提出一种基于样本的最优确定等价风险优化方法,提升金融与机器学习中的风险控制能力。
Optimized Certainty Equivalent Risk Minimization Using Samples: Algorithms, Convergence Rates, and Applications

- 通过效用型不足风险关联,将OCE转化为可样本估计的形式
- 推导出非渐近均方误差上界,证明算法收敛性与稳定性
- 适用于投资组合优化与不确定性量化,适合关注风险建模的研究者
本文研究最优确定等价(OCE)风险的优化问题,应用于金融领域的投资组合优化,以及机器学习中的不确定性量化、分类和回归。涵盖熵风险、均值-方差风险及平滑条件风险价值等典型情形。文章明确了将OCE推广至无界随机变量的条件,并建立了OCE与效用型不足风险(UBSR)之间的联系。基于此,我们从UBSR的经典样本平均逼近(SAA)出发构造了OCE估计器,推导出其均方误差(MSE)上界。进一步,利用该关联推导出OCE梯度表达式,构建梯度估计器,并给出其非渐近MSE上界。将该梯度估计器融入随机梯度(SG)算法中,实现对OCE的优化,并以非渐近界量化其收敛速率。最后,通过三个实验验证该算法在投资组合优化与不确定性量化任务中的有效性。
原文摘要 · Abstract (English)
We consider the optimization of the Optimized Certainty Equivalent (OCE) risk, with applications including portfolio optimization in finance, and uncertainty quantification, classification, and regression in machine learning. Our contributions cover popular special cases of OCE, such as entropic risk, mean-variance risk, and smooth variants of Conditional Value-at-Risk. Our treatment sets out the conditions that facilitate the extension of OCE to unbounded r.v.s.. We provide a useful characterization of OCE that links OCE to utility-based shortfall risk (UBSR). Our characterization enables us to form an OCE estimator from the classic sample-average approximation (SAA) of UBSR. We derive mean-squared error (MSE) bounds for our proposed OCE estimator. For OCE optimization, we first derive an expression for the OCE gradient using the characterization linking OCE to UBSR. This expression serves as the basis for a gradient estimator for the OCE. We derive non-asymptotic bounds on the MSE for the proposed OCE gradient estimator. We incorporate the aforementioned gradient estimator into a stochastic gradient (SG) algorithm to optimize OCE and quantify its convergence rate using non-asymptotic bounds that we derive. Finally, we present three experiments that use our OCE optimization algorithm to solve portfolio optimization and uncertainty quantification problems.
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