On Affine Homotopy between Language Encoders

Thank you for taking the time to read and review our work – we are grateful for all the suggestions. Further, we are happy to hear that our usage of novel non-trivial mathematical tools for the analysis of the space of encoders is appreciated. Below, we address the concerns raised in the review:

• “It is unclear what is the contribution of this work to the field of NeurIPS”: Although we recognize that our usage of non-standard mathematical tools for NeurIPS, this is an attempt to more rigorously characterize the space of encoders as what they are – functions between countably infinite sets – and derive formal bounds that have concrete practical implications on downstream transfer learning behavior – and to that end, such tools are required. We further note that this paper follows a line of work, published at NeurIPS, on the similarity of neural representations (e.g., Boix-Adsera et al. (2022), Ding et al. (2021)) with extensions to the formalization of the problem as well as the objective of providing guarantees for extrinsic, downstream task behavior.

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• format, “hard to read and understand”: We retain only the relevant interim results in the main text that contribute to the main results of the paper, leaving the proofs to interested readers in the appendix. We acknowledge the resulting density of §2-5 and would use any additional space to add examples and proof outlines to support better understandability for the broader ML community.

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• “Only various versions of the BERT models are compared with each other”, “What's the point of caring about [MultiBERTs’] comparison in the first place?”. As found in the original paper and as seen in the experiments in §6, MultiBERTs may behave very differently among themselves. More concretely, MultiBERTs show vastly different downstream task performance (Sellam et al. 2022) and produce representation matrices of different ranks to precision $\epsilon$ (cf. §6). These are important attributes of encoders that we aim to capture in our notion of similarity, which are often disregarded (as discussed in §8). As such, MultiBERTs offer themselves well to investigate which task-independent (intrinsic) relation between the encoders best captures extrinsic dissimilarity in practice – the subject of our study in §7. Still, we acknowledge that a study across different architectures would yield broader comparisons, and we plan on including such a study as well as more extensive dataset coverage to complement our current experimental section.

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• Connection to homotopy in algebraic geometry: In classical algebraic topology, homotopy describes the continuous deformation of one function into another, typically parameterized by the interval $[0, 1]$. However, in this paper, we consider the concept of homotopy by using a broader set of parameterizations, denoted by $S$. This allows for more flexible transformations suited to specific classification problems. For instance, $S$ could be an affine space $Aff(V), Aff(V,W)$ or others, enabling more adaptable frameworks for continuous transformations.

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• “Universality of BERT”: From Theorem 4.1, we derive the influence of the encoder rank on affine mappability. We confirm that affine mappability is strongly influenced by the encoder rank in our experiments in Appendix G. This leads to the conclusion discussed in §7, where we find that some encoders seem to learn lower rank representation matrices (to precision $\epsilon$), and this correlate significantly with how easily one can affinely map to that encoder, which influences their concrete utility for transfer learning by Lemma 5.1.1 and Def. 5.2. Considerations about the asymmetry are discussed separately in §8.

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• “How is Ð approximated”: Ð is implemented as a gradient descent over affine maps (as mentioned in §6), where the loss is simply the max loss over all strings in the dataset. The gradient over the max is handled through subgradients in standard pytorch.

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• “Task-dependence of experiments”: This is a valid concern and a limitation of the work, which we partially acknowledge in the “Limitations” section. We note, however, that the formalism of a language encoder is by definition a map between countably infinite sets (strings to vectors), and our work makes a first attempt to derive properties of such a construction, which, evidently, also have practical implications for the encoders' use in transfer learning (cf. Discussion and §7). The approximation to the finite-string setting both shows the applicability of theoretical results on real datasets and allows us to draw comparisons to existing methods in representational similarity.

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• “Purpose of Figure 2”: As described in §3, Figure 2 plots intrinsic and extrinsic similarity, and, on a larger scale, shows an empirical approximation to the theoretical linear bound derived in Lemma 5.1. This does not tell us much about the encoders individually, but may act as a visual aid in seeing the empirical implications of the connections between our notions of intrinsic and extrinsic similarity.

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• “Purpose of Table 1”: Correct. It shows the correlation between affine intrinsic similarity measures (cols) and extrinsic measures of similarity (rows). As in the experimental setup in Ding et al. (2021), this measures how strongly differences in extrinsic similarity are picked up by the intrinsic similarity measures. We find that Ð (as well as other linear alignment methods) tend to be strongly (and significantly) indicative of the extrinsic, downstream behavior, as theorized by the upper bound in Lemma 5.1.1.

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• “What are Ex and Ey”: equivalent to the set E defined in Def. 3.3, Ex and Ey are non-empty subsets of X, and Y, respectively. We will add a comment, thanks!

We again thank the reviewer for their suggestions and hope that we could address most concerns and open questions about the soundness and overall contribution of the paper.

We thank the reviewer for their thorough review and helpful feedback. We are pleased to read that you appreciate the novelty of the approach and the breadth of the ideas introduced in the paper. Below, we address the specific concerns raised in the review:

1. “Elaborate your Motivation of using Hemi-Metrics”: Although some previous work has explored measuring similarity in proper metric spaces, we posit that purely from a machine learning perspective, there is an asymmetry to the problem of measuring the similarity between encoders. More concretely, when we assess the similarity of two encoders in terms of how closely we can affinely map their classification probabilities on a task, we find this to be asymmetric theoretically as well as practically. Some encoders may be more powerful than others (e.g., think of a lower-rank encoder, such as the ones generated in Appendix G). A symmetric distance, therefore, is not practical as it does not capture this phenomenon. This motivates using Hemi-Metrics.

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2. “Extrinsic Homotopy does not seem like a mathematical definition”: We agree that Definition 5.1 is somewhat informal. We will explicitly state this in the next revision – thank you for pointing this out. We note, however, that we provide a formal mathematical definition of extrinsic homotopy in Lemma 5.2 similar to intrinsic affine homotopy in Def. 4.2.

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3. “More experiments are needed to support the claim”: We would like to highlight that most theoretical results/claims (Thm. 4.1, Lemma 5.1, and Thm. 5.1) were evaluated empirically across a large number of encoders that exhibit significant differences in properties and downstream task behavior (cf. §6, [1], [2]). Still, in light of this and other reviews, we acknowledge that an evaluation across more datasets may help support the claim, which we will include for a next revision – we thank the reviewer for the suggestion!

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4. “Such method seems not practical”: Although we acknowledge the gap between some properties derived in §2-5 with the experimental results (see “Limitations” Appendix A), our derivation of intrinsic similarity measures that upper bound extrinsic dissimilarity (Lemma 5.1, Def. 5.2) is purely practically driven. Namely, we show that our measure of intrinsic similarity is indicative of how an encoder may perform across downstream tasks in the transfer learning setting. Further, our discussion of the algebraic properties of affine homotopy gives us a rich theoretical foundation useful to understand phenomena we observe empirically (cf. §7-8), such as:

• The theoretical upper linear bounds of intrinsic similarity on task performance (Lemma 5.1, Def. 5.2) surface empirically, and therefore show us how our intrinsic measures of similarity can indicate downstream task performance similarity across arbitrary tasks. This is of significant practical interest for encoders’ utility in transfer learning, as this may reduce the need to do task-specific evaluation, as motivated in §1.

• Our proof about how affine mappability is affected by the encoder rank surfaces in our experiments §7 - “The Influence of Encoder Rank deficiency” as well as in our additional experiments in Appendix G.

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Questions

1. See point 1. above

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2. “Explain the connection between intrinsic homotopy and extrinsic homotopy [Thm. 5.1]”: We prove in Lemma 5.1 that for some fixed linear classifier $\psi’$, the extrinsic dissimilarity (i.e., how closely we can map the output probabilities of representations of encoder h with the ones from encoder g) is linearly bounded by the intrinsic similarity measure (Eq. 9b). Theorem 5.1 makes a stronger statement and states that the Haussdorff-Hoare variant of the intrinsic distance upper bounds the extrinsic dissimilarity over all possible linear classifiers. This shows that our methods of measuring intrinsic homotopy linearly upper bounds, and is therefore indicative of extrinsic homotopy. The practical implications of this are significant, as this shows we can derive encoder similarity measures that are indicative of their downstream task behavior, without the need for task-specific evaluation.

We again thank you for the insightful suggestions and hope that our clarifications address your concerns about the overall contributions of our work.

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[1] Jesse Dodge, Gabriel Ilharco, Roy Schwartz, Ali Farhadi, Hannaneh Hajishirzi, and Noah Smith. 2020. Fine-tuning pretrained language models: Weight initializations, data orders, and earlystopping.

[2] Thibault Sellam, Steve Yadlowsky, Jason Wei, Naomi Saphra, Alexander D’Amour, Tal Linzen, Jasmijn Bastings, Iulia Turc, Jacob Eisenstein, Dipanjan Das, et al. "The MultiBERTs: BERT reproductions for robustness analysis." In ICLR, 2022.

We thank the reviewer for taking the time to read our work, as well as for their thorough review and helpful feedback. We are very happy to hear the positive comments about our formalization of the problem and our writing. We address the open points and concerns raised in the review below:

• "The work should ideally study other encoder architectures and aim to establish how the notion of similarity changes over the spectrum of 'expected model distance' and what implications it might have" We fully agree that such study across encoder architectures and additional tasks would complement our current experiments and we plan to add this for the next revision of the paper – thank you for the suggestion!

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• "The only studied architecture is the BERT architecture [...] which just use different random seeds": Although we fully agree with the point raised in b) to get a broader coverage of results by further studying different encoder architectures and running experiments across more datasets (which was solely limited by page limits), we would like to restate the motivation and significance of the found results mentioned in §1. Namely, as found in the original paper and as seen in the experiments in §6, MultiBERTs may behave very differently among themselves. More concretely, MultiBERTs have been shown to produce embeddings that yield significantly different downstream task performance (Dodge et al. 2021 [1], Sellam et al. 2022, [2]) and produce representation matrices of different ranks to precision $\epsilon$ (cf. §6). These are important attributes of encoders that we aim to capture in our notion of similarity, which are often disregarded (as discussed in §8). As such, MultiBERTs offer themselves well to investigate which task-independent (intrinsic) relation between the encoders best captures extrinsic dissimilarity in practice – the subject of our study in §7.

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• “A positive correlation between intrinsic and extrinsic similarity is very much expected for this group of models”: our derivation exactly shows that specifically our notions of intrinsic similarity will be correlated with the extrinsic similarity independent of the considered group of models. The linear upper bound on the extrinsic dissimilarity by our notion of intrinsic similarity does not make any assumptions about the nature of the encoder, and we therefore expect such correlations to hold across encoder families.

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• “Have the authors considered running the similarity analyses also with BERT models having different hyper-parameters?” As mentioned above, we did not make assumptions about the underlying encoder structure to derive the upper bound on the extrinsic similarity. Although we expect gaps in similarity to become more prominent, we do not expect these to affect the correlation between intrinsic and extrinsic similarity.

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• “I am missing the reason why certain encoders end up being more informative than others.” We refer to the fact that in theory (cf. Thm. 4.1) higher-rank encoders are more powerful, as we can exactly affinely map from their image into the image of a lower-rank encoder, whereas the inverse does not hold. We also find this to surface empirically (cf. §7, “The Influence of Encoder Rank Deficiency”), and it has a significant impact on the affine mappability between the representation spaces (intrinsic similarity), and, as a result, task performance and the extrinsic similarity.

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• “Would it be possible to derive similar measures for decoder-only LMs in the future” Although our method discusses encoder functions motivated by their usage for transfer learning, we do not in principle make assumptions about the nature of the function. In that sense, decoder representations may be evaluated equivalently.

We again thank you for the insightful suggestions and hope that our clarifications address any concerns about the overall contributions of our work.

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[1] Jesse Dodge, Gabriel Ilharco, Roy Schwartz, Ali Farhadi, Hannaneh Hajishirzi, and Noah Smith. 2020. Fine-tuning pretrained language models: Weight initializations, data orders, and earlystopping.

[2] Thibault Sellam, Steve Yadlowsky, Jason Wei, Naomi Saphra, Alexander D’Amour, Tal Linzen, Jasmijn Bastings, Iulia Turc, Jacob Eisenstein, Dipanjan Das, et al. "The MultiBERTs: BERT reproductions for robustness analysis." In ICLR, 2022.

We thank the reviewer for their thorough review and the helpful feedback and ideas they provided! We are happy to hear that our theoretical results as well as practical implications are appreciated. In the following, we address the open points and concerns:

• “It is difficult to see what the richer properties are, but for a reader (that does not necessarily have theoretically expertise), it is difficult to see what these richer properties look like”: What is meant by “richer properties” in the introduction refers to our much more general theoretical results and the empirical implications that result from formalizing and studying encoders as what they are – functions between countably infinite sets. For instance, our derivation for Remark 3.2 finds that only one-sided affine alignment (Eq. 9.2) is non-trivial, and further, defining hemi-metric spaces allows us to derive proper mathematical preorders and equivalence relations on the space of encoders, which (e.g., Thm. 4.1) has concrete empirical implications (cf. §7, “The Influence of encoder Rank Deficiency”). Further, our formalization allows us to explicitly construct the linear bounds of extrinsic dissimilarity not just for a fixed classifier (Lemma 5.1.1), but for the “worst-case dissimilarity” (cf. Thm. 5.1) -- this highly practical, as it shows our intrinsic similarity measure to be indicative of how an encoder may perform across downstream tasks in the transfer learning setting. Still, we acknowledge the density of the theoretical sections, and as per recommendation by this reviewer and reviewer 6x9V we plan to create a distilled overview paragraph of the theoretical contributions in a next revision to appeal to the broader ML community.

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• “There is also very little discussion of practical aspects of running this analysis, The appendix mentions that intrinsic alignment is more expensive to compute than extrinsic alignment; given that the aim of the work is to set upper bounds, why not just compute the extrinsic scores?”: The practicality comes exactly from the fact that we show that the intrinsic similarity measures can be already indicative of task performance – not just for a fixed task (Lemma 5.1.1.), but for the worst case task across all tasks (see Thm 5.1., as well as §8; “Implications of §5”). In other words, we show that computing our intrinsic measures of similarity may already indicate the overall utility of an encoder for any downstream task, thus potentially eliminating the need to do task-specific evaluation of extrinsic scores.

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• “Lacking a discussion of what kinds of encoders can be compared”, “Can these be run across encoders with different architectures?”: Generally, we do not make any assumptions about the structure of the encoder in the derivation of our theoretical results, resulting in wide generalizability of our method. In other words, our method can be applied to measure the (intrinsic) similarity between any two encoders and evaluate how this affects how (extrinsically) similar they can be in terms of their output probabilities.

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• “What is the benefit of defining an order over encoders”, “alignment metric into a measure of "badness" for a given encoder”: Purely mathematically speaking, a proper order over a set is a powerful structure that allows us to define proper equivalence relations over encoders. Beyond equivalence, the order between encoders (cf. Lemma 4.1.) indicates that we may be able to exactly affinely map from one image into the image of a lower-rank encoder, whereas the inverse does not hold. A higher-rank encoder is therefore more powerful, and intrinsic affine homotopy is indicative of this. We find this rank deficiency to surface empirically (cf. §7, “The Influence of Encoder Rank Deficiency”). By Lemma 5.1.1., this also strongly affects extrinsic similarity, i.e., how closely we can affinely reach the output probabilities of an encoder for a specific task from another encoder – yielding a measure of the “goodness”/”badness” of an encoder, in your words.

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• “these alignment measures can somehow be used to improve pre-training”: this is an interesting suggestion, especially in light of the nuances in fine-tuning that have shown to have large effects on downstream task performance of the pretrained encoders [1]. We can add this as possible future work or motivation, thank you for the suggestion.

We again thank the reviewer for the insightful suggestions and hope that our clarifications address their concerns about the overall contributions of our work!

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[1] Jesse Dodge, Gabriel Ilharco, Roy Schwartz, Ali Farhadi, Hannaneh Hajishirzi, and Noah Smith. 2020. Fine-tuning pretrained language models: Weight initializations, data orders, and earlystopping.