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Stochastic Bilevel optimization usually involves minimizing an upper-level(UL) function that is dependent on the arg-min of a strongly-convex lower-level(LL) function. Several algorithms utilize Neumann series to approximate certainmatrix inverses involved in estimating the implicit gradient of the UL function(hypergradient). The state-of-the-art StOchastic Bilevel Algorithm (SOBA) [16]instead uses stochastic gradient descent steps to solve the linear systemassociated with the explicit matrix inversion. This modification enables SOBAto match the lower bound of sample complexity for the single-level counterpartin non-convex settings. Unfortunately, the current analysis of SOBA relies onthe assumption of higher-order smoothness for the UL and LL functions toachieve optimality. In this paper, we introduce a novel fully single-loop andHessian-inversion-free algorithmic framework for stochastic bileveloptimization and present a tighter analysis under standard smoothnessassumptions (first-order Lipschitzness of the UL function and second-orderLipschitzness of the LL function). Furthermore, we show that by a slightmodification of our approach, our algorithm can handle a more generalmulti-objective robust bilevel optimization problem. For this case, we obtainthe state-of-the-art oracle complexity results demonstrating the generality ofboth the proposed algorithmic and analytic frameworks. Numerical experimentsdemonstrate the performance gain of the proposed algorithms over existing ones.

Optimal Algorithms for Stochastic Bilevel Optimization under Relaxed Smoothness Conditions