# Nesterov acceleration in optimizing over probability measures

arXiv: https://arxiv.org/abs/2607.23008 · HTML: https://mlsanity.com/paper/2607.23008/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 43 references that are in the map.
- First posted on arXiv 25 Jul 2026.

## Builds on (most cited first, 19 of 19)

- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints](https://mlsanity.com/paper/1408.3595/) (2014)
- [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022)
- [Understanding the Acceleration Phenomenon via High-Resolution Differential Equations](https://mlsanity.com/paper/1810.08907/) (2018)
- [Mean-Field Langevin Dynamics: Exponential Convergence and Annealing](https://mlsanity.com/paper/2202.01009/) (2022)
- [Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics](https://mlsanity.com/paper/2211.00450/) (2022)
- [Projected Wasserstein gradient descent for high-dimensional Bayesian inference](https://mlsanity.com/paper/2102.06350/) (2021)
- [Overparameterization of deep ResNet: zero loss and mean-field analysis](https://mlsanity.com/paper/2105.14417/) (2021)
- [Sampling via Gradient Flows in the Space of Probability Measures](https://mlsanity.com/paper/2310.03597/) (2023)
- [Accelerating optimization over the space of probability measures](https://mlsanity.com/paper/2310.04006/) (2023)
- [Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps](https://mlsanity.com/paper/2411.07154/) (2024)
- [Inverse Problems Over Probability Measure Space](https://mlsanity.com/paper/2504.18999/) (2025)
- [Mean Field Games Master Equations with Non-separable Hamiltonians and Displacement Monotonicity](https://mlsanity.com/paper/2101.12362/) (2021)
- [Building Population-Informed Priors for Bayesian Inference Using Data-Consistent Stochastic Inversion](https://mlsanity.com/paper/2407.13814/) (2024)

## Abstract (arXiv)

Optimization over probability measures has become an increasingly important paradigm in modern machine learning, scientific computing, and uncertainty quantification. Motivated by Nesterov's accelerated gradient method in Euclidean space, we develop Heavy-ball and Nesterov acceleration methods over the probability measure space P2 and establish non-asymptotic convergence guarantees that match their Euclidean counterparts. In particular, we derive convergence rates with respect to both the number of iterations and the number of particles used to represent the underlying probability distributions. Extending accelerated optimization from Euclidean space to probability measures is challenging. The natural notion of momentum requires concepts such as tangent bundles of the set of probability space and they are hard to operate numerically. To overcome these difficulties, we introduce two complementary lifting procedures. The first lifts probability measures to phase space through a Hamiltonian formulation, introducing momentum variables into the dynamics. The second lifts probability measures to a common Hilbert space, restoring the linear structure required for convergence analysis while simultaneously yielding executable particle dynamics. Together, these two complementary lifting procedures provide a systematic methodology for designing, analyzing, and implementing momentum-based accelerated optimization methods over probability measure spaces.