Implicit Sparse Regularization: The Impact of Depth and Early Stopping
This addresses the problem of sparse model recovery in machine learning for practitioners, providing theoretical insights into training dynamics, though it is incremental by extending prior results to more realistic conditions.
The paper tackles the implicit bias of gradient descent for sparse regression in deep networks under noisy and correlated settings, showing that early stopping is crucial for achieving minimax optimal sparse recovery, with depth enlarging the initialization scale and early-stopping window.
In this paper, we study the implicit bias of gradient descent for sparse regression. We extend results on regression with quadratic parametrization, which amounts to depth-2 diagonal linear networks, to more general depth-N networks, under more realistic settings of noise and correlated designs. We show that early stopping is crucial for gradient descent to converge to a sparse model, a phenomenon that we call implicit sparse regularization. This result is in sharp contrast to known results for noiseless and uncorrelated-design cases. We characterize the impact of depth and early stopping and show that for a general depth parameter N, gradient descent with early stopping achieves minimax optimal sparse recovery with sufficiently small initialization and step size. In particular, we show that increasing depth enlarges the scale of working initialization and the early-stopping window so that this implicit sparse regularization effect is more likely to take place.