Title: On the Curse of Memory in Recurrent Neural Networks: Approximation and Optimization Analysis. Abstract: We study the approximation properties and optimization dynamics of recurrent neural networks (RNNs) when applied to learn input-output relationships in temporal data. We consider the simple but representative setting of using continuous-time linear RNNs to learn from data generated by linear relationships. Mathematically, the latter can be understood as a sequence of linear functionals.  We prove a universal approximation theorem of such linear functionals and characterize the approximation rate.  Moreover, we perform a fine-grained dynamical analysis of training linear RNNs by gradient methods.  A unifying theme uncovered is the non-trivial effect of memory, a notion that can be made precise in our framework, on both approximation and optimization: when there is long-term memory in the target, it takes a large number of neurons to approximate it. Moreover, the training process will suffer from slow downs.  In particular, both of these effects become exponentially more pronounced with increasing memory - a phenomenon we call the “curse of memory”.  These analyses represent a basic step towards a concrete mathematical understanding of new phenomenons that may arise in learning temporal relationships using recurrent architectures.