Quadratic Convergence of Smoothing Newton's Method for 0/1 Loss Optimization
研究了直接优化0/1损失函数的牛顿法，首次证明其在合理条件下具有局部二次收敛性，数值实验显示性能优越，适用于支持向量机和1比特压缩感知等分类问题。
It has been widely recognized that the 0/1 loss function is one of the most natural choices for modeling classification errors, and it has a wide range of applications including support vector machines and 1-bit compressed sensing. Due to the combinatorial nature of the 0/1 loss function, methods based on convex relaxations or smoothing approximations have dominated the existing research and are often able to provide approximate solutions of good quality. However, those methods are not optimizing the 0/1 loss function directly and hence no optimality has been established for the original problem. This paper aims to study the optimality conditions of the 0/1 function minimization, and for the first time to develop Newton's method that directly optimizes the 0/1 function with a local quadratic convergence under reasonable conditions. Extensive numerical experiments demonstrate its superior performance as one would expect from Newton-type methods. Extensive numerical experiments demonstrate its superior performance as one would expect from Newton-type methods.