Title: Spurious Local Minima Provably Exist for Deep Convolutional Neural Networks. Abstract: In this paper, we prove that a general family of spurious local minima exist in the loss landscape of deep convolutional neural networks with squared loss or cross-entropy loss.  For this purpose, we develop some new techniques to solve the challenges introduced by convolutional layers. We solve a combinatorial problem which considers the limited receptive fields of hidden neurons, and possible distinct activation status for different samples and different locations in feature maps, to show that a differentiation of data samples is always possible somewhere in feature maps. Training loss is then decreased by perturbation of network parameters that can affect different samples in different ways. Despite filters and biases are tied in each feature map, we give a construction in which this perturbation only affects the output of a single ReLU neuron and keeps the outputs at other locations unchanged. Finally, we give an example of nontrivial spurious local minimum in which different activation patterns of samples are explicitly constructed. Experimental results verify our theoretical findings. 