We propose a new outer–inner linearization method for non-convex statistical learning problems involving non-convex structural penalties and non-convex loss. Many important statistical problems fall in this category, including the robust M-estimators, generalized linear models, and different types of structured learning problems. Our method exploits the fact that many such loss and penalty functions can be represented as compositions of smooth concave functions and nonsmooth convex functions. It linearizes the outer concave functions and solves the resulting convex, but still nonsmooth, subproblems by a special alternating linearization method. We provide proof of convergence to a stationary point of the method under mild conditions. Finally, numerical examples involving non-convex structural penalties and non-convex loss functions demonstrate the efficacy and accuracy of the method.

我们为涉及非凸结构惩罚和非凸损失的非凸统计学习问题提出了一种新的外-内线性化方法。许多重要的统计问题都属于这一类，包括稳健的 M 估计量、广义线性模型和不同类型的结构化学习问题。我们的方法利用了这样一个事实，即许多此类损失和惩罚函数可以表示为平滑凹函数和非平滑凸函数的组合。它线性化外凹函数并通过特殊的交替线性化方法解决由此产生的凸但仍然不光滑的子问题。我们提供了在温和条件下收敛到该方法驻点的证明。最后，