We'll introduce a new formulation for pruning convolutional kernels in neural networks to enable efficient inference. The approach is based on interleaving greedy criteria-based pruning with fine-tuning by backpropagation -- a computationally efficient procedure that maintains good generalization in the pruned network. We'll propose a new criterion based on Taylor expansion that approximates the change in the cost function induced by pruning network parameters. We'll focus on transfer learning, where large pretrained networks are adapted to specialized tasks. The proposed criterion demonstrates superior performance compared to other criteria, for example, the norm of kernel weights or feature map activation, for pruning large CNNs after adaptation to fine-grained classification tasks (Birds-200 and Flowers-102) relaying only on the first order gradient information. We'll also show that pruning can lead to more than 10x theoretical reduction in adapted 3D-convolutional filters.