The Fenchel Game of Underdamped Langevin Dynamics: Insights into Accelerated Convergence
Abstract
For (Qt,Pt) governed by suitably damped underdamped Langevin dynamics, we quantify the convergence in KL divergence of the positional marginal to a σ-strongly log-concave target π(dq) = 1/Ze−V(q)dq as KL(Law(Qt) | π) ≤ e−√σ tKL(Law(Q0, P0) | Π0), where Π0 denotes an appropriately selected reference measure. When π is merely log-concave, the estimate KL(Law(Qt) | π) ≤ τ2/t2KL(Law(Qτ, Pτ) | Πτ) is derived, where Πτ denotes another correspondingly chosen reference measure at time τ> 0. Both rates match precisely the canonical rates of the corresponding accelerated gradient flows in ℝd. They are achieved by the novel interpretation of the underdamped Langevin dynamics as a combination of strategies in an online sampling game and by estimating the KL divergence using a cost function informed by fictitious competitors.
Related
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- Initialization Stability and Convergence of Projected Langevin Dynamics
BibTeX
@misc{borkowski2026fenchel,
title = {{The Fenchel Game of Underdamped Langevin Dynamics: Insights into Accelerated Convergence}},
author = {Alexandra Borkowski and Nikolas Nüsken},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07812},
eprint = {2609.07812},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07812},
}
RIS
TY - UNPB
TI - The Fenchel Game of Underdamped Langevin Dynamics: Insights into Accelerated Convergence
AU - Alexandra Borkowski
AU - Nikolas Nüsken
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07812
UR - https://arxiv.org/abs/2609.07812
AB - For (Q_t,P_t) governed by suitably damped underdamped Langevin dynamics, we quantify the convergence in KL divergence of the positional marginal to a σ-strongly log-concave target π(dq) = 1/Ze^−V(q)dq as KL(Law(Q_t) | π) ≤ e^−√σ tKL(Law(Q_0, P_0) | Π_0), where Π_0 denotes an appropriately selected reference measure. When π is merely log-concave, the estimate KL(Law(Q_t) | π) ≤ τ^2/t^2KL(Law(Q_τ, P_τ) | Π_τ) is derived, where Π_τ denotes another correspondingly chosen reference measure at time τ> 0. Both rates match precisely the canonical rates of the corresponding accelerated gradient flows in ℝ^d. They are achieved by the novel interpretation of the underdamped Langevin dynamics as a combination of strategies in an online sampling game and by estimating the KL divergence using a cost function informed by fictitious competitors.
ER -