  Bilevel optimization is a popular hierarchical model in machine learning, and
has been widely applied to many machine learning tasks such as meta learning,
hyperparameter learning and policy optimization. Although many bilevel
optimization algorithms recently have been developed, few adaptive algorithm
focuses on the bilevel optimization under the distributed setting. It is well
known that the adaptive gradient methods show superior performances on both
distributed and non-distributed optimization. In the paper, thus, we propose a
novel adaptive federated bilevel optimization algorithm (i.e.,AdaFBiO) to solve
the distributed bilevel optimization problems, where the objective function of
Upper-Level (UL) problem is possibly nonconvex, and that of Lower-Level (LL)
problem is strongly convex. Specifically, our AdaFBiO algorithm builds on the
momentum-based variance reduced technique and local-SGD to obtain the best
known sample and communication complexities simultaneously. In particular, our
AdaFBiO algorithm uses the unified adaptive matrices to flexibly incorporate
various adaptive learning rates to update variables in both UL and LL problems.
Moreover, we provide a convergence analysis framework for our AdaFBiO
algorithm, and prove it needs the sample complexity of
$\tilde{O}(\epsilon^{-3})$ with communication complexity of
$\tilde{O}(\epsilon^{-2})$ to obtain an $\epsilon$-stationary point.
Experimental results on federated hyper-representation learning and federated
data hyper-cleaning tasks verify efficiency of our algorithm.
