Efficient construction of checkpoints/snapshots is a critical tool for training and diagnosing deep learning models. In this paper, we propose a lossy compression scheme for checkpoint constructions (called LC-Checkpoint). LC-Checkpoint simultaneously maximizes the compression rate and optimizes the recovery speed, under the assumption that SGD is used to train the model. LC-Checkpointuses quantization and priority promotion to store the most crucial information for SGD to recover, and then uses a Huffman coding to leverage the non-uniform distribution of the gradient scales. Our extensive experiments show that LC-Checkpoint achieves a compression rate up to $28\times$ and recovery speedup up to $5.77\times$ over a state-of-the-art algorithm (SCAR).