Formalizing Consistency and Coherence of Representation Learning
Authors: Harald Stromfelt
Luke Dickens
Artur Garcez
Alessandra Russo
Venue: NeurIPS 2022 Accept (2022)
Keywords: autoencoders
neuro-symbolic
consistency
coherence
transfer learning

Abstract: In the study of reasoning in neural networks, recent efforts have sought to improve consistency and coherence of sequence models, leading to important developments in the area of neuro-symbolic AI. In symbolic AI, the concepts of consistency and coherence can be defined and verified formally, but for neural networks these definitions are lacking. The provision of such formal definitions is crucial to offer a common basis for the quantitative evaluation and systematic comparison of connectionist, neuro-symbolic and transfer learning approaches. In this paper, we introduce formal definitions of consistency and coherence for neural systems. To illustrate the usefulness of our definitions, we propose a new dynamic relation-decoder model built around the principles of consistency and coherence. We compare our results with several existing relation-decoders using a partial transfer learning task based on a novel data set introduced in this paper. Our experiments show that relation-decoders that maintain consistency over unobserved regions of representation space retain
coherence across domains, whilst achieving better transfer learning performance.

Formalizing Consistency and Coherence of Representation Learning
Harald Strömfelt Department of Computing Imperial College London
Luke Dickens
Department of Information Studies University College London London, WC1E 6BT ddickens@ucl ac uk
London, SW7 2AZ wtw.omfll:::@immerraaaa ac. uk
Artur d'Avila Garcez
Alessandra Russo Department of Computing Imperial College London
Department of Computer Science City, University London
London, ECIV OHB a. gwrceeecity.....om ac uk
London, SW7 2AZ a. rrsso@iimeeeiil.... ac uk
Abstract
In the study of reasoning neural networks, recent efforts have sought improve consistency and coherence of sequence models, leading to important developments in the area of neuro-symbolic AI. symbolic AI, the concepts of consistency and coherence can be defined and verified formally, but for neural networks these definitions are lacking. The provision of such formal definitions is crucial offer a common basis for the quantitative evaluation and systematic comparison connectionist, neuro-symbolic and transfer learning approaches. this paper, we introduce formal definitions of consistency and coherence for neural systems. To illustrate the usefulness our definitions, we propose a new dynamic relation- decoder model built around the principles of consistency and coherence. We compare our results with several existing relation-decode learning task based novel data set introduced in this paper. Our experiments show that of representation space retain coherence across domains, whilst achieving better transfer learning performance.
using a partial transfer
that maintain consistency over unobserved regions
Introduction
Humans are capable of learning concepts that can be applied to many different scenarios important principle is that human-like concepts remain coherent across contexts 611
18 34 23
an example, consider the concept of ordinality, e.g. "A is larger than B", which allows comparisons to be made between ordered sets. Ordinality should apply equally whether A and are digits or a tower of blocks. It is said that concept may pertain to multitude of properties: position, volume, reach, etc. As long as one of these properties can be attributed to an object, a set of objects can be compared on that basis. All in all, if the concept of ordinality was be learned in its most general form, its use should be consistent across objects and coherent across object properties.
In empirical results story generation and instruction-followii have shown that intuitive use of consistency and coherence can increase the accuracy neural networks. Following a neuro- symbolic perspective [12), it is argued in 51 that System approaches, fast and capable of learning
FFunding in direct support of this work: EPSRC Training Grant, project reference EP/L550486/1.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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patterns efficiently from data, "are often inconsistent and incoherent", and that "adding System 2-inspired logical reasoning" as a logical consistency, training-free -feee module allows for an improved selection of candidate stories generated by System 1. While [31] makes an important contribution by exploring several variations on the theme, in this paper we offer a formal definition for consistency and coherence in the context of neural networks, in particular neuro-symbolic autoencoders. We also apply and evaluate consistency and coherence in transfer learning tasks, where we believe that the theme will have its most practical impact.
We argue that for concept be useful during transfer learning. the system of relations that define the concept in the source domain must be coherent with the target domain, whereby logical consistency achieved in the source is retained in the target domain. This to say that the concept-specift relations learned in the source ought to be consistent with logical theory that defines their semantics, and that such consistency must extend beyond the representations learned in the source domain and, in particular, hold for the embeddings learned in the target domain.
In this paper, we offer formal definition for consistency and coherence of sub-symbolic represen- tation learners, inspired by analogous definitions from symbolic AI. This is expected to define the conditions that make a learned concept transfer well across properties and objects. To evaluate the practical value of these definitions in a real setting, we derive a simple neuro-symbolic autoencoder architecture consisting of neural encoder for objects coupled with consistent modular object relation- decoders. Relations such as isGreater, isEqual... are evaluated on a proposed Partial Relation Transfer (PRT) learning task, between new CLEVR-style BlockStacks data set and the MNIST handwritten digits data set, such that the learning of ordinality among the MNIST digits is evaluated against the learning of the relative position of a red block in a stack of multi-colored blocks. Our evaluation includes comparison with several existing relation-decoder models and results show that relation-decoders which maintain consistency over unobserved regions of representation space retain coherence across domains whilst achieving better transfer learning performance In summary, the contributions of this paper are:
A formal definition of consistency and coherence for sub-symbolic learners offering a practical evaluation score for concept coherence;
A derived model implementation and PRT data set and experimental setup used to evaluate the interplay between concept coherence and concept transfer;
A comprehensive critical evaluation of results and comparison of multiple gllaton-decoder models, showing that improvements in concept coherence, as defined in this paper, corre- spond with improved concept transfer.
In Section we provide the notation and required logic background. Section 3 formally defines coherence and consistency. Section 4 Deftines a practical consistency loss and Section 3 outlines our neuro-symbolic autoencoder. After detailing the PRT task and introducing the data set in Section b comparative experimental results are discussed in Section 7 We provide an overview of the related work in Section and Section F concluues the paper with discussion, including limitations and future work. We expand on the experimental results and setup, together with data set characteristics, model details and parameterization the Appendices P
2 Preliminaries
Notation: We reserve uppercase calligraphic letters to denote sets, and lowercase versions of the same letter to denote their elements, e.g. S = (s1...., } } is a set S of n elements We indicate with |S] = n the cardinality of S. We use uppercase roman letters denote a random variable (e.g. S), and use the uppercase calligraphic version of the same letter (S) to denote the set from which the random variable takes values according to some corresponding probability distribution ps, over the
elements of the set, such that ===1 ps (*;) = for a discrete S. For brevity, we may write ps (8;) as p(s;), where the random variable is implied by the argument. We use bold font lowercase letters to denote vector elements, e.g. s; E IRd is an d-dimensional vector element from the set S = Rdd
'This paper formalizes the theory and extends the empirical results first reported Thh codebase for this paper can be found at httpss: http://gttthboooogem wwwm://Strtoffee//eeueeeeeeeeeee www.://mswl.eeeeerpe222/222
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Logic and model-theoretic background: our proposed theory is based upon logic and model theoretic primitives. To avoid making this paper overly dense, we defer the details of the logic background to Appendix E and include here only the most important definitions supported by an illustrative example.
Definition 2.1 (Signature, Arity, Domain, Interpretation, Structure). The signature of a language is a set of relations a = (r E f) whose elements have arity given by ar + *N, where N is the set of natural numbers. Given a signature a and a non-empty domain S = {81, 82,.../ ., an interpretation Is, over elements of assigns to each relation r a set Is, (r) C _arrr)) A structure is a tuple S, (S, Is,).
We construct universally quantified first-order logic formulae (called sentences) using the signature of L. A set of sentences form theory TI and when a sentence T is true in structure Sz, we say that the structure satisfies т, denoted as S. F r. This allows us to define a model of theory:
Definition 2.2 (Model of a theory). Let T be a theory written in a language L and let Sr = (S, Is.) be structure, where is the signature of L. S, is model T if and only if Sg }= for every sentence т E T.
Example 1. Let S is a domain of images of handwritten digits and a the signature of binary relations (isGretter isEqual, isLess, isSuccessor, isPredecessor), for short = {G, E, L, S, P). Let T be the theory that defines ordinality including, for instance, the sentence ji G(i. - 5) (if digit is greater than another then they are not equal). Any structure S, = (S, Is.)) with interpretations Is, of a that captures a total order over the elements of S is model of T.
3 A Formalization of Consistency and Coherence
In this section, we turn our attention to the challenge of learning a model of a theory (Def. (.2) over a real-world domain S given signature Here, learner must determine an appropriate interpretation over real-world data, such as images or other perceptions. This can be challenging because, firstly, we may only have partial description of the interpretation, and secondly data may be noisy and contain information that is not relevant to the theory. For example, the handwritten digits in the MNIST data set contain stylistic details such as line thickness and digit skew that are irrelevant the notion of ordinality, which makes learning the structure from Example I non-trivial.
17 16, we make Following the convention from the autoencoder disentanglement literature 4 the assumption that real-world observations S are drawn from some conditional distribution Psiz- where Z is a latent random variable, itself drawn from prior pz. It is therefore useful to define domain encoding of the form:
ss + Z,
tasked with approximating the conditional expectation of the posterior, i.e. wis(s) = Epz [[]s]. Since obtaining an interpretation from domain encodings for given signature may require dealing with noise, we express the interpretation of relations over real-world data by belief functions [33 32 over the space 2, and refer to these as relation-ecooden
6 : zarr)) + (0,
with ( o}. Concretely, for binary relation and ordered pair (st, 8j) S (s:)) (s)) describes the belief that (s,, sy), E Is. (r). A belief or (rssss); ss,)s = signifies a strong belief that ss,, E Is, (r) and ø, (v)(Ss), ();(s)))=0 signifies strong belief that 88),,)) & Ts. (r). Together, tis and á allow us to define a belief-hased analogue to structure. Definition 3.1 2 and relation-decoders Φ. soft-structure is a tuple 1. = (2, 6). For a finite domain S and encoding vs= S + Z, S, = (Vs(S), Φ) is called finite sofl-substructure of 2., with sub-domain (Ss(S) (1)(()) 8} C 2.
SeructurrSSff-Suuustuuttrr)) Given a signature o, a (possibly infinite) set
A soft-structure can be used to learn logic structure over real-world domain through learning whs and & Clearly, a finite soft- coft-substructure a soft-structure. real-world domain, there may be only partial information about the values of an interpretation, and there may be errors in that partial
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interpretation. To determine the degree to which a soft-structure supports any given structure, we introduce the following measure:
p(S) = II II rEo OESS(H)
f(er. vs, O,
(3)
with
f(ór, vs. 0,,,0,,,.,)) -) (6.
(vss(((s(O))) (6- (1- (1)
(4)
where T6.s. 1 if O E Is, and 0 otherwise; we use o, (b)((()) as shorthand for (1)((s))) ....,........................... (s,(s,) for = ar(r). Eqn. B expresses the assumption that, given finite soft-structure, the beliefs in what constitutes the interpretations of different relations are independent of one another. It is straightforward to show that Es, p(S; = (summed over all possible structures with domain S and signature σ) and so Eqn. can be treated as a probability measure, where (S,) se means that there is a high probability that the interpretation sampled from S, will be Is.- If we have theory T over a then it natural ask with what weight S., supports any given structure that is model of TT In the following, we use model weight, I to describe the support given by S, models of T:
E p(S.)S)) S.EMG
(5)
where M3 is the set of all structures with domain S that are models of T. This lets us compare soft-structures, wherein a good soft-structure will be one that has a high model weight.
Definition 3.2 (e-Consistency of soft-structures). Given finite soft-structure S, and an arbitrarily small number e, if - S T < then we say that S, is e-consistent with theory T.
We propose e-consistency as an appropriate measure of the notion of consistency presented in 31 A consistent soft-structure S₂ ensures that è gives high belief only to interpretations that satisfy, and therefore are logically consistent with, theory T. As expected, consistency pertains to the domain encodings of Sg, i.e. (S). For a concept be learned in a manner comparable to how a human might learn, we would expect consistency to carry over to new domains with their corresponding soft- structures, which motivates our definition of coherence between ofttstruucture, as follows. Consider situation where a deep network has already learned from the MNIST data set a soft-structure that has high model weight, given the relations (G, E, L, S, P} from Example [ Now, consider a new domain of images, V. showing single block stacks of different heights, and we wish to re-use the signature of ordinal relations and T from Example [ Let ly, be an interpretation in the new domain that orders images according to block stack height and that is model of T. We can summarise this with the following two structures:
X₂ = )x, M and y -0 E M3,
(6)
where Xo is the structure from Example with a domain of handwritten digits and Va is our new structure, with a domain of block stack images. These can learned by soft-structures:
X = (xx(()), Φ) and )), = (vy()),
(7)
which use domain-specifir encoders, vx and thy, but share the same relation- As we know that & has high model weight and since ó is shared with Vi, a natural question to ask is: under what conditions will a ó that is consistent over domain-encone eenudinnn vxx(x)) also be consistent over ()?)? Concretely, are interested in specifying when the following coherence condition holds. Definition 3.3 (e-Coherence across soft-structures). Two soft-structures, X' and Di that share relation-decoders o, are said to be e-coherent with respect to a theory T. if X; is consissttt with T. V. is co-consist with T, S and 62 S
Coherence between X and )², as defined above means that the concept of ordinality that applies to digit ordering can also be applied to block stack height ordering. It is desirable that learning ordinality on the domain of digits produces a coherent concept of ordinality with respect to other ordinal properties, such as height. Since it is possible that (X'()') and ()s produce unique encodings, coherence relies on é's ability to generalize over possibly disjoint subsets of 2
'If soft- structure E., defined over the full space Z is consistent then coherence is guaranteed between all possible soft susttrubtuuc......

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4 Measuring Consistency and Coherence
Calculating Eqn. can be computationally too expensive for larger domains. An efficient approach to measuring the consistency of soft-structures is therefore required. In this section, we introduce proxy measure for soft-structure's c-consistency and c-coherence with a given theory when access to every logical model is not available or is computationally intractable.
Suppose that there is fixed domain S and theory T whose sentences use relations from a signature a. Let & E {1,.... K denote the index associated with each unique ground instance of the sentences in T. Take By to be a Boolean random variable. For the k-th grounding in T. the probability of the theory being satisfied under soft-structure S, expressed as p(bT 1| k). Conversely, the probability of non satisfaffaatiinn is given by p(6y .)S..k). For a model of (universally- quantified) theory T. S E M any grounding & of the T is always satisfied (by definition), and thus p(67 1|S,, k) = 1. When S, is consistent with T then should also find that p = 1( ~ 1 for all k. Hence, we define consistency loss function the expectation over randomly-chosen grounding k of the binary cross-entropy between p [B, k) and p(BT|S,, k) for any S E .yg. This in turn simplifies to produce the expected negative log- log-likelihood of satisfying a random grounding of T, as follows:
L(T, S.) = Ex = 118..))) k)].
(8)
where p(k) K. is the uniform distribution over the set of unique groundings. A measure based on this loss is required to enable practical evaluation of consistency, acting approximation for consistency. More precisely, we define - = exp(-L(T, 8.)) as a proxy measure of 1 and say that soft-structure Sr is é-proxy consistent with T if
In 2 L(T,
where E 2 1 1-1 Due to the relationship between and ST) we take the proxy measure of coherence be the smallest satisfiable value of T.(T, between domains
Although our treatment of consistency has thus far focused on a particular theory T, notice that subset of the sentences of T form partial theory, which is itself a theory. This means that consistency can be evaluated given a partial (even single sentence) theory, allowing us to exam- ine consistency losses for any partial specifications of given domain of interest. In this paper, we evaluate the proposed consistency loss against two partial specifications within the theory of ordinality. These are named Consistency-Across (Con-A) and Consistency- ConiistenyyInddividual the experiments that will follow f Con-A includes the sentences that determine inter-relation be- havior, for instance the sentence Vi, (isGreater(i, - -isEquul(i, j) A -isLess(i, (i)ess(i,))), stating that if i is greater than j then i must not be equal to or less than j. Con-T includes the sentences that are about single relation, describing any property of an individual relation over objects (or images in the case of our experiments). Each relation may satisfy a number of properties, for ex- ample Vi, j. k. (isGreater(i.) j) isGreater(j, k) + isGreater(i, k)) represents transitivity of the isGreater relation. Transitivity is true for isGreater, but is false for other relations investigated in this paper, e.g. isSuccessor. We will evaluate consistency loss of transitivity (Con-I-T), asymmetry (Con-I-A) and reflexivity (Con-I-R) for the relations in Example ! The evaluation of consistency loss for any available partial theory will be shown to provide more nuanced perspective on model performance than accuracy results and disentanglement pressure alone during transfer learning.
Individual (Con-I) in
5 A Consistent and Coherent Neuro-symbolic Autoencoder
In order to ground our definitions of consistency (Def. 3.2) and coherence (Def. 3.3) into a real system and evaluate their practical value, in this section we derive simple neuro-symbolic autoencoder architecture which offers one of many possible implementations of the theory defined in Section outlines the main components of our autoencoder: domain-encoder iss and modular
Figure D
'The complete derivation of loss function and bounds is presented in Appendix G
"Truth-table for each consistency formula given in Appendix F
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S
2i
$;
5
r E 0
or
or
Relation Decoding Yo.s.
or (Z:, Bz)
Yo,s,
ELBO L BVAE
enc 8
3
ooitt
or
or
L
S
Sj
j
Sj
Figure 1: Network architecture used for PRT tasks. In our experiments 81/j are either MNIST
(domain X) or BlockStacks (domain Y) images. Relational learning is performed on the source
(S -S=A) X) MNIST domain (to learn e.g. that digit 5 is greater than 3). Moving the target (S = y) domain (stacks of blocks) involves training a new image autoencoder together with subset of the relation-decodes from MNIST with fixed parameters. The remaining relations are held-out to evaluate zero-shot transfer learning performance. Yo.s, provides the ground truth for the given
structure, which or predicts as S. As in Section 3 O is used to abbreviate (185" (8:),, ( (8z)).
relation-decoders & form an autoencoding architecture that, given a domain of images S C CXWH (C color channels, width W and height H) and a d-dimensional latent space sub-symbolic encodings from ths into a modular relational representation via decoding for each Ø,, r E a. Additionally, to retain information in pertaining S which is beyond the requirements of σ. a Nomainndeccode produces domain reconstructions S. In Figure I we use vigie to refer the domain-encoder and vidlec to the domain-decoder Although in this paper we opt for an autoencoding architecture, our definitions of consistency and coherence are applicable to a wider range of neural architectures. For instance, multi-layer perception network can viewed as a set of encoding and decoding layers 400 As long as the architecture offers explicit soft relation decodings, provided we can define a partial theory over them, we can define consistency loss over the outputs.
12d converts
To train the model, ground-truth interpretations Is,, are provided, allowing us to maximize directly Eqn. via the negative likelieeehood loss:
S = =- log (ea).
(10)
To obtain informative latent representations for S. we use a Variational Autoencoder (VAF), specifi- cally the 3-VAE 66 17 21 due to its simplicity and demonstrated ability to separate distinct factors in the latent representation (known as disentanglement, although disentanglement not a requirement for consistency and coherence). We therefore take the Evidence Lower Bound (ELBO) objective with an additional 3 scalar hyperparameter from [17], that seeks to achieve disentanglement (Z5180)) and
combine it with S. to obtain the following aggregate objective
L i
L
BL --VEE
A1.
(11)
where À is a scalar weighting parameter.
Together with the ELBO L--VAE- the choice of relation-decoder can shape the domain encodings [5] In our evaluation, the following choices are made. We propose a Dynamic Comparator (DC) composed of two modes, a distance-based measure, Φf. to measure the distance between two inputs relative to reference point, and a step-function, of. that determines the sign of the difference between two points, optionally with an offset. Although any function could be used that has the required characteristics for of and o², in this paper we use the following implementation:
DC (zi, ²,) a,.o of. + or,1 0F
(12)
a more detailed derivation of ELBO is included in the Appendix

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where,
1 fa( Tr.00 |,,, @ (z: 23 + b;)]12]]
(13)
0f fi (1r, u, (z, - =; 2 b.)).
(14)
Here, a, Sof coftmax((A4 A,) (0. 1)a is an attention weighting between the two modes, of and o!: fo and fi are an exponential and sigmoid function, respectively; u, = Softmax(U,) E (0, 1)th is an attention mask which is applied to m-dimensional embeddings: bj. by E IRm are learnable bias terms that enable an offset to each mode; тh,0 IR are non-negative and 7/r,1 E are any-valued scalar terms, respectively. Lastly, @ denotes the Hadamard product and || - 1/22 is the Euclidean norm. The key innovation behind DC is its ability model each of the ordinality relations whilst encouraging generalized consistency across the full latent subspace, as defined by each u,, This is achieved without explicit weight sharing, wherein relation-decoders discover parametric relationships from the data. Further details are provided in Appendix D.I.
6 Experiment Design: Partial Relation Transfer
We now describe an experimental design to compare coherence of different relation-decoders.
Partial Relation Transfer (PRT): We evaluate novel PRT task across two soft-structures X, and V,. The soft-structures share a common signature a and celation-deodders é, but have disjoint domains X' and V/ respectively. The experimental design involves first learning & on source domain X. together with its domain-specific autoencoder. Then, a new domain-specific autoencoder is trained on the target domain ), alongside selection of the now learned e relation-decoders with ixed-parameters. The selection of relation-decoders is expected to help guide training of whyr (see Fig D. Held-out relation-decoders are then evaluated in y, i.e. zero-shot transfer learning task. For domain X' we use the MNIST handwritten digits data set (24). and for domain y we use the proposed BtockStacks data set, consisting of single stack of nulti-colored cubes of differing heights, each randomly-positioned containing one randomly positioned red cube (see Appendix B for details and examples). The shared signature includes the ordinal relations =G., E. L. S. PT. and it is applied digit ordering in MNIST and to red cube position ordering in BlockStacks, We provide results with respect to a theory of ordinality, as explored in Example D formal specification of the theory is provided in Appendix Fl When transferring relations from oe to w: one could use the full set ø of elation-decoders However, this is not necessary from logical standpoint because the entire system of relations can be expressed in terms of the isSuccessor relation S (e.g. the successor of a number is larger than that number). We therefore only employ the isSuccessor relation-decoder as the fixed-parameter selection to guide the learning of wapes. If coherence, as defined in this paper, is carried across domains, we would expect the transferring of isSuccessor to produce an improved performance on the remaining relations in the target domain.
Neural model components and by hyperparameters: Together with DC, existing relation-decoder
models evaluated here are: Transk (25). HolE 30 NTN 41]. We additionally include a basic all experiments use a 3-VAE [77. We provide further details for all models, including training regimen, parameterization and implementation in Appendix D In the source domain, we explore 3 values in ( [1, 4, 8, 12] and set
feedforward neural network (NN). To produce comain-encoding
x 10S. In the target domain, we first normalise losses and set 3 10-1 and À = 10-2 these
produced good image reconstructions while optimising 1,3'= all experiments we fix 2 10
7 Experimental Results and Discussion
In this section, experimental results show that transfer learning performance is positively correlated with our measures for consistency and coherence. This holds particularly true for embeddings that are different but near in space to source domain embeddings. As we have argued, for a neural model to perform well on concept transfer, its representations must maintain high probability of consistency with a theory that provides a semantics for the concept. The most robust way of doing this is to maintain consistency across regions of embedding space, rather than relying exclusively on the specific data-points observed at training time in the source domain. In our analysis, consistency losses are evaluated when sampling from different regions of latent space 2. We evaluate: data- embeddings, where all inputs are encodings of domain's 's test data; interpolation, when we derive an
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Source Domain Relation Prediction Accuracy
PRT Accuracy Per Relation and Model
1.0
Accuraly
0.8
0.6
Mean Relation Accuracy (MNIST)
Moan Held Out Relation Accuracy (BlockStacks)
1.0
Accuracy
DC
NN
NTN
HolE
TranoR
DC
NN
NTN
HolE
TransR
12.0
Figure 2: [Top] Relation-decoder prediction accuracy per model (DC, NN, NTN, HolE, TransR) and relation (abbreviated on the z-axis as in Example DT. in the source domain (MNIST, left) and target domain (BlockStack, right). red highlighted and dotted line (top right) indicates that relation isSuccessor is included in training the target domain autoencoder, but none of the other relations are. Both DC and NN retain a good performance while all other models show a decrease of accuracy in the target domain for one more of the relations not included in training. [Bottom] Impact of different values of 3 E {1,4, 8, 12} for each relation-decoder averaged across all relations in the source domain (left) and held-out relations EP L} in the target domain (right). It can be that DC is not impacted by changes in /8 and it maintains performance in the target domain. All other models show decrease of accuracy for the held-out relations in the target domain.
empirical mean and variance for the domain's embeedinbsd and sample from a corresponding Gaussian distribution; and extrapolation, when sample from regions strictly outside the smallest, axis-aligned, hyper-rectangle that encloses all data-points.
Figure 2 top provides celatio-ddecodr prediction accuracies in both the source (MNIST, left) and target (BlockStacks, right) domains P The relations are S, P. E, G, and relation transferred to the target domain. Key observations are that DC produces excellent PRT performance, whilst NN, NTN and HolE all see some degradation from their source-domain accuracies for relations other than isSuccessor (S). TransR maintains target-domain accuracies similar to its performance in the source domain, but this is significantly below the performance of other models in the source domain. We include the impact of adjusting 3 (disentanglement pressure) in Figure 2 bottom. Barring DC which has little discernible change in either source or target domains, PRT performance is significantly impacted by 3 for all models in the target domain, but has little effect in the source domain. TransR shows strong positive correlation between target domain accuracy and ,3 values, whereas the remaining models produce their best PRT performance with medium disentanglement pressure.
To investigate the broad trends that run across all | values and relation-decoder models, we ran a Spearman rank correlation analysis between consistency losses and PRT performance. Separate coefficients are produced for each combination of consistency loss: Con-A and Con-I further divided into Con-I-T (transitivityy) Con-I-A (asymmetry) and Con-I-R (reflexivity), and regions of latent space: data-embedding, interpolation and extrapolation (see Appendix H for the full table). Lower consistency losses are expected to produce higher PRT performance Spearman rank coefficient. The coefficients show that the consistency losses of data-embeddings in the source domain are weakly rank correlated with PRT. The consistency losses in the case of interpolation are, in most cases, strongly rank correlated with PRT. The consistency losses in the case of extrapolation lie in between and are generally moderately rank correlated with PRT. This supports our thesis that consistency can facilitate reliable transfer. Furthermore, consistency of certain partial theories may matter more. Here, Con-A, Con-I-T and Con-I-A interpolation are the most relevant partial theories for transfer learning performance. As we shall see, DC outperforms all other models on these losses and this result is mirrored by its PRT performance.
indicated by negative
To gain a deeper insight as to which underlying characteristics can explain the observed PRT accuracy profiles, Figure and Figure 4 present Con-A and Con-I loss profiles, respectively, for varied and regions of latent space (for data-embeddings in blue, interpolation in green and extrapolation in
We take dh inferences of 0.5 or above to signify true, and otherwise false. An alternative, left as future work, would be sample the space of values produce confidence measure.

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10y
Con-A for different regions of 2: referenced on source domain
Con-A for target domain data embeddings only
101
109
10-1
DC
NN
NTN
HOBE
TransR
DC
NN
NTN
HolE
TransR
Figure 3: Consistency-Acrosy (Con-A) losses (lower values are better) for the models (DC, NN, NTN, HolE, TransR) using the MNIST data set (source domain X)) [left] and BlockStacks (target domain }) [right]. The blue bars show the consistency loss of the data embeddings, with darker shades corresponding to models trained with higher 3 (disentanglement pressure). The green bars show the results for interpolation. The red bars show the results for extrapolation.
10a
Con-I for different regions of 2: referenced on source domalnn
Con-I for target domain data-embeddings only
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HoiE
TransR
Figure 4: Consistency-Indiie Individual (Con-l) losses (lower values are better) for the models (DC, NN, NTN, HolE, TransR) using the MNIST data set (source domain X) [left] and BlockStacks (target domain y) [right]. From top to bottom (following the same colour schematic as Figure 3 Con-I-T (Transitivity), Con-I-A (Asymmetry) and Con-I-R (Reflexivity).
red). Results refer to both source (left) and target domain embeddings (right), Firstly, we note that DC retains excellent Con-A for all regions of latent space. TransR retains consistency from data- embeddings to the interpolated regions, but not to the extrapolated regions. The remaining models show degradation of consistency between data-embeddings and interpolation and extrapolation regions, with extrapolation often being worse than interpolation. Looking at 3 trends, aside from DC, increasing 3 appears have positive but limited effect on interpolation and extrapolation performance. Considering the Con-A performance of data-embeddings in the target domain, DC shows the best performance. The Con-A performance in the target domain PRT accuracies. For all the models, Con-A performance in the target domain appears match the interpolation or extrapolation Con-A performance the source domain. This points to the possibility of anticipating transfer learning performance by evaluating the consistency of partial theories.
agreement with the
Many of the same trends can be seen the results for Con-I (Con-I-T, Con-I-A and Con-I-R) in Figure E Results are averaged over individual relations. As in Figure results are presented with respect to source domain (left) and target domain (right). We firstly observe that DC and NN share the best overall Con-I performance profiles, with TransR following closely. DC and TransR again show comparable data-embedding versus interpolation/extraplaaii performance, whereas NN, NTN and HolE suffer from degradation. With regards to 33' impact, DC is not affected by 3, while NN and NTN show a negative correlation between B and Con-I losses with comparable results for cach underlying partial theory.
Finally, Table M provides comparison between optimal coherences achieved for each relation- decoder model, as defined in Section g Results are partitioned according to each consistency type and aggregate value. DC clearly outperforms aggregate coherence, followed by TransR and NTN, with HolE performing generally worse. Looking at 3" profiles, we see that most models achieve optimum aggregate coherence at 3 8, apart from DC and HolE which perform better at 3 = 1. Overall, this is in broad agreement with the / profiles given by Figure bottom (right). However, we can see that 3 profiles for Con-A coherence are in more direct agreement TransR achieves its best at 3 = 12 and HolE 8. This suggests that
other models on coherence. NN achieves strong

---

Table I: Coherence comparison with respect source and target data-embeddings. Results reported with the corresponding 3 = 3 value (in parenthesis). The consistency loss abbreviations refer to: (A)eross, T(ransitivity), A(symmetry). R(eflexivity) and Aggn(egate), which gives the best obtained aggregate consistencies. DC outperforms all other approaches across most coherence scores.
Aggr.
(8*)
Con-A
(3*)
Con-I-T
(8*)
Con-I-A
(8")
Con-I-R
(3*)
HolE
12.12
(1)
6,61
(8)
4.30
(1)
0.52
(12)
0.08
(8)
NTN
4.11
(8)
1.92
(8)
1.50
(12)
0.22
(12)
0.09
(12)
TransR
2.51
(8)
1.02
(12)
0.71
(12)
0.18
(4)
0.55
(8)
NN
1.71
(8)
0.82
(8)
0.44
(12)
0.18
(4)
0.05
(4)
DC
0.53
(1)
0.29
(1)
0.11
(1)
0.04
(1)
0.09
(1)
Con-A is more indicative of PRT performance, which is to be expected since PRT relies inductive transfer across relations.
All in all, these results paint picture where source domain accuracy alone is not strong enough indicator of concept transfer. Instead, it may be possible to anticipate transfer performance by evaluating consistency in regions beyond the source domain's 's data-embeddings. Depending on the task at hand, certain partial theories may be more relevant than others in this analysis.
8 Related Work
Relational representations play a prominent role in Knowledge Graph Embedding (KGE), wherein sets of relodion are jointly learned obtain a semantic latent representation from data 3 Although KGE approaches typically do not use shared autoencoder done in this paper, 77 an autoencoding framework is adopted, where graph neural network is used as the encoder. However, kw did not work with visual data and the model was only applied to single data sets rather than transfer learning. Similarly, disentanglement is concerned with semantic representation learning 4.. and it has been explored using variety of methods including both Generative Adversarial Networks [0 and VAEs K
41 45
5
29
47
11 20
3
3
17
9 36
22 2
Disentangled
representations have been evaluated on their transferability
A bridge between these two VAEs, can be found in 19 S 77. fields, with relation-decoded employed in the semi-supervision
46
42
27
In 19), multiple relation-decoden are used, but to compute a triplet comparison based query. In g only a single binary relation is studied using functional forms that are not sufficient to model the full set of relations considered in this paper. Lastly, we note that our experimental setup is most remnant of domain adaptation, e.g. 15 To the best of our knowledge, this paper the first to present a comprchensive analysis of the resulting concept coherence. No previous work has compared relation-decoders on their ability to learn consistently and coherently, as measured in this paper.
Conclusion and Future Work
This paper introduced formal definitions of consistency and coherence for representation learning. As result, sub-symbolic model have its consistency and coherence measured with respect to logical theory. The paper specified a neuro-symbolic model based on domain-encoders coupled with modular relation-decoders, and an experimental procedure that, together, allowed for the investigation of how concept coherence differs for various implementations of relation-decoders applied to transfer learning. Finally, consistency and coherence results showed that the models that can retain consistency (i.e. be coherent) across regions of latent space beyond the source data-ember data-embeddings are more likely to perform better at PRT learning tasks. The empirical evaluations in this paper only considered binary relations and fixed signature which is learned "all at once" in source domain. In practical applications, however, it should be possible to discover concepts gradually, e.g. as part of a curriculum and through gradual refinement of pre-learned relations after exposure to different contexts. This necessitates an adaptation of the approach presented here and further evaluations, as part of future work. Further evaluations of the formalization introduced here should consider the use of different models, theories (such as specifying periodic, e.g. rotation, and unordered categorical, e.g. shape, properties) and scenarios/data sets in the evaluation of consistency and coherence of neural models.
10

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References
[1] Ralph Abboud, ismail Ilkan Ceylan, Thomas Lukasiewicz, and Tommaso Salvatori. Boxe: A box embedding model for knowledge base completion. In Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, 2020.
[2] Masataro Asai. Photo-Realistic Blocksworld Dataset. arXiv preprint arXiv: 1812.0181,.0 2018.
[3] Samy Badreddine, Artur d' Avila Garcez, Luciano Serafini, and Mike Spranger. Logic tensor networks. Artif. Intell., 303: 2022.
[4] Yoshua Bengio, Aaron Courville, and Pascal Vincent. Representation learning: A review and new perspectives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(8):1798- 1828, 2013.
[5] Antoine Bordes, Nicolas Usunier, Alberto Garcia-Duran, Jason Weston, and Oksana Yakhnenko. Translating Embeddings for Modeling Multi-relational Data. In Advances in Neural Information Processing Systems 26: 27th Annual Conference on Neural Information Processing Systems, pages 2787-2795. Curran Associates, Inc., Lake Tahoe, USA, 2013.
[6] Christopher P. Burgess, Irina Higgins, Arka Pal, Loic Matthey, Nick Watters, Guillaume Desjardins, and Alexander Lerchner. Understanding disentangling in 3-VAE. Advances in Neural Information Processing Systems 30, number Nips, Long Beach, CA, USA, 2017.
[7] Junxiang Chen and Kayhan Batmanghelich. Robust ordinal VAE: employing noisy pairwise comparisons for disentanglement. CoRR, abs/1910.05588,, 2019.
[8] Junxiang Chen and Kayhan Batmanghelich. Weakly Supervised Disentanglement by Pairwise Similarities. In Proceedings of the 32nd AAAI Conference Artificial Intelligence, AAAI, New York, NY, USA, 2020.
[9] Ricky T Q Chen, Xuechen Li, Roger B. Grosse, and David Duvenaud. Isolating Sources of Disentanglement in Variational Autoencoders. In Advances in Neural Information Processing Systems 31: Annual Conference Neural Information Processing Systems, pages -2625, Montreal, Quebec, Canada, 2018.
[10] Xi Chen, Yan Duan, Rein Houthooft, John Schulman, Ilya Sutskever, and Pieter Abbeel. Infogan: Interpretable representation learning by information maximizing generative adversarial nets. In Advances in Neural Information Processing Systems 29: Annual Conference Neural Information Processing Systems 2016, December 5-10, 2016, Barcelona, Spain, pages 2172- 2180, 2016
[11] Yuanfei Dai, Shiping Wang, Neal N Xiong, and Wenzhong Guo. A Survey on Knowledge Graph Embedding: Approaches, Applications and Benchmarks. Electronics, 955::122,, 2020.
[12] Artur d' Avila Garcez and Luis C. Lamb. Neurosymbolic Al: the 3rd wave. CoRR, abs/2012.05876. 2020.
[13] Ivan Donadello, Luciano Serafini, and Artur d' Avila Garcez, Logic Tensor Networks for Se- mantic Image Interpretation. In Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, pages -1662,160, 2017.
[14] Cian Eastwood and Christopher I Williams. framework for the quantitative evaluation of disentangled representations. In 6th International Conference on Learning Representations, Vancouver, BC, Canada, 2018.
[15] Victor Gutiérrez-Basulto Basulto and Steven Schockaert. From knowledge graph embedding to ontology embedding? an analysis of the compatibility between vector space representations and rules. In Principles of Knowledge Representation and Reasoning: Proceedings of the Sixleenth International Conference, 2018, Tempe, Arizona, 30 October November 2018, pages 379-388. AAAI Press, 2018
11

---

[16] Irina Higgins, David Amos, David Pfau, Sébastien Racanière, Loic Matthey, Danilo J. Rezende, and Alexander Lerchner. Towards a definition of disentangled representations. CoRR, abs/1812.0224h 2018.
[17] Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner. beta- VAE: Learning Basic Visual Concepts with a Constrained Variational Framework. In 5th International Conference Learning Represenia- tions, Toulon, France, 2017.
[18] B. Inhelder and J. Piaget. The early growth of logic in the child: classification and seriation. Routledge and Kegan Paul, London, 1964.
[19] Theofanis Karaletsos, Serge Belongie, and Gunnar Rätsch. When crowds hold privileges: Bayesian unsupervised representation learning with oracle constraints. In 4th International Conference on Learning Representations, pages 1-16, San Juan, Puerto Rico, 2016.
[20] Seyed Mehran Kazemi and David Poole. Simple embedding for link prediction in knowledge graphs. Advances in Neural Information Processing Systems, (egB-eccebbr((iisss225555 2018.
[21] Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. In 2nd Interna- tional Conference on Learning Representations, 2014, Banff, AB, Canada, April 14-16, 2014, Conference Track Proceedings, 2014.
[22] Abhishek Kumar, Prasanna Sattigeri, and Avinash Balakrishnan. Variational inference of disentangled latent concepts from unlabeled observations. In 6th International Conference Learning Representations, Vancouver, BC, Canada, 2018.
[23] Brenden M. Lake, Tomer D. Ullman, Joshua B. Tenenbaum, and Samuel J. Gershman. Building Machines That Learn and Think Like People. Behavioral and Brain Sciences, 40, 2017.
[24] Yann LeCun and Corinna Cortes. MNIST handwritten digit database. 2010.
[25] Yankai Lin, Zhiyuan Liu, Maosong Sun, Yang Liu, and Xuan Zhu. Learning entity and relation embeddings for knowledge graph completion. In Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence, January 25-30, 2015, Austin, Texas, USA, pages 2181-2187. AAAI Press, 2015.
[26] Francesco Locatello, Stefan Bauer, Mario Lucic, Gunnar Raetsch, Sylvain Gelly, Bernhard Schoelkopf, and Olivier Bachem. Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations. In Proceedings of the 36th International Conference on Machine Learning: pages 4114 4124, Long Beach, California, USA, 2019.
[27] Francesco Locatello, Ben Poole, Gunnar Rätsch, Bernhard Schölkopf, Olivier Bachem, and Michael Tschannen. Weakly-Supervise Disentanglement Without Compronises. CoRR, abs/2002.0, 2020.
[28] Vinod Nair and Geoffrey E. Hinton. Rectified linear units improve restricted boltzmann machines. In Proceedings of the 27th International Conference Machine Learning ICML, pages 807-814, 2010.
[29] Maximilian Nickel, Kevin Murphy, Volker Tresp. and Evgeniy Gabrilovich. A review of relational machine learning for knowledge graphs. Proc. IEEE, 104(1):11-3,, 2016.
[30] Maximilian Nickel, Lorenzo Rosasco, and Tomaso A. Poggio. Holographic embeddings of knowledge graphs. In Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, February 12-17, 2016, Phoenix, Arizona, USA, pages 1955-1961. AAAI Press, 2016
[31] Maxwell Nye, Michael Henry Tessler, Joshua B. Tenenbaum, and Brenden M. Lake. Improv- ing coherence and consistency in neural sequence models with dual-system, neuro-symbolic reasoning. In Proc. NeurIPS 2021, 0ass/2077.02799 2021.
[32] Jeffrey Paris and Alena Vencovská. Pure Inductive Logic. Perspectives Logic. Cambridge University Press, 2015.
12

---

[33] Jeffrey B. Paris. The Uncertain Reasoner's Companion: A Mathematiiaal Perspective. Cam- bridge University Press, 1994.
[34] Jean Piaget. The Psychology of Intelligence. Routledge and Kegan Paul, 2005.
[35] levgen Redko, Amaury Habrard, Emilie Morvant, Marc Sebban, and Younès Bennani. Advances in Domain Adaptation Theory. Elsevier, 2019.
[36] Karl Ridgeway and Michael C Mozer. Learning Deep Disentangled Embeddings With the F-Statistic Loss. In Advances in Newral Information Processing Systems 31: Annual Conference on Neural Information Processing Systems, pages 185-194, Montreal, Quebec, Canada, 2018.
[37] Michael Sejr Schlichtkrull, Thomas N. Kipf, Peter Bloem, Rianne van den Berg, Ivan Titov, and Max Welling. Modeling relational data with graph convolutional networks. 10843:593-607, 2018.
[38] Luciano Serafini and Artur Avila Garcez. Logic Tensor Networks: Deep learning and logical reasoning from data and knowledge. ArXiv, (bs/1600.04422) 2016.
[39] Stewart Shapiro and Teresa Kouri Kissel. Classical Logic. In The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Spring 2021 edition, 2021.
[40] Ravid Shwartz-Ziv and Naftali Tishby. Opening the black box of deep neural networks via information. CoRR, abs/ 100.000810, 2017.
[41] Richard Socher, Danqi Chen, Christopher Manning, Danqi Chen, and Andrew Ng. Reason- ing With Neural Tensor Networks for Knowledge Base Completion. In Advances in Neural Information Processing Systems 26: 27th Annual Conference on Neural Information Processing Systems, pages 926 934, 2013.
[42] Xander Steenbrugge, Sam Leroux, Tim Verbelen, and Bart Dhoedt. Improving Generalization for Abstract Reasoning Tasks Using Disentangled Feature Representations. In Neural Informa- tion Processing Systems (NeurIPS) Workshop on Relational Representation Learning, Montreal, Canada, 2018.
[43] Harald Strömfelt, Luke Dickens, Artur d' Avila Garcez, and Alessandra Russo. Coherent and consistent relational transfer learning with auto-encoders. In Artur d' Avila Garcez and Ernesto Jiménez-Raus Ruiz, editors, Proceedings of the 15th International Workshop on Neural-Symbolk Learning and Reasoning as part of the Ist International Joint Conference on Learning & Reasoning (IJCLR 2021), Virtual conference, October 25-27, 2021, volume 2986 of CEUR Workshop Proceedings, pages 176-192. LEUR-WS.org. 2021.
[44] Théo Trouillon, Éric Gaussier, Christopher R. Dance, and Guillaume Bouchard. On inductive abilities latent factor models for relational learning. Journal of Artificial Intelligence Research, 64:21-53, 2019.
[45] Théo Trouillon, Johannes Welbi, Sehastian Riedel, Éric Gaussier, and Guillaume Bouchard. Complex Embeddings for Simple Link Prediction. In Proceedings of the 33nd International Conference Machine Learning, pages 2071 2071-2080, New York, NY, USA, 2016.
[46] Sjoerd Steenkiste, Francesco Locatello, Jürgen Schmidhuber, and Olivier Bachem. Are Disentangled Representations Helpful for Abstract Visual Reasoning? In Advances in Neural Information Processing Systems 32: Annual Conference Neural Information Processing Systems, pages 14222 14235, Vancouver, BC, Canada, 2019.
[47] Quan Wang, Zhendong Mao, Bin Wang, and Li Guo. Knowledge graph embedding: A survey of approaches and applications. IEEE Transactions on Knowledge and Data Engineering. 29(12))2224-27444 2017.
13


Reviews:
[{"review_id": "oV1rtQBQfL", "reviewer": "Reviewer_jjf1", "summary": "This paper is motivated by the formal concepts of coherence and consistency in neuro-symbolic AI, and the fact that such concepts have not been formalized in the connectionist paradigm. It claims that these concepts are important for assessing transfer learning performance. This is demonstrated by formalization of these concepts for neural networks, proposal of the \"relation-decoder\" model, and transfer learning experiments to show that good transfer performance is associated with coherence being maintained across domains and can be predicted by ability to maintain consistency over unseen inputs. ", "questions": "- In the space of transfer learning architectures, and NN architectures in general, what all falls under this formalization paradigm?", "limitations": "The paper is upfront about its technical limitations. Based on the last couple sentences of sec7, it is likely that the authors and I disagree on the importance of having other architectures considered in this paper, but the acknowledgment is valuable nonetheless. \n\nThe discussion of social limitations leaves something to be desired, but I commend the authors on engaging with it at all when a lot of core theoretical papers get away with a generic sentence. The mention of logic, biases, and disentanglement are interesting and merit development; the claim about trustworthiness of coherent concepts is also intriguing and would benefit from examples and some investigation. \n\nTo go deeper, there are theoretical and empirical STS-type works that would engage further with the ideas of logical validity and absolutism as well. However, that would be getting beyond the scope of this type of statement. Ultimately these are **not** knocks on the paper - I only have words for this section because the authors chose to say something (anything) more than \"N/A\". ", "rating": 7, "confidence": 2, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "*Originality*: to my knowledge, this paper is original. Beyond the labels, I'm not familiar with these specific formulations being used as metrics for neural networks. This paper spends time formally defining predicate logic concepts that may not need defining (at least not in the main paper) as they are very fundamental, but the soft structure formalization is technically new to me. \n\n*Quality*: this paper takes a well-scoped problem and lays out a clear strategy to show it. The goals seem to be to present formal definitions of consistency and coherence that are meaningful in the connectionist paradigm, create a transfer learning framework from which these metrics can be calculated, and show that they are useful indicators of transfer learning ability - presumably, that they predict standard performance metrics while also providing more insights. I think this is done successfully - the setup and experiments are very much called for by the aims. \n\nThe only area for improvement I would bring up is usefulness beyond the relation-decoder framework - if the metrics only work for not only a narrow set of architectures but specifically the architectures in this paper, then it seems the paper should be more about the architecture than the metrics (though this would require different experiment analysis and to some degree different experiments). \n\n*Clarity*: This paper is quite well-written. Unlike most papers, the paper roadmap is clear early on in reading. I felt I had a pretty good mental model for the contributions after the intro, and could consider the rest of the paper from that perspective.\n\nSome areas for improvement include a less dense and better-illustrated formalization section - this may be more about the paper than the research, but a lot of space is devoted to formalization steps prior to the structure and soft-structure definitions that may not be necessary for someone familiar with basic predicate logic, which is likely most of the audience. Some of the more composite concepts (signature, interpretation) would benefit from examples and intuitive explanation, since they seem to be more familiar and easy to grasp than they are presented as being. \n\n*Significance*: This paper presents novel (to my knowledge) metrics for assessing transfer performance. As machine learning moves toward scale and ubiquity, transfer performance and robust evaluation are both important areas in my opinion, so these are posed to be valuable contributions. My primary concerns about significance are related to my comments in quality - it's unclear to me how far these metrics can apply. While other architectures may fall under the same framework mathematically, I think the burden of assessing scope is on the paper since the message is ultimately about the value of the metrics. ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "KLK4EibLH13", "reviewer": "Reviewer_kLPG", "summary": "This paper addresses the problem of how logical predicates (e.g. isGreater, isSuccessor, etc.) can be learned from real-world data in a manner that transfers from one domain (e.g. images of handwritten digits) to another (e.g. images of stacked blocks). Focusing on ordinal relations, the paper first proposes a formal theory that relates first-order logic to 'soft structures' which can be learned directly from real data. This theory is then leveraged to derive formal definitions of 'consistency' (consistency of learned logical concepts within a domain) and 'coherence' (consistency of learned concepts between domains). Finally, a neural network architecture and training procedure are proposed. A particular variant of the architecture, with a novel comparator function, performs well on the transfer evaluation, and performance on this transfer evaluation is generally shown to align with the proposed definitions of consistency and coherence.", "questions": "My questions are outlined in the 'strengths & weaknesses' section above.", "limitations": "I do not envision any potential negative societal impact from this work.", "rating": 5, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "I have reviewed this paper before, and after reading through the new manuscript in detail, it appears that the authors have not made any changes to address the concerns that I raised in my previous review. Therefore, I am including that review below. I look forward to hearing the author's responses:\n\nI am not familiar with any other work that attempts to build a rigorous theory connecting first-order logic with concepts learned from real-world data, so this aspect of the work was highly novel. The proposed neural architecture is well-motivated on the basis of the preceding theoretical considerations, but in the end is not particularly novel -- it amounts to an autoencoder paired with supervised training of relational concepts. The proposed comparator function is novel, but I worry that it is over-engineered for ordinal relations studied in this work, and the generic neural network relation decoder performs surprisingly well by comparison.\n\nOverall, the novelty and significance are primarily in terms of the theoretical foundation sketched in sections 2-4. I found these sections deeply stimulating, but in the end I have significant worries about the practical relevance of the overall paradigm. The general worry is that the approach seems well-suited to the ordinal concepts studied in this work, where the formal theory governing these concepts can be neatly defined. It's unclear how the proposed approach will scale to more complex, fuzzy, real-world concepts (such as those that figure in natural language), where the specification of such a formal theory is likely impossible.\n\nThe authors may argue that their approach doesn't actually require a formal theory to be specified for the concepts in question. Strictly speaking, all that is required for training the proposed neural network model is to have training examples corresponding to positive and negative examples of these concepts that can be used for supervised learning. This doesn't necessarily require a formal theory of how these concepts relate, and therefore is perhaps more easily scalable to more complex concepts whose meaning can't be neatly defined. But this then leads me to wonder, what is the purpose of connecting this approach to a theory in terms of first-order logic? Beyond being an interesting intellectual exercise (which it certainly is), does this suggest any particular directions for learning real-world concepts in a manner that transfers better between domains? On its face, the proposed neural network architecture and training procedure constitute a rather straightforward way of learning concepts through supervised learning, and it's not clear that their development depends in any way on the accompanying theory.\n\nAnother concern is that it's not even clear to me whether concepts (in the sense of logical predicates) are needed at all, and it isn't clear what we'll do with these concepts once we have a neural system that can obtain them from real-world data. It would be helpful if the authors could sketch out how they envision this approach fitting into the broader program of AI research, and in particular how these learned concepts can be useful for downstream functionality, e.g. applied problems in computer vision or NLP.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "--9nMNXFp4", "reviewer": "Reviewer_pBe5", "summary": "The paper formalizes coherence and consistency measures in ANN models that consist of an encoder ($\\psi$) and relation ($\\phi$) networks. To demonstrate the usefulness of these measures, they train autoencoders to additionally be coherent and consistent across two domains with respect to a single relation that was learned in only the source domain. They show that more consistent models, which implement a proposed DC model as a relation model $\\phi$, are also better in zero-shot transfer to unlearned relations in the target domain. Thus, the paper demonstrates that coherence and consistency, as they formalize it, are useful for evaluating relation models.", "questions": "Request: the supplementary materials, and relating the work to Relation Networks.", "limitations": "The paper addresses its limitations well.", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 4, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n1) Clear writing.\n\n2) The DC relation-decoder seems promising for other applications.\n\n3) The formal structure that was used to define coherence and consistency is interesting because it formally describes fundamental concepts.\n\n4) An interesting main result, of consistency leading to better zero-shot transfer to other relations.\n\n\nWeaknesses:\n1) Missing supplementary materials don’t allow for a full paper evaluation.\n\n2) The loss of eq. (9) is very similar to the established Relation Network (RN) loss [1, 2]. The paper ignores this similarity. Recognizing this similarity would have enabled comparison with results that were obtained using RN and its derivatives.\n\n3) Following the previous point, I think that the formalism presented in the paper might not be justified. The paper could have been rewritten using only common machine learning terms such as the RN loss values and generalization errors.\n\n\n\n[1] - Sung, F., Yang, Y., Zhang, L., Xiang, T., Torr, P. H., & Hospedales, T. M. (2018). Learning to compare: Relation network for few-shot learning. In Proceedings of the IEEE conference on computer vision and pattern recognition (pp. 1199-1208).\n\n‏\n[2] - https://en.wikipedia.org/wiki/Relation_network\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept