Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Input-to-State Stability of a Bilevel Proximal Gradient Descent Algorithm

Torbjørn Cunis Ilya Kolmanovsky

TL;DR

The paper addresses the convergence of a bilevel optimization problem where an inner loop solves a parameterized discrete-time optimal control problem and is executed inexactly. It introduces an interconnection of finite-step proximal gradient algorithms for the outer and inner problems and analyzes stability using $\omega$-Input-to-State Stability and a small-gain theorem to derive sufficient conditions on stepsizes and inner iterations for convergence to the bilevel optimum. The authors show that both the inner and outer subsystems are $\omega$ISS with respect to appropriate measurement functions and prove that, under the small-gain condition, the overall scheme converges to the minimizer set $\mathcal{P}_\star$ and the inner solution $\bar{\mathbf{u}}(p)$, i.e., $\mathbf{u}_\kappa^\ell \to \bar{\mathbf{u}}(p^\ell)$. A special case with additive linear parameters illustrates convexity of the outer problem and confirms convergence for sufficiently many inner iterations; the framework provides robust convergence guarantees for control-aware design optimization under computational limits.

Abstract

This paper studies convergence properties of inexact iterative solution schemes for bilevel optimization problems. Bilevel optimization problems emerge in control-aware design optimization, where the system design parameters are optimized in the outer loop and a discrete-time control trajectory is optimized in the inner loop, but also arise in other domains including machine learning. In the paper an interconnection of proximal gradient algorithms is proposed to solve the inner loop and outer loop optimization problems in the setting of control-aware design optimization and robustness is analyzed from a control-theoretic perspective. By employing input-to-state stability arguments, conditions are derived that ensure convergence of the interconnected scheme to the optimal solution for a class of the bilevel optimization problem.

Input-to-State Stability of a Bilevel Proximal Gradient Descent Algorithm

TL;DR

The paper addresses the convergence of a bilevel optimization problem where an inner loop solves a parameterized discrete-time optimal control problem and is executed inexactly. It introduces an interconnection of finite-step proximal gradient algorithms for the outer and inner problems and analyzes stability using -Input-to-State Stability and a small-gain theorem to derive sufficient conditions on stepsizes and inner iterations for convergence to the bilevel optimum. The authors show that both the inner and outer subsystems are ISS with respect to appropriate measurement functions and prove that, under the small-gain condition, the overall scheme converges to the minimizer set and the inner solution , i.e., . A special case with additive linear parameters illustrates convexity of the outer problem and confirms convergence for sufficiently many inner iterations; the framework provides robust convergence guarantees for control-aware design optimization under computational limits.

Abstract

This paper studies convergence properties of inexact iterative solution schemes for bilevel optimization problems. Bilevel optimization problems emerge in control-aware design optimization, where the system design parameters are optimized in the outer loop and a discrete-time control trajectory is optimized in the inner loop, but also arise in other domains including machine learning. In the paper an interconnection of proximal gradient algorithms is proposed to solve the inner loop and outer loop optimization problems in the setting of control-aware design optimization and robustness is analyzed from a control-theoretic perspective. By employing input-to-state stability arguments, conditions are derived that ensure convergence of the interconnected scheme to the optimal solution for a class of the bilevel optimization problem.
Paper Structure (12 sections, 10 theorems, 59 equations)

This paper contains 12 sections, 10 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Statement
  4. Size functions
  5. $\omega$-Input-to-State Stability
  6. Sensitivity of Optimal Control
  7. Stability Analysis
  8. Parametrized Gradient Descent
  9. Perturbed Gradient Descent
  10. Interconnected Gradient Descent
  11. Special Case
  12. Conclusion

Key Result

lemma 1

Let $\omega_1: \mathbb X \to \mathbb R_{\geq 0}$ be a size function for $\mathcal{A}$; then any measurement function $\omega_2$ on $\mathbb X$ is a size function (for $\mathcal{A}$) if and only if there exists $\alpha_1, \alpha_2 \in \mathcal{K}_\infty$ satisfying for all $x \in \mathbb X$. $\lhd$

Theorems & Definitions (20)

  • definition 1
  • lemma 1: Sontag2022, Corollary 2.7
  • definition 2
  • remark 1
  • theorem 1: Noroozi2018, Theorem 10
  • lemma 2
  • proof
  • proposition 1
  • proof
  • remark 2
  • ...and 10 more