Minimizing the difference of two submodular (DS) functions is a problem that naturally occurs in various machine learning problems. Although it is well known that a DS problem can be equivalently formulated as the minimization of the difference of two convex (DC) functions, existing algorithms do not fully exploit this connection. A classical algorithm for DC problems is called the DC algorithm (DCA). We introduce variants of DCA and its complete form (CDCA) that we apply to the DC program corresponding to DS minimization. We extend existing convergence properties of DCA, and connect them to convergence properties on the DS problem. Our results on DCA match the theoretical guarantees satisfied by existing DS algorithms, while providing a more complete characterization of convergence properties. In the case of CDCA, we obtain a stronger local minimality guarantee. Our numerical results show that our proposed algorithms outperform existing baselines on two applications: speech corpus selection and feature selection.

翻译：两个次模函数之差（DS）的最小化是各类机器学习问题中自然出现的一类问题。虽然已知DS问题可以等价转化为两个凸函数之差（DC）的最小化，但现有算法并未充分利用这一联系。处理DC问题的经典算法称为DC算法（DCA）。本文针对DS最小化对应的DC规划问题，提出了DCA及其完整形式（CDCA）的变体。我们扩展了DCA的现有收敛性质，并将其与DS问题的收敛性质建立联系。我们的DCA结果与现有DS算法满足的理论保证相匹配，同时提供了更完整的收敛特性刻画。对于CDCA，我们获得了更强的局部极小性保证。数值实验表明，所提算法在语音语料库选择和特征选择两个应用中优于现有基准方法。