Convergence of SBBP to a bilevel solution in the inconsistent case

Establish a rigorous convergence analysis proving that the stochastic block Bregman projection (SBBP) iteration x_{k+1}^* = x_k^* − t_k Σ_{i∈J_k} w_{i,k} ∇f_i(x_k), x_{k+1} = ∇ψ*(x_{k+1}^*), converges to a solution of the bilevel optimization problem minimize ψ(x) subject to x ∈ argmin_x F(x) with F(x) = E[f_i(x)], in the setting of inconsistent convex feasibility problems where the intersection of the constraint sets is empty.

Background

The paper proves that SBBP with Polyak-like and projective stepsizes converges (linearly in expectation under suitable assumptions) to the inner minimizer set of the proximity function F, and—under additional smoothness and strong convexity assumptions—converges to the unique solution of the bilevel problem in the consistent case.

However, when the convex feasibility problem is inconsistent, the authors do not provide a proof that SBBP converges to the bilevel solution and explicitly defer this analysis. Since the bilevel formulation selects a regularized solution from the set of minimizers of F, establishing convergence to this solution in the inconsistent regime would complete the theoretical guarantees of the proposed framework.