Linear Transformers are Versatile In-Context Learners

Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-15 19:40:10

Abstract: Recent research has demonstrated that transformers, particularly linear attention models, implicitly execute gradient-descent-like algorithms on data provided in-context during their forward inference step. However, their capability in handling more complex problems remains unexplored. In this paper, we prove that each layer of a linear transformer maintains a weight vector for an implicit linear regression problem and can be interpreted as performing a variant of preconditioned gradient descent. We also investigate the use of linear transformers in a challenging scenario where the training data is corrupted with different levels of noise. Remarkably, we demonstrate that for this problem linear transformers discover an intricate and highly effective optimization algorithm, surpassing or matching in performance many reasonable baselines. We analyze this algorithm and show that it is a novel approach incorporating momentum and adaptive rescaling based on noise levels. Our findings show that even linear transformers possess the surprising ability to discover sophisticated optimization strategies.

Main paper: 
  
 Appendix:  


Decision: This paper provides a theoretical analysis of the performance of linear transformers under in-context learning for linear regression tasks.  The authors show that each layer of the transformer maintains an implicit linear regression model and can be interpreted as conducting a preconditioned gradient descent.  The authors also show that even using diagonal attention matrices in this case the linear transformers can still offer good expressive capability.  When dealing with mixed linear regression problems with various levels of noise, the linear transformers can yield superior performance over baselines, which can be attributed to an intrinsic optimization process with noise-aware stepsize adjustments and rescaling. The paper introduces an interesting theoretical interpretation of linear transformers under in-context learning.  The rebuttal and following discussions have cleared most of the concerns raised by the reviewers and authors also put up additional results on attention matrices.  The authors are encouraged to improve the empirical investigation to make the paper stronger.

Reviews: review: paper_summary: paper_summary: The paper tries to understand the reasons of the strong performance of Transformers. The authors study linear Transformers trained on linear regression problems. Moreover, the authors explores the problem of regression where the labels have variable noise levels.
strength_weakness: strength_weakness: - Although the focus is on analyzing Transformers, the paper could have benefited from adding more empirical analysis.
questions: questions: See above
limitations: limitations: NA
paper_summary: paper_summary: This paper demonstrates that linear transformers maintain a weight vector for an implicit linear regression problem. The authors provide theoretical analysis showing that (1) linear transformers maintain a linear regression model at every block and (2) a diagonal parameterization of attention heads does not compromise the expressive power of the model. The authors support their theoretical findings with experimental validation.
strength_weakness: strength_weakness: 1. Limited empirical validation of Section 5.3: The results in Section 5.3 suggest that the update of y occurs every two steps. However, the paper lacks empirical studies to illustrate this phenomenon, particularly in a diagonal parameterization scenario. Can the authors provide:
a) Concrete experiments demonstrating this two-step update pattern? 
b) Visualizations or quantitative analyses that show how this behavior manifests in practice?
questions: questions: 1. Comparison of attention matrices in full and diagonal parameterizations:
Given the similar loss patterns shown in Figures 2 and 3 for full and diagonal parameterizations, it raises questions about the final structure of the attention matrices. Specifically, is it possible that the full matrices converge to a near-diagonal structure during training? If so, what implications does this have for the choice between full and diagonal parameterizations in practice? If not, how do the differences in attention matrices reconcile with the similar performance observed?
limitations: limitations: N/A
paper_summary: paper_summary: In this paper, the authors study linear transformers trained on linear regression problems and prove that each layer of every linear transformer maintains a weight vector for an underlying linear regression problem. Furthermore, the authors consider the mixed linear regression problem with varying noise levels and empirically demonstrate the flexibility of linear transformers for this problem.
strength_weakness: strength_weakness: **Incremental Nature**

The authors build on top of a lot of existing literature. While this is not necessarily bad, the delta compared to previous work might be too small. Since I'm not an expert in this domain, I'll have to rely on the opinion of the other reviewers to confirm my understanding.

**Unclear Baselines**

I'm not sure what the baselines are for. I understand that it is useful to give an idea how well the transformer does in the task. However, the author make it look as if it was important to beat them. Furthermore, the authors claim that these methods have an advantage given that they can use matrix inversion. In my understanding, the transformer is trained on much more data which could technically used to improve the noise-level estimation for the linear models and improve their performance as well. To me it is not really clear why the authors are so defensive in this experimentation.

The authors highlight that the transformer does so much better in the varying noise levels scenario compared to the linear models. However, there was little to no effort to make the linear models good in this scenario in the first place.
questions: questions: What are potential practical uses of this insight?
limitations: limitations: n/a
paper_summary: paper_summary: This paper proves that linear transformer layers maintain a weight vector for implicit linear regressions, including a more challenging scenario where data is corrupted with different levels of noise. In the theoretical analysis, this paper shows the intrinsic mechanism of the  gradient scent in linear attention, where implicit variants play various roles in parameter update. Besides, this paper shows that simplified linear attention with diagonal transformation also maintains a powerful GD behavior. The experiments demonstrate the flexibility of linear transformers which outperforms GD++ as well as other linear regression solutions. This research promotes the understanding of Transformer weights and implicit learning capabilities of attention-based models.
strength_weakness: strength_weakness: The analysis framework is still on linear regression and linear attention with simplified modifications, which is similar to previous references. The theory can not be generalized to other architectures and context settings.
questions: questions: 1. How do you think about different linear attention variants in theory? For example, the most trivial one is $QK^TV$, but there are many improved versions, such as RetNet, SSM, and GLA. Do these architectures bring stronger modeling capability in theory?
2. With linear regression layer-wisely, does your claim that linear Transformer with FFN only maintains linear regression still work? For example, in Theorem 4.2, it is easy to build a quadratic FFN to make more complex computations.
limitations: limitations: The limitation is discussed in the conclusion section.

Comments: title: please respond to rebuttal
Response to Rebuttal
comment: Dear reviewers,

The author-reviewer discussion period will end soon. Please make sure you read the authors' rebuttal and respond to it. If you have additional questions after reading the rebuttal please discuss with the authors. For those who have done so, thank you!

AC
Thank you for the clarifications.
Thanks for your response! I'd like to see your future work to explore the sophisticated modeling nature of the whole linear models, which is closer to real data and more valuable for the interpretation of LLMs. I will keep my score.
Thank you for the response and additional experiment. Adding more discussion or hypotheses about how the full parameterization converges with the diagonal ones would be great. Nevertheless, I like this paper and I am keeping my score.