Towards Understanding Inductive Bias in Transformers: A View From Infinity (2402.05173v2)

Abstract: We study inductive bias in Transformers in the infinitely over-parameterized Gaussian process limit and argue transformers tend to be biased towards more permutation symmetric functions in sequence space. We show that the representation theory of the symmetric group can be used to give quantitative analytical predictions when the dataset is symmetric to permutations between tokens. We present a simplified transformer block and solve the model at the limit, including accurate predictions for the learning curves and network outputs. We show that in common setups, one can derive tight bounds in the form of a scaling law for the learnability as a function of the context length. Finally, we argue WikiText dataset, does indeed possess a degree of permutation symmetry.

• The paper demonstrates that the Gaussian process limit reveals how permutation symmetry enhances function learnability in Transformer models.
• It employs representation theory to connect symmetry properties with scaling laws, clarifying the impact of sequence length on learning efficiency.
• Empirical results on datasets like WikiText-2 show that Transformer models with inductive bias generalize robustly to out-of-distribution tasks.

Understanding Inductive Bias in Transformers Through Permutation Symmetry

The paper "Towards Understanding Inductive Bias in Transformers: A View From Infinity" presents a sophisticated analysis of the inductive biases inherent in Transformer models, specifically within the framework of infinite-width neural networks using Gaussian processes (GPs). The authors employ the representation theory of symmetric groups to quantitatively address the bias towards permutation symmetry in sequence space, providing insights that are both theoretical and empirical.

Transformers, with their self-attention mechanisms, have become the backbone of numerous state-of-the-art models in various domains, including NLP and vision. They often deal with data that can exhibit permutation symmetry, such as language sequences where the order can sometimes be shuffled. This paper focuses on how these models prioritize learning tasks aligned with permutation symmetry when viewed through the lens of over-parameterized Gaussian processes—a theoretical model corresponding to infinitely wide neural networks.

Key Results and Methodology

1. Gaussian Process Limit:
   • The paper relies on understanding the behavior of Transformers in the GP limit, where the network's behavior can be analyzed analytically. The Transformer is simplified to a basic linear attention model to derive these insights.
   • The GP limit provides a tractable way to explore inductive biases, linking them directly to the Bayesian prior over functions induced by network initialization.
2. Inductive Bias and Symmetry:
   • The authors demonstrate that when datasets exhibit permutation symmetry, the learnability of functions by Transformers improves. This is quantitatively described by the irreducible representations (irreps) of the symmetric group, enabling a characterization of how many samples are required to effectively learn a target function.
3. Scaling Laws:
   • A critical finding is the scaling law derived for learning in relation to sequence length (context length). The research establishes that the required number of samples to learn a task scales with the context length, dictated by the symmetry properties and the kernel's eigenspectrum.
4. Application to Natural Language Processing:
   • By analyzing datasets like WikiText-2, the paper illustrates how natural language data possesses a permutation symmetry to some extent, supporting the practical applicability of their theoretical predictions.
5. Model Generalization and OOD:
   • The paper also extends its results to Out-of-Distribution (OOD) generalization, showing that models trained with a permutation bias manage to generalize in unforeseen contexts, indicating robustness beyond the training distribution.

Implications

The implications are multifold. Theoretically, this paper contributes to the understanding of how architecture-driven biases in neural networks, particularly Transformers, emerge and how they can be quantitatively analyzed and predicted using tools from representation theory. Practically, it offers potential guidance for model design, informing techniques for improving learnability and generalization capabilities of Transformer networks on permutationally symmetric data.

Future Developments

Speculating on future developments, this research paves the way for leveraging permutation symmetry property in real-world applications such as NLP. It opens new avenues for designing initialization schemes or architectural modifications that could exploit these biases for more efficient learning.

Moreover, there are opportunities to extend this framework to finite-width networks and explore how training dynamics (such as those involving finite learning rates) might interact with these inherent biases. This could involve developing new analytic tools to bridge gaps between infinite and finite neural network behaviors, providing a more comprehensive understanding of deep neural networks in practical scenarios.

Overall, the paper is a substantial step in demystifying the inductive biases of Transformers, providing clearer guidance on how these biases can be understood, measured, and potentially harnessed for improved model performance.