What Languages are Easy to Language-Model? A Perspective from Learning Probabilistic Regular Languages (2406.04289v4)

Abstract: What can LLMs learn? By definition, LLMs (LM) are distributions over strings. Therefore, an intuitive way of addressing the above question is to formalize it as a matter of learnability of classes of distributions over strings. While prior work in this direction focused on assessing the theoretical limits, in contrast, we seek to understand the empirical learnability. Unlike prior empirical work, we evaluate neural LMs on their home turf-learning probabilistic languages-rather than as classifiers of formal languages. In particular, we investigate the learnability of regular LMs (RLMs) by RNN and Transformer LMs. We empirically test the learnability of RLMs as a function of various complexity parameters of the RLM and the hidden state size of the neural LM. We find that the RLM rank, which corresponds to the size of linear space spanned by the logits of its conditional distributions, and the expected length of sampled strings are strong and significant predictors of learnability for both RNNs and Transformers. Several other predictors also reach significance, but with differing patterns between RNNs and Transformers.

• The paper demonstrates that PFSA rank and expected string length significantly predict neural LM performance, evidenced by KL divergence metrics.
• The paper finds that RNNs outperform Transformers in modeling probabilistic languages due to their sequential processing aligning with automata dynamics.
• The paper reveals that higher PFSA entropy correlates with lower KL divergence, suggesting that increased randomness aids language model learnability.

Investigating the Learnability of Probabilistic Formal Languages by Neural LLMs

This essay provides an expert analysis of a paper exploring the capacity of neural LLMs (LMs), specifically Recurrent Neural Networks (RNNs) and Transformers, to learn probabilistic regular languages as defined by probabilistic finite-state automata (PFSA). The central objective is the empirical assessment of these models' ability to learn distributions over strings, contrasting theoretical potential with practical learnability.

Core Investigation and Methodology

The paper differentiates itself by focusing on probabilistic language learning using neural LMs, rather than treating them as traditional classifiers of formal languages. By deriving and analyzing specific theoretical limitations, it emphasizes the necessity to go beyond coverage of formal language properties in isolation — particularly aiming to understand the significance of empirical factors that influence learning within these established boundaries. Concretely, the paper investigates how learnability is affected by variables such as the number of states, symbols, transitions, rank of the emission matrix, and expected string length.

Approximately 2100 PFSAs were generated and from these, 15k RNN and Transformer models were trained using corpora derived from random PFSAs deployed with varying complexity. The learning performance was quantitatively measured using the Kullback-Leibler (KL) divergence between the model and PFSA distributions.

Major Findings and Analysis

The paper confirms several key insights:

1. Learnability Predictors: The PFSA rank and expected string length consistently emerge as significant predictors of neural LM performance for both RNNs and Transformers. The rank of an automaton directly correlates to the complexity and consequently affects the required capacity of the neural model to approximate its language.
2. Model Comparison: Empirical findings suggest that RNNs outperform Transformers in modeling formal languages, a trend attributed to RNNs' sequential processing nature, which aligns more closely with the operation of automata. This is reflected in quantitative differences like lower average KL divergence for RNNs.
3. Impact of Entropy: Interestingly, higher entropy in the underlying PFSA proved to correlate with reduced KL divergence, suggesting that models are better at learning distributions from PFSAs that have more intrinsic randomness.

The statistical analysis through linear regression models highlights that while theoretical capacity constraints are crucial, practical learnability is influenced by a broader set of factors. These findings elucidate that both architectural capacity and the inherent properties of the language being modeled must be considered in tandem.

Theoretical and Practical Implications

From a theoretical standpoint, the restriction that hidden state size must scale with PFSA rank offers a softmax bottleneck limitation on representation in neural LMs. In effect, this underlines the trade-offs between parameter sharing and exact representation of formal models, pertinent for understanding length generalization issues.

Practically, insights suggest design considerations for LLMs: ensuring sufficient hidden state dimensionality to encompass automata-derived complexities for tasks simulating human language dynamics. Moreover, this paper opens avenues for using probabilistic formal languages to benchmark neural LLM capacitation.

Future Directions

The paper lays a foundation for further interrogation into non-deterministic automata and exploring context-free models like probabilistic pushdown automata. Moreover, investigating empirical complexities including dataset size could provide profound implications on how neural LMs can better capture the probabilistic nature of human language beyond theoretical limits.

Overall, this paper significantly advances our understanding of the intricate interplay between theoretical representational limits and empirical capabilities of neural LMs in probabilistic language tasks.