In this work, we investigate a particular implicit bias in gradient descent training, which we term “Feature Averaging,” and argue that it is one of the principal factors contributing to the non-robustness of deep neural networks. We show that, even when multiple discriminative features are present in the input data, neural networks trained by gradient descent tend to rely on an average (or a certain combination) of these features for classification, rather than distinguishing and leveraging each feature individually. Specifically, we provide a detailed theoretical analysis of the training dynamics of two-layer ReLU networks on a binary classification task, where the data distribution consists of multiple clusters with mutually orthogonal centers. We rigorously prove that gradient descent biases the network towards feature averaging, where the weights of each hidden neuron represent an average of the cluster centers (each corresponding to a distinct feature), thereby making the network vulnerable to input perturbations aligned with the negative direction of the averaged features. On the positive side, we demonstrate that this vulnerability can be mitigated through more granular supervision. In particular, we prove that a two-layer ReLU network can achieve optimal robustness when trained to classify individual features rather than merely the original binary classes. Finally, we validate our theoretical findings with experiments on synthetic datasets, MNIST, and CIFAR-10, and confirm the prevalence of feature averaging and its impact on adversarial robustness. We hope these theoretical and empirical insights deepen the understanding of how gradient descent shapes feature learning and adversarial robustness, and how more detailed supervision can enhance robustness.

Large Language Models (LLMs) have demonstrated remarkable capabilities across numerous tasks, yet principled explanations for their underlying mechanisms and several phenomena, such as scaling laws, hallucinations, and related behaviors, remain elusive. In this work, we revisit the classical relationship between compression and prediction, grounded in Kolmogorov complexity and Shannon information theory, to provide deeper insights into LLM behaviors. By leveraging the Kolmogorov Structure Function and interpreting LLM compression as a two-part coding process, we offer a detailed view of how LLMs acquire and store information across increasing model and data scales -- from pervasive syntactic patterns to progressively rarer knowledge elements. Motivated by this theoretical perspective and natural assumptions inspired by Heap’s and Zipf’s laws, we introduce a simplified yet representative hierarchical data-generation framework called the Syntax-Knowledge model. Under the Bayesian setting, we show that prediction and compression within this model naturally lead to diverse learning and scaling behaviors of LLMs. In particular, our theoretical analysis offers intuitive and principled explanations for both data and model scaling laws, the dynamics of knowledge acquisition during training and fine-tuning, factual knowledge hallucinations in LLMs. The experimental results validate our theoretical predictions.