Nonlinear activation functions are widely recognized for enhancing the expressivity of neural networks, which is the primary reason for their widespread implementation. In this work, we focus on ReLU activation and reveal a novel and intriguing property of nonlinear activations. By comparing enabling and disabling the nonlinear activations in the neural network, we demonstrate their specific effects on wide neural networks: (a) better feature separation, i.e., a larger angle separation for similar data in the feature space of model gradient, and (b) better NTK conditioning, i.e., a smaller condition number of neural tangent kernel (NTK). Furthermore, we show that the network depth (i.e., with more nonlinear activation operations) further amplifies these effects; in addition, in the infinite-width-then-depth limit, all data are equally separated with a fixed angle in the model gradient feature space, regardless of how similar they are originally in the input space. Note that, without the nonlinear activation, i.e., in a linear neural network, the data separation remains the same as for the original inputs and NTK condition number is equivalent to the Gram matrix, regardless of the network depth. Due to the close connection between NTK condition number and convergence theories, our results imply that nonlinear activation helps to improve the worst-case convergence rates of gradient based methods.

One of the chronic problems of deep-learning models is shortcut learning. In a case where the majority of training data are dominated by a certain feature, neural networks prefer to learn such a feature even if the feature is not generalizable outside the training set. Based on the framework of Neural Tangent Kernel (NTK), we analyzed the case of linear neural networks to derive some important properties of shortcut learning. We defined a "feature" of a neural network as an eigenfunction of NTK. Then, we found that shortcut features correspond to features with larger eigenvalues when the shortcuts stem from the imbalanced number of samples in the clustered distribution. We also showed that the features with larger eigenvalues still have a large influence on the neural network output even after training, due to data variances in the clusters. Such a preference for certain features remains even when a margin of a neural network output is controlled, which shows that the max-margin bias is not the only major reason for shortcut learning. These properties of linear neural networks are empirically extended for more complex neural networks as a two-layer fully-connected ReLU network and a ResNet-18.

Existing analyses of optimization in deep learning are either continuous, focusing on (variants of) gradient flow, or discrete, directly treating (variants of) gradient descent. Gradient flow is amenable to theoretical analysis, but is stylized and disregards computational efficiency. The extent to which it represents gradient descent is an open question in the theory of deep learning.

In transfer learning, the learner leverages auxiliary data to improve generalization on a main task. However, the precise theoretical understanding of when and how auxiliary data help remains incomplete. We provide new insights on this issue in two canonical linear settings: ordinary least squares regression and under-parameterized linear neural networks. For linear regression, we derive exact closed-form expressions for the expected generalization error with bias-variance decomposition, yielding necessary and sufficient conditions for auxiliary tasks to improve generalization on the main task. We also derive globally optimal task weights as outputs of solvable optimization programs, with consistency guarantees for empirical estimates. For linear neural networks with shared representations of width $q \leq K$, where $K$ is the number of auxiliary tasks, we derive a non-asymptotic expectation bound on the generalization error, yielding the first non-vacuous sufficient condition for beneficial auxiliary learning in this setting, as well as principled directions for task weight curation. We achieve this by proving a new column-wise low-rank perturbation bound for random matrices, which improves upon existing bounds by preserving fine-grained column structures. Our results are verified on synthetic data simulated with controlled parameters.

The most forceful driver of advancements in the field of Machine Learning over the past decades has been the success of deep neural networks.

It is motivated by avoiding stable manifolds of saddle points.

We study the convergence rate of first-order methods for rectangular matrix factorization, which is a canonical nonconvex optimization problem.