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，并利用均值-半偏差风险度量将其表述为随机双层组合优化问题。在此设定下，我们采用单时间尺度算法，包含两种内函数值跟踪策略：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。我们以莫罗包络梯度的范数作为平稳性度量，证明在两种情况下均可实现$\mathcal{O}(\varepsilon^{-3})$的样本复杂度，仅第二种情况的常数项更大。最后，通过鲁棒学习示例和弱凸非光滑回归示例验证了算法的性能。