Model-Free Output Feedback Stabilization via Policy Gradient Methods

Motivated by the challenge of stabilizing a general unknown linear dynamical system (LDS) from observations, we study the natural prerequisite of online prediction. Our goal is to achieve sublinear regret with a memory footprint that adapts to the intrinsic complexity of the dynamics rather than the full hidden-state dimension. We focus on the practically central regime of systems with low instability complexity -- eigenvalues outside the real stable interval that do not decay rapidly, together with non-semisimple modes -- potentially embedded in an otherwise stable real spectrum of much higher dimension; we write $k$ for this count. This regime is the primary setting in which stabilization is plausible: we show that many systems with high instability complexity cannot be stabilized without exponentially large controls. Thus, prediction is meaningful for stabilization precisely when the instability complexity is small. Within this regime, we introduce a unified online algorithm that handles every LDS (including non-diagonalizable systems with complex or exploding modes) with a learnable parameter count of $\widetilde{O}(k)$. Finally, we prove a lower bound showing that $k$ is a valid complexity measure: any filter-based predictor needs at least $k$ filters. Experiments corroborate our theory: on a high-dimensional system, our predictor sharply outperforms prior methods at an equal parameter budget.

Reward shaping is fundamental to modern robotic control with deep reinforcement learning (RL), yet practitioners still rely heavily on heuristic principles borrowed from classical optimal control and trajectory optimization. Existing methods rarely distinguish reward terms that are intrinsic to the control objective from numerical regularizers, leading to brittle hyperparameter tuning. To determine which quantities a reward must contain, we study the stabilization control problem with a focus on zeroth-order (configuration) and first-order (velocity) information. We theoretically and empirically demonstrate that policy gradient methods can successfully solve stabilization tasks without first-order reward terms, adding such terms can instead introduce severe sensitivity as their scale grows. Conversely, our findings confirm that reward functions must be zeroth-order complete over goal-relevant coordinates, while the first-order state remains necessary in the policy observation under our low-dissipation assumptions. Overall, these results provide actionable and principled guidance for reward design in robotic RL.

Inspired by REINFORCE, we introduce a novel receding-horizon algorithm for the Linear Quadratic Regulator (LQR) problem with unknown dynamics. Unlike prior methods, our algorithm avoids reliance on two-point gradient estimates while maintaining the same order of sample complexity. Furthermore, it eliminates the restrictive requirement of starting with a stable initial policy, broadening its applicability. Beyond these improvements, we introduce a refined analysis of error propagation through the contraction of the Riccati operator under the Riemannian distance. This refinement leads to a better sample complexity and ensures improved convergence guarantees.