Mathematical Introduction to Deep Learning: Methods, Implementations, and Theory

I think people in my camp (which often are confused with the Gary Marcus camp), aren't saying you shouldn't be blown away. Those people wouldn't say this

> And don't confuse this criticism as saying LLMs suck, because I use them almost every day and love them. I just don't get why we can't be realistic about their limits and can only believe they are a golden goose or pile of shit. It's, again, neither.

Fwiw, I give those people an ever harder time. They deny utility that is quite apparent. They also have these silly contrived doomer arguments that don't make any sense, as if one day AGI is just going to unexpectedly appear out of nowhere and, like you suggest, somehow jump the airgap and get control of the world's nuclear weapons without anyone noticing. What an insane hypothesis that doesn't have anything substantial evidence and is entirely based on "but what if!" It is conspiratorial and a distraction from the real harm these systems can do which is far more subtle and not really an existential crisis (at least arguably in the same way, but let's not get into that). Some of these people are shills and some are useful idiots/true believers. You're right to not pay attention to them.

I'll also mention that I too am blown away. But you can be blown away and still have criticism and be wary of a thing too. The answer is quite impressive, without a doubt. I mean we are literally putting lightning into rocks and making them capable of doing math and speaking human languages. If you're not blown away by any single one of those things then it is simply a lack of imagination.

> When's the last time you saw a human mind that didn't work that way?

Quite frequently. Same with even my cat, and she's dumb as shit. Probably ran into too many walls while chasing toys but I think that's just a feedback loop lol. She's dumb as shit but I'm also absolutely blown away by her brilliance. It may be hard to see that both those can be true, but that's the true state. But I disagree that there isn't anything special about stochastic parrots, any animal, or humans. They are all mind mindbogglingly impressive, just our brains are designed to normalize things to not be overburdened by the computational load (which itself is impressive!).

You are absolutely right though that there's a ton of exploitation that humans do (referring to exploration vs exploitation). I said memorization is incredibly useful. But creativity is far more subtle. I should put it this way, chimps (very impressive creatures), are far better at memorization than most humans. But they are nowhere near as creative. Certainly some creativity is leveraging prior works for inspiration. But a subtle aspect of this is that often when this form is considered brilliant it crosses domains, which is something no ML seems to even have the capacity to perform. This can be hard to know though because unless you have domain knowledge you may not have heard about how people like Einstein was called a mathematician and not a physicist or how Nash was said to "just used topology". This type of lore is important if we're going to discuss actual intelligence but not important for tools or our every day lives. The devil is in the details when we care about details.

It can be really hard to understand these distinctions. You have to look REALLY close at details. One thing I'll mention is that I know I have looked at the datasets we use in our group far more than anyone else that I know. This is unsurprisingly an uncommon thing because it is boring to look at the raw data and investigating things like LAION takes herculean efforts (something I haven't even approached). But your example is actually remarkably relevant to this topic. You couldn't have done anything better! Because most people rely on measurements of distance like cosine similarity or L2 to determine duplicates or near duplicates. But ask GPT this (you should get the right answer): "How does the curse of dimensionality relate to distance measurements in higher dimensions? Are there any problems this creates?" Or ask it another one, which even the fact blew me away the first time I heard it despite being absolutely obvious after I took just a moment to think about it: "If I have a n dimensional space, where n is very large, what is the expected angle between any two random tensors? What is it as n approaches infinity?" I'm positive it will again give you the correct answer.

But you also have to realize that this is frequently written about and without a doubt in the training data. You can absolutely overfit models and have them be incredibly useful. But the difference is that this won't be generalization and will be brittle. For a long time GPT was not able to correctly answer "Which weighs more, a pound of feathers or a kilogram of bricks" because it was too sensitive to the expected answer (it'll work now btw). It still has problems with a variation of the corn, goose, fox river crossing puzzle if you change it to allow all items in the boat at once (at least when I checked a month ago). But this is not the actions of sentient creatures. Ones that can think and comprehend. You're going to have to think really hard about how you think and especially how you think really hard to get a good understanding of this. But it comes down to the reason why someone can be absolutely brilliant while shockingly idiotic. This is not the quip from iRobot with the "can you?" about art and symphonies. There is something deeper and truth be told, many animals do things for no good reason (one that can't be clearly defined by our perceived loss functions, which may accurately be called emergent behaviors). Every mammal also is able to run complex simulations in their minds, at incredibly low computational costs. Even the small rat will twitch its legs while it sleeps or your dog may bark, being unable to distinguish reality from a dream, just as you do. That is truly a world model. Something we aren't remotely close to in AI, but that's okay. Why would it not be okay?

But in some way you are being gaslit, but not by what I intended to say (but maybe from how you read it. Which I apologize, I am trying to work on communicating better, but it is hard when we have a diverse global audience with many different base assumptions and knowing which type of imputation I need to direct my message at). There are plenty of people with highly invested interest to sell you these tools as far more than they are. I've written a few comments before that what's going on is as if we made a chocolate factory. One that sells the best god damn chocolate you're ever tasted. But then they started selling the chocolate as a cure for cancer. At that point, it doesn't matter how good the chocolate is, people will feel disenfranchised. Some people are responding by saying that the chocolate tastes like shit while others are saying it cured their cancer. But neither of these are true. It's damn good chocolate, but it isn't going to cure cancer. (ML certainly will be a very useful tool for tackling cancer. That was not the intent of this analogy) I just think there's this fear that people have that if something isn't a literal gift from god then it is a pile of shit, and I don't get it. Nothing we have fits that description but we have done and created so many incredible things as humans and made such leaps and bounds with these half baked incomplete things. There is nothing wrong with just okay chocolate, but the chocolate we have is without a doubt, better than just okay.

Does that make more sense?

My main point consistently has been that GPT4 can be an invaluable resource specifically for learning math subjects.

I am not aware of any papers, studying people using it as a conversational tutor for learning math and having problems with hallucinations.

From page 15:

> The one piece of new notation is the exponent on R^2. This means "pairs" of real numbers.

Your interpretation of this quote is uncharitable at best. Using it to make a blanket assertion about the book is just silly, and quite out of the spirit of mathematics.

In particular, page 19 has an example of the kind of things that my book has that other books don't: a discussion of the soft skills of learning math and the cultural acclimation process:

> Though it sometimes makes me cringe to say it, give the author the benefit of the doubt. When things are ambiguous, pick the option that doesn’t break the math. In this respect, you have to act as both the tester, the compiler, and the bug fixer when you’re reading math. The best default assumption is that the author is far smarter than we are, and if you don’t understand something, it’s likely a user error and not a bug. In the occasional event that the author is wrong, it’s often a simple mistake or typo, to which an experienced reader would say, “The author obviously meant ‘foo’ because otherwise none of this makes sense,” and continue unscathed.

The course you suggested is the sort of "grab bag of topics" course, meant to cram the basics of every topic a CS major might want to know for doing the kind of CS theory research that MIT cares about. If you find math hard, I doubt that will make it much easier, but it could be good to do alongside a book like mine if you find my book too easy.

It looks like it might be traceable to "the Golden Bough" (1890):

“The cat, which represented the devil, could never suffer enough.” From https://www.gutenberg.org/cache/epub/3623/pg3623-images.html