Convergence of Policy Gradient for Entropy Regularized MDPs with Neural Network Approximation in the Mean-Field Regime
James-Michael Leahy, Bekzhan Kerimkulov, David Siska, Lukasz Szpruch
Abstract
We study the global convergence of policy gradient for infinite-horizon, continuous state and action space, and entropy-regularized Markov decision processes (MDPs). We consider a softmax policy with (one-hidden layer) neural network approximation in a mean-field regime. Additional entropic regularization in the associated mean-field probability measure is added, and the corresponding gradient flow is studied in the 2-Wasserstein metric. We show that the objective function is increasing along the gradient flow. Further, we prove that if the regularization in terms of the mean-field measure is sufficient, the gradient flow converges exponentially fast to the unique stationary solution, which is the unique maximizer of the regularized MDP objective. Lastly, we study the sensitivity of the value function along the gradient flow with respect to regularization parameters and the initial condition. Our results rely on the careful analysis of the non-linear Fokker–Planck–Kolmogorov equation and extend the pioneering work of \cite{mei2020global} and \cite{agarwal2020optimality}, which quantify the global convergence rate of policy gradient for entropy-regularized MDPs in the tabular setting.
BibTeX
@InProceedings{pmlr-v162-leahy22a,
title = {Convergence of Policy Gradient for Entropy Regularized {MDP}s with Neural Network Approximation in the Mean-Field Regime},
author = {Leahy, James-Michael and Kerimkulov, Bekzhan and Siska, David and Szpruch, Lukasz},
booktitle = {Proceedings of the 39th International Conference on Machine Learning},
pages = {12222--12252},
year = {2022},
editor = {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan},
volume = {162},
series = {Proceedings of Machine Learning Research},
month = {17--23 Jul},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v162/leahy22a/leahy22a.pdf},
url = {https://proceedings.mlr.press/v162/leahy22a.html},
abstract = {We study the global convergence of policy gradient for infinite-horizon, continuous state and action space, and entropy-regularized Markov decision processes (MDPs). We consider a softmax policy with (one-hidden layer) neural network approximation in a mean-field regime. Additional entropic regularization in the associated mean-field probability measure is added, and the corresponding gradient flow is studied in the 2-Wasserstein metric. We show that the objective function is increasing along the gradient flow. Further, we prove that if the regularization in terms of the mean-field measure is sufficient, the gradient flow converges exponentially fast to the unique stationary solution, which is the unique maximizer of the regularized MDP objective. Lastly, we study the sensitivity of the value function along the gradient flow with respect to regularization parameters and the initial condition. Our results rely on the careful analysis of the non-linear Fokker–Planck–Kolmogorov equation and extend the pioneering work of \cite{mei2020global} and \cite{agarwal2020optimality}, which quantify the global convergence rate of policy gradient for entropy-regularized MDPs in the tabular setting.}
}