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Variable Stepsize Distributed Forward-Backward Splitting Methods as Relocated Fixed-Point Iterations

Felipe Atenas, Minh N. Dao, Matthew K. Tam

TL;DR

The paper addresses solving structured monotone inclusions of the form $0 \in \sum_{i=1}^n A_i x + \sum_{j=1}^p B_j x$ in a Hilbert space by developing relocated fixed-point iterations for conically averaged operators to support variable stepsizes without increasing per-iteration cost. It introduces fixed-point relocators $Q_{\delta\gets\gamma}$ that map fixed points across stepsizes and applies this framework to distributed forward-backward methods and graph-based splittings, including a variable-stepsize Davis–Yin method for three-operator problems. The authors prove convergence of the relocated iterates to a fixed point of the limiting operator and show that the associated shadow sequences solve the original inclusion, supported by numerical experiments on constrained LASSO and nonnegative elastic net. This work enables efficient, scalable distributed optimization with adaptive stepsizes, offering a principled route to step-size control in splitting methods and paving the way for further theory-guided design of relocation-aware algorithms.

Abstract

We present a family of distributed forward-backward methods with variable stepsizes to find a solution of structured monotone inclusion problems. The framework is constructed by means of relocated fixed-point iterations, extending the approach introduced in arXiv:2507.07428 to conically averaged operators, thus including iteration operators for methods of forward-backward type devised by graphs. The family of methods we construct preserve the per-iteration computational cost and the convergence properties of their constant stepsize counterparts. Specifically, we show that the resulting methods generate a sequence that converges to a fixed-point of the underlying iteration operator, whose shadow sequences converge to a solution of the problem. Numerical experiments illustrate the behaviour of our framework in structured sparse optimisation problems.

Variable Stepsize Distributed Forward-Backward Splitting Methods as Relocated Fixed-Point Iterations

TL;DR

The paper addresses solving structured monotone inclusions of the form in a Hilbert space by developing relocated fixed-point iterations for conically averaged operators to support variable stepsizes without increasing per-iteration cost. It introduces fixed-point relocators that map fixed points across stepsizes and applies this framework to distributed forward-backward methods and graph-based splittings, including a variable-stepsize Davis–Yin method for three-operator problems. The authors prove convergence of the relocated iterates to a fixed point of the limiting operator and show that the associated shadow sequences solve the original inclusion, supported by numerical experiments on constrained LASSO and nonnegative elastic net. This work enables efficient, scalable distributed optimization with adaptive stepsizes, offering a principled route to step-size control in splitting methods and paving the way for further theory-guided design of relocation-aware algorithms.

Abstract

We present a family of distributed forward-backward methods with variable stepsizes to find a solution of structured monotone inclusion problems. The framework is constructed by means of relocated fixed-point iterations, extending the approach introduced in arXiv:2507.07428 to conically averaged operators, thus including iteration operators for methods of forward-backward type devised by graphs. The family of methods we construct preserve the per-iteration computational cost and the convergence properties of their constant stepsize counterparts. Specifically, we show that the resulting methods generate a sequence that converges to a fixed-point of the underlying iteration operator, whose shadow sequences converge to a solution of the problem. Numerical experiments illustrate the behaviour of our framework in structured sparse optimisation problems.
Paper Structure (14 sections, 19 theorems, 98 equations, 2 figures, 1 algorithm)

This paper contains 14 sections, 19 theorems, 98 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Keywords:
  2. Mathematics Subject Classification (MSC 2020):
  3. Introduction and motivation
  4. Preliminaries
  5. Relocated fixed-point iterations of conically averaged operators
  6. Variable stepsize distributed forward-backward methods
  7. Graph-based distributed forward-backward methods
  8. Relocated forward-backward algorithms devised by graphs
  9. A variable stepsize forward-backward three-operator splitting method
  10. Numerical experiments
  11. Constrained LASSO
  12. Nonnegative elastic net
  13. Conclusion
  14. Proof of Theorem \ref{['t:dist-FB-convergence']}

Key Result

Lemma 2.1

Let $A$ be a maximally monotone operator on $X$. Then

Figures (2)

  • Figure 1: Relative error of iterates and function values generated by Algorithm \ref{['a:DY']} for different stepsize regimes to solve problem \ref{['eq:ct-LASSO']}, where FPR 1 is the stepsize rule \ref{['eq:nonstat']}, FPR 2 is \ref{['eq:accDY']} and FPR 3 is \ref{['eq:Halpern']}.
  • Figure 2: Relative error of iterates and function values generated by Algorithm \ref{['a:DY']} for different stepsize regimes to solve problem \ref{['e:elastic-net-obj']}, where FPR is the stepsize rule \ref{['eq:nonstat']}.

Theorems & Definitions (44)

  • Lemma 2.1: Properties of resolvents atenas2025relocated
  • Definition 2.2: Parametric demiclosedness
  • Theorem 2.3: Parametric demiclosedness principleatenas2025relocated
  • Remark 2.4
  • Corollary 2.5: Parametric demiclosedness principle for conically averaged operators
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7: Robbins--Siegmund, polyak1987introduction
  • Definition 3.1: Fixed-point relocator
  • ...and 34 more