Estimating LASSO Risk and Noise Level

We study the fundamental problems of variance and risk estimation in high dimensional statistical modeling. In particular, we consider the problem of learning a coefficient vector θ0 ∈ ℝp from noisy linear observations y = Xθ0 + w ∈ ℝn (p > n) and the popular estimation procedure of solving the l1-penalized least squares objective known as the LASSO or Basis Pursuit DeNoising (BPDN). In this context, we develop new estimators for the l2 estimation risk $\|\hat{\theta}-\theta_0\|_2$ and the variance of the noise when distributions of θ0 and w are unknown. These can be used to select the regularization parameter optimally. Our approach combines Stein's unbiased risk estimate [Ste81] and the recent results of [BM12a][BM12b] on the analysis of approximate message passing and the risk of LASSO. We establish high-dimensional consistency of our estimators for sequences of matrices X of increasing dimensions, with independent Gaussian entries. We establish validity for a broader class of Gaussian designs, conditional on a certain conjecture from statistical physics. To the best of our knowledge, this result is the first that provides an asymptotically consistent risk estimator for the LASSO solely based on data. In addition, we demonstrate through simulations that our variance estimation outperforms several existing methods in the literature.