  Recent advances have substantially improved the accuracy, memory cost, and
training speed of differentially private (DP) deep learning, especially on
large vision and language models with millions to billions of parameters. In
this work, we thoroughly study the per-sample gradient clipping style, a key
component in DP optimization. We show that different clipping styles have the
same time complexity but instantiate an accuracy-memory trade-off: while the
all-layer clipping (of coarse granularity) is the most prevalent and usually
gives the best accuracy, it incurs heavier memory cost compared to other
group-wise clipping, such as the layer-wise clipping (of finer granularity). We
formalize this trade-off through our convergence theory and complexity
analysis. Importantly, we demonstrate that the accuracy gap between group-wise
clipping and all-layer clipping becomes smaller for larger models, while the
memory advantage of the group-wise clipping remains. Consequently, the
group-wise clipping allows DP optimization of large models to achieve high
accuracy and low peak memory simultaneously.
