Despite tremendous advancements of machine learning models and algorithms in various application domains, they are known to be vulnerable to subtle, natural or intentionally crafted perturbations in future input data, known as adversarial attacks. While numerous adversarial learning methods have been proposed, fundamental questions about their statistical optimality in robust loss remain largely unanswered. In particular, the minimax rate of convergence and the construction of rate-optimal estimators under future $X$-attacks are yet to be worked out. In this paper, we address this issue in the context of nonparametric regression, under suitable assumptions on the smoothness of the regression function and the geometric structure of the input perturbation set. We first establish the minimax rate of convergence under adversarial $L_q$-risks with $1 \leq q \leq \infty$ and propose a piecewise local polynomial estimator that achieves the minimax optimality. The established minimax rate elucidates how the smoothness level and perturbation magnitude affect the fundamental limit of adversarial learning under future $X$-attacks. Furthermore, we construct a data-driven adaptive estimator that is shown to achieve, within a logarithmic factor, the optimal rate across a broad scale of nonparametric and adversarial classes.

Principal component analysis (PCA) is a key tool in the field of data dimensionality reduction. Various methods have been proposed to extend PCA to the union of subspace (UoS) setting for clustering data that come from multiple subspaces like K-Subspaces (KSS). However, some applications involve heterogeneous data that vary in quality due to noise characteristics associated with each data sample. Heteroscedastic methods aim to deal with such mixed data quality. This paper develops a heteroscedastic-focused subspace clustering method, named ALPCAHUS, that can estimate the sample-wise noise variances and use this information to improve the estimate of the subspace bases associated with the low-rank structure of the data. This clustering algorithm builds on K-Subspaces (KSS) principles by extending the recently proposed heteroscedastic PCA method, named LR-ALPCAH, for clusters with heteroscedastic noise in the UoS setting. Simulations and real-data experiments show the effectiveness of accounting for data heteroscedasticity compared to existing clustering algorithms. Code available at https://github.com/javiersc1/ALPCAHUS.

Channel identifiability (CID) refers to the ability to distinguish between individual channels in time series (TS) modeling. The absence of CID often results in producing identical outputs for identical inputs, disregarding channel-specific characteristics. In this paper, we highlight the importance of CID and propose Channel Normalization (CN), a simple yet effective normalization strategy that enhances CID by assigning distinct affine transformation parameters to each channel. We further extend CN in two ways: 1) Adaptive CN (ACN) dynamically adjusts parameters based on the input TS, improving adaptability in TS models, and 2) Prototypical CN (PCN) introduces a set of learnable prototypes instead of per-channel parameters, enabling applicability to datasets with unknown or varying number of channels and facilitating use in TS foundation models. We demonstrate the effectiveness of CN and its variants by applying them to various TS models, achieving significant performance gains for both non-CID and CID models. In addition, we analyze the success of our approach from an information theory perspective. Code is available at https://github.com/seunghan96/CN.

Single-cell RNA sequencing (scRNA-seq) has revolutionized our understanding of cellular processes by enabling gene expression analysis at the individual cell level. Clustering allows for the identification of cell types and the further discovery of intrinsic patterns in single-cell data. However, the high dimensionality and sparsity of scRNA-seq data continue to challenge existing clustering models. In this paper, we introduce JojoSCL, a novel self-supervised contrastive learning framework for scRNA-seq clustering. By incorporating a shrinkage estimator based on hierarchical Bayesian estimation, which adjusts gene expression estimates towards more reliable cluster centroids to reduce intra-cluster dispersion, and optimized using Stein's Unbiased Risk Estimate (SURE), JojoSCL refines both instance-level and cluster-level contrastive learning. Experiments on ten scRNA-seq datasets substantiate that JojoSCL consistently outperforms prevalent clustering methods, with further validation of its practicality through robustness analysis and ablation studies. JojoSCL's code is available at: https://github.com/ziwenwang28/JojoSCL.

Observational studies are a key resource for causal inference but are often affected by systematic biases. Prior work has focused mainly on detecting these biases, via sensitivity analyses and comparisons with randomized controlled trials, or mitigating them through debiasing techniques. However, there remains a lack of methodology for uncovering the underlying mechanisms driving these biases, e.g., whether due to hidden confounding or selection of participants. In this work, we show that the relationship between bias magnitude and the predictive performance of nuisance function estimators (in the observational study) can help distinguish among common sources of causal bias. We validate our methodology through extensive synthetic experiments and a real-world case study, demonstrating its effectiveness in revealing the mechanisms behind observed biases. Our framework offers a new lens for understanding and characterizing bias in observational studies, with practical implications for improving causal inference.

The growing importance of intraday electricity trading in Europe calls for improved price forecasting and tailored decision-support tools. In this paper, we propose a novel generative neural network model to generate probabilistic path forecasts for intraday electricity prices and use them to construct effective trading strategies for Germany's continuous-time intraday market. Our method demonstrates competitive performance in terms of statistical evaluation metrics compared to two state-of-the-art statistical benchmark approaches. To further assess its economic value, we consider a realistic fixed-volume trading scenario and propose various strategies for placing market sell orders based on the path forecasts. Among the different trading strategies, the price paths generated by our generative model lead to higher profit gains than the benchmark methods. Our findings highlight the potential of generative machine learning tools in electricity price forecasting and underscore the importance of economic evaluation.

Decentralized learning has emerged as a powerful approach for handling large datasets across multiple machines in a communication-efficient manner. However, such methods often face scalability limitations, as increasing the number of machines beyond a certain point negatively impacts convergence rates. In this work, we propose Decentralized Anytime SGD, a novel decentralized learning algorithm that significantly extends the critical parallelism threshold, enabling the effective use of more machines without compromising performance. Within the stochastic convex optimization (SCO) framework, we establish a theoretical upper bound on parallelism that surpasses the current state-of-the-art, allowing larger networks to achieve favorable statistical guarantees and closing the gap with centralized learning in highly connected topologies.

We introduce the attention-indexed model (AIM), a theoretical framework for analyzing learning in deep attention layers. Inspired by multi-index models, AIM captures how token-level outputs emerge from layered bilinear interactions over high-dimensional embeddings. Unlike prior tractable attention models, AIM allows full-width key and query matrices, aligning more closely with practical transformers. Using tools from statistical mechanics and random matrix theory, we derive closed-form predictions for Bayes-optimal generalization error and identify sharp phase transitions as a function of sample complexity, model width, and sequence length. We propose a matching approximate message passing algorithm and show that gradient descent can reach optimal performance. AIM offers a solvable playground for understanding learning in modern attention architectures.

Responsibly deploying artificial intelligence (AI) / machine learning (ML) systems in high-stakes settings arguably requires not only proof of system reliability, but also continual, post-deployment monitoring to quickly detect and address any unsafe behavior. Methods for nonparametric sequential testing -- especially conformal test martingales (CTMs) and anytime-valid inference -- offer promising tools for this monitoring task. However, existing approaches are restricted to monitoring limited hypothesis classes or ``alarm criteria'' (e.g., detecting data shifts that violate certain exchangeability or IID assumptions), do not allow for online adaptation in response to shifts, and/or cannot diagnose the cause of degradation or alarm. In this paper, we address these limitations by proposing a weighted generalization of conformal test martingales (WCTMs), which lay a theoretical foundation for online monitoring for any unexpected changepoints in the data distribution while controlling false-alarms. For practical applications, we propose specific WCTM algorithms that adapt online to mild covariate shifts (in the marginal input distribution), quickly detect harmful shifts, and diagnose those harmful shifts as concept shifts (in the conditional label distribution) or extreme (out-of-support) covariate shifts that cannot be easily adapted to. On real-world datasets, we demonstrate improved performance relative to state-of-the-art baselines.

Using stochastic differential equation (SDE) approximations, we study the dynamics of Distributed SGD, Distributed Compressed SGD, and Distributed SignSGD under $(L_0,L_1)$-smoothness and flexible noise assumptions. Our analysis provides insights -- which we validate through simulation -- into the intricate interactions between batch noise, stochastic gradient compression, and adaptivity in this modern theoretical setup. For instance, we show that \textit{adaptive} methods such as Distributed SignSGD can successfully converge under standard assumptions on the learning rate scheduler, even under heavy-tailed noise. On the contrary, Distributed (Compressed) SGD with pre-scheduled decaying learning rate fails to achieve convergence, unless such a schedule also accounts for an inverse dependency on the gradient norm -- de facto falling back into an adaptive method.