Malliavin Calculus with Weak Derivatives for Counterfactual Stochastic Optimization
This work addresses computational bottlenecks in counterfactual analysis for stochastic optimization, particularly in rare-event regimes, offering an incremental improvement over existing kernel smoothing and score function approaches.
The paper tackles the problem of inefficient Monte Carlo estimation for counterfactual stochastic optimization under rare events, where conditioning events have vanishing probability. It proposes a kernel-free method using Malliavin calculus and weak derivatives, achieving variance comparable to classical Monte Carlo and constant variance for gradient estimates, unlike the score function method whose variance grows linearly with sample path length.
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.