COMPUTATIONAL INTELLIGENCE (INTRODUCTION TO MACHINE LEARNING) SS16
|
|
- Blanche Haynes
- 6 years ago
- Views:
Transcription
1 COMPUTATIONAL INTELLIGENCE (INTRODUCTION TO MACHINE LEARNING) SS6 Lecture 3: Classification with Logistic Regression Advanced optimization techniques Underfitting & Overfitting Model selection (Training- & Validation- & Testset)
2 CLASSIFICATION WITH LOGISTIC REGRESSION
3 Logistic Regression Classification and not regression Classification = recognition Mogees and vibration classification: Action recognition:
4 Logistic Regression The default classification model Binary classification Extensions to multi-class later in the course Simple classification algorithm Convex cost - unique local optimum Gradient descent No more parameter than with linear regre Interpretability of parameters Fast evaluation of hypothesis for making predictions
5 LOGISTIC REGRESSION Hypothesis
6 Example (step function hypothesis) i Tumor size (mm) x Malignant? 2.3 (N) 2 5. (Y) 3.4 (N) (Y) (Y) labeled data y benign malignant decision boundary Tumor size (x) labels
7 Example (logistic function hypothesis) i Tumor size (mm) x labeled data Malignant? 2.3 (N) 2 5. (Y) y benign malignant? decision boundary Tumor size (x) 3.4 (N) (Y) (Y) labels p=.2, class? Hypothesis: Tumor is malignant with probability p Classification: if p <.5: if p.5:.5
8 Example (logistic function hypothesis) i Tumor size (mm) x labeled data Malignant? 2.3 (N) 2 5. (Y) y benign malignant? decision boundary Tumor size (x) 3.4 (N) (Y) (Y) labels p=., class? Hypothesis: Tumor is malignant with probability p Classification: if p <.5: if p.5:.5
9 Logistic (Sigmoid) function Advantages over step function for classification: Differentiable (gradient descent) Contains additional information (how certain is the prediction?)
10 Logistic regression hypothesis (one input)
11 Age (x2) Classification with multiple inputs i Tumor size (mm) Age Maligna nt? x x2 y (N) (Y) (N) (Y) Tumor size (x) (Y)
12 Age (x2) Multiple inputs and logistic hypothesis i Tumor size (mm) Age Maligna nt?. Reduce point in high-dimensional space to a scalar z 2. Apply logistic function p=.8, class x x2 y (N) (Y)? (N) (Y) decision boundary Tumor size (x)? (Y)
13 Age (x2) Classification with multiple inputs i Tumor size (mm) Age Maligna nt?. Reduce point in high-dimensional space to a scalar z 2. Apply logistic function p=.999, class x x2 y (N) (Y) (N) (Y) (Y) Tumor size (x)??
14 Logistic regression hypothesis. Reduce high-dimensional input to a scalar 2. Apply logistic function 3. Interpret output as probability and predict class: Class =
15 LOGISTIC REGRESSION Cost function
16 Age (x2) Logistic regression cost function How well does the hypothesis fit the data? Tumor size (x) Prediction:.98 Actual value y: Prediction:.6 Actual value y: Prediction:. Actual value y:
17 Age (x2) Logistic regression cost function Probabilistic model: y is with probability: Tumor size (x)
18 Logistic regression cost function Probabilistic model: y is with probability p(x,y=) = The parameters should maximize the likelihood of the data max θ log p X = x x n, y = y, y n θ) If data points are independants max θ i log p x i, y i θ) p x i, y i, x j, y j θ) = p x j, y j θ). p x i, y i θ) Separating positive and negative examples max θ y i = log p x i, θ) + y i = log p x i, θ) σ(x T θ) σ(x T θ)
19 Logistic regression cost function How well does the hypothesis Cost for predicting probability p when the real value is y: fit the data? Mean over all training examples:
20 Multiple inputs and logistic hypothesis How well does the hypothesis fit the data? Prediction:. Actual value y: Prediction:.6 Actual value y: Prediction:.98 Actual value y:
21 Comparison cost functions Linear regression Logistic regression Mean over all training examples:
22 Why not mean squared error (MSE) again? MSE with logistic hypothesis is non-convex (many local minima) Logistic regression is convex (unique minimum) Cost function can be derived from statistical principles ( maximum likelihood ) non-convex function convex function
23 LOGISTIC REGRESSION Learning from data
24 Minimizing the cost via gradient descent Gradient descent (simultaneous update for j= n) Gradient of logistic regression cost: error input (for j=: )
25 Age (x2) Linear to non-linear features linear decision boundary Tumor size (x)
26 Age (x2) Linear to non-linear features non-linear decision boundary Tumor size (x)
27 Age (x2) Age (x2) Decision boundaries linear decision boundary non-linear decision boundary Tumor size (x) Tumor size (x) Decision boundary is a property of hypothesis, not of data!
28 Linear vs. Logistic Regression Linear Regression Regression Hypothesis Cost for one training example: Logistic Regression Binary classification (!) Hypothesis Cost for one training example: Gradient Gradient error input error input Analytical: No analytical solution!
29 GRADIENT DESCENT TRICKS, AND MORE ADVANCED OPTIMIZATION TECHNIQUES For linear regression, logistic regression,.
30 GD trick #: feature scaling Feature scaling and mean normalization Bring all features into a similar range E.g.: shift and scale each feature to have mean and variance Mean of unscaled feature Standard deviation of unscaled feature (when using non-linear features) Do not apply to constant feature! Typically leads to much faster convergence
31 GD trick #2: monitoring convergence Diagnose typical issues with Gradient Descent:.75 6 cost J at iteration 3 cost J cost J iteration 5 iteration iteration slow convergence (increase learning rate?) oscillations (decrease learning rate) divergence (decrease learning rate)
32 GD trick #3: adaptive learning rate At each iteration Compare cost function value after Gradient Descent update before and 6 If cost increased: Reject update (go back to previous parameters) Multiply learning rate by.7 (for example) cost J iteration If cost decreased: Multiply learning rate by.2 (for example) cost J iteration Often eliminates slow convergence and divergence issues
33 A black box view of gradient descent Write code for computing the cost and its gradient Code to compute Gradient descent Code to compute Local/global minimum? Learning rate. Stop when J < 4
34 More advanced optimization methods Gradient methods = order Newton methods = order 2 Need Hessian matrix or approximations Avoid choosing a learning rate Conjugate gradient, BFGS, L-BFGS, Tricky to implement (numerical stability, etc.) Use available toolbox / library implementations! scipy.optimize.minimize Only use when fighting for performance
35 EVALUATION OF HYPOTHESIS Training and test set
36 Training and Test set Training set: used by learning algorithm to fit parameters and find a hypothesis. Test set: independent data set, used after learning to estimate the performance of the hypothesis on new (unseen) test examples training test Regression example y x E.g. 7% randomly chosen examples from dataset are training examples, the remaining 3% are test examples. Must be disjoint subsets!
37 Training and Test set workflow Training set Learning algorithm Training error (cost) Hypothesis h Test set Testing Test error (cost)
38 Linear regression training vs. test error MSEtrain=. MSEtest=2.2 2 training hypothesis 2 test hypothesis polynomial degree 4 y y x x
39 Classification Training / Test set Training set Test set
40 Logistic regression training vs. test error Training Error:.5 Test Error: polynomial features up to power.6.4 decision boundary
41 UNDERFITTING AND OVERFITTING
42 Polynomial regression under-/overfitting MSEtrain=.22, MSEtest=.29 training test polynomial fit degree y x
43 Polynomial regression under-/overfitting MSEtrain=., MSEtest=. training test polynomial fit degree 5 y x
44 Polynomial regression under-/overfitting MSEtrain=., MSEtest=3.7 training test polynomial fit degree 5 y x
45 -2-2 Polynomial regression under-/overfitting MSEtrain=.22, MSEtest=.29 training test polynomial fit degree MSEtrain=., MSEtest=3.7 training test polynomial fit degree 5 y y x underfitting MSEtrain=., MSEtest=. training test polynomial fit degree x overfitting y.5 just right
46 Logistic regression with polynomial terms Training set Test set
47 Logistic regression with polynomial terms Terms up to power Training Error:.33 Test Error:.32 decision boundary Training Error:.9 Test Error: Terms up to power
48 Logistic regression with polynomial terms Training Error:.7 Test Error:.44 Terms up to power
49 Training Error:.33 Test Error:.32 Training Error:.7 Test Error: underfitting.5.5 overfitting.5 Training Error:.9 Test Error: just right.5
50 Under-/ and Overfitting Training error (cost) underfitting Test error (cost) overfitting just right Model complexity (e.g. degree of polynomial terms)
51 Under- and Overfitting Underfitting: Model is too simple (often: too few parameters) High training error, high test error Training error (cost) Test error (cost) Overfitting Model is too complex (often: too many parameters relative to number of training examples) Low training error, high test error In between: Model has right complexity Moderate training error Lowest test error Model complexity
52 How to deal with overfitting Use model selection to automatically select the right model complexity Use regularization to keep parameters small (other lecture ) Collect more data (often not possible or inefficient) Manually throw out features which are unlikely to contribute (often hard to guess which ones, potentially throwing out the wrong ones) Change features vectors, use pre-processing (often not possible or inefficient, time consuming)
53 MODEL SELECTION Training, Validation and Test sets
54 Model selection Selection of learning algorithm and hyperparameters (model complexity) that are most suitable for a given learning problem Training error (cost) Test error (cost) Model complexity
55 Idea Try out different learning algorithms/variants Vary degree of polynomial Try different sets of features Training error (cost) Prediction error (cost) Select variant with best predictive performance Model complexity
56 Training, Validation, Test set Training set: used by learning algorithm to fit parameters and find a hypothesis for each learning algorithm/variant. Validation set: used to estimate predictive performance of each learning algorithm/variant. The hypothesis with lowest validation error (cost) is selected. Test set: independent data set, used after learning and model selection to estimate the performance of the final (selected) hypothesis on new (unseen) test examples. E.g. 6/2/2 % randomly chosen examples from dataset. Must be disjoint subsets!
57 Training/Validation/Test set workflow Training set LA with hyperparameter setting A LA with hyperparameter setting B LA with hyperparameter setting C For example: degree of polynome=, 5, and 5 Hypothesis h A Hypothesis h B Hypothesis h C Validation set Model selection Selected Hypothesis h Test set Testing Test error/cost
58 Training/Validation/Test set workflow Training set Learning algorithm A Learning algorithm B Learning algorithm C For example: Linear regression, Polynomial regression, Artificial Neural Network Hypothesis h A Hypothesis h B Hypothesis h C Validation set Model selection Selected Hypothesis h Test set Testing Test error/cost
59 Some questions Logistic regression is a method for regression/classification? Logistic regression hypothesis? What s the cost function used for logistic regression? Is it convex or non-convex? What does adaptive learning rate mean in the context of gradient descent? How to evaluate a hypothesis? What is under-/overfitting? What is model selection? What are training, validation and test sets? How does model selection work (procedure)?
60 What is next? Neural Networks: Perceptron Feedforward Neural Network Backpropagation
ECS171: Machine Learning
ECS171: Machine Learning Lecture 4: Optimization (LFD 3.3, SGD) Cho-Jui Hsieh UC Davis Jan 22, 2018 Gradient descent Optimization Goal: find the minimizer of a function min f (w) w For now we assume f
More informationDeep Learning & Artificial Intelligence WS 2018/2019
Deep Learning & Artificial Intelligence WS 2018/2019 Linear Regression Model Model Error Function: Squared Error Has no special meaning except it makes gradients look nicer Prediction Ground truth / target
More informationLecture 10. Neural networks and optimization. Machine Learning and Data Mining November Nando de Freitas UBC. Nonlinear Supervised Learning
Lecture 0 Neural networks and optimization Machine Learning and Data Mining November 2009 UBC Gradient Searching for a good solution can be interpreted as looking for a minimum of some error (loss) function
More informationLogistic Regression. 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 informationCSC2515 Winter 2015 Introduction to Machine Learning. Lecture 2: Linear regression
CSC2515 Winter 2015 Introduction to Machine Learning Lecture 2: Linear regression All lecture slides will be available as.pdf on the course website: http://www.cs.toronto.edu/~urtasun/courses/csc2515/csc2515_winter15.html
More 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 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 informationCSC321 Lecture 2: Linear Regression
CSC32 Lecture 2: Linear Regression Roger Grosse Roger Grosse CSC32 Lecture 2: Linear Regression / 26 Overview First learning algorithm of the course: linear regression Task: predict scalar-valued targets,
More 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 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 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 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 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 informationMachine Learning 4771
Machine Learning 4771 Instructor: Tony Jebara Topic 3 Additive Models and Linear Regression Sinusoids and Radial Basis Functions Classification Logistic Regression Gradient Descent Polynomial Basis Functions
More informationCS260: Machine Learning Algorithms
CS260: Machine Learning Algorithms Lecture 4: Stochastic Gradient Descent Cho-Jui Hsieh UCLA Jan 16, 2019 Large-scale Problems Machine learning: usually minimizing the training loss min w { 1 N min w {
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 informationComputational statistics
Computational statistics Lecture 3: Neural networks Thierry Denœux 5 March, 2016 Neural networks A class of learning methods that was developed separately in different fields statistics and artificial
More 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 informationLearning with multiple models. Boosting.
CS 2750 Machine Learning Lecture 21 Learning with multiple models. Boosting. Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Learning with multiple models: Approach 2 Approach 2: use multiple models
More informationNeural Networks, Computation Graphs. CMSC 470 Marine Carpuat
Neural Networks, Computation Graphs CMSC 470 Marine Carpuat Binary Classification with a Multi-layer Perceptron φ A = 1 φ site = 1 φ located = 1 φ Maizuru = 1 φ, = 2 φ in = 1 φ Kyoto = 1 φ priest = 0 φ
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 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 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 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 informationCSC 578 Neural Networks and Deep Learning
CSC 578 Neural Networks and Deep Learning Fall 2018/19 3. Improving Neural Networks (Some figures adapted from NNDL book) 1 Various Approaches to Improve Neural Networks 1. Cost functions Quadratic Cross
More informationLogistic Regression & Neural Networks
Logistic Regression & Neural Networks CMSC 723 / LING 723 / INST 725 Marine Carpuat Slides credit: Graham Neubig, Jacob Eisenstein Logistic Regression Perceptron & Probabilities What if we want a probability
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 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 informationYou submitted this quiz on Wed 16 Apr :18 PM IST. You got a score of 5.00 out of 5.00.
Feedback IX. Neural Networks: Learning Help You submitted this quiz on Wed 16 Apr 2014 10:18 PM IST. You got a score of 5.00 out of 5.00. Question 1 You are training a three layer neural network and would
More informationMachine Learning
Machine Learning 10-601 Maria Florina Balcan Machine Learning Department Carnegie Mellon University 02/10/2016 Today: Artificial neural networks Backpropagation Reading: Mitchell: Chapter 4 Bishop: Chapter
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 informationLeast Mean Squares Regression. Machine Learning Fall 2018
Least Mean Squares Regression Machine Learning Fall 2018 1 Where are we? Least Squares Method for regression Examples The LMS objective Gradient descent Incremental/stochastic gradient descent Exercises
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 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 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 Outlines Overview Introduction Linear Algebra Probability Linear Regression
More informationCSC 411 Lecture 6: Linear Regression
CSC 411 Lecture 6: Linear Regression Roger Grosse, Amir-massoud Farahmand, and Juan Carrasquilla University of Toronto UofT CSC 411: 06-Linear Regression 1 / 37 A Timely XKCD UofT CSC 411: 06-Linear Regression
More informationLecture 5 Neural models for NLP
CS546: Machine Learning in NLP (Spring 2018) http://courses.engr.illinois.edu/cs546/ Lecture 5 Neural models for NLP Julia Hockenmaier juliahmr@illinois.edu 3324 Siebel Center Office hours: Tue/Thu 2pm-3pm
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 informationLeast Mean Squares Regression
Least Mean Squares Regression Machine Learning Spring 2018 The slides are mainly from Vivek Srikumar 1 Lecture Overview Linear classifiers What functions do linear classifiers express? Least Squares Method
More informationLinear Regression. Robot Image Credit: Viktoriya Sukhanova 123RF.com
Linear Regression These slides were assembled by Eric Eaton, with grateful acknowledgement of the many others who made their course materials freely available online. Feel free to reuse or adapt these
More informationIntroduction to Neural Networks
CUONG TUAN NGUYEN SEIJI HOTTA MASAKI NAKAGAWA Tokyo University of Agriculture and Technology Copyright by Nguyen, Hotta and Nakagawa 1 Pattern classification Which category of an input? Example: Character
More informationMachine Learning Support Vector Machines. Prof. Matteo Matteucci
Machine Learning Support Vector Machines Prof. Matteo Matteucci Discriminative vs. Generative Approaches 2 o Generative approach: we derived the classifier from some generative hypothesis about the way
More 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 informationPattern Recognition and Machine Learning
Christopher M. Bishop Pattern Recognition and Machine Learning ÖSpri inger Contents Preface Mathematical notation Contents vii xi xiii 1 Introduction 1 1.1 Example: Polynomial Curve Fitting 4 1.2 Probability
More informationNeural Networks with Applications to Vision and Language. Feedforward Networks. Marco Kuhlmann
Neural Networks with Applications to Vision and Language Feedforward Networks Marco Kuhlmann Feedforward networks Linear separability x 2 x 2 0 1 0 1 0 0 x 1 1 0 x 1 linearly separable not linearly separable
More informationNeural Networks Learning the network: Backprop , Fall 2018 Lecture 4
Neural Networks Learning the network: Backprop 11-785, Fall 2018 Lecture 4 1 Recap: The MLP can represent any function The MLP can be constructed to represent anything But how do we construct it? 2 Recap:
More informationNeural Networks. CSE 6363 Machine Learning Vassilis Athitsos Computer Science and Engineering Department University of Texas at Arlington
Neural Networks CSE 6363 Machine Learning Vassilis Athitsos Computer Science and Engineering Department University of Texas at Arlington 1 Perceptrons x 0 = 1 x 1 x 2 z = h w T x Output: z x D A perceptron
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 informationPerceptron (Theory) + Linear Regression
10601 Introduction to Machine Learning Machine Learning Department School of Computer Science Carnegie Mellon University Perceptron (Theory) Linear Regression Matt Gormley Lecture 6 Feb. 5, 2018 1 Q&A
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 informationSample questions for Fundamentals of Machine Learning 2018
Sample questions for Fundamentals of Machine Learning 2018 Teacher: Mohammad Emtiyaz Khan A few important informations: In the final exam, no electronic devices are allowed except a calculator. Make sure
More 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 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 informationNONLINEAR CLASSIFICATION AND REGRESSION. J. Elder CSE 4404/5327 Introduction to Machine Learning and Pattern Recognition
NONLINEAR CLASSIFICATION AND REGRESSION Nonlinear Classification and Regression: Outline 2 Multi-Layer Perceptrons The Back-Propagation Learning Algorithm Generalized Linear Models Radial Basis Function
More informationLogistic Regression. Robot Image Credit: Viktoriya Sukhanova 123RF.com
Logistic Regression These slides were assembled by Eric Eaton, with grateful acknowledgement of the many others who made their course materials freely available online. Feel free to reuse or adapt these
More informationMidterm exam CS 189/289, Fall 2015
Midterm exam CS 189/289, Fall 2015 You have 80 minutes for the exam. Total 100 points: 1. True/False: 36 points (18 questions, 2 points each). 2. Multiple-choice questions: 24 points (8 questions, 3 points
More 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 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 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 informationStochastic Gradient Descent. CS 584: Big Data Analytics
Stochastic Gradient Descent CS 584: Big Data Analytics Gradient Descent Recap Simplest and extremely popular Main Idea: take a step proportional to the negative of the gradient Easy to implement Each iteration
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 Classification. CSE 6363 Machine Learning Vassilis Athitsos Computer Science and Engineering Department University of Texas at Arlington
Linear Classification CSE 6363 Machine Learning Vassilis Athitsos Computer Science and Engineering Department University of Texas at Arlington 1 Example of Linear Classification Red points: patterns belonging
More informationMachine Learning Basics: Maximum Likelihood Estimation
Machine Learning Basics: Maximum Likelihood Estimation Sargur N. srihari@cedar.buffalo.edu This is part of lecture slides on Deep Learning: http://www.cedar.buffalo.edu/~srihari/cse676 1 Topics 1. Learning
More informationLinear Models for Regression CS534
Linear Models for Regression CS534 Example Regression Problems Predict housing price based on House size, lot size, Location, # of rooms Predict stock price based on Price history of the past month Predict
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 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 informationCPSC 340: Machine Learning and Data Mining. Stochastic Gradient Fall 2017
CPSC 340: Machine Learning and Data Mining Stochastic Gradient Fall 2017 Assignment 3: Admin Check update thread on Piazza for correct definition of trainndx. This could make your cross-validation code
More informationMachine Learning: Logistic Regression. Lecture 04
Machine Learning: Logistic Regression Razvan C. Bunescu School of Electrical Engineering and Computer Science bunescu@ohio.edu Supervised Learning Task = learn an (unkon function t : X T that maps input
More informationNeural Networks. Prof. Dr. Rudolf Kruse. Computational Intelligence Group Faculty for Computer Science
Neural Networks Prof. Dr. Rudolf Kruse Computational Intelligence Group Faculty for Computer Science kruse@iws.cs.uni-magdeburg.de Rudolf Kruse Neural Networks 1 Supervised Learning / Support Vector Machines
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 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 informationCSC321 Lecture 9: Generalization
CSC321 Lecture 9: Generalization Roger Grosse Roger Grosse CSC321 Lecture 9: Generalization 1 / 27 Overview We ve focused so far on how to optimize neural nets how to get them to make good predictions
More informationArtificial Neural Networks
Artificial Neural Networks Stephan Dreiseitl University of Applied Sciences Upper Austria at Hagenberg Harvard-MIT Division of Health Sciences and Technology HST.951J: Medical Decision Support Knowledge
More informationOptimization and Gradient Descent
Optimization and Gradient Descent INFO-4604, Applied Machine Learning University of Colorado Boulder September 12, 2017 Prof. Michael Paul Prediction Functions Remember: a prediction function is the function
More informationLinear Discrimination Functions
Laurea Magistrale in Informatica Nicola Fanizzi Dipartimento di Informatica Università degli Studi di Bari November 4, 2009 Outline Linear models Gradient descent Perceptron Minimum square error approach
More informationLINEAR MODELS FOR CLASSIFICATION. J. Elder CSE 6390/PSYC 6225 Computational Modeling of Visual Perception
LINEAR MODELS FOR CLASSIFICATION Classification: Problem Statement 2 In regression, we are modeling the relationship between a continuous input variable x and a continuous target variable t. In classification,
More informationBig Data Analytics. Lucas Rego Drumond
Big Data Analytics Lucas Rego Drumond Information Systems and Machine Learning Lab (ISMLL) Institute of Computer Science University of Hildesheim, Germany Predictive Models Predictive Models 1 / 34 Outline
More informationNeural Networks: Backpropagation
Neural Networks: Backpropagation Machine Learning Fall 2017 Based on slides and material from Geoffrey Hinton, Richard Socher, Dan Roth, Yoav Goldberg, Shai Shalev-Shwartz and Shai Ben-David, and others
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 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 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 9: Generalization
CSC321 Lecture 9: Generalization Roger Grosse Roger Grosse CSC321 Lecture 9: Generalization 1 / 26 Overview We ve focused so far on how to optimize neural nets how to get them to make good predictions
More informationLinear & nonlinear classifiers
Linear & nonlinear classifiers Machine Learning Hamid Beigy Sharif University of Technology Fall 1394 Hamid Beigy (Sharif University of Technology) Linear & nonlinear classifiers Fall 1394 1 / 34 Table
More informationComparison of Modern Stochastic Optimization Algorithms
Comparison of Modern Stochastic Optimization Algorithms George Papamakarios December 214 Abstract Gradient-based optimization methods are popular in machine learning applications. In large-scale problems,
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 informationStatistical Machine Learning from Data
January 17, 2006 Samy Bengio Statistical Machine Learning from Data 1 Statistical Machine Learning from Data Multi-Layer Perceptrons Samy Bengio IDIAP Research Institute, Martigny, Switzerland, and Ecole
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 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 informationLecture 9: Large Margin Classifiers. Linear Support Vector Machines
Lecture 9: Large Margin Classifiers. Linear Support Vector Machines Perceptrons Definition Perceptron learning rule Convergence Margin & max margin classifiers (Linear) support vector machines Formulation
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 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 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 informationLecture 7. Logistic Regression. Luigi Freda. ALCOR Lab DIAG University of Rome La Sapienza. December 11, 2016
Lecture 7 Logistic Regression Luigi Freda ALCOR Lab DIAG University of Rome La Sapienza December 11, 2016 Luigi Freda ( La Sapienza University) Lecture 7 December 11, 2016 1 / 39 Outline 1 Intro Logistic
More informationData Mining Part 5. Prediction
Data Mining Part 5. Prediction 5.5. Spring 2010 Instructor: Dr. Masoud Yaghini Outline How the Brain Works Artificial Neural Networks Simple Computing Elements Feed-Forward Networks Perceptrons (Single-layer,
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 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 informationSelected Topics in Optimization. Some slides borrowed from
Selected Topics in Optimization Some slides borrowed from http://www.stat.cmu.edu/~ryantibs/convexopt/ Overview Optimization problems are almost everywhere in statistics and machine learning. Input Model
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 information