Learning to optimise for inverse problems

Inverse problems focus on recovering information from potentially noisy data. Many examples exist due to modern imaging modalities, such as MRI and PET. Commonly, inverse problems are tackled by considering variational regularisation, which consider minimising a data fit term, as well as a regularisation term. Often, the solution to such minimisation problems is found using iterative methods, for example, gradient descent, which have associated drawbacks, such as often being slow to converge.


This project will focus on learning to optimise, which seeks to leverage machine learning to improve this optimisation process. One example of improving the optimisation procedure is by achieving a certain performance more quickly. A simple example is by replacing the step size parameter in gradient descent with a parametrised function (e.g. a neural network). The parameters are then learned to minimise a loss function which incorporates the value of the variational loss over iterations, in order to reduce the variational loss more quickly.


Now learning to optimise has been introduced, the following provides initial routes of investigation in the project.


When training such an optimiser, it is often memory intensive due to the need to 'unroll' an iterative algorithm up to a certain number of iterations. This is an issue, as in many cases, learning is only performed on the first ~10-100 iterations, as memory scales linearly within unroll length. This varies greatly from iterative algorithms used in practice, which often require thousands of iterations to reach a suitable estimate. It is therefore an aim of the project to seek to lower the memory costs during training, and more generally how to better learn to optimise, i.e. improvements to make during training to increase performance of the learned optimiser.


There are also important considerations with respect to learning discrete actions in optimisation methods. For example, in very large-scale applications, such as PET imaging, stochastic optimisation algorithms (e.g. Stochastic Gradient Descent) are used. Such methods require a sampling method, which is usually chosen very simplistically, e.g. cyclically. There is evidence that these uniform sampling schemes are suboptimal. One could therefore look to learn how to sample within the stochastic algorithm, to increase the speed of converge. This could be done using Reinforcement Learning.

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.