Certification from Examples is Hard for Circuits and Transformers under Minimal Overparametrization
As state-of-the-art neural networks are deployed on reasoning and algorithmic tasks, exactness guarantees become increasingly important. However, high average-case accuracy can still mask inconsistent behaviors. This motivates exact certification, which asks for the smallest set of labeled examples needed to certify that a learned hypothesis equals the target. We show that while some hypotheses are easy to certify, even minimal overparametrization can make certification exponentially hard across several hypothesis classes. For threshold circuits of depth $\ge 2$, adding a single extra gate can force certificate sizes exponential in the input dimension. We show an analogous hardness result...
| Authors | Artur Back de Luca, Kimon Fountoulakis |
|---|---|
| Primary category | cs.LG |
| Published | 21 May 2026 |
| Last updated at source | 21 May 2026 |
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