Machine Learning Needs a Langlands Programme (bastian.rieck.me)

The author argues that machine learning needs a unifying research programme — likened to the Langlands programme in mathematics — that systematically builds deep correspondences between different ML subfields, architectures and theoretical formalisms. Right now the field is sprawling: empirical advances (new regularizers, architectures, training tricks) accumulate faster than theoretical explanations (batch normalization took years to be analyzed), and results often remain siloed (e.g., proofs for deep sets or approximation theorems that don’t connect to other architectures). A Langlands-style effort would act like a Rosetta Stone, letting researchers translate problems and theorems across domains to reuse tools, compare decision surfaces, and convert insights from one setting into provable statements in another. Technically, this means seeking formal mappings between architectures and mathematical objects so one can compare and manipulate representations — for example, recent work expressing neural network decision boundaries as tropical polynomials/tropical hypersurfaces lets researchers analyze and compare classifiers via algebraic geometry. Concrete targets include rigorous characterizations of the functions learned by networks, capacity and universality conditions across architectures, and a refreshed study of the manifold hypothesis (what data manifolds look like, when it fails, and consequences). Such cross-cutting frameworks could reduce “tool proliferation,” accelerate theory-driven design, and make empirical tricks more interpretable and transferable across ML domains.