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A Canonical Structure for Constructing Projected First-Order Algorithms With Delayed Feedback

Mengmou Li, Yu Zhou, Xun Shen, Masaaki Nagahara

Abstract

This work introduces a canonical structure for a broad class of unconstrained first-order algorithms that admit a Lur'e representation, including systems with relative degree greater than one, e.g., systems with delayed gradient feedback. The proposed canonical structure is obtained through a simple linear transformation. It enables a direct extension from unconstrained optimization algorithms to set-constrained ones through projection in a Lyapunov-induced norm. The resulting projected algorithms attain the optimal solution while preserving the convergence rates of their unconstrained counterparts.

A Canonical Structure for Constructing Projected First-Order Algorithms With Delayed Feedback

Abstract

This work introduces a canonical structure for a broad class of unconstrained first-order algorithms that admit a Lur'e representation, including systems with relative degree greater than one, e.g., systems with delayed gradient feedback. The proposed canonical structure is obtained through a simple linear transformation. It enables a direct extension from unconstrained optimization algorithms to set-constrained ones through projection in a Lyapunov-induced norm. The resulting projected algorithms attain the optimal solution while preserving the convergence rates of their unconstrained counterparts.

Paper Structure

This paper contains 13 sections, 7 theorems, 61 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convex analysis
  4. Unconstrained First-order algorithms and Canonical Structure
  5. Canonical Structure
  6. IQC Analysis and Assumption
  7. Projected Algorithm and Analysis
  8. Projected Algorithm on the Lyapunov-induced Norm
  9. Optimality and Rate-preserving Convergence
  10. Numerical Example
  11. Projected Algorithm with One-step Delay
  12. Numerical Implementation
  13. Conclusion

Key Result

Theorem 1

System eq: LTI system with relative degree $r \geq 1$ can be rewritten as where $r \geq 1$ denotes the relative degree of system eq: LTI system, and $e_1 \in \mathbb{R}^{r}$ is the first standard basis vector. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure E1: The trajectories of $\| y_k - y_*\|_2$ for the unconstrained algorithm with one-step delay in \ref{['eq: unconstrained alg example']}, where $y_*$ is the optimal solution to the unconstrained problem.
  • Figure E2: Convergence error for the proposed projected algorithm with one-step delay in \ref{['eq:alg example']}. The theoretical rate bound is given by the dotted lines.

Theorems & Definitions (11)

  • Theorem 1
  • proof
  • Corollary 1: li2025first
  • Lemma 1
  • proof
  • Lemma 2
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 1 more