Machine Learning. Linear Models. Fabio Vandin October 10, 2017
|
|
- Colleen Cross
- 5 years ago
- Views:
Transcription
1 Machine Learning Linear Models Fabio Vandin October 10,
2 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 i x i + b i=1 Note: each member of L d is a function x w, x + b b: bias 2
3 Linear Models Hypothesis class H: φ L d, where φ : R Y h H is h : R d Y φ depends on the learning problem Example binary classification, Y = { 1, 1} φ(z) = sign(z) regression, Y = R φ(z) = z 3
4 Equivalent Notation Given x X, w R d, b R, define: w = (b, w 1, w 2,..., w d ) R d+1 x = (1, x 1, x 2,..., x d ) R d+1 Then: h w,b (x) = w, x + b = w, x (1) we will consider bias term as part of w and assume x = (1, x 1, x 2,..., x d ) when needed, with h w (x) = w, x 4
5 Linear Classification X = R d, Y = { 1, 1}, 0-1 loss Hypothesis class = halfspaces HS d = sign L d = {x sign(h x,b (x)) : h w,b L d } (2) Example: X = R 2 5
6 Finding a Good Hypothesis Linear classification with hypothesis set H = halfspaces. How do we find a good hypothesis? Good = ERM rule Perceptron Algorithm (Rosenblatt, 1958) Note: if y i w, x i > 0 for all i = 1,..., m all points are classified correctly by model w 6
7 Perceptron Input: training set (x 1, y 1 ),..., (x m, y m ) initialize w (1) = (0,..., 0); for t = 1, 2,... do if i s.t. y i w (t), x i 0 then w (t+1) w (t) + y i x i ; else return w (t) ; Interpretation of update: Note that: y i w (t+1), x i = y i w (t) + y i x i, x i = y i w (t), x i + x i 2 updates guides w to be more correct on (x) i, y i ). Termination? Depends on the realizability assumption! 7
8 Perceptron with Linearly Separable Data Realizability assumption for halfspace = linearly separable data Proposition Assume that (x 1, y 1 ),..., (x m, y m ) is separable, let: B = min{ w : i, i = 1,..., m, y i w, x i }, and R = max i x i. Then the Perceptron algorithm stops after at most (RB) 2 iterations and when it stops it holds that i, i {1,..., m} : y i w (t), x i > 0. 8
9 Perceptron: Notes simple to implement for separable data convergence is guaranteed converge depends on B, which can be exponential in d... an ILP approach may be better to find ERM solution in such cases potentially multiple solutions, which one is picked depends on starting values non separable data? run for some time and keep best solution found up to that point (pocket algorithm) 9
10 10 Linear Regression X = R d, Y = R Hypothesis class: H reg = L d = {x sign(h x,b (x)) : h w,b L d } (3) Note: h H reg : R d R Commonly used loss function: squared-loss l(h, (x, y)) def = (h(x) y) 2 empirical risk function (training error): Mean Squared Error L S (h) = 1 m m (h(x i ) y i ) 2 i=1
11 Least Squares 11 How to find a good hypothesis? Least Squares algorithm Best hypothesis: arg min L 1 S(h w ) = arg min w w m m (h(x i ) y i ) 2 Equivalent formulation: w minimizing Residual Sum of Squares (RSS), i.e. m arg min (h(x i ) y i ) 2 w i=1 i=1
12 12 Let RSS: Matrix Form x 1 x 2 X =. x m X: design matrix y 1 y 2 y =. we have that RSS is m (h(x i ) y i ) 2 = (y Xw) T (y Xw) i=1 y m
13 13 Want to find w that minimizes RSS (=objective function): RSS(w = arg min (y w Xw)T (y Xw) How? Compute gradient RSS(w) w compare it to 0. of objective function w.r.t w and RSS(w) w = 2X T (y Xw) Then we need to find w such that 2X T (y Xw) = 0
14 14 2X T (y Xw) = 0 is equivalent to X T Xw = X T y If X T X is invertible solution to ERM problem is: w = (X T X) 1 X T y
15 Complexity Considerations 15 We need to compute (X T X) 1 X T y Algorithm: 1 compute X T X: product of (d + 1) m matrix and m (d + 1) matrix 2 compute (X T X) 1 inversion of (d + 1) (d + 1) matrix 3 compute (X T X) 1 X T : product of (d + 1) (d + 1) matrix and (d + 1) m matrix 4 compute (X T X) 1 X T : product of (d + 1) m matrix and m 1 matrix Most expensive operation? Inversion! done for (d + 1) (d + 1) matrix
16 X T X not invertible? How do we get w such that X T Xw = X T y if X T X is not invertible? Let A = X T X Let A + be the generalized inverse of A, i.e.: AA + A = A Proposition If A = X T X is not invertible, then ŵ = A + X T y is a solution to X T Xw = X T y. Proof: [UML]
17 Generalized Inverse of A 17 Note A = X T X is symmetric eigenvalue decomposition of A: A = VDV T with D: diagonal matrix (entries = eigenvalues of A) V: orthonormal matrix (V T V = I d d ) Define D + diagonal matrix such that: { 0 if Di,i = 0 D + i,i 1 D i,i otherwise
18 18 Let A + = VD + V T Then AA + A = VDV T VD + V T VDV T = VDD + DV T = VDV T = A A + is the generalized inverse of A.
19 Logistic Regression 19 Learn a function h from R d to [0, 1]. What can this be used for? Classification! Example: binary classification (Y = { 1, 1}) - h(x) = probability that label of x is 1. For simplicity of presentation, we consider binary classification with Y = { 1, 1}, but similar considerations apply for multiclass classification.
20 20 Logistic Regression: Model Hypothesis class H: φ sig L d, where φ : R [0, 1] is sigmoid function Sigmoid function = S-shaped function For logistic regression, the sigmoid φ sig used is the logistic regression: 1 φ sig (z) = 1 + e z
21 21 Therefore H sig = φ sig L d = {x φ sig ( w, x ) : w R d } Main difference with binary classification with halfspaces: when w, x 0 halfspace prediction is deterministically 1 or 1 φ sig ( w, x ) 1/2 uncertainty in predicted label
22 22 Loss Function Need to define how bad it is to predict h w (x) [0, 1] given that true label is y = ±1 Desiderata h w (x) large if y = 1 1 h w (x) large if y = 1 Note that 1 h w (x) = 1 = e w,x e w,x 1 + e w,x 1 = 1 + e w,x
23 23 Then reasonable loss function: increases monotonically with e y w,x reasonable loss function: increases monotonically with 1 + e y w,x Loss function for logistic regression: l(h w, (x, y)) = log (1 + e y w,x )
24 24 Therefore, given training set S = ((x 1, y 1 ),..., (x 1, y 1 )) the ERM problem for logistic regression is: 1 arg min w R d m m i=1 log (1 ) + e y i w,x i Notes: logistic loss function is a convex function ERM problem can be solved efficiently Definition may look a bit arbitrary: actually, ERM formulation is the same as the one arising from Maximum Likelihood Estimation
25 Maximum Likelihood Estimation (MLE) [UML, 24.1] 25 MLE is a statistical approach for finding the parameters that maximize the joint probability of a given dataset assuming a specific parametric probability function. Note: MLE essentially assumes a generative model for the data General approach: 1 given training set S = ((x 1, y 1 ),..., (x m, y m )), assume each (x i, y i ) is i.i.d. from some probability distribution of parameters θ 2 consider P[S θ] (likelihood of data given parameters) 3 log likelihood: L(S; θ) = log(p[s θ]) 4 maximum likelihood estimator: ˆθ = arg max θ L(S; θ)
26 Logistic Regression and MLE 26 Assuming x 1,..., x m are fixed, the probability that x i has label y i = 1 is 1 h w (x i ) = 1 + e w,x i while the probability that x i has label y i = 1 is (1 h w (x i ))) = Then the likelihood for training set S is: e w,x i m ( 1 ) 1 + e y i w,x i i=1
27 27 Therefore the log likelihood is: m i=1 log (1 ) + e y i w,x i And note that the maximum likelihood estimator for w is: arg max w m i=1 ( ) log 1 + e y i w,x i = arg min w m i=1 MLE solution is equivalent to ERM solution! log (1 ) + e y i w,x i
28 Bibliography 28 [UML] Chapter 9: no 9.1.1
Machine 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. Regularization and Feature Selection. Fabio Vandin November 13, 2017
Machine Learning Regularization and Feature Selection Fabio Vandin November 13, 2017 1 Learning Model A: learning algorithm for a machine learning task S: m i.i.d. pairs z i = (x i, y i ), i = 1,..., m,
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. Support Vector Machines. Fabio Vandin November 20, 2017
Machine Learning Support Vector Machines Fabio Vandin November 20, 2017 1 Classification and Margin Consider a classification problem with two classes: instance set X = R d label set Y = { 1, 1}. Training
More 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. Model Selection and Validation. Fabio Vandin November 7, 2017
Machine Learning Model Selection and Validation Fabio Vandin November 7, 2017 1 Model Selection When we have to solve a machine learning task: there are different algorithms/classes algorithms have parameters
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 informationCSE 417T: Introduction to Machine Learning. Lecture 11: Review. Henry Chai 10/02/18
CSE 417T: Introduction to Machine Learning Lecture 11: Review Henry Chai 10/02/18 Unknown Target Function!: # % Training data Formal Setup & = ( ), + ),, ( -, + - Learning Algorithm 2 Hypothesis Set H
More informationLecture 4: Linear predictors and the Perceptron
Lecture 4: Linear predictors and the Perceptron Introduction to Learning and Analysis of Big Data Kontorovich and Sabato (BGU) Lecture 4 1 / 34 Inductive Bias Inductive bias is critical to prevent overfitting.
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 informationMachine Learning. Regularization and Feature Selection. Fabio Vandin November 14, 2017
Machine Learning Regularization and Feature Selection Fabio Vandin November 14, 2017 1 Regularized Loss Minimization Assume h is defined by a vector w = (w 1,..., w d ) T R d (e.g., linear models) Regularization
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 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 informationLecture 5: Logistic Regression. Neural Networks
Lecture 5: Logistic Regression. Neural Networks Logistic regression Comparison with generative models Feed-forward neural networks Backpropagation Tricks for training neural networks COMP-652, Lecture
More informationClassification Logistic Regression
Announcements: Classification Logistic Regression Machine Learning CSE546 Sham Kakade University of Washington HW due on Friday. Today: Review: sub-gradients,lasso Logistic Regression October 3, 26 Sham
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 informationLogistic regression and linear classifiers COMS 4771
Logistic regression and linear classifiers COMS 4771 1. Prediction functions (again) Learning prediction functions IID model for supervised learning: (X 1, Y 1),..., (X n, Y n), (X, Y ) are iid random
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 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 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 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 informationECS171: Machine Learning
ECS171: Machine Learning Lecture 3: Linear Models I (LFD 3.2, 3.3) Cho-Jui Hsieh UC Davis Jan 17, 2018 Linear Regression (LFD 3.2) Regression Classification: Customer record Yes/No Regression: predicting
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 informationNotes on the framework of Ando and Zhang (2005) 1 Beyond learning good functions: learning good spaces
Notes on the framework of Ando and Zhang (2005 Karl Stratos 1 Beyond learning good functions: learning good spaces 1.1 A single binary classification problem Let X denote the problem domain. Suppose we
More informationEngineering Part IIB: Module 4F10 Statistical Pattern Processing Lecture 5: Single Layer Perceptrons & Estimating Linear Classifiers
Engineering Part IIB: Module 4F0 Statistical Pattern Processing Lecture 5: Single Layer Perceptrons & Estimating Linear Classifiers Phil Woodland: pcw@eng.cam.ac.uk Michaelmas 202 Engineering Part IIB:
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 informationMachine 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 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 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 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 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 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 informationWarm up. Regrade requests submitted directly in Gradescope, do not instructors.
Warm up Regrade requests submitted directly in Gradescope, do not email instructors. 1 float in NumPy = 8 bytes 10 6 2 20 bytes = 1 MB 10 9 2 30 bytes = 1 GB For each block compute the memory required
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 information1 Machine Learning Concepts (16 points)
CSCI 567 Fall 2018 Midterm Exam DO NOT OPEN EXAM UNTIL INSTRUCTED TO DO SO PLEASE TURN OFF ALL CELL PHONES Problem 1 2 3 4 5 6 Total Max 16 10 16 42 24 12 120 Points Please read the following instructions
More informationSupport vector machines Lecture 4
Support vector machines Lecture 4 David Sontag New York University Slides adapted from Luke Zettlemoyer, Vibhav Gogate, and Carlos Guestrin Q: What does the Perceptron mistake bound tell us? Theorem: The
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 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 (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 information6.867 Machine learning: lecture 2. Tommi S. Jaakkola MIT CSAIL
6.867 Machine learning: lecture 2 Tommi S. Jaakkola MIT CSAIL tommi@csail.mit.edu Topics The learning problem hypothesis class, estimation algorithm loss and estimation criterion sampling, empirical and
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 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 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 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 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 informationNeural Network Training
Neural Network Training Sargur Srihari Topics in Network Training 0. Neural network parameters Probabilistic problem formulation Specifying the activation and error functions for Regression Binary classification
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 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 informationQualifying Exam in Machine Learning
Qualifying Exam in Machine Learning October 20, 2009 Instructions: Answer two out of the three questions in Part 1. In addition, answer two out of three questions in two additional parts (choose two parts
More informationMachine Learning for NLP
Machine Learning for NLP Linear Models Joakim Nivre Uppsala University Department of Linguistics and Philology Slides adapted from Ryan McDonald, Google Research Machine Learning for NLP 1(26) Outline
More informationStatistical Machine Learning (BE4M33SSU) Lecture 5: Artificial Neural Networks
Statistical Machine Learning (BE4M33SSU) Lecture 5: Artificial Neural Networks Jan Drchal Czech Technical University in Prague Faculty of Electrical Engineering Department of Computer Science Topics covered
More informationThe Perceptron Algorithm
The Perceptron Algorithm Greg Grudic Greg Grudic Machine Learning Questions? Greg Grudic Machine Learning 2 Binary Classification A binary classifier is a mapping from a set of d inputs to a single output
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 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 informationKernelized Perceptron Support Vector Machines
Kernelized Perceptron Support Vector Machines Emily Fox University of Washington February 13, 2017 What is the perceptron optimizing? 1 The perceptron algorithm [Rosenblatt 58, 62] Classification setting:
More informationIFT Lecture 7 Elements of statistical learning theory
IFT 6085 - Lecture 7 Elements of statistical learning theory This version of the notes has not yet been thoroughly checked. Please report any bugs to the scribes or instructor. Scribe(s): Brady Neal and
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 informationSCMA292 Mathematical Modeling : Machine Learning. Krikamol Muandet. Department of Mathematics Faculty of Science, Mahidol University.
SCMA292 Mathematical Modeling : Machine Learning Krikamol Muandet Department of Mathematics Faculty of Science, Mahidol University February 9, 2016 Outline Quick Recap of Least Square Ridge Regression
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 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 informationLinear Classifiers. Michael Collins. January 18, 2012
Linear Classifiers Michael Collins January 18, 2012 Today s Lecture Binary classification problems Linear classifiers The perceptron algorithm Classification Problems: An Example Goal: build a system that
More informationAnnouncements. Proposals graded
Announcements Proposals graded Kevin Jamieson 2018 1 Bayesian Methods Machine Learning CSE546 Kevin Jamieson University of Washington November 1, 2018 2018 Kevin Jamieson 2 MLE Recap - coin flips Data:
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 informationLinear smoother. ŷ = S y. where s ij = s ij (x) e.g. s ij = diag(l i (x))
Linear smoother ŷ = S y where s ij = s ij (x) e.g. s ij = diag(l i (x)) 2 Online Learning: LMS and Perceptrons Partially adapted from slides by Ryan Gabbard and Mitch Marcus (and lots original slides by
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 informationFeed-forward Network Functions
Feed-forward Network Functions Sargur Srihari Topics 1. Extension of linear models 2. Feed-forward Network Functions 3. Weight-space symmetries 2 Recap of Linear Models Linear Models for Regression, Classification
More informationLecture Support Vector Machine (SVM) Classifiers
Introduction to Machine Learning Lecturer: Amir Globerson Lecture 6 Fall Semester Scribe: Yishay Mansour 6.1 Support Vector Machine (SVM) Classifiers Classification is one of the most important tasks in
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 informationComputational and Statistical Learning Theory
Computational and Statistical Learning Theory TTIC 31120 Prof. Nati Srebro Lecture 17: Stochastic Optimization Part II: Realizable vs Agnostic Rates Part III: Nearest Neighbor Classification Stochastic
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 informationAnnouncements - Homework
Announcements - Homework Homework 1 is graded, please collect at end of lecture Homework 2 due today Homework 3 out soon (watch email) Ques 1 midterm review HW1 score distribution 40 HW1 total score 35
More informationLogistic Regression Introduction to Machine Learning. Matt Gormley Lecture 8 Feb. 12, 2018
10-601 Introduction to Machine Learning Machine Learning Department School of Computer Science Carnegie Mellon University Logistic Regression Matt Gormley Lecture 8 Feb. 12, 2018 1 10-601 Introduction
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 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 Regression. Aarti Singh. Machine Learning / Sept 27, 2010
Linear Regression Aarti Singh Machine Learning 10-701/15-781 Sept 27, 2010 Discrete to Continuous Labels Classification Sports Science News Anemic cell Healthy cell Regression X = Document Y = Topic X
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 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 informationPart of the slides are adapted from Ziko Kolter
Part of the slides are adapted from Ziko Kolter OUTLINE 1 Supervised learning: classification........................................................ 2 2 Non-linear regression/classification, overfitting,
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 informationCSC321 Lecture 4 The Perceptron Algorithm
CSC321 Lecture 4 The Perceptron Algorithm Roger Grosse and Nitish Srivastava January 17, 2017 Roger Grosse and Nitish Srivastava CSC321 Lecture 4 The Perceptron Algorithm January 17, 2017 1 / 1 Recap:
More informationMachine Learning: A Statistics and Optimization Perspective
Machine Learning: A Statistics and Optimization Perspective Nan Ye Mathematical Sciences School Queensland University of Technology 1 / 109 What is Machine Learning? 2 / 109 Machine Learning Machine learning
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 informationCSE 417T: Introduction to Machine Learning. Final Review. Henry Chai 12/4/18
CSE 417T: Introduction to Machine Learning Final Review Henry Chai 12/4/18 Overfitting Overfitting is fitting the training data more than is warranted Fitting noise rather than signal 2 Estimating! "#$
More informationClassification. Sandro Cumani. Politecnico di Torino
Politecnico di Torino Outline Generative model: Gaussian classifier (Linear) discriminative model: logistic regression (Non linear) discriminative model: neural networks Gaussian Classifier We want to
More informationReading Group on Deep Learning Session 1
Reading Group on Deep Learning Session 1 Stephane Lathuiliere & Pablo Mesejo 2 June 2016 1/31 Contents Introduction to Artificial Neural Networks to understand, and to be able to efficiently use, the popular
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 information18.6 Regression and Classification with Linear Models
18.6 Regression and Classification with Linear Models 352 The hypothesis space of linear functions of continuous-valued inputs has been used for hundreds of years A univariate linear function (a straight
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 informationECE 5424: Introduction to Machine Learning
ECE 5424: Introduction to Machine Learning Topics: Ensemble Methods: Bagging, Boosting PAC Learning Readings: Murphy 16.4;; Hastie 16 Stefan Lee Virginia Tech Fighting the bias-variance tradeoff Simple
More informationNeural Networks: Backpropagation
Neural Networks: Backpropagation Seung-Hoon Na 1 1 Department of Computer Science Chonbuk National University 2018.10.25 eung-hoon Na (Chonbuk National University) Neural Networks: Backpropagation 2018.10.25
More informationSPSS, University of Texas at Arlington. Topics in Machine Learning-EE 5359 Neural Networks
Topics in Machine Learning-EE 5359 Neural Networks 1 The Perceptron Output: A perceptron is a function that maps D-dimensional vectors to real numbers. For notational convenience, we add a zero-th dimension
More informationKernel Machines. Pradeep Ravikumar Co-instructor: Manuela Veloso. Machine Learning
Kernel Machines Pradeep Ravikumar Co-instructor: Manuela Veloso Machine Learning 10-701 SVM linearly separable case n training points (x 1,, x n ) d features x j is a d-dimensional vector Primal problem:
More informationFrom Binary to Multiclass Classification. CS 6961: Structured Prediction Spring 2018
From Binary to Multiclass Classification CS 6961: Structured Prediction Spring 2018 1 So far: Binary Classification We have seen linear models Learning algorithms Perceptron SVM Logistic Regression Prediction
More informationESS2222. Lecture 4 Linear model
ESS2222 Lecture 4 Linear model Hosein Shahnas University of Toronto, Department of Earth Sciences, 1 Outline Logistic Regression Predicting Continuous Target Variables Support Vector Machine (Some Details)
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 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 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 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 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 information