How Far is Adam from Natural Gradient Descent?
Adam is the standard optimizer in deep learning, yet its geometric relationship to natural gradient descent (NGD) contains unresolved questions. We study Adam's full update rule, including momentum, as a diagonal empirical Fisher approximation subject to diagonal truncation, empirical label substitution, and temporal lag. Using the scale-invariant $γ(Δθ)$ metric, we measure Adam's geometric deviation from true NGD across four loss landscapes: well-conditioned linear regression, ill-conditioned linear regression, logistic regression, and a non-convex small neural network. Adam's geometric trajectory is context-dependent. Deviation remains low in well-conditioned settings but rises signific...
| Authors | Vihaan Paka-Hegde |
|---|---|
| Primary category | cs.LG |
| Published | 30 May 2026 |
| Last updated at source | 30 May 2026 |
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