The Slogan Strikes Again
One of the slogans heard quite frequently in Information Theory educated circles is: Compression is Intelligence. You aren’t supposed to take this too literally. The point is that it is a useful intuition about why (for example) we might expect large machine learning models to possess something that looks to us like intelligence with respect to their training domain.
The unreasonable effectiveness of LLMs is perhaps the foremost embodiment of the slogan today. One can view the training goal of the LLM as the compression of the human text corpus into its weights. If the model were sufficiently large, it could simply memorize its training data to achieve perfect loss. For the moment, though, the model is somewhat too small to do this, and so it must settle for learning something instead.
To get an intuition for how this learning process works, consider the problem of compressing mathematics. Random text is difficult to compress, so the situation would not be good if mathematical writing looked like this:
The translucent argument galloped beneath several yesterday, while gentle equations devoured the patient silence of forgotten triangles. Punctual sorrow whispered toward the hexagonal mountain, since brittle laughter cannot inhabit the velvet hypothesis. Therefore, the seventh ocean apologized quietly, and three reluctant Tuesdays married the indifferent square that had been dreaming of soluble thunder.[1]
Thankfully, it does not. Instead, it looks rather more like this:
Let
be an element of order in , and let be the cyclic group generated by . Since is contained in , it is normal. Let be the canonical map. Let be the highest power of dividing . Then divides the order of . Let be a -Sylow subgroup of (by induction) and let . Then and maps onto . Hence we have an isomorphism .[2]
Even readers unfamiliar with the language of the second paragraph should see that it contains much more structure. We declare objects and manipulate them according to well-defined rules.
A compressor will take advantage of this structure. It will catalog common objects, understand what operations on them are permitted, and identify objects that often appear together. To a great extent, this is exactly the same process that human students of mathematics go through in their early education. One learns about some object and asks:
What is my object? What can be done with it? What structure do I always know is present? Which instances of it appear “in nature”? Which ones behave in unintuitive ways? What other objects are its fellow travelers and why?
The answer to each of these questions represents a little piece of structure that a compressor would be foolish not to exploit.
Conceptual revolutions happen in mathematics when someone realizes there is some frequently appearing object that we cannot answer the above questions about.
Groups, for example, arose in the study of symmetric polynomials as an abstraction of the symmetry being studied. However, once the abstraction had been made, mathematicians in far-flung areas realized that many things they wanted to talk about were in fact groups, and so the field was advanced.
The challenging part in this is noticing that there is some interesting relevant structure in your problem space that lacks an abstraction.
This general pattern of identifying latent information about your problem and producing an abstraction to capture that information exists across all fields of science and is a major driver of progress in each.
If we want AI models to one day be effective researchers, they must be able to make good abstractions. Claude Mythos appears to be capable of doing this in a limited capacity, which is enormously exciting.
This article notices that some extremely strange Mythos CoT outputs are, in fact, quite legible when you take the time to learn the language that Mythos has taught itself for that particular problem.
Mythos teaches itself new languages for particular problems. Holy shit! How cool is that!
The model inspected the problem space of the game it was asked to play and successfully identified the relevant objects and the permitted operations on those objects. It then invented a language capture that abstraction and used this language to play much more efficiently than it could have done in English text.
The above paragraph is probably anthropomorphizing Mythos too much. The model likely learned this behavior in response to limits on reasoning tokens in reinforcement training. When you have a buffer smaller than the information you want to put into it, you must compress. Nonetheless, this behavior is exactly what is necessary to do really significant work in mathematics and other disciplines.
My current instinctual feeling is that the ability to conjure up new languages is a major contributing factor in the incredible performance of Mythos. The ability to compress a problem space in situ like this seems terribly underrated to me and, like everything else, it will only get better from here. The slogan gave us LLMs, and in this funny Mythos CoT output we are seeing it strike again.
Thanks for writing this, I think it’s a useful framing to keep in mind!
Compressing the web has certainly taken us quite far, however note that LLMs already far outperform the best humans at the Shannon
guessing game[1], and yet we are still much more generally intelligent, meaning our compressor is much more universally applicable.
Perhaps the RLVR objective is more useful for getting more general intelligence juice out of compressing bits, though so far it doesn’t seem to generalize well[2]
To me, and keep in mind I’m not a mathematician, the distribution that mathematical abstractions compress in this framing has always been a
bit mysterious. It’s certainly useful for compressing sensory information / empirical reality, yet I don’t think that’s all it comes down to.
You can estimate this via the Chinchilla-optimal loss for a realistic amount of frontier compute↩︎
″ Agents seemed much weaker in domains where
hill-climbing was difficult or risky, often making critical judgment errors that competent
humans would have been unlikely to make”, from page 17 of the METR Report↩︎
Yes, I certainly agree that our compressor is currently more universally applicable, which is exactly why I am excited about seeing this sort of auto-compressing behavior in models like Mythos (and [now Sonnet 5 also](https://thezvi.substack.com/i/204364347/illegible-thinking-645)).
The basic idea, as I think of it, is this: in mathematics we have objects and we have facts about those objects. For example, if you consider the integers,
is something called a normal subgroup of
If you didn’t know what a group was, you would have produce obscure domain specific language to communicate these same ideas (as, for example, the founding fathers of number theory did, since they worked before groups). This would put a great mental burden on you, especially if you are interested in several subjects. Instead, we compress all this domain specific language into a single abstraction called a group and instead of saying “I employ this fact about object A that I spent 5 years proving” one instead says “This is a group and by standard facts about groups, such and such follows”. I think this is the practical sense in which abstraction is compression in mathematics, although I’m sure if you knew more information theory than I do you could make a nicer, more theoretical, argument.