  We consider monotone inclusions defined on a Hilbert space where the operator
is given by the sum of a maximal monotone operator $T$ and a single-valued
monotone, Lipschitz continuous, and expectation-valued operator $V$. We draw
motivation from the seminal work by Attouch and Cabot on relaxed inertial
methods for monotone inclusions and present a stochastic extension of the
relaxed inertial forward-backward-forward (RISFBF) method. Facilitated by an
online variance reduction strategy via a mini-batch approach, we show that
(RISFBF) produces a sequence that weakly converges to the solution set.
Moreover, it is possible to estimate the rate at which the discrete velocity of
the stochastic process vanishes. Under strong monotonicity, we demonstrate
strong convergence, and give a detailed assessment of the iteration and oracle
complexity of the scheme. When the mini-batch is raised at a geometric
(polynomial) rate, the rate statement can be strengthened to a linear (suitable
polynomial) rate while the oracle complexity of computing an
$\epsilon$-solution improves to $O(1/\epsilon)$. Importantly, the latter claim
allows for possibly biased oracles, a key theoretical advancement allowing for
far broader applicability. By defining a restricted gap function based on the
Fitzpatrick function, we prove that the expected gap of an averaged sequence
diminishes at a sublinear rate of $O(1/k)$ while the oracle complexity of
computing a suitably defined $\epsilon$-solution is $O(1/\epsilon^{1+a})$ where
$a>1$. Numerical results on two-stage games and an overlapping group Lasso
problem illustrate the advantages of our method compared to stochastic
forward-backward-forward (SFBF) and SA schemes.
