Optimizing proper loss functions is popularly believed to yield predictors with good calibration properties; the intuition being that for such losses, the global optimum is to predict the ground-truth probabilities, which is indeed calibrated.

Understanding intrinsic differences between adversarial examples and clean samples is key to enhancing DNN robustness and detection against adversarial attacks. This study first empirically finds that image-based adversarial examples are notably sensitive to occlusion. Controlled experiments on CIFAR-10 used nine canonical attacks (e.g., FGSM, PGD) to generate adversarial examples, paired with original samples for evaluation. We introduce Sliding Mask Confidence Entropy (SMCE) to quantify model confidence fluctuation under occlusion. Using 1800+ test images, SMCE calculations supported by Mask Entropy Field Maps and statistical distributions show adversarial examples have significantly higher confidence volatility under occlusion than originals. Based on this, we propose Sliding Window Mask-based Adversarial Example Detection (SWM-AED), which avoids catastrophic overfitting of conventional adversarial training. Evaluations across classifiers and attacks on CIFAR-10 demonstrate robust performance, with accuracy over 62% in most cases and up to 96.5%.

Calibration of predicted probabilities is critical for reliable machine learning, yet it is poorly understood how standard training procedures yield well-calibrated models. This work provides the first theoretical proof that canonical $L_{2}$-regularized empirical risk minimization directly controls the smooth calibration error (smCE) without post-hoc correction or specialized calibration-promoting regularizer. We establish finite-sample generalization bounds for smCE based on optimization error, regularization strength, and the Rademacher complexity. We then instantiate this theory for models in reproducing kernel Hilbert spaces, deriving concrete guarantees for kernel ridge and logistic regression. Our experiments confirm these specific guarantees, demonstrating that $L_{2}$-regularized ERM can provide a well-calibrated model without boosting or post-hoc recalibration. The source code to reproduce all experiments is available at https://github.com/msfuji0211/erm_calibration.

We study calibration measures in a sequential prediction setup.

Probabilistic predictions are at the heart of modern data science.

Optimizing proper loss functions is popularly believed to yield predictors with good calibration properties; the intuition being that for such losses, the global optimum is to predict the ground-truth probabilities, which is indeed calibrated.

Calibration is a critical requirement for reliable probabilistic prediction, especially in high-risk applications. However, the theoretical understanding of which learning algorithms can simultaneously achieve high accuracy and good calibration remains limited, and many existing studies provide empirical validation or a theoretical guarantee in restrictive settings. To address this issue, in this work, we focus on the smooth calibration error (CE) and provide a uniform convergence bound, showing that the smooth CE is bounded by the sum of the smooth CE over the training dataset and a generalization gap. We further prove that the functional gradient of the loss function can effectively control the training smooth CE. Based on this framework, we analyze three representative algorithms: gradient boosting trees, kernel boosting, and two-layer neural networks. For each, we derive conditions under which both classification and calibration performances are simultaneously guaranteed. Our results offer new theoretical insights and practical guidance for designing reliable probabilistic models with provable calibration guarantees.

We initiate the study of the truthfulness of calibration measures in sequential prediction. A calibration measure is said to be truthful if the forecaster (approximately) minimizes the expected penalty by predicting the conditional expectation of the next outcome, given the prior distribution of outcomes. Truthfulness is an important property of calibration measures, ensuring that the forecaster is not incentivized to exploit the system with deliberate poor forecasts. This makes it an essential desideratum for calibration measures, alongside typical requirements, such as soundness and completeness. We conduct a taxonomy of existing calibration measures and their truthfulness. Perhaps surprisingly, we find that all of them are far from being truthful. That is, under existing calibration measures, there are simple distributions on which a polylogarithmic (or even zero) penalty is achievable, while truthful prediction leads to a polynomial penalty. Our main contribution is the introduction of a new calibration measure termed the Subsampled Smooth Calibration Error (SSCE) under which truthful prediction is optimal up to a constant multiplicative factor.