When can isotropy help adapt LLMs' next word prediction to numerical domains?
Contextual embedding space;clusters;isotropy;language model representation;numeric downstream task
Yes
No ethics review needed.
- Can the authors provide more details on the datasets that are used in this paper, as well as their application to real-world use cases?
- The paper provides a new perspective on LLM embedding structure in the context of time series analyses. The method of analyzing performance relative to isotropy is a new and potentially interesting avenue of exploration. 
- The interdisciplinary nature of the paper (exploring LLM applications in settings outside of the scope of NLP) is timely.
- The paper is well written.
This paper explores how structure in pretrained LLM embeddings is important for using these models for numerical domains tasks (such as time series forecasting). Specifically, the authors propose that when isotropy in the embeddings preserves the structure in LLM representations, it resolves the shift-invariance problem in the model's softmax function (i.e., the softmax output remains unchanged when all logits are shifted by a constant). They validate this hypothesis through examples with synthetic datasets, demonstrating cases where isotropy improves or hinders downstream performance.
- The paper is very limited in its scope of analysis. It presents results on only two synthetic datasets that limits the generalizability of findings. Moreover, the apparent difference in performance on the two datasets is purported to be because of a difference in isotropy in embeddings, but the causal link is not actually shown. Attributing poor performance on Dataset 2 to low isotropy seems speculative without exploring other possible causes.
- The theoretical claims of the paper are largely qualitative, lacking rigorous, quantitative backing, and due to the limited scope of analysis, it is hard to know exactly when and where these results hold.
-Too much space is dedicated to background ideas (such as the noisy data generation process of time series data), thereby limiting room for original contributions.
Yes
No ethics review needed.
1. As to the two datasets used (TDD & FDD), what are their settings and how are they collected? Citations and discussions are needed for these information. Since these two seems to be wireless network datasets, which is not common knowledge for most ICLR audiance. 

2. This paper seems to target the numerical domain application in general (in the introduction, it lists several domains including finance, energy, retail, climate, etc.), is there evidence that the setting described in this paper is widely used in these domains? If so, I'm wondering whether you can add more experiments from these domains as well? 

3. The choice of many terminology in this paper is not very popular and is even a bit confusing. For example, why it is "Student Model" in line-165? Note that "student model" is widely used in knowledge distillation.
The analyses and the intuition (for this particularly scenario) are reasonable.
This paper studies that, after training a LLMs on next-word prediction, if an application needs to adapt the pre-softmax LLM output for a numerical task, whether isotropy helps. The authors identify that the shift-invariance property of the softmax function would cause problems for such applications. They conduct analyses and experiments on two datasets (TDD and FDD) and the results seem to match their intuition.
1. The paper is not well written; 
2. Based on my knowledge, the setting studied in this paper is not very popular, and its not well argued that this is an important problem or may impact a wide range of applications. 
3. The experiments are conducted with relatively weak model (GPT-2) and no experiments are conducted on popular benchmarks, making it hard to judge the significance of the observations discussed in this paper.
Yes
No ethics review needed.
-  I am not very familiar with the numerical data setting -- Why is the shift-invariance of softmax specifically a problem for numerical data as compared to other downstream tasks?
- Can experimental results be shown across more extensive settings? Ie, with more empirical evidence, can isotropy be shown to be a causal factor in performance? Or otherwise, is there some property of the data that can predict whether the model will learn an isotropic representation?
- The study is well motivated in trying to understand when next token prediction capabilities of LLMs will extend to numerical data
- The authors provide a plausible argument for the role of isotropy in adapting LLMs to numerical data which builds on prior work
The authors address the following question: when can next-token capabilities of LLMs be effectively adapted to numerical domains? 
The authors respond by claiming that next-token capabilities can be extended to numerical domains when there there is isotropy in the LM contextual embeddings. Essentially, if your numerical data is a function of the logits, the model performance might not reflect an accurate representation of the numerical data since the post-softmax results may lose information due to the shift-invariance of the softmax function. The authors demonstrate experimental results with GPT on two time series datasets.
The experimental results are not very extensive; the main experiment compares performance of a GPT model on two time series datasets and shows that the model performs better on Dataset 1, in which case the model learns isotropic representations, than Dataset 2, in which case the model does not learn isotropic representations. This seems to suggest there is some underlying property of the data that is determining the performance (the existence of isotropic representations does not necessarily seem causal). It would be interesting to see a setting in which multiple models (with some variation such as architecture) are trained on the same dataset and learn representations with varying levels of isotropy, and testing whether the level of isotropy corresponds to performance.
Yes
None
No ethics review needed.
Please refer to weakness.
This paper introduces a novel explanatory perspective by exploring the role of isotropy in the embedding spaces of LLMs adapted for numerical predictions. And the author attempts to give a theoretical proof for his idea.
The paper addresses the adaptation of LLMs for numerical downstream tasks through the lens of isotropy in embedding representations. It provides a theoretical framework to understand when and how LLMs' next-word prediction capabilities can be effectively harnessed for numerical data. This helps improve the interpretability of this field.
1. The paper suffers from poor readability and overcomplication of concepts. For instance, Contribution 2 introduces a theorem (Theorem 1.) to prove "why the shift-invariance problem needs to be addressed," which I believe is entirely unnecessary.

2. I do not understand the rationale for framing regression problems as classification tasks, even though it is technically feasible. Additionally, the authors did not provide any code, and it is unclear how the time series vocabulary (mentioned in line 161) is constructed. If it uses the GPT-2 vocabulary, which is composed of numbers, how are the basic units of the time series vocabulary determined?

3. It is also unclear why a "Student Model" is suddenly introduced without explaining what the "Teacher Model" is.

4. Regarding the experimental section, I am puzzled as to why the authors chose to simulate data for wireless communications, especially since the paper mentions "applications in finance, energy, retail, climate science, wireless networks, and synthetic tabular generation, among others". Why not analyze using open-source datasets readily available for these applications?

5. The paper always uses $T_l+1$, but I think it should be $T_{l+1}$. Concerning Eq.4, I believe there is a significant issue: KL divergence (relative entropy) is inherently an expectation, so why is there an additional summation outside? Also, is there a problem with the entropy term H?

6. Section 3.2, which I initially thought would be the most valuable, is not clearly explained by the authors. It quickly transitions to measuring isotropy and clustering, which largely reuses the analytical methods proposed by Cai[1].

7. Furthermore, there are formatting issues such as improper line breaks at line 403 and incorrect spacing between lines 431-433. Additionally, all figures seem to lack clear annotations, giving the impression that the paper was hastily prepared.

[1]Cai X, Huang J, Bian Y, et al. Isotropy in the contextual embedding space: Clusters and manifolds[C]//International conference on learning representations. 2021.
@inproceedings{
anonymous2024when,
title={When can isotropy help adapt {LLM}s' next word prediction to numerical domains?},
author={Anonymous},
booktitle={Submitted to The Thirteenth International Conference on Learning Representations},
year={2024},
url={https://openreview.net/forum?id=L39yPOGCma},
note={under review}
}
Recent studies have shown that vector representations of embeddings learned by pre-trained large language models (LLMs) are effective in various downstream tasks in numerical domains.  Despite their significant benefits, the tendency of LLMs to hallucinate in such domains can have severe consequences in applications like finance, energy, retail, climate science,  wireless networks, synthetic tabular generation, among others. To guarantee prediction reliability and accuracy in numerical domains, it is necessary to have performance guarantees through explainability. However, there is little theoretical understanding of when pre-trained language models help solve numeric downstream tasks. This paper seeks to bridge this gap by understanding when the next-word prediction capability of LLMs can be adapted to numerical domains through the lens of isotropy. Specifically, we first provide a general numeric data generation process that captures the core characteristics of numeric data across various numerical domains. Then, we consider a log-linear model for LLMs in which numeric data can be predicted from its context through a network with softmax as its last layer. We demonstrate that, in order to achieve state-of-the-art performance in numerical domains, the hidden representations of the LLM embeddings must possess a structure that accounts for the shift-invariance of the softmax function. We show how the isotropic property of LLM embeddings preserves the underlying structure of representations, thereby resolving the shift-invariance problem problem of softmax function. In other words, isotropy allows numeric downstream tasks to effectively leverage pre-trained representations, thus providing performance guarantees in the numerical domain. Experiments show that different characteristics of numeric data could have different impacts on isotropy.
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Contextual embedding space clusters isotropy language model representation numeric downstream task
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learning on time series and dynamical systems
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When can isotropy help adapt LLMs' next word prediction to numerical domains?
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