Provable Convergence and Limitations of Geometric Tempering for Langevin Dynamics
Authors: Omar Chehab, Anna Korba, Austin Stromme, Adrien Vacher
ICLR 2025 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Theoretical | In this paper, we theoretically investigate the soundness of this approach when the sampling algorithm is Langevin dynamics, proving both upper and lower bounds. Our upper bounds are the first analysis in the literature under functional inequalities. They assert the convergence of tempered Langevin in continuous and discrete-time, and their minimization leads to closed-form optimal tempering schedules for some pairs of proposal and target distributions. Our lower bounds demonstrate a simple case where the geometric tempering takes exponential time, and further reveal that the geometric tempering can suffer from poor functional inequalities and slow convergence, even when the target distribution is well-conditioned. |
| Researcher Affiliation | Academia | Omar Chehab, Anna Korba, Austin Stromme & Adrien Vacher Department of Statistics CREST, ENSAE, IP Paris France EMAIL |
| Pseudocode | No | The paper describes algorithms (Unadjusted Langevin Algorithm, Tempered Langevin Dynamics) using mathematical equations (e.g., Eq. 3, Eq. 5, Eq. 9, Eq. 11) but does not present them in structured pseudocode or algorithm blocks. |
| Open Source Code | No | The paper does not provide concrete access to source code for the methodology described in this paper. It mentions "Black JAX: A sampling library for JAX" (Cabezas et al., 2023) in the related work section as a tool that uses geometric tempering, but this is a third-party resource, not the authors' own code release. |
| Open Datasets | No | The paper uses theoretical probability distributions such as Gaussian (e.g., N(0,1), N(m,1)) and mixtures of Gaussians for its theoretical analysis and numerical illustrations (e.g., Figure 4: "Geometric path from a Gaussian to a Gaussian mixture"). These are mathematical constructs for modeling and analysis, not publicly available datasets in the traditional sense for empirical studies. |
| Dataset Splits | No | The paper focuses on theoretical analysis and numerical validations of theoretical bounds using synthetic distributions. Therefore, there are no real-world datasets used, and consequently, no mention of training/test/validation splits. |
| Hardware Specification | No | The paper mentions numerical validations and simulations (e.g., in the caption for Figure 3: "full lines are from simulations of the process using 10 000 particles"), but it does not specify any hardware details such as GPU models, CPU types, or cloud computing resources used for these simulations. |
| Software Dependencies | No | The paper references "Black JAX: A sampling library for JAX" (Cabezas et al., 2023) as an example of a sampling library in the related work, but it does not specify any software dependencies (e.g., programming languages, libraries, or frameworks with version numbers) that were used to implement or validate the authors' own work. |
| Experiment Setup | No | The paper presents theoretical findings and numerical illustrations. While Figure 3 mentions "simulations of the process using 10 000 particles" and specifies properties of the simulated distributions (e.g., "two-dimensional Gaussians, with zero mean and covariance matrices that have a constant diagonal, equal to one for the proposal and 10 for the target"), this is a description of the simulated process rather than a detailed experimental setup for a machine learning model, including hyperparameters, optimizer settings, or training configurations. |