Retire KingDanger array

[hxim]

Replaced KingDanger array with direct calculation. Function is designed so that
a) in interval 0..QuadUnits quadratic function f1(x) = a * x * x is used
b) in interval QuadUnits..MaxUnits linear function f2(x) = x - b is used

Parameters a and b are must be a = 1 / (QuadUnits * 2) and b = QuadUnits / 2
because we need for x := QuadUnits both

1.) derivations same
f1'(QuadUnits) = f2'(QuadUnits)
=>
2 * a * QuadUnits = 1
=>
a = 1 / (QuadUnits * 2)

2.) values same
f1(QuadUnits) = f2(QuadUnits)
=>
a * QuadUnits * QuadUnits = QuadUnits - b
=>
b = QuadUnits - a * QuadUnits * QuadUnits
=>
b = QuadUnits / 2

STC:
LLR: 2.95 (-2.94,2.94) [-3.00,1.00]
Total: 41445 W: 7762 L: 7677 D: 26006

LTC:
LLR: 2.96 (-2.94,2.94) [-3.00,1.00]
Total: 70258 W: 9381 L: 9326 D: 51551

bench 8507177

[Rocky640]

So it's about the same shape as before.
But was it necessary to multiply all the parameters in evaluate.cpp by 11 ?
Could this be done only at the end of the calculation instead ?

[hxim]

@Rocky640 Yes, shape is same as before. I multiplied the parameters because it allows to have 1 less parameters in formula (now we have only 2 parameters - MaxUnits 2543 and QuadUnits 1786 instead of 3 parameters before MaxSlope 322, Peak 47410 and Weight 268 / 7700). Also i find nice that one attackUnit equals Value(1) of middlegame score in the linear part of function (it is less in quadratic part)

[snicolet]

So the function is a little bit like a poor man's approximation of a sigmoid with only polynomials of degree <= 2. The question is how we could further simplify it in line 483-484 after the patch is commited, if any. Maybe just continue the red function up to the green level ?

[mcostalba]

First of all, congratulations for this patch: it is not easy to have something working in king safety, especially such drastic idea is a really surprise that is proved not weaker than master.

If this patch would have been a [0, 5] or even a [0, 4] I would have gladly applied. But I am a bit dubious to totally replace a more or less easy to understand, easy to tweak table with something that is more difficult and less immediate. Your patch it is a simplification just because it removes KingDanger[] init code, but actual running code in evaluate_king() is not easier than current master, actually it is even a bit more complex, so I am wondering if it makes sense to replace something easy and well proven with something totally new just because it is not weaker.

Can you please run a SPRT[0, 4] at LTC?

[hxim]

@mcostalba sure test submited. I dont consider this change drastic because the calculation is basically same as before only array is not used - thats why it passed non regression test i think. Also i would think new code is easier to understand because it is immediately visible in code how attackUnits quantity is transformed into score. But I agree in the future it may be more difficult to tweak or replace with some more complex function than with KingDanger array.

[snicolet]

Just out of curiosity, I have pushed a test to see if the blue part of the graph is necessary (the test uses the same quadratic as the red part of the graph, up to the green level): http://tests.stockfishchess.org/tests/view/57cdcecb0ebc59030fbe4f2c

[hxim]

@snicolet removing blue part is same as seting MaxSlope to infinity in current code and will fail quickly i think

[snicolet]

I should have been more cautious with wording: I didn't mean adding a step at the end of the red function to the green level, but rather continue the red function to the right, up to the point where it reaches the green level.

I did a bench run where I computed the two versions of the king danger

k = using red + blue + green
k' = using red + red continuation + green

and then adding dbg_mean_of(abs( k - k')) resulted in:

Total 4045734 Mean 1.51005

meaning that k and k' differ by about 1.51 on average (maybe because the blue part of the graph is relatively rarely used, in only 4% of all the evaluation calls).

[hxim]

@snicolet i understand what you mean but if you make calculation of KingDanger array items you will see it is the same as removing MaxSlope parameter. That is why i would be surprised if it passed because i think this parameter was tuned in the past along with other parameters in king safety.

[snicolet]

@hxim: I don't know, let's just wait for the test result :-)

In fact your simplification and the discussion we are having just makes me realize that the version with only the red+green parts if equivalent (same bench) to

int A = attackUnits > 0 ? attackUnits : 0;
score -= make_score(std::min( A * A / 3572 , 1650), 0);

which proves that the king danger malus can never be more than two bishops (two bishops are 1652 exactly): honestly, I had never been aware of this by reading the current master code, so thanks for your insightful patch !!

So in fact my patch could almost be rewritten as :

int A = attackUnits > 0 ? attackUnits : 0;
score -= make_score(std::min( A * A / 3572 , 2 * BishopValueMg), 0);

[joergoster]

@hxim Sorry to say, but I fail to see the real benefit. A simple lookup is much easier to follow than your formula, imho. And the calculation of KingDanger array comes with no cost because is done at startup. Just my noobish 2 cents, of course.
@snicolet Thank you for the nice graph and the explanation.

[mcostalba]

@hxim It seems SPRT[0, 4] test failed and this patch really boils down to a parameter tweak: you replace KingDanger[] values with another set of parameters.

So the bad news is that I am going to close this PR.

The good news is that the road is open and eventually tuning/tweaking your formula will result in a real improvement compared to current master and I will gladly apply.