Is there a standard and accepted method for selecting the number of layers, and the number of nodes in each layer, in a FF NN? I'm interested in automated ways of building neural networks.
I realize this question has been answered, but I don't think the extant answer really engages the question beyond pointing to a link generally related to the question's subject matter. In particular, the link describes one technique for programmatic network configuration, but that is not a "[a] standard and accepted method" for network configuration.
By following a small set of clear rules, one can programmatically set a competent network architecture (i.e., the number and type of neuronal layers and the number of neurons comprising each layer). Following this schema this will give you a competent architecture but probably not an optimal one.
But once this network is initialized, you can iteratively tune the configuration during training using a number of ancillary algorithms; one family of these works by pruning nodes based on (small) values of the weight vector after a certain number of training epochs--in other words, eliminating unnecessary/redundant nodes (more on this below).
So every NN has three types of layers: input, hidden, and output.
Creating the NN architecture therefore means coming up with values for the number of layers of each type and the number of nodes in each of these layers.
The Input Layer
Simple--every NN has exactly one of them--no exceptions that I'm aware of.
With respect to the number of neurons comprising this layer, this parameter is completely and uniquely determined once you know the shape of your training data. Specifically, the number of neurons comprising that layer is equal to the number of features (columns) in your data. Some NN configurations add one additional node for a bias term.
The Output Layer
Like the Input layer, every NN has exactly one output layer. Determining its size (number of neurons) is simple; it is completely determined by the chosen model configuration.
Is your NN going running in Machine Mode or Regression Mode (the ML convention of using a term that is also used in statistics but assigning a different meaning to it is very confusing). Machine mode: returns a class label (e.g., "Premium Account"/"Basic Account"). Regression Mode returns a value (e.g., price).
If the NN is a regressor, then the output layer has a single node.
If the NN is a classifier, then it also has a single node unless softmax is used in which case the output layer has one node per class label in your model.
The Hidden Layers
So those few rules set the number of layers and size (neurons/layer) for both the input and output layers. That leaves the hidden layers.
How many hidden layers? Well if your data is linearly separable (which you often know by the time you begin coding a NN) then you don't need any hidden layers at all. Of course, you don't need an NN to resolve your data either, but it will still do the job.
Beyond that, as you probably know, there's a mountain of commentary on the question of hidden layer configuration in NNs (see the insanely thorough and insightful NN FAQ for an excellent summary of that commentary). One issue within this subject on which there is a consensus is the performance difference from adding additional hidden layers: the situations in which performance improves with a second (or third, etc.) hidden layer are very small. One hidden layer is sufficient for the large majority of problems.
So what about size of the hidden layer(s)--how many neurons? There are some empirically-derived rules-of-thumb, of these, the most commonly relied on is 'the optimal size of the hidden layer is usually between the size of the input and size of the output layers'. Jeff Heaton, author of Introduction to Neural Networks in Java offers a few more.
In sum, for most problems, one could probably get decent performance (even without a second optimization step) by setting the hidden layer configuration using just two rules: (i) number of hidden layers equals one; and (ii) the number of neurons in that layer is the mean of the neurons in the input and output layers.
Optimization of the Network Configuration
Pruning describes a set of techniques to trim network size (by nodes not layers) to improve computational performance and sometimes resolution performance. The gist of these techniques is removing nodes from the network during training by identifying those nodes which, if removed from the network, would not noticeably affect network performance (i.e., resolution of the data). (Even without using a formal pruning technique, you can get a rough idea of which nodes are not important by looking at your weight matrix after training; look weights very close to zero--it's the nodes on either end of those weights that are often removed during pruning.) Obviously, if you use a pruning algorithm during training then begin with a network configuration that is more likely to have excess (i.e., 'prunable') nodes--in other words, when deciding on a network architecture, err on the side of more neurons, if you add a pruning step.
Put another way, by applying a pruning algorithm to your network during training, you can approach optimal network configuration; whether you can do that in a single "up-front" (such as a genetic-algorithm-based algorithm) I don't know, though I do know that for now, this two-step optimization is more common.
I am working on an empirical study of this at the moment (approching a processor-century of simulations on our HPC facility!). My advice would be to use a "large" network and regularisation, if you use regularisation then the network architecture becomes less important (provided it is large enough to represent the underlying function we want to capture), but you do need to tune the regularisation parameter properly.
One of the problems with architecture selection is that it is a discrete, rather than continuous, control of the complexity of the model, and therefore can be a bit of a blunt instrument, especially when the ideal complexity is low.
However, this is all subject to the "no free lunch" theorems, while regularisation is effective in most cases, there will always be cases where architecture selection works better, and the only way to find out if that is true of the problem at hand is to try both approaches and cross-validate.
If I were to build an automated neural network builder, I would use Radford Neal's Hybrid Monte Carlo (HMC) sampling-based Bayesian approach, and use a large network and integrate over the weights rather than optimise the weights of a single network. However that is computationally expensive and a bit of a "black art", but the results Prof. Neal achieves suggests it is worth it!
As far as I know there is no way to select automatically the number of layers and neurons in each layer. But there are networks that can build automatically their topology, like EANN (Evolutionary Artificial Neural Networks, which use Genetic Algorithms to evolved the topology).
There are several approaches, a more or less modern one that seemed to give good results was NEAT (Neuro Evolution of Augmented Topologies). You can get more info:
I have tried to use Alyuda neurointelligence software to select the best network topology and it seemed to produce some good results, at least for the project I was doing. It uses genetic algorithm to select an optimal network topology
@doug's answer has worked for me. One additional rule of thumb for supervised learning networks, the upperbound on the number of hidden neurons that won't result in over-fitting is:
N_i = number of input neurons.
Others recommend setting
For an automated procedure you'd start with an alpha of 2 (twice as many degrees of freedom in your training data as your model) and work your way up to 10 if the error for training data is significantly smaller than for the cross-validation data set.