Computational Learning Theory (VC Dimension)
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1 Computational Learning Theory (VC Dimension) 1 Difficulty of machine learning problems 2 Capabilities of machine learning algorithms 1
2 Version Space with associated errors error is the true error, r is the training error Hypothesis Space H error=3 r = 1 error=1 r = 2 VS H,D error=1 r = 0 error=2 r = 0 error=3 r = 4 error=2 r = 3 2
3 Number of training samples required m 1 (ln ε H + ln(1/ δ ) With probability 1 δ, every hypothesis in H having zero training error will have a true error of at most ε Sample complexity for PAC learning grows as the logarithm of the size of the hypothesis space 3
4 Disadvantage of Sample Complexity for finite hypothesis spaces Bound in terms of H has two disadvantages Weak bounds δ H e εm Bound grows linearly with H For large H, δ > 1, or Probability at least (1-δ) is negative! 1 m (ln H + ln(1/ δ ) ε Can overestimate number of samples required Cannot be applied in case of infinite hypotheses 4
5 Sample Complexity for infinite hypothesis spaces Another measure of the complexity of H called Vapnik-Chervonenkis dimension, or VC(H) We will use VC(H) instead of H Results in tighter bounds Allows characterizing sample complexity of infinite hypothesis spaces and is fairly tight 5
6 VC Dimension VC Dimension is a property of a set of functions { f (α) } It can be defined for various classes of functions Bounds that relate the capacity of a learning machine and its performance 6
7 Shattering a Set of Instances Complexity of hypothesis space is measured not by no of distinct hypotheses H but by no of distinct instances from X that can be completely discriminated using H Definition: A set of instances S is shattered by hypothesis space H if and only if for every dichotomy of S there exists some hypothesis in H consistent with this dichotomy 7
8 Shattering a Set of A set of three instances by eight hypotheses Instance Space X 8
9 Shattering a Set of Instances S is a subset of X ( three instances in example below ) into two subsets Hypothesis h Instance Space X Ability of H to shatter a set of instances is its capacity to represent target concepts defined over these instances 9
10 Vapnik-Chervonenkis Dimension Definition: 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 of X can be shattered by H, then VC(H) = infinity 10
11 VC Dimension 11
12 Examples to illustrate VC Dimension 1 Instance space: X = set of real numbers R H is the set of intervals on the real number line 2 Instance space: X = points on the x-y plane H is the set of all linear decision surfaces in the plane 3 Instance space: X = three Boolean literals H is conjunction a b c 12
13 Example 1: VC dimension of 1-dimensional intervals X = R (eg, heights of people) H is the set of hypotheses of the form a < x <b Subset containing two instances S ={31, 57} Can S be shattered by H? Yes, eg, (1<x<2), (1<x<4),(4<x<7),(1<x<7) Since we have found a set of two that can be shattered, VC(H) is at least two However, no subset of size three can be shattered Therefore VC(H) =2 Here H is infinite but VC(H) is finite 13
14 Example 2: VC Dimension of linear discriminants on a plane
15 VC Dimension of linear discriminants on a plane
16 VC Dimension of linear discriminants on a plane
17 VC Dimension of linear discriminants on a plane
18 VC Dimension of linear discriminants on a plane
19 VC Dimension of linear discriminants on a plane
20 VC Dimension of linear discriminants on a plane
21 VC Dimension of linear discriminants on a plane
22 VC Dimension of points in 2-d space and perceptron 22
23 VC Dimension of single perceptron with 2 input units (or points are in 2-d space) VC(H) is at least 3 since 3 non-collinear points can be shattered It is not 4 since we cannot find a set of four points that can be shattered Therefore VC(H)= 3 More generally, VC dimension of linear decision surfaces in r-dimensional space is r+1 23
24 24
25 Capacity of a hyperplane Fraction of dichotomies of n points in d dimensions that are linearly separable f (n,d) = 2 n d 1LLL n d + 1 i= 0 n 1 LLn i LL > d + 1 At n=2(d+1), called the capacity of the hyperplane nearly one half of the dichotomies are still linearly separable Hyperplane is not overdetermined Until number of samples is several times the dimensionality 25
26 Example 3: VC Dimension of three Boolean literals and each hypothesis in H is conjunction instance 1 = 100 instance 2 = 010 instance 3 = 001 Exclude instance i: ~ l i Example: include instance 2 but exclude instances 1 and 3: use hypothesis ~ l ^ ~ l 1 3 VC dimension is at least 3 VC dimension of conjunction of n Boolean literals is exactly n (proof is more difficult) 26
27 Sample Complexity and the VC dimension How many randomly drawn samples are sufficient to PAC-learn any target concept C? Using VC(H) as the measure of complexity of H 1 (4log2(2 / δ ) + 8VC(H) log (13/ ε ) ε m 2 Analogous to the finite hypothesis case: m 1 (ln ε H + ln(1/ δ ) 27
28 Lower bound on sample complexity If C is a concept class such that VC( C) >2 then observing fewer than 1 VC( C) 1 max log(1/ δ ), ε 32ε samples, then L outputs a hypothesis h having error D (h)>e 1 Provides number of samples necessary to PAC learn 2 Defined in terms of concept class C rather than H 28
29 VC Dimension for Neural Networks G-composition of C is the class of all functions that can be implemented on the network G, ie, it is the hypothesis space representable by network G For acyclic layered networks containing s perceptrons, each with r inputs perceptrons VC( C ) 2( r + 1) s log( es) G 29
30 VC Dimension for Neural Networks Bound on number of samples sufficient to learn with probability at least (1-d) any target concept from Cg to within error e 1 m (4log2(2 / δ ) + 16( r + 1) s log( es)log (13/ ε) ε Does not apply to Backprop since the units are not sigmoid VC dimension of sigmoid units will at least as great as that of perceptrons 30
31 VC Dimension of SVMs 31
32 32
33 33
34 MISTAKE BOUND MODEL OF LEARNING How many mistakes will learner make in its predictions before it learns the target concept Significant in practical settings where learning must be done while the system is in actual use, rather than in an off-line training stage Example, system to learn to approve credit card purchases based on data collected during use 34
35 Mistake bound for Find-S algorithm Find-S Initialize h to the most specific hypothesis l 1 l 1 l 2 l 2 L l n l n For each positive training instance x Remove from h any literal that is not satisfied by x Output hypothesis h Total no of mistakes can be at most n+1 35
36 Mistake bound for Halving Algorithm Candidate Elimination Algorithm maintains a description of the version space, incrementally refining the version space as each new sample is encountered Assume majority vote is used among all hypotheses in the current version space Total no of mistakes can be at most log 2 H 36
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