Bregman Proximal Langevin Monte Carlo via Bregman-Moreau Envelopes
Authors: Tim Tsz-Kit Lau, Han Liu
ICML 2022 | Conference PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | We perform numerical experiments of sampling anisotropic Laplace distributions which have nonsmooth potentials. Other additional numerical experiments are given in Appendix D. In this section, we use bold lower case letters θ = (θi) 1 i d Rd to denote vectors. All numerical implementations can be found at https://github.com/ timlautk/bregman_prox_langevin_mc. ... The marginal empirical densities are given in Figure 1 (Figure D.4 for BMMMLA in Appendix B). |
| Researcher Affiliation | Academia | 1Department of Statistics and Data Science, Northwestern University, Evanston, IL, USA 2Department of Computer Science, Northwestern University, Evanston, IL, USA. |
| Pseudocode | Yes | Algorithm 1 The Bregman Moreau Mirrorless Mirror-Langevin Algorithm (BMMMLA) |
| Open Source Code | Yes | All numerical implementations can be found at https://github.com/ timlautk/bregman_prox_langevin_mc. |
| Open Datasets | Yes | For such a nonsmooth sampling task, inspired by Vorstrup Goldman et al. (2021); Bouchard-Cˆot e et al. (2018), we consider the case where f = 0 and g(θ) = α θ 1 = Pd i=1 αi|θi| with α = (1, 2, . . . , d) . ... We take d = 100, N = 1000 and θ = (0 10, 0.1 1 10, 0.2 1 10, . . . , 0.9 1 10) R100 as the ground truth. Then, each xn,i is generated from a standard Gaussian distribution and each yn is sampled following (D.5) with θ = θ . |
| Dataset Splits | No | The paper focuses on sampling algorithms and does not describe experiments with traditional training, validation, and test splits of a fixed dataset. Instead, it generates samples and evaluates their properties against a known distribution or estimated posteriors. |
| Hardware Specification | No | The paper does not specify any hardware details like CPU or GPU models used for the experiments. |
| Software Dependencies | No | The paper does not provide specific version numbers for any software dependencies used. |
| Experiment Setup | Yes | We consider d = 100, draw K = 105 samples, with a tight Bregman Moreau envelope using a small smoothing parameter λ = 10 5 and a small step size γ = λ/2. The parameter of the hyperbolic entropy is β = (2 d i + 1) 1 i d. ... In the experiment, we consider the case where ai = i and bi = i for all i Jd K. We use γ = 0.01, λ = 1 and β = (2 d i + 1) 1 i d. ... In addition, we choose α1 = (10 1 10, 9 1 10, . . . , 1 1 10) and α2 = 0.1. Again, we use the hypentropy functions ϕβ (for the mirror map) and ψσ (for the Bregman Moreau envelope), with β = (2i 1/4 1 10) 1 i 10 and σ = (α2 1,i) 1 i d. We also use a step size γ = 5 10 4 and a smoothing parameter λ = 0.01. |