Sketching Algorithms and Lower Bounds for Ridge Regression

We give a sketching-based iterative algorithm that computes a 1 + epsilon approximate solution for the ridge regression problem min(x) ||Ax-b||(2)(2) +lambda||x||(2)(2) where A is an element of R-nxd with d >= n. Our algorithm, for a constant number of iterations (requiring a constant number of passes over the input), improves upon earlier work (Chowdhury et al., 2018) by requiring that the sketching matrix only has a weaker Approximate Matrix Multiplication (AMM) guarantee that depends on epsilon, along with a constant subspace embedding guarantee. The earlier work instead requires that the sketching matrix has a subspace embedding guarantee that depends on e. For example, to produce a 1+ epsilon approximate solution in 1 iteration, which requires 2 passes over the input, our algorithm requires the OSNAP embedding to have m = O(n sigma(2)/lambda epsilon) rows with a sparsity parameter s = O(log(n)), whereas the earlier algorithm of Chowdhury et al. (2018) with the same number of rows of OSNAP requires a sparsity s = O(root sigma(2)/lambda epsilon center dot log(n)), where sigma =||A||(2) is the spectral norm of the matrix A. We also show that this algorithm can be used to give faster algorithms for kernel ridge regression. Finally, we show that the sketch size required for our algorithm is essentially optimal for a natural framework of algorithms for ridge regression by proving lower bounds on oblivious sketching matrices for AMM. The sketch size lower bounds for AMM may be of independent interest.