This is a task in which the output is completely determined by the first bit of the input. Sound easy? Unfortunately, there are 999 other distractor inputs. And the network only has 26 patterns on which it can train.

While it is very easy for the network to nail the training patterns, it doesn't always do so well on the 4 testing patterns. Open the Link Viewer to see what has happened to the weights.

Is it reasonable to expect the network to learn this simple rule? With 1000 inputs and only 26 patterns, there are many simple rules that can explain the output. The network might learn to rely on a combination of a few different bits that together explain the answer. With only 26 patterns, who's to say which rule is the best one?

So if we really want the network to reach the right solution, we are going to have to give it some a priori (well, actually sort of a posteriori in this case) bias to favor the simple pick-one rule.

The first thing you might think of is using weight decay. However, this penalizes weights in proportion to how large they are. That means the largest weight is penalized the most. What we'd really like is for the weight from the first unit to grow large and for the rest to die off. Weight decay is counteracting this, to some extent.

A more effective solution in this case is to constrain the network to only consider a certain class of solutions. In particular, only those solutions with positive weights from the inputs.

We can force the weights from the input layer to be positive with the command:

lens> setLinkValue min 0 -t input

However, we should also change the weight initialization so that it doesn't produce initially negative weights. We can do that like this:

lens> setLinkValue randMean 0.1 -t input lens> setLinkValue randRange 0.1 -t input lens> resetNet

Now when you train, all of the input-to-output weights will be positive. If a weight falls to 0, it will remain at 0 forever, and thus be effectively pruned. Try training again. With any luck, all the weights have shrunk toward 0 except those from the first input and the bias. Now does it get the testing patterns right?

Assuming your answer was "yes", is there a moral to this? I suppose it is that neural networks cannot solve every sort of problem right out of the box. Your role as designer is sometimes to lead the network to a good solution by applying the right biases.

Douglas Rohde Last modified: Thu Nov 16 14:32:45 EST 2000