Point processes are widely used statistical models for uncovering the temporal patterns in dependent event data. In many applications, the event time cannot be observed exactly, calling for the incorporation of time uncertainty into the modeling of point process data. In this work, we introduce a framework to model time-uncertain point processes possibly on a network. We start by deriving the formulation in the continuous-time setting under a few assumptions motivated by application scenarios. After imposing a time grid, we obtain a discrete-time model that facilitates inference and can be computed by first-order optimization methods such as Gradient Descent or Variation inequality (VI) using batch-based Stochastic Gradient Descent (SGD). The parameter recovery guarantee is proved for VI inference at an $O(1/k)$ convergence rate using $k$ SGD steps. Our framework handles non-stationary processes by modeling the inference kernel as a matrix (or tensor on a network) and it covers the stationary process, such as the classical Hawkes process, as a special case. We experimentally show that the proposed approach outperforms previous General Linear model (GLM) baselines on simulated and real data and reveals meaningful causal relations on a Sepsis-associated Derangements dataset.

In contrast to binary and multi-class classification, where each observation in the data is assigned to a single class, an observation in a multi-label classification task can have multiple labels. This type of problem arises in different fields, such as object detection in images, text analysis, bioinformatics, and recommendation systems (Tsoumakas et al., 2010). Consequently, numerous methods have been developed to handle multi-labeled outcomes. In contrast to existing methods, which often focus on modeling each outcome variable separately, our proposed method jointly models all labels to capture dependencies between them. In this study, we also refer to these dependencies between labels as label interdependencies. A frequently used method for a classification task with multi-labeled outcomes is to decompose the task into separate independent binary classifications (e.g., Boutell et al., 2004; Luaces et al., 2012). This approach is typically referred to as binary relevance. A limitation of binary relevance is the fact that it does not exploit potential correlations between the different labels (Godbole and Sarawagi, 2004; Zhang and Zhou, 2014).

Most ost Big Data datasets suffer from selection bias. For example, X (Twitter) training observations differ largely from the testing offline observations as individuals on Twitter are generally more educated, democratic or left-leaning. Therefore, one major obstacle to reliable estimation is the differences between training and testing data. How can researchers make use of such data even in the presence of non-ignorable selection mechanisms? A number of methods have been developed for this issue, such as distributionally robust optimization (DRO) or learning fairness. A possible avenue to reducing the effect of bias is meta-information. Researchers, being field exerts, might have prior information on the form and extent of selection bias affecting their dataset, and in which direction the selection might cause the estimate to change, e.g. over or under estimation. At the same time, there is no direct way to leverage these types of information in learning. I propose a loss function which takes into account two types of meta data information given by the researcher: quantity and direction (under or over sampling) of bias in the training set. Estimation with the proposed loss function is then implemented through a neural network, the directional Rockafellar-Uryasev (dRU) regression model. I test the dRU model on a biased training dataset, a Big Data online drawn electoral poll. I apply the proposed model using meta data information coherent with the political and sampling information obtained from previous studies. The results show that including meta information improves the electoral results predictions compared to a model that does not include them.

Transformers can efficiently learn in-context from example demonstrations. Most existing theoretical analyses studied the in-context learning (ICL) ability of transformers for linear function classes, where it is typically shown that the minimizer of the pretraining loss implements one gradient descent step on the least squares objective. However, this simplified linear setting arguably does not demonstrate the statistical efficiency of ICL, since the pretrained transformer does not outperform directly solving linear regression on the test prompt. In this paper, we study ICL of a nonlinear function class via transformer with nonlinear MLP layer: given a class of \textit{single-index} target functions $f_*(\boldsymbol{x}) = \sigma_*(\langle\boldsymbol{x},\boldsymbol{\beta}\rangle)$, where the index features $\boldsymbol{\beta}\in\mathbb{R}^d$ are drawn from a $r$-dimensional subspace, we show that a nonlinear transformer optimized by gradient descent (with a pretraining sample complexity that depends on the \textit{information exponent} of the link functions $\sigma_*$) learns $f_*$ in-context with a prompt length that only depends on the dimension of the distribution of target functions $r$; in contrast, any algorithm that directly learns $f_*$ on test prompt yields a statistical complexity that scales with the ambient dimension $d$. Our result highlights the adaptivity of the pretrained transformer to low-dimensional structures of the function class, which enables sample-efficient ICL that outperforms estimators that only have access to the in-context data.

This research proposes a novel drift detection methodology for machine learning (ML) models based on the concept of ''deformation'' in the vector space representation of data. Recognizing that new data can act as forces stretching, compressing, or twisting the geometric relationships learned by a model, we explore various mathematical frameworks to quantify this deformation. We investigate measures such as eigenvalue analysis of covariance matrices to capture global shape changes, local density estimation using kernel density estimation (KDE), and Kullback-Leibler divergence to identify subtle shifts in data concentration. Additionally, we draw inspiration from continuum mechanics by proposing a ''strain tensor'' analogy to capture multi-faceted deformations across different data types. This requires careful estimation of the displacement field, and we delve into strategies ranging from density-based approaches to manifold learning and neural network methods. By continuously monitoring these deformation metrics and correlating them with model performance, we aim to provide a sensitive, interpretable, and adaptable drift detection system capable of distinguishing benign data evolution from true drift, enabling timely interventions and ensuring the reliability of machine learning systems in dynamic environments. Addressing the computational challenges of this methodology, we discuss mitigation strategies like dimensionality reduction, approximate algorithms, and parallelization for real-time and large-scale applications. The method's effectiveness is demonstrated through experiments on real-world text data, focusing on detecting context shifts in Generative AI. Our results, supported by publicly available code, highlight the benefits of this deformation-based approach in capturing subtle drifts that traditional statistical methods often miss. Furthermore, we present a detailed application example within the healthcare domain, showcasing the methodology's potential in diverse fields. Future work will focus on further improving computational efficiency and exploring additional applications across different ML domains.

In a supervised classification split conformal/inductive framework with $K$ classes, a calibration sample of $n$ labeled examples is observed for inference on the label of a new unlabeled example. In this work, we explore the case where a "batch" of $m$ independent such unlabeled examples is given, and a multivariate prediction set with $1-\alpha$ coverage should be provided for this batch. Hence, the batch prediction set takes the form of a collection of label vectors of size $m$, while the calibration sample only contains univariate labels. Using the Bonferroni correction consists in concatenating the individual prediction sets at level $1-\alpha/m$ (Vovk 2013). We propose a uniformly more powerful solution, based on specific combinations of conformal $p$-values that exploit the Simes inequality (Simes 1986). Intuitively, the pooled evidence of fairly "easy" examples of the batch can help provide narrower batch prediction sets. We also introduced adaptive versions of the novel procedure that are particularly effective when the batch prediction set is expected to be large. The theoretical guarantees are provided when all examples are iid, as well as more generally when iid is assumed only conditionally within each class. In particular, our results are also valid under a label distribution shift since the distribution of the labels need not be the same in the calibration sample and in the new `batch'. The usefulness of the method is illustrated on synthetic and real data examples.

This paper presents Sparse Gradient Descent as a solution for variable selection in convex piecewise linear regression where the model is given as $\mathrm{max}\langle a_j^\star, x \rangle + b_j^\star$ for $j = 1,\dots,k$ where $x \in \mathbb R^d$ is the covariate vector. Here, $\{a_j^\star\}_{j=1}^k$ and $\{b_j^\star\}_{j=1}^k$ denote the ground-truth weight vectors and intercepts. A non-asymptotic local convergence analysis is provided for Sp-GD under sub-Gaussian noise when the covariate distribution satisfies sub-Gaussianity and anti-concentration property. When the model order and parameters are fixed, Sp-GD provides an $\epsilon$-accurate estimate given $\mathcal{O}(\max(\epsilon^{-2}\sigma_z^2,1)s\log(d/s))$ observations where $\sigma_z^2$ denotes the noise variance. This also implies the exact parameter recovery by Sp-GD from $\mathcal{O}(s\log(d/s))$ noise-free observations. Since optimizing the squared loss for sparse max-affine is non-convex, an initialization scheme is proposed to provide a suitable initial estimate within the basin of attraction for Sp-GD, i.e. sufficiently accurate to invoke the convergence guarantees. The initialization scheme uses sparse principal component analysis to estimate the subspace spanned by $\{ a_j^\star\}_{j=1}^k$ then applies an $r$-covering search to estimate the model parameters. A non-asymptotic analysis is presented for this initialization scheme when the covariates and noise samples follow Gaussian distributions. When the model order and parameters are fixed, this initialization scheme provides an $\epsilon$-accurate estimate given $\mathcal{O}(\epsilon^{-2}\max(\sigma_z^4,\sigma_z^2,1)s^2\log^4(d))$ observations. Numerical Monte Carlo results corroborate theoretical findings for Sp-GD and the initialization scheme.

Machine Learning (ML) models offer high-quality predictions for complex data structures and have become indispensable across various fields, including civil engineering (Lu et al., 2023), sociology (Molina and Garip, 2019), and archaeology (Bickler, 2021), due to their versatility and predictive power. However, due to their complexity, ML models present an absence of interpretability for their internal mechanism (Hooker and Hooker, 2017; Hooker et al., 2021; Freiesleben et al., 2024). To address this issue, many researchers proposed interpretable machine learning tools (IML) to provide post hoc interpretability of ML models. Among these tools, variable importance, which measures the contribution of individual covariates to the response variable, is a widely adopted measure in IML (Molnar, 2020). Traditionally, it has been applied to assess the performance of fixed models, such as random forests (Breiman, 2001) and linear models (Grömping, 2007). Additionally, efforts have been made to create model-specific uncertainty quantification methods, as seen in Gan et al. (2022).

While overparameterization is known to benefit generalization, its impact on Out-Of-Distribution (OOD) detection is less understood. This paper investigates the influence of model complexity in OOD detection. We propose an expected OOD risk metric to evaluate classifiers confidence on both training and OOD samples. Leveraging Random Matrix Theory, we derive bounds for the expected OOD risk of binary least-squares classifiers applied to Gaussian data. We show that the OOD risk depicts an infinite peak, when the number of parameters is equal to the number of samples, which we associate with the double descent phenomenon. Our experimental study on different OOD detection methods across multiple neural architectures extends our theoretical insights and highlights a double descent curve. Our observations suggest that overparameterization does not necessarily lead to better OOD detection. Using the Neural Collapse framework, we provide insights to better understand this behavior. To facilitate reproducibility, our code will be made publicly available upon publication.

Many risk-sensitive applications require well-calibrated prediction sets over multiple, potentially correlated target variables, for which the prediction algorithm may report correlated non-conformity scores. In this work, we treat the scores as random vectors and aim to construct the prediction set accounting for their joint correlation structure. Drawing from the rich literature on multivariate quantiles and semiparametric statistics, we propose an algorithm to estimate the $1-\alpha$ quantile of the scores, where $\alpha$ is the user-specified miscoverage rate. In particular, we flexibly estimate the joint cumulative distribution function (CDF) of the scores using nonparametric vine copulas and improve the asymptotic efficiency of the quantile estimate using its influence function. The vine decomposition allows our method to scale well to a large number of targets. We report desired coverage and competitive efficiency on a range of real-world regression problems, including those with missing-at-random labels in the calibration set.