We propose CVaR-penalized Generative Particle Algorithm (CVaR-GPA), a robust, tail-agnostic algorithm for fine-tuning generative models to learn heavy-tailed distributions and capture extreme events, requiring no prior knowledge or estimation of the target's tail characteristics. The method is the Wasserstein gradient flow of the Lipschitz-regularized Kullback-Leibler (KL) divergence penalized by a Conditional Value-at-Risk (CVaR) discrepancy term: the Lipschitz-regularized KL divergence enables robust learning under minimal assumptions on the target distribution, while the CVaR penalty restores the velocity that otherwise vanishes prematurely in the under-sampled tails. The penalized flow admits a bounded but non-Lipschitz velocity field. This departs from the Lipschitz transport maps of standard generators, which preserve the tail behavior of a light-tailed source, and enables transport toward heavier-tailed targets. To define this flow on empirical measures, we derive the first-variation subgradients of CVaR from its Rockafellar-Uryasev representation, valid precisely where the classical density-based formula fails. The particle algorithm CVaR-GPA fine-tunes the output samples of any pre-trained model, without access to its architecture, and runs on an adaptive time horizon set by a kinetic-energy stopping criterion rather than a preset depth. On synthetic isotropic and anisotropic Student-$t$ target distributions, Neal's funnel distribution, and the real-world high-dimensional Fama-French 25 portfolio dataset, CVaR-GPA dramatically improves global and tail accuracy on heavy-tailed targets over the pre-trained baseline.
Generating realistic time series is essential for scientific research and real-world applications. However, existing methods often emphasize overall distributional fidelity while failing to faithfully capture extreme events. To advance existing research, we propose E4GEN, an explainable diffusion framework for extreme event-aware time-series generation. E4GEN provides systematic insights into when, what, and how to control extreme-event generation through three key components. First, E-Activator learns the dataset-adaptive extreme-control signal activation step during the denoising process without interfering with regular temporal components, including trend and seasonality. Second, E-Predictor determines what control signal to enforce through Self-Driven Semantic Prediction, where each sample derives its own control signal by inferring latent extreme-event information during generation. It also includes a novel Data-Conditioned Training, Noise-Initiated Sampling mechanism to address the issue of unavailable training labels. Third, E-Control specifies how to control extreme-event generation through a trainable Extreme Control Network, which transforms the semantic control signal into layer-wise signals and injects it into the denoising process. We evaluate E4GEN on six datasets with 17 metrics, and extensive experiments show that E4GEN outperforms state-of-the-art models across multiple dimensions, including overall fidelity, extreme-event fidelity, and downstream utility.