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.

One of the most efficient methods to solve L2 -regularized primal problems, such as logistic regression and linear support vector machine (SVM) classification, is the widely used trust region Newton algorithm, TRON. While TRON has recently been shown to enjoy substantial speedups on shared-memory multi-core systems, exploiting graphical processing units (GPUs) to speed up the method is significantly more difficult, owing to the highly complex and heavily sequential nature of the algorithm. In this work, we show that using judicious GPU-optimization principles, TRON training time for different losses and feature representations may be drastically reduced. For sparse feature sets, we show that using GPUs to train logistic regression classifiers in LIBLINEAR is up to an order-of-magnitude faster than solely using multithreading. For dense feature sets–which impose far more stringent memory constraints–we show that GPUs substantially reduce the lengthy SVM learning times required for state-of-the-art proteomics analysis, leading to dramatic improvements over recently proposed speedups.

Adversarial training can achieve robustness against adversarial perturbations and has been widely used in machine-learning models. This paper delivers a non-asymptotic consistency analysis of the adversarial training procedure under \ell_\infty -perturbation in high-dimensional linear regression. It will be shown that, under the restricted eigenvalue condition, the associated convergence rate of prediction error can achieve the minimax rate up to a logarithmic factor in the high-dimensional linear regression on the class of sparse parameters. Additionally, the group adversarial training procedure is analyzed. Compared with classic adversarial training, it will be proved that the group adversarial training procedure enjoys a better prediction error upper bound under certain group-sparsity patterns.

We investigate the theoretical foundations of data poisoning attacks in machine learning models. Our analysis reveals that the Hessian with respect to the input serves as a diagnostic tool for detecting poisoning, exhibiting spectral signatures that characterize compromised datasets. We use random matrix theory (RMT) to develop a theory for the impact of poisoning proportion and regularisation on attack efficacy in linear regression. Through QR stepwise regression, we study the spectral signatures of the Hessian in multi-output regression. We perform experiments on deep networks to show experimentally that this theory extends to modern convolutional and transformer networks under the cross-entropy loss. Based on these insights we develop preliminary algorithms to determine if a network has been poisoned and remedies which do not require further training.

Peptide compounds demonstrate considerable potential as therapeutic agents due to their high target affinity and low toxicity, yet their drug development is constrained by their low membrane permeability. Molecular weight and peptide length have significant effects on the logD of peptides, which in turn influences their ability to cross biological membranes. However, accurate prediction of peptide logD remains challenging due to the complex interplay between sequence, structure, and ionization states. This study introduces LengthLogD, a predictive framework that establishes specialized models through molecular length stratification while innovatively integrating multi-scale molecular representations. We constructed feature spaces across three hierarchical levels: atomic (10 molecular descriptors), structural (1024-bit Morgan fingerprints), and topological (3 graph-based features including Wiener index), optimized through stratified ensemble learning. An adaptive weight allocation mechanism specifically developed for long peptides significantly enhances model generalizability. Experimental results demonstrate superior performance across all categories: short peptides (R^2=0.855), medium peptides (R^2=0.816), and long peptides (R^2=0.882), with a 34.7% reduction in prediction error for long peptides compared to conventional single-model approaches. Ablation studies confirm: 1) The length-stratified strategy contributes 41.2% to performance improvement; 2) Topological features account for 28.5% of predictive importance. Compared to state-of-the-art models, our method maintains short peptide prediction accuracy while achieving a 25.7% increase in the coefficient of determination (R^2) for long peptides. This research provides a precise logD prediction tool for peptide drug development, particularly demonstrating unique value in optimizing long peptide lead compounds.

We aim to approximate a continuously differentiable function $u:\mathbb{R}^d \rightarrow \mathbb{R}$ by a composition of functions $f\circ g$ where $g:\mathbb{R}^d \rightarrow \mathbb{R}^m$, $m\leq d$, and $f : \mathbb{R}^m \rightarrow \mathbb{R}$ are built in a two stage procedure. For a fixed $g$, we build $f$ using classical regression methods, involving evaluations of $u$. Recent works proposed to build a nonlinear $g$ by minimizing a loss function $\mathcal{J}(g)$ derived from Poincaré inequalities on manifolds, involving evaluations of the gradient of $u$. A problem is that minimizing $\mathcal{J}$ may be a challenging task. Hence in this work, we introduce new convex surrogates to $\mathcal{J}$. Leveraging concentration inequalities, we provide sub-optimality results for a class of functions $g$, including polynomials, and a wide class of input probability measures. We investigate performances on different benchmarks for various training sample sizes. We show that our approach outperforms standard iterative methods for minimizing the training Poincaré inequality based loss, often resulting in better approximation errors, especially for rather small training sets and $m=1$.

Differentially private (DP) linear regression has received significant attention in the recent theoretical literature, with several works aimed at obtaining improved error rates. A common approach is to set the clipping constant much larger than the expected norm of the per-sample gradients. While simplifying the analysis, this is however in sharp contrast with what empirical evidence suggests to optimize performance. Our work bridges this gap between theory and practice: we provide sharper rates for DP stochastic gradient descent (DP-SGD) by crucially operating in a regime where clipping happens frequently. Specifically, we consider the setting where the data is multivariate Gaussian, the number of training samples $n$ is proportional to the input dimension $d$, and the algorithm guarantees constant-order zero concentrated DP. Our method relies on establishing a deterministic equivalent for the trajectory of DP-SGD in terms of a family of ordinary differential equations (ODEs). As a consequence, the risk of DP-SGD is bounded between two ODEs, with upper and lower bounds matching for isotropic data. By studying these ODEs when $n / d$ is large enough, we demonstrate the optimality of aggressive clipping, and we uncover the benefits of decaying learning rate and private noise scheduling.

The success of unsupervised learning raises the question of whether also supervised models can be trained without using the information in the output $y$. In this paper, we demonstrate that this is indeed possible. The key step is to formulate the model as a smoother, i.e. on the form $\hat{f}=Sy$, and to construct the smoother matrix $S$ independently of $y$, e.g. by training on random labels. We present a simple model selection criterion based on the distribution of the out-of-sample predictions and show that, in contrast to cross-validation, this criterion can be used also without access to $y$. We demonstrate on real and synthetic data that $y$-free trained versions of linear and kernel ridge regression, smoothing splines, and neural networks perform similarly to their standard, $y$-based, versions and, most importantly, significantly better than random guessing.

Modern biomedical technologies have transformed how we diagnose, treat, and prevent diseases in the past decade. In particular, we see surging needs in combining gene expression data with traditional clinical measurements. However, since gene expression data is usually high-dimensional, while the total number of available clinical measurements is rather limited, naive approaches such as pooling all features together will not work well. Particularly, the microarray features would dominate in inference and variable selection, resulting in negligence of valuable information from clinical data or failure of type I error control. To address the issue caused by imbalance in feature dimensions, Tibshirani & Efron (2002) proposed the pre-validation procedure to make a fairer comparison between the two sets of predictors. In the setting of gene expression data and clinical measurements, the procedure includes two steps: first we use high-dimensional gene expression data to train the leave-one-out fits of the response, then we use the fitted values and clinical data to build a final prediction model. It turns out that the pre-validation procedure not only enables us to test whether the gene expression data have predictive power with type-I error control, but also gives a good estimate of the prediction error. Figure 1 illustrates the two-stage pre-validation procedure.

Graph-structured data is a type of data to be obtained associated with a graph structure where vertices and edges describe some kind of data correlation. This paper proposes a regression method on graph-structured data, which is based on multi-output Gaussian processes (MOGP), to capture both the correlation between vertices and the correlation between associated data. The proposed formulation is built on the definition of MOGP. This allows it to be applied to a wide range of data configurations and scenarios. Moreover, it has high expressive capability due to its flexibility in kernel design. It includes existing methods of Gaussian processes for graph-structured data as special cases and is possible to remove restrictions on data configurations, model selection, and inference scenarios in the existing methods. The performance of extensions achievable by the proposed formulation is evaluated through computer experiments with synthetic and real data.