We provide a statistical analysis of regularization-based continual learning on a sequence of linear regression tasks, with emphasis on how different regularization terms affect the model performance. We first derive the convergence rate for the oracle estimator obtained as if all data were available simultaneously. Next, we consider a family of generalized $\ell_2$-regularization algorithms indexed by matrix-valued hyperparameters, which includes the minimum norm estimator and continual ridge regression as special cases. As more tasks are introduced, we derive an iterative update formula for the estimation error of generalized $\ell_2$-regularized estimators, from which we determine the hyperparameters resulting in the optimal algorithm. Interestingly, the choice of hyperparameters can effectively balance the trade-off between forward and backward knowledge transfer and adjust for data heterogeneity. Moreover, the estimation error of the optimal algorithm is derived explicitly, which is of the same order as that of the oracle estimator. In contrast, our lower bounds for the minimum norm estimator and continual ridge regression show their suboptimality. A byproduct of our theoretical analysis is the equivalence between early stopping and generalized $\ell_2$-regularization in continual learning, which may be of independent interest. Finally, we conduct experiments to complement our theory.

Due to their complex structures, these models are generally accessed as black boxes. To assess their reliability and safeguard against potential errors, it is important to quantify the uncertainty in their outputs. Predictive inference is a popular methodology for this purpose. It takes as input a prediction algorithm and calibration data, and outputs a prediction set that contains the true outcome with a prescribed probability. The validity of the prediction set hinges on the assumption that the calibration data truthfully represents the underlying environment. However, this assumption is frequently violated in practice, where the data distribution may drift over time. Integrating data from both current and historical periods to construct faithful prediction sets remains a significant challenge. Despite a large body of literature on learning under distribution drift over the past two decades (Hazan and Seshadhri, 2009; Mohri and Muñoz Medina, 2012; Besbes et al., 2015; Hanneke et al., 2015; Mazzetto and Upfal, 2023; Huang and Wang, 2023), statistical inference within this context is much less explored.

Modern data-driven control applications call for flexible nonlinear models that are amenable to principled controller synthesis and realtime feedback. Many nonlinear dynamical systems of interest are control affine. We propose two novel classes of nonlinear feature representations which capture control affine structure while allowing for arbitrary complexity in the state dependence. Our methods make use of random features (RF) approximations, inheriting the expressiveness of kernel methods at a lower computational cost. We formalize the representational capabilities of our methods by showing their relationship to the Affine Dot Product (ADP) kernel proposed by Casta\~neda et al. (2021) and a novel Affine Dense (AD) kernel that we introduce. We further illustrate the utility by presenting a case study of data-driven optimization-based control using control certificate functions (CCF). Simulation experiments on a double pendulum empirically demonstrate the advantages of our methods.

Fine-resolution spatial patterns of water table depth (WTD) play a crucial role in shaping ecological resilience, hydrological connectivity, and anthropocentric objectives. Generally, a large-scale (e.g., continental or global) spatial map of static WTD can be simulated using either physically-based (PB) or machine learning-based (ML) models. We construct three fine-resolution (500 m) ML simulations of WTD, using the XGBoost algorithm and more than 20 million real and proxy observations of WTD, across the United States and Canada. The three ML models were constrained using known physical relations between WTD's drivers and WTD and were trained by sequentially adding real and proxy observations of WTD. We interpret the black box of our physically constrained ML models and compare it against available literature in groundwater hydrology. Through an extensive (pixel-by-pixel) evaluation, we demonstrate that our models can more accurately predict unseen real and proxy observations of WTD across most of North America's ecoregions compared to three available PB simulations of WTD. However, we still argue that large-scale WTD estimation is far from being a solved problem. We reason that due to biased observational data mainly collected from low-elevation floodplains, the misspecification of equations within physically-based models, and the over-flexibility of machine learning models, verifiably accurate simulations of WTD do not yet exist. Ultimately, we thoroughly discuss future directions that may help hydrogeologists decide how to proceed with WTD estimations, with a particular focus on the application of machine learning and the use of proxy satellite data.

We study the problem of symmetric positive semi-definite low-rank matrix completion (MC) with deterministic entry-dependent sampling. In particular, we consider rectified linear unit (ReLU) sampling, where only positive entries are observed, as well as a generalization to threshold-based sampling. We first empirically demonstrate that the landscape of this MC problem is not globally benign: Gradient descent (GD) with random initialization will generally converge to stationary points that are not globally optimal. Nevertheless, we prove that when the matrix factor with a small rank satisfies mild assumptions, the nonconvex objective function is geodesically strongly convex on the quotient manifold in a neighborhood of a planted low-rank matrix. Moreover, we show that our assumptions are satisfied by a matrix factor with i.i.d. Gaussian entries. Finally, we develop a tailor-designed initialization for GD to solve our studied formulation, which empirically always achieves convergence to the global minima. We also conduct extensive experiments and compare MC methods, investigating convergence and completion performance with respect to initialization, noise level, dimension, and rank.

Domain adaptation is often hampered by exceedingly small target datasets and inaccessible source data. These conditions are prevalent in speech verification, where privacy policies and/or languages with scarce speech resources limit the availability of sufficient data. This paper explored techniques of sourcefree domain adaptation unto a limited target speech dataset for speaker verificationin data-scarce languages. Both language and channel mis-match between source and target were investigated. Fine-tuning methods were evaluated and compared across different sizes of labeled target data. A novel iterative cluster-learn algorithm was studied for unlabeled target datasets.

2D irregular packing is a classic combinatorial optimization problem with various applications, such as material utilization and texture atlas generation. This NP-hard problem requires efficient algorithms to optimize space utilization. Conventional numerical methods suffer from slow convergence and high computational cost. Existing learning-based methods, such as the score-based diffusion model, also have limitations, such as no rotation support, frequent collisions, and poor adaptability to arbitrary boundaries, and slow inferring. The difficulty of learning from teacher packing is to capture the complex geometric relationships among packing examples, which include the spatial (position, orientation) relationships of objects, their geometric features, and container boundary conditions. Representing these relationships in latent space is challenging. We propose GFPack++, an attention-based gradient field learning approach that addresses this challenge. It consists of two pivotal strategies: \emph{attention-based geometry encoding} for effective feature encoding and \emph{attention-based relation encoding} for learning complex relationships. We investigate the utilization distribution between the teacher and inference data and design a weighting function to prioritize tighter teacher data during training, enhancing learning effectiveness. Our diffusion model supports continuous rotation and outperforms existing methods on various datasets. We achieve higher space utilization over several widely used baselines, one-order faster than the previous diffusion-based method, and promising generalization for arbitrary boundaries. We plan to release our source code and datasets to support further research in this direction.

Electronic Health Records (EHRs) contain rich patient information and are crucial for clinical research and practice. In recent years, deep learning models have been applied to EHRs, but they often rely on massive features, which may not be readily available for all patients. We propose HTP-Star, which leverages hypergraph structures with a pretrain-then-finetune framework for modeling EHR data, enabling seamless integration of additional features. Additionally, we design two techniques, namely (1) Smoothness-inducing Regularization and (2) Group-balanced Reweighting, to enhance the model's robustness during fine-tuning. Through experiments conducted on two real EHR datasets, we demonstrate that HTP-Star consistently outperforms various baselines while striking a balance between patients with basic and extra features.

Mental health constitutes a complex and pervasive global challenge, affecting millions of lives and often leading to severe consequences. In this paper, we conduct a thorough survey to explore the intersection of data science, artificial intelligence, and mental healthcare, focusing on the recent developments of mental disorder detection through online social media (OSM). A significant portion of the population actively engages in OSM platforms, creating a vast repository of personal data that holds immense potential for mental health analytics. The paper navigates through traditional diagnostic methods, state-of-the-art data- and AI-driven research studies, and the emergence of explainable AI (XAI) models for mental healthcare. We review state-of-the-art machine learning methods, particularly those based on modern deep learning, while emphasising the need for explainability in healthcare AI models. The experimental design section provides insights into prevalent practices, including available datasets and evaluation approaches. We also identify key issues and challenges in the field and propose promising future research directions. As mental health decisions demand transparency, interpretability, and ethical considerations, this paper contributes to the ongoing discourse on advancing XAI in mental healthcare through social media. The comprehensive overview presented here aims to guide researchers, practitioners, and policymakers in developing the area of mental disorder detection.

The performance of supervised semantic segmentation methods highly relies on the availability of large-scale training data. To alleviate this dependence, few-shot semantic segmentation (FSS) is introduced to leverage the model trained on base classes with sufficient data into the segmentation of novel classes with few data. FSS methods face the challenge of model generalization on novel classes due to the distribution shift between base and novel classes. To overcome this issue, we propose a class-shared memory (CSM) module consisting of a set of learnable memory vectors. These memory vectors learn elemental object patterns from base classes during training whilst re-encoding query features during both training and inference, thereby improving the distribution alignment between base and novel classes. Furthermore, to cope with the performance degradation resulting from the intra-class variance across images, we introduce an uncertainty-based feature augmentation (UFA) module to produce diverse query features during training for improving the model's robustness. We integrate CSM and UFA into representative FSS works, with experimental results on the widely-used PASCAL-5$^i$ and COCO-20$^i$ datasets demonstrating the superior performance of ours over state of the art.