The softmax representation of probabilities for categorical variables plays a prominent role in modern machine learning with numerous applications in areas such as large scale classification, neural language modeling and recommendation systems. However, softmax estimation is very expensive for large scale inference because of the high cost associated with computing the normalizing constant. Here, we introduce an efficient approximation to softmax probabilities which takes the form of a rigorous lower bound on the exact probability. This bound is expressed as a product over pairwise probabilities and it leads to scalable estimation based on stochastic optimization. It allows us to perform doubly stochastic estimation by subsampling both training instances and class labels. We show that the new bound has interesting theoretical properties and we demonstrate its use in classification problems.

Radio Frequency Fingerprinting (RFF) has evolved as an effective solution for authenticating devices by leveraging the unique imperfections in hardware components involved in the signal generation process. In this work, we propose a Convolutional Neural Network (CNN) based framework for detecting rogue devices and identifying genuine ones using softmax probability thresholding. We emulate an attack scenario in which adversaries attempt to mimic the RF characteristics of genuine devices by training a Generative Adversarial Network (GAN) using In-phase and Quadrature (IQ) samples from genuine devices. The proposed approach is verified using IQ samples collected from ten different ADALM-PLUTO Software Defined Radios (SDRs), with seven devices considered genuine, two as rogue, and one used for validation to determine the threshold.

Post Training Quantization (PTQ) is widely adopted due to its high compression capacity and speed with minimal impact on accuracy. However, we observed that disparate impacts are exacerbated by quantization, especially for minority groups. Our analysis explains that in the course of quantization there is a chain of factors attributed to a disparate impact across groups during forward and backward passes. We explore how the changes in weights and activations induced by quantization cause cascaded impacts in the network, resulting in logits with lower variance, increased loss, and compromised group accuracies. We extend our study to verify the influence of these impacts on group gradient norms and eigenvalues of the Hessian matrix, providing insights into the state of the network from an optimization point of view. To mitigate these effects, we propose integrating mixed precision Quantization A ware Training (QA T) with dataset sampling methods and weighted loss functions, therefore providing fair deployment of quantized neural networks. With the onset of edge devices running deep neural networks for various tasks ranging across several domains, the demand for faster computation and model lightness has become more pronounced.

Modern neural networks, despite their high accuracy, often produce poorly calibrated confidence scores, limiting their reliability in high-stakes applications. Existing calibration methods typically post-process model outputs without interrogating the internal consistency of the predictions themselves. In this work, we introduce a novel, non-parametric statistical probe, the Bag-of-Coins (BoC) test, that examines the internal consistency of a classifier's logits. The BoC test reframes confidence estimation as a frequentist hypothesis test: does the model's top-ranked class win 1-v-1 contests against random competitors at a rate consistent with its own stated softmax probability? When applied to modern deep learning architectures, this simple probe reveals a fundamental dichotomy. On Vision Transformers (ViTs), the BoC output serves as a state-of-the-art confidence score, achieving near-perfect calibration with an ECE of 0.0212, an 88% improvement over a temperature-scaled baseline. Conversely, on Convolutional Neural Networks (CNNs) like ResNet, the probe reveals a deep inconsistency between the model's predictions and its internal logit structure, a property missed by traditional metrics. We posit that BoC is not merely a calibration method, but a new diagnostic tool for understanding and exposing the differing ways that popular architectures represent uncertainty.

To deploy machine learning models in the real world, researchers have proposed many OOD detection algorithms to help models identify unknown samples during the inference phase and prevent them from making untrustworthy predictions. Unlike methods that rely on extra data for outlier exposure training, post hoc methods detect Out-of-Distribution (OOD) samples by developing scoring functions, which are model agnostic and do not require additional training. However, previous post hoc methods may fail to capture the geometric cues embedded in network representations. Thus, in this study, we propose a novel score function based on the optimal transport theory, named OTOD, for OOD detection. We utilize information from features, logits, and the softmax probability space to calculate the OOD score for each test sample. Our experiments show that combining this information can boost the performance of OTOD with a certain margin. Experiments on the CIFAR-10 and CIFAR-100 benchmarks demonstrate the superior performance of our method. Notably, OTOD outperforms the state-of-the-art method GEN by 7.19% in the mean FPR@95 on the CIFAR-10 benchmark using ResNet-18 as the backbone, and by 12.51% in the mean FPR@95 using WideResNet-28 as the backbone. In addition, we provide theoretical guarantees for OTOD. The code is available in https://github.com/HengGao12/OTOD.

Uncertainty quantification (UQ) is essential for assessing the reliability of Earth observation (EO) products. However, the extensive use of machine learning models in EO introduces an additional layer of complexity, as those models themselves are inherently uncertain. While various UQ methods do exist for machine learning models, their performance on EO datasets remains largely unevaluated. A key challenge in the community is the absence of the ground truth for uncertainty, i.e. how certain the uncertainty estimates are, apart from the labels for the image/signal. This article fills this gap by introducing three benchmark datasets specifically designed for UQ in EO machine learning models. These datasets address three common problem types in EO: regression, image segmentation, and scene classification. They enable a transparent comparison of different UQ methods for EO machine learning models. We describe the creation and characteristics of each dataset, including data sources, preprocessing steps, and label generation, with a particular focus on calculating the reference uncertainty. We also showcase baseline performance of several machine learning models on each dataset, highlighting the utility of these benchmarks for model development and comparison. Overall, this article offers a valuable resource for researchers and practitioners working in artificial intelligence for EO, promoting a more accurate and reliable quality measure of the outputs of machine learning models. The dataset and code are accessible via https://gitlab.lrz.de/ai4eo/WG_Uncertainty.

Modern over-parameterized neural networks can be trained to fit the training data perfectly while still maintaining a high generalization performance. This "benign overfitting" phenomenon has been studied in a surge of recent theoretical work; however, most of these studies have been limited to linear models or two-layer neural networks. In this work, we analyze benign overfitting in the token selection mechanism of the attention architecture, which characterizes the success of transformer models. We first show the existence of a benign overfitting solution and explain its mechanism in the attention architecture. Next, we discuss whether the model converges to such a solution, raising the difficulties specific to the attention architecture. We then present benign overfitting cases and not-benign overfitting cases by conditioning different scenarios based on the behavior of attention probabilities during training. To the best of our knowledge, this is the first study to characterize benign overfitting for the attention mechanism.

Hallucinations is a major challenge for large language models (LLMs), prevents adoption in diverse fields. Uncertainty estimation could be used for alleviating the damages of hallucinations. The skeptical emotion of human could be useful for enhancing the ability of self estimation. Inspirited by this observation, we proposed a new approach called Skepticism Modeling (SM). This approach is formalized by combining the information of token and logits for self estimation. We construct the doubt emotion aware data, perform continual pre-training, and then fine-tune the LLMs, improve their ability of self estimation. Experimental results demonstrate this new approach effectively enhances a model's ability to estimate their uncertainty, and validate its generalization ability of other tasks by out-of-domain experiments.

The softmax representation of probabilities for categorical variables plays a prominent role in modern machine learning with numerous applications in areas such as large scale classification, neural language modeling and recommendation systems. However, softmax estimation is very expensive for large scale inference because of the high cost associated with computing the normalizing constant. Here, we introduce an efficient approximation to softmax probabilities which takes the form of a rigorous lower bound on the exact probability. This bound is expressed as a product over pairwise probabilities and it leads to scalable estimation based on stochastic optimization. It allows us to perform doubly stochastic estimation by subsampling both training instances and class labels. We show that the new bound has interesting theoretical properties and we demonstrate its use in classification problems.

Although large language models (LLMs) perform impressively on many tasks, overconfidence remains a problem. We hypothesized that on multiple-choice Q&A tasks, wrong answers would be associated with smaller maximum softmax probabilities (MSPs) compared to correct answers. We comprehensively evaluate this hypothesis on ten open-source LLMs and five datasets, and find strong evidence for our hypothesis among models which perform well on the original Q&A task. For the six LLMs with the best Q&A performance, the AUROC derived from the MSP was better than random chance with p < 10^{-4} in 59/60 instances. Among those six LLMs, the average AUROC ranged from 60% to 69%. Leveraging these findings, we propose a multiple-choice Q&A task with an option to abstain and show that performance can be improved by selectively abstaining based on the MSP of the initial model response. We also run the same experiments with pre-softmax logits instead of softmax probabilities and find similar (but not identical) results.