# Convergence Analysis of Schr"odinger-F"ollmer Sampler without Convexity

arXiv: https://arxiv.org/abs/2107.04766 · HTML: https://mlsanity.com/paper/2107.04766/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 41 references that are in the map.
- First posted on arXiv 10 Jul 2021.

## Builds on (most cited first, 16 of 16)

- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017)
- [Theoretical guarantees for sampling and inference in generative models with latent diffusions](https://mlsanity.com/paper/1903.01608/) (2019)
- [Convergence of Langevin MCMC in KL-divergence](https://mlsanity.com/paper/1705.09048/) (2017)
- [Sharp convergence rates for Langevin dynamics in the nonconvex setting](https://mlsanity.com/paper/1805.01648/) (2018)
- [Sampling Can Be Faster Than Optimization](https://mlsanity.com/paper/1811.08413/) (2018)
- [Improved Bounds for Discretization of Langevin Diffusions: Near-Optimal Rates without Convexity](https://mlsanity.com/paper/1907.11331/) (2019)
- [Nonasymptotic estimates for Stochastic Gradient Langevin Dynamics under local conditions in nonconvex optimization](https://mlsanity.com/paper/1910.02008/) (2019)
- [Coupling and Convergence for Hamiltonian Monte Carlo](https://mlsanity.com/paper/1805.00452/) (2018)
- [The Bouncy Particle Sampler: A Non-Reversible Rejection-Free Markov Chain Monte Carlo Method](https://mlsanity.com/paper/1510.02451/) (2015)
- [On stochastic gradient Langevin dynamics with dependent data streams in the logconcave case](https://mlsanity.com/paper/1812.02709/) (2018)
- [On stochastic gradient Langevin dynamics with dependent data streams: the fully non-convex case](https://mlsanity.com/paper/1905.13142/) (2019)

## Built on by (most cited first, 7 of 7)

- [Probabilistic Forecasting with Stochastic Interpolants and F"ollmer Processes](https://mlsanity.com/paper/2403.13724/) (2024)
- [Latent Schr"odinger Bridge Diffusion Model for Generative Learning](https://mlsanity.com/paper/2404.13309/) (2024)
- [A Practical Diffusion Path for Sampling](https://mlsanity.com/paper/2406.14040/) (2024)
- [Sampling from multi-modal distributions with polynomial query complexity in fixed dimension via reverse diffusion](https://mlsanity.com/paper/2501.00565/) (2024)
- [An Efficient On-Policy Deep Learning Framework for Stochastic Optimal Control](https://mlsanity.com/paper/2410.05163/) (2024)
- [Variational Optimality of F"ollmer Processes in Generative Diffusions](https://mlsanity.com/paper/2602.10989/) (2026)
- [Gradient-Free Score-Based Sampling Methods with Ensembles](https://mlsanity.com/paper/2401.17539/) (2024)

## Abstract (arXiv)

Schr"odinger-F"ollmer sampler (SFS) is a novel and efficient approach for sampling from possibly unnormalized distributions without ergodicity. SFS is based on the Euler-Maruyama discretization of Schr"odinger-F"ollmer diffusion process d Xt=-nabla Uleft(Xt, tright) d t+d Bt, quad t ∈[0,1],quad X0=0 on the unit interval, which transports the degenerate distribution at time zero to the target distribution at time one. In citesfs21, the consistency of SFS is established under a restricted assumption that %the drift term b(x,t) the potential U(x,t) is uniformly (on t) strongly %concave convex (on x). In this paper we provide a nonasymptotic error bound of SFS in Wasserstein distance under some smooth and bounded conditions on the density ratio of the target distribution over the standard normal distribution, but without requiring the strongly convexity of the potential.