Occupation Measures in the ODE Method: Wasserstein Convergence and Rates
Abstract
We study occupation measures associated with stochastic approximation recursions within the ordinary differential equation (ODE) method. Under Lyapunov conditions ensuring that the limiting ODE has a globally exponentially stable equilibrium, we establish that the continuous-time occupation measures associated with the piecewise linear interpolation of the iterates converge almost surely in the $2$-Wasserstein topology to the Dirac mass at the equilibrium, the unique invariant probability measure of the limiting semiflow. The choice of the $2$-Wasserstein distance is naturally aligned with our continuous- and discrete-time Lyapunov drift analysis. We also consider two discrete-time sequences of occupation measures, induced by the algorithmic time scale and the iteration count, respectively, and we establish their almost sure convergence to the same invariant measure in the $2$-Wasserstein distance. We then derive corresponding convergence rates in expectation, providing a nonasymptotic complement to the almost sure results. These expectation bounds also imply some tail probability estimates. Finally, we verify that our results apply to a canonical class of linear recursions.
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BibTeX
@misc{katselis2026occupation,
title = {{Occupation Measures in the ODE Method: Wasserstein Convergence and Rates}},
author = {Dimitrios Katselis},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05921},
eprint = {2611.05921},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05921},
}
RIS
TY - UNPB
TI - Occupation Measures in the ODE Method: Wasserstein Convergence and Rates
AU - Dimitrios Katselis
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05921
UR - https://arxiv.org/abs/2611.05921
AB - We study occupation measures associated with stochastic approximation recursions within the ordinary differential equation (ODE) method. Under Lyapunov conditions ensuring that the limiting ODE has a globally exponentially stable equilibrium, we establish that the continuous-time occupation measures associated with the piecewise linear interpolation of the iterates converge almost surely in the $2$-Wasserstein topology to the Dirac mass at the equilibrium, the unique invariant probability measure of the limiting semiflow. The choice of the $2$-Wasserstein distance is naturally aligned with our continuous- and discrete-time Lyapunov drift analysis. We also consider two discrete-time sequences of occupation measures, induced by the algorithmic time scale and the iteration count, respectively, and we establish their almost sure convergence to the same invariant measure in the $2$-Wasserstein distance. We then derive corresponding convergence rates in expectation, providing a nonasymptotic complement to the almost sure results. These expectation bounds also imply some tail probability estimates. Finally, we verify that our results apply to a canonical class of linear recursions.
ER -