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**1 - 5**of**5**### On the rate of convergence in the CLT with respect to the Kantorovich metric

- In Probability in Banach Spaces 9 193–207. Birkhäuser
, 1994

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### Optimal Stopping and Effective Machine Complexity in Learning

- Advances in Neural Information Processing Systems 6
, 1994

"... We study the problem of when to stop learning a class of feedforward networks -- networks with linear outputs neuron and fixed input weights -- when they are trained with a gradient descent algorithm on a finite number of examples. Under general regularity conditions, it is shown that there are in g ..."

Abstract
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We study the problem of when to stop learning a class of feedforward networks -- networks with linear outputs neuron and fixed input weights -- when they are trained with a gradient descent algorithm on a finite number of examples. Under general regularity conditions, it is shown that there are in general three distinct phases in the generalization performance in the learning process, and in particular, the network has better generalization performance when learning is stopped at a certain time before the global minimum of the empirical error is reached. A notion of effective size of a machine is defined and used to explain the trade-off between the complexity of the machine and the training error in the learning process. The study leads naturally to a network size selection criterion, which turns out to be a generalization of Akaike's Information Criterion for the learning process. It is shown that stopping learning before the global minimum of the empirical error has the effect of ne...

### Asymptotically optimal Berry-Esseen-type bounds for distributions with an absolutely continuous part

"... Recursive and closed form upper bounds are o¤ered for the Kolmogorov and the total variation distance between the standard normal distribution and the distribution of a standardized sum of n independent and identically distributed random variables. The approximation error in the CLT obtained from th ..."

Abstract
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Recursive and closed form upper bounds are o¤ered for the Kolmogorov and the total variation distance between the standard normal distribution and the distribution of a standardized sum of n independent and identically distributed random variables. The approximation error in the CLT obtained from these new bounds vanishes at a rate O(n k=2+1); provided that the common distribution of the summands possesses an absolutely continuous part, and shares the same k 1 (k 3) …rst moments with the standard normal distribution. Moreover, for the …rst time, these new uniform Berry-Esseen-type bounds are asymptotically optimal, that is, the ratio of the true distance to the respective bound converges to unity for a large class of distributions of the summands. Thus, apart from the correct rate, the proposed error estimates incorporate an optimal asymptotic constant (factor). Finally, three illustrative examples are presented along with numerical comparisons revealing that the new bounds are sharp enough even to be used in practical statistical applications.