We introduce Multistage Conditional Compositional Optimization (MCCO) as a new paradigm for decision-making under uncertainty that combines aspects of multistage stochastic programming and conditional stochastic optimization. MCCO minimizes a nest of conditional expectations and nonlinear cost functions. It has numerous applications and arises, for example, in optimal stopping, linear-quadratic regulator problems, distributionally robust contextual bandits, as well as in problems involving dynamic risk measures. The naïve nested sampling approach for MCCO suffers from the curse of dimensionality familiar from scenario tree-based multistage stochastic programming, that is, its scenario complexity grows exponentially with the number of nests. We develop new multilevel Monte Carlo techniques for MCCO whose scenario complexity grows only polynomially with the desired accuracy.

翻译：我们提出多阶段条件复合优化（Multistage Conditional Compositional Optimization, MCCO）作为一种在不确定性下进行决策的新范式，它融合了多阶段随机规划与条件随机优化的特点。MCCO旨在最小化条件期望与非线性成本函数的嵌套结构。该方法具有广泛的应用场景，例如最优停止问题、线性二次型调节器问题、分布鲁棒上下文强盗问题，以及涉及动态风险度量的各类问题。基于朴素嵌套抽样的MCCO方法面临维度灾难的挑战（这与基于场景树的多阶段随机规划类似），其场景复杂度随嵌套层数呈指数增长。我们针对MCCO开发了新颖的多层级蒙特卡洛技术，使得场景复杂度仅随所需精度呈多项式增长。