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This research investigates the geometric relationship between the standard Adam optimizer and natural gradient descent (NGD), revealing that Adam's effectiveness arises from a balance of structural approximation errors and momentum smoothing rather than strict adherence to the natural gradient path. The study finds that while Adam exhibits significant geometric deviation in ill-conditioned scenarios, this drift correlates with slower initial optimization without compromising final loss minimization. These findings offer critical insights into the theoretical underpinnings of widely used optimizers and their behavior across different loss landscapes.
How Far is Adam from Natural Gradient Descent?
arXiv:2610.00004v1 Announce Type: new Abstract: 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 $\gamma(\Delta\theta)$ 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 significantly under ill-conditioning, reaching misalignments of $\approx 10^3$ in the neural network. Higher geometric drift correlates with slower initial optimization but does not degrade final objective minimization; Adam consistently reaches low loss. Furthermore, the improved empirical Fisher (iEF) tracks more stable paths than the standard empirical Fisher (EF), which frequently oscillates or diverges. Our results suggest Adam's practical optimization power may stem from a balance of structural approximation errors and momentum smoothing rather than close tracking of the natural gradient path.
Title:How Far is Adam from Natural Gradient Descent?
Abstract: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 $\gamma(\Delta\theta)$ 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 significantly under ill-conditioning, reaching misalignments of $\approx 10^3$ in the neural network. Higher geometric drift correlates with slower initial optimization but does not degrade final objective minimization; Adam consistently reaches low loss. Furthermore, the improved empirical Fisher (iEF) tracks more stable paths than the standard empirical Fisher (EF), which frequently oscillates or diverges. Our results suggest Adam's practical optimization power may stem from a balance of structural approximation errors and momentum smoothing rather than close tracking of the natural gradient path.
Machine Learning (cs.LG); Neural and Evolutionary Computing (cs.NE)
arXiv:2610.00004 [cs.LG]
(or arXiv:2610.00004v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.00004 Focus to learn more arXiv-issued DOI via DataCite
https://doi.org/10.5281/zenodo.20466393 Focus to learn more DOI(s) linking to related resources
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