D. Pollard. Rates of uniform almost-sure convergence for empirical processes indexed by unbounded classes of functions. Manuscript, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....PAC model, the malicious error PAC model and the classification noise PAC model. In each case, we determine the complexity of the PAC algorithm as a function of the complexity of the SQ algorithm. In order to prove sample complexity bounds for these simulations, we make use of the d metric [17, 13]. Haussler gives sample complexity bounds sufficient for uniform convergence, with respect to the d metric, of probabilities and their estimates based on a sample. We relate this uniform convergence to uniform convergence with respect to ( accuracy, from which we then determine sample ....

....infinite set of probabilities through the use of a single sample of data. We therefore look to take advantage of uniform convergence results based on the Vapnik Chervonenkis dimension of the class of queries to be estimated. Consider the d metric defined over the non negative reals as follows [17, 13]: d (r; s) jr Gamma sj r s Haussler ( 13] Definition 3 and Theorems 1 and 7) effectively proves the following theorem on the sample size sufficient to ensure uniform convergence, with respect to the d metric, of empirical estimates to true probabilities. Theorem 2 (Haussler) Let G be a ....

D. Pollard. Rates of uniform almost-sure convergence for empirical processes indexed by unbounded classes of functions. Manuscript, 1986.

....yield upper bounds for other familiar distance metrics. The pseudo dimension [18] of a class F of [0; 1] valued random variables is a generalization of the Vapnik Chervonenkis dimension [25] to sets of realvalued random variables, and is a measure of the richness of F . Haussler [9] and Pollard [19] showed that, for any class F of random variables whose pseudo dimension is d, if we draw O Gamma 1 ff 2 Gamma d log 1 ff d log 1 log 1 ffi Delta Delta examples, then with probability 1 Gamma ffi , the d distance between the sample average and the true expectation will ....

D. Pollard. Rates of uniform almost-sure convergence for empirical processes indexed by unbounded classes of functions, 1986. Manuscript.

....we also define close to using a measure of relative difference (the d metric) similar to the standard multiplicative measure of approximation used in combinatorial optimization. This allows us to state the relevant uniform convergence bounds as generalized Chernoff style [11] bounds, as in [105], 25] chapter 5 For general regression with the negative log likelihood loss function, the principle of minimizing empirical loss is the same as the principle of maximum likelihood [20, 67] 12) rather than Hoeffding style bounds (as in Pollard s results [104] giving better bounds on ....

....bounds, in that they only bound the probability that the empirical mean is significantly smaller than the true mean. While extremely useful, as we mentioned in the previous section, these measures of deviation suffer from a discontinuity at E(f) 0, and lack of convenient metric properties. As in [105], we will give bounds on the sample size needed so that Pr i 9f 2 F : d ( b E z (f) E(f) ff j = Pr 9f 2 F : j b E z (f) Gamma E(f)j b E z (f) E(f) ff ffi; i.e. the deviation measured using the d metric 12 . By setting and ff appropriately, we obtain results ....

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D. Pollard. Rates of uniform almost-sure convergence for empirical processes indexed by unbounded classes of functions. manuscript, 1986.

....f. Then for any ffl 0, M(2ffl; F ) N (ffl; F ) M(ffl; F ) Theorem 3.3. Pol84, Hau92] Let F be a family of functions from a set S into [0; K] where dim(F ) d 1. Let M be a probability measure on S. Then for all 0 ffl K, M(ffl; F ) 2 2eK ffl ln 2eK ffl d : Theorem 3.4. Pol86] Let F be a permissible family of functions from a set S into [0; K] and let M be a probability measure on S. Assume 0; 0 ff 1, and m 1. Suppose that x 2 S m is generated by m independent random draws from S according to M . Then the probability that there exists f 2 F such that d ....

Pollard, D. (1986), Rates of Uniform Almost-Sure Convergence for Empirical Processes Indexed by Unbounded Classes of Functions, manuscript.

....PAC model, the malicious error PAC model and the classification noise PAC model. In each case, we determine the complexity of the PAC algorithm as a function of the complexity of the SQ algorithm. In order to prove sample complexity bounds for these simulations, we make use of the d metric [15, 11]. Haussler gives sample complexity bounds sufficient for uniform convergence, with respect to the d metric, of probabilities and their estimates based on a sample. We relate this uniform convergence to uniform convergence with respect to ( accuracy, from which we then determine sample ....

....infinite set of probabilities through the use of a single sample of data. We therefore look to take advantage of uniform convergence results based on the Vapnik Chervonenkis dimension of the class of queries to be estimated. Consider the d metric defined over the non negative reals as follows [15, 11]: d (r; s) jr Gamma sj r s Haussler ( 11] Definition 3 and Theorems 1 and 7) effectively proves the following theorem on the sample size sufficient to ensure uniform convergence, with respect to the d metric, of empirical estimates to true probabilities. Theorem 2 (Haussler) Let G ....

D. Pollard. Rates of uniform almost-sure convergence for empirical processes indexed by unbounded classes of functions. Manuscript, 1986.

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Pollard, D., 1986. Rates of uniform almost--sure convergence for empirical processes

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