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 ℓ₁-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.

这项工作着眼于变量之间存在二次交互的学习优化问题，这超出了传统线性学习的加性模型。我们将更具体地研究文献中遇到的两种不同方法来解决此问题：“ hierNet”和结构稀疏正则化，并研究它们的联系。我们提出了一种基于对射投影的原对偶近邻算法，以优化这些学习问题的一般表述。与基于快速迭代收缩阈值算法（即FISTA）或乘数的交替方向方法（即ADMM）的最新方法相比，实验设置首先强调了所提出程序的改进。灵活的优化框架，ℓ ₁范数惩罚。实验是在合成数据和真实数据上进行的，它们清楚地表明，所提出的基于表象投影的原始-双重近端算法对于解决和研究分层交互学习问题是有效而有效的。