François Charton

Pierre-Alexandre Kamienny, Stéphane d'Ascoli, Guillaume Lample et al. · meta-ai

Symbolic regression, the task of predicting the mathematical expression of a function from the observation of its values, is a difficult task which usually involves a two-step procedure: predicting the "skeleton" of the expression up to the choice of numerical constants, then fitting the constants by optimizing a non-convex loss function. The dominant approach is genetic programming, which evolves candidates by iterating this subroutine a large number of times. Neural networks have recently been tasked to predict the correct skeleton in a single try, but remain much less powerful. In this paper, we challenge this two-step procedure, and task a Transformer to directly predict the full mathematical expression, constants included. One can subsequently refine the predicted constants by feeding them to the non-convex optimizer as an informed initialization. We present ablations to show that this end-to-end approach yields better results, sometimes even without the refinement step. We evaluate our model on problems from the SRBench benchmark and show that our model approaches the performance of state-of-the-art genetic programming with several orders of magnitude faster inference.

Chris Lengerich, Gabriel Synnaeve, Amy Zhang et al. · meta-ai

Traditional approaches to RL have focused on learning decision policies directly from episodic decisions, while slowly and implicitly learning the semantics of compositional representations needed for generalization. While some approaches have been adopted to refine representations via auxiliary self-supervised losses while simultaneously learning decision policies, learning compositional representations from hand-designed and context-independent self-supervised losses (multi-view) still adapts relatively slowly to the real world, which contains many non-IID subspaces requiring rapid distribution shift in both time and spatial attention patterns at varying levels of abstraction. In contrast, supervised language model cascades have shown the flexibility to adapt to many diverse manifolds, and hints of self-learning needed for autonomous task transfer. However, to date, transfer methods for language models like few-shot learning and fine-tuning still require human supervision and transfer learning using self-learning methods has been underexplored. We propose a self-supervised loss policy called contrastive distillation which manifests latent variables with high mutual information with both source and target tasks from weights to tokens. We show how this outperforms common methods of transfer learning and suggests a useful design axis of trading off compute for generalizability for online transfer. Contrastive distillation is improved through sampling from memory and suggests a simple algorithm for more efficiently sampling negative examples for contrastive losses than random sampling.

Matthew Leigh, Samuel Klein, François Charton et al.

In this work, we significantly enhance masked particle modeling (MPM), a self-supervised learning scheme for constructing highly expressive representations of unordered sets relevant to developing foundation models for high-energy physics. In MPM, a model is trained to recover the missing elements of a set, a learning objective that requires no labels and can be applied directly to experimental data. We achieve significant performance improvements over previous work on MPM by addressing inefficiencies in the implementation and incorporating a more powerful decoder. We compare several pre-training tasks and introduce new reconstruction methods that utilize conditional generative models without data tokenization or discretization. We show that these new methods outperform the tokenized learning objective from the original MPM on a new test bed for foundation models for jets, which includes using a wide variety of downstream tasks relevant to jet physics, such as classification, secondary vertex finding, and track identification.

Emily Wenger, Mingjie Chen, François Charton et al.

Currently deployed public-key cryptosystems will be vulnerable to attacks by full-scale quantum computers. Consequently, "quantum resistant" cryptosystems are in high demand, and lattice-based cryptosystems, based on a hard problem known as Learning With Errors (LWE), have emerged as strong contenders for standardization. In this work, we train transformers to perform modular arithmetic and combine half-trained models with statistical cryptanalysis techniques to propose SALSA: a machine learning attack on LWE-based cryptographic schemes. SALSA can fully recover secrets for small-to-mid size LWE instances with sparse binary secrets, and may scale to attack real-world LWE-based cryptosystems.

Samy Jelassi, Stéphane d'Ascoli, Carles Domingo-Enrich et al.

We examine how transformers cope with two challenges: learning basic integer arithmetic, and generalizing to longer sequences than seen during training. We find that relative position embeddings enable length generalization for simple tasks, such as addition: models trained on $5$-digit numbers can perform $15$-digit sums. However, this method fails for multiplication, and we propose train set priming: adding a few ($10$ to $50$) long sequences to the training set. We show that priming allows models trained on $5$-digit $\times$ $3$-digit multiplications to generalize to $35\times 3$ examples. We also show that models can be primed for different generalization lengths, and that the priming sample size scales as the logarithm of the training set size. Finally, we discuss potential applications of priming beyond arithmetic.

François Charton

The predictions of small transformers, trained to calculate the greatest common divisor (GCD) of two positive integers, can be fully characterized by looking at model inputs and outputs. As training proceeds, the model learns a list $\mathcal D$ of integers, products of divisors of the base used to represent integers and small primes, and predicts the largest element of $\mathcal D$ that divides both inputs. Training distributions impact performance. Models trained from uniform operands only learn a handful of GCD (up to $38$ GCD $\leq100$). Log-uniform operands boost performance to $73$ GCD $\leq 100$, and a log-uniform distribution of outcomes (i.e. GCD) to $91$. However, training from uniform (balanced) GCD breaks explainability.

François Charton

This paper investigates the failure cases and out-of-distribution behavior of transformers trained on matrix inversion and eigenvalue decomposition. I show that incorrect model predictions still retain deep mathematical properties of the solution (e.g. correct eigenvalues, unit norm of eigenvectors), and that almost all model failures can be attributed to, and predicted from, properties of the problem or solution. This demonstrates that, when in doubt, math transformers do not hallucinate absurd solutions (as was sometimes proposed) but remain ``roughly right''. I also show that the careful choice of a training dataset can accelerate training, while allowing the model to generalize out of its training distribution, invalidating the idea that transformers ``merely interpolate'' from memorized examples.

François Charton, Ashvni Narayanan

We investigate transformer prediction of long Collatz steps, a complex arithmetic function that maps odd integers to their distant successors in the Collatz sequence ( $u_{n+1}=u_n/2$ if $u_n$ is even, $u_{n+1}=(3u_n+1)/2$ if $u_n$ is odd). Model accuracy varies with the base used to encode input and output. It can be as high as $99.7\%$ for bases $24$ and $32$, and as low as $37$ and $25\%$ for bases $11$ and $3$. Yet, all models, no matter the base, follow a common learning pattern. As training proceeds, they learn a sequence of classes of inputs that share the same residual modulo $2^p$. Models achieve near-perfect accuracy on these classes, and less than $1\%$ for all other inputs. This maps to a mathematical property of Collatz sequences: the length of the loops involved in the computation of a long Collatz step can be deduced from the binary representation of its input. The learning pattern reflects the model learning to predict inputs associated with increasing loop lengths. An analysis of failure cases reveals that almost all model errors follow predictable patterns. Hallucination, a common feature of large language models, almost never happens. In over $90\%$ of failures, the model performs the correct calculation, but wrongly estimates loop lengths. Our observations give a full account of the algorithms learned by the models. They suggest that the difficulty of learning such complex arithmetic function lies in figuring the control structure of the computation -- the length of the loops. We believe that the approach outlined here, using mathematical problems as tools for understanding, explaining, and perhaps improving language models, can be applied to a broad range of problems and bear fruitful results.

François Charton

This paper documents Int2Int, an open source code base for using transformers on problems of mathematical research, with a focus on number theory and other problems involving integers. Int2Int is a complete PyTorch implementation of a transformer architecture, together with training and evaluation loops, and classes and functions to represent, generate and decode common mathematical objects. Ancillary code for data preparation, and Jupyter Notebooks for visualizing experimental results are also provided. This document presents the main features of Int2Int, serves as its user manual, and provides guidelines on how to extend it. Int2Int is released under the MIT licence, at https://github.com/f-charton/Int2Int.

François Charton, Jordan S. Ellenberg, Adam Zsolt Wagner et al.

We introduce PatternBoost, a flexible method for finding interesting constructions in mathematics. Our algorithm alternates between two phases. In the first ``local'' phase, a classical search algorithm is used to produce many desirable constructions. In the second ``global'' phase, a transformer neural network is trained on the best such constructions. Samples from the trained transformer are then used as seeds for the first phase, and the process is repeated. We give a detailed introduction to this technique, and discuss the results of its application to several problems in extremal combinatorics. The performance of PatternBoost varies across different problems, but there are many situations where its performance is quite impressive. Using our technique, we find the best known solutions to several long-standing problems, including the construction of a counterexample to a conjecture that had remained open for 30 years.

Samuel Stevens, Emily Wenger, Cathy Li et al.

Learning with Errors (LWE) is a hard math problem underlying recently standardized post-quantum cryptography (PQC) systems for key exchange and digital signatures. Prior work proposed new machine learning (ML)-based attacks on LWE problems with small, sparse secrets, but these attacks require millions of LWE samples to train on and take days to recover secrets. We propose three key methods -- better preprocessing, angular embeddings and model pre-training -- to improve these attacks, speeding up preprocessing by $25\times$ and improving model sample efficiency by $10\times$. We demonstrate for the first time that pre-training improves and reduces the cost of ML attacks on LWE. Our architecture improvements enable scaling to larger-dimension LWE problems: this work is the first instance of ML attacks recovering sparse binary secrets in dimension $n=1024$, the smallest dimension used in practice for homomorphic encryption applications of LWE where sparse binary secrets are proposed.

Angelica Babei, François Charton, Edgar Costa et al.

We apply transformer models and feedforward neural networks to predict Frobenius traces $a_p$ from elliptic curves given other traces $a_q$. We train further models to predict $a_p \bmod 2$ from $a_q \bmod 2$, and cross-analysis such as $a_p \bmod 2$ from $a_q$. Our experiments reveal that these models achieve high accuracy, even in the absence of explicit number-theoretic tools like functional equations of $L$-functions. We also present partial interpretability findings.

Anja Butter, François Charton, Javier Mariño Villadamigo et al.

Generative networks are an exciting tool for fast LHC event generation. Usually, they are used to generate configurations with a fixed number of particles. Autoregressive transformers allow us to generate events with variable numbers of particles, very much in line with the physics of QCD jet radiation. We show how they can learn a factorized likelihood for jet radiation and extrapolate in terms of the number of generated jets. For this extrapolation, bootstrapping training data and training with modifications of the likelihood loss can be used.

Jacky H. T. Yip, Charles Arnal, François Charton et al.

Fine, regular, and star triangulations (FRSTs) of four-dimensional reflexive polytopes give rise to toric varieties, within which generic anticanonical hypersurfaces yield smooth Calabi-Yau threefolds. We introduce CYTransformer, a deep learning model based on the transformer architecture, to automate the generation of FRSTs. We demonstrate that CYTransformer efficiently and unbiasedly samples FRSTs for polytopes across a range of sizes, and can self-improve through retraining on its own output. These results lay the foundation for AICY: a community-driven platform designed to combine self-improving machine learning models with a continuously expanding database to explore and catalog the Calabi-Yau landscape.

Eshika Saxena, Alberto Alfarano, François Charton et al.

AI-powered attacks on Learning with Errors (LWE), an important hard math problem in post-quantum cryptography, rival or outperform "classical" attacks on LWE under certain parameter settings. Despite the promise of this approach, a dearth of accessible data limits AI practitioners' ability to study and improve these attacks. Creating LWE data for AI model training is time- and compute-intensive and requires significant domain expertise. To fill this gap and accelerate AI research on LWE attacks, we propose the TAPAS datasets, a Toolkit for Analysis of Post-quantum cryptography using AI Systems. These datasets cover several LWE settings and can be used off-the-shelf by AI practitioners to prototype new approaches to cracking LWE. This work documents TAPAS dataset creation, establishes attack performance baselines, and lays out directions for future work.

Tianji Cai, Garrett W. Merz, François Charton et al.

We pursue the use of deep learning methods to improve state-of-the-art computations in theoretical high-energy physics. Planar N = 4 Super Yang-Mills theory is a close cousin to the theory that describes Higgs boson production at the Large Hadron Collider; its scattering amplitudes are large mathematical expressions containing integer coefficients. In this paper, we apply Transformers to predict these coefficients. The problem can be formulated in a language-like representation amenable to standard cross-entropy training objectives. We design two related experiments and show that the model achieves high accuracy (> 98%) on both tasks. Our work shows that Transformers can be applied successfully to problems in theoretical physics that require exact solutions.

Stéphane d'Ascoli, Pierre-Alexandre Kamienny, Guillaume Lample et al.

Symbolic regression, i.e. predicting a function from the observation of its values, is well-known to be a challenging task. In this paper, we train Transformers to infer the function or recurrence relation underlying sequences of integers or floats, a typical task in human IQ tests which has hardly been tackled in the machine learning literature. We evaluate our integer model on a subset of OEIS sequences, and show that it outperforms built-in Mathematica functions for recurrence prediction. We also demonstrate that our float model is able to yield informative approximations of out-of-vocabulary functions and constants, e.g. $\operatorname{bessel0}(x)\approx \frac{\sin(x)+\cos(x)}{\sqrt{πx}}$ and $1.644934\approx π^2/6$. An interactive demonstration of our models is provided at https://symbolicregression.metademolab.com.

François Charton, Amaury Hayat, Sean T. McQuade et al.

We show that deep learning models, and especially architectures like the Transformer, originally intended for natural language, can be trained on randomly generated datasets to predict to very high accuracy both the qualitative and quantitative features of metabolic networks. Using standard mathematical techniques, we create large sets (40 million elements) of random networks that can be used to train our models. These trained models can predict network equilibrium on random graphs in more than 99% of cases. They can also generalize to graphs with different structure than those encountered at training. Finally, they can predict almost perfectly the equilibria of a small set of known biological networks. Our approach is both very economical in experimental data and uses only small and shallow deep-learning model, far from the large architectures commonly used in machine translation. Such results pave the way for larger use of deep learning models for problems related to biological networks in key areas such as quantitative systems pharmacology, systems biology, and synthetic biology.

François Charton

Transformers can learn to perform numerical computations from examples only. I study nine problems of linear algebra, from basic matrix operations to eigenvalue decomposition and inversion, and introduce and discuss four encoding schemes to represent real numbers. On all problems, transformers trained on sets of random matrices achieve high accuracies (over 90%). The models are robust to noise, and can generalize out of their training distribution. In particular, models trained to predict Laplace-distributed eigenvalues generalize to different classes of matrices: Wigner matrices or matrices with positive eigenvalues. The reverse is not true.

François Charton, Amaury Hayat, Guillaume Lample

Using transformers over large generated datasets, we train models to learn mathematical properties of differential systems, such as local stability, behavior at infinity and controllability. We achieve near perfect prediction of qualitative characteristics, and good approximations of numerical features of the system. This demonstrates that neural networks can learn to perform complex computations, grounded in advanced theory, from examples, without built-in mathematical knowledge.

Guillaume Lample, François Charton

Neural networks have a reputation for being better at solving statistical or approximate problems than at performing calculations or working with symbolic data. In this paper, we show that they can be surprisingly good at more elaborated tasks in mathematics, such as symbolic integration and solving differential equations. We propose a syntax for representing mathematical problems, and methods for generating large datasets that can be used to train sequence-to-sequence models. We achieve results that outperform commercial Computer Algebra Systems such as Matlab or Mathematica.