Synthetic data can improve generalization when real data is scarce, but excessive reliance may introduce distributional mismatches that degrade performance. In this paper, we present a learning-theoretic framework to quantify the trade-off between synthetic and real data. Our approach leverages algorithmic stability to derive generalization error bounds, characterizing the optimal synthetic-to-real data ratio that minimizes expected test error as a function of the Wasserstein distance between the real and synthetic distributions. We motivate our framework in the setting of kernel ridge regression with mixed data, offering a detailed analysis that may be of independent interest. Our theory predicts the existence of an optimal ratio, leading to a U-shaped behavior of test error with respect to the proportion of synthetic data. Empirically, we validate this prediction on CIFAR-10 and a clinical brain MRI dataset. Our theory extends to the important scenario of domain adaptation, showing that carefully blending synthetic target data with limited source data can mitigate domain shift and enhance generalization. We conclude with practical guidance for applying our results to both in-domain and out-of-domain scenarios.

翻译：当真实数据稀缺时，合成数据能够提升泛化性能，但过度依赖可能引入分布失配而降低模型表现。本文提出一种学习理论框架，用于量化合成数据与真实数据间的权衡关系。我们的方法利用算法稳定性推导泛化误差界，通过真实分布与合成分布之间的Wasserstein距离函数，刻画了最小化期望测试误差的最优合成-真实数据比例。我们在核岭回归的混合数据场景中构建理论框架，提供了可能具有独立价值的详细分析。理论预测存在最优比例，导致测试误差随合成数据占比呈现U型变化规律。我们在CIFAR-10和临床脑部MRI数据集上实证验证了这一预测。本理论可扩展至领域自适应的重要场景，表明将合成目标数据与有限源数据精心混合能够缓解领域偏移并增强泛化能力。最后，我们为在域内和域外场景中应用本研究成果提供了实践指导。