Penalized Langevin dynamics with vanishing penalty for smooth and log-concave targets
Authors: Avetik Karagulyan, Arnak Dalalyan
NeurIPS 2020 | Conference PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Theoretical | We study the problem of sampling from a probability distribution on Rp defined via a convex and smooth potential function. We first 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 goes to infinity. An upper bound on the Wasserstein-2 distance between the distribution of the PLD at time t and the target is established. This upper bound highlights the influence of the speed of decay of the penalty on the accuracy of approximation. As a consequence, considering the low-temperature limit we infer a new nonasymptotic guarantee of convergence of the penalized gradient flow for the optimization problem. ... The main result of this work is an upper bound on W2(νPLD t , π) that is valid for every continuously differentiable and decreasing penalty factor α. Optimizing over α, we show that the choice α(t) 1/(2t), when t , leads to a simple upper bound of the order 1/√t. ... This paper is purely theoretical, thus we expect no direct or indirect ethical risks from it. |
| Researcher Affiliation | Academia | Avetik Karagulyan CREST, ENSAE, IP Paris avetik.karagulyan@ensae.fr Arnak S. Dalalyan CREST, ENSAE, IP Paris arnak.dalalyan@ensae.fr |
| Pseudocode | No | The paper does not contain any pseudocode or algorithm blocks. The processes are described using stochastic differential equations. |
| Open Source Code | No | The paper does not provide any information about open-source code for the described methodology. |
| Open Datasets | No | The paper is theoretical and does not use or refer to any specific publicly available datasets for training or evaluation. It discusses probability distributions in general terms. |
| Dataset Splits | No | The paper is theoretical and does not involve empirical experiments with dataset splits. Therefore, it does not provide train/validation/test split information. |
| Hardware Specification | No | The paper does not mention any specific hardware used for computations or simulations. |
| Software Dependencies | No | The paper does not provide specific software names with version numbers that would be necessary to replicate any computational aspects. |
| Experiment Setup | No | The paper is theoretical and does not include details on an experimental setup, such as hyperparameters or system-level training settings. The figures presented are illustrative of theoretical properties. |