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Re: [Bug-gnubg] Neural network symmetry question

From: Mark Higgins
Subject: Re: [Bug-gnubg] Neural network symmetry question
Date: Sat, 10 Dec 2011 07:10:29 -0500

You have an input that represents whose turn it is (one input for white, one 
for black, value one if that player is on turn and zero otherwise). I think 
that's in the original Tesauro setup isn't it?

On Dec 10, 2011, at 1:10 AM, Joseph Heled <address@hidden> wrote:

> Well, I am not sure how you flip the position, since it matters who is
> on the move.
> -Joseph
> On 10 December 2011 16:17, Mark Higgins <address@hidden> wrote:
>> 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?
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