Modern deep learning techniques focus on extracting intricate information from data to achieve accurate predictions. However, the training datasets may be crowdsourced and include sensitive information, such as personal contact details, financial data, and medical records. As a result, there is a growing emphasis on developing privacy-preserving training algorithms for neural networks that maintain good performance while preserving privacy. In this paper, we investigate the generalization and privacy performances of the differentially private gradient descent (DP-GD) algorithm, which is a private variant of the gradient descent (GD) by incorporating additional noise into the gradients during each iteration. Moreover, we identify a concrete learning task where DP-GD can achieve superior generalization performance compared to GD in training two-layer Huberized ReLU convolutional neural networks (CNNs). Specifically, we demonstrate that, under mild conditions, a small signal-to-noise ratio can result in GD producing training models with poor test accuracy, whereas DP-GD can yield training models with good test accuracy and privacy guarantees if the signal-to-noise ratio is not too small. This indicates that DP-GD has the potential to enhance model performance while ensuring privacy protection in certain learning tasks. Numerical simulations are further conducted to support our theoretical results.

This work presents a probabilistic digital twin framework for response prediction in dynamical systems governed by misspecified physics. The approach integrates Gaussian Process Latent Force Models (GPLFM) and Bayesian Neural Networks (BNNs) to enable end-to-end uncertainty-aware inference and prediction. In the diagnosis phase, model-form errors (MFEs) are treated as latent input forces to a nominal linear dynamical system and jointly estimated with system states using GPLFM from sensor measurements. A BNN is then trained on posterior samples to learn a probabilistic nonlinear mapping from system states to MFEs, while capturing diagnostic uncertainty. For prognosis, this mapping is used to generate pseudo-measurements, enabling state prediction via Kalman filtering. The framework allows for systematic propagation of uncertainty from diagnosis to prediction, a key capability for trustworthy digital twins. The framework is demonstrated using four nonlinear examples: a single degree of freedom (DOF) oscillator, a multi-DOF system, and two established benchmarks -- the Bouc-Wen hysteretic system and the Silverbox experimental dataset -- highlighting its predictive accuracy and robustness to model misspecification.

We study the problem of selecting a small, representative action subset from an extremely large action space shared across a family of reinforcement learning (RL) environments -- a fundamental challenge in applications like inventory management and recommendation systems, where direct learning over the entire space is intractable. Our goal is to identify a fixed subset of actions that, for every environment in the family, contains a near-optimal action, thereby enabling efficient learning without exhaustively evaluating all actions. This work extends our prior results for meta-bandits to the more general setting of Markov Decision Processes (MDPs). We prove that our existing algorithm achieves performance comparable to using the full action space. This theoretical guarantee is established under a relaxed, non-centered sub-Gaussian process model, which accommodates greater environmental heterogeneity. Consequently, our approach provides a computationally and sample-efficient solution for large-scale combinatorial decision-making under uncertainty.

Support vector machines are widely used in machine learning classification tasks, but traditional SVM models suffer from sensitivity to outliers and instability in resampling, which limits their performance in practical applications. To address these issues, this paper proposes a novel rescaled Huberized pinball loss function with asymmetric, non-convex, and smooth properties. Based on this loss function, we develop a corresponding SVM model called RHPSVM (Rescaled Huberized Pinball Loss Support Vector Machine). Theoretical analyses demonstrate that RHPSVM conforms to Bayesian rules, has a strict generalization error bound, a bounded influence function, and controllable optimality conditions, ensuring excellent classification accuracy, outlier insensitivity, and resampling stability. Additionally, RHPSVM can be extended to various advanced SVM variants by adjusting parameters, enhancing its flexibility. We transform the non-convex optimization problem of RHPSVM into a series of convex subproblems using the concave-convex procedure (CCCP) and solve it with the ClipDCD algorithm, which is proven to be convergent. Experimental results on simulated data, UCI datasets, and small-sample crop leaf image classification tasks show that RHPSVM outperforms existing SVM models in both noisy and noise-free scenarios, especially in handling high-dimensional small-sample data.

Uncertainty quantification is vital for decision-making and risk assessment in machine learning. Mean-variance regression models, which predict both a mean and residual noise for each data point, provide a simple approach to uncertainty quantification. However, overparameterized mean-variance models struggle with signal-to-noise ambiguity, deciding whether prediction targets should be attributed to signal (mean) or noise (variance). At one extreme, models fit all training targets perfectly with zero residual noise, while at the other, they provide constant, uninformative predictions and explain the targets as noise. We observe a sharp phase transition between these extremes, driven by model regularization. Empirical studies with varying regularization levels illustrate this transition, revealing substantial variability across repeated runs. To explain this behavior, we develop a statistical field theory framework, which captures the observed phase transition in alignment with experimental results. This analysis reduces the regularization hyperparameter search space from two dimensions to one, significantly lowering computational costs. Experiments on UCI datasets and the large-scale ClimSim dataset demonstrate robust calibration performance, effectively quantifying predictive uncertainty.

Observational data is widely utilized when randomized experiments are infeasible or fail to adequately represent target populations. A key challenge in observational analysis is the lack of overlap between treatment and control groups. Even when a nominally large dataset is collected, the effective sample size may be prohibitively small when there is a region with little overlap between treated and control populations. As an example, if the treatment of interest is rarely observed among older citizens, estimating their counterfactual (treated) outcome becomes inherently unreliable. This challenge is further exacerbated in modern operational contexts, where high-dimensional covariate representations [15] increase data sparsity, making causal identification particularly difficult in regions of the covariate space with small effective sample size.

Score-based generative models require guidance in order to generate plausible, on-manifold samples. The most popular guidance method, Classifier-Free Guidance (CFG), is only applicable in settings with labeled data and requires training an additional unconditional score-based model. More recently, Auto-Guidance adopts a smaller, less capable version of the original model to guide generation. While each method effectively promotes the fidelity of generated data, each requires labeled data or the training of additional models, making it challenging to guide score-based models when (labeled) training data are not available or training new models is not feasible. We make the surprising discovery that the positive curvature of log density estimates in saddle regions provides strong guidance for score-based models. Motivated by this, we develop saddle-free guidance (SFG) which maintains estimates of maximal positive curvature of the log density to guide individual score-based models. SFG has the same computational cost of classifier-free guidance, does not require additional training, and works with off-the-shelf diffusion and flow matching models. Our experiments indicate that SFG achieves state-of-the-art FID and FD-DINOv2 metrics in single-model unconditional ImageNet-512 generation. When SFG is combined with Auto-Guidance, its unconditional samples achieve general state-of-the-art in FD-DINOv2 score. Our experiments with FLUX.1-dev and Stable Diffusion v3.5 indicate that SFG boosts the diversity of output images compared to CFG while maintaining excellent prompt adherence and image fidelity.

Research increasingly relies on computational methods to analyze experimental data and predict molecular properties. Current approaches often require researchers to use a variety of tools for statistical analysis and machine learning, creating workflow inefficiencies. We present an integrated platform that combines classical statistical methods with Random Forest classification for comprehensive data analysis that can be used in the biological sciences. The platform implements automated hyperparameter optimization, feature importance analysis, and a suite of statistical tests including t tests, ANOVA, and Pearson correlation analysis. Our methodology addresses the gap between traditional statistical software, modern machine learning frameworks and biology, by providing a unified interface accessible to researchers without extensive programming experience. The system achieves this through automatic data preprocessing, categorical encoding, and adaptive model configuration based on dataset characteristics. Initial testing protocols are designed to evaluate classification accuracy across diverse chemical datasets with varying feature distributions. This work demonstrates that integrating statistical rigor with machine learning interpretability can accelerate biological discovery workflows while maintaining methodological soundness. The platform's modular architecture enables future extensions to additional machine learning algorithms and statistical procedures relevant to bioinformatics.

Interpretable topic modeling is essential for tracking how research interests evolve within co-author communities. In scientific corpora, where novelty is prized, identifying underrepresented niche topics is particularly important. However, contemporary models built from dense transformer embeddings tend to miss rare topics and therefore also fail to capture smooth temporal alignment. We propose a geometric method that integrates multimodal text and co-author network data, using Hellinger distances and Ward's linkage to construct a hierarchical topic dendrogram. This approach captures both local and global structure, supporting multiscale learning across semantic and temporal dimensions. Our method effectively identifies rare-topic structure and visualizes smooth topic drift over time. Experiments highlight the strength of interpretable bag-of-words models when paired with principled geometric alignment.

The abundance of fine-grained spatio-temporal data, such as traffic sensor networks, offers vast opportunities for scientific discovery. However, inferring causal relationships from such observational data remains challenging, particularly due to unobserved confounders that are specific to units (e.g., geographical locations) yet influence outcomes over time. Most existing methods for spatio-temporal causal inference assume that all confounders are observed, an assumption that is often violated in practice. In this paper, we introduce Spatio-Temporal Hierarchical Causal Models (ST-HCMs), a novel graphical framework that extends hierarchical causal modeling to the spatio-temporal domain. At the core of our approach is the Spatio-Temporal Collapse Theorem, which shows that a complex ST-HCM converges to a simpler flat causal model as the amount of subunit data increases. This theoretical result enables a general procedure for causal identification, allowing ST-HCMs to recover causal effects even in the presence of unobserved, time-invariant unit-level confounders, a scenario where standard non-hierarchical models fail. We validate the effectiveness of our framework on both synthetic and real-world datasets, demonstrating its potential for robust causal inference in complex dynamic systems.