  This work focuses on learning optimization problems with quadratical
interactions between variables, which go beyond the additive models of
traditional linear learning. We investigate more specifically two different
methods encountered in the literature to deal with this problem: "hierNet" and
structured-sparsity regularization, and study their connections. We propose a
primal-dual proximal algorithm based on an epigraphical projection to optimize
a general formulation of these learning problems. The experimental setting
first highlights the improvement of the proposed procedure compared to
state-of-the-art methods based on fast iterative shrinkage-thresholding
algorithm (i.e. FISTA) or alternating direction method of multipliers (i.e.
ADMM), and then, using the proposed flexible optimization framework, we provide
fair comparisons between the different hierarchical penalizations and their
improvement over the standard $\ell_1$-norm penalization. The experiments are
conducted both on synthetic and real data, and they clearly show that the
proposed primal-dual proximal algorithm based on epigraphical projection is
efficient and effective to solve and investigate the problem of hierarchical
interaction learning.
