论文标题
使用多级蒙特卡洛法对条件价值的基于梯度的优化
Gradient-based optimisation of the conditional-value-at-risk using the multi-level Monte Carlo method
论文作者
论文摘要
在这项工作中，我们使用基于基于梯度的方法与多级蒙特卡洛（MLMC）方法结合使用的是随机输入数据的复杂差分模型的输出数量的有条件值 - at风险（CVAR）的问题。特别是，我们考虑了多级蒙特卡洛的框架，以进行参数期望，并提出对MLMC估计器的修改，误差估计过程和自适应MLMC参数选择，以确保具有规定准确性的给定设计的CVAR和敏感性的估计。然后，我们提出将MLMC框架与交替的不进行最小化梯度下降算法相结合，为此，我们证明了在强凸性和Lipschitz的梯度连续性的假设下的优化迭代中指数收敛。我们在两个实际相关性的数值示例上证明了方法的性能，这证明了与标准MLMC方法相同的最佳渐近成本耐受性行为，用于固定的输出期望计算。
In this work, we tackle the problem of minimising the Conditional-Value-at-Risk (CVaR) of output quantities of complex differential models with random input data, using gradient-based approaches in combination with the Multi-Level Monte Carlo (MLMC) method. In particular, we consider the framework of multi-level Monte Carlo for parametric expectations and propose modifications of the MLMC estimator, error estimation procedure, and adaptive MLMC parameter selection to ensure the estimation of the CVaR and sensitivities for a given design with a prescribed accuracy. We then propose combining the MLMC framework with an alternating inexact minimisation-gradient descent algorithm, for which we prove exponential convergence in the optimisation iterations under the assumptions of strong convexity and Lipschitz continuity of the gradient of the objective function. We demonstrate the performance of our approach on two numerical examples of practical relevance, which evidence the same optimal asymptotic cost-tolerance behaviour as standard MLMC methods for fixed design computations of output expectations.
