Optimal M-estimation in high-dimensional regression.

We consider, in the modern setting of high-dimensional statistics, the classic problem of optimizing the objective function in regression using M-estimates when the error distribution is assumed to be known. We propose an algorithm to compute this optimal objective function that takes into account the dimensionality of the problem. Although optimality is achieved under assumptions on the design matrix that will not always be satisfied, our analysis reveals generally interesting families of dimension-dependent objective functions.