  We study a class of algorithms for solving bilevel optimization problems in
both stochastic and deterministic settings when the inner-level objective is
strongly convex. Specifically, we consider algorithms based on inexact implicit
differentiation and we exploit a warm-start strategy to amortize the estimation
of the exact gradient. We then introduce a unified theoretical framework
inspired by the study of singularly perturbed systems (Habets, 1974) to analyze
such amortized algorithms. By using this framework, our analysis shows these
algorithms to match the computational complexity of oracle methods that have
access to an unbiased estimate of the gradient, thus outperforming many
existing results for bilevel optimization. We illustrate these findings on
synthetic experiments and demonstrate the efficiency of these algorithms on
hyper-parameter optimization experiments involving several thousands of
variables.
