The results at https://www.mattmahoney.net/dc/text.html explicitly add the size of the compressor itself to the result. Note the "enwik9+prog" column. That's what it's ranked on.
The reason to do this is that it's trivial to create a compressor that 'compresses' a file to 0 bytes. Just have an executable with a dictionary of enwik9 that writes that out given any input. So we always measure what is effectively the Kolmogorov complexity. The data+program as a whole that produces the result we want.
So those results add in the compressor size. The programs there generally have no dictionary built in or in the case of LLM based compressors, no pre-trained data. They effectively build the model as they process data. Not compressing much at all at the start and slowly compressing better and better as they go. This is why these programs do better and better with larger data sets. They start with 0 knowledge. After a GB or so they have very good knowledge of the corpus of human language.
This program here however is pre-trained and shipped with a model. It's 150MB in size! This means it has 150MB of extra starting knowledge over those models in that list. The top models in that list are the better compressors, they'll quickly out learn and overtake this compressor but they just don't have that headstart.
Of course measuring fairly this should be listed with that 150MB program size added to the results when doing a comparison.
Reminded me of pi filesystem (https://github.com/philipl/pifs), with enough digits of pi precalculated you might be able to do a decent compression program. The trick is in the amount of reasonable digits for that, if it’s smaller or bigger than that trained LLM.
I suspect that the length of the offset of your input data in pi is equal to the length of the input data itself, plus or minus a few bytes at most, regardless of the size of the input data.
That is: no compression, but it won't make things worse either.
Unless the input data is the digits of pi, obviously, or the result of some computation involving pi.
I'm going to be the nerd that points out that it has not been mathematically proven that pi contains every substring, so the pifs might not work even in theory (besides being utterly impractical, of course).
On a more serious note, as far as I understand these compression competitions require that static data is included in the size computation. So if you compress 1000 MB into 500 MB, but to decompress you need a 1 MB binary and a 100 MB initial dictionary, your score would be 500 + 100 + 1 = 601 MB, not 500 MB.
The relevance to this discussion is that the LLM weights would have to be included as static data, since the only way to regenerate them is from the initial training data, which is much larger than the resulting model. By comparison, pi based compression is the other way around: since pi is a natural constant, if your decompressor requires (say) a trillion digits of pi, you could write a relatively small program (a few kb) to generate them. It would be terribly slow, but it wouldn't affect your compression ratio much.
This is something I have been curious about in terms of how an LLM's achieves compression.
I would like to know what deviations are in the output as this almost feels like a game of telephone where each re-compression results in a loss of data which is then incorrectly reconstructed. Sort of like misremembering a story and as you tell it over time the details change slightly.
When LLMs predict the next token, they actually produce a distribution of the probability of each of the possible next tokens, and the sampler chooses one of them, and not necessarily the most likely one!
If instead you run LLM prediction and then encode the probability of the next token of the input text you want to encode (from the cumulative distribution, a number in [0, 1]) using arithmetic coding, you can run the same operation in reverse to achieve lossless compression.
The tricky part is ensuring that your LLM executes absolutely deterministically, because you need to make sure that the encoder and decoder have the same probability distribution map at each step.
Arithmetic coding takes a prediction of the next bit and writes out exactly as many bits as needed to correct that prediction. The amazing part is that you can write out fractional bits. Eg. You predict the next bit is '1' with 75% probability? If it is 1 you only need to write out 1/2 of a bit (correcting that 25% portion). If it's 0 you need to write out 2.4bits. It may seem strange to work with 1/2 a bit but it works! (essentially the other half of the bit represents other future correction required). You might have heard of huffman coding which can't deal with fractional bits, arithmetic coding is a generalization of huffman coding that can deal with this.
Arithmetic coding is mathematically perfect at what it does. You will not waste a single bit using this algorithm to encode data given a prediction of that data.
So the entirety of modern compression techniques don't deal with the encoding/decoding side at all. What they deal with is modelling the data so far and making the most accurate prediction possible on the next bit of data (next byte also works, but working 1 bit at a time is easier to comprehend when learning arithmetic coding).
Incidentally the encoders and decoders essentially work exactly the same. Given the data read or data decoded so far predict the next bit. This part is exactly the same either way. The decoder would read the compressed file for the correction and the encoder would read the input file and write out the correction. The important part is "predict the next bit". This is what separates all the different compressors.
This is also where those of us experienced in this area try to correct people on the wrong track. A compression algorithm is never about the encoding side but instead 100% always about the prediction of the data. Can you build a model that can accurately predict what the next data to come is? That's what you need to do to make a better file compressor. The entropy encoding part is a completely solved problem already, don't bother re-solving that.
Lossy and lossless are essentially fundamentally the same actually. All state of the art lossy compressors use arithmetic coding as an example and they still do prediction. Eg. your favourite video codecs predict not only the next bit in the 2D frame, but also the next bit when modelling past frames (becomes a 3D cube of data at that point) and they also do things like motion prediction of individual objects in frame to help make a better prediction. They all use arithmetic encoders to encode the data.
Where the lossy part comes in is the point at which humans notice/don't notice data being thrown away. Got a bit that was waaay out of line in the prediction and going to cost you 10bits to correct? Perhaps to humans it doesn't matter? Can we throw it away? This throwing away of data is often done before the prediction+compression stage (eg. maybe quantizing the color space to 8bits from 32bits?) but from there on it's the same thing.
I love this because it gets to the heart of information theory. Shannon's foundational insight was that information is surprise. A random sequence is incompressible by definition. But what counts as surprise depends on context, and for text, we know a large amount of it is predictable slop. I suspect there's a lot of room to go along this style of compression. For example, maybe you could store an upfront summary that makes prediction more accurate. Or perhaps you could encode larger sequences or some kind of hierarchical encoding. But this is great.
Another fun application of combining LLMs with arithmetic coding is steganography. Here's a project I worked on a while back which effectively uses the opposite technique of what's being done here, to construct a steganographic transformation: https://github.com/shawnz/textcoder
"compressed size" does not seem to include the size of the model and the code to run it. According to the rules of Large Text Compression Benchmark, total size of those must be counted, otherwise a 0-byte "compressed" file with a decompressor containing the plaintext would win.
Technically correct, but a better benchmark would be a known compressor with an unknown set of inputs (that come from a real-world population, e.g. coherent English text).
Yeah, but the xz algorithm is also not counted in the bytes... Here the "program" is the LLM, much like your brain remembers things by coding them compressed and then reconstructs them. It is a different type of compression: compression by "understanding", which requires the whole corpus of possible inputs in some representation. The comparison is not fair to classical algorithms yet that's how you can compress a lot more (given a particular language): by having a model of it.
True for competitions, but if your compression algorithm is general purpose then this matters less (within reason - no one wants to lug around a 1TB compression program).
Current leader of the Large Text Compression Benchmark is NNCP (compression using neural networks), also by Fabrice Bellard:
https://bellard.org/nncp/
Also, nncp-2024-06-05.tar.gz is just 1180969 bytes, unlike ts_zip-2024-03-02.tar.gz (159228453 bytes, which is bigger than uncompressed enwiki8).
Looks like it beats everything in the large text compression benchmark for enwik8, but loses to several programs for enwik9. I wonder why that is.
It's actually not the best at enwik8 or 9.
The results at https://www.mattmahoney.net/dc/text.html explicitly add the size of the compressor itself to the result. Note the "enwik9+prog" column. That's what it's ranked on.
The reason to do this is that it's trivial to create a compressor that 'compresses' a file to 0 bytes. Just have an executable with a dictionary of enwik9 that writes that out given any input. So we always measure what is effectively the Kolmogorov complexity. The data+program as a whole that produces the result we want.
So those results add in the compressor size. The programs there generally have no dictionary built in or in the case of LLM based compressors, no pre-trained data. They effectively build the model as they process data. Not compressing much at all at the start and slowly compressing better and better as they go. This is why these programs do better and better with larger data sets. They start with 0 knowledge. After a GB or so they have very good knowledge of the corpus of human language.
This program here however is pre-trained and shipped with a model. It's 150MB in size! This means it has 150MB of extra starting knowledge over those models in that list. The top models in that list are the better compressors, they'll quickly out learn and overtake this compressor but they just don't have that headstart.
Of course measuring fairly this should be listed with that 150MB program size added to the results when doing a comparison.
As an aside, I wonder how to account for the information content embedded in the hardware itself.
A Turing Machine compressor program would likely have more bytes than the amd64 binary. So how to evaluate KolmogorovComplexity(amd64)?
The laws of physics somehow need to be accounted for too, probably.
Kolmogorov Complexity is only defined up to a constant, which represents Turing machine translation length.
Reminded me of pi filesystem (https://github.com/philipl/pifs), with enough digits of pi precalculated you might be able to do a decent compression program. The trick is in the amount of reasonable digits for that, if it’s smaller or bigger than that trained LLM.
I suspect that the length of the offset of your input data in pi is equal to the length of the input data itself, plus or minus a few bytes at most, regardless of the size of the input data.
That is: no compression, but it won't make things worse either.
Unless the input data is the digits of pi, obviously, or the result of some computation involving pi.
I'm going to be the nerd that points out that it has not been mathematically proven that pi contains every substring, so the pifs might not work even in theory (besides being utterly impractical, of course).
On a more serious note, as far as I understand these compression competitions require that static data is included in the size computation. So if you compress 1000 MB into 500 MB, but to decompress you need a 1 MB binary and a 100 MB initial dictionary, your score would be 500 + 100 + 1 = 601 MB, not 500 MB.
The relevance to this discussion is that the LLM weights would have to be included as static data, since the only way to regenerate them is from the initial training data, which is much larger than the resulting model. By comparison, pi based compression is the other way around: since pi is a natural constant, if your decompressor requires (say) a trillion digits of pi, you could write a relatively small program (a few kb) to generate them. It would be terribly slow, but it wouldn't affect your compression ratio much.
This only does 1 byte, so you only have to prove it contains the bits for 0-255.
>> The ts_zip utility can compress (and hopefully decompress) text files
Hopefully :-)
Reading data is overrated. I highly recommend S4:
http://www.supersimplestorageservice.com/
Compression and intelligence reminded me of the https://www.hutter1.net/prize
I've encountered it >10 years ago and it felt novel that compression is related to intelligence and even AGI.
So did beat his own leading program from 2019, nncp, finally.
This is something I have been curious about in terms of how an LLM's achieves compression.
I would like to know what deviations are in the output as this almost feels like a game of telephone where each re-compression results in a loss of data which is then incorrectly reconstructed. Sort of like misremembering a story and as you tell it over time the details change slightly.
When LLMs predict the next token, they actually produce a distribution of the probability of each of the possible next tokens, and the sampler chooses one of them, and not necessarily the most likely one!
If instead you run LLM prediction and then encode the probability of the next token of the input text you want to encode (from the cumulative distribution, a number in [0, 1]) using arithmetic coding, you can run the same operation in reverse to achieve lossless compression.
The tricky part is ensuring that your LLM executes absolutely deterministically, because you need to make sure that the encoder and decoder have the same probability distribution map at each step.
Yes. The secret is in understanding arithmetic coding. https://en.wikipedia.org/wiki/Arithmetic_coding
Arithmetic coding takes a prediction of the next bit and writes out exactly as many bits as needed to correct that prediction. The amazing part is that you can write out fractional bits. Eg. You predict the next bit is '1' with 75% probability? If it is 1 you only need to write out 1/2 of a bit (correcting that 25% portion). If it's 0 you need to write out 2.4bits. It may seem strange to work with 1/2 a bit but it works! (essentially the other half of the bit represents other future correction required). You might have heard of huffman coding which can't deal with fractional bits, arithmetic coding is a generalization of huffman coding that can deal with this.
Arithmetic coding is mathematically perfect at what it does. You will not waste a single bit using this algorithm to encode data given a prediction of that data.
So the entirety of modern compression techniques don't deal with the encoding/decoding side at all. What they deal with is modelling the data so far and making the most accurate prediction possible on the next bit of data (next byte also works, but working 1 bit at a time is easier to comprehend when learning arithmetic coding).
Incidentally the encoders and decoders essentially work exactly the same. Given the data read or data decoded so far predict the next bit. This part is exactly the same either way. The decoder would read the compressed file for the correction and the encoder would read the input file and write out the correction. The important part is "predict the next bit". This is what separates all the different compressors.
This is also where those of us experienced in this area try to correct people on the wrong track. A compression algorithm is never about the encoding side but instead 100% always about the prediction of the data. Can you build a model that can accurately predict what the next data to come is? That's what you need to do to make a better file compressor. The entropy encoding part is a completely solved problem already, don't bother re-solving that.
That's true for lossless compression. Is there a generalization for lossy compression?
Lossy and lossless are essentially fundamentally the same actually. All state of the art lossy compressors use arithmetic coding as an example and they still do prediction. Eg. your favourite video codecs predict not only the next bit in the 2D frame, but also the next bit when modelling past frames (becomes a 3D cube of data at that point) and they also do things like motion prediction of individual objects in frame to help make a better prediction. They all use arithmetic encoders to encode the data.
Where the lossy part comes in is the point at which humans notice/don't notice data being thrown away. Got a bit that was waaay out of line in the prediction and going to cost you 10bits to correct? Perhaps to humans it doesn't matter? Can we throw it away? This throwing away of data is often done before the prediction+compression stage (eg. maybe quantizing the color space to 8bits from 32bits?) but from there on it's the same thing.
When Jeff Dean gets stuck, he asks Bellard for help...
I love this because it gets to the heart of information theory. Shannon's foundational insight was that information is surprise. A random sequence is incompressible by definition. But what counts as surprise depends on context, and for text, we know a large amount of it is predictable slop. I suspect there's a lot of room to go along this style of compression. For example, maybe you could store an upfront summary that makes prediction more accurate. Or perhaps you could encode larger sequences or some kind of hierarchical encoding. But this is great.
Yes! information is surprise, and that's why a measure of intelligence is the ability to predict.
Another fun application of combining LLMs with arithmetic coding is steganography. Here's a project I worked on a while back which effectively uses the opposite technique of what's being done here, to construct a steganographic transformation: https://github.com/shawnz/textcoder
"compressed size" does not seem to include the size of the model and the code to run it. According to the rules of Large Text Compression Benchmark, total size of those must be counted, otherwise a 0-byte "compressed" file with a decompressor containing the plaintext would win.
Technically correct, but a better benchmark would be a known compressor with an unknown set of inputs (that come from a real-world population, e.g. coherent English text).
Yeah, but the xz algorithm is also not counted in the bytes... Here the "program" is the LLM, much like your brain remembers things by coding them compressed and then reconstructs them. It is a different type of compression: compression by "understanding", which requires the whole corpus of possible inputs in some representation. The comparison is not fair to classical algorithms yet that's how you can compress a lot more (given a particular language): by having a model of it.
“Compressors are ranked by the compressed size of enwik9 (10^9 bytes) plus the size of a zip archive containing the decompresser.” [0]
[0] https://www.mattmahoney.net/dc/text.html
True for competitions, but if your compression algorithm is general purpose then this matters less (within reason - no one wants to lug around a 1TB compression program).
I propose the name tokables for the compressed data produced by this. A play on tokens and how wild it is.
please pass the tokables to the left hand side
so barely 2 or 3 times better than xz
not really worth it
Bellard finally working with his true colleague.