PAC Learning Introduction to Machine Learning. Matt Gormley Lecture 14 March 5, 2018
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1 Introduction to Machine Learning Machine Learning Department School of Computer Science Carnegie Mellon University PAC Learning Matt Gormley Lecture 14 March 5,
2 ML Big Picture Learning Paradigms: What data is available and when? What form of prediction? supervised learning unsupervised learning semi-supervised learning reinforcement learning active learning imitation learning domain adaptation online learning density estimation recommender systems feature learning manifold learning dimensionality reduction ensemble learning distant supervision hyperparameter optimization Theoretical Foundations: What principles guide learning? q probabilistic q information theoretic q evolutionary search q ML as optimization Problem Formulation: What is the structure of our output prediction? boolean Binary Classification categorical Multiclass Classification ordinal Ordinal Classification real Regression ordering Ranking multiple discrete Structured Prediction multiple continuous (e.g. dynamical systems) both discrete & (e.g. mixed graphical models) cont. Facets of Building ML Systems: How to build systems that are robust, efficient, adaptive, effective? 1. Data prep 2. Model selection 3. Training (optimization / search) 4. Hyperparameter tuning on validation data 5. (Blind) Assessment on test data Application Areas Key challenges? NLP, Speech, Computer Vision, Robotics, Medicine, Search Big Ideas in ML: Which are the ideas driving development of the field? inductive bias generalization / overfitting bias-variance decomposition generative vs. discriminative deep nets, graphical models PAC learning distant rewards 2
3 LEARNING THEORY 3
4 Questions For Today 1. Given a classifier with zero training error, what can we say about generalization error? (Sample Complexity, Realizable Case) 2. Given a classifier with low training error, what can we say about generalization error? (Sample Complexity, Agnostic Case) 3. Is there a theoretical justification for regularization to avoid overfitting? (Structural Risk Minimization) 4
5 PAC/SLT models for Supervised Learning PAC / SLT Model Data Source Distribution D on X Learning Algorithm Expert / Oracle Labeled Examples (x 1,c*(x 1 )),, (x m,c*(x m )) Alg.outputs h : X! Y x 1 > 5 x 6 > c* : X! Y Slide from Nina Balcan 6
6 Two Types of Error True Error (aka. expected risk) Train Error (aka. empirical risk) 7
7 PAC / SLT Model 8
8 Three Hypotheses of Interest 9
9 PAC LEARNING 10
10 Probably Approximately Correct Whiteboard: (PAC) Learning PAC Criterion Meaning of Probably Approximately Correct PAC Learnable Consistent Learner Sample Complexity 11
11 Generalization and Overfitting Whiteboard: Realizable vs. Agnostic Cases Finite vs. Infinite Hypothesis Spaces 12
12 PAC Learning 13
13 SAMPLE COMPLEXITY RESULTS 14
14 Sample Complexity Results Four Cases we care about Realizable We ll start with the finite case Agnostic 15
15 Sample Complexity Results Four Cases we care about Realizable Agnostic 16
16 Example: Conjunctions In-Class Quiz: Suppose H = class of conjunctions over x in {0,1} M If M = 10,! = 0.1, δ = 0.01, how many examples suffice? Realizable Agnostic 17
17 Sample Complexity Results Four Cases we care about Realizable Agnostic 18
18 1. Bound is inversely linear in epsilon (e.g. halving the error requires double the examples) Sample Complexity Results 2. Bound is only logarithmic in H (e.g. quadrupling the hypothesis space only requires double the examples) Four Cases we care about 1. Bound is inversely quadratic in epsilon (e.g. halving the error requires 4x the examples) 2. Bound is only logarithmic in H (i.e. same as Realizable case) Realizable Agnostic 19
19 Generalization and Overfitting Whiteboard: Sample Complexity Bounds (Agnostic Case) Corollary (Agnostic Case) Empirical Risk Minimization Structural Risk Minimization Motivation for Regularization 20
20 Sample Complexity Results Four Cases we care about Realizable Agnostic We need a new definition of complexity for a Hypothesis space for these results (see VC Dimension) 21
21 Sample Complexity Results Four Cases we care about Realizable Agnostic 22
22 VC DIMENSION 23
23 What if H is infinite? E.g., linear separators in R d E.g., thresholds on the real line w + E.g., intervals on the real line a b 24
24 Definition: Shattering, VC-dimension H[S] the set of splittings of dataset S using concepts from H. H shatters S if H S = 2 S. A set of points S is shattered by H is there are hypotheses in H that split S in all of the 2 S possible ways; i.e., all possible ways of classifying points in S are achievable using concepts in H. Definition: VC-dimension (Vapnik-Chervonenkis dimension) The VC-dimension of a hypothesis space H is the cardinality of the largest set S that can be shattered by H. If arbitrarily large finite sets can be shattered by H, then VCdim(H) = 25
25 Shattering, VC-dimension Definition: VC-dimension (Vapnik-Chervonenkis dimension) The VC-dimension of a hypothesis space H is the cardinality of the largest set S that can be shattered by H. If arbitrarily large finite sets can be shattered by H, then VCdim(H) = To show that VC-dimension is d: there exists a set of d points that can be shattered there is no set of d+1 points that can be shattered. Fact: If H is finite, then VCdim (H) log ( H ). 26
26 Shattering, VC-dimension If the VC-dimension is d, that means there exists a set of d points that can be shattered, but there is no set of d+1 points that can be shattered. E.g., H= Thresholds on the real line VCdim H = w + - E.g., H= Intervals on the real line + - VCdim H =
27 Shattering, VC-dimension If the VC-dimension is d, that means there exists a set of d points that can be shattered, but there is no set of d+1 points that can be shattered. E.g., H= Union of k intervals on the real line VCdim H = 2k VCdim H 2k A sample of size 2k shatters (treat each pair of points as a separate case of intervals) VCdim H < 2k
28 Shattering, VC-dimension E.g., H= linear separators in R 2 VCdim H 3 29
29 Shattering, VC-dimension E.g., H= linear separators in R 2 VCdim H < 4 Case 1: one point inside the triangle formed by the others. Cannot label inside point as positive and outside points as negative. Case 2: all points on the boundary (convex hull). Cannot label two diagonally as positive and other two as negative. Fact: VCdim of linear separators in R d is d+1 30
30 Sample Complexity Results Four Cases we care about Realizable Agnostic 32
31 Questions For Today 1. Given a classifier with zero training error, what can we say about generalization error? (Sample Complexity, Realizable Case) 2. Given a classifier with low training error, what can we say about generalization error? (Sample Complexity, Agnostic Case) 3. Is there a theoretical justification for regularization to avoid overfitting? (Structural Risk Minimization) 37
32 Learning Theory Objectives You should be able to Identify the properties of a learning setting and assumptions required to ensure low generalization error Distinguish true error, train error, test error Define PAC and explain what it means to be approximately correct and what occurs with high probability Apply sample complexity bounds to real-world learning examples Distinguish between a large sample and a finite sample analysis Theoretically motivate regularization 38
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