Understanding LLM Behaviors via Compression: Data Generation, Knowledge Acquisition and Scaling Laws

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.

Feature Averaging: An Implicit Bias of Gradient Descent Leading to Non-Robustness in Neural Networks

In this work, we investigate a particular implicit bias in the gradient descent training process, which we term "Feature Averaging", and argue that it is one of the principal factors contributing to non-robustness of deep neural networks. Despite the existence of multiple discriminative features capable of classifying data, neural networks trained by gradient descent exhibit a tendency to learn the average (or certain combination) of these features, rather than distinguishing and leveraging each feature individually. In particular, we provide a detailed theoretical analysis of the training dynamics of gradient descent in a two-layer ReLU network for a binary classification task, where the data distribution consists of multiple clusters with orthogonal cluster center vectors. We rigorously prove that gradient descent converges to the regime of feature averaging, wherein the weights associated with each hidden-layer neuron represent an average of the cluster centers (each center corresponding to a distinct feature). It leads the network classifier to be non-robust due to an attack that aligns with the negative direction of the averaged features. Furthermore, we prove that, with the provision of more granular supervised information, a two-layer multi-class neural network is capable of learning individual features, from which one can derive a binary classifier with the optimal robustness under our setting. Besides, we also conduct extensive experiments using synthetic datasets, MNIST and CIFAR-10 to substantiate the phenomenon of feature averaging and its role in adversarial robustness of neural networks. We hope the theoretical and empirical insights can provide a deeper understanding of the impact of the gradient descent training on feature learning process, which in turn influences the robustness of the network, and how more detailed supervision may enhance model robustness.