As the size of deep learning models gets larger and larger, training takes longer time and more resources, making fault tolerance more and more critical. Existing state-of-the-art methods like CheckFreq and Elastic Horovod need to back up a copy of the model state (i.e., parameters and optimizer states) in memory, which is costly for large models and leads to non-trivial overhead. This paper presents SWIFT, a novel recovery design for distributed deep neural network training that significantly reduces the failure recovery overhead without affecting training throughput and model accuracy. Instead of making an additional copy of the model state, SWIFT resolves the inconsistencies of the model state caused by the failure and exploits the replicas of the model state in data parallelism for failure recovery. We propose a logging-based approach when replicas are unavailable, which records intermediate data and replays the computation to recover the lost state upon a failure. The re-computation is distributed across multiple machines to accelerate failure recovery further. We also log intermediate data selectively, exploring the trade-off between recovery time and intermediate data storage overhead. Evaluations show that SWIFT significantly reduces the failure recovery time and achieves similar or better training throughput during failure-free execution compared to state-of-the-art methods without degrading final model accuracy. SWIFT can also achieve up to 1.16x speedup in total training time compared to state-of-the-art methods.