Near-Linear Time Projection onto the $\ell_{1,\infty}$ Ball; Application to Sparse Neural Networks

Looking for sparsity is nowadays crucial to speed up the training of large-scale neural networks. Projections onto the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\ell_{1}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\ell_{1,\propto}$</tex> are among the most efficient techniques to sparsify and reduce the overall cost of neural networks. In this paper, we introduce a new projection algorithm for the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\ell_{1,\infty}$</tex> norm ball. Its worst-case time complexity is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathcal{O}(nm+J\log(nm))$</tex> for a matrix in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{R}^{n\times m}. J$</tex> is a term that tends to 0 when the sparsity is high, and to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n\times m$</tex> in the worst case. The algorithm is easy to implement and it is guaranteed to converge to the exact solution in finite time. Moreover, we propose to incorporate the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\ell_{1,\infty}$</tex> ball projection while training an auto encoder to enforce feature selection and sparsity of the weights. Sparsification appears in the encoder to primarily do feature selection due to our application in biology, where only a very small part (<tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$&lt;$</tex> 2%) of the data is relevant. We show that in both the biological and general cases of sparsity, our method is the fastest.