How abundant are good interpolators?

Researchers establish formal bounds on how frequently random interpolating classifiers generalize well across realistic data distributions. Using large deviation theory, the work quantifies the exponential proportion of solutions in the margin-constrained classifier space that achieve target error rates as dimension scales. This addresses a foundational question in modern machine learning: why overparameterized models that fit training data often still generalize. The result bridges statistical physics and learning theory, offering theoretical scaffolding for understanding when and why interpolation works in high dimensions, a phenomenon central to deep learning's empirical success.

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Explainer

The paper quantifies not just that interpolators can generalize, but the exponential proportion of the solution space that does so. This shifts the question from 'does it happen?' to 'how common is it?', which is a different kind of theoretical contribution than prior work.

This connects directly to the continual learning and parameter-efficient finetuning work from early June (TailLoR, ProtoAda, PEFT scaling). Those papers assume overparameterized models can adapt to new tasks without forgetting because the parameter space is large enough to route learning into unused directions. This interpolation abundance result provides theoretical scaffolding for why that assumption holds: in high dimensions, there are exponentially many solutions that fit training data and still generalize, so practitioners have room to maneuver. The work doesn't solve the routing problem those papers tackle, but it explains why the problem is tractable at all.

If follow-up work uses these bounds to predict which real datasets (CIFAR-10, ImageNet, etc.) should exhibit high interpolator abundance versus scarcity, and those predictions match empirical margin distributions, the theory moves from existence proof to predictive tool. If the bounds remain loose enough that they don't constrain practice, the contribution stays academic.

Coverage we drew on

TailLoR: Protecting Principal Components in Parameter-Efficient Continual Learning · arXiv cs.LG

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