In this work, we consider constrained stochastic optimization problems under hidden convexity, i.e., those that admit a convex reformulation via non-linear (but invertible) map $c(\cdot)$. A number of non-convex problems ranging from optimal control, revenue and inventory management, to convex reinforcement learning all admit such a hidden convex structure. Unfortunately, in the majority of applications considered, the map $c(\cdot)$ is unavailable or implicit; therefore, directly solving the convex reformulation is not possible. On the other hand, the stochastic gradients with respect to the original variable are often easy to obtain. Motivated by these observations, we examine the basic projected stochastic (sub-) gradient methods for solving such problems under hidden convexity. We provide the first sample complexity guarantees for global convergence in smooth and non-smooth settings. Additionally, in the smooth setting, we improve our results to the last iterate convergence in terms of function value gap using the momentum variant of projected stochastic gradient descent.

翻译：本文研究了一类在隐藏凸性下的约束随机优化问题，即通过非线性（但可逆）的映射$c(\cdot)$可以转化为凸优化的问题。从最优控制、收益与库存管理到凸强化学习等一系列非凸问题，均具有此类隐藏凸结构。然而，在大多数实际应用中，映射$c(\cdot)$通常是未知或隐式的，因此无法直接求解其凸重构形式。另一方面，原始变量的随机梯度往往易于获取。基于上述观察，我们研究了用于求解此类隐藏凸性问题的基本投影随机（次）梯度方法，首次在光滑与非光滑场景下给出了全局收敛的样本复杂度保证。此外，在光滑设定下，我们采用投影随机梯度下降的动量变体，进一步改进了函数值差距上的最终迭代收敛结果。