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However, those momentum-based algorithms do not achieve provably better computational complexity than $\\mathcal{\\widetilde O}(\\epsilon^{-2})$ of the SGD-based algorithm. In this paper, we propose two new algorithms for bilevel optimization, where the first algorithm adopts momentum-based recursive iterations, and the second algorithm adopts recursive gradient estimations in nested loops to decrease the variance. We show that both algorithms achieve the complexity of $\\mathcal{\\widetilde O}(\\epsilon^{-1.5})$, which outperforms all existing algorithms by the order of magnitude. 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