Simon Frieder

Ryan-Rhys Griffiths, Leo Klarner, Henry B. Moss et al. · cambridge

We introduce GAUCHE, a library for GAUssian processes in CHEmistry. Gaussian processes have long been a cornerstone of probabilistic machine learning, affording particular advantages for uncertainty quantification and Bayesian optimisation. Extending Gaussian processes to chemical representations, however, is nontrivial, necessitating kernels defined over structured inputs such as graphs, strings and bit vectors. By defining such kernels in GAUCHE, we seek to open the door to powerful tools for uncertainty quantification and Bayesian optimisation in chemistry. Motivated by scenarios frequently encountered in experimental chemistry, we showcase applications for GAUCHE in molecular discovery and chemical reaction optimisation. The codebase is made available at https://github.com/leojklarner/gauche

Simon Frieder, Luca Pinchetti, Alexis Chevalier et al. · cambridge

We investigate the mathematical capabilities of two iterations of ChatGPT (released 9-January-2023 and 30-January-2023) and of GPT-4 by testing them on publicly available datasets, as well as hand-crafted ones, using a novel methodology. In contrast to formal mathematics, where large databases of formal proofs are available (e.g., the Lean Mathematical Library), current datasets of natural-language mathematics, used to benchmark language models, either cover only elementary mathematics or are very small. We address this by publicly releasing two new datasets: GHOSTS and miniGHOSTS. These are the first natural-language datasets curated by working researchers in mathematics that (1) aim to cover graduate-level mathematics, (2) provide a holistic overview of the mathematical capabilities of language models, and (3) distinguish multiple dimensions of mathematical reasoning. These datasets also test whether ChatGPT and GPT-4 can be helpful assistants to professional mathematicians by emulating use cases that arise in the daily professional activities of mathematicians. We benchmark the models on a range of fine-grained performance metrics. For advanced mathematics, this is the most detailed evaluation effort to date. We find that ChatGPT can be used most successfully as a mathematical assistant for querying facts, acting as a mathematical search engine and knowledge base interface. GPT-4 can additionally be used for undergraduate-level mathematics but fails on graduate-level difficulty. Contrary to many positive reports in the media about GPT-4 and ChatGPT's exam-solving abilities (a potential case of selection bias), their overall mathematical performance is well below the level of a graduate student. Hence, if your goal is to use ChatGPT to pass a graduate-level math exam, you would be better off copying from your average peer!

Katherine M. Collins, Albert Q. Jiang, Simon Frieder et al. · cambridge

There is much excitement about the opportunity to harness the power of large language models (LLMs) when building problem-solving assistants. However, the standard methodology of evaluating LLMs relies on static pairs of inputs and outputs, and is insufficient for making an informed decision about which LLMs and under which assistive settings can they be sensibly used. Static assessment fails to account for the essential interactive element in LLM deployment, and therefore limits how we understand language model capabilities. We introduce CheckMate, an adaptable prototype platform for humans to interact with and evaluate LLMs. We conduct a study with CheckMate to evaluate three language models (InstructGPT, ChatGPT, and GPT-4) as assistants in proving undergraduate-level mathematics, with a mixed cohort of participants from undergraduate students to professors of mathematics. We release the resulting interaction and rating dataset, MathConverse. By analysing MathConverse, we derive a taxonomy of human behaviours and uncover that despite a generally positive correlation, there are notable instances of divergence between correctness and perceived helpfulness in LLM generations, amongst other findings. Further, we garner a more granular understanding of GPT-4 mathematical problem-solving through a series of case studies, contributed by expert mathematicians. We conclude with actionable takeaways for ML practitioners and mathematicians: models that communicate uncertainty respond well to user corrections, and are more interpretable and concise may constitute better assistants. Interactive evaluation is a promising way to navigate the capability of these models; humans should be aware of language models' algebraic fallibility and discern where they are appropriate to use.

Emanuele La Malfa, Aleksandar Petrov, Simon Frieder et al. · oxford

Some of the most powerful language models currently are proprietary systems, accessible only via (typically restrictive) web or software programming interfaces. This is the Language-Models-as-a-Service (LMaaS) paradigm. In contrast with scenarios where full model access is available, as in the case of open-source models, such closed-off language models present specific challenges for evaluating, benchmarking, and testing them. This paper has two goals: on the one hand, we delineate how the aforementioned challenges act as impediments to the accessibility, replicability, reliability, and trustworthiness of LMaaS. We systematically examine the issues that arise from a lack of information about language models for each of these four aspects. We conduct a detailed analysis of existing solutions and put forth a number of considered recommendations, and highlight the directions for future advancements. On the other hand, it serves as a comprehensive resource for existing knowledge on current, major LMaaS, offering a synthesized overview of the licences and capabilities their interfaces offer.

Luca Pinchetti, Chang Qi, Oleh Lokshyn et al.

In this work, we tackle the problems of efficiency and scalability for predictive coding networks (PCNs) in machine learning. To do so, we propose a library, called PCX, that focuses on performance and simplicity, and use it to implement a large set of standard benchmarks for the community to use for their experiments. As most works in the field propose their own tasks and architectures, do not compare one against each other, and focus on small-scale tasks, a simple and fast open-source library and a comprehensive set of benchmarks would address all these concerns. Then, we perform extensive tests on such benchmarks using both existing algorithms for PCNs, as well as adaptations of other methods popular in the bio-plausible deep learning community. All this has allowed us to (i) test architectures much larger than commonly used in the literature, on more complex datasets; (ii)~reach new state-of-the-art results in all of the tasks and datasets provided; (iii)~clearly highlight what the current limitations of PCNs are, allowing us to state important future research directions. With the hope of galvanizing community efforts towards one of the main open problems in the field, scalability, we release code, tests, and benchmarks. Link to the library: https://github.com/liukidar/pcx

Katherine M. Collins, Simon Frieder, Jonas Bayer et al.

For centuries, human mathematicians have written proofs to substantiate their mathematical arguments; yet, the ability to automatically verify the validity of proofs has long been a challenge. Advances in AI systems' ability to generate code and engage in increasingly high-level mathematical reasoning promise to transform people's ability to formalize and thereby verify proofs. While many works focus on benchmarking the current frontier, we instead study how people use these tools. We conduct a mixed-methods analysis into the initial impact of AI on people's formalization workflows: what people claim they want, what they see as the barriers to those visions, and how they actually use and adapt AI in practice. A qualitative survey shows that people's preferences are diverse, but with a general desire for AI assistance in formalization that preserves high-level human control over the proof discovery process. To assess how people actually engage with AI for formalization under such limitations, we conduct a controlled user study in which participants formalize informal math problems and their proofs, with and without AI, across a range of mathematical problems at varying levels of difficulty and domains. Despite limitations of the tools at the time for autoformalization, participants tend to attain higher formalization accuracy when allowed access to AI tools than when formalizing on their own, with most participants flexibly choosing to use multiple different AI tools. Taken together, our work sheds light on the early stages of AI integration into formalization workflows, involving an intimate interplay of human and AI engagement.

Alexis Chevalier, Jiayi Geng, Alexander Wettig et al. · princeton

NLP has recently made exciting progress toward training language models (LMs) with strong scientific problem-solving skills. However, model development has not focused on real-life use-cases of LMs for science, including applications in education that require processing long scientific documents. To address this, we introduce TutorEval and TutorChat. TutorEval is a diverse question-answering benchmark consisting of questions about long chapters from STEM textbooks, written by experts. TutorEval helps measure real-life usability of LMs as scientific assistants, and it is the first benchmark combining long contexts, free-form generation, and multi-disciplinary scientific knowledge. Moreover, we show that fine-tuning base models with existing dialogue datasets leads to poor performance on TutorEval. Therefore, we create TutorChat, a dataset of 80,000 long synthetic dialogues about textbooks. We use TutorChat to fine-tune Llemma models with 7B and 34B parameters. These LM tutors specialized in math have a 32K-token context window, and they excel at TutorEval while performing strongly on GSM8K and MATH. Our datasets build on open-source materials, and we release our models, data, and evaluations.

Andres Felipe Lerma-Pineda, Philipp Petersen, Simon Frieder et al.

We study the problem of approximating and estimating classification functions that have their decision boundary in the $RBV^2$ space. Functions of $RBV^2$ type arise naturally as solutions of regularized neural network learning problems and neural networks can approximate these functions without the curse of dimensionality. We modify existing results to show that every $RBV^2$ function can be approximated by a neural network with bounded weights. Thereafter, we prove the existence of a neural network with bounded weights approximating a classification function. And we leverage these bounds to quantify the estimation rates. Finally, we present a numerical study that analyzes the effect of different regularity conditions on the decision boundaries.

Vladmir Sicca, Tianxiang Xia, Mathïs Fédérico et al.

We introduce a new symbolic solver for geometry, called Newclid, which is based on AlphaGeometry. Newclid contains a symbolic solver called DDARN (derived from DDAR-Newclid), which is a significant refactoring and upgrade of AlphaGeometry's DDAR symbolic solver by being more user-friendly - both for the end user as well as for a programmer wishing to extend the codebase. For the programmer, improvements include a modularized codebase and new debugging and visualization tools. For the user, Newclid contains a new command line interface (CLI) that provides interfaces for agents to guide DDARN. DDARN is flexible with respect to its internal reasoning, which can be steered by agents. Further, we support input from GeoGebra to make Newclid accessible for educational contexts. Further, the scope of problems that Newclid can solve has been expanded to include the ability to have an improved understanding of metric geometry concepts (length, angle) and to use theorems such as the Pythagorean theorem in proofs. Bugs have been fixed, and reproducibility has been improved. Lastly, we re-evaluated the five remaining problems from the original AG-30 dataset that AlphaGeometry was not able to solve and contrasted them with the abilities of DDARN, running in breadth-first-search agentic mode (which corresponds to how DDARN runs by default), finding that DDARN solves an additional problem. We have open-sourced our code under: https://github.com/LMCRC/Newclid

Simon Frieder, William Hart

We show, contrary to the optimism about LLM's problem-solving abilities, fueled by the recent gold medals that were attained, that a problem exists -- Yu Tsumura's 554th problem -- that a) is within the scope of an IMO problem in terms of proof sophistication, b) is not a combinatorics problem which has caused issues for LLMs, c) requires fewer proof techniques than typical hard IMO problems, d) has a publicly available solution (likely in the training data of LLMs), and e) that cannot be readily solved by any existing off-the-shelf LLM (commercial or open-source).

Simon Frieder, Julius Berner, Philipp Petersen et al.

Large language models (LLMs) such as ChatGPT have received immense interest for their general-purpose language understanding and, in particular, their ability to generate high-quality text or computer code. For many professions, LLMs represent an invaluable tool that can speed up and improve the quality of work. In this note, we discuss to what extent they can aid professional mathematicians. We first provide a mathematical description of the transformer model used in all modern language models. Based on recent studies, we then outline best practices and potential issues and report on the mathematical abilities of language models. Finally, we shed light on the potential of LLMs to change how mathematicians work.

Simon Frieder, Jonas Bayer, Katherine M. Collins et al.

The suite of datasets commonly used to train and evaluate the mathematical capabilities of AI-based mathematical copilots (primarily large language models) exhibit several shortcomings. These limitations include a restricted scope of mathematical complexity, typically not exceeding lower undergraduate-level mathematics, binary rating protocols and other issues, which makes comprehensive proof-based evaluation suites difficult. We systematically explore these limitations and contend that enhancing the capabilities of large language models, or any forthcoming advancements in AI-based mathematical assistants (copilots or "thought partners"), necessitates a paradigm shift in the design of mathematical datasets and the evaluation criteria of mathematical ability: It is necessary to move away from result-based datasets (theorem statement to theorem proof) and convert the rich facets of mathematical research practice to data LLMs can train on. Examples of these are mathematical workflows (sequences of atomic, potentially subfield-dependent tasks that are often performed when creating new mathematics), which are an important part of the proof-discovery process. Additionally, we advocate for mathematical dataset developers to consider the concept of "motivated proof", introduced by G. Pólya in 1949, which can serve as a blueprint for datasets that offer a better proof learning signal, alleviating some of the mentioned limitations. Lastly, we introduce math datasheets for datasets, extending the general, dataset-agnostic variants of datasheets: We provide a questionnaire designed specifically for math datasets that we urge dataset creators to include with their datasets. This will make creators aware of potential limitations of their datasets while at the same time making it easy for readers to assess it from the point of view of training and evaluating mathematical copilots.

Luca Pinchetti, Simon Frieder, Thomas Lukasiewicz et al.

Research aimed at scaling up neuroscience inspired learning algorithms for neural networks is accelerating. Recently, a key research area has been the study of energy-based learning algorithms such as predictive coding, due to their versatility and mathematical grounding. However, the applicability of such methods is held back by the large computational requirements caused by their iterative nature. In this work, we address this problem by showing that the choice of initialization of the neurons in a predictive coding network matters significantly and can notably reduce the required training times. Consequently, we propose a new initialization technique for predictive coding networks that aims to preserve the iterative progress made on previous training samples. Our approach suggests a promising path toward reconciling the disparities between predictive coding and backpropagation in terms of computational efficiency and final performance. In fact, our experiments demonstrate substantial improvements in convergence speed and final test loss in both supervised and unsupervised settings.

Tommaso Salvatori, Yuhang Song, Yujian Hong et al.

Associative memories in the brain receive and store patterns of activity registered by the sensory neurons, and are able to retrieve them when necessary. Due to their importance in human intelligence, computational models of associative memories have been developed for several decades now. They include autoassociative memories, which allow for storing data points and retrieving a stored data point $s$ when provided with a noisy or partial variant of $s$, and heteroassociative memories, able to store and recall multi-modal data. In this paper, we present a novel neural model for realizing associative memories, based on a hierarchical generative network that receives external stimuli via sensory neurons. This model is trained using predictive coding, an error-based learning algorithm inspired by information processing in the cortex. To test the capabilities of this model, we perform multiple retrieval experiments from both corrupted and incomplete data points. In an extensive comparison, we show that this new model outperforms in retrieval accuracy and robustness popular associative memory models, such as autoencoders trained via backpropagation, and modern Hopfield networks. In particular, in completing partial data points, our model achieves remarkable results on natural image datasets, such as ImageNet, with a surprisingly high accuracy, even when only a tiny fraction of pixels of the original images is presented. Furthermore, we show that this method is able to handle multi-modal data, retrieving images from descriptions, and vice versa. We conclude by discussing the possible impact of this work in the neuroscience community, by showing that our model provides a plausible framework to study learning and retrieval of memories in the brain, as it closely mimics the behavior of the hippocampus as a memory index and generative model.