We will respectfully disagree that it "makes more sense to take a decaying step-size." Indeed, the improved error bound of DTDT comes at the price of doubling the commu-45 nication requirements of DTDL. Codes for49 the presented experiments are straightforward, so they were not included.

We thank all reviewers for their time and effort in reviewing our paper. We set up experiments on PyTorch with ResNet18 (He et al., 2016) on CIFAR10 (Krizhevsky, 2009). Figure 1: Evaluations on CIFAR10: training loss ( 1 st column), test accuracy ( 2 nd column) and total number of transmitted bits. MB) on CIFAR10 are shown in Figure 1 above. We will release our code on GitHub in the final version.

We would like to thank the reviewers for their comments and feedback. Janzing et al. [9] write down the same equation, but We will follow the reviewer's The decomposition for conditional SVs follows by replacing "conditioning The decomposition is introduced in Section 3 to assist our illustration of how the different SVs attribute a model's SVs. Unlike conditional (asymmetric) SVs, causal SVs provide the right intuition in the case of common confounding. See also the previous paragraph. SVs appear to fare better than the reviewer suggests.

Large language models have recently demonstrated advanced capabilities in solving IMO and Putnam problems; yet their role in research mathematics has remained fairly limited. The key difficulty is verification: suggested proofs may look plausible, but cannot be trusted without rigorous checking. We present a framework, called LLM+CAS, and an associated tool, O-Forge, that couples frontier LLMs with a computer algebra systems (CAS) in an In-Context Symbolic Feedback loop to produce proofs that are both creative and symbolically verified. Our focus is on asymptotic inequalities, a topic that often involves difficult proofs and appropriate decomposition of the domain into the "right" subdomains. Many mathematicians, including Terry Tao, have suggested that using AI tools to find the right decompositions can be very useful for research-level asymptotic analysis. In this paper, we show that our framework LLM+CAS turns out to be remarkably effective at proposing such decompositions via a combination of a frontier LLM and a CAS. More precisely, we use an LLM to suggest domain decomposition, and a CAS (such as Mathematica) that provides a verification of each piece axiomatically. Using this loop, we answer a question posed by Terence Tao: whether LLMs coupled with a verifier can be used to help prove intricate asymptotic inequalities. More broadly, we show how AI can move beyond contest math towards research-level tools for professional mathematicians.

Again, thank you all for giving us the opportunity to strengthen our paper with your valuable comments and queries.

In the revised version, we will make this clearer in the "Related Work" section. Figure 2 illustrates how the classification model (i.e. ( t 1) The parameter σ corresponds to the width of Gaussian kernel, which is fixed to be 1 in this paper (pp.3, footnote 1).

We thank all the reviewers for their time and effort to evaluate our paper. We will discuss the assumptions in the context of the SDE in Eq. 4, which is our main The boundedness assumption requires the solutions to the SDE to be'non-explosive' in the sense that for On the other hand, in Thm.S3, we already proved a bound which does not require H4, but This assumption is common in statistics (see [Bra83]) but hard to verify in practice. We agree that Prop.1 can be difficult to grasp at a first sight; hence, we provided BN is not fully understood. Our main focus is the relationship between generalization and intrinsic dimensionality. Relationship between BN and dimensionality is an interesting future direction which would complement our work.

We would like to thank the reviewers for their comments and feedback. Janzing et al. [9] write down the same equation, but We will follow the reviewer's The decomposition for conditional SVs follows by replacing "conditioning The decomposition is introduced in Section 3 to assist our illustration of how the different SVs attribute a model's SVs. Unlike conditional (asymmetric) SVs, causal SVs provide the right intuition in the case of common confounding. See also the previous paragraph. SVs appear to fare better than the reviewer suggests.

Please see response 3 under reviewer 4. Final-version plots will be made printer friendly.

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