Synthetic Data Augmentation for Functional Data: Wasserstein Risk Bounds and Diffusion-Based Conditional Generation

Generative data augmentation is widely used to mitigate class imbalance, yet its theoretical effect on downstream generalization remains poorly understood. In this work, we develop a statistical framework for conditional generative augmentation and analyze its impact on classification risk. We formalize augmentation as a distribution-mixing process and show that the resulting risk distortion is controlled by both the augmentation strength and the class-conditional Wasserstein discrepancy between real and generated distributions. We further derive a capacity-dependent generalization bound based on Rademacher complexity, revealing an explicit trade-off between hypothesis complexity, augmentation intensity, and generative fidelity. Empirically, we evaluate the framework on binary and multiclass imbalanced classification tasks using Conditional GAN and Conditional WGAN-GP augmentation. Across datasets, CWGAN-GP consistently achieves lower Wasserstein discrepancies than CGAN, indicating improved distributional fidelity. However, improved fidelity does not necessarily translate into superior classification performance, with classical oversampling methods often remaining competitive. These findings support the central theoretical prediction that augmentation reliability is governed by distributional approximation error rather than predictive performance alone. Overall, this work establishes generative augmentation as a distributional perturbation process whose reliability can be quantified through Wasserstein-based measures and supported by finite-sample generalization guarantees. The proposed framework provides a principled foundation for evaluating synthetic data quality beyond classification accuracy alone.

Synthetic data augmentation is widely used to mitigate class imbalance, but its theoretical effects on score-based classification remain poorly understood. This paper develops a framework for characterizing when synthetic minority augmentation can improve threshold-integrated and threshold-optimized metrics, including AUROC, AUPRC, best-threshold balanced accuracy, and best-threshold $\F_1$ score. We separate the effect of augmentation into two components: a change in effective class weighting and a discrepancy between the synthetic and true minority distributions. Under well-specified score models, the raw estimator already targets the likelihood-ratio ordering, which is population-optimal for the metrics considered. Consequently, augmentation cannot provide a fundamental population-level improvement beyond possible finite-sample variance reduction, and may introduce additional bias through synthetic distributional error. We further establish minimax lower bounds showing that the raw estimator already achieves the optimal metric-regret rate in the well-specified regime. Under misspecification, however, augmentation can play a qualitatively different role: by changing the effective class balance, it can alter the restricted-class projection and correct ranking errors induced by the raw imbalanced objective. We provide explicit improvement bounds quantifying the roles of approximation error, finite-sample estimation error, and synthetic distributional error. Simulation studies corroborate the theory, demonstrating limited gains under well-specification and nontrivial but nonmonotone improvements under misspecification.

Synthetic augmentation is increasingly used to mitigate data scarcity in financial machine learning, yet its statistical role remains poorly understood. We formalize synthetic augmentation as a modification of the effective training distribution and show that it induces a structural bias--variance trade-off: while additional samples may reduce estimation error, they may also shift the population objective whenever the synthetic distribution deviates from regions relevant under evaluation. To isolate informational gains from mechanical sample-size effects, we introduce a size-matched null augmentation and a finite-sample, non-parametric block permutation test that remains valid under weak temporal dependence. We evaluate this framework in both controlled Markov-switching environments and real financial datasets, including high-frequency option trade data and a daily equity panel. Across generators spanning bootstrap, copula-based models, variational autoencoders, diffusion models, and TimeGAN, we vary augmentation ratio, model capacity, task type, regime rarity, and signal-to-noise. We show that synthetic augmentation is beneficial only in variance-dominant regimes, such as persistent volatility forecasting-while it deteriorates performance in bias-dominant settings, including near-efficient directional prediction. Rare-regime targeting can improve domain-specific metrics but may conflict with unconditional permutation inference. Our results provide a structural perspective on when synthetic data improves financial learning performance and when it induces persistent distributional distortion.