Large-scale methods for bilevel learning

Variational regularisation methods are standard in the field of image reconstruction, image processing, machine vision, and clustering. Typically, these methods depend on several parameters to be set manually. Even in variational problems with only one parameter, choosing the parameter by hand is quite challenging. State-of-the-art variational methods seek to learn these parameters from training data using various machine learning techniques, such as bilevel learning. Bilevel learning provides an end-to-end scheme for learning parameters best suited to a specific task from training data. From a mathematical point of view, this scheme leads to a nested optimisation problem.
The corresponding mathematical problem of a wide range of applications such as MRI (Magnetic Resonance Imaging) and PET (Positron Emission Tomography), denoising/deblurring, inpainting, and motion estimation and classification can be formulated in a variational regularisation framework. Consequently, investigating bilevel learning for learning the suitable parameters of them is relevant.
The nested optimisation problem which arises in the bilevel learning scheme is computationally difficult to handle. It consists of an upper-level and a lower-level optimisation problem. As finding the exact solution of the lower-level problem is practically infeasible, numerical methods are used to solve it with a tolerance. Choosing a very small tolerance leads to a huge computational cost, and taking a higher tolerance, may cause divergence. Thus, taking the accuracy in a dynamic approach is a challenge. Moreover, modern machine learning models and data-adaptive neural-net-based regularisers involve millions of parameters. One needs a principled mathematical framework that scales well to the large model regime to make the most of these models. Hence, our bilevel methods for learning the parameters in machine learning and the tasks of interest should be scalable.
Based on the mentioned perspective and challenges, this PhD project pursues developing new computationally feasible and theoretically sound methods for bilevel learning. The new findings will be scalable and useful for large-scale problems like the ones that arise in machine learning and under the usage of data-adaptive regularisers.

Combining specialised modelling techniques with complex data analysis in order to deliver prediction with quantified uncertainties lies at the heart of many of the major challenges facing UK industry and society over the next decades. Indeed, the recent Government Office for Science report "Computational Modelling, Technological Futures, 2018" specifies putting the UK at the forefront of the data revolution as one of their Grand Challenges.

The beneficiaries of our research portfolio will include a wide range of UK industrial sectors such as the pharmaceutical industry, risk consultancy, telecommunications and advanced materials, as well as government bodies, including the NHS, the Met Office and the Environment Agency.

Examples of current impactful projects pursued by students and in collaboration with stake-holders include:

- Using machine learning techniques to develop automated assessment of psoriatic arthritis from hand X-Rays, freeing up consultants' time (with the NHS).

- Uncertainty quantification for the Neutron Transport Equation improving nuclear reactor safety (co-funded by Wood).

- Optimising the resilience and self-configuration of communication networks with the help of random graph colouring problems (co-funded by BT).

- Risk quantification of failure cascades on oil platforms by using Bayesian networks to improve safety assessment for certification (co-funded by DNV-GL).

- Krylov regularisation in a Bayesian framework for low-resolution Nuclear Magnetic Resonance to assess properties of porous media for real-time exploration (co-funded by Schlumberger).

- Machine learning methods to untangle oceanographic sound data for a variety of goals in including the protection of wildlife in shipping lanes (with the Department of Physics).

Future committed partners for SAMBa 2.0 are: BT, Syngenta, Schlumberger, DNV GL, Wood, ONS, AstraZeneca, Roche, Diamond Light Source, GKN, NHS, NPL, Environment Agency, Novartis, Cytel, Mango, Moogsoft, Willis Towers Watson.

SAMBa's core mission is to train the next generation of academic and industrial researchers with the breadth and depth of skills necessary to address these challenges. SAMBa's most sustained impact will be through the contributions these researchers make over the longer term of their careers. To set the students up with the skills needed to maximise this impact, SAMBa has developed a bespoke training experience in collaboration with industry, at the heart of its activities. Integrative Think Tanks (ITTs) are week-long workshops in which industrial partners present high-level research challenges to students and academics. All participants work collaboratively to formulate mathematical
models and questions that address the challenges. These outputs are meaningful both to the non-academic partner, and as a mechanism for identifying mathematical topics which are suitable for PhD research. Through the co-ownership of collaboratively developed projects, SAMBa has the capacity to lead industry in capitalising on recent advances in mathematics. ITTs occur twice a year and excel in the process of problem distillation and formulation, resulting in an exemplary environment for developing impactful projects.

SAMBa's impact on the student experience will be profound, with training in a broad range of mathematical areas, in team working, in academic-industrial collaborations, and in developing skills in communicating with specialist and generalist audiences about their research. Experience with current SAMBa students has proven that these skills are highly prized: "The SAMBa approach was a great template for setting up a productive, creative and collaborative atmosphere. The commitment of the students in getting involved with unfamiliar areas of research and applying their experience towards producing solutions was very impressive." - Dr Mike Marsh, Space weather researcher, Met Office.