In this paper, we present SwiftLearn, a data-efficient approach to accelerate training of deep learning models using a subset of data samples selected during the warm-up stages of training. This subset is selected based on an importance criteria measured over the entire dataset during warm-up stages, aiming to preserve the model performance with fewer examples during the rest of training. The importance measure we propose could be updated during training every once in a while, to make sure that all of the data samples have a chance to return to the training loop if they show a higher importance. The model architecture is unchanged but since the number of data samples controls the number of forward and backward passes during training, we can reduce the training time by reducing the number of training samples used in each epoch of training. Experimental results on a variety of CV and NLP models during both pretraining and finetuning show that the model performance could be preserved while achieving a significant speed-up during training. More specifically, BERT finetuning on GLUE benchmark shows that almost 90% of the data can be dropped achieving an end-to-end average speedup of 3.36x while keeping the average accuracy drop less than 0.92%.