Federated Learning (FL) enables multiple clients to collaboratively train models without sharing raw data, but is vulnerable to Byzantine attacks and data heterogeneity, which can severely degrade performance. Existing Byzantine-robust approaches tackle data heterogeneity, but incur high computational overhead during gradient aggregation, thereby slowing down the training process. To address this issue, we propose a simple yet effective Federated Normalized Gradients Algorithm (Fed-NGA), which performs aggregation by merely computing the weighted mean of the normalized gradients from each client. This approach yields a favorable time complexity of O(pM), where p is the model dimension and M is the number of clients. We rigorously prove that Fed-NGA is robust to both Byzantine faults and data heterogeneity. For non-convex loss functions, Fed-NGA achieves convergence to a neighborhood of stationary points under general assumptions, and further attains zero optimality gap under some mild conditions, which is an outcome rarely achieved in existing literature.

Besides standard latent subsymbolic variables, our model exploits a probabilistic logic program to define a further structured representation, which is used for logical reasoning. The entire process is end-to-end differentiable. Once trained, VAEL can solve new unseen generation tasks by (i) leveraging the previously acquired knowledge encoded in the neural component and (ii) exploiting new logical programs on the structured latent space. Our experiments provide support on the benefits of this neuro-symbolic integration both in terms of task generalization and data efficiency. To the best of our knowledge, this work is the first to propose a general-purpose end-to-end framework integrating probabilistic logic programming into a deep generative model.

While both classical and neural network classifiers can achieve high accuracy, they fall short on offering uncertainty bounds on their predictions, making them unfit for safety-critical applications. Existing kernel-based classifiers that provide such bounds scale with $\mathcal O (n^{\sim3})$ in time, making them computationally intractable for large datasets. To address this, we propose a novel, computationally efficient classification algorithm based on the Nadaraya-Watson estimator, for whose estimates we derive frequentist uncertainty intervals. We evaluate our classifier on synthetically generated data and on electrocardiographic heartbeat signals from the MIT-BIH Arrhythmia database. We show that the method achieves competitive accuracy $>$\SI{96}{\percent} at $\mathcal O(n)$ and $\mathcal O(\log n)$ operations, while providing actionable uncertainty bounds. These bounds can, e.g., aid in flagging low-confidence predictions, making them suitable for real-time settings with resource constraints, such as diagnostic monitoring or implantable devices.

However, the performance of the ONN model is often diminished by the gap between the ideal simulated system and the actual physical system. To bridge the gap, this work conducts extensive experiments to investigate systematic errors in the optical physical system within the context of image classification tasks.

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