A Generalized Alternating Method for Bilevel Optimization under the Polyak-Łojasiewicz Condition
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Bilevel optimization has recently regained interest owing to its applications in emerging machine learning fields such as hyperparameter optimization, meta-learning, and reinforcement learning. Recent results have shown that simple alternating (implicit) gradient-based algorithms can achieve the same convergence rate of single-level gradient descent (GD) for bilevel problems with a strongly convex lower-level objective. However, it remains unclear whether this result can be generalized to bilevel problems beyond this basic setting. In this paper, we propose a Generalized ALternating mEthod for bilevel opTimization (GALET) with a nonconvex lower-level objective that satisfies the Polyak-Łojasiewicz (PL) condition. We first introduce a stationary metric for the considered bilevel problems, which generalizes the existing metric. We then establish that GALET achieves an ϵ-stationary metric for the considered problem within Õ(ϵ^-1) iterations, which matches the iteration complexity of GD for smooth nonconvex problems.
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