Transformers Handle Endogeneity in In-Context Linear Regression

Authors: Haodong Liang, Krishna Balasubramanian, Lifeng Lai
Primary area: learning theory
Keywords: endogeneity, instrumental variables, transformers, bi-level gradient descent

Abstract:
We explore the capability of transformers to address endogeneity in in-context linear regression. Our main finding is that transformers inherently possess a mechanism to handle endogeneity effectively using instrumental variables (IV). First, we demonstrate that the transformer architecture can emulate a gradient-based bi-level optimization procedure that converges to the widely used two-stage least squares (2SLS) solution at an exponential rate. Next, we propose an in-context pretraining scheme and provide theoretical guarantees showing that the global minimizer of the pre-training loss achieves a small excess loss. Our extensive experiments validate these theoretical findings, showing that the trained transformer provides more robust and reliable in-context predictions and coefficient estimates than the 2SLS method, in the presence of endogeneity.

Decision: accept (poster)
Reviewer scores: 6, 6, 6

Reviews and discussion:
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creation date:
1730470473556

thread_type:
official_review

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creator: reviewer_xftg

<content>
summary:
The paper investigates the ability of transformers to address endogeneity in in-context linear regression, proposing that transformers can emulate the two-stage least squares (2SLS) method through gradient-based optimization. Key contributions include demonstrating how transformers can handle endogeneity using instrumental variables, proposing an in-context pretraining scheme with theoretical guarantees for minimal excess loss, and showcasing robust performance in various endogeneity scenarios, including weak and non-linear instrumental variables​.

strengths:
(1) This paper creatively combines transformer architectures with econometric techniques, specifically instrumental variables, to address endogeneity—a novel approach that extends transformers' applicability beyond traditional machine learning domains.

(2) The authors provide rigorous theoretical backing, including a bi-level optimization framework, and offer non-asymptotic error bounds, supporting their claims with comprehensive experiments that validate the model's performance against standard benchmarks like 2SLS.

(3)  By demonstrating that transformers can not only handle endogeneity but also generalize to complex scenarios (e.g., weak and non-linear IV), this work highlights the potential of transformers as a robust tool in econometrics, broadening the scope of in-context learning applications.

weaknesses:
(1) The paper provides strong theoretical foundations but lacks practical guidance for implementation. More details on the parameter settings, model configurations, and optimization process would enhance reproducibility and help readers better understand how to apply the proposed method.

(2) While the theoretical contributions are thorough, the presentation is complex and could benefit from simplification or visual aids. This would make the bi-level optimization framework and convergence properties more accessible to a broader audience.

(3)  The paper does not address the scalability of the proposed transformer model for handling endogeneity. Including an analysis of computational efficiency or memory requirements would help clarify its suitability for larger datasets or higher-dimensional problems.

questions:
(1) How does the proposed transformer model handle cases with highly correlated instruments? In many practical applications, instrumental variables may exhibit multicollinearity, which could impact the model's stability and coefficient estimates. Could the authors clarify whether the model’s training or architecture includes mechanisms to address such cases?

(2) What led to the choice of hyperparameters in the pretraining scheme? The paper presents a theoretical guarantee of minimal excess loss but does not detail the rationale behind the selection of specific hyperparameters, such as learning rates or number of transformer layers. Could the authors provide insights or guidelines on choosing these parameters for optimal performance?

(3) How sensitive is the model to changes in the strength of endogeneity? While the experiments demonstrate robustness to varying IV strengths, it would be valuable to understand if there is a threshold or particular cases where the transformer’s performance degrades. Could the authors elaborate on the model's sensitivity to different degrees of endogeneity?
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</thread>
<thread>
creation date:
1730687107631

thread_type:
official_review

<thread_object>
creator: reviewer_aqce

<content>
summary:
This paper first analyzes how to use IV regression for endogeneity, where the IV regression is estimated with 2SLS. Then, an error bound is proposed for this process. Furthermore, the paper shows that the transformer can achieve the gradient-based 2SLS for in-context linear regression.

strengths:
I am not familiar with IV regression and endogeneity, so I haven't reviewed the mathematical accuracy of the theorem. My comments are based solely on a basic understanding of the motivation, overall contribution, and presentation. Please consider them with low weights.

Overall, I find the paper well-motivated, with clear writing. The background information and literature review appear thorough.

Understanding the mechanism of the Transformer could be valuable for advancing future research in this area.

weaknesses:
I understand that theoretical analysis requires specific assumptions. I am, however, curious whether it might be possible to extend the theoretical analysis to the non-linear case, as most real-world scenarios tend to be non-linear.

Additionally, could you provide an example of real-world applications where the proposed analysis could be beneficial? If such examples exist, is it feasible to validate the analysis experimentally?

questions:
Please refer to the weaknesses section.
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</thread>
<thread>
creation date:
1730805695430

thread_type:
official_review

<thread_object>
creator: reviewer_fena

<content>
summary:
The paper extends the theoretical analysis done in previous work that looks at the class of functions that transformers can learn (e.g. simple linear regression) to a more complex set -- those with endogeneity and corresponding instrument variables. The authors show that transformers can learn this function and do as well as the direct solvers in most cases and potentially better in more challenging cases. They supplement this with a theoretical analysis.

strengths:
I want to caveat this review by saying that I have flagged to the AC that this is not my area of expertise.

The paper seems interesting: it extends the theoretical analysis done in previous work that looks at the class of functions that transformers can learn (e.g. simple linear regression) to a more complex set -- those with endogeneity and corresponding instrument variables. The authors show that transformers can learn this function and do as well as the direct solvers in most cases and potentially better in more challenging cases. They supplement this with a theoretical analysis.

weaknesses:
I want to caveat this review by saying that I have flagged to the AC that this is not my area of expertise.

I do not know enough about this area to understand the potential flaws in their statements / etc or more subtle points.
At a broad level, their intro / overview makes sense and seems convincing.

questions:
I want to caveat this review by saying that I have flagged to the AC that this is not my area of expertise.

I wonder how this work helps one to understand what capabilities a transformer should / should not be capable of? Can we say something more concrete / general around what class of functions we can argue that a transformer should be able to solve? How does this relate to larger models ?
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creation date:
1732295308041

thread_type:
official_comment

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creator: authors

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title:
General Response

comment:
We thank the reviewers for their positive evaluation and constructive feedback. In this general response, we would like to highlight the growing attention from the ML and AI communities toward the problem of instrumental variable (IV) regression, as evidenced by several recent publications in top machine learning conferences (e.g., Singh et al., 2019, NeurIPS; Muandet et al., 2020, NeurIPS; Bennett et al., 2023, COLT; Harris et al., 2022, ICML; Hartford et al., 2017, ICML; Cheng et al., 2024, ICLR; Wu et al., 2022, ICML). This increasing interest is driven by the connection of IV regression to various problems in causal inference and double machine learning.

Given this context, we believe our contribution is both timely and significant. Specifically, we demonstrate that transformers not only are theoretically capable of in-context IV regression but also empirically outperform traditional IV regression baselines (e.g., 2SLS) in addressing challenges such as weak instruments, multicollinearity, and certain forms of non-linearity.

Our work bridges modern AI methods like transformers with fields such as econometrics and sociology (which are consumers of IV regression methodology), while also deepening our understanding of the capabilities of these advanced models. We are encouraged by the broader implications of this research and its potential to catalyze further investigation and advancements in this area.

We would also like to point out that, based on the reviewer feedback, we have made the following updates to the draft.

1. **Multi-collinearity**: We have added new sections, Appendix C.3 and C.4 showing theoretically and empirically that transformers are capable of handling certain levels of Multi-collinearity.
2. **Non-linear extension**: Please note that our original draft contained empirical verification of the case of quadratic model for **x** and **z**. In  Appendix C.5, we have now added another case when the relationship between **x** and **z** is given by a two-layer neural network. 
3. **Varying Endogeneity**: In Appendix C.6, we have now added an experiment showing the performance of transformers on varying levels of endogeneity.
4. **Real-data experiments**: In Appendix C.7, we have now added real-data experiments on a canonical dataset for IV regression evaluation, showing the competitive performance of transformers.







Hartford, Jason, et al. "Deep IV: A flexible approach for counterfactual prediction." International Conference on Machine Learning. PMLR, 2017.

Muandet, Krikamol, et al. "Dual instrumental variable regression." Advances in Neural Information Processing Systems 33 (2020): 2710-2721.

Singh, Rahul, Maneesh Sahani, and Arthur Gretton. "Kernel instrumental variable regression." Advances in Neural Information Processing Systems 32 (2019).

Bennett, Andrew, et al. "Minimax Instrumental Variable Regression and $ L_2 $ Convergence Guarantees without Identification or Closedness." The Thirty Sixth Annual Conference on Learning Theory. PMLR, 2023.

Harris, Keegan, et al. "Strategic instrumental variable regression: Recovering causal relationships from strategic responses." International Conference on Machine Learning. PMLR, 2022.

Cheng, Debo, et al. "Conditional Instrumental Variable Regression with Representation Learning for Causal Inference." The Twelfth International Conference on Learning Representations.

Wu, Anpeng, et al. "Instrumental variable regression with confounder balancing." International Conference on Machine Learning. PMLR, 2022.
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Metareview:
metareview:
The paper investigates the ability of transformers to address endogeneity in in-context linear regression, proposing that transformers can emulate the two-stage least squares (2SLS) method through gradient-based optimization. All reviewers gave positive feedback while the confidence scores are relatively low. The paper's quality is high and authors engaged with rebuttals, so I see no clear reason to reject this work. I recommended an acceptance.