基于MCP正则的最小一乘回归问题研究

On Least Absolute Deviation Regression Problems with MCP Regularization

针对损失函数为最小一乘函数的回归问题,研究其基数罚问题与MCP (minimax concave penalty)松弛问题之间解的关系.首先,证明了 MCP罚松弛模型解的下界性质,以此为基础分析了基数罚问题与松弛问题之间解的等价性,证明了在一定条件下两个问题具有相同的全局最优解以及最优值,此外,还证明了松弛模型的局部最优解是基数罚问题的局部最优解,在局部极小值点处松弛模型与基数罚问题的最优值是相等的.文章结果为求解稀疏最小一乘回归问题提供了理论依据和可行途径.

For the regression problem where the loss function is least absolute deviation,this paper investigates the relationship between the solutions of the problem with cardinality penalty and the problem with minimax concave penalty(MCP).The lower bound property of MCP relaxation problem is proved at first.Based on this lower bound property,the equivalence between the cardinality penalized problem and the relaxed problem is analyzed.It is proved that the two problems have the same global optimal solutions and the same optimal value under some mild conditions.In addition,it is proved that the local optimal solutions of the relaxed problem are the local optimal solutions of the cardinality penalized problem,and that the two problems have the same objective function values at any local minimum.Our results provide the theoretical basis and feasible approach to solving the sparse least absolute deviation regression problems.