Breaking the Convergence Barrier: Optimization via Fixed-Time Convergent Flows

• URL: http://arxiv.org/abs/2112.01363v1
• Date: Thu, 2 Dec 2021 16:04:40 GMT
• Title: Breaking the Convergence Barrier: Optimization via Fixed-Time Convergent Flows
• Authors: Param Budhraja, Mayank Baranwal, Kunal Garg, Ashish Hota
• Abstract summary: We introduce a Poly-based optimization framework for achieving acceleration, based on the notion of fixed-time stability dynamical systems. We validate the accelerated convergence properties of the proposed schemes on a range of numerical examples against the state-of-the-art optimization algorithms.
• Score: 4.817429789586127
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Accelerated gradient methods are the cornerstones of large-scale, data-driven optimization problems that arise naturally in machine learning and other fields concerning data analysis. We introduce a gradient-based optimization framework for achieving acceleration, based on the recently introduced notion of fixed-time stability of dynamical systems. The method presents itself as a generalization of simple gradient-based methods suitably scaled to achieve convergence to the optimizer in a fixed-time, independent of the initialization. We achieve this by first leveraging a continuous-time framework for designing fixed-time stable dynamical systems, and later providing a consistent discretization strategy, such that the equivalent discrete-time algorithm tracks the optimizer in a practically fixed number of iterations. We also provide a theoretical analysis of the convergence behavior of the proposed gradient flows, and their robustness to additive disturbances for a range of functions obeying strong convexity, strict convexity, and possibly nonconvexity but satisfying the Polyak-{\L}ojasiewicz inequality. We also show that the regret bound on the convergence rate is constant by virtue of the fixed-time convergence. The hyperparameters have intuitive interpretations and can be tuned to fit the requirements on the desired convergence rates. We validate the accelerated convergence properties of the proposed schemes on a range of numerical examples against the state-of-the-art optimization algorithms. Our work provides insights on developing novel optimization algorithms via discretization of continuous-time flows.

Related papers

• From exponential to finite/fixed-time stability: Applications to optimization [0.0]
  Given an exponentially stable optimization algorithm, can it be modified to obtain a finite/fixed-time stable algorithm? We provide an affirmative answer, demonstrate how the solution can be computed on a finite-time interval via a simple scaling of the right-hand-side of the original dynamics. We certify the desired properties of the modified algorithm using the Lyapunov function that proves exponential stability of the original system.
  arXiv  Detail & Related papers  (2024-09-18T05:43:22Z)
• ODE-based Learning to Optimize [28.380622776436905]
  We present a comprehensive framework integrating the inertial systems with Hessian-driven damping equation (ISHD) We formulate a novel learning to optimize (L2O) problem aimed at minimizing the stopping time subject to the convergence and stability condition. Empirical validation of our framework is conducted through extensive numerical experiments across a diverse set of optimization problems.
  arXiv  Detail & Related papers  (2024-06-04T06:39:45Z)
• Accelerated First-Order Optimization under Nonlinear Constraints [73.2273449996098]
  We exploit between first-order algorithms for constrained optimization and non-smooth systems to design a new class of accelerated first-order algorithms. An important property of these algorithms is that constraints are expressed in terms of velocities instead of sparse variables.
  arXiv  Detail & Related papers  (2023-02-01T08:50:48Z)
• Generalized Gradient Flows with Provable Fixed-Time Convergence and Fast Evasion of Non-Degenerate Saddle Points [8.452349885923507]
  Gradient-based first-order convex optimization algorithms find widespread applicability in a variety of domains, including machine learning tasks. Motivated by the recent advances in fixed-time theory of optimal time, we introduce a framework for designing accelerated optimization algorithms. For functions that admit non-de saddle-points, we show that the time required to evade these saddle-points is uniformly bounded for all initial conditions.
  arXiv  Detail & Related papers  (2022-12-07T16:36:23Z)
• Faster Algorithm and Sharper Analysis for Constrained Markov Decision Process [56.55075925645864]
  The problem of constrained decision process (CMDP) is investigated, where an agent aims to maximize the expected accumulated discounted reward subject to multiple constraints. A new utilities-dual convex approach is proposed with novel integration of three ingredients: regularized policy, dual regularizer, and Nesterov's gradient descent dual. This is the first demonstration that nonconcave CMDP problems can attain the lower bound of $mathcal O (1/epsilon)$ for all complexity optimization subject to convex constraints.
  arXiv  Detail & Related papers  (2021-10-20T02:57:21Z)
• Optimization on manifolds: A symplectic approach [127.54402681305629]
  We propose a dissipative extension of Dirac's theory of constrained Hamiltonian systems as a general framework for solving optimization problems. Our class of (accelerated) algorithms are not only simple and efficient but also applicable to a broad range of contexts.
  arXiv  Detail & Related papers  (2021-07-23T13:43:34Z)
• On the Convergence of Stochastic Extragradient for Bilinear Games with Restarted Iteration Averaging [96.13485146617322]
  We present an analysis of the ExtraGradient (SEG) method with constant step size, and present variations of the method that yield favorable convergence. We prove that when augmented with averaging, SEG provably converges to the Nash equilibrium, and such a rate is provably accelerated by incorporating a scheduled restarting procedure.
  arXiv  Detail & Related papers  (2021-06-30T17:51:36Z)
• A Dynamical Systems Approach for Convergence of the Bayesian EM Algorithm [59.99439951055238]
  We show how (discrete-time) Lyapunov stability theory can serve as a powerful tool to aid, or even lead, in the analysis (and potential design) of optimization algorithms that are not necessarily gradient-based. The particular ML problem that this paper focuses on is that of parameter estimation in an incomplete-data Bayesian framework via the popular optimization algorithm known as maximum a posteriori expectation-maximization (MAP-EM) We show that fast convergence (linear or quadratic) is achieved, which could have been difficult to unveil without our adopted S&C approach.
  arXiv  Detail & Related papers  (2020-06-23T01:34:18Z)
• Optimization with Momentum: Dynamical, Control-Theoretic, and Symplectic Perspectives [97.16266088683061]
  The article rigorously establishes why symplectic discretization schemes are important for momentum-based optimization algorithms. It provides a characterization of algorithms that exhibit accelerated convergence.
  arXiv  Detail & Related papers  (2020-02-28T00:32:47Z)
• Convergence and sample complexity of gradient methods for the model-free linear quadratic regulator problem [27.09339991866556]
  We show that ODE searches for optimal control for an unknown computation system by directly searching over the corresponding space of controllers. We take a step towards demystifying the performance and efficiency of such methods by focusing on the gradient-flow dynamics set of stabilizing feedback gains and a similar result holds for the forward disctization of the ODE.
  arXiv  Detail & Related papers  (2019-12-26T16:56:59Z)

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.