Sketch-and-Project Meets Newton Method: Global $O(1/k^2)$ Convergence with Low-Rank Updates

In this paper, we propose the first sketch-and-project Newton method with the fast $O(1/k^2$) global convergence rate for self-concordant functions. Our method, SGN, can be viewed in three ways: i) as a sketch-and-project algorithm projecting updates of the Newton method, ii) as a cubically regularized Newton method in the sketched subspaces, and iii) as a damped Newton method in the sketched subspaces. SGN inherits the best of all three worlds: the cheap iteration costs of the sketch-and-project methods, the state-of-the-art $O(1/k^2)$ global convergence rate of the full-rank Newton-like methods, and the algorithm simplicity of the damped Newton methods. Finally, we demonstrate its comparable empirical performance to the baseline algorithms.