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The Mirror Langevin Algorithm Converges with Vanishing Bias [article]

Ruilin Li and Molei Tao and Santosh S. Vempala and Andre Wibisono
2021 arXiv   pre-print
The Mirror Langevin Diffusion (MLD) is a sampling analogue of mirror flow in continuous time, and it has nice convergence properties under log-Sobolev or Poincare inequalities relative to the Hessian metric  ...  Here we study the basic Mirror Langevin Algorithm and show it indeed has a vanishing bias.  ...  Wasserstein control of Mirror Langevin Monte Carlo.  ... 
arXiv:2109.12077v2 fatcat:s376fgfoize3pa5mcfn2nsgrdi

Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices [article]

Santosh S. Vempala, Andre Wibisono
2022 arXiv   pre-print
We study the Unadjusted Langevin Algorithm (ULA) for sampling from a probability distribution ν = e^-f on ℝ^n.  ...  We also prove a bound on the bias of the limiting distribution of ULA assuming third-order smoothness of f, without requiring isoperimetry.  ...  The authors thank Kunal Talwar for explaining the privacy motivation and application of Rényi divergence to data privacy; Yu Cao, Jianfeng Lu, and Yulong Lu for alerting us to their work [14] on Rényi  ... 
arXiv:1903.08568v4 fatcat:v2v5fnz4jvfufpywyut5zf2aom

Fast Convergence of Langevin Dynamics on Manifold: Geodesics meet Log-Sobolev [article]

Xiao Wang, Qi Lei, Ioannis Panageas
2020 arXiv   pre-print
One approach to sample from a high dimensional distribution e^-f for some function f is the Langevin Algorithm (LA).  ...  Recently, there has been a lot of progress in showing fast convergence of LA even in cases where f is non-convex, notably [53], [39] in which the former paper focuses on functions f defined in ℝ^n and  ...  Monte Carlo by [15, 57, 40] .  ... 
arXiv:2010.05263v2 fatcat:nu2cmgr3wfa3dddeqyita6npma

The query complexity of sampling from strongly log-concave distributions in one dimension [article]

Sinho Chewi, Patrik Gerber, Chen Lu, Thibaut Le Gouic, Philippe Rigollet
2021 arXiv   pre-print
We establish the first tight lower bound of Ω(loglogκ) on the query complexity of sampling from the class of strongly log-concave and log-smooth distributions with condition number κ in one dimension.  ...  ACKNOWLEDGMENTS Sinho Chewi was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program.  ...  Proximal Langevin algorithm: rapid convergence under isoperimetry. arXiv e- prints, art. arXiv:1911.01469, 2019. .  ... 
arXiv:2105.14163v2 fatcat:py6j5e2aynaitis6uv63u35prq

Composite Logconcave Sampling with a Restricted Gaussian Oracle [article]

Ruoqi Shen, Kevin Tian, Yin Tat Lee
2020 arXiv   pre-print
We conduct experiments showing our algorithm vastly improves upon the hit-and-run algorithm for sampling the restriction of a (non-diagonal) Gaussian to the positive orthant.  ...  , through the abstraction of a restricted Gaussian oracle.  ...  Acknowledgments We thank Yair Carmon for suggesting the experiment in Section 4.  ... 
arXiv:2006.05976v1 fatcat:3wqsaa627vgjxnnc5m7pgmto7i

Penalized Langevin dynamics with vanishing penalty for smooth and log-concave targets [article]

Avetik Karagulyan, Arnak S. Dalalyan
2020 arXiv   pre-print
We consider a continuous-time diffusion-type process, termed Penalized Langevin dynamics (PLD), the drift of which is the negative gradient of the potential plus a linear penalty that vanishes when time  ...  This upper bound highlights the influence of the speed of decay of the penalty on the accuracy of the approximation.  ...  Gradient based MCMC methods such as the Langevin MC, the underdamped Langevin Monte Carlo, the Hamiltonian Monte Carlo and their Metropolis adjusted counterparts were shown to have attractive features  ... 
arXiv:2006.13998v1 fatcat:ffkgkl4pfbed3larncmr5sx2jq

Structured Logconcave Sampling with a Restricted Gaussian Oracle [article]

Yin Tat Lee, Ruoqi Shen, Kevin Tian
2021 arXiv   pre-print
we also show a zeroth-order algorithm attains the same query complexity.  ...  For composite densities exp(-f(x) - g(x)), where f has condition number κ and convex (but possibly non-smooth) g admits an RGO, we obtain a mixing time of O(κ d log^3κ d/ϵ), matching the state-of-the-art  ...  RS and KT would like to thank Sinho Chewi for his extremely generous help, in particular insightful conversations which led to our discovery of the gap in Section 3, as well as his suggested fix.  ... 
arXiv:2010.03106v4 fatcat:qgtfuq5jxjbdrjtb2iez4cmu34

Convex Analysis of the Mean Field Langevin Dynamics [article]

Atsushi Nitanda, Denny Wu, Taiji Suzuki
As an example of the nonlinear Fokker-Planck equation, the mean field Langevin dynamics attracts attention due to its connection to (noisy) gradient descent on infinitely wide neural networks in the mean  ...  In this work, we give a simple and self-contained convergence rate analysis of the mean field Langevin dynamics with respect to the (regularized) objective function in both continuous and discrete time  ...  In addition, Bou-Rabee and Schuh (2020); Bou-Rabee and Eberle (2021) studied the dis-crete time convergence of Hamiltonian Monte Carlo with interaction potential.  ... 
doi:10.48550/arxiv.2201.10469 fatcat:myhxm6le7zfq5pl4dqcu4egsgy