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：双层优化问题是指两个优化问题嵌套的问题，在机器学习中应用日益广泛。在许多实际案例中，上下层目标函数对应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算法的双层扩展版本。我们证明该算法仅需$\mathcal{O}((n+m)^{\frac12}\varepsilon^{-1})$次梯度计算即可达到$\varepsilon$-平稳性，其中$n+m$为总样本数，这一结果优于所有现有双层优化算法。此外，我们给出了获取双层问题目标函数近似平稳点所需的最小Oracle调用次数下界，该下界被我们的算法所达到，因此在样本复杂度上具有最优性。