Machine Learning: Logistic Regression. Lecture 04
|
|
- Elaine Caldwell
- 6 years ago
- Views:
Transcription
1 Machine Learning: Logistic Regression Razvan C. Bunescu School of Electrical Engineering and Computer Science
2 Supervised Learning Task = learn an (unkon function t : X T that maps input instances x Î X to output targets t(x Î T: Classification: The output t(x Î T is one of a finite set of discrete categories. Regression: The output t(x Î T is continuous, or has a continuous component. Target function t(x is knon (only through (noisy set of training examples: (x 1,t 1, (x 2,t 2, (x n,t n
3 Supervised Learning Training Training Examples (x k, t k Learning Algorithm Model h Testing Test Examples (x, t Model h Generalization Performance
4 Parametric Approaches to Supervised Learning Task = build a function h(x such that: h matches t ell on the training data: => h is able to fit data that it has seen. h also matches t ell on test data: => h is able to generalize to unseen data. Task = choose h from a nice class of functions that depend on a vector of parameters : h(x º h (x º h(,x hat classes of functions are nice?
5 Neurons Soma is the central part of the neuron: here the input signals are combined. Dendrites are cellular extensions: here majority of the input occurs. Axon is a fine, long projection: carries nerve signals to other neurons. Synapses are molecular structures beteen axon terminals and other neurons: here the communication takes place.
6 Neuron Models
7 Spiking/LIF Neuron Function
8 Neuron Models
9 McCulloch-Pitts Neuron Function x activation / output function x 1 x Σ i x i f h (x x 3 Algebraic interpretation: The output of the neuron is a linear combination of inputs from other neurons, rescaled by the synaptic eights. eights i correspond to the synaptic eights (activating or inhibiting. summation corresponds to combination of signals in the soma. It is often transformed through an activation / output function.
10 Activation Functions " $ unit step f (z = # %$ Perceptron 0 if z < 0 1 if z logistic f (z = 1+ e z Logistic Regression identity f (z = z Linear Regression 0
11 Linear Regression x activation / output function x Σ f x 2 3 i x i f (z = z h (x = i x i x 3 Polynomial curve fitting is Linear Regression: x = φ(x = [1, x, x 2,..., x M ] T h(x = T x
12 McCulloch-Pitts Neuron Function x activation / output function x 1 x Σ i x i f h (x x 3 Algebraic interpretation: The output of the neuron is a linear combination of inputs from other neurons, rescaled by the synaptic eights. eights i correspond to the synaptic eights (activating or inhibiting. summation corresponds to combination of signals in the soma. It is often transformed through a monotonic activation / output function.
13 Logistic Regression x 0 x 1 x 2 x Σ activation function f i x i 1 h (x = 3 1 f (z = 1+ exp( T x 1+ exp( z Training set is (x 1,t 1, (x 2,t 2, (x n,t n. x = [1, x 1, x 2,..., x k ] T h(x = σ( T x Can be used for both classification and regression: Classification: T = {C 1, C 2 } = {1, 0}. Regression: T = [0, 1] (i.e. output needs to be normalized.
14 Logistic Regression for Binary Classification Model output can be interpreted as posterior class probabilities: p(c 1 x = σ ( T x = 1 1+ exp( T x p(c 2 x =1 σ ( T x = exp( T x 1+ exp( T x Ho do e train a logistic regression model? What error/cost function to minimize?
15 Logistic Regression Learning Learning = finding the right parameters T = [ 0, 1,, k ] Find that minimizes an error function E( hich measures the misfit beteen h(x n, and t n. Expect that h(x, performing ell on training examples x n Þ h(x, ill perform ell on arbitrary test examples x Î X. Least Squares error function? E( = 1 2 N n=1 {h(x n, t n } 2 Differentiable => can use gradient descent Non-convex => not guaranteed to find the global optimum
16 Maximum Likelihood Training set is D = {áx n, t n ñ t n Î {0,1}, n Î 1 N} Let h n = p(c 1 x n h n = p(t n =1 x n = σ ( T x n Maximum Likelihood (ML principle: find parameters that maximize the likelihood of the labels. The likelihood function is N p(t = h n t n (1 h n (1 t n n=1 The negative log-likelihood (cross entropy error function: N { } E( = ln p(t x = t n lnh n + (1 t n ln(1 h n n=1
17 Maximum Likelihood Learning for Logistic Regression The ML solution is: ML = argmax p(t = argmin E( convex in ML solution is given by ÑE( = 0. Cannot solve analytically => solve numerically ith gradient based methods: (stochastic gradient descent, conjugate gradient, L-BFGS, etc. Gradient is (prove it: E( = N n=1 (h n t n x n T
18 Regularized Logistic Regression Use a Gaussian prior over the parameters: = [ 0, 1,, M ] T Bayes Theorem: MAP solution: þ ý ü î í ì - ø ö ç è æ = = + - I 0 T M Ν p 2 exp 2, ( ( 2 1 / ( 1 a p a a ( ( ( ( ( ( t t t t p p p p p p µ = ( max arg t p MAP =
19 Regularized Logistic Regression MAP solution: = MAP arg max p( t = arg max p ( t p ( = arg min- ln p( t p( = arg min- ln p( t - ln p( = arg min E D ( - ln p( a T = arg min E D ( + 2 = arg mine D ( + E ( E E D ( ( = - N å{ tn ln yn + (1 - tnln(1 - yn } n= 1 a = 2 T regularization term data term
20 Regularized Logistic Regression MAP solution: MAP = arg min E D ( + E ( still convex in ML solution is given by ÑE( = 0. ÑE( = ÑE D ( + ÑE ( Cannot solve analytically => solve numerically: N = (h n t n x n T +α T n=1 (stochastic gradient descent [PRML 3.1.3], Neton Raphson iterative optimization [PRML 4.3.3], conjugate gradient, LBFGS. here h n = σ ( T x n
21 Softmax Regression = Logistic Regression for Multiclass Classification Multiclass classification: T = {C 1, C 2,..., C K } = {1, 2,..., K}. Training set is (x 1,t 1, (x 2,t 2, (x n,t n. x = [1, x 1, x 2,..., x M ] t 1, t 2, t n Î {1, 2,..., K} One eight vector per class [PRML 4.3.4]: p(c k x = exp( k T x exp( j T x j
22 Softmax Regression (K ³ 2 Inference: C* = arg max p( Ck x C k = argmax C k exp( T k x exp( T j x j Z(x a normalization constant Training using: = argmax C k exp( k T x = argmax C k k T x Maximum Likelihood (ML Maximum A Posteriori (MAP ith a Gaussian prior on.
23 Softmax Regression The negative log-likelihood error function is: E D ( = 1 N ln p(t n x n = 1 N = 1 N N n=1 N N n=1 ln exp( T t n x n Z(x n K n=1 k=1 δ k (t n ln exp( T k x n Z(x n convex in here d ( x t = ì1 í î0 x x = ¹ t t is the Kronecker delta function.
24 Softmax Regression The ML solution is: = arg min ML E D ( The gradient is (prove it: k E D ( = 1 N = 1 N N n=1 N n=1 ( δ k (t n p(c k x n x n δ k (t n exp( T k x n Z(x n x n E D ( = " # T E 1 D (, T 2 E D (,, T K E D ( $ % T
25 Regularized Softmax Regression The ne cost function is: E( = E D (+ E ( = 1 N N K δ k (t n ln exp( T k x n + α Z(x n 2 n=1 k=1 K k=1 T k k The ne gradient is (prove it: k E( = 1 N N n=1 T ( δ k (t n p(c k x n x n +α k T
26 Softmax Regression ML solution is given by ÑE D ( = 0. Cannot solve analytically. Solve numerically, by pluging [cost, gradient] = [E D (, ÑE D (] values into general convex solvers: L-BFGS Neton methods conjugate gradient (stochastic / minibatch gradient-based methods. gradient descent (ith / ithout momentum. AdaGrad, AdaDelta RMSProp ADAM,...
27 Implementation Need to compute [cost, gradient]: cost = 1 N gradient k N δ k (t n ln p(c k x n + α 2 n=1 k=1 => need to compute, for k = 1,..., K: K = 1 N N n=1 K k=1 T k k T T ( δ k (t n p(c k x n x n +α k output p(c k x n = exp( T k x n exp( T j x n j Overflo hen kt x n are too large.
28 Implementation: Preventing Overflos Subtract from each product kt x n the maximum product: c = max 1 k K k T x n p(c k x n = exp( k T x n c exp( j T x n c j
29 Implementation: Gradient Checking Want to minimize J(θ, here θ is a scalar. Mathematical definition of derivative: d J(θ +ε J(θ ε J(θ = lim dθ ε 2ε Numerical approximation of derivative: d J(θ +ε J(θ ε J(θ dθ 2ε here ε =
30 Implementation: Gradient Checking If θ is a vector of parameters θ i, Compute numerical derivative ith respect to each θ i. Aggregate all derivatives into numerical gradient G num (θ. Compare numerical gradient G num (θ ith implementation of gradient G imp (θ: G num (θ G imp (θ G num (θ+ G imp (θ 10 6
Machine Learning CS 4900/5900. Lecture 03. Razvan C. Bunescu School of Electrical Engineering and Computer Science
Machine Learning CS 4900/5900 Razvan C. Bunescu School of Electrical Engineering and Computer Science bunescu@ohio.edu Machine Learning is Optimization Parametric ML involves minimizing an objective function
More informationMachine Learning: The Perceptron. Lecture 06
Machine Learning: he Perceptron Razvan C. Bunescu School of Electrical Engineering and Computer Science bunescu@ohio.edu 1 McCulloch-Pitts Neuron Function 0 1 w 0 activation / output function 1 w 1 w w
More informationMachine Learning: Fisher s Linear Discriminant. Lecture 05
Machine Learning: Fisher s Linear Discriinant Lecture 05 Razvan C. Bunescu chool of Electrical Engineering and Coputer cience bunescu@ohio.edu Lecture 05 upervised Learning ask learn an (unkon) function
More informationLogistic Regression. Machine Learning Fall 2018
Logistic Regression Machine Learning Fall 2018 1 Where are e? We have seen the folloing ideas Linear models Learning as loss minimization Bayesian learning criteria (MAP and MLE estimation) The Naïve Bayes
More informationLecture 10. Neural networks and optimization. Machine Learning and Data Mining November Nando de Freitas UBC. Nonlinear Supervised Learning
Lecture 0 Neural networks and optimization Machine Learning and Data Mining November 2009 UBC Gradient Searching for a good solution can be interpreted as looking for a minimum of some error (loss) function
More informationStochastic gradient descent; Classification
Stochastic gradient descent; Classification Steve Renals Machine Learning Practical MLP Lecture 2 28 September 2016 MLP Lecture 2 Stochastic gradient descent; Classification 1 Single Layer Networks MLP
More informationMachine Learning: Logistic Regression. Lecture 04
Machie Learig: Logistic Regressio Razva C. Buescu School of Electrical Egieerig ad Computer Sciece buescu@ohio.edu Supervised Learig ask = lear a uko fuctio t : X that maps iput istaces x Î X to output
More informationMulticlass Logistic Regression
Multiclass Logistic Regression Sargur. Srihari University at Buffalo, State University of ew York USA Machine Learning Srihari Topics in Linear Classification using Probabilistic Discriminative Models
More informationGRADIENT DESCENT. CSE 559A: Computer Vision GRADIENT DESCENT GRADIENT DESCENT [0, 1] Pr(y = 1) w T x. 1 f (x; θ) = 1 f (x; θ) = exp( w T x)
0 x x x CSE 559A: Computer Vision For Binary Classification: [0, ] f (x; ) = σ( x) = exp( x) + exp( x) Output is interpreted as probability Pr(y = ) x are the log-odds. Fall 207: -R: :30-pm @ Lopata 0
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 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 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 informationNeural networks and support vector machines
Neural netorks and support vector machines Perceptron Input x 1 Weights 1 x 2 x 3... x D 2 3 D Output: sgn( x + b) Can incorporate bias as component of the eight vector by alays including a feature ith
More informationDeep Feedforward Networks
Deep Feedforward Networks Liu Yang March 30, 2017 Liu Yang Short title March 30, 2017 1 / 24 Overview 1 Background A general introduction Example 2 Gradient based learning Cost functions Output Units 3
More informationLecture 3a: The Origin of Variational Bayes
CSC535: 013 Advanced Machine Learning Lecture 3a: The Origin of Variational Bayes Geoffrey Hinton The origin of variational Bayes In variational Bayes, e approximate the true posterior across parameters
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 informationCS60021: Scalable Data Mining. Large Scale Machine Learning
J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 1 CS60021: Scalable Data Mining Large Scale Machine Learning Sourangshu Bhattacharya Example: Spam filtering Instance
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 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 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 informationIterative Reweighted Least Squares
Iterative Reweighted Least Squares Sargur. University at Buffalo, State University of ew York USA Topics in Linear Classification using Probabilistic Discriminative Models Generative vs Discriminative
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 informationDay 4: Classification, support vector machines
Day 4: Classification, support vector machines Introduction to Machine Learning Summer School June 18, 2018 - June 29, 2018, Chicago Instructor: Suriya Gunasekar, TTI Chicago 21 June 2018 Topics so far
More informationLecture 16: Modern Classification (I) - Separating Hyperplanes
Lecture 16: Modern Classification (I) - Separating Hyperplanes Outline 1 2 Separating Hyperplane Binary SVM for Separable Case Bayes Rule for Binary Problems Consider the simplest case: two classes are
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 informationMachine Learning. 7. Logistic and Linear Regression
Sapienza University of Rome, Italy - Machine Learning (27/28) University of Rome La Sapienza Master in Artificial Intelligence and Robotics Machine Learning 7. Logistic and Linear Regression Luca Iocchi,
More informationLinear models: the perceptron and closest centroid algorithms. D = {(x i,y i )} n i=1. x i 2 R d 9/3/13. Preliminaries. Chapter 1, 7.
Preliminaries Linear models: the perceptron and closest centroid algorithms Chapter 1, 7 Definition: The Euclidean dot product beteen to vectors is the expression d T x = i x i The dot product is also
More informationPreliminaries. Definition: The Euclidean dot product between two vectors is the expression. i=1
90 8 80 7 70 6 60 0 8/7/ Preliminaries Preliminaries Linear models and the perceptron algorithm Chapters, T x + b < 0 T x + b > 0 Definition: The Euclidean dot product beteen to vectors is the expression
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 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 informationNeural Networks and Deep Learning
Neural Networks and Deep Learning Professor Ameet Talwalkar November 12, 2015 Professor Ameet Talwalkar Neural Networks and Deep Learning November 12, 2015 1 / 16 Outline 1 Review of last lecture AdaBoost
More informationApril 9, Depto. de Ing. de Sistemas e Industrial Universidad Nacional de Colombia, Bogotá. Linear Classification Models. Fabio A. González Ph.D.
Depto. de Ing. de Sistemas e Industrial Universidad Nacional de Colombia, Bogotá April 9, 2018 Content 1 2 3 4 Outline 1 2 3 4 problems { C 1, y(x) threshold predict(x) = C 2, y(x) < threshold, with threshold
More informationCOMPUTATIONAL INTELLIGENCE (INTRODUCTION TO MACHINE LEARNING) SS16
COMPUTATIONAL INTELLIGENCE (INTRODUCTION TO MACHINE LEARNING) SS6 Lecture 3: Classification with Logistic Regression Advanced optimization techniques Underfitting & Overfitting Model selection (Training-
More 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 informationArtificial Neural Networks. Part 2
Artificial Neural Netorks Part Artificial Neuron Model Folloing simplified model of real neurons is also knon as a Threshold Logic Unit x McCullouch-Pitts neuron (943) x x n n Body of neuron f out Biological
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 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 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 informationComputer Vision Group Prof. Daniel Cremers. 3. Regression
Prof. Daniel Cremers 3. Regression Categories of Learning (Rep.) Learnin g Unsupervise d Learning Clustering, density estimation Supervised Learning learning from a training data set, inference on the
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 informationy(x) = x w + ε(x), (1)
Linear regression We are ready to consider our first machine-learning problem: linear regression. Suppose that e are interested in the values of a function y(x): R d R, here x is a d-dimensional vector-valued
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 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 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 informationIntelligent Systems Discriminative Learning, Neural Networks
Intelligent Systems Discriminative Learning, Neural Networks Carsten Rother, Dmitrij Schlesinger WS2014/2015, Outline 1. Discriminative learning 2. Neurons and linear classifiers: 1) Perceptron-Algorithm
More informationLecture 6. Regression
Lecture 6. Regression Prof. Alan Yuille Summer 2014 Outline 1. Introduction to Regression 2. Binary Regression 3. Linear Regression; Polynomial Regression 4. Non-linear Regression; Multilayer Perceptron
More informationLogistic Regression. William Cohen
Logistic Regression William Cohen 1 Outline Quick review classi5ication, naïve Bayes, perceptrons new result for naïve Bayes Learning as optimization Logistic regression via gradient ascent Over5itting
More informationNeed for Deep Networks Perceptron. Can only model linear functions. Kernel Machines. Non-linearity provided by kernels
Need for Deep Networks Perceptron Can only model linear functions Kernel Machines Non-linearity provided by kernels Need to design appropriate kernels (possibly selecting from a set, i.e. kernel learning)
More informationDeep Feedforward Networks
Deep Feedforward Networks Liu Yang March 30, 2017 Liu Yang Short title March 30, 2017 1 / 24 Overview 1 Background A general introduction Example 2 Gradient based learning Cost functions Output Units 3
More informationPATTERN RECOGNITION AND MACHINE LEARNING
PATTERN RECOGNITION AND MACHINE LEARNING Chapter 1. Introduction Shuai Huang April 21, 2014 Outline 1 What is Machine Learning? 2 Curve Fitting 3 Probability Theory 4 Model Selection 5 The curse of dimensionality
More informationprobability of k samples out of J fall in R.
Nonparametric Techniques for Density Estimation (DHS Ch. 4) n Introduction n Estimation Procedure n Parzen Window Estimation n Parzen Window Example n K n -Nearest Neighbor Estimation Introduction Suppose
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 informationMark Gales October y (x) x 1. x 2 y (x) Inputs. Outputs. x d. y (x) Second Output layer layer. layer.
University of Cambridge Engineering Part IIB & EIST Part II Paper I0: Advanced Pattern Processing Handouts 4 & 5: Multi-Layer Perceptron: Introduction and Training x y (x) Inputs x 2 y (x) 2 Outputs x
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 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 informationGradient-Based Learning. Sargur N. Srihari
Gradient-Based Learning Sargur N. srihari@cedar.buffalo.edu 1 Topics Overview 1. Example: Learning XOR 2. Gradient-Based Learning 3. Hidden Units 4. Architecture Design 5. Backpropagation and Other Differentiation
More informationMachine Learning. Neural Networks. Le Song. CSE6740/CS7641/ISYE6740, Fall Lecture 7, September 11, 2012 Based on slides from Eric Xing, CMU
Machine Learning CSE6740/CS7641/ISYE6740, Fall 2012 Neural Networks Le Song Lecture 7, September 11, 2012 Based on slides from Eric Xing, CMU Reading: Chap. 5 CB Learning highly non-linear functions f:
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 informationCS-E3210 Machine Learning: Basic Principles
CS-E3210 Machine Learning: Basic Principles Lecture 3: Regression I slides by Markus Heinonen Department of Computer Science Aalto University, School of Science Autumn (Period I) 2017 1 / 48 In a nutshell
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 informationAd Placement Strategies
Case Study : Estimating Click Probabilities Intro Logistic Regression Gradient Descent + SGD AdaGrad Machine Learning for Big Data CSE547/STAT548, University of Washington Emily Fox January 7 th, 04 Ad
More informationChapter ML:VI. VI. Neural Networks. Perceptron Learning Gradient Descent Multilayer Perceptron Radial Basis Functions
Chapter ML:VI VI. Neural Networks Perceptron Learning Gradient Descent Multilayer Perceptron Radial asis Functions ML:VI-1 Neural Networks STEIN 2005-2018 The iological Model Simplified model of a neuron:
More informationArtificial Neural Networks
Artificial Neural Networks 鮑興國 Ph.D. National Taiwan University of Science and Technology Outline Perceptrons Gradient descent Multi-layer networks Backpropagation Hidden layer representations Examples
More informationPart 8: Neural Networks
METU Informatics Institute Min720 Pattern Classification ith Bio-Medical Applications Part 8: Neural Netors - INTRODUCTION: BIOLOGICAL VS. ARTIFICIAL Biological Neural Netors A Neuron: - A nerve cell as
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 informationLecture 3: Multiclass Classification
Lecture 3: Multiclass Classification Kai-Wei Chang CS @ University of Virginia kw@kwchang.net Some slides are adapted from Vivek Skirmar and Dan Roth CS6501 Lecture 3 1 Announcement v Please enroll in
More informationApplication: Can we tell what people are looking at from their brain activity (in real time)? Gaussian Spatial Smooth
Application: Can we tell what people are looking at from their brain activity (in real time? Gaussian Spatial Smooth 0 The Data Block Paradigm (six runs per subject Three Categories of Objects (counterbalanced
More informationNeed for Deep Networks Perceptron. Can only model linear functions. Kernel Machines. Non-linearity provided by kernels
Need for Deep Networks Perceptron Can only model linear functions Kernel Machines Non-linearity provided by kernels Need to design appropriate kernels (possibly selecting from a set, i.e. kernel learning)
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 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 informationLECTURE NOTE #NEW 6 PROF. ALAN YUILLE
LECTURE NOTE #NEW 6 PROF. ALAN YUILLE 1. Introduction to Regression Now consider learning the conditional distribution p(y x). This is often easier than learning the likelihood function p(x y) and the
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 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 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 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 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 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 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 informationKernel Methods and Support Vector Machines
Kernel Methods and Support Vector Machines Oliver Schulte - CMPT 726 Bishop PRML Ch. 6 Support Vector Machines Defining Characteristics Like logistic regression, good for continuous input features, discrete
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 informationExpectation Maximization Algorithm
Expectation Maximization Algorithm Vibhav Gogate The University of Texas at Dallas Slides adapted from Carlos Guestrin, Dan Klein, Luke Zettlemoyer and Dan Weld The Evils of Hard Assignments? Clusters
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 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 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 informationArtificial Neural Networks. Q550: Models in Cognitive Science Lecture 5
Artificial Neural Networks Q550: Models in Cognitive Science Lecture 5 "Intelligence is 10 million rules." --Doug Lenat The human brain has about 100 billion neurons. With an estimated average of one thousand
More informationThe classifier. Theorem. where the min is over all possible classifiers. To calculate the Bayes classifier/bayes risk, we need to know
The Bayes classifier Theorem The classifier satisfies where the min is over all possible classifiers. To calculate the Bayes classifier/bayes risk, we need to know Alternatively, since the maximum it is
More informationThe classifier. Linear discriminant analysis (LDA) Example. Challenges for LDA
The Bayes classifier Linear discriminant analysis (LDA) Theorem The classifier satisfies In linear discriminant analysis (LDA), we make the (strong) assumption that where the min is over all possible classifiers.
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 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 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 informationEngineering Part IIB: Module 4F10 Statistical Pattern Processing Lecture 6: Multi-Layer Perceptrons I
Engineering Part IIB: Module 4F10 Statistical Pattern Processing Lecture 6: Multi-Layer Perceptrons I Phil Woodland: pcw@eng.cam.ac.uk Michaelmas 2012 Engineering Part IIB: Module 4F10 Introduction In
More informationSequence Modelling with Features: Linear-Chain Conditional Random Fields. COMP-599 Oct 6, 2015
Sequence Modelling with Features: Linear-Chain Conditional Random Fields COMP-599 Oct 6, 2015 Announcement A2 is out. Due Oct 20 at 1pm. 2 Outline Hidden Markov models: shortcomings Generative vs. discriminative
More informationSTA141C: Big Data & High Performance Statistical Computing
STA141C: Big Data & High Performance Statistical Computing Lecture 8: Optimization Cho-Jui Hsieh UC Davis May 9, 2017 Optimization Numerical Optimization Numerical Optimization: min X f (X ) Can be applied
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 informationMACHINE LEARNING ADVANCED MACHINE LEARNING
MACHINE LEARNING ADVANCED MACHINE LEARNING Recap of Important Notions on Estimation of Probability Density Functions 22 MACHINE LEARNING Discrete Probabilities Consider two variables and y taking discrete
More informationFeed-forward Networks Network Training Error Backpropagation Applications. Neural Networks. Oliver Schulte - CMPT 726. Bishop PRML Ch.
Neural Networks Oliver Schulte - CMPT 726 Bishop PRML Ch. 5 Neural Networks Neural networks arise from attempts to model human/animal brains Many models, many claims of biological plausibility We will
More informationMachine Learning Basics III
Machine Learning Basics III Benjamin Roth CIS LMU München Benjamin Roth (CIS LMU München) Machine Learning Basics III 1 / 62 Outline 1 Classification Logistic Regression 2 Gradient Based Optimization Gradient
More informationPattern Recognition and Machine Learning. Bishop Chapter 6: Kernel Methods
Pattern Recognition and Machine Learning Chapter 6: Kernel Methods Vasil Khalidov Alex Kläser December 13, 2007 Training Data: Keep or Discard? Parametric methods (linear/nonlinear) so far: learn parameter
More informationProbabilistic Graphical Models & Applications
Probabilistic Graphical Models & Applications Learning of Graphical Models Bjoern Andres and Bernt Schiele Max Planck Institute for Informatics The slides of today s lecture are authored by and shown with
More information