New PDF release: Algorithmic Learning Theory: 11th International Conference,

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By William W. Cohen (auth.), Hiroki Arimura, Sanjay Jain, Arun Sharma (eds.)

ISBN-10: 3540409920

ISBN-13: 9783540409922

ISBN-10: 3540412379

ISBN-13: 9783540412373

This e-book constitutes the refereed court cases of the eleventh overseas convention on Algorithmic studying concept, ALT 2000, held in Sydney, Australia in December 2000.
The 22 revised complete papers provided including 3 invited papers have been conscientiously reviewed and chosen from 39 submissions. The papers are geared up in topical sections on statistical studying, inductive common sense programming, inductive inference, complexity, neural networks and different paradigms, help vector machines.

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Additional info for Algorithmic Learning Theory: 11th International Conference, ALT 2000 Sydney, Australia, December 11–13, 2000 Proceedings

Example text

Though it is simplified, the adaptive sampling part is essentially the same as the original 34 Osamu Watanabe Adaptive Sampling begin m ← 0; n ← 0; while m < A do get x uniformly at random from D; m ← m + B(x); n ← n + 1; output m/n as an approximation of pB ; end. Fig. 2. Adaptive Sampling one. As we can see, the structure of the algorithm is simple. It runs until it sees more than A examples x with B(x) = 1. To complete the description of the algorithm, we have to specify the way to determine A.

Now these two bounds are stated as follows. ) Theorem 1. (The Hoeffding Bound) For any , 0 < < 1, we have the following relations. Pr X >p+ n ≤ exp(−2n 2 ), Pr X (1 + ε)p ≤ exp − n 3 , Pr X pnε2 < (1 − ε)p ≤ exp − n 2 . By using these bounds, we calculate “safe” sample size, the number n of examples, so that Batch Sampling satisfies our approximation goals. , bounding the absolute estimation error.

Here we explain the Hoeffding bound [13] and the Chernoff bound [2] that have been used in computer science. , smaller) sample size. But the Cen- 30 Osamu Watanabe tral Limit Theorem holds only asymptotically, and furthermore, the difference is within a constant factor. ) For explaining these bounds, let us prepare some notations. , Xn be independent trials, which are called Bernoulli trials, such that, for 1 ≤ i ≤ n, we have Pr[Xi = 1] = p and Pr[Xi = 0] = 1 − p for some p, 0 < p < 1. Let X be a random variable defined by X = ni=1 Xi .

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Algorithmic Learning Theory: 11th International Conference, ALT 2000 Sydney, Australia, December 11–13, 2000 Proceedings by William W. Cohen (auth.), Hiroki Arimura, Sanjay Jain, Arun Sharma (eds.)


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