Interesting insights, good luck on reaching LGM!

Obvious one-liner "idea" conveyed through a wall of a text. Downvote.

Nice blog! Thanks for the mention — in the blog post you linked, I did mention taking feedback from the knowledge graph (which also implies that people would also take feedback from how they grow their knowledge graph) as well as concrete ways to develop it, but this blog makes the point more explicit.

I have some of my own comments on the content on this blog, and I am curious about what people other than me think about them. Apologies if it looks like I am picking out sentences without context — I urge the reader to go and read the line in the blog and the context around it before reading my comments. I agree with all these points in moderation, however it is only a discussion of when these things become impractical/counterproductive/completely false. (Also, some of these points might look like they're already addressed in the blog, but I implore the reader to look a bit deeper).

you should learn to recognize these signs yourself

This is true — introspection is very underrated, and almost everyone would benefit from introspecting for 10 more minutes a day than they already do (and people who are learning something new seriously would need to spend way more time introspecting). However, I feel that there is an exploration-exploitation tradeoff in all aspects of real life, and only in the limit that we have infinite time (i.e., asymptotically) does n-level introspection make sense for $$$n > 2$$$ or maybe even $$$n > 1$$$. Our brains need time to adapt to certain processes. We also find it harder to do things hierarchically (for a concrete example, most people I talked to find it harder to deal with segment trees compared to simple arrays) — simply due to very limited working memory, so adding more meta stuff tends to bog people down, making anything more than 2nd order pretty cumbersome. Practically speaking, my solution around higher order meta things (which I find is much easier) is to pick someone's brain about things that they do well (often concerning completely different fields), collaborate with them if possible, and use the second-hand experience to improve the meta parts of how I think, and do it myself for a while to make it subconscious — this reduces the exploration part by a great deal (and is probably what Newton meant by standing on the shoulders of giants) and gives me sufficient experience to be able to make confident decisions. This is because if there is a best (or near-optimal) way to learn for humans, it should be successful across fields, and if it is not, then you're probably spending too much time on things that don't matter. It has the added benefit of giving you access to a vast ocean of knowledge from others, too, so you prevent yourself from generalizing things "incorrectly" at worst and have a much more cohesive knowledge data structure in your head in the average case.

That's the scientific method, and you (probably) won't get anywhere without it!

This makes sense in the context of the blog, but it also assumes that the scientific method is the only correct way to go about these things, which is a bit misleading. In my opinion, it makes sense to take feedback about obvious red flags about your learning methodology but not treat it as a perfect source of truth (even as a model) as the scientific method dictates. (Here, by the scientific method, I mean coming up with a hypothesis, treating it as the truth, deducing things to make it testable (optional), testing it out, and then accepting/rejecting the hypothesis). For example, if you go down the path of higher-order introspection, you would suffer from self-observational errors (as in, things you perceive are affected by the act of perceiving them — as lower-level concrete examples, think of self-deception, stage fright, performance anxiety, or even the difference in performance that comes from having a different mindset when you have nothing to lose). The scientific method would here give you biases that would compound (using it is invalid anyway since accounting for errors here is impossible, but let's say you use it for the sake of using it). If you think of your 1st order strategy as gradient descent (as in you decide to go in the direction that makes the most sense, which is the gradient of the utility function), then self-observational errors correspond to not being able to compute even the loss function accurately (since observing changes things), and hence the partial derivative with respect to your strategy (and derivatives of noisy things are pretty useless, especially if you only have one realization of the given random variable). Now, if your 2nd order strategy is to tweak your gradient descent algorithm, these errors will compound and lead to larger errors. One can argue that if there is a global optimum, it is feasible to run this no matter what (unless this algorithm is unstable, which is quite likely by itself), but even then, it makes sense to stop after a few iterations as you would most likely be bouncing around in a neighborhood of the global optima. Note how this also sounds similar to what Goodhart's law implies.

I think what's more important is to first be accurate in self-assessment as well as build intuition (and not strong biases) about how things affect you. These come only after a decent amount of experience — first-hand or second-hand — with the method, which again brings us to exploration-exploitation. You also don't want derivatives that correspond to the person you were before you had all the knowledge you now have — you want derivatives that account for all the knowledge you have right now (and also a sufficiently informative description of how this derivative will evolve with time in different scenarios, which is sometimes asking for a lot) — this becomes especially important/hard if you try to be very systematic and learn only from your own mistakes. Conventional wisdom and second-hand experience come to the rescue in such cases and give you much more robust priors and gradient computing tools. I think the reason people turn to writing blogs is also guided by such logic subconsciously since second-hand experience works decently pretty much everywhere.

Most of the fastest learners I know are quite good at assessing themselves and making confident decisions based on that knowledge that they get in a smaller amount of time compared to others (and most of the time, without the scientific method), and that knowledge is almost always augmented by knowledge sourced from other people. We still don't understand how learning works, and most self-experimentation in terms of learning has too many confounding factors, so using the scientific method is the same as expecting reliable conclusions without sufficient data (which any data scientist can tell you is a rookie mistake). Applying it to the data you get from others is also incomplete because, as mentioned in the blog, only you can have perfect knowledge of what is going on in your head (and combining that with the confounding factors, it is not very reliable either). So you have to, at some point, start looking at some sort of aggregates and try out things without any apparent rigorous scientific basis. As again, I'd like to emphasize that when I say "scientific method" here, it means to build a hypothesis, treat it as the absolute truth and test it out, as opposed to having uncertainty around the truth of the hypothesis — the major flaw here is to assume that things are perfectly describable and, as a corollary, not infinitely complex (why is it not possible for nature to have a different set of laws of physics for every squared centimeter of the universe? And why is it not possible for different/seemingly opposite laws of learning to exist for a person at the same time?). It is missing this uncertainty that tunnel-visions us into a fake sense of knowledge, and I think it's pretty dangerous when it comes to something as critical as building your thought process.

For example, here's something that none of these address — what if you think about problem-solving wrong?

I have been re-evaluating my biases about this over the past year since I wrote my blog — and now I believe that there are no "wrong" ways to learn, just less correct ways to do so. Let's keep aside practicalities for a moment (I agree that there are some opinions around learning that are generally more practical and better for most people, as can be seen from the fact that I wrote a blog on learning explaining my own opinion). For instance, it is possible that you are fed with knowledge from different fields of study/walks of life, and when you sleep, you're forced to reconcile those pieces of information, coming to realizations that are completely novel and deeper than ones that all of us have been thinking about (emergence). I think it is more about what we know works rather than what does not work. IMO, we should be a bit more careful in claiming that certain things don't work at all since, in complex systems, there are more surprises than one would expect from their naturally simplistic intuition, which favors Occam's razor. I also think about it in this way — whenever we reason about something mathematically, we assume certain simple axioms. In those systems, there are things you can't prove/disprove, and to do that, you would need stronger axioms. In the end, things boil down to what set of axioms you need to model real life well enough. Given that it is easier to rule out impossibility (just show an instance of the thing claimed to be impossible) than rule out possibility, it makes sense that things in real life that are absolutely (or even practically) impossible are very few in number. And this is not accounting for weird things like quantum entanglement that were thought not to be observable macroscopically but turned out to be. Simple axioms seldom work in modeling real life, and those that work are probably already known. Similarly, well-accepted theories are good enough until they are not. But that's a discussion for another day. This goes into a discussion on what truth is, when something is the truth, when something should be perceived as the truth, and so on, but the main point is that what the truth would be for someone is not the same for someone else. A computer would probably learn better in different ways than a human would, and a modern computer definitely learns better in different ways than an old computing device did.

what do you suggest for peeks and drops in the journey? Recently, I was able to solve a div 2-C which was 1600 rated in the contest but I struggle solving 1300's 1400's normally. After that contest, I have been going through 3 bad contests. What would you do in this situation when you know that you're better than what you've performed recently?

I agree with the general idea you should run some gradient descent on your practice strategy (and any strategy), this is an important general thing to learn in life, and it is all you really need. However, maybe in line with nor's comment, it can take long time to get to that point from scratch and hard to analyze without sufficient knowledge, and you can end up bouncing around global optimum at some point.

So, especially in beginning for a relatively constrained task like cp, I think it is best to give pretty generic guidelines (for yourself or to give to others as advice) that make it as easy as possible to comprehend the idea of how to get a next problem and to start analyzing fundamental similarities of problems (getting out of plug and play mindset). As long as it's not completely wrong, following extremely simple strategy along this line without too much meta analysis on how to improve is probably best in beginning because you need exposure more than anything, and good generic advice should work for anyone to get towards their global optimum at the start without wasting time overthinking. Humans are pattern recognition machines and will get better most just with more data in beginning. I think one will likely naturally begin to increase meta analysis as necessary when they get up to level where it's worth thinking about, and then gradient descent on practice method once they have gained sufficient knowledge where it will make significant difference and they are likely to tweak in right direction.

Anyway, that is basis behind my practice method (i love self advertisement), and I think it is best generic advice to give someone before they're able to tweak it (hence why I wrote it).

I was recently at ICPC Luxor and discussed similar topics with some GMs and above. Most of them said "eww" or "ugh" when I said my training strategy involved reflecting about my weak points and about my practice strategy. I found this really surprising.

What do you think explains this? My hypotheses were that they are either so talented that this was never necessary (they do whatever and quickly improve anyways) for them or so biased to the point they dont realize they're already doing it subconciously.

This is something that I had thought about, but quickly, I realised the blog basically said exactly nothing. There are some words "have most people formalised their philosophy to this extent?" -- which is confusing to me, as I cannot see what there is to formalise. Here are what I believed have been conveyed through the blog

• That your training strategy can be changed, depending on training results, actual performance, your targets, etc.
• You gave a few examples of what types of trainings could have what effects
• You can gain some positive value if you think "If it is broken, just fix it"

The first point could be new to some people: If you are not aware of this, then realising its consequences could be helpful, as it gives you an entire new space of methods to explore. It is also objectively true, but for the discussions we shall assume this to be an obvious fact.

The second point is likely true. It is also objectively useful: More understandings = more precise decisions.

Now for the third point.

I know you did not explicitly say "Find broken thing and fix it". You said "we should look for signs that strategies are not working". However, what to do with such signs? Throughout the blog, I seemed to only find "do whatever training necessary to improve this component" as the only solution, all your suggestions seems to be based on this basic idea. I afraid that you assumed this is the universal truth, and so everything from it is logical.

To me, this is just a basic 800 rated starting heuristics: The first thing you should try. If you did not think about this serious, use this strategy, it is fine. From experience, this have some positive gains. However, note that I said "from experience", there is no obvious reasons why this must be true, and I shall present som reasons why.

Let suppose there are three critical components to CP. For example, it could be {DP, Constructives, Data Structures}, or {Core Ideas, Implementation Ideas, Implementations}. Let your skills be $$$x_1,x_2,x_3$$$, a reasonable model is to assume your skill $$$R = min(x_1,x_2,x_3)$$$. (There are lots of reasons why this is min, not sum but this comment will be very long to explore this).

There is a simple strategy: Let's improve the weakest aspect.

This is equivalent to "find broken thing and fix it". In this blog, you are choosing something that you failed on and improve it, I argue this is the same thing as whatever you failed in practice is likely the weakest aspect.

Assume x_1 is your weakest aspect and now you have improved it by $$$\epsilon$$$, now provided that $$$x_1$$$ must be the weakest aspect by at least $$$epsilon$$$, then your rating have been increased by $$$\epsilon$$$, where as if you try to improve other aspects by $$$\epsilon$$$, there will be no increase in $$$R$$$, so this must be a good strategy right?

The trouble lies in considering the long run. In the long run, you will always improve each of the aspects. If training $$$A$$$ always improves component 1 by $$$\delta$$$, and training $$$B$$$ always improves component 2 by $$$\sigma$$$, then doing it in order $$$A,B$$$ and order $$$B,A$$$ will be the same result, even if one of them will give you a bigger partial skill, yet partial skill should not have been something too important.

Therefore, in order to show that this strategy truly have some gains, you must justify why $$$(A,B) > (B,A)$$$, that is, there is some advantage of learning $$$A$$$ first before $$$B$$$, be it speed and efficiency or something else. This is something that highly depends on what the components actually are, and just because component $$$A$$$ is weaker may not be enough reason.

Experienced CPers will immediately spot that the above only give you a locally optimal solution for adjacent swaps. A global optimal, even with perfect information, is even harder to imagine.

Improve the weakest aspect is a safe, baseline strategy, but there could well be other much more powerful strategies that are more efficient. Any strategies that could take note of this non-commutativity is far more useful.

This is why I think "fix whatever broken" is bad because it does not look at the full picture. However, there is a far more important issue that makes it directly wrong: Cross-Strategies.

There exist (this is an opinion), some strategies that can improve all aspects at once. Such strategies would not have been desirable if you are in the mindset of "Fix whatever is the weakest", since it is slow to fix your precise target. However, it will once again be better in the long run. If you have three strategies that that improve each components at rate of $$$(1,0,0), (0,1,0), (0,0,1)$$$,then a fourth strategy of $$$(0.4, 0.4, 0.4)$$$ would have been much more useful. Unfortunately, cross strategies are even harder to detect and verify, because it had less impact on overall skill $$$R$$$.

What I wish to conclude is that , if

1. There have been no nontrivial non-commutatitivty , that some order (A,B) is better than (B,A)
2. That your training methods are diverse, and could improve every component reliably and semi-efficiently
3. That you are a sensible human being, and will improve on components that are significantly behind

Then what dominates your rate of improvements will be the efficiency of each of those individual training methods (and diversity). Under the scope of this blog, we assume you will not be getting new training methods, or improving the efficiency of each of these methods. In this case, any attempts to optimise the meta-strategy would be futile.

Now I want to make some direct deductions, that are basically my opinions,

Finally, here are some opinions. I don't have a serious justifications for each of these. I am only 70% to 80% sure about these points. I don't want to make such assertions, since I don't think my rating is good enough to imply it is a universal truth for everyone (I may do this when I am 3600 rating or similar).

• You can reach LGM without ever thinking about meta-training strategies or training strategies, and only be moderately slower than if you had. Of course, this is under a moderate assumptions of point 2 and 3 from above.

• Generally, improving the weakest aspects is slightly better: This serves as the non-commutative justifications for simple strategy of "Fix the weakest". This is by experience and from a few cases but each of them are pretty minor. A simple example is improving implementation tells allows me to be more aware of edge cases during ideas.

• Nontrivial non-commutativity exists, but this highly depends on what you are good at. However, I won't share example until I absolutely proved it to be true for myself.

• Cross strategies exists, that there exist ways to improve many aspects at once and be overall more efficient. Similarly, I could not share an example because I still haven't completely convinced myself.

The blog is unreasonably long, and after I used gpt to summarize it I see nothing new.

For people saying the blog is a waste of time and that it could have been wrapped up in a few paras then my friend you missed the entire point, stuff like this where you got to analyze your way of practicing or lets say self-inspection needs heart to heart talks where the writer has to pour his deep thinking into words and it can't be just.. you know..summarized into bullet points because if you do that then it wont hit you as hard as it might have if you read the entire blog. Now I will show you what I mean by that.

I copy pasted the entire blog to GPT and asked it to summarize it. This is what it throws at me

"The blog explores various questions commonly asked by individuals looking to improve their problem-solving skills, particularly in competitive programming. It encourages a meta-improvement perspective, where individuals learn to answer their questions by analyzing their practice and feedback.

Identifying Necessary Topics: Rather than asking what topics are needed to solve specific problems, the author suggests practicing problems and noting unfamiliar topics. If a topic consistently appears in solutions, it's likely important. Random Practice vs. Focused Topics: Practice should balance random problem-solving with focused topic practice. If failures stem from knowledge gaps, focus on learning. If failures are due to missed ideas, more practice is needed. Choosing Problem Difficulty: The advantages and disadvantages of practicing easier versus harder problems are discussed. The decision should be based on personal feedback from problems. Time Spent on Problems and Editorials: The time spent on problems and reading editorials depends on practice goals. The aim is not to memorize solutions but to understand problem-solving approaches. Practice Environment: Simulating contest conditions, without additional information like tags or difficulty, prepares individuals for real competitions. The blog emphasizes critical self-analysis and suggests that effective improvement involves not just finding the best strategy but understanding why strategies are effective or not. It also touches on overcoming biases and suggests treating learning objectively. Finally, it suggests that many successful individuals in competitive programming likely engage in some form of meta-improvement process, even if not as deeply as outlined in the blog."

Whoever has read the entire blog and GPT summary will understand that it doesn't hit as you as the blog did.