  *Data Synthesis* is a promising way to train a small model with very little
labeled data. One approach for data synthesis is to leverage the rich knowledge
from large language models to synthesize pseudo training examples for small
models, making it possible to achieve both data and compute efficiency at the
same time. However, a key challenge in data synthesis is that the synthesized
dataset often suffers from a large distributional discrepancy from the *real
task* data distribution. Thus, in this paper, we propose *Synthesis Step by
Step* (**S3**), a data synthesis framework that shrinks this distribution gap
by iteratively extrapolating the errors made by a small model trained on the
synthesized dataset on a small real-world validation dataset using a large
language model. Extensive experiments on multiple NLP tasks show that our
approach improves the performance of a small model by reducing the gap between
the synthetic dataset and the real data, resulting in significant improvement
compared to several baselines: 9.48% improvement compared to ZeroGen and 2.73%
compared to GoldGen, and at most 15.17% improvement compared to the small model
trained on human-annotated data.
