[Paper Review] A constant step Forward-Backward algorithm involving random maximal monotone operators

Pascal Bianchi, Walid Hachem+1 more|arXiv (Cornell University)|Feb 14, 2017

Optimization and Variational Analysis30 references4 citations

TL;DR

This paper proposes a constant-step stochastic Forward-Backward algorithm involving random maximal monotone operators, analyzing its convergence via a differential inclusion formed from the mean operators. It establishes narrow convergence of the interpolated process to the solution of the differential inclusion and proves that, under demipositivity, the iterates remain close to the zero set of the sum operator in Cesàro mean, with ergodic behavior characterized via invariant measures of the associated Markov chain.

ABSTRACT

A stochastic Forward-Backward algorithm with a constant step is studied. At each time step, this algorithm involves an independent copy of a couple of random maximal monotone operators. Defining a mean operator as a selection integral, the differential inclusion built from the sum of the two mean operators is considered. As a first result, it is shown that the interpolated process obtained from the iterates converges narrowly in the small step regime to the solution of this differential inclusion. In order to control the long term behavior of the iterates, a stability result is needed in addition. To this end, the sequence of the iterates is seen as a homogeneous Feller Markov chain whose transition kernel is parameterized by the algorithm step size. The cluster points of the Markov chains invariant measures in the small step regime are invariant for the semiflow induced by the differential inclusion. Conclusions regarding the long run behavior of the iterates for small steps are drawn. It is shown that when the sum of the mean operators is demipositive, the probabilities that the iterates are away from the set of zeros of this sum are small in Ces\\`aro mean. The ergodic behavior of these iterates is studied as well. Applications of the proposed algorithm are considered. In particular, a detailed analysis of the random proximal gradient algorithm with constant step is performed.

Motivation & Objective

• To study the dynamical behavior of a stochastic Forward-Backward algorithm with fixed step size under random maximal monotone operators.
• To establish convergence of the interpolated iterates to the solution of a differential inclusion driven by the mean operators.
• To characterize the long-term behavior of the iterates using invariant measures of the associated Feller Markov chain.
• To show that, under demipositivity of the sum of mean operators, the iterates remain close to the zero set in Cesàro mean.
• To analyze the ergodic properties of the algorithm and apply the framework to the random proximal gradient method with constant step.

Proposed method

• The algorithm uses a constant step size γ and at each iteration selects independent copies of random maximal monotone operators A(ξn) and B(ξn) from a probability space (Ξ, G) with distribution μ.
• The mean operators are defined as A = ∫A(s)μ(ds) and B = ∫B(s)μ(ds), where the integrals are selection integrals, forming a differential inclusion governed by A + B.
• The interpolated process of the iterates is shown to converge narrowly to the solution of the differential inclusion in the small step regime.
• The sequence of iterates is modeled as a homogeneous Feller Markov chain with transition kernel parameterized by γ, enabling analysis of invariant measures.
• Cluster points of the invariant measures as γ → 0 are shown to be invariant for the semiflow induced by the differential inclusion.
• The ergodic behavior is studied by analyzing weak convergence of empirical measures and using Mazur’s theorem to extract convergent subsequences in L1+ε/2 spaces.

Experimental results

Research questions

• RQ1 Does the interpolated process of the constant-step stochastic Forward-Backward iterates converge to the solution of a differential inclusion in the small step regime?
• RQ2 What is the long-term behavior of the iterates when the step size is fixed and small, particularly in terms of proximity to the zero set of the sum of mean operators?
• RQ3 How do the invariant measures of the Markov chain associated with the algorithm behave as the step size γ approaches zero?
• RQ4 Under what conditions does the algorithm ensure that the iterates remain close to the solution set in Cesàro mean?
• RQ5 Can the ergodic properties of the iterates be characterized, and how do they relate to the semiflow of the differential inclusion?

Key findings

• The interpolated process converges narrowly to the solution of the differential inclusion defined by the sum of the mean operators A + B in the small step regime.
• The cluster points of the invariant measures of the Feller Markov chain are invariant for the semiflow induced by the differential inclusion.
• When A + B is demipositive, the probability that the iterates are far from the zero set of A + B is small in Cesàro mean.
• The ergodic behavior of the iterates is characterized by weak convergence of empirical measures to invariant measures of the semiflow.
• The random proximal gradient algorithm with constant step is shown to satisfy the same convergence and ergodicity properties under the same assumptions.
• The analysis establishes tightness and convergence in distribution via compactness in L1+ε/2 spaces and Mazur’s theorem for subsequence extraction.