Accelerating Inexact HyperGradient Descent for Bilevel Optimization
This work addresses a fundamental challenge in bilevel optimization for machine learning and optimization researchers, offering improved theoretical guarantees.
They tackled bilevel optimization problems by proposing the Restarted Accelerated HyperGradient Descent (RAHGD) method, achieving an oracle complexity of $ ilde{\mathcal{O}}(κ^{3.25}ε^{-1.75})$ for finding an $ε$-first-order stationary point and setting a new state-of-the-art benchmark for second-order stationary points.
We present a method for solving general nonconvex-strongly-convex bilevel optimization problems. Our method -- the \emph{Restarted Accelerated HyperGradient Descent} (\texttt{RAHGD}) method -- finds an $ε$-first-order stationary point of the objective with $\tilde{\mathcal{O}}(κ^{3.25}ε^{-1.75})$ oracle complexity, where $κ$ is the condition number of the lower-level objective and $ε$ is the desired accuracy. We also propose a perturbed variant of \texttt{RAHGD} for finding an $\big(ε,\mathcal{O}(κ^{2.5}\sqrtε\,)\big)$-second-order stationary point within the same order of oracle complexity. Our results achieve the best-known theoretical guarantees for finding stationary points in bilevel optimization and also improve upon the existing upper complexity bound for finding second-order stationary points in nonconvex-strongly-concave minimax optimization problems, setting a new state-of-the-art benchmark. Empirical studies are conducted to validate the theoretical results in this paper.