An Upper Limit of Decaying Rate with Respect to Frequency in Deep Neural Network
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Deep neural network (DNN) usually learns the target function from low to high frequency, which is called frequency principle or spectral bias. This frequency principle sheds light on a high-frequency curse of DNNs – difficult to learn high-frequency information. Inspired by the frequency principle, a series of works are devoted to develop algorithms for overcoming the high-frequency curse. A natural question arises: what is the upper limit of the decaying rate w.r.t. frequency when one trains a DNN? In this work, our theory, confirmed by numerical experiments, suggests that there is a critical decaying rate w.r.t. frequency in DNN training. Below the upper limit of the decaying rate, the DNN interpolates the training data by a function with a certain regularity. However, above the upper limit, the DNN interpolates the training data by a trivial function, i.e., a function is only non-zero at training data points. Our results indicate a better way to overcome the high-frequency curse is to design a proper pre-condition approach to shift high-frequency information to low-frequency one, which coincides with several previous developed algorithms for fast learning high-frequency information. More importantly, this work rigorously proves that the high-frequency curse is an intrinsic difficulty of DNNs.
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