Asymptotic Analysis of an Abstract Stochastic Scheme for Solving Monotone Inclusions

We propose an abstract stochastic scheme for solving a broad range of monotone operator inclusion problems in Hilbert spaces. This framework allows for the introduction of stochasticity at several levels in monotone operator splitting methods: approximation of operators, selection of coordinates and operators in block-iterative implementations, and relaxation parameters. The analysis involves an abstract reduced inclusion model with two operators. At each iteration of the proposed scheme, stochastic approximations to points in the graphs of these two operators are used to form the update. The results are applied to derive the almost sure and $L^2$ convergence of stochastic versions of the proximal point algorithm, as well as of randomized block-iterative projective splitting methods for solving systems of coupled inclusions involving a mix of set-valued, cocoercive, and Lipschitzian monotone operators combined via various monotonicity-preserving operations.

💡 Research Summary

The paper addresses the problem of solving monotone inclusions of the form 0 ∈ W x + C x in a real separable Hilbert space H, where W is maximally monotone and C is α‑cocoercive, assuming the solution set Z = zer(W + C) is non‑empty. Classical forward‑backward methods require explicit access to the resolvent of W, which is often unavailable in large‑scale or composite settings. To overcome this limitation, the authors propose an abstract stochastic scheme that works with stochastic approximations of points in the graphs of W and C, and allows random relaxation parameters beyond the usual (0, 2) interval.

The deterministic template (Algorithm 1.2) selects a pair (wₙ, wₙ*) ∈ gra W and a point qₙ ∈ H, forms tₙ* = wₙ* + C qₙ, computes a scalar Δₙ and a weight θₙ, and updates xₙ₊₁ = xₙ − λₙ dₙ with a step size λₙ ∈ (0, 2). This template underlies many splitting algorithms.

The stochastic extension (Algorithm 1.3) replaces the exact graph points by random variables wₙ, wₙ*, eₙ, eₙ* such that (wₙ + eₙ, wₙ* + eₙ*) ∈ gra W almost surely, and similarly approximates C qₙ by cₙ* + fₙ* with cₙ* + fₙ* = C qₙ a.s. The relaxation parameter λₙ is taken from L^∞(Ω;]0, ρ]) with ρ ≥ 2, thus permitting values larger than 2. The algorithm only requires stochastic access to the operators, making it suitable for block‑iterative or distributed implementations.

The core convergence analysis is presented in Theorem 3.1. Under the assumptions that (i) λₙ is independent of the σ‑algebra generated by past iterates and satisfies E