arXiv:2606. 04804v1 Announce Type: new Abstract: Generative models -- diffusion and flow matching -- are increasingly used to solve partial differential equation (PDE) inverse problems, enforcing the governing physics as a \emph{hard constraint} (via projection or guidance) and reporting the resulting samples as a Bayesian posterior with calibrated uncertainty.
By Jian Xu, Delu Zeng, John Paisley, Qibin Zhao
Generative models -- diffusion and flow matching -- are increasingly used to solve partial differential equation (PDE) inverse problems, enforcing the governing physics as a \emph{hard constraint} (via projection or guidance) and reporting the resulting samples as a Bayesian posterior with calibrated uncertainty. We show that this widely adopted recipe samples the wrong distribution.
The paper introduces D$^3$Opt, a derivative‑free diffusion‑based framework that separates constraint modeling from objective optimization in chance‑constrained programs. It learns a risk‑conditioned diffusion model once on constraint‑filtered decisions, then uses an annealed, particle‑based Feynman–Kac correction to optimize any post‑specified objective without retraining. The method is proven to preserve feasibility and provides an error bound, and experiments show effective optimization across smooth, non‑smooth, and non‑convex objectives, including objective transfer tasks.
By Ziwen Liu, Yan Liu, Congying Han, Tiande Guo, Yao Yan, Weichen Zhao
The paper introduces PDE‑Residual‑Informed Kriging (PRIK), a Gaussian process framework that embeds global PDE residual information directly into its covariance structure via a PDE‑residual kernel. PRIK yields a posterior mean with minimal integrated squared PDE residual and offers statistical interpretations linking variance to residual energy. The authors prove quasi‑optimality and convergence rates, and demonstrate the method’s PDE‑solving capability through numerical experiments.
By Yu Tan, Shichang Du, Yongxiang Li
arXiv:2603. 00393v2 Announce Type: replace-cross Abstract: Inverse problems constrained by partial differential equations are often ill-conditioned due to noisy, incomplete data or inherent non-uniqueness.
By Ali Siahkoohi, Kamal Aghazade, Ali Gholami
arXiv:2606. 00703v1 Announce Type: cross Abstract: Low-precision pretraining (FP8, MXFP4, NVFP4) is now standard for frontier language models, yet the literature is almost entirely achievability -- algorithms and empirical scaling laws -- with no matching characterization of what is information-theoretically possible.
By Munsik Kim
The paper studies contextual bilateral trade with full feedback, showing that action-independent observations eliminate the usual polynomial adaptation penalty seen in heavy-tailed bandits. It presents fully parameter-free algorithms that achieve oracle minimax regret rates without knowing the moment order or scale, and derives new regret bounds for both parametric and nonparametric settings. The key technical insight is a paired squared‑loss statistic whose noise cancels, enabling model selection and yielding regret rates that interpolate between classical nonparametric and linear extremes.
By Hangyi Zhao
arXiv:2606. 08028v1 Announce Type: new Abstract: We study high-probability regret bounds for online convex optimization (OCO) with strongly convex losses and establish three results that resolve open questions at the intersection of noise adaptivity, feedback structure, and constraint satisfaction.
By Wentao Zhang, Yutong Zhang, Wentao Mo
arXiv:2509. 20114v3 Announce Type: replace Abstract: We study \emph{online episodic Constrained Markov Decision Processes} (CMDPs) under both stochastic and adversarial constraints.
By Francesco Emanuele Stradi, Eleonora Fidelia Chiefari, Matteo Castiglioni, Alberto Marchesi, Nicola Gatti
arXiv:2606. 28281v1 Announce Type: cross Abstract: PAC-Bayesian bounds provide finite-sample guarantees for data-dependent randomized predictors, but applying them to learning-based control is difficult because the natural objective is a quadratic trajectory cost.
By Domagoj Herceg
arXiv:2606. 07931v1 Announce Type: cross Abstract: We prove a variance-aware pointwise majorizing-measure theorem for centered Gaussian processes.
By Yunbei Xu
The paper discusses the data processing inequality (DPI) in statistics, which states that a stochastically modified experiment cannot have a lower Bayes risk than the original. It shows that this classical DPI does not hold for constrained learning problems common in machine learning, where the model class is limited. The authors propose a generalized DPI that applies to constrained Bayes risks, linking it to a set containment condition on a superprediction set, and provide sufficient conditions for this containment.
By Laura Iacovissi, Rabanus Derr, Robert C. Williamson