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1. Mirrorless Mirror Descent: A Natural Derivation of Mirror Descent

   Gunasekar, Suriya; Woodworth, Blake; Srebro, Nathan (January 2021, Proceedings of Machine Learning Research)

   null (Ed.)

   We present a direct (primal only) derivation of Mirror Descent as a “partial” discretization of gradient flow on a Riemannian manifold where the metric tensor is the Hessian of the Mirror Descent potential function. We contrast this discretization to Natural Gradient Descent, which is obtained by a “full” forward Euler discretization. This view helps shed light on the relationship between the methods and allows generalizing Mirror Descent to any Riemannian geometry in Rd, even when the metric tensor is not a Hessian, and thus there is no “dual.” 

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2. A Riemannian Smoothing Steepest Descent Method for Non-Lipschitz Optimization on Embedded Submanifolds of Rn

   https://doi.org/10.1287/moor.2022.0286  

   Zhang, Chao; Chen, Xiaojun; Ma, Shiqian (August 2023, Mathematics of Operations Research)

   In this paper, we study the generalized subdifferentials and the Riemannian gradient subconsistency that are the basis for non-Lipschitz optimization on embedded submanifolds of [Formula: see text]. We then propose a Riemannian smoothing steepest descent method for non-Lipschitz optimization on complete embedded submanifolds of [Formula: see text]. We prove that any accumulation point of the sequence generated by the Riemannian smoothing steepest descent method is a stationary point associated with the smoothing function employed in the method, which is necessary for the local optimality of the original non-Lipschitz problem. We also prove that any accumulation point of the sequence generated by our method that satisfies the Riemannian gradient subconsistency is a limiting stationary point of the original non-Lipschitz problem. Numerical experiments are conducted to demonstrate the advantages of Riemannian [Formula: see text] [Formula: see text] optimization over Riemannian [Formula: see text] optimization for finding sparse solutions and the effectiveness of the proposed method. Funding: C. Zhang was supported in part by the National Natural Science Foundation of China [Grant 12171027] and the Natural Science Foundation of Beijing [Grant 1202021]. X. Chen was supported in part by the Hong Kong Research Council [Grant PolyU15300219]. S. Ma was supported in part by the National Science Foundation [Grants DMS-2243650 and CCF-2308597], the UC Davis Center for Data Science and Artificial Intelligence Research Innovative Data Science Seed Funding Program, and a startup fund from Rice University. 

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3. Output-feedback Synthesis Orbit Geometry: Quotient Manifolds and LQG Direct Policy Optimization

   Kraisler; Spencer; Mesbahi, Mehran (April 2024, arxiv)

   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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4. Curvature-Independent Last-Iterate Convergence for Games on Riemannian Manifolds

   Cai, Y; Jordan, M I; Lin, T; Oikonomou, A; Vlatakis-Gkaragkounis, E (July 2023, arXiv)

   Numerous applications in machine learning and data analytics can be formulated as equilibrium computation over Riemannian manifolds. Despite the extensive investigation of their Euclidean counterparts, the performance of Riemannian gradient-based algorithms remain opaque and poorly understood. We revisit the original scheme of Riemannian gradient descent (RGD) and analyze it under a geodesic monotonicity assumption, which includes the well-studied geodesically convex-concave min-max optimization problem as a special case. Our main contribution is to show that, despite the phenomenon of distance distortion, the RGD scheme, with a step size that is agnostic to the manifold's curvature, achieves a curvature-independent and linear last-iterate convergence rate in the geodesically strongly monotone setting. To the best of our knowledge, the possibility of curvature-independent rates and/or last-iterate convergence in the Riemannian setting has not been considered before. 

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5. Curvature-Independent Last-Iterate Convergence for Games on RiemannianManifolds

   Yai Cai; Michael I. Jordan; Tianyi Lin; Argyris Oikonomou; Emmanouil V. Vlatakis-Gkaragkounis (June 2023, arXivorg)

   Numerous applications in machine learning and data analytics can be formulated as equilibrium computation over Riemannian manifolds. Despite the extensive investigation of their Euclidean counterparts, the performance of Riemannian gradient-based algorithms remain opaque and poorly understood. We revisit the original scheme of Riemannian gradient descent (RGD) and analyze it under a geodesic monotonicity assumption, which includes the well-studied geodesically convex-concave min-max optimization problem as a special case. Our main contribution is to show that, despite the phenomenon of distance distortion, the RGD scheme, with a step size that is agnostic to the manifold’s curvature, achieves a curvature-independent and linear last-iterate convergence rate in the geodesically strongly monotone setting. To the best of our knowledge, the possibility of curvature-independent rates and/or last-iterate convergence in the Riemannian setting has not been considered before. 

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