reviews: 
Comments: This paper studies how chain-of-thought (CoT) impacts in-context learning (ICL) for transformer models. There are two main results. First, the paper shows that a linear-depth transformer model can implement a learning algorithm A_CoT that chains together k gradient descent learning algorithms A_i that run for t steps each for 1 <= i <= k (Lemma 1). An upper bound on error is given for A_CoT, thereby showing that a composite task is efficiently learnable by a transformer model if each subtask is efficiently learnable by each of the k individual gradient learning algorithms (Lemma 2), where the hardness of learning is bounded by the difficulty of the most difficult subtask (Theorem 3). Second, the authors lower-bound the error of A_CoT in the case of finite hypothesis class for A_1 as a function of the size of that hypothesis class (Theorem 4), which suggests that there may be functions that are learnable with CoT that are not learnable without CoT. A concrete witness is identified for learning parities (Corollary 5).
Strengths: The paper studies an important problem in trying to theoretically elucidate CoT and ICL for transformer models. There is also preliminary experimental results that corroborate the theoretical results.
Weaknesses: The presentation is quite dense could be improved to explain the motivation and intuitions behind definitions/results.
- For instance, the definition of in-context learning (Definition 1) makes a distinction between predictors and a learner (around line 195-196). From my understanding, the definition of Model(Data, query) = LearningAlg(Data)(query) = predictor_D(query) is just partial evaluation, and so the distinction that is drawn between predictors and learners is somewhat artificial. Rather, the main idea seems to me to formulate ICL, and so we just define a function that takes a query that is a tuple (demonstrations, x) as opposed to x. Moreover, it would be helpful to explain why PAC learning is chosen and to give pointers (presumably to Theorem 4) on how exactly PAC learning will be used to help you demonstrate your results and why it is sensible.
- Why does Definition 2 use multisets?
- Some of the material in the appendix on how your Transformers implement gradient descent result differs from Cheng et al. and Von Oswald et al.'s would be helpful in the main text to give more context.
- Could you offer more clarification on how to extend the results to handle k > 1 in lines 294-305 which is from what I understand is the main point of innovation of the paper. However, I can't seem to find more details in the main text or the appendix.
- The references for this paper do not use ICLR's reference style ...
- The statement of Lemma 1 might be helped by stating that t steps of gradient descent map to t layers of self-attention which is mentioned in the appendix.
- I find the statement of Theorem 4 to be somewhat strange. If I understood correctly, without CoT, we can essentially run for t gradient steps on a neural network that predicts from a representation that has t self-attention layers. With CoT for k steps, we can essentially run for kt gradient descent steps with kt self-attention layers and k prediction layers. How does Theorem 4 control for the same number of layers across reasoning steps k? Intuitively, I expect a theorem showing that there functions that are learnable with CoT that are not learnable without CoT to (1) show that a kt layer network run for kt gradient descent steps can not learn the task purely with an identity (trivial) task decomposition and (2) that a task decomposition (T in theorem 4) used by CoT can be constructed in polynomial time *uniformly* in the task with a Hypothesis class H_1'. However, there are no limits on the task decomposition T in the theorem and it's currently not clear to me how the size of the Transformer is controlled in the current statement of Theorem 4. Perhaps I misunderstood and would be curious to get more insight into what is actually going on.
Recommendation: 6
Reviewer confidence: 2
Soundness/correctness: 3
Clarity: 3