# Exponential Convergence and stability of Howards's Policy Improvement Algorithm for Controlled Diffusions

## TL;DR

This paper proves exponential convergence rates and stability for Howard's policy improvement algorithm applied to controlled diffusions, using backward stochastic differential equations to analyze the algorithm's robustness.

## Contribution

It establishes the first global convergence rate and stability results for the continuous-time policy improvement algorithm in controlled diffusions.

## Key findings

- Proves exponential convergence rate of the policy improvement algorithm.
- Shows stability under perturbations in PDE solutions and maximization accuracy.
- Introduces a novel proof technique using backward stochastic differential equations.

## Abstract

Optimal control problems are inherently hard to solve as the optimization must be performed simultaneously with updating the underlying system. Starting from an initial guess, Howard's policy improvement algorithm separates the step of updating the trajectory of the dynamical system from the optimization and iterations of this should converge to the optimal control. In the discrete space-time setting this is often the case and even rates of convergence are known. In the continuous space-time setting of controlled diffusion the algorithm consists of solving a linear PDE followed by maximization problem. This has been shown to converge, in some situations, however no global rate of is known. The first main contribution of this paper is to establish global rate of convergence for the policy improvement algorithm and a variant, called here the gradient iteration algorithm. The second main contribution is the proof of stability of the algorithms under perturbations to both the accuracy of the linear PDE solution and the accuracy of the maximization step. The proof technique is new in this context as it uses the theory of backward stochastic differential equations.