In this work, we investigate the existence and effect of percolation in training deep Neural Networks (NNs) with dropout. Dropout methods are regularisation techniques for training NNs, first introduced by G. Hinton et al. (2012). These methods temporarily remove connections in the NN, randomly at each stage of training, and update the remaining subnetwork with Stochastic Gradient Descent (SGD). The process of removing connections from a network at random is similar to percolation, a paradigm model of statistical physics. If dropout were to remove enough connections such that there is no path between the input and output of the NN, then the NN could not make predictions informed by the data. We study new percolation models that mimic dropout in NNs and characterise the relationship between network topology and this path problem. The theory shows the existence of a percolative effect in dropout. We also show that this percolative effect can cause a breakdown when training NNs without biases with dropout; and we argue heuristically that this breakdown extends to NNs with biases.

AI models like GPT-5 are an increasingly valuable tool for scientists, but many remain unaware of the capabilities of frontier AI. We present a collection of short case studies in which GPT-5 produced new, concrete steps in ongoing research across mathematics, physics, astronomy, computer science, biology, and materials science. In these examples, the authors highlight how AI accelerated their work, and where it fell short; where expert time was saved, and where human input was still key. We document the interactions of the human authors with GPT-5, as guiding examples of fruitful collaboration with AI. Of note, this paper includes four new results in mathematics (carefully verified by the human authors), underscoring how GPT-5 can help human mathematicians settle previously unsolved problems. These contributions are modest in scope but profound in implication, given the rate at which frontier AI is progressing.

Uncovering hidden graph structures underlying real-world data is a critical challenge with broad applications across scientific domains. Recently, transformer-based models leveraging the attention mechanism have demonstrated strong empirical success in capturing complex dependencies within graphs. However, the theoretical understanding of their training dynamics has been limited to tree-like graphs, where each node depends on a single parent. Extending provable guarantees to more general directed acyclic graphs (DAGs) -- which involve multiple parents per node -- remains challenging, primarily due to the difficulty in designing training objectives that enable different attention heads to separately learn multiple different parent relationships. In this work, we address this problem by introducing a novel information-theoretic metric: the kernel-guided mutual information (KG-MI), based on the $f$-divergence. Our objective combines KG-MI with a multi-head attention framework, where each head is associated with a distinct marginal transition kernel to model diverse parent-child dependencies effectively. We prove that, given sequences generated by a $K$-parent DAG, training a single-layer, multi-head transformer via gradient ascent converges to the global optimum in polynomial time. Furthermore, we characterize the attention score patterns at convergence. In addition, when particularizing the $f$-divergence to the KL divergence, the learned attention scores accurately reflect the ground-truth adjacency matrix, thereby provably recovering the underlying graph structure. Experimental results validate our theoretical findings.

Ensemble methods for stream mining necessitate managing multiple models and updating them as data distributions evolve. Considering the calls for more sustainability, established methods are however not sufficiently considerate of ensemble members' computational expenses and instead overly focus on predictive capabilities. To address these challenges and enable green online learning, we propose heterogeneous online ensembles (HEROS). For every training step, HEROS chooses a subset of models from a pool of models initialized with diverse hyperparameter choices under resource constraints to train. We introduce a Markov decision process to theoretically capture the trade-offs between predictive performance and sustainability constraints. Based on this framework, we present different policies for choosing which models to train on incoming data. Most notably, we propose the novel $ζ$-policy, which focuses on training near-optimal models at reduced costs. Using a stochastic model, we theoretically prove that our $ζ$-policy achieves near optimal performance while using fewer resources compared to the best performing policy. In our experiments across 11 benchmark datasets, we find empiric evidence that our $ζ$-policy is a strong contribution to the state-of-the-art, demonstrating highly accurate performance, in some cases even outperforming competitors, and simultaneously being much more resource-friendly.

We extend the ReLU Transition Graph (RTG) framework into a comprehensive graph-theoretic model for understanding deep ReLU networks. In this model, each node represents a linear activation region, and edges connect regions that differ by a single ReLU activation flip, forming a discrete geometric structure over the network's functional behavior. We prove that RTGs at random initialization exhibit strong expansion, binomial degree distributions, and spectral properties that tightly govern generalization. These structural insights enable new bounds on capacity via region entropy and on generalization via spectral gap and edge-wise KL divergence. Empirically, we construct RTGs for small networks, measure their smoothness and connectivity properties, and validate theoretical predictions. Our results show that region entropy saturates under overparameterization, spectral gap correlates with generalization, and KL divergence across adjacent regions reflects functional smoothness. This work provides a unified framework for analyzing ReLU networks through the lens of discrete functional geometry, offering new tools to understand, diagnose, and improve generalization.

In this paper, we consider nonconvex decentralised optimisation and learning over a network of distributed agents. We develop an ADMM algorithm based on the Randomised Block Coordinate Douglas-Rachford splitting method which enables agents in the network to distributedly and asynchronously compute a set of first-order stationary solutions of the problem. To the best of our knowledge, this is the first decentralised and asynchronous algorithm for solving nonconvex optimisation problems with convergence proof. The numerical examples demonstrate the efficiency of the proposed algorithm for distributed Phase Retrieval and sparse Principal Component Analysis problems.

Score-based methods have recently seen increasing popularity in modeling and generation. Methods have been constructed to perform hypothesis testing and change-point detection with score functions, but these methods are in general not as powerful as their likelihood-based peers. Recent works consider generalizing the score-based Fisher divergence into a diffusion-divergence by transforming score functions via multiplication with a matrix-valued function or a weight matrix. In this paper, we extend the score-based hypothesis test and change-point detection stopping rule into their diffusion-based analogs. Additionally, we theoretically quantify the performance of these diffusion-based algorithms and study scenarios where optimal performance is achievable. We propose a method of numerically optimizing the weight matrix and present numerical simulations to illustrate the advantages of diffusion-based algorithms.

--This work studies the problem of sparse signal recovery with automatic grouping of variables. T o this end, we investigate sorted nonsmooth penalties as a regularization approach for generalized linear models. These penalties are designed to promote clustering of variables due to their sorted nature, while the nonconvexity reduces the shrinkage of coefficients. Our goal is to provide efficient ways to compute their proximal operator, enabling the use of popular proximal algorithms to solve composite optimization problems with this choice of sorted penalties. We distinguish between two classes of problems: the weakly convex case where computing the proximal operator remains a convex problem, and the nonconvex case where computing the proximal operator becomes a challenging nonconvex combinatorial problem. We demonstrate the practical interest of using such penalties on several experiments. R is a data-fidelity term and the penalty Ψ is a regularization term that should embed some properties of the solution. Among them, sparsity and structure are particularly useful for a model as they improve its in-terpretability and decrease its complexity. Sparsity is most usually enforced through a penalty term favoring variable selection, i.e. solutions that use only a subset of features.

Due to the structural limitations of Graph Neural Networks (GNNs), in particular with respect to conventional neighborhoods, alternative aggregation strategies have recently been investigated. This paper investigates graph structure in message passing, aimed to incorporate topological characteristics. While the simplicity of neighborhoods remains alluring, we propose a novel perspective by introducing the concept of 'cover' as a generalization of neighborhoods. We design the Grothendieck Graph Neural Networks (GGNN) framework, offering an algebraic platform for creating and refining diverse covers for graphs. This framework translates covers into matrix forms, such as the adjacency matrix, expanding the scope of designing GNN models based on desired message-passing strategies. Leveraging algebraic tools, GGNN facilitates the creation of models that outperform traditional approaches. Based on the GGNN framework, we propose Sieve Neural Networks (SNN), a new GNN model that leverages the notion of sieves from category theory. SNN demonstrates outstanding performance in experiments, particularly on benchmarks designed to test the expressivity of GNNs, and exemplifies the versatility of GGNN in generating novel architectures.

Bi-level optimization has achieved considerable success in contemporary machine learning applications, especially for given proper hyperparameters. However, due to the two-level optimization structure, commonly, researchers focus on two types of bi-level optimization methods: approximate implicit differentiation (AID)-based and iterative differentiation (ITD)-based approaches. ITD-based methods can be readily transformed into single-level optimization problems, facilitating the study of their generalization capabilities. In contrast, AID-based methods cannot be easily transformed similarly but must stay in the two-level structure, leaving their generalization properties enigmatic. In this paper, although the outer-level function is nonconvex, we ascertain the uniform stability of AID-based methods, which achieves similar results to a single-level nonconvex problem. We conduct a convergence analysis for a carefully chosen step size to maintain stability. Combining the convergence and stability results, we give the generalization ability of AID-based bi-level optimization methods. Furthermore, we carry out an ablation study of the parameters and assess the performance of these methods on real-world tasks. Our experimental results corroborate the theoretical findings, demonstrating the effectiveness and potential applications of these methods.