Fine-Tuning Generative Models for Extreme Events via CVaR-Penalized Wasserstein Gradient Flows

In many high-stakes domains, extreme events carry substantial consequences, yet learning the heavy-tailed distributions that govern them from finite samples remains challenging: the quantities of interest are driven by a few extreme observations, so the tail region is under-sampled relative to its importance. Even generative models tailored for heavy tails capture the tail region inadequately in practice. We propose the Conditional Value-at-Risk (CVaR)-penalized Generative Particle Algorithm (CVaR-GPA), a tail-agnostic algorithm for fine-tuning generative models toward heavy-tailed targets, built as a time discretization of the Wasserstein gradient flow of the Lipschitz-regularized KL divergence penalized by a CVaR discrepancy term. Such a flow can be initialized from the output samples of the pre-trained model, which are then transported along the gradient descent of the loss functional, without requiring access to the pre-trained model's internal architecture. The Lipschitz-regularized KL divergence requires minimal assumptions on the target, while the CVaR penalty focuses the flow on the tail. The CVaR penalty depends on the target only through a scalar tail statistic, inducing a velocity field that remains active in the under-sampled tail region at a dimension-free estimation cost. The resulting velocity field, and hence the depth of the transport map, implicitly adapts to the target distribution, without any target-specific modifications. Across four targets, including two real-world, high-dimensional datasets (daily streamflow in the Ohio River basin ($d=64$) and the Fama–French portfolios ($d=25$) with tail indices ranging from $1.05$ to $3.34$), fine-tuning with CVaR-GPA reduces global and tail errors by geometric-mean factors of $14.0\times$ and $9.8\times$, respectively, across seven pre-trained models spanning GANs, diffusion models, and other generative flows, with a single set of hyperparameters.