Implicit Regularization for Group Sparsity
This work addresses the challenge of structured sparsity in machine learning, offering a novel implicit regularization approach that could benefit applications requiring efficient feature selection, though it is incremental as it builds on existing implicit regularization research.
The paper tackles the problem of achieving group sparsity in linear regression without explicit regularization by introducing a diagonally grouped linear neural network, showing that gradient descent on this model biases towards group-sparse solutions and achieves minimax-optimal error rates with improved sample complexity compared to prior methods.
We study the implicit regularization of gradient descent towards structured sparsity via a novel neural reparameterization, which we call a diagonally grouped linear neural network. We show the following intriguing property of our reparameterization: gradient descent over the squared regression loss, without any explicit regularization, biases towards solutions with a group sparsity structure. In contrast to many existing works in understanding implicit regularization, we prove that our training trajectory cannot be simulated by mirror descent. We analyze the gradient dynamics of the corresponding regression problem in the general noise setting and obtain minimax-optimal error rates. Compared to existing bounds for implicit sparse regularization using diagonal linear networks, our analysis with the new reparameterization shows improved sample complexity. In the degenerate case of size-one groups, our approach gives rise to a new algorithm for sparse linear regression. Finally, we demonstrate the efficacy of our approach with several numerical experiments.