We consider a distributionally robust stochastic optimization problem and formulate it as a stochastic two-level composition optimization problem with the use of the mean--semideviation risk measure. In this setting, we consider a single time-scale algorithm, involving two versions of the inner function value tracking: linearized tracking of a continuously differentiable loss function, and SPIDER tracking of a weakly convex loss function. We adopt the norm of the gradient of the Moreau envelope as our measure of stationarity and show that the sample complexity of $\mathcal{O}(\varepsilon^{-3})$ is possible in both cases, with only the constant larger in the second case. Finally, we demonstrate the performance of our algorithm with a robust learning example and a weakly convex, non-smooth regression example.

翻译：我们考虑一个分布鲁棒随机优化问题，并借助均值-半偏差风险度量将其转化为一个随机双层组合优化问题。在此框架下，我们提出一种单时间尺度算法，其中包含两种内部函数值跟踪方式：连续可微损失函数的线性化跟踪，以及弱凸损失函数的SPIDER跟踪。我们采用Moreau包络梯度的范数作为稳定性的度量指标，并证明在两种情况下样本复杂度均可达到$\mathcal{O}(\varepsilon^{-3})$，仅在第二种情形中常数项更大。最后，我们通过一个鲁棒学习实例和一个弱凸非光滑回归实例验证了算法的性能。