On Limitation of Transformer for Learning HMMs

On the Limitation of Transformers for Learning HMMs

The paper "On Limitation of Transformer for Learning HMMs" investigates the ability of Transformer models to learn Hidden Markov Models (HMMs), comparing their performance to Recurrent Neural Networks (RNNs). Despite the notable success of Transformers in various sequential modeling tasks, the paper highlights the limitations of Transformers when applied to learning HMMs.

Summary of Findings

The findings presented in this paper are derived from extensive empirical experiments on both random and structured HMM instances. The key outcomes can be summarized as follows:

1. Effectiveness in Learning HMM Models:

   • Random HMMs: Transformers can effectively learn belief state inference tasks when the training data includes true beliefs at each step. However, for next-observation prediction tasks, there are challenging instances where Transformers exhibit high prediction loss.
   • Structured HMMs: In structured HMMs with slow mixing speed and long history dependency, Transformers underperform compared to RNNs. Interestingly, Transformers tend to require considerably more depth to achieve comparable accuracy.

2. Comparison with RNNs:

   • Training Speed: RNNs consistently outperform Transformers in training speed across all tested models.
   • Testing Accuracy: RNNs achieve lower testing errors and exhibit greater robustness in hyperparameter tuning and curriculum scheduling.

3. Depth vs. Sequence Length:

   • Constant Depth: Simple HMMs like random HMMs and Linear Dynamical Systems (LDS) can be effectively learned by Transformers of constant depth, independent of sequence length.
   • Logarithmic Scaling: For more complex models like structured HMMs, the minimal depth required for Transformers shows an approximate logarithmic dependency on the sequence length.
   • Hard Instances: Certain HMM instances, constructed intentionally to be difficult, challenge Transformers even at constant sequence lengths.

Methodological Insights

The paper proposes a variant of the Chain-of-Thought (CoT) prompting, termed block CoT, to mitigate some limitations of Transformers:

• Block CoT Training: Training with block CoT involves feeding the output of Transformers back to themselves every $b$ tokens. This method integrates a recursive inductive bias within the architecture, significantly reducing evaluation error and enabling shallow Transformers to handle longer sequences. However, this approach increases the computational demands, thus slowing down the training process.

Theoretical Contributions

Complementary to the empirical findings, the paper provides theoretical results that elucidate the expressiveness of Transformers in approximating HMMs:

• Expressiveness of Transformers: Theoretical findings prove that an $L$-layer finite precision Transformer can fit any HMMs of at least $2^L$ sequence length. This conclusion aligns with the experimental observation of logarithmic scaling between Transformer depth and sequence length for more complex HMMs.

Implications and Future Work

Practical Implications:

• The limitations of Transformers in learning highly structured HMMs suggest potential challenges in applying Transformer-based architectures in reinforcement learning environments, particularly those characterized by slow mixing speeds and uninformative observations.
• The enhanced performance of RNNs in these contexts reaffirms the importance of selecting appropriate architectures based on the specific characteristics of the sequential data being modeled.

Theoretical Implications:

• Understanding the depth vs. sequence length trade-off in Transformers provides valuable insights into their architectural design and potential areas for improvement.
• The development of block CoT demonstrates a viable approach to enhance Transformer performance at a computational cost, highlighting a trade-off between model complexity and training efficiency.

Future Developments:

• Future research could focus on further refining CoT techniques to balance performance gains with computational efficiency.
• Exploring hybrid architectures that combine the strengths of RNNs and Transformers may offer a promising direction to address the limitations observed in purely Transformer-based models.

In conclusion, the paper underscores the inherent limitations of Transformer models in learning HMMs, providing both empirical evidence and theoretical insights. The proposed block CoT technique and detailed comparisons with RNNs offer a roadmap for future research to optimize Transformer performance in HMM-like data and beyond.