RuyTune's hyperbolic tanh based Sigmoid

RuyTune,
an open source framework for tuning evaluation function parameters, written by Álvaro Begué in C++, released on Bitbucket ^[1] as introduced in November 2016 ^[2]. RuyTune applies logistic regression using a limited-memory BFGS, a quasi-Newton method that approximates the Broyden–Fletcher–Goldfarb–Shanno algorithm with limited amount of memory. It uses the libLBFGS library ^[3] along with reverse-mode automatic differentiation and requires that the evaluation function is converted to a C++ template function where the score type is a template parameter, and a database of quiescent positions with associated results ^[4].

Method

The function to minimize the mean squared error of the prediction is:

where:

N is the number of test positions.
Ri is the result of the game corresponding to position i; -1* for black win, 0 for draw and +1 for white win.
qi is corresponding to position i, the value returned by the chess engine evaluation function. (Computing the gradient on the QS is a waste of time - it is much faster to run the QS saving the PV and then compute the gradient using the evaluation function of the end-of-PV position - and not worry too much about the fact that tweaking the evaluation function could result in a different position being picked ^[5]).
Sigmoid is implemented by hyperbolic tangent to convert centipawn scores into an expected result in [-1,1] ^[6].

Sigmoid(s) = tanh(0.0043s)

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