Download Algorithmic Learning Theory: 11th International Conference, by William W. Cohen (auth.), Hiroki Arimura, Sanjay Jain, Arun PDF

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

This publication 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 awarded 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 good judgment programming, inductive inference, complexity, neural networks and different paradigms, aid vector machines.

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

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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 . Then its expectation E[X] = np; hence, the expected value of X/n is p. The above three bounds respectively give an upper bound of the probability that X/n differs from p, say, . Below we use exp(x) to denote ex , where e is the base of the natural logarithm.

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 . Then its expectation E[X] = np; hence, the expected value of X/n is p. The above three bounds respectively give an upper bound of the probability that X/n differs from p, say, . Below we use exp(x) to denote ex , where e is the base of the natural logarithm. Now these two bounds are stated as follows.

Since the Chernoff bound is stated in terms of relative error, it is immediate to obtain the following sample size bound. ) Theorem 4. For any δ > 0 and ε, 0 < ε < 1, if Batch Sampling uses sample size n satisfying the following inequality, then it satisfies (4). n > 3 ε2 p ln B 2 δ . (5) The above size bound is similar to (3). But it does not seem easy to use because pB , the probability what we want to estimate, is in the denominator of the bound. (Cf. ) Nevertheless, there are some cases where a relative error bound is easier to use and the above size bound (5) provides a better analysis to us.

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