Speaker
Description
The Lanczos process is a well-known Krylov subspace method for the orthogonal tridiagonalization of a hermitian matrix $\mathbf{Z}$. Equipped with a suitable function $f$, Lanczos function approximation (LFA) can be used as a powerful tool for approximating the matrix-function $f(\mathbf{Z})$ or matrix-function-vector products $f(\mathbf{Z})\mathbf{\omega}$.
We discuss an application of LFA in computing iterates of the form
$$\mathbf{x}_{(k+1)} = \mathbf{x}_{(k)} - \eta_{(k)}\left(\left(\nabla^2g(\mathbf{x}_{(k)})\right)^2 + \lambda_{(k)}\mathbf{I}\right)^{-1/2} \nabla g(\mathbf{x}_{(k)})$$ arising in the regularized saddle-free Newton (R-SFN) algorithm for $g\in\mathcal{C}^2$. LFA facilitates efficient update steps yielding competitive performance against adaptive regularization with cubics (ARC), which is among the best available algorithms for non-convex optimization.
Additionally, we will detail ongoing work for improving LFA in the general setting, motivated by our application in optimization. We will discuss residual estimation for iterative refinement and block Lanczos for implicit preconditioning.