基于样本的优化确定等价风险最小化：算法、收敛速率及应用

中文摘要 AI 辅助

我们研究优化确定等价（OCE）风险的问题，其应用涵盖金融中的投资组合优化、机器学习中的不确定性量化、分类及回归任务。本文的贡献包括OCE的常见特殊情形，如熵风险、均值-方差风险及条件风险价值（CVaR）的平滑变体。我们阐述了使OCE可扩展至无界随机变量的条件，并给出OCE与基于效用的短缺风险（UBSR）关联的有用刻画，该刻画可用于从UBSR的经典样本平均近似（SAA）构造OCE估计量。我们推导了所提OCE估计量的均方误差（MSE）界；对于OCE优化，我们利用OCE与UBSR的关联推导OCE梯度表达式，该表达式是OCE梯度估计量的基础，我们还推导了该梯度估计量MSE的非渐近界。我们将上述梯度估计量整合到随机梯度（SG）算法中以优化OCE，并通过推导的非渐近界量化其收敛速率。最后，我们开展三项实验，利用所提OCE优化算法解决投资组合优化及不确定性量化问题。

英文摘要

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.

发表机构

• TCS Research（塔塔咨询服务公司研究院）
• Indian Institute of Technology Madras（印度理工学院马德拉斯分校）

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