Inducing Uniform Asymptotic Stability in Non-Autonomous Accelerated Optimization Dynamics via Hybrid Regularization

Abstract

There have been many recent efforts to study accelerated optimization algorithms from the perspective of dynamical systems. In this paper, we focus on the robustness properties of the time-varying continuous-time version of these dynamics. These properties are critical for the implementation of accelerated algorithms in feedback-based control and optimization architectures. We show that a family of dynamics related to the continuous-time limit of Nesterov's accelerated gradient method can be rendered unstable under arbitrarily small bounded disturbances. Indeed, while solutions of these dynamics may converge to the set of optimizers, in general, this set may not be uniformly asymptotically stable. To induce uniformity, and robustness as a byproduct, we propose a framework where we regularize the dynamics by using resetting mechanisms that are modeled by well-posed hybrid dynamical systems. For these hybrid dynamics, we establish uniform asymptotic stability and robustness properties, as well as convergence rates that are similar to those of the nonhybrid dynamics. We finish by characterizing a family of discretization mechanisms that retain the main stability and robustness properties of the hybrid algorithms. J. I. Poveda is with the

Data‐enabled extremum seeking: A cooperative concurrent learning‐based approach

“…Typical choices of b j include polynomial functions, radial basis functions, or sigmoid functions, see References 46-48 for details on universal approximation properties of different types of functions. We shall need the following technical assumption on the approximation (8). Using the smoothness of , b, and , we can compute the gradient of ∇ as follows:…”

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“…Traditionally, extremum seeking (ES) dynamics have been designed under the paradigm of exploration vs exploitation , using an external dither signal to guarantee enough exploration on the cost function, and to facilitate the optimization process via gradient approximations based on parameterizations, averaging techniques, or sampled‐data reconstructions. In most of these approaches, the injection of the external signal is needed mainly to guarantee a uniform convergence property in the closed‐loop system, that is, to avoid closed‐loop systems with rates of convergence that depend heavily on the initial conditions and which may not be able to recover from external disturbances and/or slows changes in the cost function 7,8 . To avoid these issues, adaptive ES dynamics based on parametric approximations usually rely on persistence of excitation (PE) conditions of the form where the mapping t ↦ b ( t ) usually depends on the trajectories of the system.…”

“…In most of these approaches, the injection of the external signal is needed mainly to guarantee a uniform convergence property in the closed-loop system, that is, to avoid closed-loop systems with rates of convergence that depend heavily on the initial conditions and which may not be able to recover from external disturbances and/or slows changes in the cost function. 7,8 To avoid these issues, adaptive ES dynamics based on parametric approximations usually rely on persistence of excitation (PE) conditions of the form…”

Online Optimization of Switched LTI Systems Using Continuous-Time and Hybrid Accelerated Gradient Flows

“…When the target optimization problem is time-varying -due to time-varying disturbances or time-varying cost functions -the problem at hand becomes that of tracking an optimal trajectory for the switched system, preserving suitable stability properties in the closed-loop system. We propose two design strategies: (i) an online gradient method (or gradient flow), similar to those considered in [3,4]; and, (ii) a novel hybrid feedback controller based on Nesterov's accelerated gradient flows [17][18][19], which incorporates dynamic momentum in order to induce acceleration in the closed-loop system, without sacrificing stability and robustness properties that are fundamental in feedback control. The two alternatives offer a number of trade-offs between implementation complexity, achievable convergence rates, and conditions for exponential stability.…”