Fitting high-dimensional statistical models often requires the use of non-linear parameter estimation procedures. As a consequence, it is generally impossible to obtain an exact characterization of the probability distribution of the parameter estimates. This in turn implies that it is extremely challenging to quantify the uncertainty associated with a certain parameter estimate. Concretely, no commonly accepted procedure exists for computing classical measures of uncertainty and statistical significance as confidence intervals or pvalues for these models.

We consider here high-dimensional linear regression problem, and propose an efficient algorithm for constructing confidence intervals and p-values. The resulting confidence intervals have nearly optimal size. When testing for the null hypothesis that a certain parameter is vanishing, our method has nearly optimal power. Our approach is based on constructing a ‘de-biased’version of regularized M-estimators. The new construction improves over recent work in the field in that it does not assume a special structure on the design matrix. We test our method on synthetic data and a highthroughput genomic data set about riboflavin production rate, made publicly available by Bühlmann et al.(2014).