Language Modeling is Monotone Compression

summary

The gist

The gist: LLMs are equivalent to monotone compression algorithms up to an additive gap of 2, and this equivalence requires the existence of infinitely-often one-way functions.

In short

The paper investigates the relationship between Language Models (LLMs), which predict next tokens, and monotone compression algorithms. The main finding is that LLMs are equivalent to monotone compression schemes, meaning one can be constructed from the other with a small error gap of 2. This equivalence depends on whether infinitely-often one-way functions exist in cryptography.

Key concepts

Autoregressive Models: These are models like LLMs that predict the next item in a sequence based on all previous items. They generate data step-by-step, making them inherently sequential. The paper treats these as formal next-token predictors for comparison with compression methods.
Monotone Compression: This refers to compression algorithms where the encoding process preserves the order of the input data. If you have a sequence of items, the compressed representation must maintain that original ordering, which is a key constraint studied in this work.
Infinitely-Often One-Way Functions: These are cryptographic tools that are easy to compute in one direction but incredibly hard to reverse. The existence of these functions is crucial because it determines whether the equivalence between LLMs and monotone compression holds strictly or only approximately.
Next-Bit Pseudoentropy: This is a computational measure used to quantify the incompressibility of a data distribution. It relates how well a distribution can be compressed to the performance limits of autoregressive models, providing a formal link between information theory and modeling.

Terminology used across episodes

Reinforcement Learning (RL) — explained in 25 episodes
Retrieval-Augmented Generation (RAG) — explained in 16 episodes
Interpretability — explained in 15 episodes
Sample Complexity — explained in 15 episodes
Federated Learning — explained in 14 episodes
Generalization — explained in 13 episodes

This episode discusses

Language Modeling is Monotone Compression · Paper Radio

The paper

Language Modeling is Monotone Compression · Read on arXiv

Noam Mazor, Andrew Morgan, Rafael Pass

New York University · Cornell Tech

A long-standing hypothesis in artificial intelligence and neuroscience posits that intelligence is closely related to compression: the ability to compress information efficiently intuitively reflects capacities associated with intelligence and learning. Indeed, recent experimental works verify this intuition by showing connections between the capabilities of large language models (LLMs) and their ability as compressors: for instance, Deletang et al. (ICLR'24) demonstrate that LLMs can be used as powerful compressors, and Huang et al. (COLM'24) show that the compression ability of LLMs is highly correlated with their performance on benchmarks for knowledge and reasoning. In this work, we initiate a theoretical study of this connection. Our main result is that LLMs (formally modeled as next-token predictors) are equivalent to monotone (a.k.a. order-preserving) compression algorithms---namely, compression algorithms where the encoding process preserves the ordering of the inputs---in the sense that the one can be constructed from the other while preserving the same error up to an additive gap of 2. We next show that the monotonicity is required for this equivalence to hold if and only if cryptographic (infinitely-often) one-way functions exist. As a direct corollary, we get a cryptographic result of independent interest: the notion of next-bit pseudoentropy (a computational analogue of entropy) of a distribution is equivalent to monotone incompressibility of the distribution. (Previously, it was only known (Haitner et al., ITCS'23) that incompressibility implies next-bit pseudoentropy.)

Transcript

Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "Language Modeling is Monotone Compression".

Jane: The gist: LLMs are equivalent to monotone compression algorithms up to an additive gap of 2, and this equivalence requires the existence of infinitely-often one-way functions.

Tom: First, who's behind it and why it matters.

Paper summary: Tom: So we're moving on to summarizing "Language Modeling is Monotone Compression." The main thrust of this paper is establishing that language models—which are essentially next-token predictors—are equivalent to monotone compression schemes.

Jane: They claim this equivalence means you can construct the compression scheme from the model, and you can build a model from the compression scheme while keeping errors within a gap of two >

Lu: The paper formally states that given any distribution D and any error level delta, if you have an efficiently computable delta-optimal autoregressive model, you can make a monotone compression scheme that is at most delta plus two optimal for D >

Meng: It sounds like they're showing a direct construction route: model to compression and compression back to model, both with controlled accuracy >

Lalam: This moves the idea from just observing that LLMs are good compressors to mathematically proving *how* they are equivalent in terms of their predictive power on data distributions >

Tom: And the paper highlights that this equivalence specifically requires infinitely often one-way functions to hold true for the exact relationship to be maintained >

Jane: This is significant because it sets a condition for when we can trust the intuitive link between what an AI learns and how efficiently it can encode information >

Lu: It links next-bit pseudoentropy, which is a measure of incompressibility, directly to monotone incompressibility of the distribution D >

Meng: So if you can't compress something efficiently and monotonically infinitely often, the paper suggests that the AI model won't be able to predict that data very well either >

Lalam: This has implications for understanding limits on what any language model can achieve in terms of capturing complex data structures, not just simple patterns >

Conclusion: Tom: So wrapping up this discussion on "Language Modeling is Monotone Compression," the core finding is that the relationship between language modeling and compression isn't just a coincidence but a tight mathematical equivalence >

Jane: It shows that when we look at the right assumptions, like those about one-way functions, we can have a very precise quantitative link between how well an AI predicts data and how well it compresses it >

Lu: The authors show that this connection is tied to the existence of infinitely often one-way functions, which is a deep constraint on what's possible in this area >

Meng: For practical application, if those functions are assumed to exist, we have a strong theoretical guarantee that good language models translate directly into good compression tools >

Lalam: It suggests that the next step in this area is figuring out how to work around the requirement of these one-way functions if they don't exist without losing all this mathematical structure >

Tom: Right. The paper by Mazor and Morgan really solidifies that language modeling and compression are fundamentally linked, but it puts a clear constraint on the strength of that link >

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