This research analyzes, models and develops a novel Digital Learning Environment (DLE) fortified by the innovative Private Learning Intelligence (PLI) framework. The proposed PLI framework leverages federated machine learning (FL) techniques to autonomously construct and continuously refine personalized learning models for individual learners, ensuring robust privacy protection. Our approach is pivotal in advancing DLE capabilities, empowering learners to actively participate in personalized real-time learning experiences. The integration of PLI within a DLE also streamlines instructional design and development demands for personalized teaching/learning. We seek ways to establish a foundation for the seamless integration of FL into learning systems, offering a transformative approach to personalized learning in digital environments. Our implementation details and code are made public.

Multi-task learning (MTL) is a learning paradigm that enables the simultaneous training of multiple communicating algorithms. Although MTL has been successfully applied to ether regression or classification tasks alone, incorporating mixed types of tasks into a unified MTL framework remains challenging, primarily due to variations in the magnitudes of losses associated with different tasks. This challenge, particularly evident in MTL applications with joint feature selection, often results in biased selections. To overcome this obstacle, we propose a provable loss weighting scheme that analytically determines the optimal weights for balancing regression and classification tasks. This scheme significantly mitigates the otherwise biased feature selection. Building upon this scheme, we introduce MTLComb, an MTL algorithm and software package encompassing optimization procedures, training protocols, and hyperparameter estimation procedures. MTLComb is designed for learning shared predictors among tasks of mixed types. To showcase the efficacy of MTLComb, we conduct tests on both simulated data and biomedical studies pertaining to sepsis and schizophrenia.

Stochastic programming models can lead to very large-scale optimization problems for which it may be impossible to enumerate all possible scenarios. In such cases, one adopts a sampling-based solution methodology in which case the reliability of the resulting decisions may be suspect. For such instances, it is advisable to adopt methodologies that promote variance reduction. One such approach goes under a framework known as "compromise decision", which requires multiple replications of the solution procedure. This paper studies the reliability of stochastic programming solutions resulting from the "compromise decision" process. This process is characterized by minimizing an aggregation of objective function approximations across replications, presumably conducted in parallel. We refer to the post-parallel-processing problem as the problem of "compromise decision". We quantify the reliability of compromise decisions by estimating the expectation and variance of the "pessimistic distance" of sampled instances from the set of true optimal decisions. Such pessimistic distance is defined as an estimate of the largest possible distance of the solution of the sampled instance from the "true" optimal solution set. The Rademacher average of instances is used to bound the sample complexity of the compromise decision.

In the field of heterogeneous federated learning (FL), the key challenge is to efficiently and collaboratively train models across multiple clients with different data distributions, model structures, task objectives, computational capabilities, and communication resources. This diversity leads to significant heterogeneity, which increases the complexity of model training. In this paper, we first outline the basic concepts of heterogeneous federated learning and summarize the research challenges in federated learning in terms of five aspects: data, model, task, device, and communication. In addition, we explore how existing state-of-the-art approaches cope with the heterogeneity of federated learning, and categorize and review these approaches at three different levels: data-level, model-level, and architecture-level. Subsequently, the paper extensively discusses privacy-preserving strategies in heterogeneous federated learning environments. Finally, the paper discusses current open issues and directions for future research, aiming to promote the further development of heterogeneous federated learning.

We propose a new positional encoding method for a neural network architecture called the Transformer. Unlike the standard sinusoidal positional encoding, our approach is based on solid mathematical grounds and has a guarantee of not losing information about the positional order of the input sequence. We show that the new encoding approach systematically improves the prediction performance in the time-series classification task.

In this paper, we show two new variants of multi-view k-means (MVKM) algorithms to address multi-view data. The general idea is to outline the distance between $h$-th view data points $x_i^h$ and $h$-th view cluster centers $a_k^h$ in a different manner of centroid-based approach. Unlike other methods, our proposed methods learn the multi-view data by calculating the similarity using Euclidean norm in the space of Gaussian-kernel, namely as multi-view k-means with exponent distance (MVKM-ED). By simultaneously aligning the stabilizer parameter $p$ and kernel coefficients $\beta^h$, the compression of Gaussian-kernel based weighted distance in Euclidean norm reduce the sensitivity of MVKM-ED. To this end, this paper designated as Gaussian-kernel multi-view k-means (GKMVKM) clustering algorithm. Numerical evaluation of five real-world multi-view data demonstrates the robustness and efficiency of our proposed MVKM-ED and GKMVKM approaches.

We consider the problem of estimating an unknown function f* and its partial derivatives from a noisy data set of n observations, where we make no assumptions about f* except that it is smooth in the sense that it has square integrable partial derivatives of order m. A natural candidate for the estimator of f* in such a case is the best fit to the data set that satisfies a certain smoothness condition. This estimator can be seen as a least squares estimator subject to an upper bound on some measure of smoothness. Another useful estimator is the one that minimizes the degree of smoothness subject to an upper bound on the average of squared errors. We prove that these two estimators are computable as solutions to quadratic programs, establish the consistency of these estimators and their partial derivatives, and study the convergence rate as n increases to infinity. The effectiveness of the estimators is illustrated numerically in a setting where the value of a stock option and its second derivative are estimated as functions of the underlying stock price.

With the proliferation of screening tools for chemical testing, it is now possible to create vast databases of chemicals easily. However, rigorous statistical methodologies employed to analyse these databases are in their infancy, and further development to facilitate chemical discovery is imperative. In this paper, we present conditional Gaussian process models to predict ordinal outcomes from chemical experiments, where the inputs are chemical compounds. We implement the Tanimoto distance, a metric on the chemical space, within the covariance of the Gaussian processes to capture correlated effects in the chemical space. A novel aspect of our model is that the kernel contains a scaling parameter, a feature not previously examined in the literature, that controls the strength of the correlation between elements of the chemical space. Using molecular fingerprints, a numerical representation of a compound's location within the chemical space, we show that accounting for correlation amongst chemical compounds improves predictive performance over the uncorrelated model, where effects are assumed to be independent. Moreover, we present a genetic algorithm for the facilitation of chemical discovery and identification of important features to the compound's efficacy. A simulation study is conducted to demonstrate the suitability of the proposed methods. Our proposed methods are demonstrated on a hazard classification problem of organic solvents.

As machine learning models are increasingly deployed in dynamic environments, it becomes paramount to assess and quantify uncertainties associated with distribution shifts. A distribution shift occurs when the underlying data-generating process changes, leading to a deviation in the model's performance. The prediction interval, which captures the range of likely outcomes for a given prediction, serves as a crucial tool for characterizing uncertainties induced by their underlying distribution. In this paper, we propose methodologies for aggregating prediction intervals to obtain one with minimal width and adequate coverage on the target domain under unsupervised domain shift, under which we have labeled samples from a related source domain and unlabeled covariates from the target domain. Our analysis encompasses scenarios where the source and the target domain are related via i) a bounded density ratio, and ii) a measure-preserving transformation. Our proposed methodologies are computationally efficient and easy to implement. Beyond illustrating the performance of our method through a real-world dataset, we also delve into the theoretical details. This includes establishing rigorous theoretical guarantees, coupled with finite sample bounds, regarding the coverage and width of our prediction intervals. Our approach excels in practical applications and is underpinned by a solid theoretical framework, ensuring its reliability and effectiveness across diverse contexts.

A recent study by De et al. (2022) has reported that large-scale representation learning through pre-training on a public dataset significantly enhances differentially private (DP) learning in downstream tasks, despite the high dimensionality of the feature space. To theoretically explain this phenomenon, we consider the setting of a layer-peeled model in representation learning, which results in interesting phenomena related to learned features in deep learning and transfer learning, known as Neural Collapse (NC). Within the framework of NC, we establish an error bound indicating that the misclassification error is independent of dimension when the distance between actual features and the ideal ones is smaller than a threshold. Additionally, the quality of the features in the last layer is empirically evaluated under different pre-trained models within the framework of NC, showing that a more powerful transformer leads to a better feature representation. Furthermore, we reveal that DP fine-tuning is less robust compared to fine-tuning without DP, particularly in the presence of perturbations. These observations are supported by both theoretical analyses and experimental evaluation. Moreover, to enhance the robustness of DP fine-tuning, we suggest several strategies, such as feature normalization or employing dimension reduction methods like Principal Component Analysis (PCA). Empirically, we demonstrate a significant improvement in testing accuracy by conducting PCA on the last-layer features.