Coherent and Consistent Relational Transfer Learning with Autoencoders

Abstract: Human defined concepts are inherently transferable, but it is not clear under what conditions they can be modelled effectively by non-symbolic artificial learners.
This paper argues that for a transferable concept to be learned, the system of relations that define it must be coherent across domains.
This is to say that the learned concept-specific relations ought to be consistent with respect to a theory that constrains their semantics and that such consistency must extend beyond the representations encountered in the source domain.
To demonstrate this, we first present formal definitions for consistency and coherence, and a proposed Dynamic Comparator relation-decoder model designed around these principles. 
We then perform a proposed Partial Relation Transfer learning task on a novel data set, using a neural-symbolic autoencoder architecture that combines sub-symbolic representations with modular relation-decoders.
By comparing against several existing relation-decoder models, our experiments show that relation-decoders which maintain consistency over unobserved regions of representational space retain coherence across domains, whilst achieving better transfer learning performance.

Reviews and author responses:

 **[S2] The proposed method is straightforward + short of novelty**

On the first point (regarding the model being straightforward), we are in agreement. 
Our purpose was indeed to show that a relatively standard method, critically combining a data encoder with a set of modular classifiers (relation-decoders) can implement the theory introduced in the paper. Where others have referred to consistency and coherence loosely [1], this paper is the first to provide formal definitions.
However, being standard, the method is comparable to a great many other (more complex) methods and as such the theoretical and empirical contributions of this work are applicable to a wide range of research. We believe that they can therefore help improve the way classifier performance is assessed, wherein consistency generalisation could help diagnose the transferability of the associated concept.

On the second point, whilst the overall model is straightforward, we believe that DC is innovative and designed with a clear purpose. We appreciate that this was not expressed well in the previous revision and hope that the revisions we have made to this effect have made this more clear.

**[S3] the task presented in the paper seems trivial.**

As we discussed in **[S1]**, the PRT task was developed to precisely evaluate the main hypothesis. In this regard, the current empirical analysis achieves its purpose within the context of this paper; and that this has now been strengthened via the clearer link between the consistency/coherence definitions and the empirical analysis.
Though the task outline is straightforward, we believe it is not trivial. This is a common trend within transfer learning, where seemingly simple tasks (from a human perspective) are most definitely not simple for a non-symbolic learner.

We hope that these points improve your impression of our work and we welcome any further feedback.

____
[1] Maxwell I. Nye, Michael Henry Tessler, Joshua B. Tenenbaum, Brenden M. Lake:
Improving Coherence and Consistency in Neural Sequence Models with Dual-System, Neuro-Symbolic Reasoning. CoRR abs/2107.02794 (2021)


 **[C5] Suggestion for other experiments: *What if using relation other than isSuccessor for transferring?***

Thank you for your suggestion. Logic would tell us that isPredecessor should work as well as isSuccessor when used as the single supervised relation. We suspect that isEqual will be problematic, as it is not sufficient on its own to define an ordering. isGreater and isLess could both prove to be quite effective, but it is not clear how suitable the resulting embeddings will be for the other relations. It is quite likely that each relation-decoder model will produce a different behaviour, but in principle the logical relationships between the supervised and unsupervised relations should dictate their effectiveness.
This would be very interesting to confirm and we have begun setting up these experiments. We aim to include the results in the camera ready version of the manuscript, if accepted.

___

Lastly, we wish to respectfully respond to each of your closing statements in the “Summary Of The Review”, as we feel they do not give an entirely fair assessment of the contributions of this manuscript.


**[S1] The motivation of the work is not clear**

This is in disagreement with reviewer v3U9 in particular, who stated that 
> The paper is very well written and the organization is clear

For completeness, the motivation of this work is that although humans are able to learn transferable concepts, such that they can be applied in multiple domains and to different properties, it is not clear under what conditions we can say the same for non-symbolic learners.
Our hypothesis is that for a concept to be transferable, its consistency should be maintained regardless of domain.
We define this condition formally, and introduce $\epsilon$-coherence as a measure for the domain-invariance of consistency. 
In doing so, we present a neural-symbolic theoretic foundation that contrasts symbolic and non-symbolic learning in terms of structures and proposed soft-structures. This foundation gives a formal perspective regarding what a neural learner tries to achieve when learning the relations over real-world data.
We use this foundation to define a purposeful neural model and experimental setup, to precisely evaluate the hypothesis.

To address any ambiguity regarding the motivation (also addressing the comments by reviewers **v3U9** and **8JLA**, and strengthening the purpose of the definitions and connection with the empirical analysis), we have extended our consistency loss to derive an $\epsilon$-proxy measure that lets us calculate a tractable proxy for
$\epsilon$-consistency/coherence. We provide a clear numerical comparison of
$\epsilon$-proxy coherence across relation-decoder models in Table 1 and have modified the results and discussion sections accordingly. The derivation outline is provided in Section 4 and the expanded derivation is given in Appendix H.

We hope that this improves the clarity regarding the motivation of this work. 

 Thank you for the time and effort you have dedicated to your review. Your comments have been very helpful in improving the manuscript.
____

In the following, we address each of the main comments/questions in turn.

**[C1] Possible misunderstanding regarding use of sentences to train relation prediction**

We would like to firstly clarify a possible misunderstanding, specifically in response to Weakness 1: 
> What are the benefits of using the sentences in the theorem to train relation prediction, compared to independent training of each relation? From Equation 8, the loss of a sentence seems to be the product of the loss of each relation in the sentence. So why not directly train the loss of each relation independently?

We would like to address a possible misconception regarding how we train the overall neural model. Specifically, we do not explicitly use any sentences in the loss function. Instead, in $p(\mathcal{S}\_\sigma | \tilde{\mathcal{S}}\_\sigma)$, for a signature consisting of only binary relations, $\mathcal{S}\_\sigma$ can be decomposed into a set: (($s_i$, $s_j$), $\gamma\^{r}\_{(i, j),\mathcal{S}\_\sigma}$), where each  $\gamma\^{r}\_{(i,j),\mathcal{S}\_\sigma}$ is a relation specific label that determines whether $r(s_i, s_j)$ is true or false given structure $\mathcal{S}\_\sigma$. We train each relation independently using training data with this form, making no use of any sentence level labels. 
Relation-decoders are therefore required to discover parametric relationships from the data, without any additional training biases.

This brings us to Weakness 2, regarding the loss function complexity

**[C2] What is the computational complexity of the loss function?**

Since we do not require sentences in the loss function, the computational complexity is only determined by the number of relations and the number of respective training data per relation. We set $R = \frac{|\mathcal{B}|}{\mathcal{|S|}}$ to be the overall ratio of training data versus domain size. For MNIST we set this ratio to $R=1$ and $R=3$ for BlockStacks. The total amount of training data is then divided equally between the relations, so $\mathcal{B} = \cup\_{r \in \sigma} \mathcal{B}\_r$ for relation specific training sets $\mathcal{B}\_r$. In summary, each relation has up to $\frac{|\mathcal{B}|}{|\sigma|}$ unique training examples to train on. We provide these details in Appendix D.4.

We hope that this clarifies the training methodology in the paper and we welcome any further questions, should you have any.

**[C3] In Equation 9, why do we need to calculate expectation over s?**

Thank you for bringing our attention to this. 
We should have conditioned both $p(B\_{\tau} | I\_{\mathcal{S}\_\sigma} \in \mathcal{M}^{\mathcal{T}}\_{\mathcal{S}})$ and $p(B\_{\tau} | \tilde{\mathcal{S}}\_\sigma)$ on $s\_{i,j,k}$, since their underlying definitions both depend on $s_{i,j,k}$. We have rewritten section 4 to be more concise, following reviewer **v3U9**’s suggestion, and shifted the expanded technical details to the appendices (Appendix H). We have made sure to correct this error in these updates.

**[C4] The paper lacks an explanation for the motivation of DC. Why is DC better?**

We realise that our explanation of DC was not as clear as it could be. 
We developed DC around the intuition that a relation-decoder would be more suitable for concept transfer if it was able to generalise logical consistency across its input space. 
To make this motivation clearer, we have added the following to the end of Section 5, directly after DC is defined:

> The key innovation behind DC is its ability to model each of the ordinal relations whilst encouraging generalised consistency across the full latent subspace, as defined by each $\mathbf{u}_r$. This is achieved without explicit weight sharing, wherein relation-decoders discover parametric relationships between relations from the data. We provide an additional depiction and analysis of DC in Appendix D.1, which further illustrates these effects.

As referred to in the excerpt, we provide a visual illustration (Figure 5 - Appendix D.1) that demonstrates how DC models each of the ordinal relations and shows how generalisation of logical consistency can be achieved. 
We have also updated the results/discussion to refer more clearly to the fact that the empirical evidence suggests that DC's strong consistency generalisation corresponds with its excellent PRT performance.

We hope that these changes improve the clarity regarding DC and why it is able to achieve strong consistency generalisation and transfer.

 **[C1.3] Continued from ***"Response to reviewer 8JLA [1/2]"*****

To make the link between sections 2-4 and sections 5 onwards stronger, we firstly extended our consistency loss and derived an $\epsilon$-proxy measure that lets us calculate a proxy for $\epsilon$-consistency and coherence (which are themselves intractable). We present the derivation outline in the revised Section 4, and defer the expanded derivation and technical details to Appendix H.
Following your suggestion regarding a numerical comparison of Figure 2/3, we include a numerical comparison over $\epsilon$-proxy coherence across relation-decoder models, in Table 1, and have modified the results and discussion accordingly. 
We felt that Figures 2 and 3 are better presented as bar plots, since they expose important trends, but we agree that a numerical analysis over their content was missing. We feel that Table 1 addresses this, whilst linking the two principle parts of the paper more strongly.

We hope that these changes improve the link between sections 2-4 and 5 onwards, whilst providing an appropriate and clear numerical analysis.

**[C1.4] relation-decoders only use distance-based similarity measurement**

We wish to address a possible small misunderstanding, in reference to 
> the neural model proposed in this work is simply learning a tied latent representation, where the latent representations are constrained by some distance-based similarity measurement.

Each relation-decoder model produces its decoding using a different mathematical computation and this influences how the data embeddings are shaped. It is true that some use a distance-based measurement, namely TransR and DC (when it employs the $\phi_r^\dagger$ mode). However, NN, NTN and HolE use interaction terms (similar to dot products) to produce their relation decoding, and when DC instead uses the $\phi_r^{\ddagger}$ mode, it looks at which input argument is greater/lower on a specific latent subspace and uses this to produce its decoding. 
We had observed that if DC only uses a distance-based measurement, it could not model each relation whilst ensuring good consistency generalisation. We aim to include an associated ablation study to expose this in the camera ready version (if accepted) as per reviewer **v3U9**’s suggestion. 
This is all to say that distance-based similarity is only one of the possible methods, and that the underlying method plays an important role in each model's ability to represent each relation type. These differences are ultimately what cause each relation-decoder to perform better or worse in the PRT task.

**[C2] Missing footnote**

Thank you for bringing our attention to the broken footnote. We have fixed this in the updated section 4. 

**[C3] Why does DC do better in the target domain?**

We were surprised by this as well. We noted that the main improvements are when DC uses to $\phi_r^\dagger$ mode (which uses the distance-based similarity form). This modality can produce erroneous outputs if the data encodings are not precise, since it can only produce a true (1) output for a very small region of input space (this is demonstrated by Appendix D.1 - Figure 5). It is possible that using a single pretrained isSuccessor relation-decoder, to guide the target encoder, encourages the BlockStacks encodings to be more precise than the MNIST encodings, such that each red-block height group is clustered tightly on a latent subspace. 
If so, this would explain the improvements, as the corresponding improved performances on other held-out relation-decoders will carry over via the strong parametric consistency associations that DC learns. 
We will compute the relative dispersion of data encodings to confirm this and add it to the appendix for the camera ready paper, if accepted.

Once again we would like to thank you for your review. Your comments were very insightful and helpful in improving the manuscript, and we welcome any further feedback.

 Thank you for your additional comments regarding typos and text clarity.
In particular:

**[C3] Comments on Section 3**

> In Definition 3: $\psi_{\mathcal{S}}(\mathcal{S})$ is undefined, I supposed it means $\psi_{\mathcal{S}}(s) | s \in \mathcal{S}$

Yes, you are correct. We have made sure to explicitly define this in the updated revision.

> It could be helpful to add a sentence explaining the measure on substructures

We have now included the sentence "where $p( \mathcal{S}_\sigma | \tilde{\mathcal{S}}_\sigma)$ means that there is a high probability that the interpretation sampled from $\tilde{\mathcal{S}}_\sigma$ will be $I\_{\mathcal{S}\_\sigma}$".

> "It is straightforward to show that  $\sum_{\mathcal{S}_\sigma}$" --> what are we summing over here?

Thank you for bringing this ambiguity to our attention. The sum is over all possible structures with domain $\mathcal{S}$ and signature $\sigma$. We have included a sentence to make this clear in the revision.

> "If we have a theory $\tau$ over $\sigma$" --> I don't get this, isn't a theory defined over the language which contains the relations?

Yes, you are right. We corrected the sentence by: "Suppose we have a theory $\mathcal{T}$, whose sentences use relations from a signature $\sigma$ ..."

**[C4] Comments on Section 4**

> Overall I found this section quite difficult to follow and I wasn't sure the details were relevant (but I could be wrong here). You could consider removing some technical details here (and moving them to the appendix), so that you have more space for possible additional experiments.

Thank you for this suggestion. We took your advice and slimmed down section 4 to be more concise. We deferred the technical details to Appendix H, which provides the expanded derivation for the new $\epsilon$-proxy. This gave us extra space to include Table 1. in the main results section. 

> The grounding $s_{ijk}$ is used but only in the next paragraph it is explained what it means, so please define it beforehand.

In the updated Section 4, we have made sure to define all terms clearly before their use.

> It is not explained what an "atomic subformula" is

An atomic sub-formula is an atomic formula. The “sub” refers to the fact that they are a part of another given formula. This was an unnecessary complication and since atomic formula are defined in Section 2, we now only refer to atomic formula in Section 4 and we make clear from the context when they belong to a given formula.

**[C5] Comments on section 5**
> The explanation on the DC is extremely dense, and I'd like to understand better why you developed it this way and why you think it works well.

We hope that the updates we have made to Section 5, and the additional DC illustration in Appendix D.1, have resolved this comment.

We would like to thank you once again for your review and comments. We welcome any further feedback or questions.

 Thank you for your review and for your helpful suggestions which have helped us improve the manuscript. We are happy to hear that you found our contributions interesting and that the paper was easy to follow.
___
In the following, we answer each of your main comments in turn:

**[C1] Experiments feel nascent and it’s not clear whether the same trends will hold for more complex datasets / relations.**

The purpose of the empirical analysis is to evaluate the paper’s hypothesis, which is that the transferability of a concept relies on the underlying coherence of the concept. The manuscript has at its core a two part contribution: first, a theoretical foundation that characterises what a neural-learner learns when modelling relations over real-world data, thereby acting as a neural-symbolic theoretic foundation; and second, an empirical evaluation of the hypothesis based on a purposeful experimental setup.
Following comments from reviewers **v3U9** and **8JLA**, we have refined the empirical analysis by making the link between the definitions for consistency/coherence and the empirical analysis stronger, via the specification of an $\epsilon$-proxy, which provides a calculable proxy for $\epsilon$-consistency/coherence. We present the corresponding $\epsilon$-proxy coherence results in Table 1. to give a direct numerical comparison between relation-decoder coherence results.
We feel that the current empirical analysis achieves its purpose within the context of this paper, particularly with the now stronger link to the definitions via the $\epsilon$-proxy and Table 1.

That said, we do take the point that additional experiments can help consolidate the trends. 
Following reviewer **v3U9**’s suggestion, we have begun mirrored experiments where we learn the relation-decoders on BlockStacks and then transfer to MNIST. Following reviewer **TRpG**’s suggestion, we are planning to also experiment with changing which relation-decoder(s) is(are) used as the target-encoder guide. We are aiming to incorporate the corresponding results into the camera ready version of the paper, if accepted.


**[C2] Minor comments: Repetition of background can be reduced/moved + broken footnote**

Thank you for pointing out the broken footnote, we have fixed it in the revised Section 4.
Thank you also for the suggestion to move some of the background to the appendices. We reduced Section 2's size by removing some of the redundancy. This helped us make space for the newly added $\epsilon$-proxy coherence table. 


Thank you again for your review and comments. We welcome any further feedback or questions.

 We would like to sincerely thank you for your thorough review and for your insightful comments. They have helped us improve the manuscript, as we detail below. We greatly appreciate your positive comments and are very happy to hear that you found the manuscript interesting to read.

Thank you also for pointing out each typo, these have been corrected in the updated revision.

___

Responding to each of your main comments in turn:

**[C1] Importance of coherence is not clear in empirical analysis + what is the use of the formal definitions:**

We would like to thank you for bringing our attention to this. We believe this is related to a similar comment that was expressed by reviewer **8JLA**. We can see that the relationship between coherence and the empirical analysis was not as clear as it should have been.

The hypothesis of the paper is that for a concept to be transferable, it should remain logically consistent regardless of domain. By defining $\epsilon$-consistency and $\epsilon$-coherence, we formally express this condition. However, producing the exact $\epsilon$-coherence value relied on us computing $\Gamma_{\mathcal{T}}^{\tilde{\mathcal{S}}_\sigma}$ and this quickly becomes intractable. Since we could not then produce an exact $\epsilon$-coherence comparison between relation decoders, we propose our practical consistency loss and use this to compare relation-decoder consistency on different data splits. By contrasting these results against PRT accuracy, we are able to determine a trend wherein models that achieve better consistency loss retention across domains do indeed perform better at PRT, which supports the hypothesis.

However, that said, we realise that the analysis was missing a direct use of the definitions, namely we did not detail a clear comparison between relation-decoder coherences. We feel that what was missing was a numerical comparison between relation-decoder $\epsilon$-coherence values, but as discussed, producing the exact $\epsilon$ value is not straightforward.
To make the link between the definitions and empirical analysis stronger, we have extended our consistency loss and derived an $\epsilon$-proxy measure that lets us calculate a proxy for $\epsilon$-consistency/coherence. We provide a clear numerical comparison of $\epsilon$-proxy coherence across relation-decoder models, in Table 1. and have modified the results and discussion sections accordingly. Following your suggestion, we made Section 4 more concise and moved the technical details to Appendix H, which now includes the expanded derivation for the proposed $\epsilon$-proxy.

We feel that these changes make the link stronger and we welcome any further feedback.

**[C2] DC explanation too short + discussion over its strong performance**

We developed DC around the intuition that a relation-decoder would be more suitable for concept transfer if it is able to generalise logical consistency across its input space. We realise that we did not explicitly express this intuition and have now added the following to the end of Section 5, directly after DC is defined:

>The key innovation behind DC is its ability to model each of the ordinal relations whilst encouraging generalised consistency across the full latent subspace, as defined by each $\mathbf{u}_r$. This is achieved without explicit weight sharing, wherein relation-decoders discover parametric relationships between relations from the data. We provide an additional depiction and analysis of DC in Appendix D.1, which further illustrates these effects.

As referred to in the excerpt, we provide a visual illustration (Figure 5. Appendix D.1) that demonstrates how DC models each of the ordinal relations and shows how generalisation of logical consistency is achieved.
We have also updated the results/discussion to refer more clearly to the fact that the empirical evidence suggests that DC's strong consistency generalisation corresponds with its excellent PRT performance.

**[C2.1] Recommendation for an ablation study**

Thank you for your suggestion to perform an ablation study, this would indeed expose the inner workings of DC.
During the development of DC we had experimented with many variations and eventually found the reported configuration to be the one that carried the best tradeoff between model capacity (ability to model each of the relation types) whilst ensuring good consistency generalisation. We aim to reproduce these intermediary results for the camera ready paper version, if accepted.

 Dear Reviewers and AC,

We would like to sincerely thank all of the reviewers for their detailed and insightful reviews. Their comments have helped us improve the manuscript, in particular through producing a clear continuity between the theoretical contributions of sections 2-4, and the empirical evaluation in the latter sections.

In the following, we first detail the changes made in the revised manuscript in response to shared reviewer comments, where all references to reviewers are made in bold e.g. **0a2C**.
The specific answers to the individual reviewers will be posted soon.

**1. Consistency/coherence are a key focus in the manuscript's theoretical contribution, but are lost in the empirical evaluation** [**v3U9** + **8JLA**] 

We have extended our consistency loss to produce an $\\epsilon$-proxy measure that allows us to practically evaluate a proxy for $\\epsilon$-consistency/coherence.
As suggested by **v3U9**, we reduced Section 4, such that it provides the core derivation outline for the new $\\epsilon$-proxy, and deferred the expanded derivation to Appendix H.
In response to reviewer **8JLA**’s comment that the manuscript would benefit from a numerical comparison between relation-decoders, we produced the following table (Table 1. in the manuscript):

*[Table 1 has been updated. The revised table is in comment titled "Summary of updates made in latest revision"]*

which provides a clear numerical comparison between relation-decoder concept coherence, together with the associated additional Con-I results on target data-embeddings (Figure 3 bottom-right).
We feel that these revisions provide the previously missing link between Sections 2-4 and the empirical analysis.

**2. The motivation and details regarding our proposed DC is not clear** [**v3U9** + **TRpG**]

Thank you for this comment. We realise that we may not have introduced our DC model as clearly as we should. To address the reviewers’ concerns, we have now included the following text at the end of Section 5: 
> The key innovation behind DC is its ability to model each of the ordinal relations whilst encouraging generalised consistency across the full latent subspace, as defined by each $\\mathbf{u}_r$.  This is achieved without explicit weight sharing, wherein relation-decoders discover parametric relationships between relations from the data. We provide an additional depiction and analysis of DC in Appendix D.1, which further illustrates these effects.

and follow up with a visual illustration (Appendix D.1 - Figure 5) that demonstrates the effects of these parametric relationships between DC modelled relations. 

**3. The learning task and empirical results are one-sided/underdeveloped/trivial** [**v3U9** + **LAq9** + **TRpG**]

The purpose of the empirical analysis is to evaluate the paper’s hypothesis, which is that concept transferability relies on coherence: retention of consistency across domains. The manuscript has at its core a two part contribution: first, a theoretical foundation that characterises what a neural-learner learns when modelling relations over real-world data with intermediary vector representations, thereby acting as a neural-symbolic theoretic foundation; second, an empirical evaluation of the hypothesis which includes a purposeful experimental setup.
We feel that the empirical analysis is sufficient to evaluate the hypothesis and has been improved much further with the inclusion of the $\\epsilon$-proxy. 
But we take onboard the reviewers' suggestion for additional experiments as they can help consolidate the observed trends. Specifically, reviewer **v3U9** suggested running the reverse source-to-target setup, where we perform relation-decoder learning on BlockStacks, and transfer to MNIST. 
We have begun these experiments and plan to incorporate these into the camera ready version of the paper, should it be accepted.
Additionally, reviewer **TRpG** suggested running the experimental setup using other relations/groups of relations, instead of using isSuccessor as the target encoder guide. We suspect that isPredecessor will work similarly to isSuccessor when used as the only guide, but it is not clear how others will perform due to incomplete logical associations.
These results would be no doubt interesting to confirm and we are aiming to also incorporate these into the camera ready revision. 

We would like to thank the reviewers once again for their important comments. We believe that the resulting updates to the manuscript have had a significant positive impact on its overall quality.

Thank you,

Authors

 Dear Reviewers and AC,

We wish to outline the changes we have made in the latest manuscript revision.

- We updated Table 1 as the first version still had stacked Con-I values (which were used to create the plot in Figure 3-bottom (right)). 
These have now been unstacked. We additionally include the corresponding best $\beta$ values for which each $\epsilon$-proxy type was obtained, and include an (aggr)egated $\epsilon$-proxy which gives the best summed $\epsilon$-proxy across consistency types.
The updated version of Table 1 is: 
\\begin{array}{l|rc|rc|rc|rc|rc}
   \\hline
    \\phi & \\text{Aggr.} & (\\beta^*) & \\text{Con-A} & (\\beta^*) & \\text{Con-I-tr }& (\\beta^*) & \\text{Con-I-asym} & (\\beta^*) & \\text{Con-I-refl} & (\\beta^*) \\\\
    \\hline
    \\text{TransR} &     90.33  &   (8)  &          44.34 & (12)&    35.30  & (8)  &       9.94  & (8)  &       0.55  & (8)  \\\\
    \\text{HolE} &     82.06  &   (8)  &          41.18  & (8) &    32.15  & (4)  &       5.96  & (1)  &       0.07  & (8)  \\\\
    \\text{NTN} &     79.54  &   (8)  &          38.91  & (8) &    30.08 & (12) &       4.49 & (12) &       0.09 & (12) \\\\
    \\text{NN} &     34.09  &   (8)  &          24.78  & (8) &     7.24  & (8)  &       3.88  & (8)  &       \textbf{0.04}  & (4) \\\\
   \\hline 
    \\text{DC (ours)} &     \textbf{0.34}   &   (1)  &           \textbf{0.07}  & (1) &     \textbf{0.18}  & (1)  &       \textbf{0.00}  & (1)  &       0.09  & (1)  \\\\
   \\hline
\\end{array}

- We made improvements to the main text, to ensure that the revision fits to the page limit. We also improved overall clarity in the newly added text.
- We refined the newly added Appendix content.

Thank you.

Authors 

 We would like to sincerely thank you for your thorough review. We greatly appreciate the time and effort you have dedicated to it and your comments have helped us improve the manuscript.
____
In the following, we respond to each of your main comments in turn.

**[C1] The link between sections 2-4 and the sections thereafter is not clear**

We would like to thank you for bringing our attention to this. We believe this is related to a similar concern that was expressed by reviewer **v3U9**. We can see that the relationship between coherence and the empirical analysis was not as clear as it should have been. 

To address each of your questions in order:

**[C1.1] What are the main contribution of sections 2-4?**

The main contribution of sections 2-4 is a neural-symbolic theoretical foundation for what non-symbolic learners essentially try to accomplish in terms of a symbolic representation that expresses relationships between objects. This lets us look further than basic model accuracies and instead perform a deeper analysis over a neural model’s characteristics. We argue that these characteristics are key to developing neural models that exhibit human-like qualities (namely concept transferability).

**[C1.2] how the theoretical description in these sections can be connected to the neural model proposed in section 5? + to what extent the model proposed in section 5 can realise those concepts discussed?**

[We use bullet points to break up the arguments]

- The neural model from section 5 is relatively standard in terms of non-symbolic learners: it critically contains a raw data encoder and a set of classifiers (relation-decoders). 
- the theoretical foundation gives us an understanding regarding what this model (and the many that are like it) is attempting to learn from a symbolic perspective.
- For a signature consisting of only binary relations, $\mathcal{S}\_{\sigma}$ can be decomposed into a training set, consisting of training data taking the form (($s_i$, $s_j$), $\gamma_{rij}$). We are then training a neural model to predict the correct binary label $\gamma_{rij}$ for inputs $s_i$ and $s_j$.
- Therefore, the loss function $p(\mathcal{S}\_{\sigma} | \tilde{\mathcal{S}}\_{\sigma})$ is another way to represent the log loss of training a conventional (multi-class) classifier, where $\mathcal{\mathcal{S}}\_\sigma$ is the empirical structure present in the observed data.
- Section 3 showed that the resulting encoder and relation-decoder(s)/classifier(s) together define a soft-structure $\tilde{\mathcal{S}}\_\sigma$, which approximates the above $\mathcal{S}\_\sigma$. So the neural model is an implementation of a soft-structure and, in particular, the $\phi$ relation-decoders are together a concept learner.
- We argue that minimising $- \log p(\mathcal{S}\_\sigma | \tilde{\mathcal{S}}\_\sigma)$ does not guarantee good transfer performance, as it only relates to the structure encountered in the source domain (i.e. the classifier only sees inputs from the source data set), whereas the empirical structure for the target domain is different, due to it having a different domain. However, through theory $\mathcal{T}$, all empirical structures that have the same signature are related, in that they are all models of $\mathcal{T}$. This aspect led us to the hypothesis that the domain-invariance  of the soft-structure consistency with $\mathcal{T}$ could be the key factor in determining whether the soft-structure (and in particular concept associated with $\phi$) ends up being transferable. 

This brings us nicely to the next question:

**[C1.3] how to make sure an ϵ-consistency of sub-structure in the model?**

- In this work, our aim is to evaluate the above hypothesis using our developed consistency loss. The PRT transfer learning accuracy results demonstrate how different choices for relation-decoder affect the resulting transferability of the ordinal concept. Using our consistency loss, we investigated whether the coherence (domain-invariance of consistency) was important to this transferability, thus evaluating the hypothesis, and we found strong evidence in support of this.
- In summary, the scope of this work is to formally present and then evaluate, using our consistency loss as a measure, the hypothesis. We therefore do not explore any explicit form of encouragement or guarantee that a relation-decoder achieves a good $\epsilon$-consistency (for example through additional regularisation).
- That said, we should note that this hypothesis is what motivated us to develop our DC relation-decoder and in this way, DC is built to encourage strong $\epsilon$-consistency. Our intuition was that a relation-decoder would be more suitable for concept transfer if it was likely to generalise logical consistency across its input space and that this would depend on parametric associations. We have made this motivation clearer by including it in Section 5., together with additional details in Appendix D.1 and Figure 5. 

The main motivation of this paper is the observation that once we humans have learned the meaning of a relation, we can apply it to different contexts and scenarios effortlessly. For instance, once we understand the meaning of a concept "larger than", we can apply this in many context such as integers, person lengths, building sizes, etc. The authors argue that it is interesting to study how we can build learning systems that similarly are able to transfers learned relations across domains.

In order to express relations, a language based on first-order logic is developed, and semantics is given to this language using an interpretation function, mapping relation symbols to the tuples of elements over which these relations hold. In order to facilitate learning these relations, the authors develop a quantitative account of the logic. They develop so-called "soft structures", which can essentially learn a symbolic structure over real-world domains. This learning is done using the Variational Auto-Encoder framework, where an encoder maps the real-world input (for instance, an image), to a latent space Z. The interpretation of relations over the real world data is then implemented by "relation decoders" (one decoder for each relation), which are belief functions assigning a number in (0, 1) to relations. In order to train such models, they define a measure on how well a soft structure support a symbolic structure, and they use this as a term in the loss function. They also provide a formal account of what it means for two soft structures to be "coherent", which intuitively mean that relations that apply to one domain (e.g., ordinality in numbers) also can be applied to another domain (e.g., ordinality in block sizes).

They authors then derive an efficient consistency loss, since the original one is intractable because it requires computing beliefs over all interpretations of the relations. They then propose a Dynamic Comparator relation decoder model, and a new learning task called "Partial Relation Transfer learning". They compare against several existing relation decoder models, and empirically show that their relation decoder retains coherence across domains, while achieving better transfer learning performance. === Strength ===
* The paper is very well written and the organization is clear.
* I find the way in which the authors develop a formal framework, and then provide a quantitative account for this in terms of soft structure, which are then used to train a Variational Auto-Encoder very appealing. I believe the proposed approach is advanced and brings together  insights from both logic and machine learning. Given the different approaches of these two fields, I find this impressive.
* The topic is also interesting and relevant to AI.

=== Weaknesses / Questions for authors ===
1. "Coherence" seems to be an important topic of the paper. It is mentioned often, and Section 3.1 provide a formal definition of epsilon-coherence. However, in the empirical results this didn't seem to be mentioned anymore. It made me wonder: what is the use of the formal definitions in Section 3.1? I was expected this to be used to evaluate the models. Maybe it did, but then it wasn't clear to me, so in any case I think the relation between the experiments and Section 3.1 can be made clearer.

2. I found the explanation of the newly proposed relation decoder the "Dynamic Comparator" too short. It was difficult for me to understand what it did. More importantly: it seems to outperform all other models very convincingly, but I could not find any discussion on why you believe this is the case. So I felt the empirical results deserve more detailed reflection. Perhaps an ablation study would be interesting?

3. The empirical studies by itself are also somewhat one-sided (MNIST --> BlockStack). Did you also try the other direction? I.e., train on the BlockStack domain and then evaluate on MNIST? More generally, I was wondering whether the empirical setup is actually similar to how we humans learn. Can we really learn a relation such as "larger than" from one domain, and can we directly apply it to another? This is actually not directly clear to me. I can imagine we learn a relation in multiple domains, after which we are able to generalize to new domains.

4. All appendices seem to be missing from the PDF.

=== Detailed comments for improving the paper ===

Section 2:
- I liked the formalisation and I think it was well presented, also the example was helpful.

Section 3:
- "Z is a latent variable, itself draw from marginal p_z" --> "draw" should be "drawn". Also, I believe p_z is a prior instead of a marginal.
- In Definition 3: $\psi_S(S)$ is undefined, I supposed it means $\psi_S(S) = \{\psi_S(s) | s \in S\}$, i.e. all encoding of S?
- It could be helpful to add a sentence explaining the measure on substructures, e.g., saying that this value will be close to 1 if the soft-structure is similar to the structure.
- "It is straightforward to show that $\sum_{S_\sigma}...= 1$" --> what are we summing over here? All possible interpretations of \sigma use S? Please clarify this.
- "If we have a theory \Tau over \sigma" --> I don't get this, isn't a theory defined over the language which contains the relations?

Section 4:
- Overall I found this section quite difficult to follow and I wasn't sure the details were relevant (but I could be wrong here). You could consider removing some technical details here (and moving them to the appendix), so that you have more space for possible additional experiments.
- The grounding $s_{ijk}$ is used but only in the next paragraph it is explained what it means, so please define it beforehand.
- It is not explained what an "atomic subformula" is.

Section 5:
- "given a domain images" --> remove "a"
- "latent space ." --> remove space
- The explanation on the DC is extremely dense, and I'd like to understand better why you developed it this way and why you think it works well.

Section 8:
- "ordinaliety" --> ordinality The paper addresses an important element of human learning, namely that we are able to apply our understanding of learned relations to new domains. The paper is very well written and brings together insights from logic and deep learning in an innovative way. The main weakness is that the empirical results felt somewhat underdeveloped, so I would recommend swapping out some of the technical details (e..g, Section 4) for more analysis on the empirical studies (see main review for more details).

This work introduces a more general definition of structures namely soft structures that can be used to learn structure over richer domains in a data-driven manner. Further, they introduce suitable definitions of coherence and consistency for the proposed soft structures. Next, the paper proposes a neural architecture that is essentially, jointly learning relation encodings jointly with representations for set elements. The key argument of the paper is that transfer is to be expected across shared relations despite their overloaded usage across domains — for example, greater than (GE) can have completely different interpretations in the source / target domains.  Strengths:
- The paper makes multiple simple yet interesting contributions. The definition of soft-structure is useful and intuitive. 
- The experimental results, at least with the DC model, are consistent with the hypothesis of the work. 
- The paper is generally well-written and is easy to follow. 
Weaknesses: 
- While intuitive and backed by experimental results, the experiments feel nascent and it’s not clear whether the same trends will hold for more complex datasets / relations. 

Minor comments:
- Repetition of background / definition can be cut down. 
    - And perhaps, the space can be used to make the definitions more concrete with example
- Fix footnote 2 The paper introduces soft structures along with allied definitions. Further, it introduces a neural architecture that jointly learns element and relation representations -- the experimental results do back the hypothesis of the paper. However, the experiments have a limited scope and therefore, I am inclined but not very strongly towards acceptance. 

This work presents a new neural network model that can learn some symbolic relations in a specifically designed neural network module, which can later generalize to other unseen data. The proposed method was evaluated on datasets MNIST and BlockStacks to demonstrate its potentials.  My big concern about this work is *probably* a writing issue. Specifically, I had significant trouble identifying the relationship between sections 2-4 and section 5. The theory, definition, and discussion in sections 2 - 4 are interesting and I enjoyed reading it. But it seems to be disconnected from section 5. When I read it again, I found out that it's actually much easier to read by directly starting from section 5. To my understanding, the neural model proposed in this work is simply learning a tied latent representation, where the latent representations are constrained by some distance-based similarity measurement. To better achieve the goal of representation learning, this work further suggests using $\beta$-VAE for disentangled representations. 

To make sure my understanding is correct, I would like to ask a few questions

- what is the main contribution of sections 2 - 4? 
- how the theoretical description in these sections can be connected to the neural model proposed in section 5?
- on the other hand, if there is a connection, to what extent the model proposed in section 5 can realize those concepts discussed. For example, how to make sure an $\epsilon$-consistency of sub-structure in the model? 

Some additional questions and comments

- There is a missing reference in the footnotes on page 5
- It would be great to report the numbers in figure 2, instead of using bar plots. A similar suggestion also applies to figure 3.
- Furthermore, in figure 2, if my observation is correct, the proposed model performs much better on the out-of-domain data (nearly 100% accuracy on the BlockStacks data) than on the in-domain dataset (MNIST), how should we understand this difference?  The major concern is the disconnection between the theory and model realization, which led to the trouble of identifying the main contribution of this work. 

This paper proposes a loss based on theory for the relation prediction problem, using the sentence of the relationship as the supervision. The paper designs an relation transfer learning experiment. By supervising one relation, other relations can be transferred. strengths
1. The thesis theoretically and rigorously expressed the relation prediction problem in the framework of universal algebra, which is inspiring. 
2. This paper proposes a loss based on theory for the relation prediction problem, using the sentence of the relationship as the supervision. This loss links the different relationships together and is a more general and universal form.
3. The paper designs an relation transfer learning experiment. By supervising one relation, other relations can be transferred. It seems interesting.

Weaknesses
1. What are the benefits of using the sentences in the theorem to train relation prediction, compared to independent training of each relation? From Equation 8, the loss of a sentence seems to be the product of the loss of each relation in the sentence. So why not directly train the loss of each relation independently?
2. What is the computational complexity of the loss function? For example, isGreater can be represented by the conjunction of several isSuccessors, but it is not feasible to list all these sentences when the domain is larger.
3. About Definition of Equation 9: I don’t understand why we need to calculate expectation over s. The two terms in the formula seem to have been summed over s, and have nothing to do with s.
4. About Experimental setting: What is the purpose of comparing DC with other relation representation models? The paper lacks an explanation for the motivation of DC. Why is DC better?
5. The experiment aims to show that by supervising one relation, other relations can be transferred. The reason to use isSuccessor is “Our system of relations can all be expressed in terms of isSuccessor”. What if using relation other than isSuccessor for transferring?
 First,  the motivation of the work is not clear.  Second, the proposed method is straightforward and short of novelty. Finally, the task presented in the paper seems trivial.

Meta-review: After reading the reviews and the rebuttal I unfortunately feel the paper is not ready to be accepted. 

The reasoning for this decision is as follows:
* the empirical evaluation is somewhat weak in its current format, and even adding experiments going from BlockStacks to MNIST would have improved the results, or potentially other synthetically generated data. Or playing with which relation is used during the transfer phase. Something to give a bit more weight to the empirical section and help it connect better with the theoretical one
 * But maybe more importantly (and to some extend this is true for the formalism introduced as well), I think there needs to be a bit more context. After reading the reviews, I went and read the paper, and for example in results provided, it is not clearly explained what is the relationship between the proposed method and some of the baselines. I noticed that the related work section ended up in the appendix, which is fine, to the extent the main text can connect to the literature a bit. But while I agree that the introduction of the method seems good and clear, and this is a hard and important problem that lacks a proper framework and the proposal in the paper is quite interesting. It is also important to understand its relation to other frameworks, and to explain clearly what it tries to fix in other proposal. And to interpret the result, maybe justifying or providing some intuition of why the proposed model performs better. I think this is very crucial particularly for a topic that is still in a growing phase, which makes it harder to judge.
I know in the appendix, the author mention domain adaptation which is also something that jumps in mind when looking at this architecture. However this point is not discussed or mentioned as much in the main paper.

In current form, while the paper reads well, one is left to trying to understand whether these results are significant. I think the work is definitely very interesting and I hope the authors will resubmit it with modification. I just feel in the current format it will not have the impact it should, because of a preserved weak experimental section and not a clear grounding in the literature, making readers unsure of the significance of the work.

Decision: Reject