Transformers Simulate MLE for Sequence Generation in Bayesian Networks (2501.02547v1)

Abstract: Transformers have achieved significant success in various fields, notably excelling in tasks involving sequential data like natural language processing. Despite these achievements, the theoretical understanding of transformers' capabilities remains limited. In this paper, we investigate the theoretical capabilities of transformers to autoregressively generate sequences in Bayesian networks based on in-context maximum likelihood estimation (MLE). Specifically, we consider a setting where a context is formed by a set of independent sequences generated according to a Bayesian network. We demonstrate that there exists a simple transformer model that can (i) estimate the conditional probabilities of the Bayesian network according to the context, and (ii) autoregressively generate a new sample according to the Bayesian network with estimated conditional probabilities. We further demonstrate in extensive experiments that such a transformer does not only exist in theory, but can also be effectively obtained through training. Our analysis highlights the potential of transformers to learn complex probabilistic models and contributes to a better understanding of LLMs as a powerful class of sequence generators.

• The paper demonstrates that a simple Transformer model can simulate Maximum Likelihood Estimation for sequence generation in Bayesian networks by estimating conditional probabilities.
• Empirically validated on synthetic and real datasets, the model uses attention mechanisms to identify parent nodes for accurate conditional probability estimation.
• These findings suggest significant potential for using Transformers in probabilistic modeling and offer insights into in-context learning capabilities within structured data environments.

Overview of "Transformers Simulate MLE for Sequence Generation in Bayesian Networks"

This paper investigates the theoretical capacity of Transformer models to autoregressively generate sequences given a Bayesian network structure through maximum likelihood estimation (MLE). Despite substantial empirical successes of Transformers in tasks involving sequential data, their theoretical capabilities remain under-explored. This work aims to fill this gap by demonstrating that a simple Transformer model can inherently perform MLE for sequence generation in Bayesian networks by estimating conditional probabilities based on observed contexts.

Main Contributions

• Theoretical Foundation: The authors establish that a simple two-layer Transformer model can successfully estimate conditional probabilities within a Bayesian network, effectively leveraging its structure to generate sequences with comparable fidelity to traditional MLE methods. This model independently handles each variable in the sequence by estimating its conditional distribution given its parent nodes and already generated variables in the sequence.
• Empirical Validation: Through extensive experiments on both synthetic and real-world datasets, the paper validates the theoretical claims, showing that Transformers can be trained to achieve MLE-based sequence generation with high accuracy. The synthetic experiments involve various data scenarios, including chains, trees, and general graphs, to test the versatility of the model.

Key Technical Insights

The paper illustrates that Transformers can implicitly model the dependencies within a Bayesian network using autoregressive sampling strategies. Theoretically, they show that with appropriately designed attention mechanisms, a Transformer can identify and utilize the parent-child relationships essential for conditional probability estimation. Here are the main technical insights:

• Attention Mechanism as a Selector: The use of self-attention layers in Transformers serves as an effective tool for identifying relevant parent nodes in the Bayesian network, enabling the model to focus on critical aspects of the input context that are necessary for accurately estimating the next variable's conditional distribution.
• Inference and Sampling: Through autoregressive sampling, the Transformer iteratively estimates the distribution of each variable in the Bayesian network, based on its previously generated variables and observed context, emulating the conventional MLE process.

Implications and Future Directions

The results suggest significant implications for using Transformers in probabilistic modeling and reasoning, extending their applicability beyond conventional tasks in NLP to structured data environments such as Bayesian networks. Moreover, this foundational understanding opens avenues for further research in several key areas:

• Understanding In-Context Learning: The ability of Transformers to perform in-context MLE for Bayesian networks paves the way for exploring their role in in-context learning, potentially offering insights into how these models can internally adapt to varied contexts without explicit re-training.
• Scalability and Generalization: Future work could explore the scalability of this approach to more complex networks and larger datasets, as well as generalization capabilities of models trained under this framework, particularly when dealing with out-of-distribution data.
• Algorithmic Efficiency: Another pertinent line of inquiry is the exploration of more efficient parameterization techniques to further streamline the model’s learning process, possibly enhancing its performance and reducing computational overhead.

Conclusion

This paper significantly advances the theoretical understanding of Transformers' capabilities in probabilistic sequence generation tasks. By aligning Transformers with MLE principles in Bayesian network contexts, it highlights their potential not only in language processing but also in broader AI systems that require sophisticated understanding and generation of structured data. This work lays a substantial foundation for both extending the application of Transformers and driving future theoretical explorations.