论文标题
使用基于重要性采样的外推针对重尾目标来优化尾巴风险
Optimizing tail risks using an importance sampling based extrapolation for heavy-tailed objectives
论文作者
论文摘要
由于有条件的价值风险（CVAR）的突出性，作为受不确定性影响的设置中的尾巴风险的衡量标准，我们开发了一种新的公式，用于近似于CVAR的优化目标及其梯度来自有限的样本。限制这些优化公式的广泛实际使用的关键难度是最先进的样本平均近似值计划所需的大量数据，以使CVAR目标具有高忠诚度。与最新的样本平均近似值不同，在尾部概率区域中需要大量数据的平均近似值不同，拟议的近似方案利用了重型分布的自相似性，以从合适的较低分位数中推断数据。所产生的近似值在统计上是一致的，并且可以通过常规梯度下降来优化。近似是通过系统的重要性采样方案进行指导的，该方案的渐近方差降低属性得到了严格的检查。数值实验证明了所提出的近似值的优越性，并且实现的易于实现点与可以应用近似方案的设置的多功能性。
Motivated by the prominence of Conditional Value-at-Risk (CVaR) as a measure for tail risk in settings affected by uncertainty, we develop a new formula for approximating CVaR based optimization objectives and their gradients from limited samples. A key difficulty that limits the widespread practical use of these optimization formulations is the large amount of data required by the state-of-the-art sample average approximation schemes to approximate the CVaR objective with high fidelity. Unlike the state-of-the-art sample average approximations which require impractically large amounts of data in tail probability regions, the proposed approximation scheme exploits the self-similarity of heavy-tailed distributions to extrapolate data from suitable lower quantiles. The resulting approximations are shown to be statistically consistent and are amenable for optimization by means of conventional gradient descent. The approximation is guided by means of a systematic importance-sampling scheme whose asymptotic variance reduction properties are rigorously examined. Numerical experiments demonstrate the superiority of the proposed approximations and the ease of implementation points to the versatility of settings to which the approximation scheme can be applied.
