Title: Differentially Private Optimization on Large Model at Small Cost. Abstract: Differentially private (DP) optimization is the  standard paradigm to learn large neural networks that are accurate and privacy-preserving. The computational cost for DP deep learning, however, is notoriously heavy due to the per-sample gradient clipping. Existing DP implementations are $2-1000\times$ more costly in time and space complexity than the standard (non-private) training. In this work, we develop a novel Book-Keeping (BK) technique that implements existing DP optimizers (thus achieving the same accuracy), with a substantial improvement on the computational cost. Specifically, BK enables DP training on large models and high dimensional data to be roughly as efficient as the standard training, whereas previous DP algorithms can be inefficient or incapable of training due to memory error. The computational advantage of BK is supported by the complexity analysis as well as extensive experiments on vision and language tasks. Our implementation achieves state-of-the-art (SOTA) accuracy with very small extra cost: on GPT2 and at the same memory cost, BK has 1.0$\times$ the time complexity of the standard training (0.75$\times$ training speed in practice), and 0.6$\times$ the time complexity of the most efficient DP implementation (1.24$\times$ training speed in practice). We will open-source the codebase for the BK algorithm.