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A Generalized Alternating Method for Bilevel Learning under the Polyak-{L}ojasiewicz Condition

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arxiv 2306.02422 v4 pith:WFMMI5CJ submitted 2023-06-04 math.OC cs.LG

A Generalized Alternating Method for Bilevel Learning under the Polyak-{L}ojasiewicz Condition

classification math.OC cs.LG
keywords bilevelproblemsalternatinggeneralizedlearningoptimizationconditionconsidered
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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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 match the convergence rate of single-level gradient descent (GD) when addressing 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 first introduce a stationary metric for the considered bilevel problems, which generalizes the existing metric, for a nonconvex lower-level objective that satisfies the Polyak-{\L}ojasiewicz (PL) condition. We then propose a Generalized ALternating mEthod for bilevel opTimization (GALET) tailored to BLO with convex PL LL problem and establish that GALET achieves an $\epsilon$-stationary point for the considered problem within $\tilde{\cal O}(\epsilon^{-1})$ iterations, which matches the iteration complexity of GD for single-level smooth nonconvex problems.

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Cited by 4 Pith papers

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