In a recent study journal thread schwaipp suggested a logarithmic scale for improvement. I completely agree that improvement is non-linear. The stronger you are, the more effort required to improve another stone. I suspect understanding this is the source of aggravation for some with the "1d in 1yr" threads. As an aside, I have seen people make 1d in 1 year, so I have no problem with that goal, and that's not what this thread is really about.

I'd combine the concept of non-linear advancement, with the idea of walls, starting age, and the idea of "permanent x-rank". I know really smart people who have played go for 20 years, have had tons of pro lessons, but can't advance beyond SDK. Could their improvement curve by asymptotic, limited at SDK? Could all our curves be asymptotic? And could we all be on separate curves with different limits?

Furthermore, I'd suggest that starting age impacts the limit, giving the resounding answer of "No!" to all those twenty-somethings asking about their pro ambitions.

And please note that I described the perma-SDK's as really smart. I am confident that IQ is not the only factor in our personal limits.

I suspect that most L19'ers have asymptotic limits in the high-dan levels, so don't take this concept personally. But my painful belief is, we are all limited, to some extent, to a certain strength beyond which we will never advance.

Okay then, shoot me down. Or tell me what your asymptote is, and why.

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- Brady
Want to see videos of low-dan mistakes and what to learn from them? Brady's Blunders

Your first difficult is the difference between someone's potential asymptote and someone's actual asymptote. For instance using totally made up numbers, my theoretical could be 6d EGF but a combination of bipolar and multiple other hobbies and a wife and kids taking up a lot of my time means my actual asymptote could very well be in the SDK range. I could potentially be a very strong player but due to circumstance it's extremely unlikely that I could ever achieve my potential (or for almost anyone else who cannot devote their life totally to go). Now, honestly, I don't really care what my potential "perfect circumstances" rank is and I don't think other people should too, it's just not relevant to me as a go player and it's unknowable anyway, at best a very strong player can look through your games and take a guess but really that's all it is, a guess albeit an educated one.

I think it is more powerful as a concept than an actual number.

It helps us understand the perma-SDK phenomenon. It incorporates the non-linear aspect of improvement, it accounts for why similar effort results in different outcomes for different people (even of similar intelligence), it explains why so many of us will never be pro-strength. And you can add whatever explanations you like to the function to justify why your theoretical limit is higher than your real-world limit*.

Simply, it is the beginning of a model to understand potential and improvement. This model, if agreed upon, can help answer many questions.

But the underlying assumption is that we all have an individual limit, that we will never move beyond, no matter how hard we study. Is this idea reasonable? Or is it too uncomfortable? Are we all potential pros, no matter our background, spatial reasoning, or how old we were when we learned the game?

* which possibly shows we all don't understand the concept of asymptotic limit, but whatever.

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- Brady
Want to see videos of low-dan mistakes and what to learn from them? Brady's Blunders

I wasn't thinking of specific numbers but that you are getting at two different concepts here. One is our potential without hinderances and one is our potential given our circumstances. They'll be different. I mean as soon as you move from student life to working life, or even more so life as a parent with young kids, your potential time for any hobby becomes smaller. This will adjust the asymptote if it exists under any reasonable set of assumptions. So too, arguably will our age.

So, to lay out a few ideas:

1) Progress is subject to strongly diminishing returns. We don't know the shape of the curve but it's more hours/stone for pretty much everyone past some point.
2) Life circumstances, specifically perhaps those affecting free time for study and those affecting stress/tiredness levels, will adjust the position of the asymptote.
3) Our age as it increases brings down this asymptote. Now, the controversial part is describing this, do we have a decline until our 30s where it levels out until our 50s where it starts to decline again in many/most individuals? (I'm pulling these ages out of thin air, I just want to bring up the idea that it's not a linear relationship)


Under 2) and 3) one could argue that unless one achieves a very high rank before you either enter the workforce proper or probably especially have children and before you reach a certain age (probably around 30 +/-5) then you are unlikely to ever reach a very high rank. Just as a result of assuming the above, I'm not arguing that this is true, there are many holes to pick at here in the argument.

I suspect a number of things are going on.

1. First, the amount you need to know to consistently beat someone grows geometrically. People don't expect this, partly because they are not warned "the amount you need to know grows geometrically", they are given something vague like "it's a hard goal, and you might not be young enough/clever enough to do it, but if you work hard, maybe..." So while they are making perceptible weekly progress they think they have proved that they are young enough, clever enough, industrious enough, and then when geometric scaling moves them into the domain of imperceptible improvement, they get discouraged.

2. Second, it seems to me that people start to identify as "go players" at some point in their lives when they have an unusual amount of time to put into the hobby. They make a certain amount of progress, but then life circumstances change and they can't even increase their plays at a linear rate, let alone their rank. In fact, many people go through periods where they aren't playing enough to keep all previously acquired intuitions intact and limber.

3. Third, at least in my experience, go clubs have a membership structure where by the time you are 4k or so, you are taking white more than black. I don't think you learn as much from giving handicap as taking, and I certainly don't think you learn as much from reviewing as being reviewed. People say "teaching is learning twice" or something like that, but...

Asymptotic to what? Certainly not a line with 0 slope.

In the basically impossible case a human achieves perfect Go play, we could always increase the board size to further differentiate skill. So assuming finite minds, an asymptote has to exist. Given all the variation in genetics, life experience, etc., it's again almost certain that some variations would confer an advantage to go playing, so the asymptotes would differ from person to person as well. Age and health, at a minimum, bound the time available to achieve mastery.

I suspect, though, that this maximum possible skill for most of us is past or at least close to a professional level. As Boidhre points out, that asymptote would assume devoting your life to Go. We also have to start considering the intangibles of learning: Let's say your mind has incorrectly started assuming a particular, complicated position can always be cut but that the presence of a single, distant stone changes that. You might play 4,000 games before running into the right position that forces you to re-evaluate some small aspect of your go knowledge. In the best case, you happen to play a game that involves the right configuration immediately and your mind adjusts. In the worst case you play for years before running into the position and don't notice it, then another 4,000 games and do, but the neural pathways are well set now and very hard to fix. The asymptote would seem to involve a team of the very best Go teachers helping you day in and day out, you making lucky guesses when forming mental models of Go, you just happening to play the most beneficial games for your learning every time you sit down at the goban...

So yes there's an asymptote and yes it varies, but no it probably doesn't tell us anything about people stuck at SDK, or not able to become a pro. Instead we need to start adding conditions to the asymptote: assuming they play 4 hours a day, assuming 3 hours of teaching a week of good quality, assuming they maintain this sort of disposition, and have this sort of 'luck' in how well their opponents moves help them learn... And then we're back answering questions of 'can I become a pro?' with 'it's possible, but very unlikely, and at a minimum requires you doing x,y and z...'

It's another diminishing returns thing with a possible hidden trap. Every additional time you teach a concept the less benefit it most likely has for you barring some sudden insight that comes on teaching effort #459. The greater issue in my opinion is that you're just reinforcing what you know and unless you are very strong indeed most of this will be incomplete knowledge at best and possibly incorrectly applied knowledge at worst.

Just because someone gets stuck, perhaps permanently, at a certain rank doesn't mean that that's their actual limit. I don't believe getting stuck through one "run-through" of their life doesn't mean that they were actually bound to get stuck at that rank when they started out, nor that the rank they get stuck at is actually insurmountable. For me to believe that, I'd have to believe that if you took 1000 clones of that person and had each of them try different things to get stronger, that none of the clones would get stronger.

I think of each person as just one data point on the potential of their own life. Peaking out at certain areas in one "instance" of their life doesn't mean that they reached the actual cap of their potential in that area of their life.

_________________
We don't know who we are; we don't know where we are.
Each of us woke up one moment and here we were in the darkness.
We're nameless things with no memory; no knowledge of what went before,
No understanding of what is now, no knowledge of what will be.

Absolutely. Someone's theoretical limit is irrelevant if they're not putting go at the centre of their life for a very long time. I bet if we looked at all those people who've been stuck at SDK all their lives, the vast majority simply aren't really that interested in go (except perhaps during the hour a week they have a board in front of them).

Well, I don't think anyone is ever trapped without the ability to improve at all, but I do think that, after a certain point, it becomes very difficult to gain a stone. The reason for this is that learning go isn't like a ratchet: you do need to review concepts you've already learned in order to maintain your current strength. So if you have the knowledge of a 5 kyu player it might take you at least an hour a week of review and 3-5 games a week just to keep from losing a stone, let alone the effort it would take to get up to 4 kyu.

This seems to be mirrored in how I've advanced myself. Obviously some have risen more quickly than me and other not as fast, but I think my experience is fairly common. Here is my rise to SDK:

I learned go in middle school (we had a club, which is rare in America to say the least) but then didn't play during high school. I was probably about 20 kyu coming out of middle school.

In college I found a new go club. With no work or study I was able to get to about the 15 kyu level - I just played go a few times a week at my go club, that was all.

To get to about 10-12 kyu took some study, but I basically just learned common joseki and shape. I learned about life and death doing online go problems and figured out how to keep stones connected. My biggest improvement was in reading of cuts.

To break into SDK range required real effort - I read books, found online video lectures, and started reviewing professional matches to improve my feel for the game. For the first time ever, I took out my board and read out common situations instead of only playing go against others to improve.

I can imagine the pattern would stay the same going forward.

To achieve 1-5 kyu level I'm going to need to put in regular study, not just occasional study when I feel like it. I would also need to increase my number of games per week from the 6-8 I currently play at club and online to more like 15-20 games. This is what I hope to begin doing over the summer and it seems reasonable.

To reach 1 dan will likely take daily study and an increase in my number of games per week even more (maybe even as many as 30-40 per week). Again, that seems doable.

However, I would probably reach a point where the amount of time a day needed to gain a stone isn't feasible. Then, I guess, I would have hit an asymptote. Maybe 3-5 dan?

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"You have to walk before you can run. Black 1 was a walking move.
I blushed inwardly to recall the ignorant thoughts that had gone through
my mind before, when I had not realized the true worth of Black 1."
-Kageyama Toshiro on proper moves

I was trying to extract some data via my KGS game record to get some more statistics, however I ran into a quota limitation of the KGS archive system ; thus I added a "sleep(20);" into my perl script and will try it out within the next days...

Regarding the "asymptote" I actually think - if the gain in information with rank is really exponentially - then there is no real asymptote. It is rather like the log(x) -function, which increases incredibly slow for x -> infinity but does not have an upper limit (however - if at some point you really need 10^6 games for further improvement there would be a kind of "practical" asymptote ).

Generally IMHO, increasing in strength can be due to increase in reading strength, by removal of bad habits or by increasing the coolness (tournament experience, good time manegement etc.). The first aspect can IMHO only be done by training of tsumego etc. on one's own, there are IMHO limited possibilities, where stronger players could assist. In the second aspect I think the input from stronger players is most important; especially at low ranks removing bad habits could yield some big advancement without much training effort (On the other hand, in some situtations, "bad habits" may exceptionally work well, but then you need good reading strength to see that).

The different aspects of playing strength may partially compensate each other. Therefore I would say that the assumed logarithmic character of rank climbing is merely a zero-order approximation.

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

Read the text carefully - he fitted those lines on the assumption that the function was asymptotic.

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.)

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!

_________________
- Brady
Want to see videos of low-dan mistakes and what to learn from them? Brady's Blunders

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...

_________________
- Brady
Want to see videos of low-dan mistakes and what to learn from them? Brady's Blunders

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