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.

Neural network classifiers trained with cross-entropy loss achieve strong predictive accuracy but lack the capability to provide inherent predictive uncertainty estimates, thus requiring external techniques to obtain these estimates. In addition, softmax scores for the true class can vary substantially across independent training runs, which limits the reliability of uncertainty-based decisions in downstream tasks. Evidential Deep Learning aims to address these limitations by producing uncertainty estimates in a single pass, but evidential training is highly sensitive to design choices including loss formulation, prior regularization, and activation functions. Therefore, this work introduces an alternative Dirichlet parameter estimation strategy by applying a method of moments estimator to ensembles of softmax outputs, with an optional maximum-likelihood refinement step. This ensemble-based construction decouples uncertainty estimation from the fragile evidential loss design while also mitigating the variability of single-run cross-entropy training, producing explicit Dirichlet predictive distributions. Across multiple datasets, we show that the improved stability and predictive uncertainty behavior of these ensemble-derived Dirichlet estimates translate into stronger performance in downstream uncertainty-guided applications such as prediction confidence scoring and selective classification.

GPN applies a personalized page rank (PPR) module to diffuse the evidence among neighboring nodes.

Ensemble approaches for uncertainty estimation have recently been applied to the tasks of misclassification detection, out-of-distribution input detection and adversarial attack detection. Prior Networks have been proposed as an approach to efficiently emulate an ensemble of models for classification by parameteris-ing a Dirichlet prior distribution over output distributions.

GPN applies a personalized page rank (PPR) module to diffuse the evidence among neighboring nodes.

Hyperparameter tuning is a challenging problem especially when the system itself involves uncertainty. Due to noisy function evaluations, optimization under uncertainty can be computationally expensive. In this paper, we present a novel Bayesian optimization framework tailored for hyperparameter tuning under uncertainty, with a focus on optimizing a scale- or precision-type parameter in stochastic models. The proposed method employs a statistical surrogate for the underlying random variable, enabling analytical evaluation of the expectation operator. Moreover, we derive a closed-form expression for the optimizer of the random acquisition function, which significantly reduces computational cost per iteration. Compared with a conventional one-dimensional Monte Carlo-based optimization scheme, the proposed approach requires 40 times fewer data points, resulting in up to a 40-fold reduction in computational cost. We demonstrate the effectiveness of the proposed method through two numerical examples in computational engineering.