提出新方法解决罕见事件下条件优化的低效问题。
Malliavin Calculus with Weak Derivatives for Counterfactual Stochastic Optimization
- 用马利万微分构造无核的精确积分表示
- 弱导数估计方差恒定，优于传统方法的线性增长
- 适合罕见事件下的高效参数优化，如金融风险建模
我们研究在模型误设和噪声梯度信息下，条件损失泛函的反事实随机优化。当条件事件概率趋近于零时，朴素蒙特卡洛估计器效率极低；虽然核平滑常用，但收敛速度缓慢。本文提出两阶段无核方法：首先，利用马利万微分证明扩散过程的条件损失泛函可精确表示为斯科罗霍德积分，方差与经典蒙特卡洛相当；其次，证明关于模型参数的弱导数估计具有恒定方差，而广泛使用的得分函数法方差随样本路径长度线性增长。上述结果共同构建了罕见事件条件下高效反事实条件随机梯度算法框架。
原文摘要 · Abstract (English)
We study counterfactual stochastic optimization of conditional loss functionals under misspecified and noisy gradient information. The difficulty is that when the conditioning event has vanishing or zero probability, naive Monte Carlo estimators are prohibitively inefficient; kernel smoothing, though common, suffers from slow convergence. We propose a two-stage kernel-free methodology. First, we show using Malliavin calculus that the conditional loss functional of a diffusion process admits an exact representation as a Skorohod integral, yielding variance comparable to classical Monte-Carlo variance. Second, we establish that a weak derivative estimate of the conditional loss functional with respect to model parameters can be evaluated with constant variance, in contrast to the widely used score function method whose variance grows linearly in the sample path length. Together, these results yield an efficient framework for counterfactual conditional stochastic gradient algorithms in rare-event regimes.
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