[论文解读] Variable Stepsize Distributed Forward-Backward Splitting Methods as Relocated Fixed-Point Iterations

Felipe Atenas, Minh N. Dao+1 more|arXiv (Cornell University)|Jan 21, 2026

Optimization and Variational Analysis被引用 0

一句话总结

该论文通过重新定位的固定点迭代，提出一个变量步长的分布式前向后向分裂框架，用于结构化的单调包含问题，保持常数步长方法的每步成本与收敛性质。并推广到圆锥均匀化算子，包含基于图的多算子变体。

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.

研究动机与目标

• Motivate and solve structured monotone inclusion problems using distributed forward-backward splitting with variable stepsizes.
• Extend relocated fixed-point iterations to conically averaged operators to preserve convergence guarantees with variable parameters.
• Enable multioperator forward-backward methods in distributed settings with preserved per-iteration cost and convergence properties.
• Incorporate graph-based structures to handle multioperator inclusions in a scalable distributed framework.

提出的方法

• Define conically averaged operator framework and extend the demiclosedness principle to parametric families.
• Introduce fixed-point relocators and prove their key properties for rela- cated iterations.
• Construct variable stepsize distributed forward-backward iteration operators using matrices M, N, P, R under specific Assumptions.
• Prove convergence of relocated fixed-point iterations to fixed points and shadow sequences to problem solutions.
• Specialize to multioperator and graph-based forward-backward methods with a relocated three-operator (Davis–Yin) scheme as a special case.
• Derive conditions under which the methods maintain per-iteration cost and convergence guarantees.

(a) Relative error of $(x_{k})_{k\in\mathbbm{N}}$ .

实验结果

研究问题

• RQ1 Can variable stepsize distributed forward-backward methods be developed with convergence guarantees for structured monotone inclusion problems?
• RQ2 How can relocated fixed-point iterations be extended to conically averaged operators to encompass multioperator and graph-based forward-backward schemes?
• RQ3 What fixed-point relocator constructions preserve per-iteration cost and enable convergence for distributed multioperator inclusions?
• RQ4 How do these relocated methods relate to and generalize known forward-backward and Davis–Yin schemes in a distributed setting?

主要发现

• A parametric demiclosedness framework is established for conically averaged operators, enabling convergence analysis under variable parameters.
• A fixed-point relocator is defined and shown to map fixed points across different step sizes, enabling relocation-based iterations.
• A distributed forward-backward operator is formulated with matrices M, N, P, R satisfying structural assumptions, and a corresponding fixed-point relocator e^γ is constructed.
• Under cocoercivity assumptions and the stated matrix conditions, the relocated iterations converge to a fixed-point and the shadow sequences converge to a solution of the monotone inclusion.
• The framework encompasses graph-based forward-backward methods and includes a relocated Davis–Yin three-operator scheme as a special case.
• Per-iteration computational cost is preserved relative to constant-stepsize counterparts.

(b) Relative error of $(y_{k})_{k\in\mathbbm{N}}$ .