  In this paper, we present C-ADAM, the first adaptive solver for compositional
problems involving a non-linear functional nesting of expected values. We proof
that C-ADAM converges to a stationary point in $\mathcal{O}(\delta^{-2.25})$
with $\delta$ being a precision parameter. Moreover, we demonstrate the
importance of our results by bridging, for the first time, model-agnostic
meta-learning (MAML) and compositional optimisation showing fastest known rates
for deep network adaptation to-date. Finally, we validate our findings in a set
of experiments from portfolio optimisation and meta-learning. Our results
manifest significant sample complexity reductions compared to both standard and
compositional solvers.
