Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

 

Past events in this series


Thu, 12 Jun 2025

12:00 - 12:30
L4

Cubic-quartic regularization models for solving polynomial subproblems in third-order tensor methods

Kate Zhu
(Mathematical Institute (University of Oxford))
Abstract

High-order tensor methods for solving both convex and nonconvex optimization problems have recently generated significant research interest, due in part to the natural way in which higher derivatives can be incorporated into adaptive regularization frameworks, leading to algorithms with optimal global rates of convergence and local rates that are faster than Newton's method. On each iteration, to find the next solution approximation, these methods require the unconstrained local minimization of a (potentially nonconvex) multivariate polynomial of degree higher than two, constructed using third-order (or higher) derivative information, and regularized by an appropriate power of the change in the iterates. Developing efficient techniques for the solution of such subproblems is currently, an ongoing topic of research,  and this talk addresses this question for the case of the third-order tensor subproblem. In particular, we propose the CQR algorithmic framework, for minimizing a nonconvex Cubic multivariate polynomial with  Quartic Regularisation, by sequentially minimizing a sequence of local quadratic models that also incorporate both simple cubic and quartic terms.

The role of the cubic term is to crudely approximate local tensor information, while the quartic one provides model regularization and controls progress. We provide necessary and sufficient optimality conditions that fully characterise the global minimizers of these cubic-quartic models. We then turn these conditions into secular equations that can be solved using nonlinear eigenvalue techniques. We show, using our optimality characterisations, that a CQR algorithmic variant has the optimal-order evaluation complexity of $O(\epsilon^{-3/2})$ when applied to minimizing our quartically-regularised cubic subproblem, which can be further improved in special cases.  We propose practical CQR variants that judiciously use local tensor information to construct the local cubic-quartic models. We test these variants numerically and observe them to be competitive with ARC and other subproblem solvers on typical instances and even superior on ill-conditioned subproblems with special structure.

Thu, 19 Jun 2025

12:00 - 12:30
L4

Optimal random sampling for approximation with non-orthogonal bases

Astrid Herremans
(KU Leuven)
Abstract
Recent developments in optimal random sampling for least squares approximations have led to the identification of a (near-)optimal sampling distribution. This distribution can easily be evaluated given an orthonormal basis for the approximation space. However, many computational problems in science and engineering naturally yield building blocks that enable accurate approximation but do not form an orthonormal basis. In the first part of the talk, we will explore how numerical rounding errors affect the approximation error and the optimal sampling distribution when approximating with non-orthogonal bases. In the second part, we will demonstrate how this distribution can be computed without the need to orthogonalize the basis. This is joint work with Daan Huybrechs and Ben Adcock.