[Paper Review] A stochastic inertial forward-backward splitting algorithm for multivariate monotone inclusions

Lorenzo Rosasco, Silvia Villa+1 more|arXiv (Cornell University)|Jul 3, 2015

Optimization and Variational Analysis43 references4 citations

TL;DR

This paper proposes a stochastic inertial forward-backward splitting algorithm for solving multivariate monotone inclusion problems with stochastic errors in operator evaluations. It establishes almost sure convergence of iterates to a solution in real Hilbert spaces, extending inertial and primal-dual methods to stochastic settings with theoretical guarantees for structured composite inclusions and minimization problems.

ABSTRACT

We propose an inertial forward-backward splitting algorithm to compute the zero of a sum of two monotone operators allowing for stochastic errors in the computation of the operators. More precisely, we establish almost sure convergence in real Hilbert spaces of the sequence of iterates to an optimal solution. Then, based on this analysis, we introduce two new classes of stochastic inertial primal-dual splitting methods for solving structured systems of composite monotone inclusions and prove their convergence. Our results extend to the stochastic and inertial setting various types of structured monotone inclusion problems and corresponding algorithmic solutions. Application to minimization problems is discussed.

Motivation & Objective

• To develop a stochastic inertial forward-backward algorithm that handles noisy evaluations of monotone operators in Hilbert spaces.
• To establish almost sure convergence of the algorithm's iterates to a solution of the monotone inclusion problem.
• To extend inertial and primal-dual splitting methods to stochastic settings for structured composite monotone inclusions.
• To derive new classes of stochastic inertial primal-dual algorithms for coupled systems of monotone inclusions.
• To provide convergence guarantees under minimal assumptions, including stochastic error conditions and bounded inertial steps.

Proposed method

• The algorithm incorporates an inertial term using past iterates to accelerate convergence, with step-sizes satisfying summability conditions.
• It uses a stochastic oracle to estimate the cocoercive operator B, with errors modeled as zero-mean, square-integrable random variables.
• The method applies forward-backward splitting in a product Hilbert space framework to decouple composite monotone inclusions.
• A primal-dual formulation is derived by reformulating the problem using Fenchel-Rockafellar duality and proximal mappings.
• The algorithm alternates between updating primal variables via proximal steps and dual variables via gradient-like updates with stochastic gradients.
• Convergence is proven using a Lyapunov function and almost sure convergence analysis under assumptions on the stochastic errors and inertial parameters.

Experimental results

Research questions

• RQ1 Can inertial acceleration be effectively combined with stochastic forward-backward splitting for monotone inclusions?
• RQ2 Under what conditions does the stochastic inertial forward-backward algorithm converge almost surely to a solution?
• RQ3 How can inertial and stochastic primal-dual methods be constructed for coupled systems of composite monotone inclusions?
• RQ4 What role do stochastic errors in operator evaluations play in convergence, and how can they be controlled?
• RQ5 Can the convergence theory be extended to include uniformly convex regularizers and stronger convergence guarantees?

Key findings

• The proposed stochastic inertial forward-backward algorithm achieves almost sure convergence of iterates to a solution of the monotone inclusion problem in real Hilbert spaces.
• The algorithm converges under minimal assumptions: summable inertial steps and square-integrable stochastic errors in operator evaluations.
• For uniformly convex regularizers, the algorithm achieves strong convergence of the primal iterates almost surely.
• The method yields two new classes of stochastic inertial primal-dual splitting algorithms for structured composite monotone inclusions in duality.
• The convergence results generalize existing deterministic and stochastic splitting methods to the inertial and stochastic setting.
• The analysis applies to minimization problems and extends to cases where the cocoercivity parameter and step-sizes are chosen appropriately.