January 2026

arXiv:2510.16745v3 Announce Type: replace Abstract: We propose a kernel-based nonparametric framework for mean-variance optimization that enables inference on economically motivated shape constraints in finance, including positivity, monotonicity, and convexity. Many central hypotheses in financial econometrics are naturally expressed as shape relations on latent functions (e.g., term premia, CAPM relations, and the pricing kernel), yet enforcing such constraints during estimation can mask economically meaningful violations; our approach therefore separates learning from validation by first estimating an unconstrained solution and […]

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arXiv:2505.04613v3 Announce Type: replace Abstract: We prove that kernel covariance embeddings lead to information-theoretically perfect separation of distinct continuous probability distributions. In statistical terms, we establish that testing for the emph{equality} of two non-atomic (Borel) probability measures on a locally compact Polish space is emph{equivalent} to testing for the emph{singularity} between two centered Gaussian measures on a reproducing kernel Hilbert space. The corresponding Gaussians are defined via the notion of kernel covariance embedding of a probability measure, and […]

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arXiv:2502.06168v2 Announce Type: replace Abstract: We study an online dynamic pricing problem where the potential demand at each time period $t=1,2,ldots, T$ is stochastic and dependent on the price. However, a perishable inventory is imposed at the beginning of each time $t$, censoring the potential demand if it exceeds the inventory level. To address this problem, we introduce a pricing algorithm based on the optimistic estimates of derivatives. We show that our algorithm achieves $tilde{O}(sqrt{T})$ optimal regret even […]

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arXiv:2502.01953v3 Announce Type: replace Abstract: We consider a general model for high-dimensional empirical risk minimization whereby the data $mathbf{x}_i$ are $d$-dimensional Gaussian vectors, the model is parametrized by $mathbf{Theta}inmathbb{R}^{dtimes k}$, and the loss depends on the data via the projection $mathbf{Theta}^mathsf{T}mathbf{x}_i$. This setting covers as special cases classical statistics methods (e.g. multinomial regression and other generalized linear models), but also two-layer fully connected neural networks with $k$ hidden neurons. We use the Kac-Rice formula from Gaussian process theory […]

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arXiv:2406.00778v4 Announce Type: replace Abstract: It is increasingly common to collect data of multiple different types on the same set of samples. Our focus is on studying relationships between such multiview features and responses. A motivating application arises in the context of precision medicine where multi-omics data are collected to correlate with clinical outcomes. It is of interest to infer dependence within and across views while combining multimodal information to improve the prediction of outcomes. The signal-to-noise ratio […]

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arXiv:2601.16979v1 Announce Type: cross Abstract: Understanding the curvature evolution of the loss landscape is fundamental to analyzing the training dynamics of neural networks. The most commonly studied measure, Hessian sharpness ($lambda_{max}^H$) — the largest eigenvalue of the loss Hessian — determines local training stability and interacts with the learning rate throughout training. Despite its significance in analyzing training dynamics, direct measurement of Hessian sharpness remains prohibitive for Large Language Models (LLMs) due to high computational cost. We analyze […]

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arXiv:2601.16922v1 Announce Type: cross Abstract: The sample complexity of multi-group learning is shown to improve in the group-realizable setting over the agnostic setting, even when the family of groups is infinite so long as it has finite VC dimension. The improved sample complexity is obtained by empirical risk minimization over the class of group-realizable concepts, which itself could have infinite VC dimension. Implementing this approach is also shown to be computationally intractable, and an alternative approach is suggested […]

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arXiv:2601.16897v1 Announce Type: cross Abstract: We introduce FedSGM, a unified framework for federated constrained optimization that addresses four major challenges in federated learning (FL): functional constraints, communication bottlenecks, local updates, and partial client participation. Building on the switching gradient method, FedSGM provides projection-free, primal-only updates, avoiding expensive dual-variable tuning or inner solvers. To handle communication limits, FedSGM incorporates bi-directional error feedback, correcting the bias introduced by compression while explicitly understanding the interaction between compression noise and multi-step local […]

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arXiv:2601.16777v1 Announce Type: cross Abstract: Under the assumption that data lie on a compact (unknown) manifold without boundary, we derive finite sample bounds for kernel smoothing and its (first and second) derivatives, and we establish asymptotic normality through Berry-Esseen type bounds. Special cases include kernel density estimation, kernel regression and the heat kernel signature. Connections to the graph Laplacian are also discussed.

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arXiv:2601.16636v1 Announce Type: cross Abstract: Polynomial Chaos Expansions (PCEs) are widely recognized for their efficient computational performance in surrogate modeling. Yet, a robust framework to quantify local model errors is still lacking. While the local uncertainty of PCE prediction can be captured using bootstrap resampling, other methods offering more rigorous statistical guarantees are needed, especially in the context of small training datasets. Recently, conformal predictions have demonstrated strong potential in machine learning, providing statistically robust and model-agnostic prediction […]

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