I've been playing around a bit with neural networks for backgammon and found something interesting, and want to see whether this is already part of gnubg.
Assume a Tesauro-style network with the usual inputs, and some number of hidden nodes. And for simplicity, just one output representing the probability of win. If I take a given board and translate the position into the inputs and then evaluate the network, it gives me a probability of win. If I then flip the board's perspective (ie white vs black) and do the same, I get another probability of win. Those two probabilities should sum to 1, since one or the other player must win (or equivalently, the probability of white winning = probability of black losing = 1 - probability of black winning). But that constraint isn't satisfied with the usual TD setup. If however you make a few assumptions: * Hidden layer nodes don't include bias weight. * Hidden->input weights have a specific symmetry: weight of the i'th hidden node vs the j'th input node = w(i,j) = -w(i,j*), where j* is the index of the other player's corresponding position. * Output layer node doesn't include a bias weight. Then you can show that, for each set of output->hidden node weights, those weights sum to zero, the flip-the-perspective constraint is satisfied. This seems to reduce the number of weights by about half, since you need only half the middle weights. The network should be more accurate since a known symmetry is respected, and should converge quicker since there are fewer parameters to optimize. You can generalize to a bias weight on the output node; in that case, the constraint is on the bias weight that it = -1/2 sum( output->hidden node weights ). You can generalize as well to including a "gammon win" output node. In this case there are no constraints on the output->hidden node weights, but the probability of a gammon loss can be calculated from the probability of a gammon win weights, and you don't need to explicitly include an output node for the gammon loss. I googled around a fair bit but couldn't figure out whether this is well known or already included somewhere in gnubg. I took a look through eval.c but it's a bit daunting. :) Is there documentation somewhere that I've just not found? _______________________________________________ Bug-gnubg mailing list Bug-gnubg@gnu.org https://lists.gnu.org/mailman/listinfo/bug-gnubg