl@cs.cmu.edu SOLUTIONS Naive Bayes, 20 points Problem. Basic concepts, 0
More informationLinear Classification: Probabilistic Generative Models
Linear Classification: Probabilistic Generative Models Sargur N. University at Buffalo, State University of New York USA 1 Linear Classification using Probabilistic Generative Models Topics 1. Overview
More informationGenerative v. Discriminative classifiers Intuition
Logistic Regression Machine Learning 070/578 Carlos Guestrin Carnegie Mellon University September 24 th, 2007 Generative v. Discriminative classifiers Intuition Want to Learn: h:x a Y X features Y target
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 informationMachine Learning
Machine Learning 10-701 Tom M. Mitchell Machine Learning Department Carnegie Mellon University February 1, 2011 Today: Generative discriminative classifiers Linear regression Decomposition of error into
More informationCS534: Machine Learning. Thomas G. Dietterich 221C Dearborn Hall
CS534: Machine Learning Thomas G. Dietterich 221C Dearborn Hall tgd@cs.orst.edu http://www.cs.orst.edu/~tgd/classes/534 1 Course Overview Introduction: Basic problems and questions in machine learning.
More informationCPSC 340: Machine Learning and Data Mining
CPSC 340: Machine Learning and Data Mining MLE and MAP Original version of these slides by Mark Schmidt, with modifications by Mike Gelbart. 1 Admin Assignment 4: Due tonight. Assignment 5: Will be released
More informationProbabilistic Machine Learning
Probabilistic Machine Learning by Prof. Seungchul Lee isystes Design Lab http://isystes.unist.ac.kr/ UNIST Table of Contents I.. Probabilistic Linear Regression I... Maxiu Likelihood Solution II... Maxiu-a-Posteriori
More informationCh 4. Linear Models for Classification
Ch 4. Linear Models for Classification Pattern Recognition and Machine Learning, C. M. Bishop, 2006. Department of Computer Science and Engineering Pohang University of Science and echnology 77 Cheongam-ro,
More informationMachine Learning 2017
Machine Learning 2017 Volker Roth Department of Mathematics & Computer Science University of Basel 21st March 2017 Volker Roth (University of Basel) Machine Learning 2017 21st March 2017 1 / 41 Section
More informationParametric Models. Dr. Shuang LIANG. School of Software Engineering TongJi University Fall, 2012
Parametric Models Dr. Shuang LIANG School of Software Engineering TongJi University Fall, 2012 Today s Topics Maximum Likelihood Estimation Bayesian Density Estimation Today s Topics Maximum Likelihood
More informationUniversität Potsdam Institut für Informatik Lehrstuhl Maschinelles Lernen. Bayesian Learning. Tobias Scheffer, Niels Landwehr
Universität Potsdam Institut für Informatik Lehrstuhl Maschinelles Lernen Bayesian Learning Tobias Scheffer, Niels Landwehr Remember: Normal Distribution Distribution over x. Density function with parameters
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 informationClassification: Logistic Regression from Data
Classification: Logistic Regression from Data Machine Learning: Jordan Boyd-Graber University of Colorado Boulder LECTURE 3 Slides adapted from Emily Fox Machine Learning: Jordan Boyd-Graber Boulder Classification:
More informationSTA141C: Big Data & High Performance Statistical Computing
STA141C: Big Data & High Performance Statistical Computing Lecture 4: ML Models (Overview) Cho-Jui Hsieh UC Davis April 17, 2017 Outline Linear regression Ridge regression Logistic regression Other finite-sum
More informationCSC 411: Lecture 09: Naive Bayes
CSC 411: Lecture 09: Naive Bayes Class based on Raquel Urtasun & Rich Zemel s lectures Sanja Fidler University of Toronto Feb 8, 2015 Urtasun, Zemel, Fidler (UofT) CSC 411: 09-Naive Bayes Feb 8, 2015 1
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 informationSupport Vector Machines. CAP 5610: Machine Learning Instructor: Guo-Jun QI
Support Vector Machines CAP 5610: Machine Learning Instructor: Guo-Jun QI 1 Linear Classifier Naive Bayes Assume each attribute is drawn from Gaussian distribution with the same variance Generative model:
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 informationProbabilistic modeling. The slides are closely adapted from Subhransu Maji s slides
Probabilistic modeling The slides are closely adapted from Subhransu Maji s slides Overview So far the models and algorithms you have learned about are relatively disconnected Probabilistic modeling framework
More informationLecture 9: PGM Learning
13 Oct 2014 Intro. to Stats. Machine Learning COMP SCI 4401/7401 Table of Contents I Learning parameters in MRFs 1 Learning parameters in MRFs Inference and Learning Given parameters (of potentials) and
More informationFrom perceptrons to word embeddings. Simon Šuster University of Groningen
From perceptrons to word embeddings Simon Šuster University of Groningen Outline A basic computational unit Weighting some input to produce an output: classification Perceptron Classify tweets Written
More informationECE521 Lecture7. Logistic Regression
ECE521 Lecture7 Logistic Regression Outline Review of decision theory Logistic regression A single neuron Multi-class classification 2 Outline Decision theory is conceptually easy and computationally hard
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 informationMachine Learning. Bayesian Regression & Classification. Marc Toussaint U Stuttgart
Machine Learning Bayesian Regression & Classification learning as inference, Bayesian Kernel Ridge regression & Gaussian Processes, Bayesian Kernel Logistic Regression & GP classification, Bayesian Neural
More informationCS 195-5: Machine Learning Problem Set 1
CS 95-5: Machine Learning Problem Set Douglas Lanman dlanman@brown.edu 7 September Regression Problem Show that the prediction errors y f(x; ŵ) are necessarily uncorrelated with any linear function of
More informationEmpirical Risk Minimization
Empirical Risk Minimization Fabrice Rossi SAMM Université Paris 1 Panthéon Sorbonne 2018 Outline Introduction PAC learning ERM in practice 2 General setting Data X the input space and Y the output space
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 informationGenerative v. Discriminative classifiers Intuition
Logistic Regression Machine Learning 10701/15781 Carlos Guestrin Carnegie Mellon University September 24 th, 2007 1 Generative v. Discriminative classifiers Intuition Want to Learn: h:x a Y X features
More informationLogistic Regression. Will Monroe CS 109. Lecture Notes #22 August 14, 2017
1 Will Monroe CS 109 Logistic Regression Lecture Notes #22 August 14, 2017 Based on a chapter by Chris Piech Logistic regression is a classification algorithm1 that works by trying to learn a function
More informationCMU-Q Lecture 24:
CMU-Q 15-381 Lecture 24: Supervised Learning 2 Teacher: Gianni A. Di Caro SUPERVISED LEARNING Hypotheses space Hypothesis function Labeled Given Errors Performance criteria Given a collection of input
More informationClassification: Logistic Regression from Data
Classification: Logistic Regression from Data Machine Learning: Alvin Grissom II University of Colorado Boulder Slides adapted from Emily Fox Machine Learning: Alvin Grissom II Boulder Classification:
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 305 Part VII
More informationMachine Learning for Signal Processing Bayes Classification and Regression
Machine Learning for Signal Processing Bayes Classification and Regression Instructor: Bhiksha Raj 11755/18797 1 Recap: KNN A very effective and simple way of performing classification Simple model: For
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 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 information> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE GRAVIS 2016 BASEL. Logistic Regression. Pattern Recognition 2016 Sandro Schönborn University of Basel
Logistic Regression Pattern Recognition 2016 Sandro Schönborn University of Basel Two Worlds: Probabilistic & Algorithmic We have seen two conceptual approaches to classification: data class density estimation
More informationMLPR: Logistic Regression and Neural Networks
MLPR: Logistic Regression and Neural Networks Machine Learning and Pattern Recognition Amos Storkey Amos Storkey MLPR: Logistic Regression and Neural Networks 1/28 Outline 1 Logistic Regression 2 Multi-layer
More informationOutline. MLPR: Logistic Regression and Neural Networks Machine Learning and Pattern Recognition. Which is the correct model? Recap.
Outline MLPR: and Neural Networks Machine Learning and Pattern Recognition 2 Amos Storkey Amos Storkey MLPR: and Neural Networks /28 Recap Amos Storkey MLPR: and Neural Networks 2/28 Which is the correct
More informationCSCI-567: Machine Learning (Spring 2019)
CSCI-567: Machine Learning (Spring 2019) Prof. Victor Adamchik U of Southern California Mar. 19, 2019 March 19, 2019 1 / 43 Administration March 19, 2019 2 / 43 Administration TA3 is due this week March
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 informationσ(a) = a N (x; 0, 1 2 ) dx. σ(a) = Φ(a) =
Until now we have always worked with likelihoods and prior distributions that were conjugate to each other, allowing the computation of the posterior distribution to be done in closed form. Unfortunately,
More informationMachine Learning Tom M. Mitchell Machine Learning Department Carnegie Mellon University. September 20, 2012
Machine Learning 10-601 Tom M. Mitchell Machine Learning Department Carnegie Mellon University September 20, 2012 Today: Logistic regression Generative/Discriminative classifiers Readings: (see class website)
More informationIntroduction to Machine Learning
Introduction to Machine Learning Machine Learning: Jordan Boyd-Graber University of Maryland LOGISTIC REGRESSION FROM TEXT Slides adapted from Emily Fox Machine Learning: Jordan Boyd-Graber UMD Introduction
More informationLogistic Regression. Professor Ameet Talwalkar. Professor Ameet Talwalkar CS260 Machine Learning Algorithms January 25, / 48
Logistic Regression Professor Ameet Talwalkar Professor Ameet Talwalkar CS260 Machine Learning Algorithms January 25, 2017 1 / 48 Outline 1 Administration 2 Review of last lecture 3 Logistic regression
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 informationMaximum Likelihood, Logistic Regression, and Stochastic Gradient Training
Maximum Likelihood, Logistic Regression, and Stochastic Gradient Training Charles Elkan elkan@cs.ucsd.edu January 17, 2013 1 Principle of maximum likelihood Consider a family of probability distributions
More informationWeek 5: Logistic Regression & Neural Networks
Week 5: Logistic Regression & Neural Networks Instructor: Sergey Levine 1 Summary: Logistic Regression In the previous lecture, we covered logistic regression. To recap, logistic regression models and
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 informationMachine Learning. Lecture 3: Logistic Regression. Feng Li.
Machine Learning Lecture 3: Logistic Regression Feng Li fli@sdu.edu.cn https://funglee.github.io School of Computer Science and Technology Shandong University Fall 2016 Logistic Regression Classification
More informationMachine Learning
Machine Learning 10-601 Tom M. Mitchell Machine Learning Department Carnegie Mellon University February 4, 2015 Today: Generative discriminative classifiers Linear regression Decomposition of error into
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 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 informationUniversität Potsdam Institut für Informatik Lehrstuhl Maschinelles Lernen. Linear Classifiers. Blaine Nelson, Tobias Scheffer
Universität Potsdam Institut für Informatik Lehrstuhl Linear Classifiers Blaine Nelson, Tobias Scheffer Contents Classification Problem Bayesian Classifier Decision Linear Classifiers, MAP Models Logistic
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 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 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 informationIntroduction to Machine Learning. Maximum Likelihood and Bayesian Inference. Lecturers: Eran Halperin, Yishay Mansour, Lior Wolf
1 Introduction to Machine Learning Maximum Likelihood and Bayesian Inference Lecturers: Eran Halperin, Yishay Mansour, Lior Wolf 2013-14 We know that X ~ B(n,p), but we do not know p. We get a random sample
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 informationBayesian Learning (II)
Universität Potsdam Institut für Informatik Lehrstuhl Maschinelles Lernen Bayesian Learning (II) Niels Landwehr Overview Probabilities, expected values, variance Basic concepts of Bayesian learning MAP
More informationMachine Learning Linear Classification. Prof. Matteo Matteucci
Machine Learning Linear Classification Prof. Matteo Matteucci Recall from the first lecture 2 X R p Regression Y R Continuous Output X R p Y {Ω 0, Ω 1,, Ω K } Classification Discrete Output X R p Y (X)
More informationLinear Regression and Discrimination
Linear Regression and Discrimination Kernel-based Learning Methods Christian Igel Institut für Neuroinformatik Ruhr-Universität Bochum, Germany http://www.neuroinformatik.rub.de July 16, 2009 Christian
More informationSequence labeling. Taking collective a set of interrelated instances x 1,, x T and jointly labeling them
HMM, MEMM and CRF 40-957 Special opics in Artificial Intelligence: Probabilistic Graphical Models Sharif University of echnology Soleymani Spring 2014 Sequence labeling aking collective a set of interrelated
More informationCS 6375 Machine Learning
CS 6375 Machine Learning Nicholas Ruozzi University of Texas at Dallas Slides adapted from David Sontag and Vibhav Gogate Course Info. Instructor: Nicholas Ruozzi Office: ECSS 3.409 Office hours: Tues.
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 informationOnline Learning and Sequential Decision Making
Online Learning and Sequential Decision Making Emilie Kaufmann CNRS & CRIStAL, Inria SequeL, emilie.kaufmann@univ-lille.fr Research School, ENS Lyon, Novembre 12-13th 2018 Emilie Kaufmann Online Learning
More informationIntroduction to Bayesian Learning. Machine Learning Fall 2018
Introduction to Bayesian Learning Machine Learning Fall 2018 1 What we have seen so far What does it mean to learn? Mistake-driven learning Learning by counting (and bounding) number of mistakes PAC learnability
More informationLinear Classification
Linear Classification Lili MOU moull12@sei.pku.edu.cn http://sei.pku.edu.cn/ moull12 23 April 2015 Outline Introduction Discriminant Functions Probabilistic Generative Models Probabilistic Discriminative
More information