Policy Gradient Bounds in Multitask LQR

We analyze the performance of policy gradient in multitask linear quadratic regulation (LQR), where the system and cost parameters differ across tasks. The main goal of multitask LQR is to find a controller with satisfactory performance on every task. Prior analyses on relevant contexts fail to capture closed-loop task similarities, resulting in conservative performance guarantees. To account for such similarities, we propose bisimulation-based measures of task heterogeneity. Our measures employ new bisimulation functions to bound the cost gradient distance between a pair of tasks in closed loop with a common stabilizing controller. Employing these measures, we derive suboptimality bounds for both the multitask optimal controller and the asymptotic policy gradient controller with respect to each of the tasks. We further provide conditions under which the policy gradient iterates remain stabilizing for every system. For multiple random sets of certain tasks, we observe that our bisimulation-based measures improve upon baseline measures of task heterogeneity dramatically.