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These\nprocedures mirror two methods of computing gradients for recurrent neural\nnetworks and have different trade-offs in terms of running time and space\nrequirements. Our formulation of the reverse-mode procedure is linked to\nprevious work by Maclaurin et al. [2015] but does not require reversible\ndynamics. The forward-mode procedure is suitable for real-time hyperparameter\nupdates, which may significantly speed up hyperparameter optimization on large\ndatasets. We present experiments on data cleaning and on learning task\ninteractions. 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