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High-dimensional statistical learning (HDSL) has wide applications in dataanalysis, operations research, and decision-making. Despite the availability ofmultiple theoretical frameworks, most existing HDSL schemes stipulate thefollowing two conditions: (a) the sparsity, and (b) the restricted strongconvexity (RSC). This paper generalizes both conditions via the use of thefolded concave penalty (FCP). More specifically, we consider an M-estimationproblem where (i) the (conventional) sparsity is relaxed into the approximatesparsity and (ii) the RSC is completely absent. We show that the FCP-basedregularization leads to poly-logarithmic sample complexity; the training datasize is only required to be poly-logarithmic in the problem dimensionality.This finding can facilitate the analysis of two important classes of modelsthat are currently less understood: the high-dimensional nonsmooth learning andthe (deep) neural networks (NN). For both problems, we show that thepoly-logarithmic sample complexity can be maintained. In particular, ourresults indicate that the generalizability of NNs under over-parameterizationcan be theoretically ensured with the aid of regularization.

High-Dimensional Learning under ApproximateSparsity with Applications to Nonsmooth Estimation and Regularized Neural Networks