Iterative Hessian sketch: Fast and accurate solution approximation for constrained least-squares
This addresses the need for efficient and accurate solution approximations in large-scale optimization, particularly for constrained least-squares problems, though it appears incremental as it builds on existing sketching methods.
The paper tackles the problem of approximating solutions to constrained least-squares problems, showing that a widely used sketching method is suboptimal and introducing a new iterative Hessian sketch method that achieves approximations with a projection dimension proportional to the statistical complexity and a logarithmic number of iterations.
We study randomized sketching methods for approximately solving least-squares problem with a general convex constraint. The quality of a least-squares approximation can be assessed in different ways: either in terms of the value of the quadratic objective function (cost approximation), or in terms of some distance measure between the approximate minimizer and the true minimizer (solution approximation). Focusing on the latter criterion, our first main result provides a general lower bound on any randomized method that sketches both the data matrix and vector in a least-squares problem; as a surprising consequence, the most widely used least-squares sketch is sub-optimal for solution approximation. We then present a new method known as the iterative Hessian sketch, and show that it can be used to obtain approximations to the original least-squares problem using a projection dimension proportional to the statistical complexity of the least-squares minimizer, and a logarithmic number of iterations. We illustrate our general theory with simulations for both unconstrained and constrained versions of least-squares, including $\ell_1$-regularization and nuclear norm constraints. We also numerically demonstrate the practicality of our approach in a real face expression classification experiment.