Bridging Gaps: How Mathematics Advances Machine Learning — A Conversation with Martin Burger

Machine Learning (ML) is reshaping science and society – from language models and climate simulations to cancer diagnostics. But while these systems are driven by data and computation, they are grounded in mathematics. Concepts like gradient descent, stochastic optimization, regularization, and generalization theory are not just academic abstractions, they form the foundation of many methods used in modern machine learning applications.

The Conference on Mathematics of Machine Learning 2025 puts this foundation in the spotlight. It brings together researchers from both theory and practice to explore how mathematical tools – from optimization and geometry to probability and analysis – can help explain, guide, and improve machine learning systems.

To better understand the mission of the conference and the key issues it will address, we spoke with Martin Burger, Head of Helmholtz Imaging Research Unit at DESY and one of the organizers of the event. We asked Martin to reflect on the goals of the conference, the evolution of theory and practice in ML, and what motivates him to keep exploring the mathematical depths of this rapidly evolving field.

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What motivated you and your colleagues to organize the Conference on Mathematics of Machine Learning?

There has been a first edition of this conference four years ago with some of the organizers, which was in difficult times due to the COVID pandemic. With the second edition and possible future ones we are motivated to establish the only regular conference of this kind in Germany and make it also a leading event on the international level.

How do you see the current relationship between mathematical theory and practical ML applications?

The answer could be twofold: in a pessimistic view I would say current ML systems in practice are including a lot of engineered components and some trial-and-error solutions, with a lot of their success simply being based on the availability of enormous computational power and amounts of data. In an optimistic view I would say that all current practical ML applications build on deep mathematical developments and the turnaround time between theoretical developments and practical use is quite low. This can be seen in many celebrated generative models that are closely connected to developments in optimal transport and the theory of diffusion processes.

What are some success stories where mathematical insights have directly improved ML methods?

Basically the whole field builds on its mathematical foundations, many of which were developed decades ago but had to wait for huge data sets and computational power to become available. Gradient descent and its stochastic variants used nowadays for training models were developed and analyzed in the 1960s and 70s, with many modifications in the decades after. Automatic differentiation being a fundamental tool now had its strongest steps forward in the 1990s and early 2000s. There are several examples were mathematical insights have later improved the existing models, for example, adversarial learning, which makes models more resilient against adversarial attacks.

What are some critical open questions in the field that you hope this conference will address?

A major critical question since the beginning of machine learning is the (missing) reliability of machine learning, leading to issues such as hallucinations. Another one is the missing efficiency still inherent in deep neural networks. Compared to, for example, a human brain the carbon and water consumption of machine learning models achieving similar tasks is ridiculously high. This indicates that there could be fundamentally different models with much better performance to be developed and studied in the future.

What new mathematical questions do you see emerging from practical machine learning applications?

A partly new question arises from the understanding of transformer architectures and their inherent, so-called attention mechanisms. This gives rise to new problems in optimization, particle systems, partial differential equations and even dynamical systems. Generative models currently motivate advances in stochastic control, large-scale statistical inference and inverse problems. And issues like generalization or optimization aspects in training large neural networks keep mathematicians busy since decades.

Where do you see gaps between experimental results and theoretical understanding today?

There are still some areas where theoretical results can only partially explain effects observed in practice, for example, estimates for generalization errors are far from being sharp and thus often from explaining what is observed. Also the convergence observed in training deep neural networks cannot be underpinned fully theoretical convergence guarantees. On other hand, so far machine learning was always open to new theoretical developments and able to integrate them fast into practice in case they were promising. With the growing maturity of AI products and huge amount of investments behind them I am not sure if the openness will prevail in future.

Looking ahead, what do you hope participants will take home from this year’s conference?

I would hope they will take home a lot of new insights from mathematical approaches to ML on the one hand, and those working in theoretical fields will find many stimulating questions for their research.

On a personal note, what keeps you inspired to work on these foundational questions in such a fast-moving field?

I am strongly convinced that technological advance in this area needs a deep theoretical understanding, and indeed current advances build on foundational mathematical questions solved in the last decades. In addition a critical view from mathematics can help to highlight and identify problems inherent in large machine learning models, which developers are not aware or might prefer to put under the carpet otherwise. Even more personal I got into this field more than two decades ago due to various close connections with my core research area of inverse problems.

Thank you, Martin, for your insights! Conversations like these show how collaboration between disciplines can drive meaningful progress in machine learning.

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As machine learning technologies continue to advance, understanding their foundations becomes increasingly important. The Conference on Mathematics of Machine Learning 2025 highlights how mathematical research contributes to developing more reliable, efficient, and interpretable machine learning systems.

By fostering dialogue between theoretical and applied communities, the event provides an opportunity to reflect on what we know, identify current limitations, and shape future directions. The conference also emphasizes that collaboration across disciplines is essential for meaningful progress.