Nonsmooth low-rank matrix recovery: methodology, theory and algorithm

Many interesting problems in statistics and machine learning can be written as \(min_xF(x)=f(x)+g(x)\), where \(x\) is the model parameter, \(f\) is the loss and \(g\) is the regularizer. Examples include regularized regression in high-dimensional feature selection and low-rank matrix/tensor factorization. Sometimes the loss function and/or the regularizer is nonsmooth due to the nature of the problem, for example, \(f(x)\) could be quantile loss to induce some robustness or to put more focus on different parts of the distribution other than the mean. In this paper we propose a general framework to deal with situations when you have nonsmooth loss or regularizer. Specifically we use low-rank matrix recovery as an example to demonstrate the main idea. The framework involves two main steps: the optimal smoothing of the loss function or regularizer and then a gradient based algorithm to solve the smoothed loss. The proposed smoothing pipeline is highly flexible, computationally efficient, easy to implement and well suited for problems with high-dimensional data. Strong theoretical convergence guarantee has also been established. In the numerical studies, we used \(L_1\) loss as an example to illustrate the practicability of the proposed pipeline. Various state-of-art algorithms such as Adam, NAG and YelowFin all show promising results for the smoothed loss.