This paper introduces a new regularized version of the robust $\tau$-regression estimator for analyzing high-dimensional datasets subject to gross contamination in the response variables and covariates (explanatory variables). The resulting estimator, termed adaptive $\tau$-Lasso, is robust to outliers and high-leverage points. It also incorporates an adaptive $\ell_1$-norm penalty term, which enables the selection of relevant variables and reduces the bias associated with large true regression coefficients. More specifically, this adaptive $\ell_1$-norm penalty term assigns a weight to each regression coefficient. For a fixed number of predictors $p$, we show that the adaptive $\tau$-Lasso has the oracle property, ensuring both variable-selection consistency and asymptotic normality. Asymptotic normality applies only to the entries of the regression vector corresponding to the true support, assuming knowledge of the true regression vector support. We characterize its robustness via the finite-sample breakdown point and the influence function. We carry out extensive simulations and observe that the class of $\tau$-Lasso estimators exhibits robustness and reliable performance in both contaminated and uncontaminated data settings. We also validate our theoretical findings on robustness properties through simulation experiments. In the face of outliers and high-leverage points, the adaptive $\tau$-Lasso and $\tau$-Lasso estimators achieve the best performance or close-to-best performance in terms of prediction and variable selection accuracy compared to other competing regularized estimators for all scenarios considered in this study. Therefore, the adaptive $\tau$-Lasso and $\tau$-Lasso estimators can be effectively employed for a variety of sparse linear regression problems, particularly in high-dimensional settings and when the data is contaminated by outliers and high-leverage points.

We study the performance of empirical risk minimization on the $p$-norm linear regression problem for $p \in (1, \infty)$. We show that, in the realizable case, under no moment assumptions, and up to a distribution-dependent constant, $O(d)$ samples are enough to exactly recover the target. Otherwise, for $p \in [2, \infty)$, and under weak moment assumptions on the target and the covariates, we prove a high probability excess risk bound on the empirical risk minimizer whose leading term matches, up to a constant that depends only on $p$, the asymptotically exact rate. We extend this result to the case $p \in (1, 2)$ under mild assumptions that guarantee the existence of the Hessian of the risk at its minimizer.

Learning with rejection has been a prototypical model for studying the human-AI interaction on prediction tasks. Upon the arrival of a sample instance, the model first uses a rejector to decide whether to accept and use the AI predictor to make a prediction or reject and defer the sample to humans. Learning such a model changes the structure of the original loss function and often results in undesirable non-convexity and inconsistency issues. For the classification with rejection problem, several works develop consistent surrogate losses for the joint learning of the predictor and the rejector, while there have been fewer works for the regression counterpart. This paper studies the regression with rejection (RwR) problem and investigates a no-rejection learning strategy that uses all the data to learn the predictor. We first establish the consistency for such a strategy under the weak realizability condition. Then for the case without the weak realizability, we show that the excessive risk can also be upper bounded with the sum of two parts: prediction error and calibration error. Lastly, we demonstrate the advantage of such a proposed learning strategy with empirical evidence.

As language models (LMs) become more capable, it is increasingly important to align them with human preferences. However, the dominant paradigm for training Preference Models (PMs) for that purpose suffers from fundamental limitations, such as lack of transparency and scalability, along with susceptibility to overfitting the preference dataset. We propose Compositional Preference Models (CPMs), a novel PM framework that decomposes one global preference assessment into several interpretable features, obtains scalar scores for these features from a prompted LM, and aggregates these scores using a logistic regression classifier. CPMs allow to control which properties of the preference data are used to train the preference model and to build it based on features that are believed to underlie the human preference judgment. Our experiments show that CPMs not only improve generalization and are more robust to overoptimization than standard PMs, but also that best-of-n samples obtained using CPMs tend to be preferred over samples obtained using conventional PMs. Overall, our approach demonstrates the benefits of endowing PMs with priors about which features determine human preferences while relying on LM capabilities to extract those features in a scalable and robust way.

Traditional computational methods, such as the finite element analysis, have provided valuable insights into uncovering the underlying mechanisms of brain physical behaviors. However, precise predictions of brain physics require effective constitutive models to represent the intricate mechanical properties of brain tissue. In this study, we aimed to identify the most favorable constitutive material model for human brain tissue. To achieve this, we applied artificial neural network and multiple regression methods to a generalization of widely accepted classic models, and compared the results obtained from these two approaches. To evaluate the applicability and efficacy of the model, all setups were kept consistent across both methods, except for the approach to prevent potential overfitting. Our results demonstrate that artificial neural networks are capable of automatically identifying accurate constitutive models from given admissible estimators. Nonetheless, the five-term and two-term neural network models trained under single-mode and multi-mode loading scenarios, were found to be suboptimal and could be further simplified into two-term and single-term, respectively, with higher accuracy using multiple regression. Our findings highlight the importance of hyperparameters for the artificial neural network and emphasize the necessity for detailed cross-validations of regularization parameters to ensure optimal selection at a global level in the development of material constitutive models. This study validates the applicability and accuracy of artificial neural network to automatically discover constitutive material models with proper regularization as well as the benefits in model simplification without compromising accuracy for traditional multivariable regression.

Although most pregnancies result in a good outcome, complications are not uncommon and can be associated with serious implications for mothers and babies. Predictive modeling has the potential to improve outcomes through better understanding of risk factors, heightened surveillance for high risk patients, and more timely and appropriate interventions, thereby helping obstetricians deliver better care. We identify and study the most important risk factors for four types of pregnancy complications: (i) severe maternal morbidity, (ii) shoulder dystocia, (iii) preterm preeclampsia, and (iv) antepartum stillbirth. We use an Explainable Boosting Machine (EBM), a high-accuracy glass-box learning method, for prediction and identification of important risk factors. We undertake external validation and perform an extensive robustness analysis of the EBM models. EBMs match the accuracy of other black-box ML methods such as deep neural networks and random forests, and outperform logistic regression, while being more interpretable. EBMs prove to be robust. The interpretability of the EBM models reveals surprising insights into the features contributing to risk (e.g. maternal height is the second most important feature for shoulder dystocia) and may have potential for clinical application in the prediction and prevention of serious complications in pregnancy.

State-of-the-art machine learning models can be vulnerable to very small input perturbations that are adversarially constructed. Adversarial training is an effective approach to defend against it. Formulated as a min-max problem, it searches for the best solution when the training data were corrupted by the worst-case attacks. Linear models are among the simple models where vulnerabilities can be observed and are the focus of our study. In this case, adversarial training leads to a convex optimization problem which can be formulated as the minimization of a finite sum. We provide a comparative analysis between the solution of adversarial training in linear regression and other regularization methods. Our main findings are that: (A) Adversarial training yields the minimum-norm interpolating solution in the overparameterized regime (more parameters than data), as long as the maximum disturbance radius is smaller than a threshold. And, conversely, the minimum-norm interpolator is the solution to adversarial training with a given radius. (B) Adversarial training can be equivalent to parameter shrinking methods (ridge regression and Lasso). This happens in the underparametrized region, for an appropriate choice of adversarial radius and zero-mean symmetrically distributed covariates. (C) For $\ell_\infty$-adversarial training -- as in square-root Lasso -- the choice of adversarial radius for optimal bounds does not depend on the additive noise variance. We confirm our theoretical findings with numerical examples.

While machine learning (ML) has made significant contributions to the biopharmaceutical field, its applications are still in the early stages in terms of providing direct support for quality-by-design based development and manufacturing of biopharmaceuticals, hindering the enormous potential for bioprocesses automation from their development to manufacturing. However, the adoption of ML-based models instead of conventional multivariate data analysis methods is significantly increasing due to the accumulation of large-scale production data. This trend is primarily driven by the real-time monitoring of process variables and quality attributes of biopharmaceutical products through the implementation of advanced process analytical technologies. Given the complexity and multidimensionality of a bioproduct design, bioprocess development, and product manufacturing data, ML-based approaches are increasingly being employed to achieve accurate, flexible, and high-performing predictive models to address the problems of analytics, monitoring, and control within the biopharma field. This paper aims to provide a comprehensive review of the current applications of ML solutions in a bioproduct design, monitoring, control, and optimisation of upstream, downstream, and product formulation processes. Finally, this paper thoroughly discusses the main challenges related to the bioprocesses themselves, process data, and the use of machine learning models in biopharmaceutical process development and manufacturing. Moreover, it offers further insights into the adoption of innovative machine learning methods and novel trends in the development of new digital biopharma solutions.

Recent research has observed that in machine learning optimization, gradient descent (GD) often operates at the edge of stability (EoS) [Cohen, et al., 2021], where the stepsizes are set to be large, resulting in non-monotonic losses induced by the GD iterates. This paper studies the convergence and implicit bias of constant-stepsize GD for logistic regression on linearly separable data in the EoS regime. Despite the presence of local oscillations, we prove that the logistic loss can be minimized by GD with \emph{any} constant stepsize over a long time scale. Furthermore, we prove that with \emph{any} constant stepsize, the GD iterates tend to infinity when projected to a max-margin direction (the hard-margin SVM direction) and converge to a fixed vector that minimizes a strongly convex potential when projected to the orthogonal complement of the max-margin direction. In contrast, we also show that in the EoS regime, GD iterates may diverge catastrophically under the exponential loss, highlighting the superiority of the logistic loss. These theoretical findings are in line with numerical simulations and complement existing theories on the convergence and implicit bias of GD for logistic regression, which are only applicable when the stepsizes are sufficiently small.

Type 2 diabetes mellitus represents a prevalent and widespread global health concern, necessitating a comprehensive assessment of its risk factors. This study aimed towards learning whether there is any differential impact of age, Lifestyle, BMI and Waist to height ratio on the risk of Type 2 diabetes mellitus in males and females in Kolkata, West Bengal, India based on a sample observed from the out-patient consultation department of Belle Vue Clinic in Kolkata. Various machine learning models like Logistic Regression, Random Forest, and Support Vector Classifier, were used to predict the risk of diabetes, and performance was compared based on different predictors. Our findings indicate a significant age-related increase in risk of diabetes for both males and females. Although exercising and BMI was found to have significant impact on the risk of Type 2 diabetes in males, in females both turned out to be statistically insignificant. For both males and females, predictive models based on WhtR demonstrated superior performance in risk assessment compared to those based on BMI. This study sheds light on the gender-specific differences in the risk factors for Type 2 diabetes, offering valuable insights that can be used towards more targeted healthcare interventions and public health strategies.