Wasserstein Distributionally Robust Linear-Quadratic Estimation under Martingale Constraints
Wasserstein Distributionally Robust Linear-Quadratic Estimation under Martingale Constraints
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发表时间：
2023
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通讯作者：
Kyriakos Lotidis;N. Bambos;J. Blanchet;Jiajin Li
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作者：
Kyriakos Lotidis;N. Bambos;J. Blanchet;Jiajin Li
研究了一类离散线性随机系统不可观测状态的鲁棒估计问题。这个估计问题的标准分析假设一个基线创新模型，高斯创新，我们恢复卡尔曼滤波器。然而，在许多情况下，没有足够或损坏的数据来验证基线模型。为了科普这个问题，我们最小化在基线周围的Wasserstein邻域内选择的对抗模型的最坏情况均方估计误差。我们还限制了对抗的新息，形成一个鞅差序列。鞅约束放松了独立同分布。这些假设通常被强加于基准模型。此外，我们证明了鞅约束保证了对抗动态仍然适应自然时间生成的信息。因此，添加鞅约束可以改进过于保守的策略，这些策略也可以防止不切实际的无所不知的对手。我们建立了一个强大的对偶结果，我们使用它来开发一个有效的次梯度方法来计算分布鲁棒估计政策。如果基线创新是高斯的，我们表明，最坏情况下的对手仍然是高斯。我们的数值实验表明，鞅约束也可能有助于增加一层的鲁棒性的选择的对抗力量。
We focus on robust estimation of the unobserved state of a discrete-time stochastic system with linear dynamics. A standard analysis of this estimation problem assumes a baseline innovation model; with Gaussian innovations we recover the Kalman filter. However, in many settings, there is insufficient or corrupted data to validate the baseline model. To cope with this problem, we minimize the worst-case mean-squared estimation error of adversarial models chosen within a Wasserstein neighborhood around the baseline. We also constrain the adversarial innovations to form a martingale difference sequence. The mar-tingale constraint relaxes the i.i.d. assumptions which are often imposed on the baseline model. Moreover, we show that the martingale constraints guarantee that the adversarial dynamics remain adapted to the natural time-generated information. Therefore, adding the martingale constraint allows to improve upon over-conservative policies that also protect against unrealistic omniscient adversaries. We establish a strong duality result which we use to develop an efficient subgradient method to compute the distributionally robust estimation policy. If the baseline innovations are Gaussian, we show that the worst-case adversary remains Gaussian. Our numerical experiments indicate that the martingale constraint may also aid in adding a layer of robustness in the choice of the adversarial power.