The use of convex regularizers allows for easy optimization,
though they often produce biased estimation and inferior
prediction performance. Recently, nonconvex regularizers have
attracted a lot of attention and outperformed convex ones.
However, the resultant optimization problem is much harder. In
this paper, a popular subclass of $\ell_1$-based nonconvex
sparsity-inducing and low-rank regularizers is considered. This
includes nonconvex variants of lasso, sparse group lasso, tree-
structured lasso, nuclear norm and total variation regularizers.
We propose to move the nonconvexity from the regularizer to the
loss. The nonconvex regularizer is then transformed to a
familiar convex one, while the resultant loss function can still
be guaranteed to be smooth. Learning with the convexified
regularizer can be performed by existing efficient algorithms
originally designed for convex regularizers (such as the
proximal algorithm, Frank-Wolfe algorithm, alternating direction
method of multipliers and stochastic gradient descent). This is
further extended to consider cases where the convexified
regularizer does not have a closed-form proximal step, and when
the loss function is nonconvex nonsmooth. Extensive experiments
on a variety of machine learning application scenarios show that
optimizing the transformed problem is much faster than running
the state-of-the-art on the original problem.
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