Feature emergence via margin maximization: case studies in algebraic tasks

Understanding the internal representations learned by neural networks is a cornerstone challenge in the science of machine learning. While there have been significant recent strides in some cases towards understanding how neural networks implement specific target functions, this paper explores a complementary question -- why do networks arrive at particular computational strategies? Our inquiry focuses on the algebraic learning tasks of modular addition, sparse parities, and finite group operations. Our primary theoretical findings analytically characterize the features learned by stylized neural networks for these algebraic tasks. Notably, our main technique demonstrates how the principle of margin maximization alone can be used to fully specify the features learned by the network. Specifically, we prove that the trained networks utilize Fourier features to perform modular addition and employ features corresponding to irreducible group-theoretic representations to perform compositions in general groups, aligning closely with the empirical observations of Nanda et al. and Chughtai et al. More generally, we hope our techniques can help to foster a deeper understanding of why neural networks adopt specific computational strategies.

Beyond Implicit Bias: The Insignificance of SGD Noise in Online Learning

The success of SGD in deep learning has been ascribed by prior works to the implicit bias induced by high learning rate or small batch size ("SGD noise"). While prior works that focused on offline learning (i.e., multiple-epoch training), we study the impact of SGD noise on online (i.e., single epoch) learning. Through an extensive empirical analysis of image and language data, we demonstrate that large learning rate and small batch size do not confer any implicit bias advantages in online learning. In contrast to offline learning, the benefits of SGD noise in online learning are strictly computational, facilitating larger or more cost-effective gradient steps. Our work suggests that SGD in the online regime can be construed as taking noisy steps along the "golden path" of the noiseless gradient flow algorithm. We provide evidence to support this hypothesis by conducting experiments that reduce SGD noise during training and by measuring the pointwise functional distance between models trained with varying SGD noise levels, but at equivalent loss values. Our findings challenge the prevailing understanding of SGD and offer novel insights into its role in online learning.

Feature-Learning Networks Are Consistent Across Widths At Realistic Scales

We study the effect of width on the dynamics of feature-learning neural networks across a variety of architectures and datasets. Early in training, wide neural networks trained on online data have not only identical loss curves but also agree in their point-wise test predictions throughout training. For simple tasks such as CIFAR-5m this holds throughout training for networks of realistic widths. We also show that structural properties of the models, including internal representations, preactivation distributions, edge of stability phenomena, and large learning rate effects are consistent across large widths. This motivates the hypothesis that phenomena seen in realistic models can be captured by infinite-width, feature-learning limits. For harder tasks (such as ImageNet and language modeling), and later training times, finite-width deviations grow systematically. Two distinct effects cause these deviations across widths. First, the network output has initialization-dependent variance scaling inversely with width, which can be removed by ensembling networks. We observe, however, that ensembles of narrower networks perform worse than a single wide network. We call this the bias of narrower width. We conclude with a spectral perspective on the origin of this finite-width bias.

Simplicity Bias in 1-Hidden Layer Neural Networks

Recent works have demonstrated that neural networks exhibit extreme simplicity bias(SB). That is, they learn only the simplest features to solve a task at hand, even in the presence of other, more robust but more complex features. Due to the lack of a general and rigorous definition of features, these works showcase SB on semi-synthetic datasets such as Color-MNIST, MNIST-CIFAR where defining features is relatively easier. In this work, we rigorously define as well as thoroughly establish SB for one hidden layer neural networks. More concretely, (i) we define SB as the network essentially being a function of a low dimensional projection of the inputs (ii) theoretically, we show that when the data is linearly separable, the network primarily depends on only the linearly separable ($1$-dimensional) subspace even in the presence of an arbitrarily large number of other, more complex features which could have led to a significantly more robust classifier, (iii) empirically, we show that models trained on real datasets such as Imagenette and Waterbirds-Landbirds indeed depend on a low dimensional projection of the inputs, thereby demonstrating SB on these datasets, iv) finally, we present a natural ensemble approach that encourages diversity in models by training successive models on features not used by earlier models, and demonstrate that it yields models that are significantly more robust to Gaussian noise.

Using noise resilience for ranking generalization of deep neural networks

Recent papers have shown that sufficiently overparameterized neural networks can perfectly fit even random labels. Thus, it is crucial to understand the underlying reason behind the generalization performance of a network on real-world data. In this work, we propose several measures to predict the generalization error of a network given the training data and its parameters. Using one of these measures, based on noise resilience of the network, we secured 5th position in the predicting generalization in deep learning (PGDL) competition at NeurIPS 2020.

Inductive Bias of Gradient Descent for Exponentially Weight Normalized Smooth Homogeneous Neural Nets

We analyze the inductive bias of gradient descent for weight normalized smooth homogeneous neural nets, when trained on exponential or cross-entropy loss. Our analysis focuses on exponential weight normalization (EWN), which encourages weight updates along the radial direction. This paper shows that the gradient flow path with EWN is equivalent to gradient flow on standard networks with an adaptive learning rate, and hence causes the weights to be updated in a way that prefers asymptotic relative sparsity. These results can be extended to hold for gradient descent via an appropriate adaptive learning rate. The asymptotic convergence rate of the loss in this setting is given by $\Theta(\frac{1}{t(\log t)^2})$, and is independent of the depth of the network. We contrast these results with the inductive bias of standard weight normalization (SWN) and unnormalized architectures, and demonstrate their implications on synthetic data sets.Experimental results on simple data sets and architectures support our claim on sparse EWN solutions, even with SGD. This demonstrates its potential applications in learning prunable neural networks.