利用基于重尾目标的重要性抽样外推法优化尾部风险
Optimizing tail risks using an importance sampling based extrapolation for heavy-tailed objectives
摘要：
出于突出的条件风险价值（CVaR）作为衡量尾部风险的设置受不确定性的影响，我们开发了一个新的公式近似基于CVaR的优化目标和它们的梯度从有限的样本。一个关键的困难，限制了广泛的实际使用这些优化配方是大量的数据所需的国家的最先进的样本平均近似计划，以近似CVaR目标高保真。不同于国家的最先进的样本平均近似，需要不切实际的大量数据的尾部概率区域，建议的近似方案利用重尾分布的自相似性外推数据从合适的较低的分位数。所得到的近似被证明是统计上一致的，并适合通过传统的梯度下降优化。该近似是指导通过系统的重要性抽样方案，其渐近方差降低性能进行严格检查。数值实验表明，所提出的近似的优越性和易于实施点的多功能性设置的近似方案可以应用。
英文摘要：
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.
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