By Hans Ulrich Simon (auth.), David Helmbold, Bob Williamson (eds.)
This publication constitutes the refereed court cases of the 14th Annual and fifth eu meetings on Computational studying idea, COLT/EuroCOLT 2001, held in Amsterdam, The Netherlands, in July 2001.
The forty revised complete papers offered including one invited paper have been conscientiously reviewed and chosen from a complete of sixty nine submissions. All present facets of computational studying and its purposes in various fields are addressed.
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In a third section, u ˜1 might again be best and so forth. The pool size is small (m << n) and the best comparator switches back and forth between the few convex combinations in the pool (m << k, where k is the number of shifts). Of course, the convex combinations of the pool are not known to the master algorithm. This type of setting was popularized by an open problem posed by Yoav Freund . e. the convex combinations in the pool are unit vectors). Thus the goal is to develop bounds for the case when the comparator shifts back and forth within a pool of m out a much larger set of n experts.
Learning proceeds in trials. In each trial the master receives the predictions from n experts and uses them to form its own prediction. At the end of the trial both the master and the experts receive the true outcome and incur a loss measuring the discrepancy between their predictions and the outcome. The master maintains a weight for each of its experts. The weight of an expert is Supported by NSF grant CCR 9821087. This research was done while the ﬁrst author was visiting UC Santa Cruz D. Helmbold and B.
Corollary 3. Suppose N is a network with one hidden layer of k binary CSRF neurons and input dimension n ≥ 2, where k ≤ 2n , and assume that the output node is linear. Then N has VC dimension at least k · log 2 k 2 · n − log k 2 +1 . This even holds if the weights of the output node are not adjustable. Proof. We use Theorem 2 with h = k/2 , q = log(k/2) , and m = n − log(k/2) . The condition k ≤ 2n guarantees that m ≥ 1. Then there is a set of cardinality hq(m + 1) = k · log 2 k 2 · n − log k 2 +1 .