Hypothesis Testing and Computational Learning Theory. EECS 349 Machine Learning With slides from Bryan Pardo, Tom Mitchell
|
|
|
- Anissa Copeland
- 5 years ago
- Views:
Transcription
1 Hypothesis Testing and Computational Learning Theory EECS 349 Machine Learning With slides from Bryan Pardo, Tom Mitchell
2 Overview Hypothesis Testing: How do we know our learners are good? What does performance on test data imply/guarantee about future performance? Computational Learning Theory: Are there general laws that govern learning? Sample Complexity: How many training examples are needed to learn a successful hypothesis? Computational Complexity: How much computational effort is needed to learn a successful hypothesis?
3 Some terms X C H L D is the set of all possible instances is the set of all possible concepts c where c: X {0,1} is the set of hypotheses considered by a learner, H C is the learner is a probability distribution over X that generates observed instances
4 Definition The true error of hypothesis h, with respect to the target concept c and observation distribution D is the probability that h will misclassify an instance drawn according to D error P [ c( x) h( x)] D xd In a perfect world, we d like the true error to be 0
5 Definition The sample error of hypothesis h, with respect to the target concept c and sample S is the proportion of S that that h misclassifies: error S (h) = 1/ S xs (c(x), h(x)) where (c(x), h(x)) = 0 if c(x) = h(x), 1 otherwise
6 Problems Estimating Error
7 Example on Independent Test Set
8 Estimators
9 Confidence Intervals and n*error S (h), n*(1-error S (h)) each > 5
10 Confidence Intervals Under same conditions
11 Life Skills Convincing demonstration that certain enhancements improve performance? Use online Fisher Exact or Chi Square tests to evaluate hypotheses, e.g:
12 Overview Hypothesis Testing: How do we know our learners are good? What does performance on test data imply/guarantee about future performance? Computational Learning Theory: Are there general laws that govern learning? Sample Complexity: How many training examples are needed to learn a successful hypothesis? Computational Complexity: How much computational effort is needed to learn a successful hypothesis?
13 Computational Learning Theory Are there general laws that govern learning? No Free Lunch Theorem: The expected accuracy of any learning algorithm across all concepts is 50%. But can we still say something positive? Yes. Probably Approximately Correct (PAC) learning
14 The world isn t perfect If we can t provide every instance for training, a consistent hypothesis may have error on unobserved instances. Instance Space X Hypothesis H Training set Concept C How many training examples do we need to bound the likelihood of error to a reasonable level? When is our hypothesis Probably Approximately Correct (PAC)?
15 Definitions A hypothesis is consistent if it has zero error on training examples The version space (VS H,T ) is the set of all hypotheses consistent on training set T in our hypothesis space H (reminder: hypothesis space is the set of concepts we re considering, e.g. depth-2 decision trees)
16 Definition: e-exhausted IN ENGLISH: The set of hypotheses consistent with the training data T is e-exhausted if, when you test them on the actual distribution of instances, all consistent hypotheses have error below e IN MATH: VS H,T is e - exhausted and sampledistribution hvs, error ( h) H,T D for concept D, e if... c
17 A Theorem If hypothesisspace H is finite, & training set T contains m independent randomly drawn examples of concept c THEN,for any 0 e 1... P( VS is NOTε - exhausted) H e H,T em
18 Proof of Theorem If hypothesis h has true error e, the probability of getting a single random exampe right is : it P( h got 1example right) 1-ε Ergo the probability of h getting m examples right is : P( h got m examples right) (1-ε ) m
19 Proof of Theorem If there are k hypotheses in H with error at least e, call the probability at least one of those k hypotheses got m instances right P(at least one bad h looks good). This prob. is BOUNDED by k(1-ε ) m P at least one bad h looks good k(1-ε ) m Union bound
20 Proof of Theorem (continued) Since k H, it follows that k(1-ε ) m H (1-ε ) m If 0 e 1, then (1 e) e e Therefore... P(at least one bad h looks good) k(1-ε ) m H (1-ε ) m H e em Proof will complete! We now have a hypothsesis consistent have error e bound on the likelihood that a with the training data
21 Using the theorem Let's rearrange tosee how many training examples we need toset a bound on the likelihood our true error is e. 1 e ln 1 ln e e em em H e ln em H ln e ln ln ln ln ln H H em ln H ln H ln 1 em m H ln m
22 Probably Approximately Correct (PAC) 1 ln H ln m e The worst error we ll tolerate hypothesis space size The likelihood a hypothesis consistent with the training data will have error e number of training examples
23 Using the bound 1 ln H ln m e Plug in e,, and H to get a number of training examples m that will guarantee your learner will generate a hypothesis that is Probably Approximately Correct. NOTE: This assumes that the concept is actually IN H, that H is finite, and that your training set is drawn using distribution D
24 Think/Pair/Share Average accuracy of any learner across all concepts is 50%, but also: 1 ln H ln e m How can both be true? Think Start End 24
25 Think/Pair/Share Average accuracy of any learner across all concepts is 50%, but also: 1 ln H ln e m How can both be true? Pair Start End 25
26 Think/Pair/Share Average accuracy of any learner across all concepts is 50%, but also: 1 ln H ln e m How can both be true? Share 26
27 Problems with PAC The PAC Learning framework has 2 disadvantages: 1) It can lead to weak bounds 2)Sample Complexity bound cannot be established for infinite hypothesis spaces We introduce the VC dimension for dealing with these problems
28 Shattering Def: A set of instances S is shattered by hypothesis set H iff for every possible concept c on S there exists a hypothesis h in H that is consistent with that concept.
29 Can a linear separator shatter this? NO! The ability of H to shatter a set of instances is a measure of its capacity to represent target concepts defined over those instances
30 Can a quadratic separator shatter this?
31 Vapnik-Chervonenkis Dimension Def: The Vapnik-Chervonenkis dimension, VC(H) of hypothesis space H defined over instance space X is the size of the largest finite subset of X shattered by H. If arbitrarily large finite sets can be shattered by H, then VC(H) is infinite.
32 How many training examples needed? Lower bound on m using VC(H) m 1 e (4log (2/ ) 8VC( H)log 2 (13/ e 2 ))
33 Infinite VC dimension?
34 Think/Pair/Share What kind of classifier (that we ve talked about) has infinite VC dimension? Think Start End 34
35 Think/Pair/Share What kind of classifier (that we ve talked about) has infinite VC dimension? Pair Start End 35
36 Think/Pair/Share What kind of classifier (that we ve talked about) has infinite VC dimension? Share 36
Computational Learning Theory. CS 486/686: Introduction to Artificial Intelligence Fall 2013
Computational Learning Theory CS 486/686: Introduction to Artificial Intelligence Fall 2013 1 Overview Introduction to Computational Learning Theory PAC Learning Theory Thanks to T Mitchell 2 Introduction
Computational Learning Theory (VC Dimension)
Computational Learning Theory (VC Dimension) 1 Difficulty of machine learning problems 2 Capabilities of machine learning algorithms 1 Version Space with associated errors error is the true error, r is
Computational Learning Theory
Computational Learning Theory Pardis Noorzad Department of Computer Engineering and IT Amirkabir University of Technology Ordibehesht 1390 Introduction For the analysis of data structures and algorithms
Introduction to Machine Learning
Introduction to Machine Learning PAC Learning and VC Dimension Varun Chandola Computer Science & Engineering State University of New York at Buffalo Buffalo, NY, USA chandola@buffalo.edu Chandola@UB CSE
Machine Learning
Machine Learning 10-601 Tom M. Mitchell Machine Learning Department Carnegie Mellon University October 11, 2012 Today: Computational Learning Theory Probably Approximately Coorrect (PAC) learning theorem
Machine Learning
Machine Learning 10-601 Tom M. Mitchell Machine Learning Department Carnegie Mellon University October 11, 2012 Today: Computational Learning Theory Probably Approximately Coorrect (PAC) learning theorem
VC Dimension Review. The purpose of this document is to review VC dimension and PAC learning for infinite hypothesis spaces.
VC Dimension Review The purpose of this document is to review VC dimension and PAC learning for infinite hypothesis spaces. Previously, in discussing PAC learning, we were trying to answer questions about
CS 6375: Machine Learning Computational Learning Theory
CS 6375: Machine Learning Computational Learning Theory Vibhav Gogate The University of Texas at Dallas Many slides borrowed from Ray Mooney 1 Learning Theory Theoretical characterizations of Difficulty
CS340 Machine learning Lecture 4 Learning theory. Some slides are borrowed from Sebastian Thrun and Stuart Russell
CS340 Machine learning Lecture 4 Learning theory Some slides are borrowed from Sebastian Thrun and Stuart Russell Announcement What: Workshop on applying for NSERC scholarships and for entry to graduate
Learning Theory. Machine Learning B Seyoung Kim. Many of these slides are derived from Tom Mitchell, Ziv- Bar Joseph. Thanks!
Learning Theory Machine Learning 10-601B Seyoung Kim Many of these slides are derived from Tom Mitchell, Ziv- Bar Joseph. Thanks! Computa2onal Learning Theory What general laws constrain inducgve learning?
Computational Learning Theory
Computational Learning Theory Sinh Hoa Nguyen, Hung Son Nguyen Polish-Japanese Institute of Information Technology Institute of Mathematics, Warsaw University February 14, 2006 inh Hoa Nguyen, Hung Son
Generalization, Overfitting, and Model Selection
Generalization, Overfitting, and Model Selection Sample Complexity Results for Supervised Classification Maria-Florina (Nina) Balcan 10/03/2016 Two Core Aspects of Machine Learning Algorithm Design. How
Computational Learning Theory
Computational Learning Theory Slides by and Nathalie Japkowicz (Reading: R&N AIMA 3 rd ed., Chapter 18.5) Computational Learning Theory Inductive learning: given the training set, a learning algorithm
Machine Learning. VC Dimension and Model Complexity. Eric Xing , Fall 2015
Machine Learning 10-701, Fall 2015 VC Dimension and Model Complexity Eric Xing Lecture 16, November 3, 2015 Reading: Chap. 7 T.M book, and outline material Eric Xing @ CMU, 2006-2015 1 Last time: PAC and
Machine Learning. Computational Learning Theory. Le Song. CSE6740/CS7641/ISYE6740, Fall 2012
Machine Learning CSE6740/CS7641/ISYE6740, Fall 2012 Computational Learning Theory Le Song Lecture 11, September 20, 2012 Based on Slides from Eric Xing, CMU Reading: Chap. 7 T.M book 1 Complexity of Learning
COMP9444: Neural Networks. Vapnik Chervonenkis Dimension, PAC Learning and Structural Risk Minimization
: Neural Networks Vapnik Chervonenkis Dimension, PAC Learning and Structural Risk Minimization 11s2 VC-dimension and PAC-learning 1 How good a classifier does a learner produce? Training error is the precentage
Computational Learning Theory
1 Computational Learning Theory 2 Computational learning theory Introduction Is it possible to identify classes of learning problems that are inherently easy or difficult? Can we characterize the number
Web-Mining Agents Computational Learning Theory
Web-Mining Agents Computational Learning Theory Prof. Dr. Ralf Möller Dr. Özgür Özcep Universität zu Lübeck Institut für Informationssysteme Tanya Braun (Exercise Lab) Computational Learning Theory (Adapted)
Computational Learning Theory (COLT)
Computational Learning Theory (COLT) Goals: Theoretical characterization of 1 Difficulty of machine learning problems Under what conditions is learning possible and impossible? 2 Capabilities of machine
Computational Learning Theory. Definitions
Computational Learning Theory Computational learning theory is interested in theoretical analyses of the following issues. What is needed to learn effectively? Sample complexity. How many examples? Computational
Learning Theory. Piyush Rai. CS5350/6350: Machine Learning. September 27, (CS5350/6350) Learning Theory September 27, / 14
Learning Theory Piyush Rai CS5350/6350: Machine Learning September 27, 2011 (CS5350/6350) Learning Theory September 27, 2011 1 / 14 Why Learning Theory? We want to have theoretical guarantees about our
Computational Learning Theory
Computational Learning Theory [read Chapter 7] [Suggested exercises: 7.1, 7.2, 7.5, 7.8] Computational learning theory Setting 1: learner poses queries to teacher Setting 2: teacher chooses examples Setting
Computational Learning Theory
0. Computational Learning Theory Based on Machine Learning, T. Mitchell, McGRAW Hill, 1997, ch. 7 Acknowledgement: The present slides are an adaptation of slides drawn by T. Mitchell 1. Main Questions
Learning Theory, Overfi1ng, Bias Variance Decomposi9on
Learning Theory, Overfi1ng, Bias Variance Decomposi9on Machine Learning 10-601B Seyoung Kim Many of these slides are derived from Tom Mitchell, Ziv- 1 Bar Joseph. Thanks! Any(!) learner that outputs a
Learning Theory. Machine Learning CSE546 Carlos Guestrin University of Washington. November 25, Carlos Guestrin
Learning Theory Machine Learning CSE546 Carlos Guestrin University of Washington November 25, 2013 Carlos Guestrin 2005-2013 1 What now n We have explored many ways of learning from data n But How good
Statistical and Computational Learning Theory
Statistical and Computational Learning Theory Fundamental Question: Predict Error Rates Given: Find: The space H of hypotheses The number and distribution of the training examples S The complexity of the
Lecture Slides for INTRODUCTION TO. Machine Learning. By: Postedited by: R.
Lecture Slides for INTRODUCTION TO Machine Learning By: alpaydin@boun.edu.tr http://www.cmpe.boun.edu.tr/~ethem/i2ml Postedited by: R. Basili Learning a Class from Examples Class C of a family car Prediction:
A Tutorial on Computational Learning Theory Presented at Genetic Programming 1997 Stanford University, July 1997
A Tutorial on Computational Learning Theory Presented at Genetic Programming 1997 Stanford University, July 1997 Vasant Honavar Artificial Intelligence Research Laboratory Department of Computer Science
Computational Learning Theory
09s1: COMP9417 Machine Learning and Data Mining Computational Learning Theory May 20, 2009 Acknowledgement: Material derived from slides for the book Machine Learning, Tom M. Mitchell, McGraw-Hill, 1997
Active Learning and Optimized Information Gathering
Active Learning and Optimized Information Gathering Lecture 7 Learning Theory CS 101.2 Andreas Krause Announcements Project proposal: Due tomorrow 1/27 Homework 1: Due Thursday 1/29 Any time is ok. Office
Computational learning theory. PAC learning. VC dimension.
Computational learning theory. PAC learning. VC dimension. Petr Pošík Czech Technical University in Prague Faculty of Electrical Engineering Dept. of Cybernetics COLT 2 Concept...........................................................................................................
Computational Learning Theory: Shattering and VC Dimensions. Machine Learning. Spring The slides are mainly from Vivek Srikumar
Computational Learning Theory: Shattering and VC Dimensions Machine Learning Spring 2018 The slides are mainly from Vivek Srikumar 1 This lecture: Computational Learning Theory The Theory of Generalization
Computational Learning Theory. CS534 - Machine Learning
Computational Learning Theory CS534 Machine Learning Introduction Computational learning theory Provides a theoretical analysis of learning Shows when a learning algorithm can be expected to succeed Shows
Introduction to Machine Learning
Introduction to Machine Learning Slides adapted from Eli Upfal Machine Learning: Jordan Boyd-Graber University of Maryland FEATURE ENGINEERING Machine Learning: Jordan Boyd-Graber UMD Introduction to Machine
Introduction to Machine Learning
Introduction to Machine Learning Machine Learning: Jordan Boyd-Graber University of Maryland RADEMACHER COMPLEXITY Slides adapted from Rob Schapire Machine Learning: Jordan Boyd-Graber UMD Introduction
EECS 349: Machine Learning Bryan Pardo
EECS 349: Machine Learning Bryan Pardo Topic: Concept Learning 1 Concept Learning Much of learning involves acquiring general concepts from specific training examples Concept: subset of objects from some
Computational Learning Theory: Probably Approximately Correct (PAC) Learning. Machine Learning. Spring The slides are mainly from Vivek Srikumar
Computational Learning Theory: Probably Approximately Correct (PAC) Learning Machine Learning Spring 2018 The slides are mainly from Vivek Srikumar 1 This lecture: Computational Learning Theory The Theory
Understanding Generalization Error: Bounds and Decompositions
CIS 520: Machine Learning Spring 2018: Lecture 11 Understanding Generalization Error: Bounds and Decompositions Lecturer: Shivani Agarwal Disclaimer: These notes are designed to be a supplement to the
Generalization and Overfitting
Generalization and Overfitting Model Selection Maria-Florina (Nina) Balcan February 24th, 2016 PAC/SLT models for Supervised Learning Data Source Distribution D on X Learning Algorithm Expert / Oracle
Learning Theory Continued
Learning Theory Continued Machine Learning CSE446 Carlos Guestrin University of Washington May 13, 2013 1 A simple setting n Classification N data points Finite number of possible hypothesis (e.g., dec.
Classification: The PAC Learning Framework
Classification: The PAC Learning Framework Machine Learning: Jordan Boyd-Graber University of Colorado Boulder LECTURE 5 Slides adapted from Eli Upfal Machine Learning: Jordan Boyd-Graber Boulder Classification:
PAC-learning, VC Dimension and Margin-based Bounds
More details: General: http://www.learning-with-kernels.org/ Example of more complex bounds: http://www.research.ibm.com/people/t/tzhang/papers/jmlr02_cover.ps.gz PAC-learning, VC Dimension and Margin-based
Lecture 25 of 42. PAC Learning, VC Dimension, and Mistake Bounds
Lecture 25 of 42 PAC Learning, VC Dimension, and Mistake Bounds Thursday, 15 March 2007 William H. Hsu, KSU http://www.kddresearch.org/courses/spring2007/cis732 Readings: Sections 7.4.17.4.3, 7.5.17.5.3,
Computational Learning Theory
CS 446 Machine Learning Fall 2016 OCT 11, 2016 Computational Learning Theory Professor: Dan Roth Scribe: Ben Zhou, C. Cervantes 1 PAC Learning We want to develop a theory to relate the probability of successful
Generalization, Overfitting, and Model Selection
Generalization, Overfitting, and Model Selection Sample Complexity Results for Supervised Classification MariaFlorina (Nina) Balcan 10/05/2016 Reminders Midterm Exam Mon, Oct. 10th Midterm Review Session
Machine Learning. Computational Learning Theory. Eric Xing , Fall Lecture 9, October 5, 2016
Machine Learning 10-701, Fall 2016 Computational Learning Theory Eric Xing Lecture 9, October 5, 2016 Reading: Chap. 7 T.M book Eric Xing @ CMU, 2006-2016 1 Generalizability of Learning In machine learning
PAC Learning Introduction to Machine Learning. Matt Gormley Lecture 14 March 5, 2018
10-601 Introduction to Machine Learning Machine Learning Department School of Computer Science Carnegie Mellon University PAC Learning Matt Gormley Lecture 14 March 5, 2018 1 ML Big Picture Learning Paradigms:
Learning Theory. Aar$ Singh and Barnabas Poczos. Machine Learning / Apr 17, Slides courtesy: Carlos Guestrin
Learning Theory Aar$ Singh and Barnabas Poczos Machine Learning 10-701/15-781 Apr 17, 2014 Slides courtesy: Carlos Guestrin Learning Theory We have explored many ways of learning from data But How good
http://imgs.xkcd.com/comics/electoral_precedent.png Statistical Learning Theory CS4780/5780 Machine Learning Fall 2012 Thorsten Joachims Cornell University Reading: Mitchell Chapter 7 (not 7.4.4 and 7.5)
THE VAPNIK- CHERVONENKIS DIMENSION and LEARNABILITY
THE VAPNIK- CHERVONENKIS DIMENSION and LEARNABILITY Dan A. Simovici UMB, Doctoral Summer School Iasi, Romania What is Machine Learning? The Vapnik-Chervonenkis Dimension Probabilistic Learning Potential
Machine Learning. Gaussian Mixture Models. Zhiyao Duan & Bryan Pardo, Machine Learning: EECS 349 Fall
Machine Learning Gaussian Mixture Models Zhiyao Duan & Bryan Pardo, Machine Learning: EECS 349 Fall 2012 1 Discriminative vs Generative Models Discriminative: Just learn a decision boundary between your
Machine Learning. Lecture 9: Learning Theory. Feng Li.
Machine Learning Lecture 9: Learning Theory Feng Li fli@sdu.edu.cn https://funglee.github.io School of Computer Science and Technology Shandong University Fall 2018 Why Learning Theory How can we tell
Introduction to Algorithms / Algorithms I Lecturer: Michael Dinitz Topic: Intro to Learning Theory Date: 12/8/16
600.463 Introduction to Algorithms / Algorithms I Lecturer: Michael Dinitz Topic: Intro to Learning Theory Date: 12/8/16 25.1 Introduction Today we re going to talk about machine learning, but from an
PAC Model and Generalization Bounds
PAC Model and Generalization Bounds Overview Probably Approximately Correct (PAC) model Basic generalization bounds finite hypothesis class infinite hypothesis class Simple case More next week 2 Motivating
Relationship between Least Squares Approximation and Maximum Likelihood Hypotheses
Relationship between Least Squares Approximation and Maximum Likelihood Hypotheses Steven Bergner, Chris Demwell Lecture notes for Cmpt 882 Machine Learning February 19, 2004 Abstract In these notes, a
CS340 Machine learning Lecture 5 Learning theory cont'd. Some slides are borrowed from Stuart Russell and Thorsten Joachims
CS340 Machine learning Lecture 5 Learning theory cont'd Some slides are borrowed from Stuart Russell and Thorsten Joachims Inductive learning Simplest form: learn a function from examples f is the target
CSE 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
Cognitive Cyber-Physical System
Cognitive Cyber-Physical System Physical to Cyber-Physical The emergence of non-trivial embedded sensor units, networked embedded systems and sensor/actuator networks has made possible the design and implementation
Learning theory Lecture 4
Learning theory Lecture 4 David Sontag New York University Slides adapted from Carlos Guestrin & Luke Zettlemoyer What s next We gave several machine learning algorithms: Perceptron Linear support vector
COMS 4771 Introduction to Machine Learning. Nakul Verma
COMS 4771 Introduction to Machine Learning Nakul Verma Announcements HW2 due now! Project proposal due on tomorrow Midterm next lecture! HW3 posted Last time Linear Regression Parametric vs Nonparametric
Machine Learning
Machine Learning 10-601 Tom M. Mitchell Machine Learning Department Carnegie Mellon University January 14, 2015 Today: The Big Picture Overfitting Review: probability Readings: Decision trees, overfiting
Introduction to Machine Learning
Introduction to Machine Learning Vapnik Chervonenkis Theory Barnabás Póczos Empirical Risk and True Risk 2 Empirical Risk Shorthand: True risk of f (deterministic): Bayes risk: Let us use the empirical
An Introduction to Statistical Theory of Learning. Nakul Verma Janelia, HHMI
An Introduction to Statistical Theory of Learning Nakul Verma Janelia, HHMI Towards formalizing learning What does it mean to learn a concept? Gain knowledge or experience of the concept. The basic process
Statistical Learning Theory: Generalization Error Bounds
Statistical Learning Theory: Generalization Error Bounds CS6780 Advanced Machine Learning Spring 2015 Thorsten Joachims Cornell University Reading: Murphy 6.5.4 Schoelkopf/Smola Chapter 5 (beginning, rest
MACHINE LEARNING. Probably Approximately Correct (PAC) Learning. Alessandro Moschitti
MACHINE LEARNING Probably Approximately Correct (PAC) Learning Alessandro Moschitti Department of Information Engineering and Computer Science University of Trento Email: moschitti@disi.unitn.it Objectives:
An Introduction to No Free Lunch Theorems
February 2, 2012 Table of Contents Induction Learning without direct observation. Generalising from data. Modelling physical phenomena. The Problem of Induction David Hume (1748) How do we know an induced
Machine Learning 4771
Machine Learning 477 Instructor: Tony Jebara Topic 5 Generalization Guarantees VC-Dimension Nearest Neighbor Classification (infinite VC dimension) Structural Risk Minimization Support Vector Machines
PAC-learning, VC Dimension and Margin-based Bounds
More details: General: http://www.learning-with-kernels.org/ Example of more complex bounds: http://www.research.ibm.com/people/t/tzhang/papers/jmlr02_cover.ps.gz PAC-learning, VC Dimension and Margin-based
Online Learning. Jordan Boyd-Graber. University of Colorado Boulder LECTURE 21. Slides adapted from Mohri
Online Learning Jordan Boyd-Graber University of Colorado Boulder LECTURE 21 Slides adapted from Mohri Jordan Boyd-Graber Boulder Online Learning 1 of 31 Motivation PAC learning: distribution fixed over
Machine 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
Learning 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
Introduction to Machine Learning
Introduction to Machine Learning Machine Learning: Jordan Boyd-Graber University of Maryland SUPPORT VECTOR MACHINES Slides adapted from Tom Mitchell, Eric Xing, and Lauren Hannah Machine Learning: Jordan
FINAL: CS 6375 (Machine Learning) Fall 2014
FINAL: CS 6375 (Machine Learning) Fall 2014 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 out of room for
Computational and Statistical Learning Theory
Computational and Statistical Learning Theory TTIC 31120 Prof. Nati Srebro Lecture 8: Boosting (and Compression Schemes) Boosting the Error If we have an efficient learning algorithm that for any distribution
Machine Learning, Midterm Exam: Spring 2008 SOLUTIONS. Q Topic Max. Score Score. 1 Short answer questions 20.
10-601 Machine Learning, Midterm Exam: Spring 2008 Please put your name on this cover sheet If you need more room to work out your answer to a question, use the back of the page and clearly mark on the
Concept Learning through General-to-Specific Ordering
0. Concept Learning through General-to-Specific Ordering Based on Machine Learning, T. Mitchell, McGRAW Hill, 1997, ch. 2 Acknowledgement: The present slides are an adaptation of slides drawn by T. Mitchell
VC dimension, Model Selection and Performance Assessment for SVM and Other Machine Learning Algorithms
03/Feb/2010 VC dimension, Model Selection and Performance Assessment for SVM and Other Machine Learning Algorithms Presented by Andriy Temko Department of Electrical and Electronic Engineering Page 2 of
[Read Ch. 5] [Recommended exercises: 5.2, 5.3, 5.4]
![[Read Ch. 5] [Recommended exercises: 5.2, 5.3, 5.4]](/thumbs/87/95366673.jpg "[Read Ch. 5] [Recommended exercises: 5.2, 5.3, 5.4]")
Evaluating Hypotheses [Read Ch. 5] [Recommended exercises: 5.2, 5.3, 5.4] Sample error, true error Condence intervals for observed hypothesis error Estimators Binomial distribution, Normal distribution,
Voting (Ensemble Methods)
1 2 Voting (Ensemble Methods) Instead of learning a single classifier, learn many weak classifiers that are good at different parts of the data Output class: (Weighted) vote of each classifier Classifiers
Support Vector Machines
Support Vector Machines Jordan Boyd-Graber University of Colorado Boulder LECTURE 7 Slides adapted from Tom Mitchell, Eric Xing, and Lauren Hannah Jordan Boyd-Graber Boulder Support Vector Machines 1 of
Machine Learning: Chenhao Tan University of Colorado Boulder LECTURE 9
Machine Learning: Chenhao Tan University of Colorado Boulder LECTURE 9 Slides adapted from Jordan Boyd-Graber Machine Learning: Chenhao Tan Boulder 1 of 39 Recap Supervised learning Previously: KNN, naïve
Homework 5 SOLUTIONS CS 6375: Machine Learning
Homework 5 SOLUTIONS CS 6375: Machine Learning Spring 2012 Due date: Wednesday, May 2, 11:59 p.m. 1 Learning Theory [40 points, 10 points each] Mitchell 7.2 Solution: ( ) a. Here, H = 1 2, 2 (n+1)(n) n
VC dimension and Model Selection
VC dimension and Model Selection Overview PAC model: review VC dimension: Definition Examples Sample: Lower bound Upper bound!!! Model Selection Introduction to Machine Learning 2 PAC model: Setting A
Machine Learning
Machine Learning 10-701 Tom M. Mitchell Machine Learning Department Carnegie Mellon University January 13, 2011 Today: The Big Picture Overfitting Review: probability Readings: Decision trees, overfiting
Dan Roth 461C, 3401 Walnut
CIS 519/419 Applied Machine Learning www.seas.upenn.edu/~cis519 Dan Roth danroth@seas.upenn.edu http://www.cis.upenn.edu/~danroth/ 461C, 3401 Walnut Slides were created by Dan Roth (for CIS519/419 at Penn
ECE 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
MACHINE LEARNING. Vapnik-Chervonenkis (VC) Dimension. Alessandro Moschitti
MACHINE LEARNING Vapnik-Chervonenkis (VC) Dimension Alessandro Moschitti Department of Information Engineering and Computer Science University of Trento Email: moschitti@disi.unitn.it Computational Learning
Stochastic Gradient Descent
Stochastic Gradient Descent Machine Learning CSE546 Carlos Guestrin University of Washington October 9, 2013 1 Logistic Regression Logistic function (or Sigmoid): Learn P(Y X) directly Assume a particular
Data Mining Prof. Pabitra Mitra Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur
Data Mining Prof. Pabitra Mitra Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Lecture 21 K - Nearest Neighbor V In this lecture we discuss; how do we evaluate the
Notes on Machine Learning for and
Notes on Machine Learning for 16.410 and 16.413 (Notes adapted from Tom Mitchell and Andrew Moore.) Learning = improving with experience Improve over task T (e.g, Classification, control tasks) with respect
CS534 Machine Learning - Spring Final Exam
CS534 Machine Learning - Spring 2013 Final Exam Name: You have 110 minutes. There are 6 questions (8 pages including cover page). If you get stuck on one question, move on to others and come back to the
Probably Approximately Correct Learning - III
Probably Approximately Correct Learning - III Prof. Dan A. Simovici UMB Prof. Dan A. Simovici (UMB) Probably Approximately Correct Learning - III 1 / 18 A property of the hypothesis space Aim : a property
CS 543 Page 1 John E. Boon, Jr.
CS 543 Machine Learning Spring 2010 Lecture 05 Evaluating Hypotheses I. Overview A. Given observed accuracy of a hypothesis over a limited sample of data, how well does this estimate its accuracy over
12.1 A Polynomial Bound on the Sample Size m for PAC Learning
67577 Intro. to Machine Learning Fall semester, 2008/9 Lecture 12: PAC III Lecturer: Amnon Shashua Scribe: Amnon Shashua 1 In this lecture will use the measure of VC dimension, which is a combinatorial
COS 511: Theoretical Machine Learning. Lecturer: Rob Schapire Lecture #5 Scribe: Allen(Zhelun) Wu February 19, ). Then: Pr[err D (h A ) > ɛ] δ
![COS 511: Theoretical Machine Learning. Lecturer: Rob Schapire Lecture #5 Scribe: Allen(Zhelun) Wu February 19, ). Then: Pr[err D (h A ) > ɛ] δ](/thumbs/91/107320871.jpg "COS 511: Theoretical Machine Learning. Lecturer: Rob Schapire Lecture #5 Scribe: Allen(Zhelun) Wu February 19, ). Then: Pr[err D (h A ) > ɛ] δ")
COS 511: Theoretical Machine Learning Lecturer: Rob Schapire Lecture #5 Scribe: Allen(Zhelun) Wu February 19, 018 Review Theorem (Occam s Razor). Say algorithm A finds a hypothesis h A H consistent with
VC-dimension for characterizing classifiers
VC-dimension for characterizing classifiers Note to other teachers and users of these slides. Andrew would be delighted if you found this source material useful in giving your own lectures. Feel free to
Computational Learning Theory for Artificial Neural Networks
Computational Learning Theory for Artificial Neural Networks Martin Anthony and Norman Biggs Department of Statistical and Mathematical Sciences, London School of Economics and Political Science, Houghton
EVALUATING MISCLASSIFICATION PROBABILITY USING EMPIRICAL RISK 1. Victor Nedel ko
94 International Journal "Information Theories & Applications" Vol13 [Raudys, 001] Raudys S, Statistical and neural classifiers, Springer, 001 [Mirenkova, 00] S V Mirenkova (edel ko) A method for prediction
Lecture Learning infinite hypothesis class via VC-dimension and Rademacher complexity;
CSCI699: Topics in Learning and Game Theory Lecture 2 Lecturer: Ilias Diakonikolas Scribes: Li Han Today we will cover the following 2 topics: 1. Learning infinite hypothesis class via VC-dimension and
BITS F464: MACHINE LEARNING
BITS F464: MACHINE LEARNING Lecture-09: Concept Learning Dr. Kamlesh Tiwari Assistant Professor Department of Computer Science and Information Systems Engineering, BITS Pilani, Rajasthan-333031 INDIA Jan