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A Single-Loop Algorithm for Decentralized Bilevel Optimization

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arXiv 2023年

作者： Dong, Youran Ma, Shiqian Yang, Junfeng Yin, Chao Department of Mathematics Nanjing University Nanjing China Department of Computational Applied Mathematics and Operations Research Rice University Houston United States

Bilevel optimization has gained significant attention in recent years due to its broad applications in machine learning. This paper focuses on bilevel optimization in decentralized networks and proposes a novel single-loop algorithm for solving decentralized bilevel optimization with a strongly convex lower-level problem. Our approach is a fully single-loop method that approximates the hypergradient using only two matrix-vector multiplications per iteration. Importantly, our algorithm does not require any gradient heterogeneity assumption, distinguishing it from existing methods for decentralized bilevel optimization and federated bilevel optimization. Our analysis demonstrates that the proposed algorithm achieves the best-known convergence rate for bilevel optimization algorithms. We also present experimental results on hyperparameter optimization problems using both synthetic and MNIST datasets, which demonstrate the efficiency of our proposed algorithm. Copyright © 2023, The Authors. All rights reserved.

关键词： Machine learning

Coarse-grained spectral projection: A deep learning assisted approach to quantum unitary dynamics

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Physical Review B 2021年第2期103卷 024304-024304页

作者： Pinchen Xie Weinan E Department of Mathematics and Program in Applied and Computational Mathematics Princeton University New Jersey 08544 USA

We propose the coarse-grained spectral projection method (CGSP), a deep learning assisted approach for tackling quantum unitary dynamic problems with an emphasis on quench dynamics. We show that CGSP can extract spectral components of many-body quantum states systematically with a sophisticated neural network quantum ansatz. CGSP fully exploits the linear unitary nature of the quantum dynamics and is potentially superior to other quantum Monte Carlo methods for ergodic dynamics. Preliminary numerical results on one-dimensional XXZ models with periodic boundary conditions are carried out to demonstrate the practicality of CGSP.

关键词： Quantum quench Quantum statistical mechanics Nonequilibrium lattice models Machine learning Monte Carlo methods

Detailed discussion of discrepancy between theoretical and observed spectral lines in Kr-like W^(38+)based on advanced consideration of core electron correlations

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Chinese Physics B 2025年第5期34卷 328-337页

作者： Guo-Qing Peng Kai Wang Jun Yan Wei Kang Center for Applied Physics and Technology Peking UniversityBeijing 100871China Department of Physics and Anhui Key Laboratory of Optoelectric Materials Science and Technology Key Laboratory of Functional Molecular SolidsMinistry of EducationAnhui Normal UniversityWuhu 241000China Institute of Applied Physics and Computational Mathematics Beijing 100088China

For the observed line at 799.23°A in tungsten EBIT experiment,which was assigned to be^(3)F_(4)^(o)−^(3)F_(3)^(o)([Ar]4s^(2)4p^(5)4d)of W^(38+)ion,there were noticeable deviations for most calculated wavelengths from the measured *** clarify this issue,we carry out an extensive calculation for energy levels and transition properties of W^(38+)ion using the multi-configuration Dirac–Hartree–Fock and relativistic configuration interaction method,in which more deeper inner core electron correlations are included,and different forms of Breit interaction as well as quantum electrodynamics corrections are *** is found that the inner core electron correlations can affect the total energy of levels,while only slightly modify the excited energy of levels in 4s^(2)4p^(5)4d *** present calculated wavelengths agree with the corresponding measured values excellently except the line at 799.23Å.Thus we are strongly suspicious this line should be misidentified,and suggest that new experiment with higher resolution and spectra analysis based on more accurate atomic data should be performed for W^(38+)ion.

关键词： Kr-like W^(38+) the multi-configuration Dirac-Hartree-Fock(MCDHF)method electron correlation imbalance Breit interaction quantum electrodynamic(QED)

A Mathematical Framework for Learning Probability Distributions

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arXiv 2022年

作者： Yang, Hongkang Program in Applied and Computational Mathematics Princeton University United States

The modeling of probability distributions, specifically generative modeling and density estimation, has become an immensely popular subject in recent years by virtue of its outstanding performance on sophisticated data such as images and texts. Nevertheless, a theoretical understanding of its success is still incomplete. One mystery is the paradox between memorization and generalization: In theory, the model is trained to be exactly the same as the empirical distribution of the finite samples, whereas in practice, the trained model can generate new samples or estimate the likelihood of unseen samples. Likewise, the overwhelming diversity of distribution learning models calls for a unified perspective on this subject. This paper provides a mathematical framework such that all the well-known models can be derived based on simple principles. To demonstrate its efficacy, we present a survey of our results on the approximation error, training error and generalization error of these models, which can all be established based on this framework. In particular, the aforementioned paradox is resolved by proving that these models enjoy implicit regularization during training, so that the generalization error at early-stopping avoids the curse of dimensionality. Furthermore, we provide some new results on landscape analysis and the mode collapse *** Codes 68T07, 62G05, 60-08 Copyright © 2022, The Authors. All rights reserved.

关键词： Probability distributions

Stability of the Inviscid Power-Law Vortex

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arXiv 2024年

作者： Binz, Tim Coiculescu, Matei P. Princeton University Program in Applied & Computational Mathematics Fine Hall Washington Road PrincetonNJ08544 United States Princeton University Department of Mathematics Fine Hall Washington Road PrincetonNJ08544 United States

We prove that the power-law vortex ω¯ (x) = β|x|−α, which explicitly solves the stationary unforced incompressible Euler equations in R2 in both physical and self-similar coordinates, is exponentially linearly stable in self-similar coordinates with the natural scaling. This answers a question from [2]. Moreover, we prove that in physical coordinates, the linearization around ω¯ generates a strongly continuous group of isometries and we prove that the spectrum of the linearization is contained in the imaginary axis. Our results are valid for L2(R2) functions with a mild symmetry condition. Copyright © 2024, The Authors. All rights reserved.

关键词： Euler equations

Multiple solutions and hysteresis in the flows driven by surface with antisymmetric velocity profile

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Chinese Physics B 2021年第9期30卷 83-88页

作者： Xiao-Feng Shi Dong-Jun Ma Zong-Qiang Ma De-Jun Sun Pei Wang Institute of Applied Physical and Computational Mathematics Beijing 100094China Center for Applied Physics and Technology Peking UniversityBeijing 100871China Department of Modern Mechanics University of Science and Technology of ChinaHefei 230027China

Multiple steady solutions and hysteresis phenomenon in the square cavity flows driven by the surface with antisymmetric velocity profile are investigated by numerical simulation and bifurcation analysis.A high order spectral element method with the matrix-free pseudo-arclength technique is used for the steady-state solution and numerical *** complex flow patterns beyond the symmetry-breaking at Re≈320 are presented by a bifurcation diagram for Re<*** results of stable symmetric and asymmetric solutions are consistent with those reported in literature,and a new unstable asymmetric branch is obtained besides the stable branches.A novel hysteresis phenomenon is observed in the range of 2208

关键词： bifurcation and continuation symmetry-breaking hysteresis surface driven cavity flow

Correction: Evaluation of beta integrals of Ramanujan type and integral representations for bilateral hypergeometric series

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The Ramanujan Journal 2025年第3期67卷 1-2页

作者： Cohl, Howard S. Volkmer, Hans Applied and Computational Mathematics Division National Institute of Standards and Technology Mission Viejo USA Department of Mathematical Sciences University of Wisconsin–Milwaukee Milwaukee USA

FINITE TIME BLOWUP FOR KELLER-SEGEL EQUATION WITH LOGISTIC DAMPING IN THREE DIMENSIONS

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arXiv 2025年

作者： Liu, Jiaqi Wang, Yixuan Zhou, Tao Department of Mathematics University of Southern California Los AngelesCA90089 United States Applied and Computational Mathematics California Institute of Technology PasadenaCA91125 United States Department of Mathematics National University of Singapore Singapore119076 Singapore

The Keller-Segel equation, a classical chemotaxis model, and many of its variants have been extensively studied for decades. In this work, we focus on 3D Keller-Segel equation with a quadratic logistic damping term −µρ2 (modeling density-dependent mortality rate) and show the existence of finite-time blowup solutions with nonnegative density and finite mass for any µ ∈ [0, 1 / 3 ). This range of µ is sharp;for µ ≥ 1/3 , the logistic damping effect suppresses the blowup as shown in [40, 60]. A key ingredient is to construct a self-similar blowup solution to a related aggregation equation as an approximate solution, with subcritical scaling relative to the original model. Based on this construction, we employ a robust weighted L2 method to prove the stability of this approximate solution, where modulation ODEs are introduced to enforce local vanishing conditions for the perturbation lying in a singular-weighted L2 space. As a byproduct, we exhibit a new family of type I blowup mechanisms for the classical 3D Keller-Segel equation. Copyright © 2025, The Authors. All rights reserved.

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Riemannian Proximal Sampler for High-accuracy Sampling on Manifolds

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arXiv 2025年

作者： Guan, Yunrui Balasubramanian, Krishnakumar Ma, Shiqian Department of Computational Applied Mathematics and Operations Research Rice University United States Department of Statistics University of California Davis United States

We introduce the Riemannian Proximal Sampler, a method for sampling from densities defined on Riemannian manifolds. The performance of this sampler critically depends on two key oracles: the Manifold Brownian Increments (MBI) oracle and the Riemannian Heat-kernel (RHK) oracle. We establish high-accuracy sampling guarantees for the Riemannian Proximal Sampler, showing that generating samples with Ε-accuracy requires O(log(1/Ε)) iterations in Kullback-Leibler divergence assuming access to exact oracles and O(log2(1/Ε)) iterations in the total variation metric assuming access to sufficiently accurate inexact oracles. Furthermore, we present practical implementations of these oracles by leveraging heat-kernel truncation and Varadhan’s asymptotics. In the latter case, we interpret the Riemannian Proximal Sampler as a discretization of the entropy-regularized Riemannian Proximal Point Method on the associated Wasserstein space. We provide preliminary numerical results that illustrate the effectiveness of the proposed methodology. © 2025, CC BY.

关键词： Numerical methods