  Bilevel optimization problems are receiving increasing attention in machine
learning as they provide a natural framework for hyperparameter optimization
and meta-learning. A key step to tackle these problems is the efficient
computation of the gradient of the upper-level objective (hypergradient). In
this work, we study stochastic approximation schemes for the hypergradient,
which are important when the lower-level problem is empirical risk minimization
on a large dataset. The method that we propose is a stochastic variant of the
approximate implicit differentiation approach in (Pedregosa, 2016). We provide
bounds for the mean square error of the hypergradient approximation, under the
assumption that the lower-level problem is accessible only through a stochastic
mapping which is a contraction in expectation. In particular, our main bound is
agnostic to the choice of the two stochastic solvers employed by the procedure.
We provide numerical experiments to support our theoretical analysis and to
show the advantage of using stochastic hypergradients in practice.
