Understanding Machine Learning: From Theory to Algorithms
6.8 Exercises 81 Prove that VCdim (∪ri=1Hi)≤ 4 dlog(2d) + 2 log(r). Hint: Take a set ofkexamples and assume that they are shat ...
82 The VC-Dimension where, forx= (x 1 ....,xn),hp(x) = (^1) [p(x)≥0](the degree of a multi- variable polynomial is the maximal s ...
7 Nonuniform Learnability The notions of PAC learnability discussed so far in the book allow the sample sizes to depend on the a ...
84 Nonuniform Learnability with a low risk compared to the minimal risk achieved by hypotheses in our class (in the agnostic cas ...
7.2 Structural Risk Minimization 85 theorem7.3 LetHbe a hypothesis class that can be written as a countable union of hypothesis ...
86 Nonuniform Learnability Concretely, letHbe a hypothesis class that can be written asH= ⋃ n∈NHn. For example,Hmay be the class ...
7.2 Structural Risk Minimization 87 1 −δit holds that ∀h∈H, LD(h)≤LS(h) + minn:h∈H n n(m,w(n)·δ). (7.3) Proof For eachndefineδn ...
88 Nonuniform Learnability Proof LetAbe the SRM algorithm with respect to the weighting functionw. For every∑ h∈ H,, andδ, letm ...
7.3 Minimum Description Length and Occam’s Razor 89 the index of the first class in whichhresides. That cost increases with the ...
90 Nonuniform Learnability lemma7.6 (Kraft Inequality) IfS ⊆{ 0 , 1 }∗is a prefix-free set of strings, then ∑ σ∈S 1 2 |σ| ≤ 1. P ...
7.3 Minimum Description Length and Occam’s Razor 91 binary string obtained by running the gzip command on the program (this yiel ...
92 Nonuniform Learnability 7.4 Other Notions of Learnability – Consistency The notion of learnability can be further relaxed by ...
7.5 Discussing the Different Notions of Learnability 93 which led to overfitting, is in fact theMemorizealgorithm. In the next s ...
94 Nonuniform Learnability error term, we do not know how many more examples are needed to make the estimation error small. How ...
7.5 Discussing the Different Notions of Learnability 95 advance how many examples are needed to compete with the best hypothesis ...
96 Nonuniform Learnability will get a sample ofmi.i.d. training examples, labeled byh?, thenAis likely to return a classifier wi ...
7.7 Bibliographic Remarks 97 7.7 Bibliographic Remarks Our definition of nonuniform learnability is related to the definition of ...
98 Nonuniform Learnability Prove a bound onLD(hS)−LD(h∗B) in terms ofB, the confidence parameter δ, and the size of the training ...
7.8 Exercises 99 Prove that for everyη >0, ifnis such thatD({xi})< ηfor alli > n, then for everym∈N, P S∼Dm [∃xi: (D( ...
8 The Runtime of Learning So far in the book we have studied the statistical perspective of learning, namely, how many samples a ...
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