We study the advantages of accelerated gradient methods, specifically based on the Frank-Wolfe method and projected gradient descent, for privacy and heavy-tailed robustness. Our approaches are as follows: For the Frank-Wolfe method, our technique is based on a tailored learning rate and a uniform lower bound on the gradient of the $\ell_2$-norm over the constraint set. For accelerating projected gradient descent, we use the popular variant based on Nesterov's momentum, and we optimize our objective over $\mathbb{R}^p$. These accelerations reduce iteration complexity, translating into stronger statistical guarantees for empirical and population risk minimization. Our analysis covers three settings: non-random data, random model-free data, and parametric models (linear regression and generalized linear models). Methodologically, we approach both privacy and robustness based on noisy gradients. We ensure differential privacy via the Gaussian mechanism and advanced composition, and we achieve heavy-tailed robustness using a geometric median-of-means estimator, which also sharpens the dependency on the dimension of the covariates. Finally, we compare our rates to existing bounds and identify scenarios where our methods attain optimal convergence.

Understanding the complex interactions within dynamic multilayer networks is critical for advancements in various scientific domains. Existing models often fail to capture such networks' temporal and cross-layer dynamics. This paper introduces a novel Tensor State Space Model for Dynamic Multilayer Networks (TSSDMN), utilizing a latent space model framework. TSSDMN employs a symmetric Tucker decomposition to represent latent node features, their interaction patterns, and layer transitions. Then by fixing the latent features and allowing the interaction patterns to evolve over time, TSSDMN uniquely captures both the temporal dynamics within layers and across different layers. The model identifiability conditions are discussed. By treating latent features as variables whose posterior distributions are approximated using a mean-field variational inference approach, a variational Expectation Maximization algorithm is developed for efficient model inference. Numerical simulations and case studies demonstrate the efficacy of TSSDMN for understanding dynamic multilayer networks.

This paper presents $whittlehurst$, a Python package implementing Whittle's likelihood method for estimating the Hurst exponent in fractional Brownian motion (fBm). While the theoretical foundations of Whittle's estimator are well-established, practical and computational considerations are critical for effective use. We focus explicitly on assessing our implementation's performance across several numerical approximations of the fractional Gaussian noise (fGn) spectral density, comparing their computational efficiency, accuracy, and consistency across varying input sequence lengths. Extensive empirical evaluations show that our implementation achieves state-of-the-art estimation accuracy and computational speed. Additionally, we benchmark our method against other popular Hurst exponent estimation techniques on synthetic and real-world data, emphasizing practical considerations that arise when applying these estimators to financial and biomedical data.

The importance of clinical variables in the prognosis of the disease is explained using statistical correlation or machine learning (ML). However, the predictive importance of these variables may not represent their causal relationships with diseases. This paper uses clinical variables from a heart failure (HF) patient cohort to investigate the causal explainability of important variables obtained in statistical and ML contexts. Due to inherent regression modeling, popular causal discovery methods strictly assume that the cause and effect variables are numerical and continuous. This paper proposes a new computational framework to enable causal structure discovery (CSD) and score the causal strength of mixed-type (categorical, numerical, binary) clinical variables for binary disease outcomes. In HF classification, we investigate the association between the importance rank order of three feature types: correlated features, features important for ML predictions, and causal features. Our results demonstrate that CSD modeling for nonlinear causal relationships is more meaningful than its linear counterparts. Feature importance obtained from nonlinear classifiers (e.g., gradient-boosting trees) strongly correlates with the causal strength of variables without differentiating cause and effect variables. Correlated variables can be causal for HF, but they are rarely identified as effect variables. These results can be used to add the causal explanation of variables important for ML-based prediction modeling.

We study the recovery of multiple high-dimensional signals from two noisy, correlated modalities: a spiked matrix and a spiked tensor sharing a common low-rank structure. This setting generalizes classical spiked matrix and tensor models, unveiling intricate interactions between inference channels and surprising algorithmic behaviors. Notably, while the spiked tensor model is typically intractable at low signal-to-noise ratios, its correlation with the matrix enables efficient recovery via Bayesian Approximate Message Passing, inducing staircase-like phase transitions reminiscent of neural network phenomena. In contrast, empirical risk minimization for joint learning fails: the tensor component obstructs effective matrix recovery, and joint optimization significantly degrades performance, highlighting the limitations of naive multi-modal learning. We show that a simple Sequential Curriculum Learning strategy-first recovering the matrix, then leveraging it to guide tensor recovery-resolves this bottleneck and achieves optimal weak recovery thresholds. This strategy, implementable with spectral methods, emphasizes the critical role of structural correlation and learning order in multi-modal high-dimensional inference.

Causal Discovery plays a pivotal role in revealing relationships among observed variables, particularly in the temporal setup. While the majority of CD methods rely on synthetic data for evaluation, and recently for training, these fall short in accurately mirroring real-world scenarios; an effect even more evident in temporal data. Generation techniques depending on simplified assumptions on causal structure, effects and time, limit the quality and diversity of the simulated data. In this work, we introduce Temporal Causal-based Simulation (TCS), a robust framework for generating realistic time-series data and their associated temporal causal graphs. The approach is structured in three phases: estimating the true lagged causal structure of the data, approximating the functional dependencies between variables and learning the noise distribution of the corresponding causal model, each part of which can be explicitly tailored based on data assumptions and characteristics. Through an extensive evaluation process, we highlight that single detection methods for generated data discrimination prove inadequate, accentuating it as a multifaceted challenge. For this, we detail a Min-max optimization phase that draws on AutoML techniques. Our contributions include a flexible, model-agnostic pipeline for generating realistic temporal causal data, a thorough evaluation setup which enhances the validity of the generated datasets and insights into the challenges posed by realistic data generation. Through experiments involving not only real but also semi-synthetic and purely synthetic datasets, we demonstrate that while sampling realistic causal data remains a complex task, our method enriches the domain of generating sensible causal-based temporal data.

Quantifying uncertainties for machine learning (ML) models is a foundational challenge in modern data analysis. This challenge is compounded by at least two key aspects of the field: (a) inconsistent terminology surrounding uncertainty and estimation across disciplines, and (b) the varying technical requirements for establishing trustworthy uncertainties in diverse problem contexts. In this position paper, we aim to clarify the depth of these challenges by identifying these inconsistencies and articulating how different contexts impose distinct epistemic demands. We examine the current landscape of estimation targets (e.g., prediction, inference, simulation-based inference), uncertainty constructs (e.g., frequentist, Bayesian, fiducial), and the approaches used to map between them. Drawing on the literature, we highlight and explain examples of problematic mappings. To help address these issues, we advocate for standards that promote alignment between the \textit{intent} and \textit{implementation} of uncertainty quantification (UQ) approaches. We discuss several axes of trustworthiness that are necessary (if not sufficient) for reliable UQ in ML models, and show how these axes can inform the design and evaluation of uncertainty-aware ML systems. Our practical recommendations focus on scientific ML, offering illustrative cases and use scenarios, particularly in the context of simulation-based inference (SBI).

Estimating the distribution of outcomes under counterfactual policies is critical for decision-making in domains such as recommendation, advertising, and healthcare. We analyze a novel framework-Counterfactual Policy Mean Embedding (CPME)-that represents the entire counterfactual outcome distribution in a reproducing kernel Hilbert space (RKHS), enabling flexible and nonparametric distributional off-policy evaluation. We introduce both a plug-in estimator and a doubly robust estimator; the latter enjoys improved uniform convergence rates by correcting for bias in both the outcome embedding and propensity models. Building on this, we develop a doubly robust kernel test statistic for hypothesis testing, which achieves asymptotic normality and thus enables computationally efficient testing and straightforward construction of confidence intervals. Our framework also supports sampling from the counterfactual distribution. Numerical simulations illustrate the practical benefits of CPME over existing methods.

Amyotrophic lateral sclerosis (ALS) is a degenerative disorder of motor neurons that causes progressive paralysis in patients. Current treatment options aim to prolong survival and improve quality of life; however, due to the heterogeneity of the disease, it is often difficult to determine the optimal time for potential therapies or medical interventions. In this study, we propose a novel method to predict the time until a patient with ALS experiences significant functional impairment (ALSFRS-R<=2) with respect to five common functions: speaking, swallowing, handwriting, walking and breathing. We formulate this task as a multi-event survival problem and validate our approach in the PRO-ACT dataset by training five covariate-based survival models to estimate the probability of an event over a 500-day period after a baseline visit. We then predict five event-specific individual survival distributions (ISDs) for each patient, each providing an interpretable and meaningful estimate of when that event will likely take place in the future. The results show that covariate-based models are superior to the Kaplan-Meier estimator at predicting time-to-event outcomes. Additionally, our method enables practitioners to make individual counterfactual predictions, where certain features (covariates) can be changed to see their effect on the predicted outcome. In this regard, we find that Riluzole has little to no impact on predicted functional decline. However, for patients with bulbar-onset ALS, our method predicts considerably shorter counterfactual time-to-event estimates for tasks related to speech and swallowing compared to limb-onset ALS. The proposed method can be applied to current clinical examination data to assess the risk of functional decline and thus allow more personalized treatment planning.

Given an undirected and connected graph $G$ on $T$ vertices, suppose each vertex $t$ has a latent signal $x_t \in \mathbb{R}^n$ associated to it. Given partial linear measurements of the signals, for a potentially small subset of the vertices, our goal is to estimate $x_t$'s. Assuming that the signals are smooth w.r.t $G$, in the sense that the quadratic variation of the signals over the graph is small, we obtain non-asymptotic bounds on the mean squared error for jointly recovering $x_t$'s, for the smoothness penalized least squares estimator. In particular, this implies for certain choices of $G$ that this estimator is weakly consistent (as $T \rightarrow \infty$) under potentially very stringent sampling, where only one coordinate is measured per vertex for a vanishingly small fraction of the vertices. The results are extended to a ``multi-layer'' ranking problem where $x_t$ corresponds to the latent strengths of a collection of $n$ items, and noisy pairwise difference measurements are obtained at each ``layer'' $t$ via a measurement graph $G_t$. Weak consistency is established for certain choices of $G$ even when the individual $G_t$'s are very sparse and disconnected.