The Low-Rank Simplicity Bias in Deep Networks

Modern deep neural networks are highly over-parameterized compared to the data on which they are trained, yet they often generalize remarkably well. A flurry of recent work has asked: why do deep networks not overfit to their training data? In this work, we make a series of empirical observations that investigate the hypothesis that deeper networks are inductively biased to find solutions with lower rank embeddings. We conjecture that this bias exists because the volume of functions that maps to low-rank embedding increases with depth. We show empirically that our claim holds true on finite width linear and non-linear models and show that these are the solutions that generalize well. We then show that the low-rank simplicity bias exists even after training, using a wide variety of commonly used optimizers. We found this phenomenon to be resilient to initialization, hyper-parameters, and learning methods. We further demonstrate how linear over-parameterization of deep non-linear models can be used to induce low-rank bias, improving generalization performance without changing the effective model capacity. Practically, we demonstrate that simply linearly over-parameterizing standard models at training time can improve performance on image classification tasks, including ImageNet.