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An Asymptotic Rate for the LASSO Loss

The LASSO is a well-studied method for use in high-dimensional linear regression where one wishes to recover a sparse vector β ∈ R from noisy observations y ∈ R measured through a matrix X ∈ Rn×p with the model y = Xβ + w where w is a vector of independent, mean-zero noise. We study the linear asymptotic regime where n/p → δ for a constant δ ∈ (0,∞). Using a carefully constructed approximate message passing (AMP) algorithm that converges to the LASSO estimator and recent finite sample theoretical performance guarantees for AMP, we provide large deviations bounds between various measures of LASSO loss and their concentrating values predicted by the AMP state evolution that shows exponentially fast convergence (in n) when the measurement matrix X is i.i.d. Gaussian. This work refines previous asymptotic analysis of LASSO loss in [Bayati and Montanari, 2012].