The idea is to first develop a prediction model without concern for the downstream risk profile or robustness guarantee, and then utilize calibration (or recalibration) methods to quantify the uncertainty of the prediction.

Technically,various methods of both categories exploit the link between graph data and linear algebra by representing graphs by their (normalized) adjacency matrix. Such methods are often defined or can be interpreted in terms ofwalks. On the other hand, the Weisfeiler-Leman heuristic for graph isomorphism testing has attracted great interest in machine learning [33, 34].

In the existing literature, a common approach to approximate the metric is to first find a map that preserves Euclidean lengths instead of distances.

Decision tree ensembles are widely used in critical domains, making robustness and sensitivity analysis essential to their trustworthiness. We study the feature sensitivity problem, which asks whether an ensemble is sensitive to a specified subset of features -- such as protected attributes -- whose manipulation can alter model predictions. Existing approaches often yield examples of sensitivity that lie far from the training distribution, limiting their interpretability and practical value. We propose a data-aware sensitivity framework that constrains the sensitive examples to remain close to the dataset, thereby producing realistic and interpretable evidence of model weaknesses. To this end, we develop novel techniques for data-aware search using a combination of mixed-integer linear programming (MILP) and satisfiability modulo theories (SMT) encodings. Our contributions are fourfold. First, we strengthen the NP-hardness result for sensitivity verification, showing it holds even for trees of depth 1. Second, we develop MILP-optimizations that significantly speed up sensitivity verification for single ensembles and for the first time can also handle multiclass tree ensembles. Third, we introduce a data-aware framework generating realistic examples close to the training distribution. Finally, we conduct an extensive experimental evaluation on large tree ensembles, demonstrating scalability to ensembles with up to 800 trees of depth 8, achieving substantial improvements over the state of the art. This framework provides a practical foundation for analyzing the reliability and fairness of tree-based models in high-stakes applications.

Stochastic gradient methods are central to large-scale learning, but they treat mini-batch gradients as unbiased estimators, which classical decision theory shows are inadmissible in high dimensions. We formulate gradient computation as a high-dimensional estimation problem and introduce a framework based on Stein-rule shrinkage. We construct a gradient estimator that adaptively contracts noisy mini-batch gradients toward a stable estimator derived from historical momentum. The shrinkage intensity is determined in a data-driven manner using an online estimate of gradient noise variance, leveraging statistics from adaptive optimizers. Under a Gaussian noise model, we show our estimator uniformly dominates the standard stochastic gradient under squared error loss and is minimax-optimal. We incorporate this into the Adam optimizer, yielding SR-Adam, a practical algorithm with negligible computational cost. Empirical evaluations on CIFAR10 and CIFAR100 across multiple levels of input noise show consistent improvements over Adam in the large-batch regime. Ablation studies indicate that gains arise primarily from selectively applying shrinkage to high-dimensional convolutional layers, while indiscriminate shrinkage across all parameters degrades performance. These results illustrate that classical shrinkage principles provide a principled approach to improving stochastic gradient estimation in deep learning.

Recent works have shown that MFLD globally minimizes an entropy-regularized convex functional in the space of measures.