Corollary1statesthatf0 ismoreexpressivethan f when γ = 0, and thus the optimalf0 always outputs no higher loss, which, yet, is not true for arbitraryf0 (e.g.

Such message exchanging isparameter-free andselfadaptive, as it is dynamically controlled by the scaling factors that are determined by the training itself.

In Split Federated Learning (SFL), the clients collaboratively train a model with the help of a server by splitting the model into two parts. Part-1 is trained locally at each client and aggregated by the aggregator at the end of each round. Part-2 is trained at a server that sequentially processes the intermediate activations received from each client. We study the phenomenon of catastrophic forgetting (CF) in SFL in the presence of data heterogeneity. In detail, due to the nature of SFL, local updates of part-1 may drift away from global optima, while part-2 is sensitive to the processing sequence, similar to forgetting in continual learning (CL). Specifically, we observe that the trained model performs better in classes (labels) seen at the end of the sequence. We investigate this phenomenon with emphasis on key aspects of SFL, such as the processing order at the server and the cut layer. Based on our findings, we propose Hydra, a novel mitigation method inspired by multi-head neural networks and adapted for the SFL's setting. Extensive numerical evaluations show that Hydra outperforms baselines and methods from the literature.

For more complicated periodic functions, see Figure 11 and 12. For a larger value of a, the extrapolation improves. We use Adam as the optimizer. The region within the dashed vertical lines are the range of the training set. Learning range is indicated by the blue vertical bars.

We appreciate suggestions from R6, 7, 8 and will include these in the paper. We have included most competitive methods with comparable settings to ours at the submission time. We will include the shown Algorithm 1. S sampled such that all attributes are present? However, our framework can compose features from any set S by solving Eq (10) even with missing attributes in S . Please notice that they are different.

Estimating uncertainty from deep neural networks is a widely used approach for detecting out-of-distribution (OoD) samples, which typically exhibit high predictive uncertainty. However, conventional methods such as Monte Carlo (MC) Dropout often focus solely on either model or data uncertainty, failing to align with the semantic objective of OoD detection. T o address this, we propose the Free-Energy Posterior Network, a novel framework that jointly models distributional uncertainty and identifying OoD and misclassified regions using free energy. Our method introduces two key contributions: (1) a free-energy-based density estimator parameterized by a Beta distribution, which enables fine-grained uncertainty estimation near ambiguous or unseen regions; and (2) a loss integrated within a posterior network, allowing direct uncertainty estimation from learned parameters without requiring stochastic sampling. By integrating our approach with the residual prediction branch (RPL) framework, the proposed method goes beyond post-hoc energy thresholding and enables the network to learn OoD regions by leveraging the variance of the Beta distribution, resulting in a semantically meaningful and computationally efficient solution for uncertainty-aware segmentation.