The dominant paradigm for learning on graphs is message passing. Despite being a strong inductive bias, the local message passing mechanism faces challenges such as over-smoothing, over-squashing, and limited expressivity. To address these issues, we introduce Bundle Neural Networks (BuNNs), a novel graph neural network architecture that operates via *message diffusion* on *flat vector bundles* — geometrically inspired structures that assign to each node a vector space and an orthogonal map. A BuNN layer evolves node features through a diffusion-type partial differential equation, where its discrete form acts as a special case of the recently introduced Sheaf Neural Network (SNN), effectively alleviating over-smoothing. The continuous nature of message diffusion enables BuNNs to operate at larger scales, reducing over-squashing. We establish the universality of BuNNs in approximating feature transformations on infinite families of graphs with injective positional encodings, marking the first positive expressivity result of its kind. We support our claims with formal analysis and synthetic experiments. Empirically, BuNNs perform strongly on heterophilic and long-range tasks, which demonstrates their robustness on a diverse range of challenging real-world tasks.

Positional Encodings (PEs) are a critical component of Transformer-based Large Language Models (LLMs), providing the attention mechanism with important sequence-position information. One of the most popular types of encoding used today in LLMs are Rotary Positional Encodings (RoPE), that rotate the queries and keys based on their relative distance. A common belief is that RoPE is useful because it helps to decay token dependency as relative distance increases. In this work, we argue that this is unlikely to be the core reason. We study the internals of a trained Gemma 7B model to understand how RoPE is being used at a mechanical level. We find that Gemma learns to use RoPE to construct robust `positional' attention patterns by exploiting the highest frequencies. We also find that, in general, Gemma greatly prefers to use the lowest frequencies of RoPE, which we suspect are used to carry semantic information. We mathematically prove interesting behaviours of RoPE and conduct experiments to verify our findings, proposing a modification of RoPE that fixes some highlighted issues and improves performance. We believe that this work represents an interesting step in better understanding PEs in LLMs, which we believe holds crucial value for scaling LLMs to large sizes and context lengths.

A key property of reasoning systems is the ability to make sharp decisions on their input data. For contemporary AI systems, a key carrier of sharp behaviour is the softmax function, with its capability to perform differentiable query-key lookups. It is a common belief that the predictive power of networks leveraging softmax arises from "circuits" which sharply perform certain kinds of computations consistently across many diverse inputs. However, for these circuits to be robust, they would need to generalise well to arbitrary valid inputs. In this paper, we dispel this myth: even for tasks as simple as finding the maximum key, any learned circuitry must disperse as the number of items grows at test time. We attribute this to a fundamental limitation of the softmax function to robustly approximate sharp functions with increasing problem size, prove this phenomenon theoretically, and propose adaptive temperature as an ad-hoc technique for improving the sharpness of softmax at inference time.

Graph Neural Networks (GNNs) are models that leverage the graph structure to transmit information between nodes, typically through the message-passing operation. While widely successful, this approach is well-known to suffer from representational collapse as the number of layers increases and insensitivity to the information contained at distant and poorly connected nodes. In this paper, we present a unified view of on the appearance of these issues through the lens of vanishing gradients, using ideas from linear control theory for our analysis. We propose an interpretation of GNNs as recurrent models and empirically demonstrate that a simple state-space formulation of an GNN effectively alleviates these issues at no extra trainable parameter cost. Further, we show theoretically and empirically that (i) Traditional GNNs are by design prone to extreme gradient vanishing even after few layers; (ii) Feature collapse is directly related to the mechanism causing vanishing gradients; (iii) Long-range modeling is most easily achieved by a combination of graph rewiring and vanishing gradient mitigation. We believe our work will help bridge the gap between the recurrent and graph neural network literature and will unlock the design of new deep and performant GNNs.

While Generative AI rapidly advances in various domains, generating truly creative, aesthetic, and counter-intuitive outputs remains a challenge. This paper presents an approach to tackle these difficulties in the domain of chess puzzles. We start by benchmarking Generative AI architectures, and then introduce an RL framework with novel rewards based on chess engine search statistics to overcome some of those shortcomings. The rewards are designed to enhance a puzzle's uniqueness, counter-intuitiveness, diversity, and realism. Our RL approach dramatically increases counter-intuitive puzzle generation by 10x, from 0.22\% (supervised) to 2.5\%, surpassing existing dataset rates (2.1\%) and the best Lichess-trained model (0.4\%). Our puzzles meet novelty and diversity benchmarks, retain aesthetic themes, and are rated by human experts as more creative, enjoyable, and counter-intuitive than composed book puzzles, even approaching classic compositions. Our final outcome is a curated booklet of these novel AI-generated puzzles, which is acknowledged for creativity by three world-renowned experts.

Graph Neural Networks (GNNs) are popular models for machine learning on graphs that typically follow the message-passing paradigm, whereby the feature of a node is updated recursively upon aggregating information over its neighbors. While exchanging messages over the input graph endows GNNs with a strong inductive bias, it can also make GNNs susceptible to over-squashing, thereby preventing them from capturing long-range interactions in the given graph. To rectify this issue, graph rewiring techniques have been proposed as a means of improving information flow by altering the graph connectivity. In this work, we identify three desiderata for graph-rewiring: (i) reduce over-squashing, (ii) respect the locality of the graph, and (iii) preserve the sparsity of the graph. We highlight fundamental trade-offs that occur between spatial and spectral rewiring techniques; while the former often satisfy (i) and (ii) but not (iii), the latter generally satisfy (i) and (iii) at the expense of (ii). We propose a novel rewiring framework that satisfies all of (i)--(iii) through a locality-aware sequence of rewiring operations. We then discuss a specific instance of such rewiring framework and validate its effectiveness on several real-world benchmarks, showing that it either matches or significantly outperforms existing rewiring approaches.

We study how information propagates in decoder-only Transformers, which are the architectural foundation of most existing frontier large language models (LLMs). We rely on a theoretical signal propagation analysis---specifically, we analyse the representations of the last token in the final layer of the Transformer, as this is the representation used for next-token prediction. Our analysis reveals a representational collapse phenomenon: we prove that certain distinct pairs of inputs to the Transformer can yield arbitrarily close representations in the final token. This effect is exacerbated by the low-precision floating-point formats frequently used in modern LLMs. As a result, the model is provably unable to respond to these sequences in different ways---leading to errors in, e.g., tasks involving counting or copying. Further, we show that decoder-only Transformer language models can lose sensitivity to specific tokens in the input, which relates to the well-known phenomenon of over-squashing in graph neural networks. We provide empirical evidence supporting our claims on contemporary LLMs. Our theory points to simple solutions towards ameliorating these issues.

Message Passing Neural Networks (MPNNs) are instances of Graph Neural Networks that leverage the graph to send messages over the edges. This inductive bias leads to a phenomenon known as over-squashing, where a node feature is insensitive to information contained at distant nodes. Despite recent methods introduced to mitigate this issue, an understanding of the causes for over-squashing and of possible solutions are lacking. In this theoretical work, we prove that: (i) Neural network width can mitigate over-squashing, but at the cost of making the whole network more sensitive; (ii) Conversely, depth cannot help mitigate over-squashing: increasing the number of layers leads to over-squashing being dominated by vanishing gradients; (iii) The graph topology plays the greatest role, since over-squashing occurs between nodes at high commute time. Our analysis provides a unified framework to study different recent methods introduced to cope with over-squashing and serves as a justification for a class of methods that fall under graph rewiring.

Machine learning for malware classification shows encouraging results, but real deployments suffer from performance degradation as malware authors adapt their techniques to evade detection. This phenomenon, known as concept drift, occurs as new malware examples evolve and become less and less like the original training examples. One promising method to cope with concept drift is classification with rejection in which examples that are likely to be misclassified are instead quarantined until they can be expertly analyzed.We propose TRANSCENDENT, a rejection framework built on Transcend, a recently proposed strategy based on conformal prediction theory. In particular, we provide a formal treatment of Transcend, enabling us to refine conformal evaluation theory—its underlying statistical engine—and gain a better understanding of the theoretical reasons for its effectiveness. In the process, we develop two additional conformal evaluators that match or surpass the performance of the original while significantly decreasing the computational overhead. We evaluate TRANSCENDENT on a malware dataset spanning 5 years that removes sources of experimental bias present in the original evaluation. TRANSCENDENT outperforms state-of-the-art approaches while generalizing across different malware domains and classifiers.To further assist practitioners, we showcase optimal operational settings for a TRANSCENDENT deployment and show how it can be applied to many popular learning algorithms. These insights support both old and new empirical findings, making Transcend a sound and practical solution for the first time. To this end, we release TRANSCENDENT as open source, to aid the adoption of rejection strategies by the security community.