A Lean Dataset for International Math Olympiad: Small Steps towards Writing Math Proofs for Hard Problems

Using AI to write formal proofs for mathematical problems is a challenging task that has seen some advancements in recent years. Automated systems such as Lean can verify the correctness of proofs written in formal language, yet writing the proofs in formal language can be challenging for humans and machines. The miniF2F benchmark has 20 IMO problems in its testing set, yet formal proofs are available only for 7 of these problems (3 of which are written only by mathematicians). The model with best accuracy can only prove 4 of these 20 IMO problems, from 1950s and 60s, while its training set is a secret. In this work, we write complete, original formal proofs for the remaining 13 IMO problems in Lean along with 3 extra problems from IMO 2022 and 2023. This effort expands the availability of proof currently in the public domain by creating 5,150 lines of Lean proof. The goal of the paper is to pave the way for developing AI models that can automatically write the formal proofs for all the IMO problems in miniF2F and beyond. In this pursuit, we devise a method to decompose the proof of these problems into their building blocks, constructing a dataset of about 900 lemmas with 25,500 lines of Lean code. These lemmas are not trivial, yet they are approachable, providing the opportunity to evaluate and diagnose the failures and successes of AI models. We then evaluate the ability of GPT-4 in writing formal proofs for these lemmas with zero shot prompting, CoT reasoning and lemma retrieval. In evaluating the responses, we also analyze the confounding factor of LLM's ability to write the proofs in natural language vs Lean language.

Towards a Scalable Reference-Free Evaluation of Generative Models

While standard evaluation scores for generative models are mostly reference-based, a reference-dependent assessment of generative models could be generally difficult due to the unavailability of applicable reference datasets. Recently, the reference-free entropy scores, VENDI and RKE, have been proposed to evaluate the diversity of generated data. However, estimating these scores from data leads to significant computational costs for large-scale generative models. In this work, we leverage the random Fourier features framework to reduce the computational price and propose the Fourier-based Kernel Entropy Approximation (FKEA) method. We utilize FKEA's approximated eigenspectrum of the kernel matrix to efficiently estimate the mentioned entropy scores. Furthermore, we show the application of FKEA's proxy eigenvectors to reveal the method's identified modes in evaluating the diversity of produced samples. We provide a stochastic implementation of the FKEA assessment algorithm with a complexity $O(n)$ linearly growing with sample size $n$. We extensively evaluate FKEA's numerical performance in application to standard image, text, and video datasets. Our empirical results indicate the method's scalability and interpretability applied to large-scale generative models. The codebase is available at https://github.com/aziksh-ospanov/FKEA.

Large Language Models' Understanding of Math: Source Criticism and Extrapolation

It has been suggested that large language models such as GPT-4 have acquired some form of understanding beyond the correlations among the words in text including some understanding of mathematics as well. Here, we perform a critical inquiry into this claim by evaluating the mathematical understanding of the GPT-4 model. Considering that GPT-4's training set is a secret, it is not straightforward to evaluate whether the model's correct answers are based on a mathematical understanding or based on replication of proofs that the model has seen before. We specifically craft mathematical questions which their formal proofs are not readily available on the web, proofs that are more likely not seen by the GPT-4. We see that GPT-4 is unable to solve those problems despite their simplicity. It is hard to find scientific evidence suggesting that GPT-4 has acquired an understanding of even basic mathematical concepts. A straightforward way to find failure modes of GPT-4 in theorem proving is to craft questions where their formal proofs are not available on the web. Our finding suggests that GPT-4's ability is to reproduce, rephrase, and polish the mathematical proofs that it has seen before, and not in grasping mathematical concepts. We also see that GPT-4's ability to prove mathematical theorems is continuously expanding over time despite the claim that it is a fixed model. We suggest that the task of proving mathematical theorems in formal language is comparable to the methods used in search engines such as Google while predicting the next word in a sentence may be a misguided approach, a recipe that often leads to excessive extrapolation and eventual failures. Prompting the GPT-4 over and over may benefit the GPT-4 and the OpenAI, but we question whether it is valuable for machine learning or for theorem proving.

Should Machine Learning Models Report to Us When They Are Clueless?

The right to AI explainability has consolidated as a consensus in the research community and policy-making. However, a key component of explainability has been missing: extrapolation, which describes the extent to which AI models can be clueless when they encounter unfamiliar samples (i.e., samples outside a convex hull of their training sets, as we will explain down below). We report that AI models extrapolate outside their range of familiar data, frequently and without notifying the users and stakeholders. Knowing whether a model has extrapolated or not is a fundamental insight that should be included in explaining AI models in favor of transparency and accountability. Instead of dwelling on the negatives, we offer ways to clear the roadblocks in promoting AI transparency. Our analysis commentary accompanying practical clauses useful to include in AI regulations such as the National AI Initiative Act in the US and the AI Act by the European Commission.

To what extent should we trust AI models when they extrapolate?

Many applications affecting human lives rely on models that have come to be known under the umbrella of machine learning and artificial intelligence. These AI models are usually complicated mathematical functions that map from an input space to an output space. Stakeholders are interested to know the rationales behind models' decisions and functional behavior. We study this functional behavior in relation to the data used to create the models. On this topic, scholars have often assumed that models do not extrapolate, i.e., they learn from their training samples and process new input by interpolation. This assumption is questionable: we show that models extrapolate frequently; the extent of extrapolation varies and can be socially consequential. We demonstrate that extrapolation happens for a substantial portion of datasets more than one would consider reasonable. How can we trust models if we do not know whether they are extrapolating? Given a model trained to recommend clinical procedures for patients, can we trust the recommendation when the model considers a patient older or younger than all the samples in the training set? If the training set is mostly Whites, to what extent can we trust its recommendations about Black and Hispanic patients? Which dimension (race, gender, or age) does extrapolation happen? Even if a model is trained on people of all races, it still may extrapolate in significant ways related to race. The leading question is, to what extent can we trust AI models when they process inputs that fall outside their training set? This paper investigates several social applications of AI, showing how models extrapolate without notice. We also look at different sub-spaces of extrapolation for specific individuals subject to AI models and report how these extrapolations can be interpreted, not mathematically, but from a humanistic point of view.