Title: Cheap DNN Pruning with Performance Guarantees . Abstract: Recent DNN pruning algorithms have succeeded in reducing the number of parameters in fully connected layers often with little or no drop in classification accuracy. However most of the existing pruning schemes either have to be applied during training or require a costly retraining procedure after pruning to regain classification accuracy. In this paper we propose a cheap pruning algorithm based on difference of convex (DC) optimisation. We also provide theoretical analysis for the growth in the Generalisation Error (GE) of the new pruned network. Our method can be used with any convex regulariser and allows for a controlled degradation in classification accuracy while being orders of magnitude faster than competing approaches. Experiments on common feedforward neural networks show that for sparsity levels above 90% our method achieves 10% higher classification accuracy compared to Hard Thresholding.