Unraveling Text Generation in LLMs: A Stochastic Differential Equation Approach

TL;DR

This work treats text generation in large language models as a stochastic process governed by an SDE on the token-embedding state $X(t)$ with drift $\mu(X(t),t)$ and diffusion $\sigma(X(t),t)$. Both terms are parameterized by neural networks and learned from data, yielding an integrated model $dX(t)=\text{NN}_{\mu}(X(t),t;\theta_{\mu})dt+\text{NN}_{\sigma}(X(t),t;\theta_{\sigma})dW(t)$. The authors provide a rigorous theoretical analysis (existence/uniqueness, Lyapunov stability, and moment dynamics via Itô calculus) and validate the framework on real-world data (HelpSteer) with Drift/Diffusion nets, loss curves, and trajectory visualizations. Empirically, the drift captures deterministic linguistic structure while diffusion accounts for variability, enabling controlled generation and potential improvements in interpretability, safety, and robustness of LLM outputs. The study offers a mathematically grounded perspective on language generation that complements traditional static interpretability approaches and opens avenues for guided generation and multi-modal extensions.

Abstract

This paper explores the application of Stochastic Differential Equations (SDE) to interpret the text generation process of Large Language Models (LLMs) such as GPT-4. Text generation in LLMs is modeled as a stochastic process where each step depends on previously generated content and model parameters, sampling the next word from a vocabulary distribution. We represent this generation process using SDE to capture both deterministic trends and stochastic perturbations. The drift term describes the deterministic trends in the generation process, while the diffusion term captures the stochastic variations. We fit these functions using neural networks and validate the model on real-world text corpora. Through numerical simulations and comprehensive analyses, including drift and diffusion analysis, stochastic process property evaluation, and phase space exploration, we provide deep insights into the dynamics of text generation. This approach not only enhances the understanding of the inner workings of LLMs but also offers a novel mathematical perspective on language generation, which is crucial for diagnosing, optimizing, and controlling the quality of generated text.

Figures (7)

• Figure 1: Training and Validation Losses over Epochs for the SDE Model. The top plot shows the total loss, the middle plot shows the drift loss, and the bottom plot shows the diffusion loss. The model shows a consistent decrease in losses over the epochs, indicating improved learning and stability.
• Figure 2: Actual vs Predicted Trajectories. The left plot shows the actual trajectory of the text generation, while the right plot presents the predicted trajectory. The comparison between these trajectories highlights the alignment and discrepancies in the model's predictions.
• Figure 3: Text Generation Trajectories with Words. This plot visualizes the actual and predicted word trajectories during the text generation process. Each point represents a word in the sequence, showing the semantic movement in the PCA-reduced space. The alignment between the red (predicted) and blue (actual) paths illustrates the model's accuracy in capturing the semantic evolution.
• Figure 4: (Left) SDE Vector Field illustrating the drift magnitude during the text generation process. (Right) Diffusion Magnitude Heatmap showing the diffusion effects across different token positions.
• Figure 5: (Left) Refined SDE Vector Field highlighting drift magnitude across various components. (Right) Uncertainty Heatmap demonstrating the uncertainty across different tokens during generation.