Divergence at the Interpolation Threshold: Identifying, Interpreting & Ablating the Sources of a Deep Learning Puzzle

Rylan Schaeffer, Zachary Robertson, Akhilan Boopathy, Mikail Khona, Ila R Fiete, Andrey Gromov, Sanmi Koyejo

Keywords: mechanistic interpretability, interpretability, double descent

Abstract: Machine learning models misbehave, often in unexpected ways. One prominent misbehavior is when the test loss diverges at the interpolation threshold, perhaps best known from its distinctive appearance in double descent. While considerable theoretical effort has gone into understanding generalization of overparameterized models, less effort has been made at understanding why the test loss misbehaves at the interpolation threshold. Moreover, analytically solvable models in this area employ a range of assumptions and use complex techniques from random matrix theory, statistical mechanics, and kernel methods, making it difficult to assess when and why test error might diverge; for instance, recent work found a divergence in noise-free toy nonlinear autoencoders, surprising the authors and raising questions about whether such an outcome should have been anticipated. In this work, we analytically study the simplest supervised model - ordinary linear regression - and show intuitively and rigorously when and why a divergence occurs at the interpolation threshold using basic linear algebra. We identify three interpretable factors that, when simultaneously all present, cause double descent. We demonstrate on real data that both models' test losses diverge at the interpolation threshold and that the divergence disappears when we ablate any one of the three identified factors. We conclude by using our fresh perspective to shed light on recent observations in nonlinear models concerning superposition and double descent.

Primary Area: visualization or interpretation of learned representations

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Submission Number: 6092