Theory and Algorithms for Relation between Stochastic Control and Reinforcement Learning

Reinforcement learning (RL) is an extremely active subfield in machine learning. Its famous real-world triumphs in the area of complex decision making problems in recent years include playing perfect-information board games such as Go and driving autonomous vehicles. The basic idea is for an algorithm to learn gradually near-best strategies to solve a problem in real time. The learning is based on trial-and-error for improving control policies. It requires specifying an explicit objective score function on a black-box environment, which incorporates the environmental responses to the control actions. However, such trial-and-error exploration is in many situations prohibitively costly. The exploration-exploitation dichotomy in RL is typically mitigated by adding an entropy-regularisation term that needs to
be minimised as part of the objective function. The connection between entropy-regularized RL problems and the relaxed continuous-time stochastic control has been established recently. In particular, Linear-Quadratic (LQ) control problem has elegant solutions and is able to approximate more general problems. One of the objective is to design and implement algorithms for solving this problem, either LQ problem or general control problem, and provide corresponding convergence theorems. To this end, algorithms proposed by Szpruch et al. and Reisinger et al. are good examples and could be a starting point. In real-world problems, stochastic process with jumps is widely applied. Thus, another objective of this project is to consider more complex controlled dynamics, i.e. processes with jump, and see whether our algorithms could tackle this more complex problem. The martingale approach proposed by Hernandez-Hernandez et al. could be applied to this project. In this research, we will mainly study the theory of the problem. When the theory is relatively complete, we will try to develop corresponding algorithms and provide convergence theorems for them. With these theory and algorithms, we will proceed to the last objective of this project: applying the algorithms to different real-world problem, such as financial trading, autonomous driving and board games. Many applications of RL in finance are yet to be incorporated in the continuous-time relaxed stochastic control framework. It is envisaged that, during this PhD, theoretical advances will be applied to problems in quantitative finance (e.g. optimal
execution of portfolio transactions and the risk-management of derivative securities) with emphasis on real-worlds applicability of the algorithms.

In the 2018 Government Office for Science report, 'Computational Modelling: Technological Futures', Greg Clarke, the Secretary of State for Business Energy and Industrial Strategy, wrote "Computational modelling is essential to our future productivity and competitiveness, for businesses of all sizes and across all sectors of the economy". With its focus on computational models, the mathematics that underpin them, and their integration with complex data, the MathSys II CDT will generate diverse impacts beyond academia. This includes impacts on skills, on the economy, on policy and on society.

Impacts on skills.
MathSys II will produce a minimum of 50 PhD graduates to support the growing national demand for advanced mathematical modelling and data analysis skills. The CDT will provide each of them with broad core skills in the MSc, a deep knowledge of their chosen research specialisation in the PhD and a complementary qualification in transferable skills integrated throughout. Graduates will thus acquire the profiles needed to form the next generation of leaders in business, government and academia. They will be supported by an integrated pastoral support framework, including a diverse group of accessible leadership role models. The cohort based environment of the CDT provides a multiplier effect by encouraging cohorts to forge long-lasting professional networks whose value and influence will long outlast the CDT itself. MathSys II will seek to maximise the influence of these networks by providing topical training in Responsible Research and Innovation, by maintaining a robust Equality, Diversity & Inclusion policy, and by integration with Warwick's global network of international partnerships.

Economic impacts.
The research outputs from many MathSys II PhD projects will be of direct economic value to commercial, public sector and charitable external partners. Engagement with CDT partners will facilitate these impacts. This includes co-supervision of PhD and MSc projects, co-creation of Research Study Groups, and a strong commitment to provide placements/internships for CDT students. When commercial innovations or IP are generated, we will work with Warwick Ventures, the commercial arm of the University of Warwick, to commercialise/license IP where appropriate. Economic impact may also come from the creation of new companies by CDT graduates. MathSys II will present entrepreneurship as a viable career option to students. One external partner, Spectra Analytics, was founded by graduates of the preceding Complexity Science CDT, thus providing accessible role models. We will also provide in-house entrepreneurship training via Warwick Ventures and host events by external start-up accelerator Entrepreneur First.

Impacts on policy.
The CDT will influence policy at the national and international level by working with external partners operating in policy. UK examples include Department of Health, Public Health England and DEFRA. International examples include World Health Organisation (WHO) and the European Commission for the Control of Foot-and-mouth Disease (EuFMD). MathSys students will also utilise the recently announced UKRI policy internships scheme.

Impacts on society.
Public engagement will allow CDT students to promote the value of their research to society at large. Aside from social media, suitable local events include DataBeers, Cafe Scientifique, and the Big Bang Fair. MathSys will also promote a socially-oriented ethos of technology for the common good. Concretely, this includes the creation of open-source software, integration of software and data carpentry into our computational and data driven research training and championing open-access to research. We will also contribute to the 'innovation culture and science' strand of Coventry's 2021 City of Culture programme.