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In this paper, we propose a new Hessian inverse free Fully Single LoopAlgorithm (FSLA) for bilevel optimization problems. Classic algorithms forbilevel optimization admit a double loop structure which is computationallyexpensive. Recently, several single loop algorithms have been proposed withoptimizing the inner and outer variable alternatively. However, thesealgorithms not yet achieve fully single loop. As they overlook the loop neededto evaluate the hyper-gradient for a given inner and outer state. In order todevelop a fully single loop algorithm, we first study the structure of thehyper-gradient and identify a general approximation formulation ofhyper-gradient computation that encompasses several previous common approaches,e.g. back-propagation through time, conjugate gradient, \emph{etc.} Based onthis formulation, we introduce a new state variable to maintain the historicalhyper-gradient information. Combining our new formulation with the alternativeupdate of the inner and outer variables, we propose an efficient fully singleloop algorithm. We theoretically show that the error generated by the new statecan be bounded and our algorithm converges with the rate of $O(\epsilon^{-2})$.Finally, we verify the efficacy our algorithm empirically through multiplebilevel optimization based machine learning tasks.

A Fully Single Loop Algorithm for Bilevel Optimization without Hessian Inverse