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

arXiv: https://arxiv.org/abs/2512.23843 · HTML: https://mlsanity.com/paper/2512.23843/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Dec 2025.

## Built on by (most cited first, 1 of 1)

- [Learning with Boolean threshold functions](https://mlsanity.com/paper/2602.17493/) (2026)

## Abstract (arXiv)

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.