On the Implicit Bias of Adam
This work addresses the theoretical understanding of optimization algorithms for machine learning practitioners, but it is incremental as it extends prior analysis to other methods.
The paper investigates whether RMSProp and Adam exhibit implicit regularization like gradient descent, finding that such regularization depends on hyperparameters and training stage, and can penalize the one-norm of gradients or hinder its reduction, with numerical experiments exploring generalization effects.
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.