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arXiv:2603.01162v3 Announce Type: replace-cross Abstract: Group relative policy optimization (GRPO), a core methodological component of DeepSeekMath and DeepSeek-R1, has emerged as a cornerstone for scaling reasoning capabilities of large language models. Despite its widespread adoption and the proliferation of follow-up works, the theoretical properties of GRPO remain less studied. This paper provides a unified framework to understand GRPO through the lens of classical U-statistics. We demonstrate that the GRPO policy gradient is inherently a U-statistic, allowing us to […]

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arXiv:2602.12683v2 Announce Type: replace-cross Abstract: We reformulate Optimal Transport Conditional Flow Matching (OT-CFM), a class of dynamical generative models, showing that it admits an exact proximal formulation via an extended Brenier potential, without assuming that the target distribution has a density. In particular, the mapping to recover the target point is exactly given by a proximal operator, which yields an explicit proximal expression of the vector field. We also discuss the convergence of minibatch OT-CFM to the population […]

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arXiv:2602.08998v4 Announce Type: replace-cross Abstract: We study homology of ample groupoids via the compactly supported Moore complex of the nerve. Let $A$ be a topological abelian group. For $nge 0$ set $C_n(mathcal G;A) := C_c(mathcal G_n,A)$ and define $partial_n^A=sum_{i=0}^n(-1)^i(d_i)_*$. This defines $H_n(mathcal G;A)$. The theory is functorial for continuous ‘etale homomorphisms. It is compatible with standard reductions, including restriction to saturated clopen subsets. In the ample setting it is invariant under Kakutani equivalence. We reprove Matui type long […]

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arXiv:2602.07098v2 Announce Type: replace-cross Abstract: Modern Bayesian inference involves a mixture of computational methods for estimating, validating, and drawing conclusions from probabilistic models as part of principled workflows. An overarching motif of many Bayesian methods is that they are relatively slow, which often becomes prohibitive when fitting complex models to large data sets. Amortized Bayesian inference (ABI) offers a path to solving the computational challenges of Bayes. ABI trains neural networks on model simulations, rewarding users with rapid […]

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arXiv:2601.22481v2 Announce Type: replace-cross Abstract: This dissertation presents a general framework for changepoint detection based on L0 model selection. The core method, Iteratively Reweighted Fused Lasso (IRFL), improves upon the generalized lasso by adaptively reweighting penalties to enhance support recovery and minimize criteria such as the Bayesian Information Criterion (BIC). The approach allows for flexible modeling of seasonal patterns, linear and quadratic trends, and autoregressive dependence in the presence of changepoints. Simulation studies demonstrate that IRFL achieves accurate […]

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arXiv:2511.01137v2 Announce Type: replace-cross Abstract: We use geometric invariant theory (GIT) to study the deep linear network (DLN). The Kempf-Ness theorem is used to establish that the $L^2$ regularizer is minimized on the balanced manifold. We introduce related balancing flows using the Riemannian geometry of fibers. The balancing flow defined by the $L^2$ regularizer is shown to converge to the balanced manifold at a uniform exponential rate. The balancing flow defined by the squared moment map is computed […]

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arXiv:2510.03798v2 Announce Type: replace-cross Abstract: The batched multi-armed bandit (MAB) problem, in which rewards are collected in batches, is crucial for applications such as clinical trials. Existing research predominantly assumes light-tailed reward distributions, yet many real-world scenarios, including clinical outcomes, exhibit heavy-tailed characteristics. This paper bridges this gap by proposing robust batched bandit algorithms designed for heavy-tailed rewards, within both finite-arm and Lipschitz-continuous settings. We reveal a surprising phenomenon: in the instance-independent regime, as well as in the […]

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arXiv:2509.20721v2 Announce Type: replace-cross Abstract: Scaling laws, a defining feature of deep learning, reveal a striking power-law improvement in model performance with increasing dataset and model size. Yet, their mathematical origins, especially the scaling exponent, have remained elusive. In this work, we show that scaling laws can be formally explained as redundancy laws. Using kernel regression, we show that a polynomial tail in the data covariance spectrum yields an excess risk power law with exponent alpha = 2s […]

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arXiv:2508.14936v2 Announce Type: replace-cross Abstract: Synthetic data holds substantial potential to address practical challenges in epidemiology due to restricted data access and privacy concerns. However, many current methods suffer from limited quality, high computational demands, and complexity for non-experts. Furthermore, common evaluation strategies for synthetic data often fail to directly reflect statistical utility and measure privacy risks sufficiently. Against this background, a critical underexplored question is whether synthetic data can reliably reproduce key findings from epidemiological research while […]

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arXiv:2508.07392v3 Announce Type: replace-cross Abstract: Modern methods of generative modelling and unpaired data translation based on Schr”odinger bridges and stochastic optimal control theory aim to transform an initial density to a target one in an optimal way. In the present paper, we assume that we only have access to i.i.d. samples from the initial and final distributions. This makes our setup suitable for both generative modelling and unpaired data translation. Relying on the stochastic optimal control approach, we […]

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