Bilinear Sequence Regression: A Model for Learning from Long Sequences of High-dimensional Tokens (2410.18858v2)

Abstract: Current progress in artificial intelligence is centered around so-called LLMs that consist of neural networks processing long sequences of high-dimensional vectors called tokens. Statistical physics provides powerful tools to study the functioning of learning with neural networks and has played a recognized role in the development of modern machine learning. The statistical physics approach relies on simplified and analytically tractable models of data. However, simple tractable models for long sequences of high-dimensional tokens are largely underexplored. Inspired by the crucial role models such as the single-layer teacher-student perceptron (aka generalized linear regression) played in the theory of fully connected neural networks, in this paper, we introduce and study the bilinear sequence regression (BSR) as one of the most basic models for sequences of tokens. We note that modern architectures naturally subsume the BSR model due to the skip connections. Building on recent methodological progress, we compute the Bayes-optimal generalization error for the model in the limit of long sequences of high-dimensional tokens, and provide a message-passing algorithm that matches this performance. We quantify the improvement that optimal learning brings with respect to vectorizing the sequence of tokens and learning via simple linear regression. We also unveil surprising properties of the gradient descent algorithms in the BSR model.

• The paper quantifies Bayes-optimal generalization error using MMSE analysis, revealing phase transitions in high-dimensional token sequences.
• The paper demonstrates that preserving token sequence structure allows bilinear regression to outperform traditional ridge regression approaches.
• The paper introduces a novel GAMP-RIE message-passing algorithm that efficiently attains Bayes-optimal performance in polynomial time.

Overview of Bilinear Sequence Regression for High-Dimensional Token Sequences

The paper "Bilinear Sequence Regression: A Model for Learning from Long Sequences of High-Dimensional Tokens" introduces a prototypical model for understanding learning dynamics in the context of sequences composed of high-dimensional tokens, such as those encountered in natural language processing. Coined as the Bilinear Sequence Regression (BSR) model, this framework aligns closely with statistical physics methodologies and provides an analytically tractable approach to studying the theoretical underpinnings of such learning paradigms.

Central to the paper is the consideration of two primary parameters: the width of the regression $r$, representing the rank of the latent bilinear form, and the dimensions $L$ and $d$, depicting the length and embedding size of the token sequences, respectively. The authors operate within a high-dimensional setting, examining the asymptotic behaviors as both $L$ and $d$ grow to infinity while maintaining a fixed ratio $\beta = \max(L,d) / \min(L,d)$. This setup allows the exploration of the performance landscape for various sample complexities and width parameters.

Key Findings

1. Bayes-Optimal Estimation:
   • The paper quantifies the Bayes-optimal generalization error, termed the Minimal Mean Square Error (MMSE), in the high-dimensional limit for the BSR model. This is done for both Gaussian and non-Gaussian output channels. The optimal overlap between the true and estimated sequences is computed via replica analysis, leading to insights about phase transitions and predictive power in sequences of different widths and sequence lengths.
2. Performance of Traditional Algorithms:
   • A comparison is made between the BSR model's Bayes-optimal performance and traditional ridge regression applied to vectorized data sequences. The authors provide explicit evidence that the BSR model, by respecting the sequence structure, achieves superior learning performance compared to naive flattening approaches that disregard the token-sequence relationships.
3. Message-Passing Algorithm:
   • The paper introduces an innovative message-passing algorithm, termed GAMP-RIE, derived to efficiently meet the Bayes-optimal performance in polynomial time. Such algorithms are essential for practical application scenarios where theoretical optimality must coincide with computational feasibility.
4. Strong and Weak Recovery Thresholds:
   • The strong recovery threshold, at which the generalization error becomes zero, is analytically defined in terms of the sequence dimensions and the width of the latent representations. Notably, this threshold is always smaller for structured data, offering a benchmark for the performance of emerging neural architectures.

Theoretical and Practical Implications

Theoretically, this work paves the way for a deeper understanding of neural architectures that exploit attention mechanisms and token-sequence learning, such as transformers. By establishing a basic, mathematically-grounded model, it helps elucidate why such architectures outperform traditional vectorized approaches. Practically, this could influence the design of new architectures by emphasizing the utility of learning in token space rather than flattening data, hence leveraging structured sequence relationships inherent in the data.

Future Directions

Future work, as hinted by the authors, includes extending the BSR model to more structured inputs beyond Gaussian assumptions, exploring how attention mechanisms advantageously handle structured sequences, and providing a comprehensive analysis of gradient-based learning algorithms, including their convergence properties and generalization capabilities in practical applications. Additionally, addressing computational limits and elucidating statistical-to-computational gaps offers a promising area for continued research.

The paper effectively marries theoretical rigor with computational considerations, providing novel insights into the landscape of learning from sequences of high-dimensional tokens. With a focus on simplifying the complex interactions in token sequences, it reaffirms the importance of model-prior alignment in modern machine learning tasks.