When Does Training on Synthetic Data Provably Work?

High-quality training data are a major bottleneck in machine learning, motivating increasing use of artificially generated data. We study when a binary classifier trained on synthetic data transfers successfully to a real distribution, under a shared conditional label distribution. First, we derive finite-sample excess-error bounds when the real covariate distribution is absolutely continuous with respect to the synthetic distribution under a controlled density ratio. Second, we study the partially and fully singular case, where the real distribution has mass outside the synthetic one. If the hypothesis class is sufficiently expressive, synthetic risk alone cannot uniformly control real-world error: a classifier can be nearly optimal on synthetic data while approaching worst-case error on the uncovered real region, even when the distributions are arbitrarily close in Wasserstein distance. For function classes where hypotheses and the log-odds of the conditional label probability are Lipschitz, we recover a bound in terms of the probability that a real sample lies outside the reach of the synthetic data, restoring a transfer guarantee even in the fully singular case. Experiments confirm this in controlled low-dimensional settings and on 32,200 images generated as counterparts to real ImageNet images by four state-of-the-art generators: an adversarial classifier reaches 0.5% error on the synthetic domain but 99.25% error on the real domain, while ordinarily trained architectures show positively correlated synthetic and real test errors. Synthetic data should be budgeted at high sample counts and models validated on real data.