Generative flow networks are able to sample, via sequential construction, high-reward, complex objects according to a reward function. However, such reward functions are often estimated approximately from noisy data, leading to epistemic uncertainty in the learnt policy. We present an approach to quantify this uncertainty by constructing a surrogate model composed of a polynomial chaos expansion, fit on a small ensemble of trained flow networks. This model learns the relationship between reward functions, parametrised in a low-dimensional space, and the probability distributions over actions at each step along a trajectory of the flow network. The surrogate model can then be used for inexpensive Monte Carlo sampling to estimate the uncertainty in the policy given uncertain rewards. We illustrate the performance of our approach on a discrete and continuous grid-world, symbolic regression, and a Bayesian structure learning task.

The rapid scaling of large language models (LLMs) has made low-precision training essential for reducing memory, improving efficiency, and enabling larger models and datasets. Existing convergence theories for adaptive optimizers, however, assume all components are exact and neglect hardware-aware quantization, leaving open the question of why low-precision training remains effective. We introduce the first theoretical framework for analyzing the convergence of adaptive optimizers, including Adam and Muon, under floating-point quantization of gradients, weights, and optimizer states (e.g., moment estimates). Within this framework, we derive convergence rates on smooth non-convex objectives under standard stochastic gradient assumptions, explicitly characterizing how quantization errors from different components affect convergence. We show that both algorithms retain rates close to their full-precision counterparts provided mantissa length scales only logarithmically with the number of iterations. Our analysis further reveals that Adam is highly sensitive to weights and second-moment quantization due to its reliance on $β_2 \to 1$, while Muon requires weaker error control and is thus potentially more robust. These results narrow the gap between empirical success and theoretical understanding of low-precision training methods. Numerical experiments on synthetic and real-world data corroborate our theory.

Methods for forecasting time series adhering to linear constraints have seen notable development in recent years, especially with the advent of forecast reconciliation. This paper extends forecast reconciliation to the open question of non-linearly constrained time series. Non-linear constraints can emerge with variables that are formed as ratios such as mortality rates and unemployment rates. On the methodological side, Non-linearly Constrained Reconciliation (NLCR) is proposed. This algorithm adjusts forecasts that fail to meet non-linear constraints, in a way that ensures the new forecasts meet the constraints. The NLCR method is a projection onto a non-linear surface, formulated as a constrained optimisation problem. On the theoretical side, optimisation methods are again used, this time to derive sufficient conditions for when the NLCR methodology is guaranteed to improve forecast accuracy. Finally on the empirical side, NLCR is applied to two datasets from demography and economics and shown to significantly improve forecast accuracy relative to relevant benchmarks.

What property of the data distribution determines the excess risk of principal component analysis? In this paper, we provide a precise answer to this question. We establish a central limit theorem for the error of the principal subspace estimated by PCA, and derive the asymptotic distribution of its excess risk under the reconstruction loss. We obtain a non-asymptotic upper bound on the excess risk of PCA that recovers, in the large sample limit, our asymptotic characterization. Underlying our contributions is the following result: we prove that the negative block Rayleigh quotient, defined on the Grassmannian, is generalized self-concordant along geodesics emanating from its minimizer of maximum rotation less than $π/4$.

Selective classifiers improve model reliability by abstaining on inputs the model deems uncertain. However, few practical approaches achieve the gold-standard performance of a perfect-ordering oracle that accepts examples exactly in order of correctness. Our work formalizes this shortfall as the selective-classification gap and present the first finite-sample decomposition of this gap to five distinct sources of looseness: Bayes noise, approximation error, ranking error, statistical noise, and implementation- or shift-induced slack. Crucially, our analysis reveals that monotone post-hoc calibration -- often believed to strengthen selective classifiers -- has limited impact on closing this gap, since it rarely alters the model's underlying score ranking. Bridging the gap therefore requires scoring mechanisms that can effectively reorder predictions rather than merely rescale them. We validate our decomposition on synthetic two-moons data and on real-world vision and language benchmarks, isolating each error component through controlled experiments. Our results confirm that (i) Bayes noise and limited model capacity can account for substantial gaps, (ii) only richer, feature-aware calibrators meaningfully improve score ordering, and (iii) data shift introduces a separate slack that demands distributionally robust training. Together, our decomposition yields a quantitative error budget as well as actionable design guidelines that practitioners can use to build selective classifiers which approximate ideal oracle behavior more closely.

Edge intelligence enables low-latency inference via compact on-device models, but assuring reliability remains challenging. We study edge-cloud cascades that must preserve conditional coverage: whenever the edge returns a prediction set, it should contain the true label with a user-specified probability, as if produced by the cloud model. We formalize conditional coverage with respect to the cloud predictive distribution, and introduce a conformal alignment-based (CAb) cascading mechanism that certifies this property with user control over the risk level. Our method casts escalation from edge to cloud models as a multiple-hypothesis testing (MHT) problem, tailoring conformal alignment (CA) to select which inputs can be safely handled at the edge. The proposed CAb model cascading method yields statistical guarantees on the average fraction of edge decisions that satisfy cloud-level conditional coverage. The procedure applies to arbitrary edge prediction sets, including variants of conformal prediction (CP), and exposes a tunable trade-off among coverage, deferral rate, and set size. Experiments on CIFAR-100 image classification and the TeleQnA question-answering (QA) benchmark show that the proposed CAb cascade maintains the target conditional coverage for edge predictions while substantially reducing offloading to the cloud and incurring modest increases in prediction-set size.

Synthetic oversampling of minority examples using SMOTE and its variants is a leading strategy for addressing imbalanced classification problems. Despite the success of this approach in practice, its theoretical foundations remain underexplored. We develop a theoretical framework to analyze the behavior of SMOTE and related methods when classifiers are trained on synthetic data. We first derive a uniform concentration bound on the discrepancy between the empirical risk over synthetic minority samples and the population risk on the true minority distribution. We then provide a nonparametric excess risk guarantee for kernel-based classifiers trained using such synthetic data. These results lead to practical guidelines for better parameter tuning of both SMOTE and the downstream learning algorithm. Numerical experiments are provided to illustrate and support the theoretical findings

Differentially private (DP) decentralized Federated Learning (FL) allows local users to collaborate without sharing their data with a central server. However, accurately quantifying the privacy budget of private FL algorithms is challenging due to the co-existence of complex algorithmic components such as decentralized communication and local updates. This paper addresses privacy accounting for two decentralized FL algorithms within the $f$-differential privacy ($f$-DP) framework. We develop two new $f$-DP-based accounting methods tailored to decentralized settings: Pairwise Network $f$-DP (PN-$f$-DP), which quantifies privacy leakage between user pairs under random-walk communication, and Secret-based $f$-Local DP (Sec-$f$-LDP), which supports structured noise injection via shared secrets. By combining tools from $f$-DP theory and Markov chain concentration, our accounting framework captures privacy amplification arising from sparse communication, local iterations, and correlated noise. Experiments on synthetic and real datasets demonstrate that our methods yield consistently tighter $(ε,δ)$ bounds and improved utility compared to Rényi DP-based approaches, illustrating the benefits of $f$-DP in decentralized privacy accounting.

Multi-omics data integration is crucial for understanding complex diseases, yet limited sample sizes, noise, and heterogeneity often reduce predictive power. To address these challenges, we introduce Omics-GAN, a Generative Adversarial Network (GAN)-based framework designed to generate high-quality synthetic multi-omics profiles while preserving biological relationships. We evaluated Omics-GAN on three omics types (mRNA, miRNA, and DNA methylation) using the ROSMAP cohort for Alzheimer's disease (AD) and TCGA datasets for colon and liver cancer. A support vector machine (SVM) classifier with repeated 5-fold cross-validation demonstrated that synthetic datasets consistently improved prediction accuracy compared to original omics profiles. The AUC of SVM for mRNA improved from 0.72 to 0.74 in AD, and from 0.68 to 0.72 in liver cancer. Synthetic miRNA enhanced classification in colon cancer from 0.59 to 0.69, while synthetic methylation data improved performance in liver cancer from 0.64 to 0.71. Boxplot analyses confirmed that synthetic data preserved statistical distributions while reducing noise and outliers. Feature selection identified significant genes overlapping with original datasets and revealed additional candidates validated by GO and KEGG enrichment analyses. Finally, molecular docking highlighted potential drug repurposing candidates, including Nilotinib for AD, Atovaquone for liver cancer, and Tecovirimat for colon cancer. Omics-GAN enhances disease prediction, preserves biological fidelity, and accelerates biomarker and drug discovery, offering a scalable strategy for precision medicine applications.

We present a novel enhanced cyclic coordinate descent (ECCD) framework for solving generalized linear models with elastic net constraints that reduces training time in comparison to existing state-of-the-art methods. We redesign the CD method by performing a Taylor expansion around the current iterate to avoid nonlinear operations arising in the gradient computation. By introducing this approximation, we are able to unroll the vector recurrences occurring in the CD method and reformulate the resulting computations into more efficient batched computations. We show empirically that the recurrence can be unrolled by a tunable integer parameter, $s$, such that $s > 1$ yields performance improvements without affecting convergence, whereas $s = 1$ yields the original CD method. A key advantage of ECCD is that it avoids the convergence delay and numerical instability exhibited by block coordinate descent. Finally, we implement our proposed method in C++ using Eigen to accelerate linear algebra computations. Comparison of our method against existing state-of-the-art solvers shows consistent performance improvements of $3\times$ in average for regularization path variant on diverse benchmark datasets. Our implementation is available at https://github.com/Yixiao-Wang-Stats/ECCD.