DeepSeekMath-V2: Towards Self-Verifiable Mathematical Reasoning [pdf]

Well they do that too: https://huggingface.co/deepseek-ai/DeepSeek-Prover-V2-671B

But I suppose the bigger goal remains improving their language model, and this was an experimentation born from that. These works are symbiotic; the original DeepSeekMath resulted in GRPO, which eventually formed the backbone of their R1 model: https://arxiv.org/abs/2402.03300

More training data on advanced math. Lean is cool, but it's mostly about formalizing stuff we already know.

Natural language is a lot more, well, readable than say lean. You get a lot less intuition and understanding of what the model is attempting to do in the first place.

I think that mathematical proofs, as they are actually written, rely on natural language and on a large amount of implicit shared knowledge. They are not formalized in the Principia Mathematica sense, and they are even further from the syntax required by modern theorem provers. Even the most rigorous proofs such as those in Bourbaki are not directly translatable into a fully formal system.

Verifying math requires something like Lean which is a huge bottleneck, as the paper explains.

Plus there isn't a lot of training data in lean.

Most gains come from training on stuff already out there, not really the RLVR part which just amps it up a bit.

Maths can be super deterministic but often difficult to compute because of concepts like inferring by induction. I had to personally unlearn and rebase my understanding of math based in computation to 'get' pure maths. Another example is set building. You often don't need to compute the existence of members of sets in pure math you just need to agree that there are some members of a set that meet the criteria. How many or how many things that aren't in the set aren't meaningful often times to accept something and move on with the proof. From the computing perspective this can be difficult to put together.

I haven’t read the paper yet, but I’d imagine the issue is converting the natural language generated by the reasoner into a form where a formal verifier can be applied.

such high performance program indeed could potentially be superior, if it would exist (this area is very undeveloped, there is no existing distributed well established solution which could handle large domain) and math would be formalized in that program's dsl, which also didn't happen yet.

Advanced math solving, as the results indicate. Informal proof reasoning is advancing faster than formal proof reasoning because the latter is slow and compute intensive.

I suspect it's also because there isn't a lot of data to train on.