Bilevel optimization problems, which are problems where two optimization problems are nested, have more and more applications in machine learning. In many practical cases, the upper and the lower objectives correspond to empirical risk minimization problems and therefore have a sum structure. In this context, we propose a bilevel extension of the celebrated SARAH algorithm. We demonstrate that the algorithm requires $\mathcal{O}((n+m)^{\frac12}\varepsilon^{-1})$ oracle calls to achieve $\varepsilon$-stationarity with $n+m$ the total number of samples, which improves over all previous bilevel algorithms. Moreover, we provide a lower bound on the number of oracle calls required to get an approximate stationary point of the objective function of the bilevel problem. This lower bound is attained by our algorithm, making it optimal in terms of sample complexity.

翻译：双层优化问题，即嵌套两个优化问题的形式，在机器学习中应用日益广泛。在许多实际场景中，上下层的目标函数对应经验风险最小化问题，因而具有求和结构。基于此，我们提出了著名SARAH算法的双层扩展版本。我们证明该算法在总样本量为$n+m$时，仅需$\mathcal{O}((n+m)^{\frac12}\varepsilon^{-1})$次Oracle调用即可达到$\varepsilon$-平稳性，这一复杂度优于所有现有双层优化算法。此外，我们给出了求解双层问题目标函数近似驻点所需Oracle调用的下界。该下界被我们的算法所达到，从而在样本复杂度上实现了最优性。