Understanding Machine Learning: From Theory to Algorithms
2.4 Exercises 41 and letmbe an integer that satisfies m≥log(|H|/δ) Then, for any labeling function,f, and for any distribution ...
42 A Gentle Start LetAbe the algorithm that returns the smallest rectangle enclosing all positive examples in the training set. ...
3 A Formal Learning Model In this chapter we define our main formal learning model – the PAC learning model and its extensions. ...
44 A Formal Learning Model to reflect. Our accuracy parameter,, allows “forgiving” the learner’s classifier for making minor er ...
3.2 A More General Learning Model 45 Learning Problems beyond Binary Classification The learning task that we have been discussi ...
46 A Formal Learning Model remains the same as before, namely, LS(h) def= |{i∈[m] :h(xi) 6 =yi}| m . GivenS, a learner can compu ...
3.2 A More General Learning Model 47 Clearly, if the realizability assumption holds, agnostic PAC learning provides the same gua ...
48 A Formal Learning Model different. We may evaluate the quality of a hypothesis function,h:X →Y, by theexpected square differe ...
3.3 Summary 49 This loss function is used in regression problems. We will later see more examples of useful instantiations of lo ...
50 A Formal Learning Model not impose any restrictions on the underlying distribution over the examples. We also generalized the ...
3.5 Exercises 51 Describe an algorithm that implements the ERM rule for learningHSingleton in the realizable setup. Show thatHS ...
52 A Formal Learning Model Hint: Use the geometric-arithmetic mean inequality. LetHbe a hypothesis class of binary classifiers. ...
3.5 Exercises 53 of the two training sets, and possibly over the nondeterministic decisions made by the learning algorithm), bot ...
4 Learning via Uniform Convergence The first formal learning model that we have discussed was the PAC model. In Chapter 2 we hav ...
4.2 Finite Classes Are Agnostic PAC Learnable 55 Proof For everyh∈H, LD(hS)≤LS(hS) + 2 ≤LS(h) + 2 ≤LD(h) + 2 + 2 =LD(h) +, ...
56 Learning via Uniform Convergence i.i.d. fromDwe have that for allh∈H,|LS(h)−LD(h)|≤. That is, Dm({S:∀h∈H,|LS(h)−LD(h)|≤})≥ ...
4.2 Finite Classes Are Agnostic PAC Learnable 57 further assume that the range of`is [0,1] and thereforeθi∈[0,1]. We therefore o ...
58 Learning via Uniform Convergence classes is bounded by^128 d+2 log(2 2 /δ). This upper bound on the sample complex- ity has ...
4.5 Exercises 59 (whereES∼Dmdenotes the expectation over samplesSof sizem). 2.Bounded loss functions:In Corollary 4.6 we assumed ...
5 The Bias-Complexity Tradeoff In Chapter 2 we saw that unless one is careful, the training data can mislead the learner, and re ...
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