mlsanityData to 7 Oct 2026

Heavy-tailed Sampling via Transformed Unadjusted Langevin Algorithm

arXiv 2201.08349 ↗First posted 20 Jan 2022math.ST stat.CO stat.MLcited by 6

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Abstract (arXiv)

We analyze the oracle complexity of sampling from polynomially decaying heavy-tailed target densities based on running the Unadjusted Langevin Algorithm on certain transformed versions of the target density. The specific class of closed-form transformation maps that we construct are shown to be diffeomorphisms, and are particularly suited for developing efficient diffusion-based samplers. We characterize the precise class of heavy-tailed densities for which polynomial-order oracle complexities (in dimension and inverse target accuracy) could be obtained, and provide illustrative examples. We highlight the relationship between our assumptions and functional inequalities (super and weak Poincar'e inequalities) based on non-local Dirichlet forms defined via fractional Laplacian operators, used to characterize the heavy-tailed equilibrium densities of certain stable-driven stochastic differential equations.

Key facts. Cited by 6 papers in the map (data through 7 Oct 2026). Builds on 30 of its 58 references that are in the map. First posted on arXiv 20 Jan 2022.

Builds on

30 of 58 references.

References · 30

  1. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,315
  2. 2019Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Sufficescited by 224
  3. 2014Theoretical guarantees for approximate sampling from smooth and log-concave densitiescited by 206
  4. 2015Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithmcited by 195
  5. 2018Analysis of Langevin Monte Carlo via convex optimizationcited by 126
  6. 2018Log-concave sampling: Metropolis-Hastings algorithms are fastcited by 117
  7. 2017User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradientcited by 116
  8. 2020Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximatorscited by 85
  9. 2019The Randomized Midpoint Method for Log-Concave Samplingcited by 77
  10. 2018Sampling Can Be Faster Than Optimizationcited by 68
  11. 2018Sharp convergence rates for Langevin dynamics in the nonconvex settingcited by 67
  12. 2020Optimal dimension dependence of the Metropolis-Adjusted Langevin Algorithmcited by 57
18 more
  1. 2020On the Convergence of Langevin Monte Carlo: The Interplay between Tail Growth and Smoothnesscited by 55
  2. 2019Fast mixing of Metropolized Hamiltonian Monte Carlo: Benefits of multi-step gradientscited by 51
  3. 2018Global Non-convex Optimization with Discretized Diffusionscited by 44
  4. 2021Minimax Mixing Time of the Metropolis-Adjusted Langevin Algorithm for Log-Concave Samplingcited by 44
  5. 2017Inference via low-dimensional couplingscited by 43
  6. 2018Non-asymptotic bounds for sampling algorithms without log-concavitycited by 35
  7. 2019Stochastic Runge-Kutta Accelerates Langevin Monte Carlo and Beyondcited by 33
  8. 2020Fractional Underdamped Langevin Dynamics: Retargeting SGD with Momentum under Heavy-Tailed Gradient Noisecited by 33
  9. 2021The Langevin Monte Carlo algorithm in the non-smooth log-concave casecited by 26
  10. 2020Representational aspects of depth and conditioning in normalizing flowscited by 25
  11. 2020On the Ergodicity, Bias and Asymptotic Normality of Randomized Midpoint Sampling Methodcited by 23
  12. 2020Logsmooth Gradient Concentration and Tighter Runtimes for Metropolized Hamiltonian Monte Carlocited by 22
  13. 2019Bounding the error of discretized Langevin algorithms for non-strongly log-concave targetscited by 16
  14. 2017Fractional Langevin Monte Carlo: Exploring L'evy Driven Stochastic Differential Equations for Markov Chain Monte Carlocited by 12
  15. 2020Deep composition of tensor-trains using squared inverse Rosenblatt transportscited by 12
  16. 2019Non-Asymptotic Analysis of Fractional Langevin Monte Carlo for Non-Convex Optimizationcited by 8
  17. 2014Efficient strategy for the Markov chain Monte Carlo in high-dimension with heavy-tailed target probability distributioncited by 5
  18. 2021Unadjusted Langevin algorithm for sampling a mixture of weakly smooth potentialscited by 3

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Cited by year

Citing papers per year: 2022: 2; 2023: 3; 2024: 0; 2025: 1.

6 papers.

  1. 2022Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlocited by 63
  2. 2023Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincar'e Inequalitycited by 31
  3. 2022Fisher information lower bounds for samplingcited by 12
  4. 2023Mean-Square Analysis of Discretized Ito Diffusions for Heavy-tailed Samplingcited by 4
  5. 2025Anchored Langevin Algorithmscited by 2
  6. 2023Transport map unadjusted Langevin algorithms: learning and discretizing perturbed samplerscited by 1