In this paper, we introduce a novel discriminative loss function with large margin in the context of Deep Learning. This loss boosts the discriminative power of neural nets, represented by intra-class compactness and inter-class separability. On the one hand, the class compactness is ensured by close distance of samples of the same class to each other. On the other hand, the inter-class separability is boosted by a margin loss that ensures the minimum distance of each class to its closest boundary. All the terms in our loss have an explicit meaning, giving a direct view of the feature space obtained. We analyze mathematically the relation between compactness and margin term, giving a guideline about the impact of the hyper-parameters on the learned features. Moreover, we also analyze properties of the gradient of the loss with respect to the parameters of the neural net. Based on this, we design a strategy called partial momentum updating that enjoys simultaneously stability and consistency in training. Furthermore, we also investigate generalization errors to have better theoretical insights. Our loss function systematically boosts the test accuracy of models compared to the standard softmax loss in our experiments.

In this paper we present novel methodology for automatic anomaly and switch event filtering to improve load estimation in power grid systems. By leveraging unsupervised methods with supervised optimization, our approach prioritizes interpretability while ensuring robust and generalizable performance on unseen data. Through experimentation, a combination of binary segmentation for change point detection and statistical process control for anomaly detection emerges as the most effective strategy, specifically when ensembled in a novel sequential manner. Results indicate the clear wasted potential when filtering is not applied. The automatic load estimation is also fairly accurate, with approximately 90% of estimates falling within a 10% error margin, with only a single significant failure in both the minimum and maximum load estimates across 60 measurements in the test set. Our methodology's interpretability makes it particularly suitable for critical infrastructure planning, thereby enhancing decision-making processes.

Continuous-time approximation of Stochastic Gradient Descent (SGD) is a crucial tool to study its escaping behaviors from stationary points. However, existing stochastic differential equation (SDE) models fail to fully capture these behaviors, even for simple quadratic objectives. Built on a novel stochastic backward error analysis framework, we derive the Hessian-Aware Stochastic Modified Equation (HA-SME), an SDE that incorporates Hessian information of the objective function into both its drift and diffusion terms. Our analysis shows that HA-SME matches the order-best approximation error guarantee among existing SDE models in the literature, while achieving a significantly reduced dependence on the smoothness parameter of the objective. Further, for quadratic objectives, under mild conditions, HA-SME is proved to be the first SDE model that recovers exactly the SGD dynamics in the distributional sense. Consequently, when the local landscape near a stationary point can be approximated by quadratics, HA-SME is expected to accurately predict the local escaping behaviors of SGD.

Over the past decade, the problem of assessing the fairness of classifiers has garnered significant attention, revealing that machine learning (ML) systems not only reproduce existing biases in the data but also tend to amplify them [1, 2, 3]. Given the complexity of the ML pipeline, isolating and characterising the key drivers of this amplification is challenging. Recent studies have begun to disentangle the contributions from architectural design choices, including overparameterisation [4], model complexity, activation functions [5, 6], learning protocols [7, 8], post-processing practices such as pruning [9], and intrinsic aspects of the data like its geometrical properties [10]. Theoretical results in this area (e.g., [4, 10]) are mostly based on asymptotic analysis, leaving the transient learning regime poorly understood. Due to limitations on computational resources, a trained ML system may operate far from the asymptotic regime and hence existing results may not always apply. Insights from class imbalance literature [7, 6] indicate that classifiers converge faster for classes with more data, but how this applies to fairness, where datasets might be balanced by label but imbalanced by demographics, remains unclear.

Large amount of multidimensional data represented by multiway arrays or tensors are prevalent in modern applications across various fields such as chemometrics, genomics, physics, psychology, and signal processing. The structural complexity of such data provides vast new opportunities for modeling and analysis, but efficiently extracting information content from them, both statistically and computationally, presents unique and fundamental challenges. Addressing these challenges requires an interdisciplinary approach that brings together tools and insights from statistics, optimization and numerical linear algebra among other fields. Despite these hurdles, significant progress has been made in the last decade. This review seeks to examine some of the key advancements and identify common threads among them, under eight different statistical settings.

We study the problem of estimating a rank one signal matrix from an observed matrix generated by corrupting the signal with additive rotationally invariant noise. We develop a new class of approximate message-passing algorithms for this problem and provide a simple and concise characterization of their dynamics in the high-dimensional limit. At each iteration, these algorithms exploit prior knowledge about the noise structure by applying a non-linear matrix denoiser to the eigenvalues of the observed matrix and prior information regarding the signal structure by applying a non-linear iterate denoiser to the previous iterates generated by the algorithm. We exploit our result on the dynamics of these algorithms to derive the optimal choices for the matrix and iterate denoisers. We show that the resulting algorithm achieves the smallest possible asymptotic estimation error among a broad class of iterative algorithms under a fixed iteration budget.

Forward stagewise regression is a simple algorithm that can be used to estimate regularized models. The updating rule adds a small constant to a regression coefficient in each iteration, such that the underlying optimization problem is solved slowly with small improvements. This is similar to gradient boosting, with the essential difference that the step size is determined by the product of the gradient and a step length parameter in the latter algorithm. One often overlooked challenge in gradient boosting for distributional regression is the issue of a vanishing small gradient, which practically halts the algorithm's progress. We show that gradient boosting in this case oftentimes results in suboptimal models, especially for complex problems certain distributional parameters are never updated due to the vanishing gradient. Therefore, we propose a stagewise boosting-type algorithm for distributional regression, combining stagewise regression ideas with gradient boosting. Additionally, we extend it with a novel regularization method, correlation filtering, to provide additional stability when the problem involves a large number of covariates. Furthermore, the algorithm includes best-subset selection for parameters and can be applied to big data problems by leveraging stochastic approximations of the updating steps. Besides the advantage of processing large datasets, the stochastic nature of the approximations can lead to better results, especially for complex distributions, by reducing the risk of being trapped in a local optimum. The performance of our proposed stagewise boosting distributional regression approach is investigated in an extensive simulation study and by estimating a full probabilistic model for lightning counts with data of more than 9.1 million observations and 672 covariates.

Staged trees are probabilistic graphical models capable of representing any class of non-symmetric independence via a coloring of its vertices. Several structural learning routines have been defined and implemented to learn staged trees from data, under the frequentist or Bayesian paradigm. They assume a data set has been observed fully and, in practice, observations with missing entries are either dropped or imputed before learning the model. Here, we introduce the first algorithms for staged trees that handle missingness within the learning of the model. To this end, we characterize the likelihood of staged tree models in the presence of missing data and discuss pseudo-likelihoods that approximate it. A structural expectation-maximization algorithm estimating the model directly from the full likelihood is also implemented and evaluated. A computational experiment showcases the performance of the novel learning algorithms, demonstrating that it is feasible to account for different missingness patterns when learning staged trees.

Gaussian processes are a versatile probabilistic machine learning model whose effectiveness often depends on good hyperparameters, which are typically learned by maximising the marginal likelihood. In this work, we consider iterative methods, which use iterative linear system solvers to approximate marginal likelihood gradients up to a specified numerical precision, allowing a trade-off between compute time and accuracy of a solution. We introduce a three-level hierarchy of marginal likelihood optimisation for iterative Gaussian processes, and identify that the computational costs are dominated by solving sequential batches of large positive-definite systems of linear equations. We then propose to amortise computations by reusing solutions of linear system solvers as initialisations in the next step, providing a $\textit{warm start}$. Finally, we discuss the necessary conditions and quantify the consequences of warm starts and demonstrate their effectiveness on regression tasks, where warm starts achieve the same results as the conventional procedure while providing up to a $16 \times$ average speed-up among datasets.

Conformal prediction methodologies have significantly advanced the quantification of uncertainties in predictive models. Yet, the construction of confidence regions for model parameters presents a notable challenge, often necessitating stringent assumptions regarding data distribution or merely providing asymptotic guarantees. We introduce a novel approach termed CCR, which employs a combination of conformal prediction intervals for the model outputs to establish confidence regions for model parameters. We present coverage guarantees under minimal assumptions on noise and that is valid in finite sample regime. Our approach is applicable to both split conformal predictions and black-box methodologies including full or cross-conformal approaches. In the specific case of linear models, the derived confidence region manifests as the feasible set of a Mixed-Integer Linear Program (MILP), facilitating the deduction of confidence intervals for individual parameters and enabling robust optimization. We empirically compare CCR to recent advancements in challenging settings such as with heteroskedastic and non-Gaussian noise.