Title: On the Mysterious Optimization Geometry of Deep Neural Networks. Abstract: Understanding why gradient-based algorithms are successful in practical deep learning optimization is a fundamental and long-standing problem. Most existing works promote the explanation that deep neural networks have smooth and amenable nonconvex optimization geometries. In this work, we argue that this may be an oversimplification of practical deep learning optimization by revealing a mysterious and complex optimization geometry of deep networks through extensive experiments. Specifically, we consistently observe two distinct geometric patterns in training various deep networks: a regular smooth geometry and a mysterious zigzag geometry, where gradients computed in adjacent iterations are extremely negatively correlated. Also, such a zigzag geometry exhibits a fractal structure in that it appears over a wide range of geometrical scales, implying that deep networks can be highly non-smooth in certain local parameter regions. Moreover, our results show that a substantial part of the training progress is achieved under such complex geometry. Therefore, the existing smoothness-based explanations do not fully match the practice. 