We explore the control of stochastic systems with potentially continuous
state and action spaces, characterized by the state dynamics $X_{t+1} = f(X_t,
A_t, W_t)$. Here, $X$, $A$, and $W$ represent the state, action, and exogenous
random noise processes, respectively, with $f$ denoting a known function that
describes state transitions. Traditionally, the noise process $\{W_t, t \geq
0\}$ is assumed to be independent and identically distributed, with a
distribution that is either fully known or can be consistently estimated.
However, the occurrence of distributional shifts, typical in engineering
settings, necessitates the consideration of the robustness of the policy. This
paper introduces a distributionally robust stochastic control paradigm that
accommodates possibly adaptive adversarial perturbation to the noise
distribution within a prescribed ambiguity set. We examine two adversary
models: current-action-aware and current-action-unaware, leading to different
dynamic programming equations. Furthermore, we characterize the optimal finite
sample minimax rates for achieving uniform learning of the robust value
function across continuum states under both adversary types, considering
ambiguity sets defined by $f_k$-divergence and Wasserstein distance. Finally,
we demonstrate the applicability of our framework across various real-world
settings.

该研究论文介绍了一个分布鲁棒的随机控制范式，能够容纳对分布内可能的自适应敌对扰动的噪声进行考虑，在给定的模糊集合内。通过对两种敌对模型的研究，我们揭示了在不同的动态规划方程下的最优有限样本极小化率，以实现对连续状态下健壮价值函数的统一学习，考虑由 f_k - 散度和 Wasserstein 距离定义的模糊集合。最后，我们展示了我们的框架在各种真实世界环境中的适用性。