Two-Loop Stochastic Mirror Langevin Algorithms for Constrained Sampling
Abstract
We study the problem of sampling from a target distribution π(x)∝ e−f(x) supported on a convex set $ X\subseteq\mathbb R^d$, when the potential f is accessible only through a stochastic first-order oracle. Mirror Langevin algorithms provide a natural approach to constrained sampling by transporting the problem to an unconstrained dual space and discretizing the resulting Mirror Langevin diffusion. Existing implementations, however, typically use a fixed discretization step size and consequently retain a nonvanishing discretization bias at any fixed step size. Moreover, their direct extension to settings with noisy gradient information entails the challenge of controlling both discretization and stochastic-oracle error. We study a stochastic first-order version of the Mirror Langevin Algorithm (sFO-MLA) and, as our main contribution, develop a warm-started two-loop implementation in which an outer loop progressively decreases the step size while an inner loop runs sFO-MLA (with a fixed step size) for an appropriately chosen epoch length. The construction provides a principled schedule linking step sizes and epoch lengths, so that successive epochs warm-start from increasingly accurate distributions rather than repeatedly paying the cost of mixing from a cold start. We establish finite-time Wasserstein guarantees for sFO-MLA that explicitly separate mixing, Euler--Maruyama discretization, and stochastic-gradient errors. These bounds yield a fixed horizon rate of Õ(T−1/2) and show that the two-loop scheme removes the associated logarithmic penalty, attaining the canonical O(T−1/2) rate under a geometric step-size schedule and corresponding epoch lengths. We illustrate the methodology in two statistically distinct settings.
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BibTeX
@misc{tong2026twoloop,
title = {{Two-Loop Stochastic Mirror Langevin Algorithms for Constrained Sampling}},
author = {Ruiting Tong and Antik Chakraborty and Raghu Pasupathy},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09402},
eprint = {2610.09402},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09402},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Two-Loop Stochastic Mirror Langevin Algorithms for Constrained Sampling
AU - Ruiting Tong
AU - Antik Chakraborty
AU - Raghu Pasupathy
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09402
UR - https://arxiv.org/abs/2610.09402
AB - We study the problem of sampling from a target distribution π(x)∝ e^−f(x) supported on a convex set $ X\subseteq\mathbb R^d$, when the potential f is accessible only through a stochastic first-order oracle. Mirror Langevin algorithms provide a natural approach to constrained sampling by transporting the problem to an unconstrained dual space and discretizing the resulting Mirror Langevin diffusion. Existing implementations, however, typically use a fixed discretization step size and consequently retain a nonvanishing discretization bias at any fixed step size. Moreover, their direct extension to settings with noisy gradient information entails the challenge of controlling both discretization and stochastic-oracle error. We study a stochastic first-order version of the Mirror Langevin Algorithm (sFO-MLA) and, as our main contribution, develop a warm-started two-loop implementation in which an outer loop progressively decreases the step size while an inner loop runs sFO-MLA (with a fixed step size) for an appropriately chosen epoch length. The construction provides a principled schedule linking step sizes and epoch lengths, so that successive epochs warm-start from increasingly accurate distributions rather than repeatedly paying the cost of mixing from a cold start. We establish finite-time Wasserstein guarantees for sFO-MLA that explicitly separate mixing, Euler--Maruyama discretization, and stochastic-gradient errors. These bounds yield a fixed horizon rate of Õ(T^−1/2) and show that the two-loop scheme removes the associated logarithmic penalty, attaining the canonical O(T^−1/2) rate under a geometric step-size schedule and corresponding epoch lengths. We illustrate the methodology in two statistically distinct settings.
ER -