A proximal gradient algorithm for composite log-concave sampling

Abstract

We propose an algorithm to sample from composite log-concave distributions over ℝ^d, i.e., densities of the form π∝ e^−f−g, assuming access to gradient evaluations of α_f-convex f and a restricted Gaussian oracle (RGO) for α_g-convex g, where α_g ≥ 0. The latter requirement means that we can easily sample from the density RGO_g,h,y(x) ∝ exp(−g(x) −1/2h||y−x||²), which is the sampling analogue of the proximal operator for g. If f + g is α-convex and f is β-smooth, our sampler achieves ε error in total variation distance using 𝒪̃(κ√d log⁴(1/ε)) oracle calls where κ:= 1 ∨ β/α, with 0≤ α_g ≤ β and α:= α_f + α_g > 0, which matches prior state-of-the-art results for the case g=0. We further extend our results to cases where (1) π is non-log-concave but satisfies a Poincaré or log-Sobolev inequality, and (2) f is non-smooth but Lipschitz.

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BibTeX RIS

BibTeX

@misc{liu2026proximal,
  title = {{A proximal gradient algorithm for composite log-concave sampling}},
  author = {Linghai Liu and Sinho Chewi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.12461},
  eprint = {2605.12461},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.12461},
}

RIS

TY  - UNPB
TI  - A proximal gradient algorithm for composite log-concave sampling
AU  - Linghai Liu
AU  - Sinho Chewi
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2605.12461
UR  - https://arxiv.org/abs/2605.12461
AB  - We propose an algorithm to sample from composite log-concave distributions over ℝ^d, i.e., densities of the form π∝ e^−f−g, assuming access to gradient evaluations of α_f-convex f and a restricted Gaussian oracle (RGO) for α_g-convex g, where α_g ≥ 0. The latter requirement means that we can easily sample from the density RGO_g,h,y(x) ∝ exp(−g(x) −1/2h||y−x||^2), which is the sampling analogue of the proximal operator for g. If f + g is α-convex and f is β-smooth, our sampler achieves ε error in total variation distance using 𝒪̃(κ√d log^4(1/ε)) oracle calls where κ:= 1 ∨ β/α, with 0≤ α_g ≤ β and α:= α_f + α_g > 0, which matches prior state-of-the-art results for the case g=0. We further extend our results to cases where (1) π is non-log-concave but satisfies a Poincaré or log-Sobolev inequality, and (2) f is non-smooth but Lipschitz.
ER  -

Abstract

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