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

TL;DR

This paper proposes a canonical structure for a broad class of first-order algorithms with delayed feedback, enabling their extension to constrained problems while maintaining convergence properties.

Contribution

It introduces a simple linear transformation-based canonical structure that extends unconstrained algorithms to constrained settings via projection.

Findings

Projected algorithms achieve optimal solutions.

Convergence rates are preserved in the extension.

Applicable to systems with delayed gradient 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.