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- Artificial Neural Nets Problem Solving Methods
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Weight decay is also a commonly used technique for controlling capacity in neural networks. Early stopping is considered to be fast, but it is not well defined keep in mind the pitfalls mentioned in chapter 2. On the other hand, weight decay regularizers [5, 2] are well understood, but finding a suitable parameter A to control the strength of the weight decay term can be tediously time consuming.
Thorsteinn Rognvaldsson proposes a simple trick for estimating A by making use of the best of both worlds see p.
The Ghost in the Shell: Global Neural Network
Other penalties are also possible. The trick is speedy, since we neither have to do a complete training nor a scan of the whole A parameter space, and the accuracy of the determined A is good, as seen from some interesting simulations. Muller Tony Plate in chapter 4 treats the penalty factors for the weights hyperpa- rameters along the Bayesian framework of MacKay  and Neal .
There are two levels in searching for the best network.
Artificial Neural Nets Problem Solving Methods
The inner loop is a minimization of the training error keeping the hyperparameters fixed, whereas the outer loop searches the hyperparameter space with the goal of maximizing the evidence of having generated the data. This whole procedure is rather slow and compu- tationally expensive, since, in theory, the inner search needs to converge to a local minimum at each outer loop search step. When applied to classification networks using the cross-entropy error function the outer-loop search can be unstable with the hyperparameter values oscillating wildly or going to inappro- priate extremes.
To make this Bayesian framework work better in practice, Tony Plate proposes a number of tricks that speed and simplify the hyperparameter search strategies see p.
In particular, his search strategies center around the questions: 1 how often when should the hyperparameters be updated see p. To discuss the effects of the choices made in 1 and 2 , Tony Plate uses simulations based on artificial examples and concludes with a concise set of rules for making the hyperparameter framework work better. In chapter 5, Jan Larsen et al. The trick is simple: perform gradient descent on the validation set errors with respect to the regu- larization parameters, and iteratively use the results for updating the estimate of the regularization parameters see p.
This method holds for a variety of penalty terms e. The computational overhead is negligible for computing the gradients, however, an inverse Hessian has to be estimated. If se- cond order methods are used for training, then the inverse Hessian may already be available, so there is little additional effort. Otherwise obtaining full Hessian information is rather tedious and limits the approach to smaller applications see discussion in chapter 1.
Nevertheless approximations of the Hessian e. Jan Larsen, et al. Averaging over multiple predictors is a well known method for impro- ving generalization see e. David Horn et al. They present solutions for answering these questions by providing a method for estimating the error of an infinite number of predictors and they demonstrate the usefulness of their trick for the sunspot prediction task.
Additional theoreti- cal reasoning is given to explain their success in terms of variance minimization within the ensemble. Amari, N. Miiller, M. Finke, and H. Asymptotic sta- tistical theory of overtraining and cross-validation. Bagging predictors.
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Machine Learning, 26 2 : , Cowan, G. Tesauro, and J. Alspector, editors. Morgan Kaufman Publishers Inc. Girosi, M. Jones, and T. Regularization theory and neural networks architectures. Neural Computation, 7 2 , A bound on the error of cross validation using the approximation and estimation rates, with consequences for the training-test split. Neural Computation, 9 5 , Lincoln and J. Synergy of clustering multiple back propagation networks.
Touretzky, editor. A practical Bayesian framework for backpropagation networks. Bayesian Learning for Neural Networks.
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Number in Lecture Notes in Statistics. Springer, New York, Brown University, May Plant, S. Nowlan, and Geoffrey E. Experiments on learning by back-propagation. Technical Report Computer Science Dept. Report, Pittsburgh, PA, Wang, S. Venkatesh, and J. Optimal stopping and effective machine complexity in learning. In [f], Stacked generalization. Neural Networks, 5 2 , The exact criterion used for validation-based early stopping, however, is usually chosen in an ad-hoc fashion or training is stopped interactively.
This trick describes how to select a stopping criterion in a systematic fas- hion; it is a trick for either speeding learning procedures or improving generalization, whichever is more important in the particular situation. When training a neural network, one is usually interested in obtaining a network with optimal generalization performance. However, all standard neural network architectures such as the fully connected multi-layer perceptron are prone to overfitting : While the network seems to get better and better, i.
The idealized expectation is that during training the generalization error of the network evolves as shown in Figure 2. Typically the generalization error is estimated by a validation error, i. There are basically two ways to fight overfitting: reducing the number of di- mensions of the parameter space or reducing the effective size of each dimension.
Techniques for reducing the number of parameters are greedy constructive lear- ning , pruning [5, 12, 14], or weight sharing . Techniques for reducing the size of each parameter dimension are regularization, such as weight decay  G. Prechelt and others , or early stopping .
See also [8, 20] for an overview and  for an experimental comparison. Early stopping is widely used because it is simple to understand and imple- ment and has been reported to be superior to regularization methods in many cases, e. It is claimed to show the evolution over time of the per-example error on the training set and on a validation set not used for training the training error curve and the validation error curve.
Given this behavior, it is clear how to do early stopping using validation: Fig.
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Idealized training and validation error curves. Vertical: errors; horizontal: time 1. Split the training data into a training set and a validation set, e. Train only on the training set and evaluate the per-example error on the validation set once in a while, e. Stop training as soon as the error on the validation set is higher than it was the last time it was checked. Use the weights the network had in that previous step as the result of the training run. This approach uses the validation set to anticipate the behavior in real use or on a test set , assuming that the error on both will be similar: The validation error is used as an estimate of the generalization error.
Early Stopping - But When?