Randomized algorithms play a crucial role in efficiently solving large-scale optimization problems. In this paper, we introduce Sketching for Regularized Optimization (SRO), a fast sketching algorithm designed for least squares problems with convex or nonconvex regularization. SRO operates by first creating a sketch of the original data matrix and then solving the sketched problem. We establish minimax optimal rates for sparse signal estimation by addressing the sketched sparse convex and nonconvex learning problems. Furthermore, we propose a novel Iterative SRO algorithm, which significantly reduces the approximation error geometrically for sketched convex regularized problems. To the best of our knowledge, this work is among the first to provide a unified theoretical framework demonstrating minimax rates for convex and nonconvex sparse learning problems via sketching. Experimental results validate the efficiency and effectiveness of both the SRO and Iterative SRO algorithms.