Sketching for Convex and Nonconvex Regularized Least Squares with Sharp Guarantees

Randomized algorithms are important for solving large-scale optimization problems. In this paper, we propose a fast sketching algorithm for least square problems regularized by convex or nonconvex regularization functions, Sketching for Regularized Optimization (SRO). Our SRO algorithm first generates a sketch of the original data matrix, then solves the sketched problem. Different from existing randomized algorithms, our algorithm handles general Frechet subdifferentiable regularization functions in an unified framework. We present general theoretical result for the approximation error between the optimization results of the original problem and the sketched problem for regularized least square problems which can be convex or nonconvex. For arbitrary convex regularizer, relative-error bound is proved for the approximation error. Importantly, minimax rates for sparse signal estimation by solving the sketched sparse convex or nonconvex learning problems are also obtained using our general theoretical result under mild conditions. To the best of our knowledge, our results are among the first to demonstrate minimax rates for convex or nonconvex sparse learning problem by sketching under a unified theoretical framework. We further propose an iterative sketching algorithm which reduces the approximation error exponentially by iteratively invoking the sketching algorithm. Experimental results demonstrate the effectiveness of the proposed SRO and Iterative SRO algorithms. X ∈ R n×d is the data matrix or design matrix for regression problems, h λ : R d → R is a regularizer function and λ is a positive regularization weight. When ) is the optimization problem for ridge regression or ℓ 1 regularized least square estimation (Lasso). We study the regime that n ≫ r = rank(X) where r is the rank of X in most results of this paper, and it is a popular setting for large-scale problems such as fast least square estimation by sketching