Life In 19x19

HermanHiddema wrote:He made a hypothesis that the data would fit a certain asymptotic function, then found that the data did indeed fit that function.

That's not how statistics works. He made the hypothesis that the data would fit a certain function, then calculated which parameters would make the fit look most accurate, then plotted it - and nothing more. It looks reasonable, but a logarithmic curve or cubic polynomial curve or any non-stupid curve would look reasonable too, and it wouldn't surprise me if you could fit a sine curve to it, though that would be obvious nonsense. The fact that a curve looks good locally isn't enough evidence to say that it backs up your assumptions, because the assumptions (e.g. the existence of an asymptote) are very global statements. The data is simply too noisy to extrapolate the shape of this curve from. (Basically all real-world data suffers from the same problem.)

HermanHiddema wrote:Henric Bergsaker did some research and found an asymptotic improvement function, see http://goforbundet.se/ng/200901.pdf (page 6)

What a great paper, thanks for linking. I'm not even sad my post wasn't original thought.

L19'ers, if you have the vaguest interest in the mathematics of this topic, take the time to read it!

billywoods wrote:

HermanHiddema wrote:He made a hypothesis that the data would fit a certain asymptotic function, then found that the data did indeed fit that function.

That's not how statistics works. He made the hypothesis that the data would fit a certain function, then calculated which parameters would make the fit look most accurate, then plotted it - and nothing more. It looks reasonable, but a logarithmic curve or cubic polynomial curve or any non-stupid curve would look reasonable too, and it wouldn't surprise me if you could fit a sine curve to it, though that would be obvious nonsense. The fact that a curve looks good locally isn't enough evidence to say that it backs up your assumptions, because the assumptions (e.g. the existence of an asymptote) are very global statements. The data is simply too noisy to extrapolate the shape of this curve from. (Basically all real-world data suffers from the same problem.)

Billywoods, I agree, the paper proves nothing. But it is a reasonable model. As you know, generally before choosing a model, you should have a real world reason for believing it might be true. I think his parameter definitions and assumptions, based on real world player ratings, are pretty reasonable, with one inconsequential disagreement that I'll address below. There is not a good reason for believing your alternatives that I can think of (sine curve, really)? And, of course, the less parameters, the better, which he has effectively minimized. The author hasn't provided the goodness of fit information, so of course nothing is "proved", but the curve seems reasonable. My biggest problem is dropping 10% of the samples because they don't fit the model, but I don't care that much; it's not like it's medical efficacy data.

I find it interesting that his asymptote (R0), if I understand it correctly, is claimed to be the strength of your teacher, rather than the players own potential. I see no reason, other than not wanting to be demotivating, to believe that is true. But it makes no difference to the model how R0 is defined, strength of teacher, personal potential, whatever, the model will be the same.

Good stuff. Brings back memories of grad school, combined with my more recently found love of go. Too bad I didn't play back then...

wineandgolover wrote:I find it interesting that his asymptote (R0), if I understand it correctly, is claimed to be the strength of your teacher, rather than the players own potential.

I don't know what level I would place the teacher's strength in, but it is a very important factor.

So if you want a straighter curve over time just increase your effort exponentially. Players are disappointed because they are insufficiently lazy when they start out.

Or there is another option: stop caring about the curve and just have fun.

wineandgolover wrote:I find it interesting that his asymptote (R0), if I understand it correctly, is claimed to be the strength of your teacher, rather than the players own potential. I see no reason, other than not wanting to be demotivating, to believe that is true. But it makes no difference to the model how R0 is defined, strength of teacher, personal potential, whatever, the model will be the same.

The article is not as precise as it could be, but R0 is not claimed to be the observed strength of the teacher. It is a parameter that is designed to fit the other data. EDIT: wineandgolover understands this. I am making the point to others.

It is an interesting article, and the author is quite honest in the sense that he hints at the severe limitations of his study more than once.

Note that R0, t0, and tau are all fitted parameters NOT data. Understanding this distinction is important. They are not direct observations. Furthermore, the discussion in the article seems to suggest that these parameters are individualized to each player. If that is indeed the case, then it would be shocking if the author was not able to generate such suggestive pictures that fit these functional forms nicely. There are enough degrees of freedom there to fit almost any steadily declining rate of growth.

Furthermore, it is unclear whether t, which is an observed variable in the study, is correlated with other factors that may affect growth rates.

Conclusion: We observe that rates of growth slow with time. We do not know if time is the cause of this deceleration or if time is simply correlated with the real causes.

The exercise is interesting and intellectually stimulating, but we should treat it more like engaging parlor conversation rather than anything resembling a serious model. As far as I can see, the author seems to feel that way as well.

PS: This is perhaps a trivial thing to say, but go learning has to be asymptotic in the sense that you cannot exceed a 100% win rate. The presence or absence of an asymptote can be merely a byproduct of how ratings are measured. The rulers we use are important.

snorri wrote:Or there is another option: stop caring about the curve and just have fun.

A Dr. Strangelove reference? Yes, having fun is the best. I agree.

lemmata wrote:PS: This is perhaps a trivial thing to say, but go learning has to be asymptotic in the sense that you cannot exceed a 100% win rate. The presence or absence of an asymptote can be merely a byproduct of how ratings are measured. The rulers we use are important.

This may sound trivial but in fact it's not. If you e. g. assume that each rank improvement represents a certain reduction factor in mistakes-per-move, then a 100% win rate (-> 0% mistakes) would be represented by an infinite rating. In the hypothetic case of two players with very high ratings, a limiting factor may be also the board size since very low error rates might be not very well measureable with a finite number of moves.

Regarding the curve fits in the publication it is clear that there are many candidate functions which yield good fits. However, it seems that e. g. a logarithmic increase in rank can be excluded, since it is very difficult to fit a function like a*(1-exp(-t/tau)) against log(t):


Thus we can conclude at least that after a certain time the rank increase for typical players is slower than logarithmic.

I don't think that that last point is trivial. As discussed in another thread, you can describe a certain increase in strength as reducing your losses against a fixed set of opponents by a factor of ten (eg, .99 win rate becomes .999), or as maintaining your loss rate against that set while increasing the handicap by four stones, or as maintaining your old loss rate against a new set of opponents who are four stones stronger than the old set.

Now, the first description of getting strongeris "trivially" asymptotic, while the second one is ambiguous and the third one doesn't admit of any trivial limit. Which just goes to show you can parametrize these things however you damn well please. With the right kind of scaling, even the first description could be graphed in a way that didn't show an asymptote.

schawipp wrote:Thus we can conclude at least that after a certain time the rank increase for typical players is slower than logarithmic.

Is this time shorter or longer than the lifespan of an average human?

This theoretical bound definitely exists, I just have trouble concluding from there that "my asymptote" (as in the thread title) exists. I'm 3k - it's unlikely that I'll ever get near the top ranks of players in my lifetime, so I will always have a wealth of learning material and stronger players available to me to learn from, so the biggest factor that decides how fast I improve isn't some theoretical upper bound I'll never get near, it's my brain (along with how much time I make it spend on go). If I eventually stop improving, I probably haven't "trailed off" like people usually think of asymptotes doing, e.g. due to my brain reaching full capacity - I've probably died or given up go. And this holds as true for me as it does for all but the very strongest amateurs.

(Analogy: when you were in primary school learning about fractions, did how fast you learnt depend fundamentally on whether mathematics was finite or infinite in scope, or on how well educated your teacher was in degree-level or research-level mathematics? Probably not - you learnt at your own speed, regardless of those things, which were way out of your grasp. It's that internal speed of learning that interests me.)

billywoods wrote:(Analogy: when you were in primary school learning about fractions, did how fast you learnt depend fundamentally on whether mathematics was finite or infinite in scope, or on how well educated your teacher was in degree-level or research-level mathematics? Probably not - you learnt at your own speed, regardless of those things, which were way out of your grasp. It's that internal speed of learning that interests me.)

However if your teacher taught you the wrong way to do fractions, you might have some problems. There's a 10k I see teaching on KGS a bit, and it can make me cringe.

All other things being equal, if I had a million years to study go how strong would I get?

I'm not convinced I could beat the current pros. I might learn everything there is to learn about go but in terms of raw reading strength, would my brain be able to adapt itself in the same way a younger brain can?

Not that it matters anyway, I just play a few moves a day on DGS when my work gets too boring, I'm not going to come close to my potential limits that way.

schawipp wrote:This may sound trivial but in fact it's not. If you e. g. assume that each rank improvement represents a certain reduction factor in mistakes-per-move, then a 100% win rate (-> 0% mistakes) would be represented by an infinite rating. In the hypothetic case of two players with very high ratings, a limiting factor may be also the board size since very low error rates might be not very well measureable with a finite number of moves.

This isn't different from what I said, which is that existence of asymptotes is sensitive to the rating system used. Perhaps we mean different things by triviality. I meant that my statement is essentially a tautology. I wasn't making any statement about its importance.

Codexus wrote:All other things being equal, if I had a million years to study go how strong would I get?

I'm not convinced I could beat the current pros. .....

After a million years you have a sporting chance that your brains are in better condition than the ones of some current pros.

oren wrote:However if your teacher taught you the wrong way to do fractions, you might have some problems. There's a 10k I see teaching on KGS a bit, and it can make me cringe.

There is that. Most people equally well reach an "asymptote" in their mathematical education, though all I really mean by that is that they have a panic attack every time they see numbers, and run far away as quickly as they can.