Federico Errica

Daniele Atzeni, Federico Errica, Davide Bacciu et al.

We propose an extension of the Contextual Graph Markov Model, a deep and probabilistic machine learning model for graphs, to model the distribution of edge features. Our approach is architectural, as we introduce an additional Bayesian network mapping edge features into discrete states to be used by the original model. In doing so, we are also able to build richer graph representations even in the absence of edge features, which is confirmed by the performance improvements on standard graph classification benchmarks. Moreover, we successfully test our proposal in a graph regression scenario where edge features are of fundamental importance, and we show that the learned edge representation provides substantial performance improvements against the original model on three link prediction tasks. By keeping the computational complexity linear in the number of edges, the proposed model is amenable to large-scale graph processing.

Donald Shenaj, Federico Errica, Antonio Carta

Low Rank Adaptation (LoRA) is the de facto fine-tuning strategy to generate personalized images from pre-trained diffusion models. Choosing a good rank is extremely critical, since it trades off performance and memory consumption, but today the decision is often left to the community's consensus, regardless of the personalized subject's complexity. The reason is evident: the cost of selecting a good rank for each LoRA component is combinatorial, so we opt for practical shortcuts such as fixing the same rank for all components. In this paper, we take a first step to overcome this challenge. Inspired by variational methods that learn an adaptive width of neural networks, we let the ranks of each layer freely adapt during fine-tuning on a subject. We achieve it by imposing an ordering of importance on the rank's positions, effectively encouraging the creation of higher ranks when strictly needed. Qualitatively and quantitatively, our approach, LoRA$^2$, achieves a competitive trade-off between DINO, CLIP-I, and CLIP-T across 29 subjects while requiring much less memory and lower rank than high rank LoRA versions. Code: https://github.com/donaldssh/NotAllLayersAreCreatedEqual.

Tran Gia Bao Ngo, Zulfikar Alom, Federico Errica et al.

Adversarial learning and the robustness of Graph Neural Networks (GNNs) are topics of widespread interest in the machine learning community, as documented by the number of adversarial attacks and defenses designed for these purposes. While a rigorous evaluation of these adversarial methods is necessary to understand the robustness of GNNs in real-world applications, we posit that many works in the literature do not share the same experimental settings, leading to ambiguous and potentially contradictory scientific conclusions. In this benchmark, we demonstrate the importance of adopting fair, robust, and standardized evaluation protocols in adversarial GNN research. We perform a comprehensive re-evaluation of seven widely used attacks and eight recent defenses under both poisoning and evasion scenarios, across six popular graph datasets. Our study spans over 453,000 experiments conducted within a unified framework. We observe substantial differences in adversarial attack performance when evaluated under a fair and robust procedure. Our findings reveal that previously overlooked factors, such as target node selection and the training process of the attacked model, have a profound impact on attack effectiveness, to the extent of completely distorting performance insights. These results underscore the urgent need for standardized evaluations in adversarial graph machine learning.

Henrik Christiansen, Federico Errica, Francesco Alesiani

The performance of Hamiltonian Monte Carlo simulations crucially depends on both the integration timestep and the number of integration steps. We present an adaptive general-purpose framework to automatically tune such parameters, based on a local loss function which promotes the fast exploration of phase-space. We show that a good correspondence between loss and autocorrelation time can be established, allowing for gradient-based optimization using a fully-differentiable set-up. The loss is constructed in such a way that it also allows for gradient-driven learning of a distribution over the number of integration steps. Our approach is demonstrated for the one-dimensional harmonic oscillator and alanine dipeptide, a small protein common as a test case for simulation methods. Through the application to the harmonic oscillator, we highlight the importance of not using a fixed timestep to avoid a rugged loss surface with many local minima, otherwise trapping the optimization. In the case of alanine dipeptide, by tuning the only free parameter of our loss definition, we find a good correspondence between it and the autocorrelation times, resulting in a $>100$ fold speed up in optimization of simulation parameters compared to a grid-search. For this system, we also extend the integrator to allow for atom-dependent timesteps, providing a further reduction of $25\%$ in autocorrelation times.

Daniele Castellana, Federico Errica

In the past, the dichotomy between homophily and heterophily has inspired research contributions toward a better understanding of Deep Graph Networks' inductive bias. In particular, it was believed that homophily strongly correlates with better node classification predictions of message-passing methods. More recently, however, researchers pointed out that such dichotomy is too simplistic as we can construct node classification tasks where graphs are completely heterophilic but the performances remain high. Most of these works have also proposed new quantitative metrics to understand when a graph structure is useful, which implicitly or explicitly assume the correlation between node features and target labels. Our work empirically investigates what happens when this strong assumption does not hold, by formalising two generative processes for node classification tasks that allow us to build and study ad-hoc problems. To quantitatively measure the influence of the node features on the target labels, we also use a metric we call Feature Informativeness. We construct six synthetic tasks and evaluate the performance of six models, including structure-agnostic ones. Our findings reveal that previously defined metrics are not adequate when we relax the above assumption. Our contribution to the workshop aims at presenting novel research findings that could help advance our understanding of the field.

Henrik Christiansen, Takashi Maruyama, Federico Errica et al.

We present an end-to-end differentiable molecular simulation framework (DIMOS) for molecular dynamics and Monte Carlo simulations. DIMOS easily integrates machine-learning-based interatomic potentials and implements classical force fields including an efficient implementation of particle-mesh Ewald. Thanks to its modularity, both classical and machine-learning-based approaches can be easily combined into a hybrid description of the system (ML/MM). By supporting key molecular dynamics features such as efficient neighborlists and constraint algorithms for larger time steps, the framework makes steps in bridging the gap between hand-optimized simulation engines and the flexibility of a \verb|PyTorch| implementation. We show that due to improved linear instead of quadratic scaling as function of system size DIMOS is able to obtain speed-up factors of up to $170\times$ for classical force field simulations against another fully differentiable simulation framework. The advantage of differentiability is demonstrated by an end-to-end optimization of the proposal distribution in a Markov Chain Monte Carlo simulation based on Hamiltonian Monte Carlo (HMC). Using these optimized simulation parameters a $3\times$ acceleration is observed in comparison to ad-hoc chosen simulation parameters. The code is available at https://github.com/nec-research/DIMOS.

Federico Errica, Henrik Christiansen, Viktor Zaverkin et al.

Long-range interactions are essential for the correct description of complex systems in many scientific fields. The price to pay for including them in the calculations, however, is a dramatic increase in the overall computational costs. Recently, deep graph networks have been employed as efficient, data-driven models for predicting properties of complex systems represented as graphs. These models rely on a message passing strategy that should, in principle, capture long-range information without explicitly modeling the corresponding interactions. In practice, most deep graph networks cannot really model long-range dependencies due to the intrinsic limitations of (synchronous) message passing, namely oversmoothing, oversquashing, and underreaching. This work proposes a general framework that learns to mitigate these limitations: within a variational inference framework, we endow message passing architectures with the ability to adapt their depth and filter messages along the way. With theoretical and empirical arguments, we show that this strategy better captures long-range interactions, by competing with the state of the art on five node and graph prediction datasets.

Viktor Zaverkin, Francesco Alesiani, Takashi Maruyama et al.

The ability to perform fast and accurate atomistic simulations is crucial for advancing the chemical sciences. By learning from high-quality data, machine-learned interatomic potentials achieve accuracy on par with ab initio and first-principles methods at a fraction of their computational cost. The success of machine-learned interatomic potentials arises from integrating inductive biases such as equivariance to group actions on an atomic system, e.g., equivariance to rotations and reflections. In particular, the field has notably advanced with the emergence of equivariant message passing. Most of these models represent an atomic system using spherical tensors, tensor products of which require complicated numerical coefficients and can be computationally demanding. Cartesian tensors offer a promising alternative, though state-of-the-art methods lack flexibility in message-passing mechanisms, restricting their architectures and expressive power. This work explores higher-rank irreducible Cartesian tensors to address these limitations. We integrate irreducible Cartesian tensor products into message-passing neural networks and prove the equivariance and traceless property of the resulting layers. Through empirical evaluations on various benchmark data sets, we consistently observe on-par or better performance than that of state-of-the-art spherical and Cartesian models.

Julia Gastinger, Christian Meilicke, Federico Errica et al.

Temporal Knowledge Graph (TKG) Forecasting aims at predicting links in Knowledge Graphs for future timesteps based on a history of Knowledge Graphs. To this day, standardized evaluation protocols and rigorous comparison across TKG models are available, but the importance of simple baselines is often neglected in the evaluation, which prevents researchers from discerning actual and fictitious progress. We propose to close this gap by designing an intuitive baseline for TKG Forecasting based on predicting recurring facts. Compared to most TKG models, it requires little hyperparameter tuning and no iterative training. Further, it can help to identify failure modes in existing approaches. The empirical findings are quite unexpected: compared to 11 methods on five datasets, our baseline ranks first or third in three of them, painting a radically different picture of the predictive quality of the state of the art.

Adrian Arnaiz-Rodriguez, Federico Errica

After a renaissance phase in which researchers revisited the message-passing paradigm through the lens of deep learning, the graph machine learning community shifted its attention towards a deeper and practical understanding of message-passing's benefits and limitations. In this position paper, we notice how the fast pace of progress around the topics of oversmoothing and oversquashing, the homophily-heterophily dichotomy, and long-range tasks, came with the consolidation of commonly accepted beliefs and assumptions that are not always true nor easy to distinguish from each other. We argue that this has led to ambiguities around the investigated problems, preventing researchers from focusing on and addressing precise research questions while causing a good amount of misunderstandings. Our contribution wants to make such common beliefs explicit and encourage critical thinking around these topics, supported by simple but noteworthy counterexamples. The hope is to clarify the distinction between the different issues and promote separate but intertwined research directions to address them.

Federico Errica, Henrik Christiansen, Viktor Zaverkin et al.

For almost 70 years, researchers have mostly relied on hyper-parameter tuning to select the width of neural networks' layers. This paper challenges the status quo by introducing an easy-to-use technique to learn an unbounded width of a neural network's layer during training. The technique does not rely on alternate optimization nor hand-crafted gradient heuristics; rather, it jointly optimizes the width and the parameters of each layer via simple backpropagation. We apply the technique to a broad range of data domains such as tables, images, text, sequences, and graphs, showing how the width adapts to the task's difficulty. The method imposes a soft ordering of importance among neurons, by which it also is possible to truncate the trained network at virtually zero cost, achieving a smooth trade-off between performance and compute resources in a structured way. Alternatively, one can dynamically compress the network with no performance degradation. In light of recent foundation models trained on large datasets, believed to require billions of parameters and where hyper-parameter tuning is unfeasible due to humongous training costs, our approach stands as a viable alternative for width learning.

Joël Mathys, Federico Errica

Message-passing architectures struggle to sufficiently model long-range dependencies in node and graph prediction tasks. We propose a novel approach exploiting hierarchical graph structures and adaptive random walks to address this challenge. Our method introduces learnable transition probabilities that decide whether the walk should prefer the original graph or travel across hierarchical shortcuts. On a synthetic long-range task, we demonstrate that our approach can exceed the theoretical bound that constrains traditional approaches operating solely on the original topology. Specifically, walks that prefer the hierarchy achieve the same performance as longer walks on the original graph. These preliminary findings open a promising direction for efficiently processing large graphs while effectively capturing long-range dependencies.

Francesco Alesiani, Henrik Christiansen, Federico Errica

Kolmogorov Arnold Networks (KANs) are an emerging architecture for building machine learning models. KANs are based on the theoretical foundation of the Kolmogorov-Arnold Theorem and its expansions, which provide an exact representation of a multi-variate continuous bounded function as the composition of a limited number of univariate continuous functions. While such theoretical results are powerful, their use as a representation learning alternative to a multi-layer perceptron (MLP) hinges on the ad-hoc choice of the number of bases modeling each of the univariate functions. In this work, we show how to address this problem by adaptively learning a potentially infinite number of bases for each univariate function during training. We therefore model the problem as a variational inference optimization problem. Our proposal, called InfinityKAN, which uses backpropagation, extends the potential applicability of KANs by treating an important hyperparameter as part of the learning process.

Federico Errica, Giuseppe Siracusano, Davide Sanvito et al.

Large Language Models (LLMs) changed the way we design and interact with software systems. Their ability to process and extract information from text has drastically improved productivity in a number of routine tasks. Developers that want to include these models in their software stack, however, face a dreadful challenge: debugging LLMs' inconsistent behavior across minor variations of the prompt. We therefore introduce two metrics for classification tasks, namely sensitivity and consistency, which are complementary to task performance. First, sensitivity measures changes of predictions across rephrasings of the prompt, and does not require access to ground truth labels. Instead, consistency measures how predictions vary across rephrasings for elements of the same class. We perform an empirical comparison of these metrics on text classification tasks, using them as guideline for understanding failure modes of the LLM. Our hope is that sensitivity and consistency will be helpful to guide prompt engineering and obtain LLMs that balance robustness with performance.

Federico Errica, Mathias Niepert

We introduce Graph-Induced Sum-Product Networks (GSPNs), a new probabilistic framework for graph representation learning that can tractably answer probabilistic queries. Inspired by the computational trees induced by vertices in the context of message-passing neural networks, we build hierarchies of sum-product networks (SPNs) where the parameters of a parent SPN are learnable transformations of the a-posterior mixing probabilities of its children's sum units. Due to weight sharing and the tree-shaped computation graphs of GSPNs, we obtain the efficiency and efficacy of deep graph networks with the additional advantages of a probabilistic model. We show the model's competitiveness on scarce supervision scenarios, under missing data, and for graph classification in comparison to popular neural models. We complement the experiments with qualitative analyses on hyper-parameters and the model's ability to answer probabilistic queries.

Federico Errica

The adaptive processing of structured data is a long-standing research topic in machine learning that investigates how to automatically learn a mapping from a structured input to outputs of various nature. Recently, there has been an increasing interest in the adaptive processing of graphs, which led to the development of different neural network-based methodologies. In this thesis, we take a different route and develop a Bayesian Deep Learning framework for graph learning. The dissertation begins with a review of the principles over which most of the methods in the field are built, followed by a study on graph classification reproducibility issues. We then proceed to bridge the basic ideas of deep learning for graphs with the Bayesian world, by building our deep architectures in an incremental fashion. This framework allows us to consider graphs with discrete and continuous edge features, producing unsupervised embeddings rich enough to reach the state of the art on several classification tasks. Our approach is also amenable to a Bayesian nonparametric extension that automatizes the choice of almost all model's hyper-parameters. Two real-world applications demonstrate the efficacy of deep learning for graphs. The first concerns the prediction of information-theoretic quantities for molecular simulations with supervised neural models. After that, we exploit our Bayesian models to solve a malware-classification task while being robust to intra-procedural code obfuscation techniques. We conclude the dissertation with an attempt to blend the best of the neural and Bayesian worlds together. The resulting hybrid model is able to predict multimodal distributions conditioned on input graphs, with the consequent ability to model stochasticity and uncertainty better than most works. Overall, we aim to provide a Bayesian perspective into the articulated research field of deep learning for graphs.

Antonio Carta, Andrea Cossu, Federico Errica et al.

In this work, we study the phenomenon of catastrophic forgetting in the graph representation learning scenario. The primary objective of the analysis is to understand whether classical continual learning techniques for flat and sequential data have a tangible impact on performances when applied to graph data. To do so, we experiment with a structure-agnostic model and a deep graph network in a robust and controlled environment on three different datasets. The benchmark is complemented by an investigation on the effect of structure-preserving regularization techniques on catastrophic forgetting. We find that replay is the most effective strategy in so far, which also benefits the most from the use of regularization. Our findings suggest interesting future research at the intersection of the continual and graph representation learning fields. Finally, we provide researchers with a flexible software framework to reproduce our results and carry out further experiments.

Federico Errica, Davide Bacciu, Alessio Micheli

We introduce the Graph Mixture Density Networks, a new family of machine learning models that can fit multimodal output distributions conditioned on graphs of arbitrary topology. By combining ideas from mixture models and graph representation learning, we address a broader class of challenging conditional density estimation problems that rely on structured data. In this respect, we evaluate our method on a new benchmark application that leverages random graphs for stochastic epidemic simulations. We show a significant improvement in the likelihood of epidemic outcomes when taking into account both multimodality and structure. The empirical analysis is complemented by two real-world regression tasks showing the effectiveness of our approach in modeling the output prediction uncertainty. Graph Mixture Density Networks open appealing research opportunities in the study of structure-dependent phenomena that exhibit non-trivial conditional output distributions.

Federico Errica, Marco Giulini, Davide Bacciu et al.

The limits of molecular dynamics (MD) simulations of macromolecules are steadily pushed forward by the relentless developments of computer architectures and algorithms. This explosion in the number and extent (in size and time) of MD trajectories induces the need of automated and transferable methods to rationalise the raw data and make quantitative sense out of them. Recently, an algorithmic approach was developed by some of us to identify the subset of a protein's atoms, or mapping, that enables the most informative description of it. This method relies on the computation, for a given reduced representation, of the associated mapping entropy, that is, a measure of the information loss due to the simplification. Albeit relatively straightforward, this calculation can be time consuming. Here, we describe the implementation of a deep learning approach aimed at accelerating the calculation of the mapping entropy. The method relies on deep graph networks, which provide extreme flexibility in the input format. We show that deep graph networks are accurate and remarkably efficient, with a speedup factor as large as $10^5$ with respect to the algorithmic computation of the mapping entropy. Applications of this method, which entails a great potential in the study of biomolecules when used to reconstruct its mapping entropy landscape, reach much farther than this, being the scheme easily transferable to the computation of arbitrary functions of a molecule's structure.

Federico Errica, Ludovic Denoyer, Bora Edizel et al.

We propose a model to tackle classification tasks in the presence of very little training data. To this aim, we approximate the notion of exact match with a theoretically sound mechanism that computes a probability of matching in the input space. Importantly, the model learns to focus on elements of the input that are relevant for the task at hand; by leveraging highlighted portions of the training data, an error boosting technique guides the learning process. In practice, it increases the error associated with relevant parts of the input by a given factor. Remarkable results on text classification tasks confirm the benefits of the proposed approach in both balanced and unbalanced cases, thus being of practical use when labeling new examples is expensive. In addition, by inspecting its weights, it is often possible to gather insights on what the model has learned.

Federico Errica, Davide Bacciu, Alessio Micheli

We propose a new Graph Neural Network that combines recent advancements in the field. We give theoretical contributions by proving that the model is strictly more general than the Graph Isomorphism Network and the Gated Graph Neural Network, as it can approximate the same functions and deal with arbitrary edge values. Then, we show how a single node information can flow through the graph unchanged.

Davide Bacciu, Federico Errica, Alessio Micheli et al.

The adaptive processing of graph data is a long-standing research topic which has been lately consolidated as a theme of major interest in the deep learning community. The snap increase in the amount and breadth of related research has come at the price of little systematization of knowledge and attention to earlier literature. This work is designed as a tutorial introduction to the field of deep learning for graphs. It favours a consistent and progressive introduction of the main concepts and architectural aspects over an exposition of the most recent literature, for which the reader is referred to available surveys. The paper takes a top-down view to the problem, introducing a generalized formulation of graph representation learning based on a local and iterative approach to structured information processing. It introduces the basic building blocks that can be combined to design novel and effective neural models for graphs. The methodological exposition is complemented by a discussion of interesting research challenges and applications in the field.

Federico Errica, Marco Podda, Davide Bacciu et al.

Experimental reproducibility and replicability are critical topics in machine learning. Authors have often raised concerns about their lack in scientific publications to improve the quality of the field. Recently, the graph representation learning field has attracted the attention of a wide research community, which resulted in a large stream of works. As such, several Graph Neural Network models have been developed to effectively tackle graph classification. However, experimental procedures often lack rigorousness and are hardly reproducible. Motivated by this, we provide an overview of common practices that should be avoided to fairly compare with the state of the art. To counter this troubling trend, we ran more than 47000 experiments in a controlled and uniform framework to re-evaluate five popular models across nine common benchmarks. Moreover, by comparing GNNs with structure-agnostic baselines we provide convincing evidence that, on some datasets, structural information has not been exploited yet. We believe that this work can contribute to the development of the graph learning field, by providing a much needed grounding for rigorous evaluations of graph classification models.

Davide Bacciu, Federico Errica, Alessio Micheli

We introduce the Contextual Graph Markov Model, an approach combining ideas from generative models and neural networks for the processing of graph data. It founds on a constructive methodology to build a deep architecture comprising layers of probabilistic models that learn to encode the structured information in an incremental fashion. Context is diffused in an efficient and scalable way across the graph vertexes and edges. The resulting graph encoding is used in combination with discriminative models to address structure classification benchmarks.