A new type of neural network is more interpretable
spectrum.ieee.org
spectrum.ieee.org
My main finding was that KANs are very tricky to train compared to NNs. It's usually possible to get per-parameter loss roughly on par with NNs, but it requires a lot of hyperparameter tuning and extra tricks in the KAN architecture. In comparison, vanilla NNs were much easier to train and worked well under a much broader set of conditions.
Some people commented that we've invested an incredible amount of effort into getting really good at training NNs efficiently, and many of the things in ML libraries (optimizers like Adam, for example) are designed and optimized specifically for NNs. For that reason, it's not really a good apples-to-apples comparison.
I think there's definitely potential in KANs, but they aren't a magic bullet. I'm also a bit dubious about interpretability claims; the splines that are usually used for KANs don't really offer much more insight to me than just analyzing the output of a neuron in a lower layer of a NN.
I 100% agree with the idea that these won't be any more interpretable and I've never understood the argument that they would be. Sure, if the NN was a single neuron I can see it, but as soon as you start composing these things you lose all interpretability imo
I do think they have a lot of potential, but what has been published so far does not represent a panacea. Perhaps they will have an impact like transformers, perhaps they will only serve in a little niche. You can't really tell immediately how refinements will alter the usability.
Finding out what those refinements are and how they change things is what research is all about. I have been quite enjoying following https://github.com/mintisan/awesome-kan progress and seeing the variety of things being tried. I have a few ideas of my own I might try at sometime.
Between KANs and fixed activation function networks there is an entire continuum of activation function tuning available for research.
Buckets of simple parameter activation functions something like xsigmoid(mx) ( ReLU when m is large, GeLU at m=1.7, SiLU at m=1). This adds a small number of parameters for presumably some game
Single activation functions as above per neuron.
Multi parameterizable activation functions, in batches, or per neuron.
Many parameter function approximators, in batches, or per neuron.
Full KANs without weights.
I can see some significant acclaim being awarded to the person who can calculate a unified formula for determining where additional parameters should go for the largest impact.
The measure of parameter efficiency is the issue at hand. Given the relative newness of this approach even hitting the bar of "sometimes better, under specific circumstances" is impressive.
I am less likely to believe in biological research revealing new techniques. I feel like the inverse is more likely, that discovering new techniques will allow biologists to identify mechanisms where it was already in use.
Otherwise -- in a different world -- you and I could be brothers..
(If you judge me to be delusional.. however, erm.. frater quoque! The delusionality helps, sadly)
Will re-investigate your research pointers when i have time.
Curiously KANs are not very efficient when implemented with normal matrix multiplications in Pytorch, say. But with a custom cuda kernel, or using torch.compile they can be very fast: https://x.com/thomasahle/status/1798408687981297844
Can people this deep in the field read that visualization with all the formulas and actually grok what's going on? I'm trying to understand just how far behind I am from the average math person (obviously very very very far, but quantifiable lol)
Do I understand the Python? Somewhat. I know a relu is a rectified linear unit, which is a type of activation function. I have seen einsum before but forget what it is.
For the classical diagram I know what the nodes, edges and weights are. I have some idea what the formulas do, but not totally.
I'm unfamiliar with tensor diagrams.
So I have very little knowledge of this field, and I have a decent grasp of some of what it means, a vague grasp on other parts, and tensor diagrams I have little to no familiarity with.
I've recently been working on a library for doing automatic manipulation and differentiation of tensor diagrams (https://github.com/thomasahle/tensorgrad), and to me they are clearly a cleaner notation.
For a beautiful introduction to tensor networks, see also Jordan Taylor's blog post (https://www.lesswrong.com/posts/BQKKQiBmc63fwjDrj/graphical-...)
However, when thinking about ML architectures, I actually like that classical tensor diagrams make it harder to express non-associative architectures. E.g. RNNs are much harder to write than Transformers.
Some notes for other readers seeing this for the first time:
My favorite property of that these diagrams is that they make it easy to re-interpret a multilinear expression as a multilinear function of any of its variables. For example, in standard matrix notation you'd write x^T A x to get a quadratic form with respect to the variable x. I think most people read this either left to right or right to left: take a matrix-vector product, and then take an inner product between vectors. Tensor notation is more like prolog: the diagram
x - A - x
I still have to get a hold of all the rules for dealing with non-linearities ("bubbles") though. I'll have to take a look at your tensor cookbook :) I'm also sad that I can't write tensor diagrams easily in my digital notes.
Tensor diagrams are algebraically the same thing as factor graphs in probability theory. (Tensors correspond to factors and indices correspond to variables.) The only difference is that factors in probability theory need to be non-negative. You can define a contraction over indices for tensors taking values in any semiring though. The max-plus semiring gives you maximum log-likelihood problems, and so on.
Now this structure of functions is different in each implementations, but the type of function is quite similar - even though a large model will combine billions of those nodes and weights. Those visualizations tell you f.e. that some models connect neurons back to ones earlier in the chain to better remember a state. But the activation function is usually a weight and threshold.
KAN changes the functions on the edges to more sophisticated ones than just "multiply by 0.x" and uses known physical formulas that you can actually explain to a human instead of the result coming from 100x different weights which tell you nothing.
The language models we use currently may map how your brain works, but how strong the neurons are connected and to which others does not tell you anything. Instead a computer can chain different functions like you would chain a normal work task and explain each step to you / combine those learned routines on different tasks.
I am by no means an expert in this field, but i do a lot of category theory, especially for the reason that i wanted a more explainable neuron network. So take my pov with a grain of salt, but please don't be discouraged to learn this. If you can program a little and remember some calculus you can definitely grasp these concepts after learning the vocabulary!
I'm very tired of this... it needs to stop as it literally hinders ML progress
1) I know one (ONE) person who took multivariate calculus in high school. They did so by going to the local community college. I know zero people who took linear algebra. I just checked the listing of my old high school. Over a decade later neither multivariate calculus nor linear algebra is offered.
2) There's something I like to tell my students
You don't need math to train a good model, but you do need to know math to know why your model is wrong.
3) Math isn't just writing calculations down. That's part of it -- a big part -- but the concepts are critical. And to truly understand those concepts, you at some point need to do these calculations. Because at the end of the day, math is a language[2].
4) Just because the simplified view is not mathematically intensive does not mean math isn't important nor does it mean there isn't extremely complex mathematics under the hood. You're only explaining the mathematics in a simple way that is only about the updating process. There's a lot more to ML. And this should obviously be true since we consider them "black boxes"[3]. A lack of interpretability is not due to an immutable law, but due to our lack of understanding of a highly complex system. Yes, maybe each action in that system is simple, but if that meant the system as a whole was simple then I welcome you to develop a TOE for physics. Emergence is useful but also a pain in the ass[4].
[0] https://en.wikipedia.org/wiki/All_models_are_wrong
[1] For one, this is more accurately called a validation set. Test sets are held out. No more tuning. You're done. This is self-referential to my point.
[2] If you want to fight me on this, at least demonstrate to me you have taken an abstract algebra course and understand ideals and rings. Even better if axioms and set theory. I accept other positions, but too many argue from the basis of physics without understanding the difference between a physics and physics. Just because math is the language of physics does not mean math (or even physics) is inherently an objective principle (physics is a model).
[3] I hate this term. They are not black, but they are opaque. Which is to say that there is _some_ transparency.
[4] I am using the term "emergence" in the way a physicist would, not what you've seen in an ML paper. Why? Well read point 4 again starting at footnote [3].
Unless you are specifically dealing with intractable Bayesian integral problems, the multivariate calculus involved in NNs are primarily differentiation, not integration. The fun problems like boundary conditions and Stokes/Green that makes up the meat of multivariable calculus don't truly apply when you are dealing with differentiation only. In other words you only need the parts of calc 2/3 that can be taught in an afternoon, not the truly difficult parts.
> I'm sure many here recognize the reference[0], but being able to make a model that performs successfully on a test set[1] is not always meaningful. (sic) ...[2] If you want to fight me on this, at least demonstrate to me you have taken an abstract algebra course and understand ideals and rings. Even better if axioms and set theory.
Doesn't matter, if it creates value, it is sufficiently correct for all intents and purposes. Pray tell me how discrete math and abstract algebra has anything to do with day to day ML research. If you want to appeal to physics sure, plenty of Ising models, energy functions, and belief propagation in ML but you have lost all credibility bringing up discrete math.
Again those correlation tests you use to fact check your model are primarily linear frequentist models. Most statistics practitioners outside of graduate research will just be plugging formulas, not doing research level proofs.
> Just because the simplified view is not mathematically intensive does not mean math isn't important nor does it mean there isn't extremely complex mathematics under the hood. You're only explaining the mathematics in a simple way that is only about the updating process. There's a lot more to ML.
Are you sure? The traditional linear algebra (and similar) models never (or rarely) outperformed neural networks, except perhaps on efficiency, absent hardware acceleration and all other things being equal. A flapping bird wing is beautiful from a bioengineering point of view but the aerospace industry is powered by dumb (mostly) static airfoils. Just because something is elegant doesn't mean it solves problems. A scaled up CNN is about as boring a NN can get, yet it beats the pants off all those traditional computer vision algorithms that I am sure contain way more "discrete math and abstract algebra".
That being said, more knowledge is always a good thing, but I am not naive enough to believe that ML research can only be advanced by people with "mathematical maturity". It's still in the highly empirical stage where we experimentation (regardless of whether it's guided by mathematical intuition) dominates. I have seen plenty of interesting ML results from folks who don't know what ELBOs and KL divergences are.
> intractable Bayesian integral problems
> the multivariate calculus involved in NNs are primarily differentiation
Which as a side note, I've found this is an important point and one of the most difficult lessons to learn to be an effective math teacher: Once you understand something, it often seems obvious and it is easy to forget how much you struggled to get to that point. If you can remember the struggle, you will be a better teacher. I also encourage teaching as revisiting can reveal the holes in your knowledge and often overconfidence (but the problem repeats as you teach a course for a long time). Clearly this is something that Feynman recognized and lead to his famous studying technique.
> Doesn't matter, if it creates value
> Pray tell me how discrete math and abstract algebra has anything to do with day to day ML research.
> Most statistics practitioners outside of graduate research will just be plugging formulas
I think there is also an issue with how one analyzes value and reward. We're in a complicated enough society -- certainly a field -- that it is frequent for costs to be outsourced to others and to time. It is frequent to gain reward immediately or in the short term but have overall negative rewards in the medium to long term. It is unfortunate that these feedback signals degrade (noise) with time, but that is the reality of the world. I can even give day to day examples if you want (as well as calc), but this is long enough.
> Are you sure? The traditional linear algebra (and similar) models never (or rarely) outperformed neural networks
> A flapping bird wing... [vs] static airfoils.
I think miscommunication is happening on this point due to a difference in usage of "elegance." If we reference MW, I believe you are using it with definition 1c while I'm using it with 1d. As in, it isn't just aesthetics. There's good reason nature went down this path instead of another. It's the same reason the context matters, because all optimization problems are solved under constraints. Solution spaces are also quite large, and as we've referenced before, in these large intractable spaces, there's usually no global optima. This is often even true in highly constrained problems.
> more knowledge is always a good thing
> ML research can only be advanced by people with "mathematical maturity"
I am aware and we specifically don't directly compute those because they are intractable, hence rendering the need for low level ML practitioners to be familiar with their theoretical properties to be mostly unnecessary. MCMC exists for a reason and modern deep learning contains almost zero direct integration. There are lots of sampling but few integrals.
I have seen high schoolers use and implement VAEs without understanding what the reparametrization trick is.
> Value is too abstract and I think you should clarify
The value of LLMs and similar deep learning classifiers/generators is self evident. If your research is only good for publishing papers, you should stay in academia. You are in no position to judge or gatekeep ML research.
> I think this example better clarifies your lack of understanding in areospace engineering rather than your argument.
I am a pilot, software engineer, and a machine learning practitioner with plenty of interdisciplinary training in other scientific fields. I assure you I am more than familiar with the basics of fluid dynamics and flight principles. Granny knows how to suck eggs, no need for the lecture.
> First, I'm not sure I claimed abstract algebra was necessary, but that's a comment about if you were going to argue with me about "math being a language"
You claimed that people needed to know rings, groups and set theory to debate you on understanding ML. ̶I̶ ̶t̶h̶i̶n̶k̶ ̶y̶o̶u̶ ̶a̶r̶e̶ ̶t̶h̶e̶ ̶o̶n̶e̶ ̶w̶h̶o̶ ̶n̶e̶e̶d̶s̶ ̶t̶o̶ ̶g̶o̶ ̶b̶a̶c̶k̶ ̶t̶o̶ ̶s̶c̶h̶o̶o̶l̶ ̶a̶n̶d̶ ̶s̶t̶o̶p̶ ̶g̶a̶t̶e̶ ̶k̶e̶e̶p̶i̶n̶g̶.̶ ̶ ̶Y̶o̶u̶ ̶r̶e̶m̶i̶n̶d̶ ̶m̶e̶ ̶o̶f̶ ̶t̶h̶o̶s̶e̶ ̶f̶u̶n̶c̶t̶i̶o̶n̶a̶l̶ ̶p̶r̶o̶g̶r̶a̶m̶m̶e̶r̶s̶ ̶w̶h̶o̶ ̶w̶o̶u̶l̶d̶ ̶r̶e̶w̶r̶i̶t̶e̶ ̶n̶e̶u̶r̶a̶l̶ ̶n̶e̶t̶w̶o̶r̶k̶ ̶l̶i̶b̶r̶a̶r̶i̶e̶s̶ ̶i̶n̶ ̶H̶a̶s̶k̶e̶l̶l̶ ̶b̶e̶l̶i̶e̶v̶i̶n̶g̶ ̶c̶a̶t̶e̶g̶o̶r̶y̶ ̶t̶h̶e̶o̶r̶y̶ ̶w̶o̶u̶l̶d̶ ̶u̶n̶l̶o̶c̶k̶ ̶s̶o̶m̶e̶ ̶m̶a̶g̶i̶c̶ ̶i̶n̶s̶i̶g̶h̶t̶ ̶t̶h̶a̶t̶ ̶w̶o̶u̶l̶d̶ ̶l̶e̶a̶d̶ ̶t̶h̶e̶m̶ ̶t̶o̶w̶a̶r̶d̶s̶ ̶A̶G̶I̶.̶
̶I̶t̶ ̶m̶u̶s̶t̶ ̶b̶e̶ ̶n̶i̶c̶e̶ ̶u̶p̶ ̶t̶h̶e̶r̶e̶ ̶i̶n̶ ̶t̶h̶e̶ ̶i̶v̶o̶r̶y̶ ̶t̶o̶w̶e̶r̶ ̶o̶f̶ ̶a̶c̶a̶d̶e̶m̶i̶a̶.̶ I pity your students. Those who teach has a duty to encourage value creation and seeking out knowledge for its own sake, not constantly dangling a carrot in front of the student like leading a donkey. Don't gatekeep.
> I don't know how to address this because I'm not sure where I made this claim.
I am referring specifically to: I'm sure many here recognize the reference[0], but being able to make a model that performs successfully on a test set[1] is not always meaningful. For example, about a year ago I was working a very big tech firm and increased their model's capacity on customer data by over 200% with a model that performed worse on their "test set". No additional data was used, nor did I make any changes to the architecture. Figure that out without math. (note, I was able to predict poor generalization performance PRIOR to my changes and accurately predict my model's significantly higher generalization performance).
̶T̶h̶e̶r̶e̶ ̶a̶r̶e̶ ̶m̶a̶n̶y̶ ̶w̶a̶y̶s̶ ̶t̶o̶ ̶t̶e̶s̶t̶ ̶c̶a̶u̶s̶a̶l̶i̶t̶y̶.̶ ̶T̶h̶e̶ ̶d̶a̶t̶a̶ ̶s̶c̶i̶e̶n̶c̶e̶/̶s̶t̶a̶t̶i̶s̶t̶i̶c̶ ̶w̶a̶y̶s̶ ̶a̶r̶e̶ ̶S̶p̶e̶a̶r̶m̶a̶n̶/̶P̶e̶a̶r̶s̶o̶n̶ ̶r̶a̶n̶k̶s̶ ̶a̶n̶d̶ ̶t̶ ̶t̶e̶s̶t̶s̶.̶ ̶T̶h̶o̶s̶e̶ ̶a̶r̶e̶ ̶g̶e̶n̶e̶r̶a̶l̶l̶y̶ ̶l̶i̶n̶e̶a̶r̶.̶ ̶ ̶h̶t̶t̶p̶s̶:̶/̶/̶l̶i̶n̶d̶e̶l̶o̶e̶v̶.̶g̶i̶t̶h̶u̶b̶.̶i̶o̶/̶t̶e̶s̶t̶s̶-̶a̶s̶-̶l̶i̶n̶e̶a̶r̶/̶ ̶ ̶A̶l̶t̶e̶r̶n̶a̶t̶i̶v̶e̶l̶y̶ ̶t̶h̶e̶r̶e̶ ̶a̶r̶e̶ ̶M̶L̶ ̶m̶e̶t̶h̶o̶d̶s̶ ̶l̶i̶k̶e̶ ̶g̶r̶a̶p̶h̶i̶c̶a̶l̶ ̶m̶o̶d̶e̶l̶s̶ ̶b̶u̶t̶ ̶I̶ ̶d̶o̶n̶'̶t̶ ̶t̶h̶i̶n̶k̶ ̶t̶h̶a̶t̶'̶s̶ ̶w̶h̶a̶t̶ ̶y̶o̶u̶ ̶a̶r̶e̶ ̶r̶e̶f̶e̶r̶r̶i̶n̶g̶ ̶t̶o̶ ̶h̶e̶r̶e̶.̶ ̶F̶o̶r̶ ̶d̶e̶e̶p̶ ̶l̶e̶a̶r̶n̶i̶n̶g̶ ̶s̶p̶e̶c̶i̶f̶i̶c̶a̶l̶l̶y̶ ̶t̶h̶e̶r̶e̶ ̶a̶r̶e̶ ̶t̶r̶i̶c̶k̶s̶ ̶w̶i̶t̶h̶ ̶s̶a̶m̶p̶l̶i̶n̶g̶ ̶t̶h̶a̶t̶ ̶y̶o̶u̶ ̶c̶a̶n̶ ̶u̶s̶e̶ ̶t̶o̶ ̶e̶y̶e̶b̶a̶l̶l̶ ̶t̶h̶i̶n̶g̶s̶,̶ ̶g̶u̶i̶d̶e̶d̶ ̶b̶y̶ ̶i̶n̶t̶u̶i̶t̶i̶o̶n̶.̶ ̶ ̶H̶e̶r̶e̶'̶s̶ ̶a̶ ̶g̶o̶o̶d̶ ̶r̶e̶f̶e̶r̶e̶n̶c̶e̶ ̶o̶f̶ ̶w̶h̶a̶t̶ ̶I̶ ̶m̶e̶a̶n̶:̶ ̶ ̶h̶t̶t̶p̶s̶:̶/̶/̶m̶a̶t̶h̶e̶u̶s̶f̶a̶c̶u̶r̶e̶.̶g̶i̶t̶h̶u̶b̶.̶i̶o̶/̶p̶y̶t̶h̶o̶n̶-̶c̶a̶u̶s̶a̶l̶i̶t̶y̶-̶h̶a̶n̶d̶b̶o̶o̶k̶/̶l̶a̶n̶d̶i̶n̶g̶-̶p̶a̶g̶e̶.̶h̶t̶m̶l̶ ̶h̶t̶t̶p̶s̶:̶/̶/̶a̶r̶x̶i̶v̶.̶o̶r̶g̶/̶a̶b̶s̶/̶2̶3̶0̶5̶.̶1̶8̶7̶9̶3̶ ̶ ̶A̶g̶a̶i̶n̶ ̶m̶o̶r̶e̶ ̶o̶f̶ ̶t̶h̶e̶s̶e̶ ̶a̶r̶e̶ ̶e̶m̶p̶i̶r̶i̶c̶a̶l̶ ̶c̶o̶m̶m̶o̶n̶ ̶s̶e̶n̶s̶e̶.̶ No need for mathematical maturity or any grasp of discrete mathematics.
> Maybe I should have been clearer by adding a line that most people do not take calculus in high school. While it is offered there, these are the advance courses, and I'd be wary of being so pejorative. I know a large number of great mathematicians, computer scientists, and physicists who did not take calculus in high school. I don't think we need to discourage anyone or needlessly make them feel dumb. I'd rather encourage more to undertake further math education and I believe the lessons learned from calculus are highly beneficial in real world every day usage, without requiring explicit formula writing (as referenced in my prior post).
Okay fair you have a point. I forgot not all schools offer AP classes and advanced mathematics.
I believe we both share the view that education is important, but disagree on how much mathematical understanding is truly necessary to apply or advance ML. I suppose we will have to agree to disagree.
> but disagree on how much mathematical understanding is truly necessary to apply or advance ML
We do disagree on one thing, but it isn't about math, science, or ML. If you would like to have a real conversation, I would be happy to. But it is required that you respond in good faith and more carefully read what I've written. I expect you to respect my time as much as I've respected yours.
You should be prod of your credentials and the work you've accomplished. I intimately understand the hard work it takes to achieve each one of those things, but I don't want to have a pissing contest or try to diminish yours. You should be proud of them. But if you want to take your anger out on someone, I suggest going elsewhere. HN is not the place for that and I personally will have none of it.
So you know you're wrong as a matter of plain fact, but you're going to continue to spout your "mental model" as truth anyway?
What are you trying to say here? It doesn't matter much what "should" be high school knowledge unless you're designing curriculum. If no one actually learns it in high school then a phrase like "you only need high school knowledge" means nothing to most people.
> I know many people who did take...
In fact, the vast majority of my friends did, so my mental model is more useful to me than one that apportions a larger cut to the rest of the population. I also find it egregious that thirteen years of schooling doesn't get everyone to this level, so I want to hold the education system accountable by not loosening my standard.
> If [almost] no one actually learns it in high school then a phrase like "you only need high school knowledge" means nothing to most people.
I agree that this isn't as good at conveying information (unless the consensus changes), but that's not all I'm trying to do.
> so my mental model
> so I want to hold the education system accountable
For a frame of reference the high school I went to is (currently) in the top 20% of CA and top 10% of the country. I checked their listings and while there's a 50% participation rate in AP (they also have IB), they do not offer Linear Algebra or anything past Calc I. So I think this should help you update your model to consider what opportunities people have. I think this is especially important because we should distinguish opportunity from potential and skill. I firmly believe metrics hinder the chance of any form of meritocracy in part due to the fact that opportunity is so disproportionate (more so due to to the fact that metrics are models. And you know what they say about all models ;).
If we want to actually make a better society and smarter population, we should not be diminutive to people for the lack of opportunities that are out of their control. Instead I think we should recognize this and make sure that we are not the ones denying opportunities. If we talk about education, (with exception at the extreme ends) I think we can recognize that the difference between a top tier high school student and a bit below average, is not that huge. Post undergrad it certainly grows, but I don't think it is that large either. So I'm just saying, a bit of compassion goes a long ways. Opportunity compounds, so the earlier the better. I'm fond of the phrase "the harder I work, the luckier I get" because your hard work does contribute to your success, but it isn't the only factor[0]. We know "advanced" math, so we know nothing in real life is univariate, right? You work hard so that you may take advantage of opportunities that come your way, but the cards you are dealt are out of your control. And personally, I think we should do our best to ensure that we the dominating factor that determines outcome is due to what someone can actually control. And more importantly, we recognize how things compound (also [0]).
I'm not mad or angry with you. But I think you should take a second to reevaluate your model. I'm sure it has utility, but I'm sure you're not always in a setting where it is useful (like now). If you are, at least recognize how extreme your bubble is.
[0] I highly suggest watching, even if you've seen it before. https://www.youtube.com/watch?v=3LopI4YeC4I
I think this is where I'm coming from as well. When I got to university, I met tons of people who were just connected to the right resources, be they textbooks, summer camps, math tutors, or college counselors. My lucky break was a smart father and learning about AoPS in 4th grade, but I still wish I knew what else was out there.
It'd be great if people didn't need to get lucky to learn this stuff. There is a whole group of people paid to set standards and make people aware of what is out there. The standards filter down from the board of education to the teachers, and the teachers don't actually have much sway in what they teach (re: r/teachers). So, my ultimate goal for imposing my definition of "high school math" on everyone else is to make it common enough that the standards reflect that, rather than a slow trend of weakening standards that has happened in the past few decades[*].
But... now that I type this all out, it seems pretty far removed, and probably does more harm than good (except in my bubble). It'd be much more effective to send a few emails or get myself elected to one of these seats.
[*]: Note, standards have obviously risen since the early 1900s, but they've actually fallen in the last twenty years.
Another thing I think about a lot is Hanlon's Razor. I think people misinterpret it because the term "stupidity" is often used harsher. Maybe because we don't want to admit that we're all stupid[0]. I also despise the phrase "good enough" because many times the details do matter. It's not that we shouldn't approximate things, but that we often use this phrase to dismiss nuance. It's an unfortunate consequence of weak feedback signals. In the same way as if you fix something before it is broken you will save a lot of time and money, but the signal isn't apparent because you don't have the counterfactual experience of running around like a chicken who's lost it's head when shit hits the fan (but you probably have somewhere else).
So, we're all wrong, right? It's not like there's anything objective we can actually hold onto. But there are things that are less wrong, and that is meaningful. But the environment moves and time marches, so it's important we ensure what was once less wrong doesn't become more wrong. If our beliefs are unmoving, we'll only become more wrong with time. And that doesn't seem to be any of our goals and this fact doesn't sit well with our ego that creates that unmoving force in us.
FWIW, I'm highly in favor of younger kids learning much more difficult math. I've seen studies of how kids as young as 5 can learn core concepts of calculus (hell Terrance Tao exists). I'll settle for Middle School lol.
[0] In one framing I think our stupidity is impressive. We're bumbling chimps who can barely communicate with one another. Yet look what we've accomplished! I find this quite motivating, but it does not allow me to forget how foolish I am. Besides, if there's nowhere to move up, what would be the fun in that?
> "you only need high school knowledge" means nothing to most people.
I'll even state it now, as someone who highly advocates for learning math:
You don't even need calculus to build good models. At most, a rudimentary understanding of algebra, but I'm not sure even that. A little programming skill, which can freely and easily be obtained, is all that is necessary to begin. So if you can read and can motivate yourself, you can build good and useful models. It might just take longer if you don't have these yet.
- Very few high schools in America offer these classes. Even fewer people take them. The lie to yourself is not recognizing your bubble. You might think you're encouraging others, but you're doing the opposite. People who had those opportunities are likely not the ones that feel like ML is beyond their capabilities.
- While you can be successful in ML without math, this does not mean you should discourage its pursuit (just as you shouldn't place it as a gate keeping requirement. Even Calc and LA aren't required!).
- Math is about a way of thinking and approaching problems. These skills generalize beyond the ability to solve mathematical functions.
- The mathematical knowledge compounds and will make your models better. This may be nonobvious, especially given your suggested background, you've lived with this knowledge for quite some time. But if you haven't gone into things like statistical theory (more than ISLR), probability, metric theory, optimization, and so on, it is quite difficult to see how these help you in the same way it's hard to see what's on a shelf above you. It can also be difficult to explain how these help if you lack the language. But if you want to build good products (that work in the real world and not just in a demo), you'll find this knowledge is invaluable. If you don't understand why, let this be a signal of your overconfidence. Models aren't worth shit if they don't generalize (I'm not talking about AGI, I'm talking about generalizing to customer data)[0].
I like this page a lot: https://d2l.ai/chapter_preliminaries/calculus.html And i also learned a lot from ben eater (Youtube and this: https://eater.net/quaternions) Category theory was a godsend from Milewski, esp. his PDF is great! https://bartoszmilewski.com/2014/10/28/category-theory-for-p...
Maybe you can add a little and offer ressources from your background?
I do understand that it is difficult to find the math pathways in ML. What I do suggest is focusing on generative models[0], but the important part is to keep asking "why" and "what does this actually mean?" I think it is too easy to get stuck in knowing the answer and accepting it without understanding that there is no "right" answer, but less wrong answers. The path gets clearer if this attitude is adopted. This is where I've found teaching highly beneficial[1], as you face the basics and you'll find many questions that are easy to dismiss. To be a good teacher you have to take dumb questions seriously to determine if they're actually dumb or "dumb"[2]. Things you probably asked when learning: "how many layers?", "how many neurons per layer?", "is the optimization space actually smooth like the 2d decent graphs we saw?", "does data lie on a manifold?", "what is a manifold", "does data always lie on a lower dimensional manifold?", "is data always a distribution?", "is my data actually representative of my goals?", "what does this measurement mean?", "what does this measurement not tell me?", and so on. These are all extremely important questions that are almost universally ignored, but that all appear to have simple answers.
The reason I do like category theory (Bartosz also has video lectures for those), is because it helps connect the many different disciplines of math that are needed to answer some of these. To see the generalizations of things like surjective (epimorphic) and injective (monomorphic) functions, plays a role in answering the layer and neuron questions. It was the way I could start understanding how field theory wasn't just cool but impractical math.
But to ML, I think there's this hard gap and I'm not sure of a good resource that fills it (I'm working on one myself). That there's lots of basics (blogs like "the math behind transformers" that show the equations and little to no more) and there's also plenty at a high level by experts in cat theory, set theory, algebraic geometry, or others. The former aren't very useful and the latter can be easily found if you have the requisite knowledge but are impenetrable if you don't.
But with diffusion models and score matching being all the rage now, I highly suggest reading into Aapo Hyvärinen's[3] work. At lower levels I suggest Gelman's book and/or McElreath's. Needham has impressive illustrations and writes so well that you can only emulate because he's at a different level. I found Shao's Mathematical Statistics greatly helpful, but this is not easy to parse. Gallier and Quaintance are also worth looking into. But if you need something on the easier side, Tomczak is your friend. On the YouTube side I'll recommend some that are more easily missed: mathemaniac, EpsilonDeltaMain, jHan, ron-math, and Mutual_Information. You should find more from these too. Also search the #some{,2,3} tags, there's a lot of hidden gems. There's also a Cats4AI group btw, and many of them have now created a startup.
[0] Realistically all models are generative. The definition is ill-defined and you can even demonstrate that classifiers are EBMs. https://arxiv.org/abs/1912.03263
[1] I'm finishing a PhD program, so I do teach a ML course
[2] In either case you have to answer nicely. But it is easy to trick yourself into thinking something is simple when it is not.
That is actually correct, i miss those steps - if you can recommend anything besides d2l.ai (which i haven't finished yet) let me know! Enjoy your summer and train those smiling-muscles every once in a while.
In physics i had a lot of trouble in the beginning with the new symbols that where introduced, but they just shorten longer functions. Also chaining output of one function into another or combining them is something most coders do every day and it's written a bit different in math, but the same.
What i was missing was someone telling me: "This xyz basically is what you do here with zyx every day and already on a complex level like spacetime. You can do this!". Certainly no teacher can know every specialty of their students, but usually people uncertain about their abilities are the ones that are so deep in their field that they mistake others not understanding them as being behind - when they just miss the vocabulary to exchange ideas.
Last month i started with spinors and on one slide one term was named with a handful of different identifiers - same in category theory. The terms and names come from different fields and it can already be an eye-opener to some coder that a vector is just an array or that a scalar is a simple number, etc.
The education system and environment may have given up on coders that are sitting in some company struggling with themselves even though they do a lot of good work and i'm always trying to reach those as i know how isolating it can be.
Back to topic: One thing that bugged me in school was learning only the derivative rules like n^x => x*n^(x-1) and mostly skipping on using limits to do it in general.
https://d2l.ai/chapter_preliminaries/calculus.html <-- See Chapter 2.4.1
For AI you also need to know the power and chain rules mostly, but if you know that you are already way ahead of the bootcampers.
At 16 i coded a 3D-engine on my graphical calculator and got stuck at arbitrary rotations around a point - but at mid 30s learned you need 4 dimensions and quaternions lol. So never be hard on yourself if you have to learn something that others know at any other point in their lives.
Don't hesitate to drop me a mail if you want! I mostly hang around on #coreboot too!
There’s definitely a practice effect though. I know people who aren’t used to it will have their eyes glaze over when they read einsum notation.
relu(np.outer(x, y)) @ z.
KANs are similar. This is the forward code for KANs:
x = einsum("bi,oik->boik", x, w1) + b1
x = einsum("boik,oik->bo", relu(x), w2) + b2
x = einsum("bi,iok->bok", x, w1) + b1
x = einsum("bok,okp->bp", relu(x), w2) + b2
It's actually quite similar to flash-attention.
Personally, I think this is fine in context. Context that it is a new formulation and the difficulty and non-obviousness of optimization. Shouldn't be expected that every researcher can recognize and solve all optimization problems.
They don't even say that it might be possible to take advantage of GPUs in future. Reads like a fundamental problem with these.
A simple example of something hard to port to a GPU is a deep (24 lvls) binary tree with large leaf sizes (4kb). Particular trees can be optimized further, particular operations on trees might have further optimizations, and trees with nicer dimensionality might have tricks available, but solving that problem in the abstract is 32x slower on a GPU that "good" GPU problems. That's not a death knell, but it cuts down substantially the constraints which would make a GPU a better fit than a CPU.
Instruction branching is much worse, when required. Runtime is exponential.
As far as KANs are concerned, the problem is more with data branching. Each spline computation requires its own set of data and is only used once. The math being done on the aggregate computations is non-negligible, but fast relative to the memory loads. You quickly enter a regime where (1) you're bottlenecked on RAM bandwidth, and (2) for a given RAM load you can't efficiently use the warp allocated to it.
You can tweak the parameters a bit to alleviate that problem (smaller splines allow you to load and parallelize a few at once, larger ones allow you to do more work at once), but it's a big engineering challenge to fully utilize a GPU for that architecture. Your best bets are (1) observing something clever allowing you to represent the same result with different computations, and (2) a related idea, construct a different KAN-inspired algorithm with similar expressivity and more amenable to acceleration. My gut says (2) is more likely, but we'll see.
More succinctly: The algorithm as written is not a good fit for the GPU primitives we have. It might be possible to bridge that gap, but that isn't guaranteed.
EDIT: FourierKAN exists https://arxiv.org/html/2406.01034v1
Do any of you think that can be done cost-effectively with KAN’s? Especially using pre-trained, language models like LlaMa-3 to train the interpretable models?
"ReLU-KAN: New Kolmogorov-Arnold Networks that Only Need Matrix Addition, Dot Multiplication, and ReLU" https://arxiv.org/abs/2406.02075#
The idea of KANs sounds really exciting, but just to nitpick, you could also write any traditional NN as a closed-form "one line" expression - the line will just become very very long. I don't see how the expression itself would become less complex if you used splines instead of weights (even if this resulted in less neurons for the same decision boundary).
I guess in principle you could do something similar with MLPs but since MLP representations are sort of delocalized they might be harder to sparsify and symbolify
I understand how neural networks try to reduce their loss function to get the best result. But what's actually different about the KANs?
A regular neural network (MLP) has matrices full of floating point numbers that act as weights. A weight is a linear function y=wx, meaning if I plot the input x and output y on cartesian coordinates, it will generate a straight line. Increasing or decreasing the input also increases or decreases the output by consistent amounts. We won't have points where increasing the output suddenly has more or less effect than the previous increase, or starts sending the output in the other direction. So we train the network by having it learn multiple layers of these weights and also connecting them with some magic glue functions that are part of the design, not something that is trained up. The end result is the output can have a complex relationship with the input by being passed through all these layers.
In contrast, in a KAN rather than weights (acting as linear functions) we let the network learn other kinds of functions. These are nonlinear so it's possible that as we increase the input, the output keeps rising in an accelerating fashion, or turns around and starts decreasing. We can learn much more complex relationships between input and output, but lose some of the computational efficiency of the MLP approach (huge matrix operations are what GPUs are built for, while you need a CPU to do arbitrary math).
So with the KAN we end up with few but more complex "neurons", made up of complex functions. And if I understand what they're getting at here, the appeal of this is that you can inspect one of those neurons and get a clear formula that describes what it is doing, because all the complexity is distilled into a formula in the neuron. While with an MLP you have to track what is happening through multiple layers of weights and do more work to figure out how it all works.
Again I'm not in the space, but I imagine the functions that come out of a KAN still aren't super intuitive formulas that look like something out of Isaac Newton's notebooks, they're probably full of bizarre constants and unintuitive factors that cancel each other out.
perceptron layer is
output = simple_function( sum(many_inputs*many_weights) + extra_weight_for_bias)
a KAN layer is
output = sum(fancy_functions(many_inputs))
but I could be wrong, it's been a day.
The output of a KAN is a nice formula like exp(0.3sin(x) + 4cos(y)). This is what is meant by interpretable.
I think it'd be easy to check.
----
Anyways that's great news for differentiability. For now 'if' conditions expressed in JAX are tricky (at least for me), and are de facto an optimization barrier. If they're learnable and already into the network, I'd say that's a great thing.
Here's the recording https://youtu.be/FYYZZVV5vlY?si=ReoygVJMgY9oje3p
This seems like a big gap. Anyone know if this is a fundamental architecture mismatch, or just no one has written the required CUDA kernels yet?
I don't know why I'm interested because I'm not planning to actually do any work in the space, but I always struggle to understand when some new architecture is announced if it's a fundamental shift or if it's an optimization.
... given how, what you can "see" (ie. have visibility into) is something that regulatory stakeholders can ask you to exercise control over, or for oversight or information about ...
... whereas a "black box" they have trained and control - but few understand - can perhaps give you "plausible deniability" of the "we don't know how it works either" type.-