The Flow-Limit of Reflect-Reflect-Relax: Existence, Stability, and Discrete-Time Behavior

TL;DR

This paper analyzes the small-step regime of the Reflect-Reflect-Relax (RRR) algorithm, demonstrating exponential decay, stability, and finite-time convergence properties, and introduces a mesoscopic framework for performance analysis.

Contribution

It provides a detailed flow-limit analysis of RRR, establishing stability, convergence, and the connection to discrete-time behavior, along with a new heuristic performance framework.

Findings

Exponential decay of the gap measure in the flow limit.

Finite-time convergence to solution domain in the discrete setting.

Emergence of iteration-optimal relaxation parameters.

Abstract

We study the Reflect-Reflect-Relax (RRR) algorithm in its small-step (flow-limit) regime. In the smooth transversal setting, we show that the transverse dynamics form a hyperbolic sink, yielding exponential decay of a natural gap measure. Under uniform geometric assumptions, we construct a tubular neighborhood of the feasible manifold on which the squared gap defines a strict Lyapunov function, excluding recurrent dynamics and chaotic behavior within this basin. In the discrete setting, the induced flow is piecewise constant on W-domains and supports Filippov sliding along convergent boundaries, leading to finite-time capture into a solution domain. We prove that small-step RRR is a forward-Euler discretization of this flow, so that solution times measured in rescaled units converge to a finite limit while iteration counts diverge, explaining the emergence of iteration-optimal relaxation parameters. Finally, we introduce a heuristic mesoscopic framework based on percolation and renormalization group to organize performance deterioration near the Douglas-Rachford limit.