By Martin Anthony
This crucial paintings describes contemporary theoretical advances within the learn of synthetic neural networks. It explores probabilistic types of supervised studying difficulties, and addresses the major statistical and computational questions. Chapters survey learn on development class with binary-output networks, together with a dialogue of the relevance of the Vapnik Chervonenkis measurement, and of estimates of the measurement for a number of neural community types. additionally, Anthony and Bartlett improve a version of type via real-output networks, and exhibit the usefulness of class with a "large margin." The authors clarify the position of scale-sensitive types of the Vapnik Chervonenkis measurement in huge margin type, and in genuine prediction. Key chapters additionally speak about the computational complexity of neural community studying, describing quite a few hardness effects, and outlining effective, optimistic studying algorithms. The booklet is self-contained and available to researchers and graduate scholars in laptop technology, engineering, and arithmetic.
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Additional resources for Neural Network Learning: Theoretical Foundations
This quantity can be thought of as the approximation error of the class H, since it describes how accurately the best function in H can approximate the relationship between x and y that is determined by the probability distribution P. (Note that we take an infimum rather than simply a minimum here because the set of values that erp ranges over t The functions in H have to be measurable, and they also have to satisfy some additional, fairly weak, measurability conditions for the subsequent quantities to be well-defined.
Fig. 1. The planes Pi, P2, and P3 (defined by points xi,X2,#3 € R) divide R2 into six cells. ,ra, where zf = (xj,—l). , zm} is linearly independent. To apply the lemma, we shall set d = n + 1. ,2 m } Q Kd has every subset of no more than d points linearly independent. , m, and define C(T) = CC I Rd - | J Pi 1 . 2 The growth function 33 P2 Fig. 2. Planes Pi, P 2 , and P in R3. The intersections of Pi and P 2 with P are shown as bold lines. 2) k=0 Proof First notice that linear independence of every subset of up to d points of T is equivalent to the condition that the intersection of any 1 < k < d linear subspaces Pi is a (d —fc)-dimensionallinear subspace (a '(d - A;)-plane').
For any fixed positive 5, this probability is less than 6 provided ra : as required. 4 is the presence of 1/e rather than the larger 1/e2. We shall see this difference arise in a number of contexts. The intuitive explanation of this improvement is that less data is needed to form an accurate estimate of a random quantity if its variance is smaller. 5 Remarks Learning with respect to a touchstone class It is often useful to weaken the requirement of a learning algorithm by asking only that (with high probability) eiP(L(z)) < optp(T) + e = inf erP(*) + c 26 The Pattern Classification Problem where T, called the touchstone class, is a subset of H.