This fully funded PhD position at the University of Southampton offers an exciting opportunity to advance the mathematical foundations of deep bilevel learning—a cutting-edge area at the intersection of optimization and artificial intelligence. The project is co-supervised by Professor Alain Zemkoho (bilevel optimization expert, University of Southampton) and Professor Massimiliano Pontil (machine/deep learning expert, University College London & Italian Institute of Technology), providing a rich collaborative environment and potential for research visits to UCL and IIT. Deep learning has revolutionized AI, especially with the advent of large language models and transformer architectures. Bilevel optimization, meanwhile, has become a powerful framework for modeling complex machine learning tasks, leading to the emergence of deep bilevel learning. This research will focus on uncovering the mathematical structures underlying deep bilevel learning, aiming to design more efficient, robust, and explainable algorithms. The outcomes have the potential to shape future AI systems and contribute to theoretical advances in optimization and machine learning. The position is funded by DAS Ltd and the EPSRC Doctoral Landscape Award (Collaborative Studentships Scheme), offering a tax-free stipend of approximately £20,780 per year for four years, with additional benefits such as research travel funds and optional paid part-time teaching assistantship. Tuition fees are covered at the UK rate, and both UK and international applicants are encouraged to apply. Ideal candidates will have a strong background in optimization, applied mathematics, and/or machine learning, and be motivated to contribute to foundational research in AI. Application requirements include a personal statement (highlighting relevant experience in optimization, operational research, game theory, programming, and mathematical background), a CV, contact details for two academic referees, and all degree transcripts. The application deadline is January 9, 2026, with a start date of October 1, 2026. To apply, visit the University of Southampton's online portal, select the appropriate programme and supervisor, and upload the required documents. For further details, see the official FindAPhD listing and university application page.