The PAC Learning Framework -II
|
|
- Candace Booth
- 6 years ago
- Views:
Transcription
1 The PAC Learning Framework -II Prof. Dan A. Simovici UMB 1 / 1
2 Outline 1 Finite Hypothesis Space - The Inconsistent Case 2 Deterministic versus stochastic scenario 3 Bayes Error and Noise 2 / 1
3 Outline Universal Concept Class Let X = {0, 1} n and let U n = P(X be the concept class formed by all subsets of X. To guarantee a consistent hypothesis, the hypothesis class must include the concept class, so H U n = 2 n. We have m 1 ( 2 n log 2 + log 1, ɛ δ The number of example is exponential in n by the theorem, so PAC learnability does not follow. 3 / 1
4 Finite Hypothesis Space - The Inconsistent Case Framework If the concept class is more complex than the hypotheses space it may be the case that there is no hypothesis consistent with a labeled training sample, that is, for no h S we would have ˆR(h S = 0. We use the corollary of Hoeffding s Inequality: Corollary Let X 1,..., X n be n independent random variables such that X i [0, 1] for 1 i n and let Z n be the random variable defined by: Z n = 1 n n X i. i=1 The following inequalities hold: P(Z n E(Z n ɛ e 2nɛ2 P(Z n E(Z n ɛ e 2nɛ2. 4 / 1
5 Finite Hypothesis Space - The Inconsistent Case Framework (cont d Recall that R(h = E( ˆR(h. The Corolarry of Hoeffding s Inequality applied to ˆR(h = 1 m {x i h(x i c(x i } m i=1 implies that for ɛ > 0, any S = (x 1,..., x m size n and any hypothesis h : X {0, 1} the following inequalities hold: ( P ˆR(h R(h ɛ e 2mɛ2 ( P ˆR(h R(h ɛ e 2mɛ2 Therefore, and P P ( ˆR(h R(h ɛ 2e 2mɛ2. ( ˆR(h R(h < ɛ 1 2e 2mɛ2. (1 5 / 1
6 Finite Hypothesis Space - The Inconsistent Case Generalization Bound - Single Hypothesis Corollary For a random hypothesis h : X {0, 1} and for any δ > 0 the following inequality log R(h ˆR(h 2 δ 2m holds with probability at least 1 δ. Proof: Taking 1 2 2mɛ2 1 δ, or δ 2e 2mɛ2 in Equality (1 we obtain ( P ˆR(h R(h < ɛ 1 δ, log for ɛ = 2 δ 2m. Note: The inequality of the corollary is an inequality involving random variables not numbers because h is a randomly chosen hypothesis in H 6 / 1
7 Finite Hypothesis Space - The Inconsistent Case Tossing a Coin Example Let p be the probability that a biased coin that lands heads. Let h be the hypothesis be the one that always guesses tails. The generalization error rate is R(h = p and let ˆR(h = ˆp, where ˆp is the empirical probability of heads based on the training sample drawn iid. Thus, with a probability of at least 1 δ we have log 2 δ ˆp p 2m. 7 / 1
8 Finite Hypothesis Space - The Inconsistent Case Learning bound finite H, inconsistent case Theorem Let H be a finite hypothesis set. For any δ > 0, the inequality log H + log ( h H R(h ˆR(h 2 δ 2m holds with probability at least 1 δ. Remark: this is a uniform bound (it applies to all hypotheses in H. 8 / 1
9 Finite Hypothesis Space - The Inconsistent Case Proof Let H = {h 1,..., h H } be the set of hypotheses. We have: ( P ( h H R(h ˆR(h > ɛ ( = P ( R(h 1 ˆR(h 1 > ɛ ( R(h H ˆR(h H > ɛ ( P R(h ˆR(h > ɛ h H 2 H e 2mɛ2. 9 / 1
10 Finite Hypothesis Space - The Inconsistent Case Proof (cont d Thus, we have ( P ( h H R(h ˆR(h > ɛ 2 H e 2mɛ2. Choosing δ = 2 H e 2mɛ2 it follows that log δ = log 2 + log H 2mɛ 2, so log 2+log H log δ log H +log ɛ = 2m = 2 δ 2m. With these choices we have: ( P ( h H R(h ˆR(h > ɛ δ, which amounts to the inequality of the theorem: ( P ( h H R(h ˆR(h < ɛ 1 δ. 10 / 1
11 Finite Hypothesis Space - The Inconsistent Case Previous theorem stipulates that for a finite hypothesis set H, we have ( log2 H R(h ˆR(h + O m Note that log 2 H is the number of bits needed to represent H ; this point to Occam s principle: a smaller hypothesis space size is better; a larger sample size m guarantees better generalization; for the inconsistent size, a larger sample size is required to obtain the same guarantee as in the consistent case (R(h S 1 ɛ (log H + log 1 δ. 11 / 1
12 Deterministic versus stochastic scenario The Stochastic Scenario Example the distribution D is defined now on X Y (in the deterministic scenario it was defined just on X ; the training data is a sample S = {(x 1, y 1,..., (x m, y m }, where (x i, y i are iid random variables; the output label y i is a probabilistic function of the input. If we try to predict the gender of a person based on weight and height, the result (male, or female is not unique. 12 / 1
13 Deterministic versus stochastic scenario Agnostic PAC-algorithms Definition Let H be a hypothesis set. An algorithm A is an agnostic PAC-algorithm if there exists a polynomial function such that of any ɛ > 0 and δ > 0 we have ( P R(h S min R(h < ɛ 1 δ h H for every sample of size ( 1 m ɛ, 1 δ and for all probability distributions D over X Y. If A runs in time polynomial in 1 ɛ, 1 δ, then A is an efficient agnostic PAC-algorithm. 13 / 1
14 Bayes Error and Noise Definition Given a distribution D over X Y, the Bayes error R is R = inf{r(h his measurable }. A hypothesis h such that R(h = R is called a Bayes hypothesis and denoted by h Bayes. 14 / 1
15 Bayes Error and Noise in the deterministic case R = 0; in the stochastic case we may have R 0; using conditional probabilities the Bayes hypothesis can be defined by ( xh Bayes (x = argmax y {0,1} P(y x, which means that the class y is the most probable class a posteriori, that is, after seeing the data x; the average error made by h Bayes on x is min{p(1 x, P(0 x}. 15 / 1
16 Bayes Error and Noise Definition Given a distribution D the noise at x is noise(x = min{p(1 x, P(0 x}. The average noise at x is E(noise(x. The average noise is the Bayes error: E(noise(x = R. The noise indicates the level of difficulty of the learning task. A point x X with noise(x = 0.5 is said to be noisy. 16 / 1
17 Bayes Error and Noise Estimation and Approximation Errors R(h the error of hypothesis h; R = inf{r(h his measurable } is the Bayes error; h is the hypothesis in H with minimal error (best in class hypothesis. It always exists when H if finite; if this is not the case, instead of R(h we can use inf h H R(h. By definition, R(h R(h R. 17 / 1
18 Bayes Error and Noise Since R(h R(h R, we can define: estimation error: R(h R(h ; it depends on the hypothesis h selected; approximation error: R(h R : it measures how well the Bayes error can be approximated using H. R(h R = (R(h R(h + (R(h R. 18 / 1
19 Bayes Error and Noise Empirical Risk Minimization ERM Definition An algorithm that returns a hypothesis hs ERM error ˆR(h is said to be an ERM algorithm. with the smallest empirical We have R(h ERM S R(h = (R(h ERM ˆR(h S ERM + ( ˆR(h S ERM R(h (R(h ERM ˆR(h S ERM + ( ˆR(h R(h 2 sup ˆR(h R(h. h H Note that: Since h is the hypothesis in H with minimal error (best in class hypothesis, R(h decreases with H. log R(h ˆR(h 2 δ 2m and increases with H. 19 / 1
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
More informationLecture 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
More informationPAC Learning. prof. dr Arno Siebes. Algorithmic Data Analysis Group Department of Information and Computing Sciences Universiteit Utrecht
PAC Learning prof. dr Arno Siebes Algorithmic Data Analysis Group Department of Information and Computing Sciences Universiteit Utrecht Recall: PAC Learning (Version 1) A hypothesis class H is PAC learnable
More informationProbably 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
More informationMachine 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
More information1 The Probably Approximately Correct (PAC) Model
COS 511: Theoretical Machine Learning Lecturer: Rob Schapire Lecture #3 Scribe: Yuhui Luo February 11, 2008 1 The Probably Approximately Correct (PAC) Model A target concept class C is PAC-learnable by
More informationComputational and Statistical Learning theory
Computational and Statistical Learning theory Problem set 2 Due: January 31st Email solutions to : karthik at ttic dot edu Notation : Input space : X Label space : Y = {±1} Sample : (x 1, y 1,..., (x n,
More informationEmpirical Risk Minimization
Empirical Risk Minimization Fabrice Rossi SAMM Université Paris 1 Panthéon Sorbonne 2018 Outline Introduction PAC learning ERM in practice 2 General setting Data X the input space and Y the output space
More information1 A Lower Bound on Sample Complexity
COS 511: Theoretical Machine Learning Lecturer: Rob Schapire Lecture #7 Scribe: Chee Wei Tan February 25, 2008 1 A Lower Bound on Sample Complexity In the last lecture, we stopped at the lower bound on
More informationThe Vapnik-Chervonenkis Dimension
The Vapnik-Chervonenkis Dimension Prof. Dan A. Simovici UMB 1 / 91 Outline 1 Growth Functions 2 Basic Definitions for Vapnik-Chervonenkis Dimension 3 The Sauer-Shelah Theorem 4 The Link between VCD and
More informationMachine 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
More informationIntroduction to Bayesian Learning. Machine Learning Fall 2018
Introduction to Bayesian Learning Machine Learning Fall 2018 1 What we have seen so far What does it mean to learn? Mistake-driven learning Learning by counting (and bounding) number of mistakes PAC learnability
More informationIntroduction to Machine Learning (67577) Lecture 5
Introduction to Machine Learning (67577) Lecture 5 Shai Shalev-Shwartz School of CS and Engineering, The Hebrew University of Jerusalem Nonuniform learning, MDL, SRM, Decision Trees, Nearest Neighbor Shai
More informationIFT Lecture 7 Elements of statistical learning theory
IFT 6085 - Lecture 7 Elements of statistical learning theory This version of the notes has not yet been thoroughly checked. Please report any bugs to the scribes or instructor. Scribe(s): Brady Neal and
More informationGeneralization bounds
Advanced Course in Machine Learning pring 200 Generalization bounds Handouts are jointly prepared by hie Mannor and hai halev-hwartz he problem of characterizing learnability is the most basic question
More information1 Differential Privacy and Statistical Query Learning
10-806 Foundations of Machine Learning and Data Science Lecturer: Maria-Florina Balcan Lecture 5: December 07, 015 1 Differential Privacy and Statistical Query Learning 1.1 Differential Privacy Suppose
More informationThe sample complexity of agnostic learning with deterministic labels
The sample complexity of agnostic learning with deterministic labels Shai Ben-David Cheriton School of Computer Science University of Waterloo Waterloo, ON, N2L 3G CANADA shai@uwaterloo.ca Ruth Urner College
More informationUnderstanding 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
More information1 Review of The Learning Setting
COS 5: Theoretical Machine Learning Lecturer: Rob Schapire Lecture #8 Scribe: Changyan Wang February 28, 208 Review of The Learning Setting Last class, we moved beyond the PAC model: in the PAC model we
More informationORIE 4741: Learning with Big Messy Data. Generalization
ORIE 4741: Learning with Big Messy Data Generalization Professor Udell Operations Research and Information Engineering Cornell September 23, 2017 1 / 21 Announcements midterm 10/5 makeup exam 10/2, by
More informationIntroduction to Statistical Learning Theory
Introduction to Statistical Learning Theory Definition Reminder: We are given m samples {(x i, y i )} m i=1 Dm and a hypothesis space H and we wish to return h H minimizing L D (h) = E[l(h(x), y)]. Problem
More informationFORMULATION OF THE LEARNING PROBLEM
FORMULTION OF THE LERNING PROBLEM MIM RGINSKY Now that we have seen an informal statement of the learning problem, as well as acquired some technical tools in the form of concentration inequalities, we
More informationSupervised Machine Learning (Spring 2014) Homework 2, sample solutions
58669 Supervised Machine Learning (Spring 014) Homework, sample solutions Credit for the solutions goes to mainly to Panu Luosto and Joonas Paalasmaa, with some additional contributions by Jyrki Kivinen
More informationIntroduction to Machine Learning (67577) Lecture 3
Introduction to Machine Learning (67577) Lecture 3 Shai Shalev-Shwartz School of CS and Engineering, The Hebrew University of Jerusalem General Learning Model and Bias-Complexity tradeoff Shai Shalev-Shwartz
More informationComputational 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
More informationMachine 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
More informationComputational and Statistical Learning Theory
Computational and Statistical Learning Theory Problem set 1 Due: Monday, October 10th Please send your solutions to learning-submissions@ttic.edu Notation: Input space: X Label space: Y = {±1} Sample:
More informationComputational 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
More information12.1 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
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 informationStatistical Learning Learning From Examples
Statistical Learning Learning From Examples We want to estimate the working temperature range of an iphone. We could study the physics and chemistry that affect the performance of the phone too hard We
More informationClassification: 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:
More information10.1 The Formal Model
67577 Intro. to Machine Learning Fall semester, 2008/9 Lecture 10: The Formal (PAC) Learning Model Lecturer: Amnon Shashua Scribe: Amnon Shashua 1 We have see so far algorithms that explicitly estimate
More informationCOMS 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
More informationFoundations of Machine Learning
Introduction to ML Mehryar Mohri Courant Institute and Google Research mohri@cims.nyu.edu page 1 Logistics Prerequisites: basics in linear algebra, probability, and analysis of algorithms. Workload: about
More informationIntroduction 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
More informationMachine 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
More informationGeneralization Bounds and Stability
Generalization Bounds and Stability Lorenzo Rosasco Tomaso Poggio 9.520 Class 9 2009 About this class Goal To recall the notion of generalization bounds and show how they can be derived from a stability
More informationTHE 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
More informationComputational 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
More informationNaïve Bayes classification. p ij 11/15/16. Probability theory. Probability theory. Probability theory. X P (X = x i )=1 i. Marginal Probability
Probability theory Naïve Bayes classification Random variable: a variable whose possible values are numerical outcomes of a random phenomenon. s: A person s height, the outcome of a coin toss Distinguish
More informationAdvanced Introduction to Machine Learning CMU-10715
Advanced Introduction to Machine Learning CMU-10715 Risk Minimization Barnabás Póczos What have we seen so far? Several classification & regression algorithms seem to work fine on training datasets: Linear
More informationAn 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
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 informationComputational 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
More informationIntroduction 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
More informationLearning with Rejection
Learning with Rejection Corinna Cortes 1, Giulia DeSalvo 2, and Mehryar Mohri 2,1 1 Google Research, 111 8th Avenue, New York, NY 2 Courant Institute of Mathematical Sciences, 251 Mercer Street, New York,
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 informationEmpirical Risk Minimization, Model Selection, and Model Assessment
Empirical Risk Minimization, Model Selection, and Model Assessment CS6780 Advanced Machine Learning Spring 2015 Thorsten Joachims Cornell University Reading: Murphy 5.7-5.7.2.4, 6.5-6.5.3.1 Dietterich,
More informationComputational 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
More informationCS340 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
More informationComputational 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
More informationComputational and Statistical Learning Theory
Computational and Statistical Learning Theory TTIC 31120 Prof. Nati Srebro Lecture 4: MDL and PAC-Bayes Uniform vs Non-Uniform Bias No Free Lunch: we need some inductive bias Limiting attention to hypothesis
More informationhttp://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)
More informationCOS 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
More informationQuantitative Introduction ro Risk and Uncertainty in Business Module 5: Hypothesis Testing
Quantitative Introduction ro Risk and Uncertainty in Business Module 5: Hypothesis Testing M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu October
More informationDan 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
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 informationFoundations of Machine Learning and Data Science. Lecturer: Avrim Blum Lecture 9: October 7, 2015
10-806 Foundations of Machine Learning and Data Science Lecturer: Avrim Blum Lecture 9: October 7, 2015 1 Computational Hardness of Learning Today we will talk about some computational hardness results
More informationMachine 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
More informationGeneralization, 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
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 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 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 informationLecture : Probabilistic Machine Learning
Lecture : Probabilistic Machine Learning Riashat Islam Reasoning and Learning Lab McGill University September 11, 2018 ML : Many Methods with Many Links Modelling Views of Machine Learning Machine Learning
More informationExpectation maximization tutorial
Expectation maximization tutorial Octavian Ganea November 18, 2016 1/1 Today Expectation - maximization algorithm Topic modelling 2/1 ML & MAP Observed data: X = {x 1, x 2... x N } 3/1 ML & MAP Observed
More informationStatistical learning. Chapter 20, Sections 1 3 1
Statistical learning Chapter 20, Sections 1 3 Chapter 20, Sections 1 3 1 Outline Bayesian learning Maximum a posteriori and maximum likelihood learning Bayes net learning ML parameter learning with complete
More information12. Structural Risk Minimization. ECE 830 & CS 761, Spring 2016
12. Structural Risk Minimization ECE 830 & CS 761, Spring 2016 1 / 23 General setup for statistical learning theory We observe training examples {x i, y i } n i=1 x i = features X y i = labels / responses
More informationStatistical learning. Chapter 20, Sections 1 4 1
Statistical learning Chapter 20, Sections 1 4 Chapter 20, Sections 1 4 1 Outline Bayesian learning Maximum a posteriori and maximum likelihood learning Bayes net learning ML parameter learning with complete
More informationComputational 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...........................................................................................................
More informationMachine Learning
Machine Learning 10-601 Tom M. Mitchell Machine Learning Department Carnegie Mellon University August 30, 2017 Today: Decision trees Overfitting The Big Picture Coming soon Probabilistic learning MLE,
More informationComputational 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
More informationMachine Learning Theory (CS 6783)
Machine Learning Theory (CS 6783) Tu-Th 1:25 to 2:40 PM Hollister, 306 Instructor : Karthik Sridharan ABOUT THE COURSE No exams! 5 assignments that count towards your grades (55%) One term project (40%)
More informationMachine Learning Theory (CS 6783)
Machine Learning Theory (CS 6783) Tu-Th 1:25 to 2:40 PM Kimball, B-11 Instructor : Karthik Sridharan ABOUT THE COURSE No exams! 5 assignments that count towards your grades (55%) One term project (40%)
More informationGeneralization Bounds
Generalization Bounds Here we consider the problem of learning from binary labels. We assume training data D = x 1, y 1,... x N, y N with y t being one of the two values 1 or 1. We will assume that these
More informationGeneralization Bounds in Machine Learning. Presented by: Afshin Rostamizadeh
Generalization Bounds in Machine Learning Presented by: Afshin Rostamizadeh Outline Introduction to generalization bounds. Examples: VC-bounds Covering Number bounds Rademacher bounds Stability bounds
More informationMachine 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
More informationCOMP9444: 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
More informationoutline Nonlinear transformation Error measures Noisy targets Preambles to the theory
Error and Noise outline Nonlinear transformation Error measures Noisy targets Preambles to the theory Linear is limited Data Hypothesis Linear in what? Linear regression implements Linear classification
More informationMathematical Foundations of Supervised Learning
Mathematical Foundations of Supervised Learning (growing lecture notes) Michael M. Wolf November 26, 2017 Contents Introduction 5 1 Learning Theory 7 1.1 Statistical framework.........................
More informationIntroduction 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
More informationDomain Adaptation Can Quantity Compensate for Quality?
Domain Adaptation Can Quantity Compensate for Quality? hai Ben-David David R. Cheriton chool of Computer cience University of Waterloo Waterloo, ON N2L 3G1 CANADA shai@cs.uwaterloo.ca hai halev-hwartz
More informationActive Learning: Disagreement Coefficient
Advanced Course in Machine Learning Spring 2010 Active Learning: Disagreement Coefficient Handouts are jointly prepared by Shie Mannor and Shai Shalev-Shwartz In previous lectures we saw examples in which
More informationLecture 35: December The fundamental statistical distances
36-705: Intermediate Statistics Fall 207 Lecturer: Siva Balakrishnan Lecture 35: December 4 Today we will discuss distances and metrics between distributions that are useful in statistics. I will be lose
More informationStatistical 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
More informationLearning Theory. Sridhar Mahadevan. University of Massachusetts. p. 1/38
Learning Theory Sridhar Mahadevan mahadeva@cs.umass.edu University of Massachusetts p. 1/38 Topics Probability theory meet machine learning Concentration inequalities: Chebyshev, Chernoff, Hoeffding, and
More informationIntroduction to Machine Learning CMU-10701
Introduction to Machine Learning CMU10701 11. Learning Theory Barnabás Póczos Learning Theory We have explored many ways of learning from data But How good is our classifier, really? How much data do we
More informationCSCE 478/878 Lecture 6: Bayesian Learning
Bayesian Methods Not all hypotheses are created equal (even if they are all consistent with the training data) Outline CSCE 478/878 Lecture 6: Bayesian Learning Stephen D. Scott (Adapted from Tom Mitchell
More informationMODULE -4 BAYEIAN LEARNING
MODULE -4 BAYEIAN LEARNING CONTENT Introduction Bayes theorem Bayes theorem and concept learning Maximum likelihood and Least Squared Error Hypothesis Maximum likelihood Hypotheses for predicting probabilities
More informationThe Learning Problem and Regularization
9.520 Class 02 February 2011 Computational Learning Statistical Learning Theory Learning is viewed as a generalization/inference problem from usually small sets of high dimensional, noisy data. Learning
More informationComputational 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
More informationMachine Learning. Regularization and Feature Selection. Fabio Vandin November 13, 2017
Machine Learning Regularization and Feature Selection Fabio Vandin November 13, 2017 1 Learning Model A: learning algorithm for a machine learning task S: m i.i.d. pairs z i = (x i, y i ), i = 1,..., m,
More informationCS 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
More informationIntroduction to Statistical Learning Theory
Introduction to Statistical Learning Theory In the last unit we looked at regularization - adding a w 2 penalty. We add a bias - we prefer classifiers with low norm. How to incorporate more complicated
More informationMACHINE LEARNING - CS671 - Part 2a The Vapnik-Chervonenkis Dimension
MACHINE LEARNING - CS671 - Part 2a The Vapnik-Chervonenkis Dimension Prof. Dan A. Simovici UMB Prof. Dan A. Simovici (UMB) MACHINE LEARNING - CS671 - Part 2a The Vapnik-Chervonenkis Dimension 1 / 30 The
More informationIntroduction: MLE, MAP, Bayesian reasoning (28/8/13)
STA561: Probabilistic machine learning Introduction: MLE, MAP, Bayesian reasoning (28/8/13) Lecturer: Barbara Engelhardt Scribes: K. Ulrich, J. Subramanian, N. Raval, J. O Hollaren 1 Classifiers In this
More informationLecture 7 Introduction to Statistical Decision Theory
Lecture 7 Introduction to Statistical Decision Theory I-Hsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw December 20, 2016 1 / 55 I-Hsiang Wang IT Lecture 7
More informationBayesian Machine Learning
Bayesian Machine Learning Andrew Gordon Wilson ORIE 6741 Lecture 2: Bayesian Basics https://people.orie.cornell.edu/andrew/orie6741 Cornell University August 25, 2016 1 / 17 Canonical Machine Learning
More informationLecture 9: Bayesian Learning
Lecture 9: Bayesian Learning Cognitive Systems II - Machine Learning Part II: Special Aspects of Concept Learning Bayes Theorem, MAL / ML hypotheses, Brute-force MAP LEARNING, MDL principle, Bayes Optimal
More informationClassification objectives COMS 4771
Classification objectives COMS 4771 1. Recap: binary classification Scoring functions Consider binary classification problems with Y = { 1, +1}. 1 / 22 Scoring functions Consider binary classification
More information