Sparse Optimization on Measures with Over-parameterized Gradient Descent

Understand

Minimizing a convex function of a measure with a sparsity-inducing penalty is a typical problem arising, e.g., in sparse spikes deconvolution or two-layer neural networks training.

• We show that this problem can be solved by discretizing the measure and running non-convex gradient descent on the positions and weights of the particles.
• For measures on a $d$-dimensional manifold and under some non-degeneracy assumptions, this leads to a global optimization algorithm with a complexity scaling as $\log(1/\epsilon)$ in the desired accuracy $\epsilon$, instead of $\epsilon^{-d}$ for convex methods.
• The key theoretical tools are a local convergence analysis in Wasserstein space and an analysis of a perturbed mirror descent in the space of measures.