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 $\mathcal{O}((n+m)^{\frac12}\varepsilon^{-1})$ gradient computations 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, which is therefore optimal in terms of sample complexity.

翻译：双层优化问题，即嵌套两个优化问题的问题，在机器学习中应用日益广泛。在许多实际场景中，上下层目标函数对应于经验风险最小化问题，因此具有求和结构。在此背景下，我们提出了著名SARAH算法的双层扩展。我们证明，该算法需要$\mathcal{O}((n+m)^{\frac12}\varepsilon^{-1})$次梯度计算即可达到$\varepsilon$-平稳点，其中$n+m$为总样本数，这优于所有先前的双层算法。此外，我们给出了求解双层问题目标函数近似平稳点所需调用次数下界。该下界可由我们的算法达到，因此该算法在样本复杂度方面是最优的。