Symmetries, Flat Minima, and the Conserved Quantities of Gradient Flow

Bo Zhao, Iordan Ganev, Robin Walters, Rose Yu, Nima Dehmamy

Keywords: symmetry, gradient flow, conserved quantity, flat minima, Lie group, Lie algebra

TL;DR: We introduce a framework for finding linear and nonlinear continuous symmetries in deep learning and show how they lead to extended local minima and conserved quantities

Abstract: Empirical studies of the loss landscape of deep networks have revealed that many local minima are connected through low-loss valleys. Yet, little is known about the theoretical origin of such valleys. We present a general framework for finding continuous symmetries in the parameter space, which carve out low-loss valleys. Our framework uses equivariances of the activation functions and can be applied to different layer architectures. To generalize this framework to nonlinear neural networks, we introduce a novel set of nonlinear, data-dependent symmetries. These symmetries can transform a trained model such that it performs similarly on new samples, which allows ensemble building that improves robustness under certain adversarial attacks. We then show that conserved quantities associated with linear symmetries can be used to define coordinates along low-loss valleys. The conserved quantities help reveal that using common initialization methods, gradient flow only explores a small part of the global minimum. By relating conserved quantities to convergence rate and sharpness of the minimum, we provide insights on how initialization impacts convergence and generalizability.

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