Towards a theory of how the structure of language is acquired by deep neural networks

Weakness 1,a (context-freeness). We agree with the reviewer's remark on the context-freeness of natural languages and we will modify the text accordingly, stating specifically in both the abstract and the limitations section that the context-free description is approximate and captures many (but not all) syntactical forms observed in natural language.

Weakness 1,b (poverty of the stimulus. We agree with the reviewer's definition and we will clarify the text to make it explicit. We will replace the sentence in line 20 of the introduction with stating that the data acquired by children is not sufficient to uniquely determine the grammatical structure of their language.

Weakness 2 (non-ambiguity of RHM). We agree that the non-ambiguity of the rules and the uniformity of the production rules are not language-like features, and we will emphasize this point further both in section 2.1 and in the limitations section.

Weakness 3 (additional dataset, contemporary text). Please see comment above (author rebuttal).

Weakness 4 (ref. on LSTM). Ref. 2 is indeed relevant to our work and we thank the reviewer for pointing it to us. We will include the following sentence at the beginning of the 'additional related works' section to acknowledge this result: [ref. 2] provides evidence that LSTM language models learn short-range dependencies first and use them as a basis to build longer-range dependencies. Our results provide a theoretical framework for understanding this phenomenon.

Weakness 5 (practical impact on LM research). This paper provides a novel explanation for the scaling laws characterising the performance of LLMs, and extends them to include the context window length(see also reply to Weakness 2 of reviewer h6xP). These laws are having a huge practical impact, both in showing that success could be achieved simply by scaling up LLMs and in guiding the choice of hyper-parameters depending on the available data and compute. In addition, our work suggests a criterion for optimising the size of the context window depending on the available data and the behaviour of token-token correlations. We will emphasise this point further in the conclusion section of the revised manuscript.

Question 1. See comment above (author rebuttal).

Question 2. See reply to weakness 5 and Weakness 2 of reviewer h6xP.

In addition, we will implement the reviewer's suggestions and cite the mentioned papers in the revised manuscript.

Weakness 1. While the assumption of fixed tree geometry is unlikely to be satisfied in natural language data, we do not believe this assumption to be necessary for the power-law decay of correlations. This paper, for instance, shows other examples of text data where correlations decay as a power of the distance and argues that this is possible due to the context-free structure. This is indeed what we find in our model, where correlations decay exponentially with the distance along the tree, while the actual distance between tokens grows exponentially with the tree distance, resulting in power-law behaviour. We expect that this result will be quite generic.

Weakness 2. While we will add experiments on natural text data to the present submission, the research of effects that require other assumptions to be described is indeed out of the scope of the present work.

Weakness 1/Question 1. Please see the comment above (author rebuttal).

Weakness 2. We agree with the reviewer that this key point needs to be emphasised much more. In particular for deep generative models of data (large $L$), we indeed find that for large $P$, the learning curve can be described with a power-law $P^{−\alpha}$ as in~[17] (Kaplan et al., Scaling laws for neural language models), with exponent $\alpha =\log{ (m/v^{s−1})/(2 \log m)}$. It is interesting to note that this power-law results from a series of `emergent' phenomena where sub-portions of the data tree of different depths are learned. We will add a full paragraph on this point in section 4.1 and restate the importance of the result in the conclusions.

Question 2. The tests performed in sections 4.3 ( behaviour of hidden representations with increasing $P$) and 5 (saturation of the scaling law due to the size of the context window) can also be performed on state-of-the-art LLMs trained on larger datasets. Such a study would require a paper of its own---we are currently seeking to develop collaborations to make it possible. We also hope that the present paper can trigger concurrent works in that direction.

Weakness 1/Question 1. Please see the comment above (author rebuttal).

Weakness 2. Although the generation process is cheaper, the costs of training limit the available range of $P$ to that considered in the paper (up to a few million).

Question 2. Since this paper focuses on sample complexity (and not on optimising performance at fixed compute), we varied the model size only to guarantee overparametrisation, in the sense that the training loss approaches zero as the training time increases. Increasing, depth, number of heads or embedding dimension of the architectures beyond the values reported in the paper does not affect performance (measured by the test loss at early stopping) beyond the fluctuations due to the randomness of the initialisation. We will clarify this point further in the text and mention the possibility of studying compute-optimal scaling laws as an interesting question for the future.

Dear Reviewers,

we have completed the analysis of the Wikitext dataset and we confirm that the results are similar to those obtained for the Shakespeare dataset (as displayed in Figure 4 of the original manuscript). We will include this analysis in the revised manuscript.

We would be happy to share the relevant figure here if allowed.

Best regards, The authors