  Understanding the implicit regularization (or implicit bias) of gradient
descent has recently been a very active research area. However, the implicit
regularization in nonlinear neural networks is still poorly understood,
especially for regression losses such as the square loss. Perhaps surprisingly,
we prove that even for a single ReLU neuron, it is impossible to characterize
the implicit regularization with the square loss by any explicit function of
the model parameters (although on the positive side, we show it can be
characterized approximately). For one hidden-layer networks, we prove a similar
result, where in general it is impossible to characterize implicit
regularization properties in this manner, except for the "balancedness"
property identified in Du et al. [2018]. Our results suggest that a more
general framework than the one considered so far may be needed to understand
implicit regularization for nonlinear predictors, and provides some clues on
what this framework should be.
