Formal Mathematics Statement Curriculum Learning

We explore the use of expert iteration in the context of language modeling applied to formal mathematics. We show that at same compute budget, expert iteration, by which we mean proof search interleaved with learning, dramatically outperforms proof search only. We also observe that when applied to a collection of formal statements of sufficiently varied difficulty, expert iteration is capable of finding and solving a curriculum of increasingly difficult problems, without the need for associated ground-truth proofs. Finally, by applying this expert iteration to a manually curated set of problem statements, we achieve state-of-the-art on the miniF2F benchmark, automatically solving multiple challenging problems drawn from high school olympiads.

Unsupervised Neural Machine Translation with Generative Language Models Only

We show how to derive state-of-the-art unsupervised neural machine translation systems from generatively pre-trained language models. Our method consists of three steps: few-shot amplification, distillation, and backtranslation. We first use the zero-shot translation ability of large pre-trained language models to generate translations for a small set of unlabeled sentences. We then amplify these zero-shot translations by using them as few-shot demonstrations for sampling a larger synthetic dataset. This dataset is distilled by discarding the few-shot demonstrations and then fine-tuning. During backtranslation, we repeatedly generate translations for a set of inputs and then fine-tune a single language model on both directions of the translation task at once, ensuring cycle-consistency by swapping the roles of gold monotext and generated translations when fine-tuning. By using our method to leverage GPT-3's zero-shot translation capability, we achieve a new state-of-the-art in unsupervised translation on the WMT14 English-French benchmark, attaining a BLEU score of 42.1.

MiniF2F: a cross-system benchmark for formal Olympiad-level mathematics

We present miniF2F, a dataset of formal Olympiad-level mathematics problems statements intended to provide a unified cross-system benchmark for neural theorem proving. The miniF2F benchmark currently targets Metamath, Lean, and Isabelle and consists of 488 problem statements drawn from the AIME, AMC, and the International Mathematical Olympiad (IMO), as well as material from high-school and undergraduate mathematics courses. We report baseline results using GPT-f, a neural theorem prover based on GPT-3 and provide an analysis of its performance. We intend for miniF2F to be a community-driven effort and hope that our benchmark will help spur advances in neural theorem proving.

Proof Artifact Co-training for Theorem Proving with Language Models

Labeled data for imitation learning of theorem proving in large libraries of formalized mathematics is scarce as such libraries require years of concentrated effort by human specialists to be built. This is particularly challenging when applying large Transformer language models to tactic prediction, because the scaling of performance with respect to model size is quickly disrupted in the data-scarce, easily-overfitted regime. We propose PACT ({\bf P}roof {\bf A}rtifact {\bf C}o-{\bf T}raining), a general methodology for extracting abundant self-supervised data from kernel-level proof terms for co-training alongside the usual tactic prediction objective. We apply this methodology to Lean, an interactive proof assistant which hosts some of the most sophisticated formalized mathematics to date. We instrument Lean with a neural theorem prover driven by a Transformer language model and show that PACT improves theorem proving success rate on a held-out suite of test theorems from 32\% to 48\%.

Generative Language Modeling for Automated Theorem Proving

We explore the application of transformer-based language models to automated theorem proving. This work is motivated by the possibility that a major limitation of automated theorem provers compared to humans -- the generation of original mathematical terms -- might be addressable via generation from language models. We present an automated prover and proof assistant, GPT-f, for the Metamath formalization language, and analyze its performance. GPT-f found new short proofs that were accepted into the main Metamath library, which is to our knowledge, the first time a deep-learning based system has contributed proofs that were adopted by a formal mathematics community.