Replying to Official Comment by Reviewer cRiK
Thank you sincerely for your feedback. Here is our response:
> **Question 1: ** A 1-1 (quick) comparison against each of the five papers. [1-5].
- Oswald et al. [1] focus on the ability of the linear attention layer to perform gradient descent during ICL when faced with linear regression tasks. Their observations are based on specific assumptions, including the constructed forms for $W_Q, W_K, W_V$ and the input tokens (concatenated $[x,y]$). Our work does not target specific tasks like linear regression; therefore, we do not make detailed assumptions about the model weights (simply treated as weights after pre-training) or construct specific input forms. Instead, we aim to view the ICL inference process from the perspective of representation learning in the dual model.
- Zhang et al. [2] also theoretically analyze the gradient flow of linear self-attention modules under certain initialization conditions when dealing with linear regression tasks and provide the forms of weights at the global minimum as well as the prediction error. Similar to the comparison above, the aim of our work is not to investigate the expressive capabilities of the model's structure for these specific tasks but to interpret the ICL inference process from a representation learning lens in a more general setting, thus we do not use these assumptions.
- Shen et al. [3] experimentally illustrates that the assumptions used in previous works including Oswald et al. [1] and Zhang et al. [2] may be too strong in real LLMs. They analyze the differences between ICL inference in LLMs and the fine-tuned models in real-world scenarios from various perspectives, including weight sparsity, order stability, and output distribution. In comparison, our work: (1) for the sake of theoretical analysis, considers not a "complete" real model but a simplified one that omits structures like residual connections; and (2) our interpretation of the ICL inference is not linked to fine-tuning on the original model but rather to training on the dual model.
- Dai et al. [4] use the dual form to interpret ICL as an implicit fine-tuning (gradient descent) of the original model under the linear attention setting and this alignment is ambiguous as the specific details of the gradient descent process is not clear. Thus, our work extends this analysis to the nonlinear attention setting and delve deeper into more details of this process (exploring the specific form of the loss function). The main difference is that we consider this as training the dual model under a self-supervised representation learning loss, rather than performing supervised fine-tuning on the original model.
- Natan et al. [5] investigate potential shortcomings in the evaluation metrics used by Dai et al.[4] in real model assessments and propose a layer-causal GD variant that performs better in simulating ICL. However, similar to the comparison above [3], their study discusses complex gradients in the original model in real scenarios, whereas we turn our attention to the gradient descent of the dual model of the attention layer.
> **Question 2: **A more detailed explanation why using the dual is important. (As the dual is simply a mathematical tool).
By using the dual, or rather, by analyzing the potential gradient descent process of the dual model, we can gain new insights into the model mechanisms in reverse. Specifically, through the dual perspective,
- we transform the forward inference process into an optimization process. Since optimization processes are well-known and have established theoretical tools (for example, generalization error as mentioned in our work), this transformation can provide reverse insights into analyzing the model mechanisms.
- we can clearly observe that the dual model involves a self-supervised representation learning process. Considering that there are lots of mature works in this area, we can draw on these works to reflect on the attention mechanism, which has also inspired attention modifications as illustrated in our work.
Finally, as for "the dual is simply a mathematical tool", we want to clarify that the term "dual" we use is different from the one in optimization within the mathematical field. Instead, it follows the terminology used in previous work [4], where the forward process of the attention layer and backward process on some model are referred to as a form of "dual". We are not sure if this has also led to some misunderstandings. If there are any remaining questions, please do not hesitate to let us know!
[1] Von Oswald J, et al. Transformers learn in-context by gradient descent.
[2] Zhang R, et al. Trained transformers learn linear models in-context.
[3] Shen L, et al. Do pretrained Transformers Really Learn In-context by Gradient Descent?
[4] Dai D, et al. Why can gpt learn in-context? language models implicitly perform gradient descent as meta-optimizers
[5] Nathan T B, et al. In-context learning and gradient descent revisited