Diachronic Sample Integration Robust Tail-Risk Estimation with Generative Models

Jul 16, 2026

Deep generative models increasingly serve as simulators for downstream decisions under data scarcity, yet in risk-sensitive settings their value hinges on rare adverse tail scenarios. Conventional training objectives emphasize bulk distributional fidelity, leaving low-probability regions unstable under finite simulation budgets. Diachronic Sample Integration (DSI) introduces a test-time framework that ensembles samples across checkpoints from a stochastic training trajectory. DSI forms a checkpoint-mixture that averages tail fluctuations, supported by finite-budget bias–variance theory. Experiments on synthetic multivariate processes and high-frequency trading data show substantially reduced tail-estimation error versus single-checkpoint diffusion baselines under equal budgets, without changing the generator objective.

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Introduction

Related Work

Generative Simulators and Tail-Sensitive Objectives

Diachronic Sample Integration: Robust Tail-Risk Estimation with Generative Models Shuning Zhao 1 Patrick Wong2 Leran Zhang3 Xiaolin Hu 1 1Department of Computer Science and Technology, BNRist, Tsinghua University, China 2Department of Econometrics and Business Statistics, Monash Business School, Monash University, Australia 3 School of Mathematics and Statistics, University of Melbourne, Australia arXiv :2607 . 108 10v 1 [ cs .LG] 12 Jul 2026 Abstract Deep generative models are increasingly used as simulators for downstream decision making under data scarcity, but in risk-sensitive applications their usefulness depends on rare adverse scenarios rather than typical samples. Standard generative objectives prioritize bulk distributional fidelity, leaving low-probability tails vulnerable to localized optimization noise and making tail-dependent functionals unstable under finite simulation budgets. We introduce Diachronic Sample Integration (DSI), a test-time inference framework that ensembles generated samples across checkpoints from a stochastic training trajectory. DSI targets a checkpoint-mixture distribution that averages checkpoint-specific tail fluctuations rather than relying on a single brittle endpoint. We formalize this mechanism through a finite-budget bias-variance theory. Empirically, across multivariate synthetic processes and high-frequency trading data, DSI substantially reduces tail-estimation error compared to single-checkpoint baselines under fixed simulation budgets, outperforming standard diffusion and state-of-the-art tail-aware baselines without modifying the generative objective. 1 INTRODUCTION Deep generative models are increasingly used as simulators for downstream decision making under data scarcity. In risk-sensitive applications, however, simulator reliability depends not only on typical samples but also on rare adverse scenarios. Reliable estimation of statistical functionals defined on low-probability regions is central to risk-sensitive inference. Because empirical tail observations are exceptionally scarce, practitioners increasingly rely on deep generative models to act as approximate simulators for downstream risk estimation [Gonen et al., 2025, Tanaka et al., 2025, Liet al., 2025] .1 However, standard generative objectives, such as score matching or maximum likelihood, prioritize the “bulk” of the probability mass where data is dense. The extreme tails are consequently starved of reliable gradient signals and become dominated by localized optimization noise. Drawing synthetic samples from a single “converged” checkpoint often yields severe tail miscalibration, either arbitrarily truncating the tail (mode collapse) or hallucinating unrealistic extremes. Existing approaches attempt to resolve this during training via adversarial tail constraints or extreme-value reweighting [Huang et al., 2024, Galib et al., 2024] . While improving targeted metrics, these modeling interventionscan alter the simulator’s effective target distribution or impose additional structural assumptions. DSI addresses a different practical problem: finite-budget risk estimates without imposing any parametric tail form, distributional constraint, or additional training objective. In many risk-management settings, the immediate task is not to recover an asymptotic tail law, but to estimate risk measures such as Value-at-Risk (VaR) and Expected Shortfall (ES) under a limited simulation budget constrained by compute, latency, or validation resources. We therefore propose Diachronic Sample Integration (DSI), a test-time framework that ensembles samples across stochastic training checkpoints (Figure 1) . Rather than treating a single converged model state as the sole simulator, DSI uses the training trajectory as a source of checkpoint-level variability and aggregates generated samples to reduce tail-estimation instability while leaving the underlying generator unchanged. DSI is not guaranteed to remove

FAQs

What problem does Diachronic Sample Integration (DSI) address?

DSI addresses unstable tail-risk estimation when deep generative models are used as simulators under limited simulation budgets. Standard single-checkpoint sampling can miscalibrate low-probability tails due to optimization noise and scarce tail gradients.

How does DSI improve tail estimation compared with using a single checkpoint?

DSI ensembles generated samples across multiple checkpoints along a stochastic training trajectory. This targets a checkpoint-mixture distribution that averages checkpoint-specific tail fluctuations while leaving persistent trajectory bias as an irreducible error floor.

What do the results show about DSI’s effectiveness under fixed compute budgets?

Across controlled multivariate synthetic processes and high-frequency trading data, DSI substantially reduces tail-estimation error relative to single-checkpoint diffusion baselines. It also outperforms tail-aware baselines while not retraining or modifying the generative objective.

Jul 16, 2026

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