Implicit Riemannian Optimism with Applications to Min-Max Problems

• URL: http://arxiv.org/abs/2501.18381v1
• Date: Thu, 30 Jan 2025 14:31:28 GMT
• Title: Implicit Riemannian Optimism with Applications to Min-Max Problems
• Authors: Christophe Roux, David Martínez-Rubio, Sebastian Pokutta,
• Abstract summary: We introduce an optimistic online learning algorithm for Hadamard problems. Our method can handle in-mani-fold constraints, and matches the best known bounds on the Euclidean setting.
• Score: 23.421903887404618
• License:
• Abstract: We introduce a Riemannian optimistic online learning algorithm for Hadamard manifolds based on inexact implicit updates. Unlike prior work, our method can handle in-manifold constraints, and matches the best known regret bounds in the Euclidean setting with no dependence on geometric constants, like the minimum curvature. Building on this, we develop algorithms for g-convex, g-concave smooth min-max problems on Hadamard manifolds. Notably, one method nearly matches the gradient oracle complexity of the lower bound for Euclidean problems, for the first time.

Related papers

• Riemannian Bilevel Optimization [35.42472057648458]
  We focus in particular on batch and gradient-based methods, with the explicit goal of avoiding second-order information. We propose and analyze $mathrmRF2SA$, a method that leverages first-order gradient information. We provide explicit convergence rates for reaching $epsilon$-stationary points under various setups.
  arXiv  Detail & Related papers  (2024-05-22T20:49:01Z)
• Convergence and complexity of block majorization-minimization for constrained block-Riemannian optimization [18.425648833592312]
  Blockization-minimization (BMM) is a simple iterative gradient for non-exhaustive subspace estimation. Our analysis makes explicit use of Euclidian constraints.
  arXiv  Detail & Related papers  (2023-12-16T05:40:19Z)
• Riemannian stochastic optimization methods avoid strict saddle points [68.80251170757647]
  We show that policies under study avoid strict saddle points / submanifolds with probability 1. This result provides an important sanity check as it shows that, almost always, the limit state of an algorithm can only be a local minimizer.
  arXiv  Detail & Related papers  (2023-11-04T11:12:24Z)
• First-order Methods for Affinely Constrained Composite Non-convex Non-smooth Problems: Lower Complexity Bound and Near-optimal Methods [23.948126336842634]
  We make the first attempt to establish lower complexity bounds of first-order FOMs for solving a composite non-smooth optimization problem. We find that our method and the proposed IPG method are almostimprovable.
  arXiv  Detail & Related papers  (2023-07-14T19:59:18Z)
• Curvature-Independent Last-Iterate Convergence for Games on Riemannian Manifolds [77.4346324549323]
  We show that a step size agnostic to the curvature of the manifold achieves a curvature-independent and linear last-iterate convergence rate. To the best of our knowledge, the possibility of curvature-independent rates and/or last-iterate convergence has not been considered before.
  arXiv  Detail & Related papers  (2023-06-29T01:20:44Z)
• Decentralized Riemannian Algorithm for Nonconvex Minimax Problems [82.50374560598493]
  The minimax algorithms for neural networks have been developed to solve many problems. In this paper, we propose two types of minimax algorithms. For the setting, we propose DRSGDA and prove that our method achieves a gradient.
  arXiv  Detail & Related papers  (2023-02-08T01:42:45Z)
• The Dynamics of Riemannian Robbins-Monro Algorithms [101.29301565229265]
  We propose a family of Riemannian algorithms generalizing and extending the seminal approximation framework of Robbins and Monro. Compared to their Euclidean counterparts, Riemannian algorithms are much less understood due to lack of a global linear structure on the manifold. We provide a general template of almost sure convergence results that mirrors and extends the existing theory for Euclidean Robbins-Monro schemes.
  arXiv  Detail & Related papers  (2022-06-14T12:30:11Z)
• First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spaces [93.35384756718868]
  min-max algorithms have been analyzed in the Euclidean setting. We prove that the extraiteient (RCEG) method corrected lastrate convergence at a linear rate.
  arXiv  Detail & Related papers  (2022-06-04T18:53:44Z)
• Optimal Randomized First-Order Methods for Least-Squares Problems [56.05635751529922]
  This class of algorithms encompasses several randomized methods among the fastest solvers for least-squares problems. We focus on two classical embeddings, namely, Gaussian projections and subsampled Hadamard transforms. Our resulting algorithm yields the best complexity known for solving least-squares problems with no condition number dependence.
  arXiv  Detail & Related papers  (2020-02-21T17:45:32Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.