Artur Back de Luca

Artur Back de Luca, Guojun Zhang, Xi Chen et al.

Federated Learning (FL) is a prominent framework that enables training a centralized model while securing user privacy by fusing local, decentralized models. In this setting, one major obstacle is data heterogeneity, i.e., each client having non-identically and independently distributed (non-IID) data. This is analogous to the context of Domain Generalization (DG), where each client can be treated as a different domain. However, while many approaches in DG tackle data heterogeneity from the algorithmic perspective, recent evidence suggests that data augmentation can induce equal or greater performance. Motivated by this connection, we present federated versions of popular DG algorithms, and show that by applying appropriate data augmentation, we can mitigate data heterogeneity in the federated setting, and obtain higher accuracy on unseen clients. Equipped with data augmentation, we can achieve state-of-the-art performance using even the most basic Federated Averaging algorithm, with much sparser communication.

Artur Back de Luca, Kimon Fountoulakis

As state-of-the-art neural networks are deployed on reasoning and algorithmic tasks, exactness guarantees become increasingly important. However, high average-case accuracy can still mask inconsistent behaviors. This motivates exact certification, which asks for the smallest set of labeled examples needed to certify that a learned hypothesis equals the target. We show that while some hypotheses are easy to certify, even minimal overparametrization can make certification exponentially hard across several hypothesis classes. For threshold circuits of depth $\ge 2$, adding a single extra gate can force certificate sizes exponential in the input dimension. We show an analogous hardness result for log-precision Transformers with only constant architectural overhead. We also characterize approximate certification, showing that allowing only polynomially many mistakes still requires exponentially large certificates, whereas constant relative-error guarantees can hide exponentially many mistakes. Empirically, we study certification for constructed circuits and trained Transformers for recognizing binary addition. While the constructed circuits instantiate the exponential barrier for certification, the trained Transformer analysis shows that imperfect models can evade detection by large uniformly sampled certificate candidates.

Artur Back de Luca, Kimon Fountoulakis, Shenghao Yang

The growing interest in machine learning problems over graphs with additional node information such as texts, images, or labels has popularized methods that require the costly operation of processing the entire graph. Yet, little effort has been made to the development of fast local methods (i.e. without accessing the entire graph) that extract useful information from such data. To that end, we propose a study of local graph clustering using noisy node labels as a proxy for additional node information. In this setting, nodes receive initial binary labels based on cluster affiliation: 1 if they belong to the target cluster and 0 otherwise. Subsequently, a fraction of these labels is flipped. We investigate the benefits of incorporating noisy labels for local graph clustering. By constructing a weighted graph with such labels, we study the performance of graph diffusion-based local clustering method on both the original and the weighted graphs. From a theoretical perspective, we consider recovering an unknown target cluster with a single seed node in a random graph with independent noisy node labels. We provide sufficient conditions on the label noise under which, with high probability, using diffusion in the weighted graph yields a more accurate recovery of the target cluster. This approach proves more effective than using the given labels alone or using diffusion in the label-free original graph. Empirically, we show that reliable node labels can be obtained with just a few samples from an attributed graph. Moreover, utilizing these labels via diffusion in the weighted graph leads to significantly better local clustering performance across several real-world datasets, improving F1 scores by up to 13%.

Muhammad Fetrat Qharabagh, Artur Back de Luca, George Giapitzakis et al.

Understanding what graph neural networks can learn, especially their ability to learn to execute algorithms, remains a central theoretical challenge. In this work, we prove exact learnability results for graph algorithms under bounded-degree and finite-precision constraints. Our approach follows a two-step process. First, we train an ensemble of multi-layer perceptrons (MLPs) to execute the local instructions of a single node. Second, during inference, we use the trained MLP ensemble as the update function within a graph neural network (GNN). Leveraging Neural Tangent Kernel (NTK) theory, we show that local instructions can be learned from a small training set, enabling the complete graph algorithm to be executed during inference without error and with high probability. To illustrate the learning power of our setting, we establish a rigorous learnability result for the LOCAL model of distributed computation. We further demonstrate positive learnability results for widely studied algorithms such as message flooding, breadth-first and depth-first search, and Bellman-Ford.

Artur Back de Luca, Kimon Fountoulakis

The execution of graph algorithms using neural networks has recently attracted significant interest due to promising empirical progress. This motivates further understanding of how neural networks can replicate reasoning steps with relational data. In this work, we study the ability of transformer networks to simulate algorithms on graphs from a theoretical perspective. The architecture we use is a looped transformer with extra attention heads that interact with the graph. We prove by construction that this architecture can simulate individual algorithms such as Dijkstra's shortest path, Breadth- and Depth-First Search, and Kosaraju's strongly connected components, as well as multiple algorithms simultaneously. The number of parameters in the networks does not increase with the input graph size, which implies that the networks can simulate the above algorithms for any graph. Despite this property, we show a limit to simulation in our solution due to finite precision. Finally, we show a Turing Completeness result with constant width when the extra attention heads are utilized.

Artur Back de Luca, George Giapitzakis, Kimon Fountoulakis

Neural networks are known for their ability to approximate smooth functions, yet they fail to generalize perfectly to unseen inputs when trained on discrete operations. Such operations lie at the heart of algorithmic tasks such as arithmetic, which is often used as a test bed for algorithmic execution in neural networks. In this work, we ask: can neural networks learn to execute binary-encoded algorithmic instructions exactly? We use the Neural Tangent Kernel (NTK) framework to study the training dynamics of two-layer fully connected networks in the infinite-width limit and show how a sufficiently large ensemble of such models can be trained to execute exactly, with high probability, four fundamental tasks: binary permutations, binary addition, binary multiplication, and Subtract and Branch if Negative (SBN) instructions. Since SBN is Turing-complete, our framework extends to computable functions. We show how this can be efficiently achieved using only logarithmically many training data. Our approach relies on two techniques: structuring the training data to isolate bit-level rules, and controlling correlations in the NTK regime to align model predictions with the target algorithmic executions.