Akiyoshi Sannai

Banri Yanahama, Akiyoshi Sannai

AI-driven autoformalization of mathematics is advancing rapidly. However, the type checker of a proof assistant guarantees only the logical correctness of proofs; it does not verify whether propositions and definitions faithfully capture their intended mathematical content. Consequently, AI-generated formal proofs can exhibit semantic hallucination-passing the type checker yet failing to express the intended mathematics. We propose a human-in-the-loop approach in which human scientists and AI collaboratively produce formal proofs, with humans responsible for the semantic verification of propositions and definitions. To realize this approach, we develop Lean Atlas, a Lean 4 tool that visualizes the dependency graph of a Lean 4 project as an interactive web viewer, enabling human scientists to grasp the overall structure of a formalization efficiently. Its core feature, Lean Compass, is an algorithm that, given a selected theorem set, automatically extracts the project-specific nodes whose semantic correctness can affect those target statements, thereby reducing the candidate set for semantic review in large-scale formalizations. We further define *aligned Lean code* as formalization code that has undergone human semantic verification, and propose it as a quality standard for AI-generated formalizations. We evaluate the tool on six Lean 4 formalization projects with different structural characteristics; proof-heavy projects (PrimeNumberTheoremAnd, Carleson, Brownian Motion) achieved 94-99% average node reduction, a 6-theorem milestone subset of FLT achieved 59.8%, mixed PhysLib 69.0%, and definition-heavy XMSS 27.3%. Lean Atlas is available as open-source software at https://github.com/NyxFoundation/lean-atlas .

Ryutaro Yamauchi, Sho Sonoda, Akiyoshi Sannai et al.

In utilizing large language models (LLMs) for mathematical reasoning, addressing the errors in the reasoning and calculation present in the generated text by LLMs is a crucial challenge. In this paper, we propose a novel framework that integrates the Chain-of-Thought (CoT) method with an external tool (Python REPL). We discovered that by prompting LLMs to generate structured text in XML-like markup language, we could seamlessly integrate CoT and the external tool and control the undesired behaviors of LLMs. With our approach, LLMs can utilize Python computation to rectify errors within CoT. We applied our method to ChatGPT (GPT-3.5) to solve challenging mathematical problems and demonstrated that combining CoT and Python REPL through the markup language enhances the reasoning capability of LLMs. Our approach enables LLMs to write the markup language and perform advanced mathematical reasoning using only zero-shot prompting.

Masayuki Takayama, Tadahisa Okuda, Thong Pham et al.

In practical statistical causal discovery (SCD), embedding domain expert knowledge as constraints into the algorithm is important for reasonable causal models reflecting the broad knowledge of domain experts, despite the challenges in the systematic acquisition of background knowledge. To overcome these challenges, this paper proposes a novel method for causal inference, in which SCD and knowledge-based causal inference (KBCI) with a large language model (LLM) are synthesized through ``statistical causal prompting (SCP)'' for LLMs and prior knowledge augmentation for SCD. The experiments in this work have revealed that the results of LLM-KBCI and SCD augmented with LLM-KBCI approach the ground truths, more than the SCD result without prior knowledge. These experiments have also revealed that the SCD result can be further improved if the LLM undergoes SCP. Furthermore, with an unpublished real-world dataset, we have demonstrated that the background knowledge provided by the LLM can improve the SCD on this dataset, even if this dataset has never been included in the training data of the LLM. For future practical application of this proposed method across important domains such as healthcare, we also thoroughly discuss the limitations, risks of critical errors, expected improvement of techniques around LLMs, and realistic integration of expert checks of the results into this automatic process, with SCP simulations under various conditions both in successful and failure scenarios. The careful and appropriate application of the proposed approach in this work, with improvement and customization for each domain, can thus address challenges such as dataset biases and limitations, illustrating the potential of LLMs to improve data-driven causal inference across diverse scientific domains. The code used in this work is publicly available at: www.github.com/mas-takayama/LLM-and-SCD

Akiyoshi Sannai, Yuuki Takai, Matthieu Cordonnier

In this paper, we develop a theory about the relationship between invariant and equivariant maps with regard to a group $G$. We then leverage this theory in the context of deep neural networks with group symmetries in order to obtain novel insight into their mechanisms. More precisely, we establish a one-to-one relationship between equivariant maps and certain invariant maps. This allows us to reduce arguments for equivariant maps to those for invariant maps and vice versa. As an application, we propose a construction of universal equivariant architectures built from universal invariant networks. We, in turn, explain how the universal architectures arising from our construction differ from standard equivariant architectures known to be universal. Furthermore, we explore the complexity, in terms of the number of free parameters, of our models, and discuss the relation between invariant and equivariant networks' complexity. Finally, we also give an approximation rate for G-equivariant deep neural networks with ReLU activation functions for finite group G.

Akiyoshi Sannai, Yasunari Hikima, Ken Kobayashi et al.

In this paper, we propose a strategy to construct a multi-objective optimization algorithm from a single-objective optimization algorithm by using the Bézier simplex model. Also, we extend the stability of optimization algorithms in the sense of Probability Approximately Correct (PAC) learning and define the PAC stability. We prove that it leads to an upper bound on the generalization with high probability. Furthermore, we show that multi-objective optimization algorithms derived from a gradient descent-based single-objective optimization algorithm are PAC stable. We conducted numerical experiments and demonstrated that our method achieved lower generalization errors than the existing multi-objective optimization algorithm.

Kaito Baba, Chaoran Liu, Shuhei Kurita et al.

We present Prover Agent, a novel AI agent for automated theorem proving that integrates large language models (LLMs) with a formal proof assistant, Lean. Prover Agent coordinates an informal reasoning LLM, a formal prover model, and feedback from Lean while also generating auxiliary lemmas. These auxiliary lemmas are not limited to subgoals in the formal proof but can also include special cases or potentially useful facts derived from the assumptions, which help in discovering a viable proof strategy. It achieves an 88.1% success rate on the MiniF2F benchmark, establishing a new state-of-the-art among methods using small language models (SLMs) with a much lower sample budget than previous approaches. We also present theoretical analyses and case studies that illustrate how these generated lemmas contribute to solving challenging problems. Our code is publicly available at: https://github.com/kAIto47802/Prover-Agent.

Kazumi Kasaura, Naoto Onda, Yuta Oriike et al.

Large Language Models have demonstrated significant promise in formal theorem proving. However, previous works mainly focus on solving existing problems. In this paper, we focus on the ability of LLMs to find novel theorems. We propose Conjecturing-Proving Loop pipeline for automatically generating mathematical conjectures and proving them in Lean 4 format. A feature of our approach is that we generate and prove further conjectures with context including previously generated theorems and their proofs, which enables the generation of more difficult proofs by in-context learning of proof strategies without changing parameters of LLMs. We demonstrated that our framework rediscovered theorems with verification, which were published in past mathematical papers and have not yet formalized. Moreover, at least one of these theorems could not be proved by the LLM without in-context learning, even in natural language, which means that in-context learning was effective for neural theorem proving. The source code is available at https://github.com/auto-res/ConjecturingProvingLoop.

Masato Kamba, Hirotake Murakami, Akiyoshi Sannai

Security-critical software is routinely audited by tools that reason about vulnerabilities as repository-local code patterns. Yet specification-governed systems -- protocol stacks, consensus implementations, cryptographic libraries -- are constrained by invariants and correctness conditions defined in natural-language specifications. When a vulnerability arises from what the specification requires rather than how code is written, code-level approaches lack the representational vocabulary to detect it, and their false positives resist systematic diagnosis. We present SPECA, a specification-anchored security audit framework that derives explicit, typed security properties from natural-language specifications and audits implementations through structured proof-attempt reasoning grounded in each property. The framework yields three capabilities absent from code-driven auditing: spec-dependent detections, controlled cross-implementation comparison under a shared property vocabulary, and false positives that decompose into interpretable, pipeline-phase-traceable root causes. On the Sherlock Ethereum Fusaka Audit Contest (366 submissions, 10 implementations), SPECA recovers all 15 in-scope vulnerabilities and independently discovers 4 bugs confirmed by developer fix commits. On the RepoAudit C/C++ benchmark (15 projects), SPECA matches the best published precision (88.9\%) while surfacing 12 candidate bugs beyond the established ground truth, two confirmed by upstream maintainers. A multi-model analysis reveals that more capable models audit more faithfully within property scope, shifting the detection bottleneck from model reasoning to property generation quality. All false positives trace to three recurring root causes -- trust boundary misunderstanding, code reading errors, and specification misinterpretation -- each yielding actionable improvement targets.

Koji Hashimoto, Yuji Hirono, Akiyoshi Sannai

Understanding the inner workings of neural networks, including transformers, remains one of the most challenging puzzles in machine learning. This study introduces a novel approach by applying the principles of gauge symmetries, a key concept in physics, to neural network architectures. By regarding model functions as physical observables, we find that parametric redundancies of various machine learning models can be interpreted as gauge symmetries. We mathematically formulate the parametric redundancies in neural ODEs, and find that their gauge symmetries are given by spacetime diffeomorphisms, which play a fundamental role in Einstein's theory of gravity. Viewing neural ODEs as a continuum version of feedforward neural networks, we show that the parametric redundancies in feedforward neural networks are indeed lifted to diffeomorphisms in neural ODEs. We further extend our analysis to transformer models, finding natural correspondences with neural ODEs and their gauge symmetries. The concept of gauge symmetries sheds light on the complex behavior of deep learning models through physics and provides us with a unifying perspective for analyzing various machine learning architectures.

Toshinori Kitamura, Tadashi Kozuno, Masahiro Kato et al.

We study a primal-dual (PD) reinforcement learning (RL) algorithm for online constrained Markov decision processes (CMDPs). Despite its widespread practical use, the existing theoretical literature on PD-RL algorithms for this problem only provides sublinear regret guarantees and fails to ensure convergence to optimal policies. In this paper, we introduce a novel policy gradient PD algorithm with uniform probably approximate correctness (Uniform-PAC) guarantees, simultaneously ensuring convergence to optimal policies, sublinear regret, and polynomial sample complexity for any target accuracy. Notably, this represents the first Uniform-PAC algorithm for the online CMDP problem. In addition to the theoretical guarantees, we empirically demonstrate in a simple CMDP that our algorithm converges to optimal policies, while baseline algorithms exhibit oscillatory performance and constraint violation.

Naoto Onda, Kazumi Kasaura, Yuta Oriike et al.

We introduce LeanConjecturer, a pipeline for automatically generating university-level mathematical conjectures in Lean 4 using Large Language Models (LLMs). Our hybrid approach combines rule-based context extraction with LLM-based theorem statement generation, addressing the data scarcity challenge in formal theorem proving. Through iterative generation and evaluation, LeanConjecturer produced 12,289 conjectures from 40 Mathlib seed files, with 3,776 identified as syntactically valid and non-trivial, that is, cannot be proven by \texttt{aesop} tactic. We demonstrate the utility of these generated conjectures for reinforcement learning through Group Relative Policy Optimization (GRPO), showing that targeted training on domain-specific conjectures can enhance theorem proving capabilities. Our approach generates 103.25 novel conjectures per seed file on average, providing a scalable solution for creating training data for theorem proving systems. Our system successfully verified several non-trivial theorems in topology, including properties of semi-open, alpha-open, and pre-open sets, demonstrating its potential for mathematical discovery beyond simple variations of existing results.

Akiyoshi Sannai, Makoto Kawano, Wataru Kumagai

Invariant and equivariant networks are useful in learning data with symmetry, including images, sets, point clouds, and graphs. In this paper, we consider invariant and equivariant networks for symmetries of finite groups. Invariant and equivariant networks have been constructed by various researchers using Reynolds operators. However, Reynolds operators are computationally expensive when the order of the group is large because they use the sum over the whole group, which poses an implementation difficulty. To overcome this difficulty, we consider representing the Reynolds operator as a sum over a subset instead of a sum over the whole group. We call such a subset a Reynolds design, and an operator defined by a sum over a Reynolds design a reductive Reynolds operator. For example, in the case of a graph with $n$ nodes, the computational complexity of the reductive Reynolds operator is reduced to $O(n^2)$, while the computational complexity of the Reynolds operator is $O(n!)$. We construct learning models based on the reductive Reynolds operator called equivariant and invariant Reynolds networks (ReyNets) and prove that they have universal approximation property. Reynolds designs for equivariant ReyNets are derived from combinatorial observations with Young diagrams, while Reynolds designs for invariant ReyNets are derived from invariants called Reynolds dimensions defined on the set of invariant polynomials. Numerical experiments show that the performance of our models is comparable to state-of-the-art methods.

Akinori Tanaka, Akiyoshi Sannai, Ken Kobayashi et al.

Bézier simplex fitting algorithms have been recently proposed to approximate the Pareto set/front of multi-objective continuous optimization problems. These new methods have shown to be successful at approximating various shapes of Pareto sets/fronts when sample points exactly lie on the Pareto set/front. However, if the sample points scatter away from the Pareto set/front, those methods often likely suffer from over-fitting. To overcome this issue, in this paper, we extend the Bézier simplex model to a probabilistic one and propose a new learning algorithm of it, which falls into the framework of approximate Bayesian computation (ABC) based on the Wasserstein distance. We also study the convergence property of the Wasserstein ABC algorithm. An extensive experimental evaluation on publicly available problem instances shows that the new algorithm converges on a finite sample. Moreover, it outperforms the deterministic fitting methods on noisy instances.

Makoto Kawano, Wataru Kumagai, Akiyoshi Sannai et al.

We present the group equivariant conditional neural process (EquivCNP), a meta-learning method with permutation invariance in a data set as in conventional conditional neural processes (CNPs), and it also has transformation equivariance in data space. Incorporating group equivariance, such as rotation and scaling equivariance, provides a way to consider the symmetry of real-world data. We give a decomposition theorem for permutation-invariant and group-equivariant maps, which leads us to construct EquivCNPs with an infinite-dimensional latent space to handle group symmetries. In this paper, we build architecture using Lie group convolutional layers for practical implementation. We show that EquivCNP with translation equivariance achieves comparable performance to conventional CNPs in a 1D regression task. Moreover, we demonstrate that incorporating an appropriate Lie group equivariance, EquivCNP is capable of zero-shot generalization for an image-completion task by selecting an appropriate Lie group equivariance.

Wataru Kumagai, Akiyoshi Sannai

Group symmetry is inherent in a wide variety of data distributions. Data processing that preserves symmetry is described as an equivariant map and often effective in achieving high performance. Convolutional neural networks (CNNs) have been known as models with equivariance and shown to approximate equivariant maps for some specific groups. However, universal approximation theorems for CNNs have been separately derived with individual techniques according to each group and setting. This paper provides a unified method to obtain universal approximation theorems for equivariant maps by CNNs in various settings. As its significant advantage, we can handle non-linear equivariant maps between infinite-dimensional spaces for non-compact groups.

Yuuki Takai, Akiyoshi Sannai, Matthieu Cordonnier

The classical approach to measure the expressive power of deep neural networks with piecewise linear activations is based on counting their maximum number of linear regions. This complexity measure is quite relevant to understand general properties of the expressivity of neural networks such as the benefit of depth over width. Nevertheless, it appears limited when it comes to comparing the expressivity of different network architectures. This lack becomes particularly prominent when considering permutation-invariant networks, due to the symmetrical redundancy among the linear regions. To tackle this, we propose a refined definition of piecewise linear function complexity: instead of counting the number of linear regions directly, we first introduce an equivalence relation among the linear functions composing a piecewise linear function and then count those linear functions relative to that equivalence relation. Our new complexity measure can clearly distinguish between the two aforementioned models, is consistent with the classical measure, and increases exponentially with depth.

Akiyoshi Sannai, Masaaki Imaizumi, Makoto Kawano

Numerous invariant (or equivariant) neural networks have succeeded in handling invariant data such as point clouds and graphs. However, a generalization theory for the neural networks has not been well developed, because several essential factors for the theory, such as network size and margin distribution, are not deeply connected to the invariance and equivariance. In this study, we develop a novel generalization error bound for invariant and equivariant deep neural networks. To describe the effect of invariance and equivariance on generalization, we develop a notion of a \textit{quotient feature space}, which measures the effect of group actions for the properties. Our main result proves that the volume of quotient feature spaces can describe the generalization error. Furthermore, the bound shows that the invariance and equivariance significantly improve the leading term of the bound. We apply our result to specific invariant and equivariant networks, such as DeepSets (Zaheer et al. (2017)), and show that their generalization bound is considerably improved by $\sqrt{n!}$, where $n!$ is the number of permutations. We also discuss the expressive power of invariant DNNs and show that they can achieve an optimal approximation rate. Our experimental result supports our theoretical claims.

Akinori Tanaka, Akiyoshi Sannai, Ken Kobayashi et al.

The Bezier simplex fitting is a novel data modeling technique which exploits geometric structures of data to approximate the Pareto front of multi-objective optimization problems. There are two fitting methods based on different sampling strategies. The inductive skeleton fitting employs a stratified subsampling from each skeleton of a simplex, whereas the all-at-once fitting uses a non-stratified sampling which treats a simplex as a whole. In this paper, we analyze the asymptotic risks of those Bézier simplex fitting methods and derive the optimal subsample ratio for the inductive skeleton fitting. It is shown that the inductive skeleton fitting with the optimal ratio has a smaller risk when the degree of a Bezier simplex is less than three. Those results are verified numerically under small to moderate sample sizes. In addition, we provide two complementary applications of our theory: a generalized location problem and a multi-objective hyper-parameter tuning of the group lasso. The former can be represented by a Bezier simplex of degree two where the inductive skeleton fitting outperforms. The latter can be represented by a Bezier simplex of degree three where the all-at-once fitting gets an advantage.

Akiyoshi Sannai, Yuuki Takai, Matthieu Cordonnier

In this paper, we develop a theory about the relationship between $G$-invariant/equivariant functions and deep neural networks for finite group $G$. Especially, for a given $G$-invariant/equivariant function, we construct its universal approximator by deep neural network whose layers equip $G$-actions and each affine transformations are $G$-equivariant/invariant. Due to representation theory, we can show that this approximator has exponentially fewer free parameters than usual models.

Akiyoshi Sannai

This paper presents a new mathematical framework to analyze the loss functions of deep neural networks with ReLU functions. Furthermore, as as application of this theory, we prove that the loss functions can reconstruct the inputs of the training samples up to scalar multiplication (as vectors) and can provide the number of layers and nodes of the deep neural network. Namely, if we have all input and output of a loss function (or equivalently all possible learning process), for all input of each training sample $x_i \in \mathbb{R}^n$, we can obtain vectors $x'_i\in \mathbb{R}^n$ satisfying $x_i=c_ix'_i$ for some $c_i \neq 0$. To prove theorem, we introduce the notion of virtual polynomials, which are polynomials written as the output of a node in a deep neural network. Using virtual polynomials, we find an algebraic structure for the loss surfaces, called semi-algebraic sets. We analyze these loss surfaces from the algebro-geometric point of view. Factorization of polynomials is one of the most standard ideas in algebra. Hence, we express the factorization of the virtual polynomials in terms of their active paths. This framework can be applied to the leakage problem in the training of deep neural networks. The main theorem in this paper indicates that there are many risks associated with the training of deep neural networks. For example, if we have N (the dimension of weight space) + 1 nonsmooth points on the loss surface, which are sufficiently close to each other, we can obtain the input of training sample up to scalar multiplication. We also point out that the structures of the loss surfaces depend on the shape of the deep neural network and not on the training samples.