Delay Embedding Theory of Neural Sequence Models (2406.11993v1)

Abstract: To generate coherent responses, LLMs infer unobserved meaning from their input text sequence. One potential explanation for this capability arises from theories of delay embeddings in dynamical systems, which prove that unobserved variables can be recovered from the history of only a handful of observed variables. To test whether LLMs are effectively constructing delay embeddings, we measure the capacities of sequence models to reconstruct unobserved dynamics. We trained 1-layer transformer decoders and state-space sequence models on next-step prediction from noisy, partially-observed time series data. We found that each sequence layer can learn a viable embedding of the underlying system. However, state-space models have a stronger inductive bias than transformers-in particular, they more effectively reconstruct unobserved information at initialization, leading to more parameter-efficient models and lower error on dynamics tasks. Our work thus forges a novel connection between dynamical systems and deep learning sequence models via delay embedding theory.

• The paper reveals that state-space models (SSMs) inherently produce high-quality delay embeddings that lead to lower prediction errors.
• It demonstrates that transformers, though initially weaker, improve embedding quality with training and enhanced positional encoding.
• Results indicate that metrics like decoding accuracy and smoothness of embeddings critically impact performance on noisy time-series tasks.

Delay Embedding Theory of Neural Sequence Models: An Expert Overview

The paper "Delay Embedding Theory of Neural Sequence Models" inspects the capabilities of neural sequence models in dynamically reconstructing latent variables from time-series data. The authors, Mitchell Ostrow, Adam Eisen, and Ila Fiete, explore the application of delay embedding theory within the context of transformers and state-space models (SSMs), focusing on their performance in time-series prediction tasks.

Core Concept and Motivation

Neural sequence models such as transformers and SSMs have achieved significant performance in various tasks within NLP. These models operate on ordered sequences of data, which inherently positions them as candidates for solving temporal prediction problems. However, transformers are noted for their underwhelming performance in continuous time-series forecasting compared to SSMs. This performance gap has catalyzed efforts to understand the differences in their predictive capabilities and to refine transformer architectures for better temporal predictions.

The crux of the paper lies in leveraging delay embedding theory—a well-established concept in dynamical systems—to empirically evaluate the embedding quality of neural sequence models. Delay embedding theory posits that the unobserved dimensions of a dynamical system can be accurately reconstructed from a series of observed variables with appropriate delays. The paper specifically examines how well transformers and SSMs can generate delay embeddings that capture the underlying dynamics of chaotic systems, using the Lorenz attractor as a benchmark.

Methodology and Experimental Setup

To quantify the sequence models' effectiveness in constructing delay embeddings, the researchers employ the following methodological approach:

1. Sequence Models: The paper focuses on a simplified version of transformers and a structured SSM—the Linear Recurrent Unit (LRU). Both models are compared based on their performance on a noisy, partially observed Lorenz system.
2. Training: Models are trained on next-step prediction tasks over time series data with varying levels of Gaussian noise to evaluate robustness.
3. Embedding Metrics: Three main metrics are used to analyze the embeddings:
   • Decoding Hidden Variables: The ability to predict unobserved dimensions using decoders.
   • Measuring Smoothness: Assessing the overlap of nearest neighbors in the embedding and original space.
   • Measuring Unfolding: Evaluating the conditional variance of future predictions given the embedding.

Findings and Insights

The paper reveals several key insights:

• Inductive Bias of SSMs: SSMs demonstrate a pronounced inductive bias towards generating effective delay embeddings right from initialization. This results in better reconstruction of the underlying system and lower prediction errors compared to transformers.
• Performance Metrics: SSMs display superior performance in terms of the Mean Absolute Standardized Error (MASE) on noisy time series data. However, this comes at the cost of increased sensitivity to observational noise, partly due to the lower dimensionality and higher folding of the embeddings.
• Transformers' Flexibility: While transformers initially underperform in embedding quality, they progressively improve with training and can achieve competitive performance, albeit with a higher parameter count due to the inclusion of positional embeddings.

Implications and Future Directions

This work contributes to a deeper understanding of how neural sequence models learn temporal structures. The identification of SSMs' strong inductive biases suggests that they are particularly useful in low-data, low-compute scenarios, where efficient learning of delay embeddings is paramount. Moreover, the paper underscores the potential for improving transformers through better architectural designs that enhance their ability to handle continuous time series.

The paper posits future research directions, including the exploration of selective SSMs and transformers' mechanisms for delay selection and memory retention. These advancements could bridge the performance gap between transformers and SSMs further and enhance their applicability across a broader range of temporal prediction tasks.

In conclusion, the paper establishes a valuable connection between dynamical systems theory and deep learning, providing mechanistic insights into the capabilities and limitations of current neural sequence models. The findings have practical implications for the deployment of these models in various real-world time-series forecasting applications and theoretical significance in advancing our understanding of neural architectures.