Smoothing the Black-Box: Signed-Distance Supervision for Black-Box Model Copying

Deployed machine learning systems must continuously evolve as data, architectures, and regulations change, often without access to original training data or model internals. In such settings, black-box copying provides a practical refactoring mechanism, i.e. upgrading legacy models by learning replicas from input-output queries alone. When restricted to hard-label outputs, copying turns into a discontinuous surface reconstruction problem from pointwise queries, severely limiting the ability to recover boundary geometry efficiently. We propose a distance-based copying (distillation) framework that replaces hard-label supervision with signed distances to the teacher's decision boundary, converting copying into a smooth regression problem that exploits local geometry. We develop an $α$-governed smoothing and regularization scheme with Hölder/Lipschitz control over the induced target surface, and introduce two model-agnostic algorithms to estimate signed distances under label-only access. Experiments on synthetic problems and UCI benchmarks show consistent improvements in fidelity and generalization accuracy over hard-label baselines, while enabling distance outputs as uncertainty-related signals for black-box replicas.

Fractional signature: a generalisation of the signature inspired by fractional calculus

In this paper, we propose a novel generalisation of the signature of a path, motivated by fractional calculus, which is able to describe the solutions of linear Caputo controlled FDEs. We also propose another generalisation of the signature, inspired by the previous one, but more convenient to use in machine learning. Finally, we test this last signature in a toy application to the problem of handwritten digit recognition, where significant improvements in accuracy rates are observed compared to those of the original signature.