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Abstract—In this paper, we consider policy optimization over the Riemannian submanifolds of stabilizing controllers arising from constrained Linear Quadratic Regulators (LQR), including output feedback and structured synthesis. In this direction, we provide a Riemannian Newton-type algorithm that enjoys local convergence guarantees and exploits the inherent geometry of the problem. Instead of relying on the exponential mapping or a global retraction, the proposed algorithm revolves around the developed stability certificate and the constraint structure, utilizing the intrinsic geometry of the synthesis problem. We then showcase the utility of the proposed algorithm through numerical examples.
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Output-feedback Synthesis Orbit Geometry: Quotient Manifolds and LQG Direct Policy Optimization
In this paper, we consider direct policy optimization for the linear-quadratic Gaussian (LQG) setting. Over the past few years, it has been recognized that the landscape of stabilizing output-feedback controllers of relevance to LQG has an intricate geometry, particularly as it pertains to the existence of spurious stationary points. In order to address such challenges, in this paper, we first adopt a Riemannian metric for the space of stabilizing full-order minimal output-feedback controllers. We then proceed to prove that the orbit space of such controllers modulo coordinate transformation admits a Riemannian quotient manifold structure. This geometric structure is then used to develop a Riemannian gradient descent for the direct LQG policy optimization. We prove a local convergence guarantee with linear rate and show the proposed approach exhibits significantly faster and more robust numerical performance as compared with ordinary gradient descent for LQG. Subsequently, we provide reasons for this observed behavior; in particular, we argue that optimizing over the orbit space of controllers is the right theoretical and computational setup for direct LQG policy optimization.
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- Award ID(s):
- 2149470
- PAR ID:
- 10516485
- Publisher / Repository:
- arxiv
- Date Published:
- Journal Name:
- arxiv
- ISSN:
- 2331-8422
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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