On the lack of gradient domination for linear quadratic Gaussian problems with incomplete state information

Policy gradient algorithms in model-free reinforcement learning have been shown to achieve global exponential convergence for the Linear Quadratic Regulator problem despite the lack of convexity. However, extending such guarantees beyond the scope of standard LQR and full-state feedback has remained open. A key enabler for existing results on LQR is the so-called gradient dominance property of the underlying optimization problem that can be used as a surrogate for strong convexity. In this paper, we take a step further by studying the convergence of gradient descent for the Linear Quadratic Gaussian problem and demonstrate through examples that LQG does not satisfy the gradient dominance property. Our study shows the non-uniqueness of equilibrium points and thus disproves the global convergence of policy gradient methods for LQG.