The supplementary material is organized as follows. We give details of the definitions and notation in Section B.1 . Then, we provide the technical details of the lower bound (Lemma 3.3). In Section D.4 we provide insights into auto-labeling using This suggests, in these settings auto-labeling using active learning followed by selective classification is expected to work well. This idea is captured by the Chow's excess risk [ Nevertheless, it would be interesting future work to explore the connections between auto-labeling and active learning with abstention.

Denoising and score estimation have long been known to be linked via the classical Tweedie's formula. In this work, we first extend the latter to a wider range of distributions often called "energy models" and denoted elliptical distributions in this work. Next, we examine an alternative view: we consider the denoising posterior $P(X|Y)$ as the optimizer of the energy score (a scoring rule) and derive a fundamental identity that connects the (path-) derivative of a (possibly) non-Euclidean energy score to the score of the noisy marginal. This identity can be seen as an analog of Tweedie's identity for the energy score, and allows for several interesting applications; for example, score estimation, noise distribution parameter estimation, as well as using energy score models in the context of "traditional" diffusion model samplers with a wider array of noising distributions.

Determining whether inputs are out-of-distribution (OOD) is an essential building block for safely deploying machine learning models in the open world. However, previous methods relying on the softmax confidence score suffer from overconfident posterior distributions for OOD data. We propose a unified framework for OOD detection that uses an energy score. We show that energy scores better distinguish in-and out-of-distribution samples than the traditional approach using the softmax scores. Unlike softmax confidence scores, energy scores are theoretically aligned with the probability density of the inputs and are less susceptible to the overconfidence issue. Within this framework, energy can be flexibly used as a scoring function for any pre-trained neural classifier as well as a trainable cost function to shape the energy surface explicitly for OOD detection. On a CIFAR-10 pre-trained WideResNet, using the energy score reduces the average FPR (at TPR 95%) by 18.03% compared to the softmax confidence score. With energy-based training, our method outperforms the state-of-the-art on common benchmarks.

Federated learning is a promising paradigm that utilizes distributed client resources while preserving data privacy. Most existing FL approaches assume clients possess labeled data, however, in real-world scenarios, client-side labels are often unavailable. Semi-supervised Federated learning, where only the server holds labeled data, addresses this issue. However, it experiences significant performance degradation as the number of labeled data decreases. To tackle this problem, we propose \textit{CATCHFed}, which introduces client-aware adaptive thresholds considering class difficulty, hybrid thresholds to enhance pseudo-label quality, and utilizes unpseudo-labeled data for consistency regularization. Extensive experiments across various datasets and configurations demonstrate that CATCHFed effectively leverages unlabeled client data, achieving superior performance even in extremely limited-label settings.

In this work, we estimate the intrinsic dimension (iD) of the Radio Galaxy Zoo (RGZ) dataset using a score-based diffusion model. We examine how the iD estimates vary as a function of Bayesian neural network (BNN) energy scores, which measure how similar the radio sources are to the MiraBest subset of the RGZ dataset. We find that out-of-distribution sources exhibit higher iD values, and that the overall iD for RGZ exceeds those typically reported for natural image datasets. Furthermore, we analyse how iD varies across Fanaroff-Riley (FR) morphological classes and as a function of the signal-to-noise ratio (SNR). While no relationship is found between FR I and FR II classes, a weak trend toward higher SNR at lower iD. Future work using the RGZ dataset could make use of the relationship between iD and energy scores to quantitatively study and improve the representations learned by various self-supervised learning algorithms.

However, these methods have significant drawbacks, such as sensitivity to token-splitting strategies and damage to informative tokens in later layers.

The Maha-lanobis score is calculated using the features of the second-to-last layer. Bold numbers are superior results. We fine-tune the models once with a fixed random seed. OE [15], reported performance for each OOD dataset is averaged over 10 random batches of samples. The Maha-lanobis scores are calculated from the features of the second-to-last layer.

OOD detection that uses an energy score .

We thank the reviewers for their insightful feedback. Can you evaluate on additional in-distribution dataset? We will definitely include these results in the final version. How to distinguish methodological differences w.r .t. prior work? The idea of using the energy score for OOD detection is novel and theoretically motivated.