Frugal Splitting Operators: Representation, Minimal Lifting, and Convergence

TL;DR

This paper introduces a new framework for frugal splitting operators in monotone inclusion problems, providing a unified convergence analysis and a method to design minimal lifting operators with practical applications.

Contribution

It presents a novel representation of frugal splitting operators via a generalized primal-dual resolvent, enabling convergence analysis and the design of minimal lifting operators.

Findings

Minimal lifting number is n-1-f, achievable with resolvent evaluations.

Unified convergence analysis for cocoercive operators.

Constructive method for designing new frugal splitting operators.

Abstract

We investigate frugal splitting operators for finite sum monotone inclusion problems. These operators utilize exactly one direct or resolvent evaluation of each operator of the sum, and the splitting operator's output is dictated by linear combinations of these evaluations' inputs and outputs. To facilitate analysis, we introduce a novel representation of frugal splitting operators via a generalized primal-dual resolvent. The representation is characterized by an index and four matrices, and we provide conditions on these that ensure equivalence between the classes of frugal splitting operators and generalized primal-dual resolvents. Our representation paves the way for new results regarding lifting numbers and the development of a unified convergence analysis for frugal splitting operator methods, contingent on the directly evaluated operators being cocoercive. The minimal lifting number is $n-1-f$ where $n$ is the number of monotone operators and $f$ is the number of direct evaluations in the splitting. Notably, this lifting number is achievable only if the first and last operator evaluations are resolvent evaluations. These results generalize the minimal lifting results by Ryu and Malitsky--Tam that consider frugal resolvent splittings. Building on our representation, we delineate a constructive method to design frugal splitting operators, exemplified in the design of a novel, convergent, and parallelizable frugal splitting operator with minimal lifting.