DOUBLE INERTIAL STOCHASTIC RELAXED FORWARD-BACKWARD-FORWARD ALGORITHM

We propose the double inertial stochastic relaxed forward-backward-forward (DI-SRIFBF) algorithm for solving monotone inclusions in Hilbert spaces. The method augments the stochastic FBF framework with two sequential inertial extrapolation steps, and it evaluates the single-valued operator via independent mini-batches at each forward step. This design avoids the correlation issues that arise when the same batch is reused, and it yields a stochastic conditioning that fits the Robbins–Siegmund template without hidden measurability assumptions. Under explicit, algebraically verifiable step-size restrictions that hold for every relaxation parameter ρ ∈ (0, 2), we establish almost-sure weak convergence provided the inertial parameters and mini-batch variances are summable. We also prove an O(1/k) rate for the discrete velocity and, under strong monotonicity with geometrically growing batch sizes, linear convergence. Choosing the growth factor τ = (1 − η/2)−1 gives an O(1/ε) oracle complexity. Finally, we derive explicit non-asymptotic bounds for biased stochastic oracles, showing that the iterates settle into an O(B2/μ2) neighborhood of the unique solution. Numerical experiments on two-stage stochastic variational inequalities and group sparse learning confirm the theoretical predictions and demonstrate a practical advantage over single-inertial and non-inertial benchmarks.