Bayesian methods lie at the heart of modern data science and provide a powerful scaffolding for estimation in data-constrained settings and principled quantification and propagation of uncertainty. Yet in many real-world use cases where these methods are deployed, there is a natural need to preserve the privacy of the individuals whose data is being scrutinized. While a number of works have attempted to approach the problem of differentially private Bayesian estimation through either reasoning about the inherent privacy of the posterior distribution or privatizing off-the-shelf Bayesian methods, these works generally do not come with rigorous utility guarantees beyond low-dimensional settings. In fact, even for the prototypical tasks of Gaussian mean estimation and linear regression, it was unknown how close one could get to the Bayes-optimal error with a private algorithm, even in the simplest case where the unknown parameter comes from a Gaussian prior. In this work, we give the first efficient algorithms for both of these problems that achieve mean-squared error $(1+o(1))\mathrm{OPT}$ and additionally show that both tasks exhibit an intriguing computational-statistical gap. For Bayesian mean estimation, we prove that the excess risk achieved by our method is optimal among all efficient algorithms within the low-degree framework, yet is provably worse than what is achievable by an exponential-time algorithm. For linear regression, we prove a qualitatively similar lower bound. Our algorithms draw upon the privacy-to-robustness framework of arXiv:2212.05015, but with the curious twist that to achieve private Bayes-optimal estimation, we need to design sum-of-squares-based robust estimators for inherently non-robust objects like the empirical mean and OLS estimator. Along the way we also add to the sum-of-squares toolkit a new kind of constraint based on short-flat decompositions.

We study contextual bandits with finitely many actions in which the reward of each arm follows a single-index model with an arm-specific index parameter and an unknown nonparametric link function. We consider a regime in which arms correspond to stable decision options and covariates evolve adaptively under the bandit policy. This setting creates significant statistical challenges: the sampling distribution depends on the allocation rule, observations are dependent over time, and inverse-propensity weighting induces variance inflation. We propose a kernelized $\varepsilon$-greedy algorithm that combines Stein-based estimation of the index parameters with inverse-propensity-weighted kernel ridge regression for the reward functions. This approach enables flexible semiparametric learning while retaining interpretability. Our analysis develops new tools for inference with adaptively collected data. We establish asymptotic normality for the single-index estimator under adaptive sampling, yielding valid confidence regions, and derive a directional functional central limit theorem for the RKHS estimator, which provides asymptotically valid pointwise confidence intervals. The analysis relies on concentration bounds for inverse-weighted Gram matrices together with martingale central limit theorems. We further obtain finite-time regret guarantees, including $\tilde{O}(\sqrt{T})$ rates under common-link Lipschitz conditions, showing that semiparametric structure can be exploited without sacrificing statistical efficiency. These results provide a unified framework for simultaneous learning and inference in single-index contextual bandits.

Learning-to-Defer routes each input to the expert that minimizes expected cost, but it assumes that the information available to every expert is fixed at decision time. Many modern systems violate this assumption: after selecting an expert, one may also choose what additional information that expert should receive, such as retrieved documents, tool outputs, or escalation context. We study this problem and call it Learning-to-Defer with advice. We show that a broad family of natural separated surrogates, which learn routing and advice with distinct heads, is inconsistent even in the smallest non-trivial setting. We then introduce an augmented surrogate that operates on the composite expert--advice action space and prove an $\mathcal{H}$-consistency guarantee together with an excess-risk transfer bound, yielding recovery of the Bayes-optimal policy in the limit. Experiments on tabular, language, and multi-modal tasks show that the resulting method improves over standard Learning-to-Defer while adapting its advice-acquisition behavior to the cost regime; a synthetic benchmark confirms the failure mode predicted for separated surrogates.

Non-Gaussian statistics are a challenge for data assimilation. Linear methods oversimplify the problem, yet fully nonlinear methods are often too expensive to use in practice. The best solution usually lies between these extremes. Triangular measure transport offers a flexible framework for nonlinear data assimilation. Its success, however, depends on how the map is parametrized. Too much flexibility leads to overfitting; too little misses important structure. To address this balance, we develop an adaptation algorithm that selects a parsimonious parametrization automatically. Our method uses P-spline basis functions and an information criterion as a continuous measure of model complexity. This formulation enables gradient descent and allows efficient, fine-scale adaptation in high-dimensional settings. The resulting algorithm requires no hyperparameter tuning. It adjusts the transport map to the appropriate level of complexity based on the system statistics and ensemble size. We demonstrate its performance in nonlinear, non-Gaussian problems, including a high-dimensional distributed groundwater model.

The topic of Multivariate Time Series Anomaly Detection (MTSAD) has grown rapidly over the past years, with a steady rise in publications and Deep Learning (DL) models becoming the dominant paradigm. To address the lack of systematization in the field, this study introduces a novel and unified taxonomy with eleven dimensions over three parts (Input, Output and Model) for the categorization of DL-based MTSAD methods. The dimensions were established in a two-fold approach. First, they derived from a comprehensive analysis of methodological studies. Second, insights from review papers were incorporated. Furthermore, the proposed taxonomy was validated using an additional set of recent publications, providing a clear overview of methodological trends in MTSAD. Results reveal a convergence toward Transformer-based and reconstruction and prediction models, setting the foundation for emerging adaptive and generative trends. Building on and complementing existing surveys, this unified taxonomy is designed to accommodate future developments, allowing for new categories or dimensions to be added as the field progresses. This work thus consolidates fragmented knowledge in the field and provides a reference point for future research in MTSAD.

Motivated by the principle of satisficing in decision-making, we study satisficing regret guarantees for nonstationary $K$-armed bandits. We show that in the general realizable, piecewise-stationary setting with $L$ stationary segments, the optimal regret is $Θ(L\log T)$ as long as $L\geq 2$. This stands in sharp contrast to the case of $L=1$ (i.e., the stationary setting), where a $T$-independent $Θ(1)$ satisficing regret is achievable under realizability. In other words, the optimal regret has to scale with $T$ even if just a little nonstationarity presents. A key ingredient in our analysis is a novel Fano-based framework tailored to nonstationary bandits via a \emph{post-interaction reference} construction. This framework strictly extends the classical Fano method for passive estimation as well as recent interactive Fano techniques for stationary bandits. As a complement, we also discuss a special regime in which constant satisficing regret is again possible.

We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem, which models linear classification. Both are fundamental problems in computational geometry and machine learning in their exact and approximate forms. However, only $O(n^d)$ and respectively $\tilde O(1/\varepsilon^d)$ upper bounds are known and complemented by polynomial lower bounds that do not support the exponential in dimension dependence. We close this gap up to polylogarithmic terms by reduction from widely-believed hardness conjectures for Affine Degeneracy testing and $k$-Sum problems. Our reductions yield matching lower bounds of $\tildeΩ(n^d)$ and respectively $\tildeΩ(1/\varepsilon^d)$ based on Affine Degeneracy testing, and $\tildeΩ(n^{d/2})$ and respectively $\tildeΩ(1/\varepsilon^{d/2})$ conditioned on $k$-Sum. The first bound also holds unconditionally if the computational model is restricted to make sidedness queries, which corresponds to a widely spread setting implemented and optimized in many contemporary algorithms and computing paradigms.

In the present paper we study the performance of linear denoisers for noisy data of the form $\mathbf{x} + \mathbf{z}$, where $\mathbf{x} \in \mathbb{R}^d$ is the desired data with zero mean and unknown covariance $\mathbfΣ$, and $\mathbf{z} \sim \mathcal{N}(0, \mathbfΣ_{\mathbf{z}})$ is additive noise. Since the covariance $\mathbfΣ$ is not known, the standard Wiener filter cannot be employed for denoising. Instead we assume we are given samples $\mathbf{x}_1,\dots,\mathbf{x}_n \in \mathbb{R}^d$ from the true distribution. A standard approach would then be to estimate $\mathbfΣ$ from the samples and use it to construct an ``empirical" Wiener filter. However, in this paper, motivated by the denoising step in diffusion models, we take a different approach whereby we train a linear denoiser $\mathbf{W}$ from the data itself. In particular, we synthetically construct noisy samples $\hat{\mathbf{x}}_i$ of the data by injecting the samples with Gaussian noise with covariance $\mathbfΣ_1 \neq \mathbfΣ_{\mathbf{z}}$ and find the best $\mathbf{W}$ that approximates $\mathbf{W}\hat{\mathbf{x}}_i \approx \mathbf{x}_i$ in a least-squares sense. In the proportional regime $\frac{n}{d} \rightarrow κ> 1$ we use the {\it Convex Gaussian Min-Max Theorem (CGMT)} to analytically find the closed form expression for the generalization error of the denoiser obtained from this process. Using this expression one can optimize over $\mathbfΣ_1$ to find the best possible denoiser. Our numerical simulations show that our denoiser outperforms the ``empirical" Wiener filter in many scenarios and approaches the optimal Wiener filter as $κ\rightarrow\infty$.

Reservoir computing is a well-established approach for processing data with a much lower complexity compared to traditional neural networks. Despite two decades of experimental progress, the core properties of reservoir computing (namely separation, robustness, and fading memory) still lack rigorous mathematical foundations. This paper addresses this gap by providing a control-theoretic framework for the analysis of time-delay-based reservoir computers. We introduce formal definitions of the separation property and fading memory in terms of functional norms, and establish their connection to well-known stability notions for time-delay systems as incremental input-to-state stability. For a class of linear reservoirs, we derive an explicit lower bound for the separation distance via Fourier analysis, offering a computable criterion for reservoir design. Numerical results on the NARMA10 benchmark and continuous-time system prediction validate the approach with a minimal digital implementation.

Anomaly and failure detection methods are crucial in identifying deviations from normal system operational conditions, which allows for actions to be taken in advance, usually preventing more serious damages. Long-lasting deviations indicate failures, while sudden, isolated changes in the data indicate anomalies. However, in many practical applications, changes in the data do not always represent abnormal system states. Such changes may be recognized incorrectly as failures, while being a normal evolution of the system, e.g. referring to characteristics of starting the processing of a new product, i.e. realizing a domain shift. Therefore, distinguishing between failures and such ''healthy'' changes in data distribution is critical to ensure the practical robustness of the system. In this paper, we propose a method that not only detects changes in the data distribution and anomalies but also allows us to distinguish between failures and normal domain shifts inherent to a given process. The proposed method consists of a modified Page-Hinkley changepoint detector for identification of the domain shift and possible failures and supervised domain-adaptation-based algorithms for fast, online anomaly detection. These two are coupled with an explainable artificial intelligence (XAI) component that aims at helping the human operator to finally differentiate between domain shifts and failures. The method is illustrated by an experiment on a data stream from the steel factory.