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On the Implicit Bias of Adam

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arxiv 2309.00079 v4 pith:DLY4FVFM submitted 2023-08-31 cs.LG cs.AImath.OCstat.COstat.ML

classification cs.LGcs.AImath.OCstat.COstat.ML
keywords adamgradientsimplicitlossodespenalizetermsanalysis
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In previous literature, backward error analysis was used to find ordinary differential equations (ODEs) approximating the gradient descent trajectory. It was found that finite step sizes implicitly regularize solutions because terms appearing in the ODEs penalize the two-norm of the loss gradients. We prove that the existence of similar implicit regularization in RMSProp and Adam depends on their hyperparameters and the training stage, but with a different "norm" involved: the corresponding ODE terms either penalize the (perturbed) one-norm of the loss gradients or, conversely, impede its reduction (the latter case being typical). We also conduct numerical experiments and discuss how the proven facts can influence generalization.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient

    cs.LG 2026-02 conditional novelty 7.0 of 10

    Mini-batch stochastic steepest descent has a batch-size-dependent margin gap; momentum or variance reduction eliminates it, while batch-size-1 sign/normalized SGD converges to a sample-frequency-biased direction rathe...

  2. Learning by solving differential equations

    cs.LG 2025-05 conditional novelty 5.0 of 10

    Runge-Kutta optimizers adapted with momentum, preconditioning, or adaptive learning rates can close the large-batch generalization gap and match Adam on small MLP workloads.

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