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A phase transition between positional and semantic learning in a solvable model of dot-product attention (2402.03902v2)

Published 6 Feb 2024 in cs.LG

Abstract: Many empirical studies have provided evidence for the emergence of algorithmic mechanisms (abilities) in the learning of LLMs, that lead to qualitative improvements of the model capabilities. Yet, a theoretical characterization of how such mechanisms emerge remains elusive. In this paper, we take a step in this direction by providing a tight theoretical analysis of the emergence of semantic attention in a solvable model of dot-product attention. More precisely, we consider a non-linear self-attention layer with trainable tied and low-rank query and key matrices. In the asymptotic limit of high-dimensional data and a comparably large number of training samples we provide a tight closed-form characterization of the global minimum of the non-convex empirical loss landscape. We show that this minimum corresponds to either a positional attention mechanism (with tokens attending to each other based on their respective positions) or a semantic attention mechanism (with tokens attending to each other based on their meaning), and evidence an emergent phase transition from the former to the latter with increasing sample complexity. Finally, we compare the dot-product attention layer to a linear positional baseline, and show that it outperforms the latter using the semantic mechanism provided it has access to sufficient data.

Citations (10)

Summary

  • The paper reveals a phase transition where dot-product attention shifts from positional to semantic learning as sample complexity rises.
  • It introduces a solvable model using tied low-rank query and key matrices to characterize the empirical loss landscape.
  • Experiments show that semantic attention outperforms positional baselines when training data is sufficiently abundant.

Overview of Attention Mechanisms in Deep Learning

Attention mechanisms have revolutionized the modeling of sequential data, particularly in the domain of natural language processing. Two distinct forms of attention, positional and semantic, cater to different aspects of the data. Positional attention leverages the order of tokens, whereas semantic attention harnesses the meaning embedded within them. A paper investigates the scenario wherein a simple architecture is capable of adapting to utilize either form of attention depending on the context and complexity of the samples it's trained on.

Theoretical Insights and Experimental Evidence

The paper presents a comprehensive exploration at both experimental and theoretical levels. On the experimental side, the authors demonstrate the flexibility of a transformer architecture to employ positional or semantic attention in accomplishing a counting task. Theoretically, they propose a solvable model of attention that captures the learning of tied low-rank query and key matrices. Notably, they find that the model displays a phase transition from positional to semantic attention as the sample complexity increases.

Sharp Phase Transitions in Attention

In the limit where both the embedding dimension and number of training samples grow large, the authors provide a closed-form characterization of the minimum of the empirical loss landscape, which reveals a sharp transition between mechanisms. This transition illustrates that under limited data, the transformer adopts a positional mechanism whereas, with ample data, it shifts to a semantic mechanism. The point at which this transition takes place increases with the strength of the positional information encoded in the task.

Practical Implications and Model Performance

When compared to a linear positional baseline model, the dot-product attention layer emerges superior provided it employs semantic attention and is trained on a sufficiently large dataset. This superior performance hinges on the model's ability to learn semantic relationships when sample complexity surpasses a certain threshold. The findings suggest a fundamental role for semantic learning in the model's accuracy, and imply the necessity for appropriate data regimes to fully exploit the potential of attention-based models.

In conclusion, this paper opens new avenues of paper for understanding and optimizing attention mechanisms in artificial intelligence. The phase transition phenomenon has implications for model design, especially in resource-constrained scenarios, and emphasizes the importance of data availability in achieving high-performing models. These insights have the potential to impact the development of more efficient and effective transformer architectures in the future.

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