How do simple rotations affect the implicit bias of Adam?

📄 Abstract

Abstract: Adaptive gradient methods such as Adam and Adagrad are widely used in machine learning, yet their effect on the generalization of learned models -- relative to methods like gradient descent -- remains poorly understood. Prior work on binary classification suggests that Adam exhibits a ``richness bias,'' which can help it learn nonlinear decision boundaries closer to the Bayes-optimal decision boundary relative to gradient descent. However, the coordinate-wise preconditioning scheme employed by Adam renders the overall method sensitive to orthogonal transformations of feature space. We show that this sensitivity can manifest as a reversal of Adam's competitive advantage: even small rotations of the underlying data distribution can make Adam forfeit its richness bias and converge to a linear decision boundary that is farther from the Bayes-optimal decision boundary than the one learned by gradient descent. To alleviate this issue, we show that a recently proposed reparameterization method -- which applies an orthogonal transformation to the optimization objective -- endows any first-order method with equivariance to data rotations, and we empirically demonstrate its ability to restore Adam's bias towards rich decision boundaries.