Extrapolatable Relational Reasoning With Comparators in Low-Dimensional Manifolds
Authors: Duo Wang
Mateja Jamnik
Pietro Liò
Venue:  (2021)
Keywords: Visual Reasoning
Relational Reasoning
Generalisation

Abstract: While modern deep neural architectures generalise well when test data is sampled from the same distribution as training data, they fail badly for cases when the test data distribution differs from the training distribution even along a few dimensions. This lack of out-of-distribution generalisation is increasingly manifested when the tasks become more abstract and complex, such as in relational reasoning. In this paper we propose a neuroscience-inspired inductive-biased module that can be readily amalgamated with current neural network architectures to improve out-of-distribution (o.o.d) generalisation performance on relational reasoning tasks. This module learns to project high-dimensional object representations to low-dimensional manifolds for more efficient and generalisable relational comparisons. We show that neural nets with this inductive bias achieve considerably better o.o.d generalisation performance for a range of relational reasoning tasks. We finally analyse the proposed inductive bias module to understand the importance of lower dimension projection, and propose an augmentation to the algorithmic alignment theory to better measure algorithmic alignment with generalisation.

Under review as a conference paper at ICLR 2021
EXTRAPOLATABLE RELATIONAL REASONING WITH COMPARATORS IN LOW-DIMENSIONAL MANIFOLDS
Anonymous authors Paper under double-bling review
ABSTRACT
While modern deep neural architectures generalise well when test data is sampled from the same distribution as training data, they fail badly for cases when the test data distribution differs from the training distribution even along few dimensions. This lack of out-of-distribution generalisation is increasingly manifested when the tasks become more abstract and complex, such in relational reasoning, In this paper we propose a ceurrosceecc-inspiree inductively biased module that can be readily amalgamated with current neural network architectures to improve out- of-distribution (o.o.d) generalisation performance on relational reasoning tasks. This module learns project high-dimensional object representations to low- dimensional manifolds for more efficient and generalisable relational comparisons. We show that neural nets with this inductive bias achieve considerably better o.o.d generalisation performance for a range of relational reasoning tasks, thus more closely models human ability to generalise even when no previous examples from that domain exist. Finally, we analyse the proposed inductive bias module to understand the importance of lower dimensional projection, and propose an augmentation to the algorithmic alignment theory to better measure algorithmic alignment with generalisation.
INTRODUCTION
The goal of Artificial Intelligence research, first proposed in the 1950s and reiterated many times, is to create machine intelligence comparable that of human being. While today's 's deep learning based systems achieve human-comparable performances in specific tasks such as object classification and natural language understanding, they often fail to generalise in out-of-distribution (o.o.d) sce- narios, where the test data distribution differs from the training data distribution (Recht et al., 2019; Trask et al., 2018; Barrett et al., 2018; Belinkov & Bisk, 2018). Moreover, it is observed that the generalisation error increases the tasks become more abstract and require more reasoning than perception. This ranges from small drops (3% to 15%) in classification accuracy on ImageNet (Recht al., 2019) to accuracy only slightly better than random chance for the Raven Progressive Matrices (RPM) test (a popular Human IQ test), when testing data are sampled completely out of the training distribution (Barrett et al., 2018).
In contrast, human brain is observed to generalise better to unseen inputs (Geirhos et al., 2018), 8), and typically requires only a small number of training samples. For example, human, when trained to recognise that there is a progression relation of circle sizes in Figure 1a, can easily recognise that the same progression relation exists for larger circles in Figure 1b, even though such size comparison has not been done between larger circles. However, today's 's state-of-the art neural networks (Barrett al., 2018; Wang et al., 2020) are not able to achieve the same. Researchers (Spelke & Kinzler, 2007; Chollet, 2019; Battaglia et al., 2018; Xu et al., 2020) argue that the human brain developed special inductive biases that adapt to the form of information processing needed for humans, thereby improving generalisation. Examples include convolution-lik: cells in the visual cortex (Hubel & Wiesel, 1959; Güçlü van Gerven, 2015) for visual information processing, and grid cells (Hafting al., 2005) for spatial information processing and relational comparison between objects (Battaglia et al., 2018).
In this work, we propose simple yet effective inductive bias which improves o.o.d generalisation for relational reasoning. We specifically focus on a the type of o.o.d called 'extrapolation'. For extrapola- tion tasks, the range of one more data attributes (e.g., object size) from training and test datasets

---

Under review as a conference paper at ICLR 2021
Small
Big
Low- cow-Dim
Low-Din Projection
Projection
(a) Small Circles
(b) Big Circles
Figure 2: Illustration of projecting object represen- tations onto a I-dimensional manifold in which size comparison can be achieved by simply measuring the difference between two projections.
Size Progression Relations for circles of different sizes.
Figure 1:
completely non-overlapping. The proposed inductive bias is inspired by neuroscience and psychology research (Fitzgerald et al., 2013; Chafee, 2013; Summerfield et al., 2020) showing that in primate brain there are neurons in the Parietal Cortex which only responds to different specific attributes of perceived entities. For examples, certain LIP neurons fire at higher rate for larger objects, while the firing rate of other neurons correlates with the horizontal position of objects the scene (left vs right) (Gong & Liu, 2019). These observations show that these neurons map objects' attributes such as size and spatial position the one-dimensional axis of the neurons' firing rate. From a computational perspective, this can be viewed as projecting object representations to different now-dimensional manifolds corresponding
different attributes. Based on these observations (Summerfield al., 2020), we hypothesise that these neurons are developed to learn low-dimensional representations of relational structure that are optimised for abstraction and generalisation, and the same inductive bias can be readily adapted for artificial neural networks to achieve similar optimisation for abstraction and generalisation.
We test this hypothesis by designing inductive bias module which projects high- nigh-dimensional object representations into low-dimensional manifolds, and make comparisons between different objects in these manifolds. We show that this module can be readily amalgamated with existing architectures to improve extrapolation performance for different relational reasoning tasks. Specifically, we performed experiments three different extrapolation tasks, including maximum of a set, visual object compar- ison on dSprites dataset (Higgins et al., 2017) and extrapolation on RPM style tasks (Barrett et al., 2018). We show that models with the proposed low-dimensional comparators perform considerably better than baseline models on all three tasks. In order to understand the effectiveness of comparing in -iw-dimesiional manifolds, of the comparator to show the importance of projection to low-dimensional manifolds in improving generalisation. Finally, we perform the analysis relating to algorithmic alignment theory (Xu et al., 2020), and propose an augmentation to the sample complexity criteria used by this theory to better measure algorithmic alignment with generalisation.
analyse the projection space and corresponding function space
2 METHOD
Here, we describe the inductive bias module we developed to test our hypothesis that the same inductive bias low-dimensional representation observed Parietal Cortex can readily adapted for artificial neural network, thus enabling it to achieve similar optimisation for abstraction and generalisation. The proposed module learns to project object representations into low-dimensional manifolds and make comparisons in these manifolds. In Section 2.1 we describe the module in detail. In Section 2.2, 2.3 and 2.4 we discuss how this module can be utilised for three different relational reasoning tasks, which are: finding the maximum of set, visual object comparisons and Raven Progressive Matrics (RPM) reasoning.
2.1 COMPARATOR IN LOW-DIMENSION
MANIFOLDS
The inductive bias module is comprised of low-dim projection functions p and comparators c. Let {o,: i E 1 1...N} N} be the set of object representations, obtained by extracting features from raw inputs such as applying Convolutional Neural Networks (CNN) on images. Pairwise comparison between object pair o; and 0, can be achieved with a function f expressed as:
f(o,, 0,) = o( ll (e
(I)
k=
2

---

Under review as a conference paper at ICLR 2021
Here Pk is the kt projection function that projects represstattnnnntiino into the kth low dimensional manifold, Cş is the 1t comparator function that compares the projected representations, || is the concatenation symbol and is function that combines the K comparison results to make a prediction. Having K parallel projection functions P& and comparators cj enables a simultaneous comparison between objects with respect to their different attributes. Figure 2 shows an example of comparing sizes of circles by projection onto a 1-dimensional manifold. Both p and can be implemented as feed forward neural networks. While the comparator c, implemented as neural network, can theoretically learn rich range of comparison metrics, we found that adding to e an additional inductive bias of distance measure for the projection, such as vector distance p(o,) p(o,) or absolute distance [p(o;) - p(o))], improves the generalisation performance.
Let re(o)) be the ground truth mapping function from 7th object's 's representation o, to its 8dh attributes (such as colour and size for visual object). If such ground truth labels of object attributes exist, f(o,, oj) can be trained to directly predict the differences in attributes by minimising the loss where d a distance function (e.g., r'z r2(o;)) rr(0,) for continuous attributes or (o,()))=,((;) for categorical attributes). However, in real-world datasets, such ground truth attribute labels seldom exist. Instead, in many relational reasoning tasks, learning signals for attribute comparison are only provided implicitly in the training objective. For example, in Visual Question Answering task, an example question might be 'Is the object behind Object A smaller?'. The learning signals for the required size and spatial position comparator are provided only through correctness of the answers to the given questions. Thus, the proposed module is only useful and scalable if it can be integrated into neural architectures for relational reasoning and still learn to compare attributes with the weaker, implicit learning signal. Next, we describe 3 examples of such integrations for different relational reasoning tasks, and show in Section that the proposed module can learn relational reasoning tasks with better generalisation capability.
L(d(ri(oi)) ((),
2.2 ARCHITECTURE: MAXIMUM OF A SET
The first task consider is finding the maximum of a set of real numbers. Formally, given set (ır; i E 1....N N} where v; is a real number represented as a scalar value, we want to train a function hmaz (.
}} that gives the maximum value in the set. Many neural architectures have been applied on this task, including Deep Sets (Zaheer et al., 2017) and Set Transformer (Lee et al., 2019), but none of them test (o.o.d) generalisation capability. In order to test o.o.d generalisation in the extrapolation scenario, we create the training and test sets such that their ranges do not overlap. We sample from the range (Vow.. Vitrúi) ) for the training set, and from the range (Vieet Viteat) for high
high
the test set, and restrict that hhihh < Vieet.
We integrate the proposed low-dim comparator module with Set Transformer (Lee et al., 2019), state-of-the-art neural architecture for sets. Set Transformer first encodes each element in the set with respect all other elements with a Multihead Attention Block (MAB), an attention module modified from self-attention used in language tasks (Vaswani et al., 2017): e; = encode(z;) = MAB (4,, 2z).. The Set Transformer then uses Pooling with Multihead Attention (PMA) to combine all encoded eN). While MAB uses query and key embeddings to generate
elements of the set as PMA(e1.... attention variables, which are then used as weights in the weight sum of value embeddings of elements, we swap the query-key attention mechanism with our low-dim comparator as:
-MLP ((I,,)))
Here is the low-dim comparator and MTP is standard Multi-Layer Perceptron. Note that we directly use the scalar input 2; object representation in Equation (1) as no feature extraction is needed. We then use attention-based pooling to combine projection of x; Σn a(e;) (x(x), where a outputs attention values while p is the 1-dim projection function. For detailed architecture configuration, please refer to Appendix A.
2.3 ARCHITECTURE: VISUAL OBJECT COMPARISON
The second task we consider is comparing visual objects for different attributes such as size and spatial position. For this task, two images 21 and ra containing single objects of randomly sampled attributes are given, and one is asked a specific attribute of the second object is larger than, equal to, or smaller than the attribute of the first object. Figure 3a shows an example of this task for comparing
3

---

Under review as a conference paper at ICLR 2021
Columns/Rows
Smaller? Same?
Larger?
CNN
P
Comparator
P
Ro Ans
P
C
CNN
P
C
CNN
CNN
From other Rows/Columns Comparisons
CNN
P
From other Rows/Columns
(a) dSprites
(b) RPM
Figure 3: Figure (a) illustrates the architecture for comparing sizes of objects sampled from dSprites dataset. "P" the projection function. Figure (b) illustrates the architecture for logical reasoning on RPM-style tasks. "P" is the projection function and "C" is the comparator (g in Equation 1).
sizes between two heart-shaped objects. To sample images of objects in the implementation, used the dSprites dataset (Higgins et al., 2017), a widely used dataset for studying latent space disentangling. To extrapolation capability, sample the training set and the test such that for the compared attribute, the training attribute range has no overlap with the test attributes. We leave details of the dataset construction Appendix
Figure 3a shows an overview of the architecture integrated with the proposed low-dimensional comparator. The image pair 21 and 22 is first passed through a CNN extract feature embeddings and The feature embeddings are then projected to low-dim manifold and compared as c(p(e1). p(e))) where p is the projector and e the comparator. The comparator has 3 output units with softmax predict probabilities that attribute of the second object a(x2) is smaller than, equal to, or larger than the attribute of the first object a(x1). )) The architecture is trained with cross entropy loss with respect to ground truth labels. While of relational reasoning. it is reasonable to expect that the visual perception module is exposed to all possible scenarios of the input distribution in unsupervised way. The same assumption also holds for humans, whose vision system has to be exposed to the world inputs sufficiently after birth before they can associate objects with semantic meaning and perform relational reasoning (Maurer, 2016). Thus, we initialise the CNN with the pretrained encoder weights of Beta- VAE (Higgins et al., 2017), a disentangled VAE model trained in the unsupervised setup on dSprites dataset. For detailed configuration of the architecture please refer to Appendix C.
testing the o.o.d generalisation
2.4 ARCHITECTURE: VISUAL REASONING FOR RAVEN PROGRESSIVE MATRICES
The third task we consider more complex visual reasoning task named 'Raven Progressive Matrices' (RPM), which is popular human IQ test. In this task one is given 8 context diagrams with logic relations present them, and asked to pick an answer that best fits with the context diagrams. In our experiment we use the PGM dataset (Barrett et al., 2018), 8), the largest RPM-style dataset available. In the PGM dataset, there special data split called 'extrapolation' which designed to test for the extrapolation scenario of o.o.d generalisation.
In this split, colour and size values of objects in the training set are sampled from the lower half of the range, while the same attributes in the test set sampled from the upper half of the range. Thus the attribute ranges of training and test sets are non-overlapping. For details and examples of the PGM dataset, please refer to Appendix D and Barrett et al. (2018).
Our architecture integrates the low-dim comparator with Multi-Layer Relation Network (Jahrens Martinetz, 2020). Figure 3b shows overview of the architecture developed for PGM tasks. We use a 2-layer relation network with the first layer encoding pairs of diagrams within a row/column and the second layer encoding pairs of rows/columns. Applying such prior knowledge that rules only exist in
4

---

Under review as a conference paper at ICLR 2021
rows and columns has been standard practice in sfate-of-the-arr methods for RPM reasoning (Wang et al., 2020; Zhang et al., 2019). Following Wang et al. (2020) we fill each of the 8 candidate answers to the third row and column to obtain in total of 16 answer rows and columns. At each layer of the relation network, we use the owwdimennsional comparator instead of the MIP in the original relation network (Santoro al., 2017) for diagram comparison. Diagram 3; is first passed through CNN to produce the embedding og. embedding pairs (o;, 0z) are then compared as eij = f(o, 0z), where is the now-dimensional comparator described in Equation (1). The comparison results from the same rows/columns are then concatenated to form row/column embedding Tijk = Cij] | lezk. The row/column embeddings are compared with the second layer comparator. The comparison results are then concatenated and input into a reasoning network Rg to predict the correct answer. Similar to the dSprites comparison task, as discussed Section 2.3, we pre-train CNN as encoders of VAE, a technique that has also been previously explored for the PGM dataset (Steenbrugge et al., 2018). For detailed configuration of the architecture, please refer to Appendix E.
2.5 ALGORITHMIC ALIGNMENT AND O.O.D GENERALISATION
Xu al. (2020) proposed measure algorithmic alignment of neural networks specific task with sample complexity Ca(g, e, 5), which is the minimum sample size M so that g. the ground truth label mapping function, is (M, c. 53-learnable with a learning algorithm A. This essentially says that a model is more algorithmically aligned with a task if it can learn the task more easily with fewer samples. However, in the original definition, both training and test data are independently and identically distributed (i.i.)) samples drawn from the same data distribution. Thus, the algorithmic alignment theory measures how well can a NN fit to a particular data distribution, but does not measure how well can NN model perform in the 0.0.d scenario. For example, for the visual object comparison task, an averpparameterised MLP can learn the following two algorithms with low complexity: (1) m(hash(o;)) hash(o;)) ) where hash is hashing function and m is memory read/write function based on the hash index; and (2) (((())).p(o)) which is our proposed comparator function. While both algorithms can fit well for the training data, the first algorithm clearly does not o.o.d generalise as the memory function does not store unseen samples. In Section 3.5 we confirm experimentally that algorithmic alignment not indicative of o.o.d generalisation.
Intuitively, a more algorithmically aligned model should generalise better as it captures better the underlying algorithm of label generation. Here we propose an augmentation to sample complexity metric (Definition 3.3 in Xu et al. (2020)) in order to measure for algorithmic alignment with generalisation (specifically extrapolation).
Definition 2.1. o.o.d metric. Fix an error parameter c>> 0 and failure probability 8 (0, 1). Suppose {at, yf J are i.i.d samples from distribution D B, u), where D is the full data distribution, T is a truncating function, / E (0, 1) is the truncation ratio, and is the set of dimensions for truncation (See Appendix K for a detailed discussion about truncation function). Let g be the underlying data function y/+ 92;;), and f = A({x, yo ) 12))) be the function learnt with the learning algorithm A. Let V be the value ranges of all dimensions of the data distribution D. Then g is (M, e, 5, V, u) learnable with A if:
P Pe~Dllf((>) g(=)] < <d/1-1
(3)
The sample complexity the minimum M for g to be (M, e, V, u) learnable with A. In Section 3.5 we show experimentally that this metric measures better NN's ability to generalise.
3
EVALUATION
3.1
MAXIMUM OF SET
For the task of finding the maximum number in a set, we randomly sample number sets of cardinality ranging from 2 to 20 for training, and 2 to 40 for testing. For number sets for training, we uniformly sample numbers in the real value range [0, 100). For testing we sample numbers in the range [100, 200). In this way we test if the model can generalise both, for sets of larger cardinality and for numbers sampled from an unseen range. We sampled 10000 sets for training and 2000 sets for testing. For hyper-parameters of this and subsequent experiments please refer to Appendix F. Table I shows the test error of our model compared to Deep Sets (Zaheer et al., 2017) and Set Transformer (Lee et al., 2019), two previous -fae--of-the-ar architectures for sets. Our model achieves much lower extrapolation error than other methods, even lower than Deep Sets with a built-in Max-Pooline function.
5

---

Under review as a conference paper at ICLR 2021
Table 1:
Extrapolation test error for learning to find the maximum number in of numbers (mean + std for 10 runs). M.S.E means Mean Squared Error.
Deep Mean)
Deep Sets(Max)
Set Transformer
OURS
Model
Zaheer et al. (2017)
Zaheer al. (2017)
Lee al. (2019)
M.S.E
73.22 17.11
0.51 + 0.29
1.62 = 0.76 0.0015 0.0008
Table 2: Extrapolation test accuracies of baseline and our proposed model for dSprites attribute comparison task (10 runs). X-Coord is horizontal position.
Attributes
Size
X-Coord
Colour
Model
Baseline
79.52 + 6.71%
66. 16.14 + 5.03%
78.45 3.34%
OURS
05.05 + 3.03%
79.11 = 1.92%
97.71 = 2.81%
Table 3: Extrapolation test accuracies for the extrapolation split of PGM dataset.
WReN
MXGNet
MLRN
MLRN-P OURS
Model
Jahrens & Mar-
Barrett et al. (2018)
Wang et al. (2020)
tinetz (2020)
Accuracy
17.2%
18.9%
14.9%
18.1%
25.9%
3.2
VISUAL. OBJECT COMPARISON
For visual object comparison task, set three sub-tasks for comparing different attributes of the object, including size, horizontal position and colour intensity. For each task we sample visual objects with different range for the compared attributes from the dSprites dataset (Higgins et al., 2017). Given the compared attribute range (Vow, Vhigh]. we sample training data from range [Vow 2
3 Virigh) and
test data from range [ Vnigh, Vhigh]. As ground truth attribute value is provided in the dSprites dataset, we can build the comparison label as (Le) (eer<ez fai=e >az>))). For all experiments we sample 60000 training pairs and 20000 test pairs. We test our proposed model against an MLP baseline, which directly processes the object representations o and 0j extracted by CNN as MLP(o, 0,). We select the best MLP by hyper-parameter search over the number of layers and layer sizes. We use I-dimensional projection as we find this gives the highest accuracy. Table shows the extrapolation test accuracies of our model compared to the baseline. Our model with low-dimensional comparator significantly outperforms baselines for all three compared attributes.
3.3 VISUAL REASONING FOR RAVEN PROGRESSIVE MATRICES
For RPM-style tasks we use the extrapolation split of the PGM dataset (Barrett et al., 2018), which is already well-defined extrapolation type of generalisation task. We compare our proposed model against all previous methods (to the best of our knowledge) that have reported results on the extrapolation data split. We additionally include baseline model named "MLRN-P", which is 2-layer MIRN (Jahrens & Martinetz, 2020) with prior knowledge of only the relations present in nows/columns and with pre-training. Table 3 shows the test accuracy comparison. Our proposed model outperforms all other baselines. We note that we used a vanilla CNN as the perception module, same as most previous methods (Barrett al., 2018; Zhang al., 2019; Jahrens & Martinetz, 2020) on RPM tasks. Multiple-object representation learning methods (Greff et al., 2019; Kosiorek et al., 2019; Wang et al., 2019a), which achieve better results for multi-object scene learning than CNN, can be investigated for potential improvement of generalisation performance. We leave this for future work.
3.4 WHY LOW DIMENSION?
While we show that the comparators in oow-dimensional manifolds improve o.o.d generalisation for a range of relational reasoning tasks, the reason behind it is still not clear. In this section we analyse the projection space and the comparator function landscape of the visual object comparison task shed light on this. We first state 3 observations invariant across different sub-tasks comparing different attributes:
1. When the ground truth attribute can be represented in a I-dimensional manifold (such as vertical position). comparators in higher dimensions learn to project the object representa- tion into the I-dimensional manifold. Figure 4a illustrates this with plot of the projection

---

Under review as a conference paper at ICLR 2021
2d projected distribution
equal
100
Test Range
20
Train Range
20
10
Dim 2
-20
-40
-60
-100
-150
150
100
50
-00
120
-140
50
-100
-50
Dim
100
150
-150
100
150
100
50
0
-50
-100
-150
(a) 2D projection space
(b) 2D comparator function
Figure 4: (a) shows a scatter plot of the 2D projected distribution of objects in the task of comparing
x-coord attributes of objects in the images. X-axis and Y-axis are 2 dimensions of the projection manifold. Latent vertical position (ranging from O) to 32) is indicated by colour. (b) plots the comparator's function landscape for "equal" output unit (before softmax) in the space of vector differences between 2D projections of objects. Green circles represent the vector difference sampled from the training set while triangles represent the vector difference sampled from the test set.
distribution for the task of comparing vertical positions. It can be observed that even though the projection space is 2-dimensional, the projected points cluster around a line.
2. The projection of the test data in the manifold is less clustered around the sub-manifold of attributes than that of training data. This can be observed from Figure 4a, where the points projected from the test set are more spread out than from the training set. There is also less order in the distribution of test points, where points of noticeably different intensity appear next to each other.
3.
The function landscape becomes less defined outside of the sub- sabbmanifold. Figure 4b plots the comparator function landscape for the "equal" (1o)=ag) output unit over the space of vector differences hetween 2D projected representation p(02) - p(01). The equality function is well defined in the sub-manifold in which training points (circles) lic, peaking close to the (0, 0) point. However, outside of the training points' sub-manifold, the function is more random, with significant region with higher function value than at (0, 0) point. Note that the vector differences of test points (triangles) may be this region.
From the above observations, we conclude that when comparators of higher dimension than the intrinsic dimension of compared attributes, the projection tends to lie in a sub-manifold of the same dimension as for the attributes, resulting in the comparator function be only well defined in that manifold. However, the projection of test data tends to escape from this sub- aub-manifold into the region where the comparator function is never trained on, resulting incorrect prediction.
In addition to the observations above, we also use Hausdorff distance as quantitative measure of discrepancies between projected repesentations from the training and test sets. Due to space limitations, this is discussed in Appendix H.
We also performed ablation study on the dimensionality of the projected embedding space. Figure shows the plot of test accuracy against different dimension size of the projection functions for the visual object comparison tasks. It can be observed that increasing dimension sizes (x-axis) reduce the test performance. This further validate that dinedemiiional embeddings are crucial in improving neural architecture's 's extrapolation performance.
3.5 ALGORITIMIIC ALIGNMENT
Figure 6a shows the o.o.d and i.d (identically distributed) test accuracies baseline and our proposed comparator model for size comparison task with different training sample sizes. The dataset, model
7

---

Under review as a conference paper at ICLR 2021
100
98
test accuracy (%)
96
94
92
90
4
8
16
32
64
128
dimension size
Figure 5: Projection dimension ablation study for visual object comparison tasks. test accuracies
Test accuracies of size comparisons
100
test accuracy (%)
0.4
comparator o.o.d
0.2
60
baseline o.o.d
0.0
comparator i.d baseline i.d
25
train samples (k)
40
15
10
15
20
25
30
0.75
10
0.5
train size (in thousands)
Beta
0.25
Comparator
(a)
(b)
Figure Figure (a) shows o.o.d and i.d. (identically distributed) test accuracy of baseline and comparator for different training sample sizes. (b) shows test accuracies of baseline and comparator for differnt training sample sizes and with different 3-rate for truncating training distribution.
and baseline configurations the observed that i.d test accuracy's sample complexity (training samples needed to achieve the same accuracy), which measures algorithmic alignment, is not indicative of o.o.d accuracies.
in Visual Object Comparison task (Section 2.3). It can be
Figure 6b shows the size comparison of the test accuracies of baseline and comparator for different training sample sizes with different rates for truncating the training distribution along the latent dimension 'size'. For example when / = 0.5, only objects with size in the range VVow. 0.5V is seen during training. This corresponds to our proposed metric Section 2.5. The new metric reflects that the model which learns better with truncated training distribution is the one with better o.o.d generalisation.
RELATED WORK AND CONCLUSION
The deep neural network's 's lack of o.o.d generalisation (sometimes termed domain generalisation or extrapolation) capability has recently come under scrutiny. Different types of approaches have been proposed to improve o.o.d generalisation, such reducing superficial domain specific statistics training data (Wang et al., 2019b; Carlucci et al., 2019), adversarially learn representations that are domain-invariant (Li et al., 2017: Albuquerque et al., 2019), disentan- gling representations to separate functional variables with spurious correlations (Heinze- -Deml & Meinshausen, 2017; Gowal et al., 2019), and constructing models with innate causal inference graphs to reduce the dependence on spurious correlations (Arjovsky et al., 2019; Bengio et al., 2020). Our work aligns more with the line of work discovering inductive bias that improves generalisation.
o.o.d Generalisation:
8

---

Under review as a conference paper at ICLR 2021
Arguably CNN (LeCun al., 1995) is such inductive bias that improves generalisation visual input, and Graph Neural Network an inductive bias on graph-structured data (Battaglia et al., 2018). Trask et al. (2018) proposed Neural Arithmetic Logic Units (NALU), an inductive bias that allows neural networks to learn simple arithmetic with improved o.o.d generalisation.
Relational Reasoning: There is wide range of relational reasoning tasks such Visual Question Answering (Antol et al., 2015; Johnson et al., 2017). Raven Progressive Matrices (Barrett et al., 2018; Zhang et al., 2019), and Inferring Physical interactions (Kipf al., 2018; Sanchez-Gonzalez et al., 2018). These tasks involve comparing entities such as visual objects to infer relations between them. A large proportion of models proposed solve relational reasoning tasks fall into the broader range of graph neural networks and relational networks (Battaglia et al., 2018). Our work is mostly orthogonal these works, and may be viewed as special type of edge layer that can be integrated into most of these models. There is another line of research on neural-symbolic models (Yi et al., 2018; Mao al., 2019). Our work differs from these approaches in not using pre-defined programs, but learnable modules for comparing representations. PrediNet (Shanahan et al., 2019) use self- attention to extract features from the input, and then compare extracted features with a comparator. While our work is similar to PrediNet in that we also use comparator module to compare extracted features, PrediNet uses a different comparator module, different ways to extract features, and does not explicitly investigate how low-dimensional projections help with o.o.d generalisation.
Low-Dimensional Embedding: While there is a rich history of producing low-dimensional embed- dings for machine learning (see Appendix J for brief discussion), the best of our knowledge, little prior work has investigated how low-dimensional embedding improves o.o.d generalisation for relational reasoning tasks, van Steenkiste al. (2019) is the most related work, but focuses on the disentanglement property rather than on low-dimensional property for relational reasoning tasks.
Conclusion: We proposed neuroscience-inspieee inductive bias for improving the generalisation ability of neural networks for relational reasoning tasks. We showcased its effectiveness on three selected tasks, but the method can readily be adapted to any other relational reasoning task.
REFERENCES
Isabela Albuquerque, João Monteiro, Tiago H Falk, and Ioannis Mitliagkas. Adversarial target- invariant representation learning for domain generalization. arXiv preprint arXiv: 1911 1.00804 2019.
Stanislaw Antol, Aishwarya Agrawal, Jiasen Lu, Margaret Mitchell, Dhruv Batra, C Lawrence Zitnick, and Devi Parikh. Vqa: Visual question answering. In Proceedings of the IEEE international conference on computer vision, pp. 2425-2433, 2015.
Martin Arjovsky, Léon Bottou, Ishaan Gulrajani, and David Lopez-Paz. Invariant risk minimization. arXiv preprint arXiv:1907.028800 2019.
David Barrett, Felix Hill, Adam Santoro, Ari Morcos, and Timothy Lillicrap. Measuring abstract reasoning neural networks. In International Conference on Machine Learning, pp. 511-520, 2018.
Peter W Battaglia, Jessica B Hamrick, Victor Bapst, Alvaro Sanchez-Gonzalez, Vinicius Zambaldi, Mateusz Malinowski, Andrea Tacchetti, David Raposo, Adam Santoro, Ryan Faulkner, al. Relational inductive biases, deep learning, and graph networks. arXiv preprint 10X.v:880.001266 2018.
Yonatan Belinkov and Yonatan Bisk. Synthetic and natural noise both break neural machine translation. In International Conference on Learning Representations, 2018 URL https:
//openreview.....m net wtt.eeewee//ooummieeeeeeeee
Yoshua Bengio, Tristan Deleu, Nasim Rahaman, Nan Rosemary Ke, Sehastien Lachapelle, Olexa Bilaniuk, Anirudh Goyal, and Christopher Pal. A meta-transfer objective for learning to disentangle causal mechanisms. In International Conference on Learning Representations, 2020. URL
www.teeeeoottiin forum? de
Fabio M Carlucci, Antonio D' D'Innocente, Silvia Bucci, Barbara Caputo, and Tatiana Tommasi. Domain generalization by solving jigsaw puzzles. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2229-2238, 2019.

---

Under review as a conference paper at ICLR 2021
Matthew V Chafee. A scalar neural code for categories in parietal cortex: Representing cognitive variables as "more" or "less". Neuron, 77(1):7-9, 2013.
François Chollet. On the measure of intelligence, 2019.
Pierre Comon, Independent component analysis, new concept? Signal processing, 36(3):287-314,
1994.
Jamie K Fitzgerald, David J Freedman, Alessandra Fanini, Sharath Bennur, Joshua I Gold, and John A Assad. Biased associative representations in parietal cortex. Neuron, 77(1): 180, 2013.
Robert Geirhos, Carlos RM Temme, Jonas Rauber, Heiko H Schütt, Matthias Bethge, and Felix Wichmann. Generalisation in humans and deep neural networks. In Advances in Neural Information Processing Systems, pp. 7538-7550, 2018.
Mengyuan Gong and Taosheng Liu. Biased neural coding of feature-based attention in human brain. bioRxiv, pp. 688226, 2019.
Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger (eds.), Advances in Neural Information Process- ing Systems 27, pp. 2672-2680. Curran Associates, Inc., 2014. URL http: /tppper/ppprssse nips. .ddf.
ecc/paper/5420 / ppaper/5423-geeeraatcc wwweerrttvvvvvvvvvvvveeeeeeeeeee aeeeeevvveerrrarae
Sven Gowal, Chongli Qin, Po-Sen Huang, Taylan Cemgil, Krishnamurthy Dvijotham, Timothy Mann, and Pushmeet Kohli. Achieving robustness in the wild via adversarial mixing with disentangled representations. arXiv preprint /rXiv:1912.031 1912.03/22. 2019.
Klaus Greff, Raphaël Lopez Kaufmann, Rishab Kabra, Nick Watters, Chris Burgess, Daniel Zoran, Loic Matthey, Matthew Botvinick, and Alexander Lerchner. Multi-object representation learning with iterative variational inference. arXiv preprint arXiv::105 1903.000900 2019.
Umut Güşlüü and Marcel AJ van Gerven. Deep neural networks reveal gradient in the complexity of neural representations across the ventral stream. Journal of Neuroscience, 15057):0000 2015.
Torkel Hafting, Marianne Fyhn, Sturla Molden, May-Britt Moser, and Edvard I Moser. Microstructure of a spatial map in the entorhinal cortex. Nature, 436(7052):001-866 2005.
Christina Heinze-Demi and Nicolai Meinshausen. Conditional variance penalties and domain shift robustness. arXiv preprint arXiv: 1469, 2017.
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. Iclr, 2(5):6, 2017.
David H Hubel and Torsten N Wiesel. Receptive fields of single neurones in the cat's striate cortex.
The Journal of physiology, 148(3)5574-511, 1959.
Marius Jahrens and Thomas Martinetz. Solving raven's 's progressive matrices with multi multi-layer relation networks. arXiv preprint arXiv:2003. 1608, 2020.
Justin Johnson, Bharath Hariharan, Laurens van der Maaten, Li Fei-Fei, C Lawrence Zitnick, and Ross Girshick. Clevr: diagnostic dataset for compositional language and elementary visual reasoning. In Proceedings of the IEEE Conference Computer Vision and Patlern Recognition, pp. 2901-2910, 2017.
Diederik P Kingma and Max Welling. arXiv:1312.61 2013.
Auto-encoding variational bayes.
arXiv preprint
Thomas Kipf, Ethan Fetaya, Kuan-Chieh Wang, Max Welling, and Richard Zemel. Neural relational inference for interacting systems. arXiv preprint arXiv:1802.046877 2018.
Adam Kosiorek, Sara Sabour, Yee Whye Tch, and Geoffrey E Hinton. Stacked capsule autoencoders. In Advances in Neural Information Processing Systems, pp. 15486-15496, 2019.
10

---

Under review as a conference paper at ICLR 2021
Yann LeCun, Yoshua Bengio, et al. Convolutional networks for images, speech, and time series. The handbook of brain theory and neural networks, 3361(10):1995, 1995.
Juho Lee, Yoonho Lee, Jungtack Kim, Adam Kosiorek, Seungjin Choi, and Yee Whye Teh. Set trans- former: framework for attention- based permutation invvrriann neural networks. In International Conference Machine Learning, pp. 7744-3753, 2019.
Da Li, Yongxin Yang, Yi-Zhe Song, and Timothy M Hospedales. Deeper, broader and artier domain generalization. In Proceedings of the IEEE international conference on computer vision, pp- 5542-5550, 2017.
Liyua Liu, Haoming Jiang, Pengcheng He, Weizhu Chen, Xiaodong Liu, Jianfeng Gao, and Jiawei Han. On the variance of the adaptive learning rate and beyond. arXiv preprint arXiv: 1908,03265, 2019.
Jiayuan Mao, Chuang Gan, Pushmeet Kohli, Joshua Tenenbaum, and Jiajun Wu. The neuro- symbolic concept learner: Interpreting scenes, words, and sentences from natural supervision. arXiv preprint arXiv: 12584, 2019.
Daphne Maurer. How the baby learns see: Donald o. hebb award lecture, canadian society for brain, behaviour, and cognitive science, ottawa, june 2015, Canadian Journal of Experimental Psychology/Revue canadienne de psychologie expérimentale, 70(3):195, 2016.
Benjamin Recht, Rebecca Roclofs, Ludwig Schmidt, and Vaishaal Shankar. Do imagenet classifiers generalize to imagenet? International Conference on Machine Learning, pp. 5389-5400, 2019.
Alvaro Sanchez-Gonzale Nicolas Heess, Jost Tobias Springenberg, Josh Merel, Martin Riedmiller, Raia Hadsell, and Peter Battaglia. Graph networks as learnable physics engines for inference and 0222, 2018.
control. arXiv preprint
Adam Santoro, David Raposo, David G Barrett, Mateusz Malinowski, Razvan Pascanu, Peter Battaglia, and Timothy Lillicrap. A simple neural network module for relational reasoning. In Advances in neural information processing systems, pp. 4967-4976, 2017.
Murray Shanahan, Kyriacos Nikiforou, Antonia Creswell, Christos Kaplanis, David Barrett, and Marta Gareeo.. An explicitly relational neural network architecture. arXiv preprint 2019.
Elizabeth S Spelke and Katherine D Kinzler. Core knowledge. Developmental science, 10(1):89-96 2007.
Xander Steenbrugge, Sam Leroux, Tim Verbelen, and Bart Dhoedt. Improving generalization for ab- stract reasoning tasks using disentangled feature representations. arXiv preprint arXiv: 1811:08811.04784 2018.
Christopher Summerfield, Fabrice Luyckx, and Hannah Sheahan. Structure learning and the posterior parietal cortex. Progress neurobiology, 184:101717, 2020.
Andrew Trask, Felix Hill, Scott E Reed, Jack Rae, Chris Dyer, and Phil Blunsom. Neural arithmetic logic units. Advances in Neural Information Prucessing Systems, pp. 8035-8044, 2018.
Sjoerd van Steenkiste, Francesco Locatello, Jürgen Schmidhuber, and Olivier Bachem. Are disen- tangled representations helpful for abstract visual reasoning? Advances in Neural Information Processing Systems, pp. 14245-14258, 2019.
Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in neural information processing systems, pp. 5998-6008, 2017.
D. Wang. M. Jamnik, and P. Lio'. Unsupervised and interpretable scene discovery with discrete- ineend-ifferrrepeatt In ICML workshop on Self-Supervised Learning, pp. 9pp, 2019a. URL 0150 06581.

---

Under review as a conference paper at ICLR 2021
Duo Wang, Mateja Jamnik, and Pietro Lio. Abstract diagrammatic reasoning with multiplex graph networks. In International Conference on Learning Representations, 2020. URL https:// openreview n wtt////rummii///// wow.mnnddByys
Haohan Wang, Zexue He, Zachary C Lipton, and Eric P Xing. Learning robust representations by projecting superficial statistics out, arXiv preprint arXiv:1903.000 06256, 2019b.
Svante Wold, Kim Esbensen, and Paul Geladi. Principal component analysis. Chemometrics and intelligent laboratory systems, 2(1-3):37-52 1987.
Keyulu Xu, Jingling Li, Mozhi Zhang, Simon S. Du, Ken ichi Kawarabayashi, and Stefanic Jegelka. What neural networks reason about? In International Conference on Learning Representations, 2020. URL httttp:://openrevvice
net forum? w
Kexin Yi, Jiajun Wu, Chuang Gan, Antonio Torralba, Pushmeet Kohli, and Josh Tenenbaum. Neural- symbolic vqa: Disentangling reasoning from vision and language understanding, Advances in Neural Information Processing Systems, pp. 1031-1042, 2018.
Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Russ R Salakhutdinov, and Alexander Smola. Deep sets. In Advances in neural information processing systems, pp. 3391-3401, 2017.
Chi Zhang, Baoxiong Jia, Feng Gao, Yixin Zhu, Hongjing Lu, and Song-Chum Zhu. Learning perceptual inference by contrasting, In Advances in Neural Information Processing Systems, pp. 1073-1085, 2019.
Shengjia Zhao, Hongyu Ren, Arianna Yuan, Jiaming Song, Noah Goodman, and Stefano Ermon. Bias and generalization in deep generative models: An empirical study. In Advances in Neural Information Processing Systems, pp. 10792-10801, 2018.
12

---

Under review as a conference paper at ICLR 2021
A
MAXIMUM OF A SET: ARCHITECTURE CONFIGURATIONS
Table 4:
Model configuration for Maximum of a Set task.
Projection Dim Comparator Type Number of Objects
Oulput Type
Loss Function M.S.E
MLP(ppoo)-pPp
20
Real-Vart Valued
The architecture for the maximum of a set task has three sub-modules, namely comparator of (T; RT), comparison summariser e; ML f(z, zy)) and pooling function Σ ((x)) For comparator f. we K, the number of parallel comparisons (Equantion in the main body of the paper) to be I because the scalar valued real numbers do not have multiple parallel attributes. We implement the projection function p as a single feed forward layer. We choose 1-dimensional comparison space as this gives the best result. The comparison function e takes the projected difference p(x;) p(x,)) as input, and is implemented as a single feed forward layer with output unit. The LP in the comparison summariser is implemented as a 2-layer MLP of hidden-size 16 1. In the pooling function, the attention function is implemented as a softmax layer, which normalises e, across I...N.
N. Table 4 shows summarised model configuration.
B VISUAL OBJECT COMPARISON: DATASET GENERATION
In this section describe details of the visual object comparison dataset. We sample images from the dSprites dataset Higgins et al. (2017) and generate comparison labels (categories include smaller than, equal to, greater than) from ground truth latent values provided in the dataset. For each image in the dSprites dataset, ground truth attribute values are provided, which are shape "category". "size", "rotation angle", "horizontal position" and "vertical position". We add "colour" the 6th attribute by randomly generating the colour intensity value from the set 1,0.000. 2,00.,00.4. 00.5,.66,..7.00 0.9.. for each image. We multiply image pixel values with the colour intensity, and add the colour intensity value to the ground truth latent values.
Algorithm 1 shows the pseudo-code for generating the visual object comparison dataset. compare_attr indicates the object attribute to be compare for the task. We pick three different attributes for experiments, which are "size", "horizontal position" and "colour intensity". We set training attribute range be the lower 60% while test attribute range to be the upper 40%.
Input: train__iaiiee test_ sist_sszee,e__ compare _ttr train_data= = Eiptyl.st()
test best_data- = Emptyl.sst)
for split in (train,test] do for io 1 to train_size do attr_range, attr attr_idx attr_ attreaatsssssscompe__ttttt
if split == train then
LO, HI 0,
else
I.O, -I 0.6, 1.0 end if
sample_range = Truncate(attr_range) LO, HI)
latent_values A = SamplelLaten((attr_id)) idx, range)
latent_ vaautt_vales_BBB B = Samplel.teen(aatr_idh _id sample_range)
image A Samplellngge(llteetttaleuee
image_B Samplelmaggellatett_adllude
less_than = latent_ < vaauest_value___
equal = filenn_vvllus_... A = valuts__aluu___
greater than = lawent.ettttttm values A > latent valuue__aauue_Be B
label Concat(les__ thann equal, greater_thaa t_ann)
if split == train then
train_data
Hammae_A.imaag_BBee
else
test_ (image_Aiimae__Blabb
end if end for
end for
Algorithm 1: Visual Object Comparison Dataset Generation
13

---

Under review as a conference paper at ICLR 2021
C VISUAL OBJECT COMPARISON: ARCHITECTURE CONFIGURATIONS
In this section we list detailed configuration of the architecture with low-dim comparator and the baseline architecture. The architecture with low-dim comparator is illustrated in Figure 3a. The CNN module is a 4-layer CNN of filter number 32- - 32 - 64 64 followed by a 2-layer MLP of hidden size 256- 10. Each CNN layer has stride value 2 and padding value 1. The output of each CNN is the object representation o of raw image input r;- The projection module is a single feed forward layer projecting to space with dimension d. We use d for reporting results except for the manifold analysis experiments in Section 3.4. The comparator takes projected vector difference p(o,) p(o,) as input, and is implemented as a 2-layer MLP of size h 3, where h is the hidden size and 3 is the output size (corresponding to 3 different output categories). We found that varying h in the range 4 to 32 has little effect on the performance. Hence we report performance with the h 4 to reduce computational costs.
The baseline architecture uses the same CNN module as the architecture with low-dim comparator. The baseline model concatenates the CNN output o and o, and feed the concatenated vector into a MLP. We performed ppper-parameter search of the MLP with number of layers ranging from I to 4, and with hidden unit sizes from 32 to 64. We found that 64 64 3 is the best performing architecture.
Table 5: Model configuration for Visual Object Comparison task.
K Projection Dim
Comparator Type MLP(p(o.) plo,))
Number of Objects
Output Type Categorical
Loss Function
2
Cross-Entropy
D PGM DATASET
In this section we give brief description of PGM dataset. For more details please refer to Barrett et al. (2018). PGM contains 8 context panels and 8 answer panels. The 8 context panels for a 3 x 3 diagram matrix. One is asked to pick the answer the logically fit with the context panels. In PGM, logic relations can exist in both rows and columns of the diagram matrix. Figure 7a and 7b show two examples from the PGM dataset(Image courtesy Barrett et al. (2018)). The first example contains a 'Progression' relation of the number of objects across diagrams in columns. The second examples contains a 'OOR' relation of position of objects across diagrams in rows. The objects in PGM datasets have different attributes such as colour and size. In total five types of relations can be present in the task: (Progression, AND, OR, XOR, ConsistentUUninn
E PGM ARCHITECTURE CONFIGURATIONS
In this section we describe detailed configuration of the PGM architecture with low-dim comparators, and the baseline model MLRN-P, which an augmented version of MLRN Jahrens Martinetz (2020). The descriptions here is supplementary to descriptions in section 2.4 of the main paper and figure 3b.
The CNN module is 4-yeyer CNN of filter number 32 - -32 - 64 - 64 followed by 2-layer MIP of hidden size 256 128. Each CNN layer has stride value 2 and padding value 1. The output of cach CNN is the object representation o of raw image input x,. Following Barrett et al. (2018) we attach to og position tag to indicate its position in the diagram matrix. The tagged object representation is then projected onto K- 512 1-dimensional manifold for parallel attribute comparison. Next we describe the comparator module f (equation 1. As shown in figure 3b, there two hierarchical projection comparison. For the first level, we found that implementing e as a simple vector difference module achieve best results, which means e = p(o:) - p(o,). This reflects the fact that comparison between diagrams directional, such as increase number of objects from one diagram the other. We implement in equation 1 as residual MLP of 4 layers with hidden size 2048- 2048 2048 - 796. The output from each pairwise comparison of diagrams in a row/column are concatenated to form the relation embedding for that row/column. For the second level we implement c as absolute difference, which means c = Ip(o,) - p(o,)) I. This works better because relation comparison is less directional. For example the difference between relation of increasing number of objects and relation of increasing sizes of objects should be the same when the compared diagram is swapped. For the second level we implement g as 3 layer MLP of hidden sizes 1024 512 - 1, which directly output the predicted similarity score between two rows/columns. For predicting the correct answer candidate we follow Barrett al. (2018) by applying a softmax function to scores produced by comparing each
14

---

Under review as a conference paper at ICLR 2021
XOR(panel panel 2)
Context Panels
+1
+1
?
Answer Panels
B
A
B
C
D
E
F
(a)
(b)
Figure 7: Two examples in PGM dataset. (a) task contains 'Progression' relation of the number of objects across diagrams in columns while (b) contains a 'XOR' relation of position objects across diagrams in rows. For both task "A" is the correct answer.
answer rows/columns with context row/columns to produce scores for each answer candidates. For all context row/column embedding and process it with 3-layer MLP of hidden size 1024 - 512 - 12, where 12 is the meta-target label size. We have attached source code in the submission for clarity.
meta-target prediction
The baseline model MLRN-P is modified from MLRN Jahrens & Martinetz (2020). We have performed three architecture modifications for fair comparison with our model. Firstly we inject the prior knowledge of relations existing only in rows/columns into the model. The first level of Relation Networks compare diagrams within row/columns and the second level of Relation Networks compare row/colum embeddings. Secondly swap MLP in the original MLRN with residual MLP, which shown to improve performance slightly in our model. Thirdly we pretrained CNN module with Beta- VAE Higgins al. (2017) for fair comparison with our model.
TRAINING DETAILS
In this section we describe the training details for all three experiments. We use PyTorch' for implementation. For gradient descent optimiser, we use RAdam Liu al. (2019), an improved version of the Adam optimiser. For all 3 experiments we use learning rate 0.001 and betas (0.9,0.999). set and visual We used 2 Nvidia Geforee Titan Xp GPUs for training all models. For Maximum object comparison,
set batch size to be 64. For PGM we found larger batch size of 512 slightly improves result. For Maximum of the set and visual object comparison we set training epochs to be 20. For PGM we trained for 50 epochs. For visual object comparison and RPM tasks pre-trained
15

---

Under review as a conference paper at ICLR 2021
CNN the encoder of a Beta-VAR VAE Higgins al. (2017). We follow standard training procedures of Beta- VAE as described in the paper, and set the value to 1.
Table 6: Model configuration for Maximum of a Set task.
K
Projection Dim
Comparator Type MLP(p(o.) -((O))
Number of Objects
Output Type
Loss Function
512
16
Categorical
Cross-Eiie Pioss-Entropy
G ADDITIONAL PLOTS
In section 3.4 of the main paper we show plots projected distribution and comparator's function landscape for position comparison task. In this section show the plots for comparison tasks for other attributes, which are sizes and colour intensity. Figure shows the plot for size comparison while figure 9 shows the plot for colour intensity comparison. Training range is the lower 60% of the colour bar while test range is the upper 40%. The observations stated section 3.4 also holds true for these attributes. For size and colour intensity comparison tasks, the projected distribution plots show that the test data is more clustered than that of spatial position comparison. This shows that size and colour intensity attributes can be learnt better with bespoke CNN perception module than spatial position attributes. This is supported by that models trained for these two attributes achieved higher o.o.d test accuracies.
equal
projected distribution
0-
100
50
-2
Dlm 2
33
-50
100
-50
-00
100
50
-1.0
-2.
00
Dim
5.0
7.5
10.0
12.5
0
-50
1000
-150
150
100
50
-50
1100
150
(a) 2D projection space
(b) 2D comparator function
Figure 8: (a) shows a scatter plot of the 2D projected distribution objects the task of comparing sizes of objects in the image. X-axis and Y-axis are 2 dimensions of the projection manifold. Latent size variable (possible values are (,,,,4,3)) is indicated by colour. (b) plots the comparator's function landscape for "equal" output unit (before softmax) in the space of vector differences between 2D projections of objects.
H QUANTITATIVE EVIDENCE FOR LOW DIMENSION
In section 3.4 and G we showed visual illustration for 1-dim and 2-dim comparator's function landscape for the visual object comparison task to illustrate how out-of-distribution test data lies outside of the training function landscape. dimensions. In this section we discuss how to use Hausdorff Distance, which measures distance between sets, to measure the differences between projected distance from training and o.o.d test set.
difficult to visualise function landscape of higher
Hausdorff distance measures how far subsets of metric space are from each other. Given two sets = (a1: ..,...; uyy } and B (b1, be....bb of the same metric space. The Hausdorff distance of A and B is computed as:
(). max sup inf d(a, b), sup inf d(a, 6) a
beB aEA
where is a distance measure of the metric space. Informally Hausdorff distance measures the greatest of all the distances from point in to the closest point in the other set.
16

---

Under review as a conference paper at ICLR 2021
equal
2d projected distribution
50
40
200
1. 150
100
50
Dm
0.6
10
02
-100
-10
150
100
0
50
50
100
150
100
150
-150
100
-50
projection space
(b) 2D comparator function
Figure 9: (a) shows a scatter plot of the 2D projected distribution of objects in the task of comparing colour intensity of objects in the image. X-axis and Y-axis are 2 dimensions of the projection manifold. Latent colour intensity variable (possible values are indicated by colour. softmax) in the space of vector differences between 2D) projections of objects.
(b) plots the comparator's function landscape for "equal" output unit (before
16
14
12
Distance
10
8
6
4
2
8
16
32
64
128
dimension size
Figure 10: Hausdorff distance between training set and test set vector differences between projected object representations (normalised) for the visual object comparison task.
We apply Hausdorff distance to measure distances between sets of vector differences from training and test set. The training set Straán contains vector difference p(e,) p(e,) for all pairs of inputs es, ey E Dtrain from the training distribution Diroin. Recall that is the projection module. The test set Stest contains vector differences of pairs from the test data distribution Drest- We measure the Hausdorff distance between training and test set du (Strains Stest) ) for different projection dimensions, and show the result in Figure 10. compare in the same scale, we normalise vector differences with the mean and standard deviation of the training set. It can be observed that as the dimensionality of the projection space increase, there is increased distance between Strain and Stest- This quantitatively shows that the observations in Section 3.4 extends beyond 2-dimensional cases.
I ABLATION STUDIES
We performed ablation studies of different hyper-parameter: our experiments. and show the results in Table 7, Table 8 and Table 9. Table 7 shows the test accuracy of architectures with different number of projection functions for PGM task. The number of projection functions indeed another hyper-parameter tune, but think the improved performance is definitely worth it. Table 8 shows the test accuracy on the extrapolation split of PGM dataset with different pre-training for the perception module. We picked the parammeass giving the best extrapolation test accuracy.
17

---

Under review as a conference paper at ICLR 2021
Table 9 shows the effectiveness of different comparator imput types the test accuracy for comparing 'colour' attributes in the Visual Object Comparison task.
Number of Projector Functions
Accuracy
128
20.1
256
24.2
512
25.9
Table 7: Ablation Study Number of Projecton/Comparator functions for PGM task.
Pre-trained Encoder
Accuracy
3-VAE (5=4, dim=64) Steenbrugge et al. (2018)
23.5
B-VAE (3=4, dim=128)
25.1
3-VAE (3=1, dim=128) (OURS)
25.9
Table 8: Ablation study on different pre-trained encoders for PGM task.
vector dist accuracy:0.9771 0.0281 I1 dist accuracy:0.692std 0.149 MLP dist accuracy:0.912
+-0.0652
Comparator Input Type
Accuracy
Vector Difference
97, 71 +: 2.81%
L1 Difference
69.2 + 14.9%
Concatenation
91.2 + 6.52%
Table 9: Ablation study on different input types of comparator functions for 'colour' comparison task.
BRIEF DISCUSSION ON LOW-DIMENSIONAL EMBEDDING
In machine learning there is a rich history of learning low-dimensional embedding, ranging from more classic methods like Principal Component Analysis (PCA) (Wold al., 1987) and Inde- pendent Component Analysis (ICA) (Comon, 1994) to recent neural-network-bar Variational Suto-EEccdders (VAE) (Kingma & Welling, 2013) and Generative Adversarial Networks (GAN) (Goodfellow et al., 2014). VAE and GAN have been frequently used to learn in an unsu- pervised manner the cww-dimensional representations from an unlabelled dataset, and use the learnt representations to boost performance of supervised downstream tasks, such image classification and natural language processing. However, most researches on VAE and GANs put more emphasis understanding how disentanglement affect downstream task performance, but rather than low- dimensions, such as Beta- VAE (Higgins et al., 2017) and its variants. Arguably disentanglement can be considered as splitting the latent distribution into multiple orthogonal distributions of lower dimensions, but to our best knowledge this view has not been discussed in the literature. Moreover, there is limited work on how oow-dimension representations learnt by neural generative models help with o.o.d generalisation. To our best knowledge, the only work discussing this is by Zhao et al. (2018), who systematically study the generalisation capability GANs and VAEs using experimental methods from cognitive psychology.
methods like
K TRUNCATION FUNCTION
The truncation function T(D, 8, u) in section 2.5 can be any functions that truncate the data distribu- tion D given truncation ratio 3 and truncation dimensions u. The truncation can be performend in the data distribution (e.g. select one dimension of the data space and truncate (0, B). In the 'Visual Object Comparison' task, we truncate the latent data distribution instead. The latent data distribution of dSprites dataset(according to the beta-VAE and dSprites dataset paper) has six latent dimensions which are: (color, shape, size, rotation, x-coord, y-coord). For example, we can select the x-coord dimension (range [1,32], and truncate the distribution with 3 0.6 so samples from the truncated distrubtion can only have x-coord in range 1,3=e1132=006]]
be in range
L CORRELATION BETWEEN PROJECTION AND GROUND TRUTH ATTRIBUTES
We analysed the projected 1 I-D embeddings by our model for the Visual Object Comparison task see if they correlate with the ground truth attributes. We measured Pearson Correlation between
18

---

Under review as a conference paper at ICLR 2021
projection and ground truth attribute values for all three comparison tasks. The results are listed in Table 10. It can seen that our model learns projections that are highly correlated with the ground truth attribute values, even though the ground truth is never used during training.
Attribute Type Correlation Score
Size
0,945
Colour
0,977
X-Coord
0.796
Table 10: Pearson Correlation between projected embedding and ground truth attribute values.
M RESULTS ON OTHER PGM SPLITS
We additionally tested our model on "neutral" and "interpolation" split of PGM dataset, and compare our model''s s results with other baselines' in Table 11. Our model performs better than other baselines for "interpolation" task, while is outperformed by MLRN Jahrens & Martinetz (2020) on "neutral"
split.
Table 11: Test accuracies for "neutral" and "interpolation" split of PGM dataset.
WReN
MXGNet
MLRN Jahrens & Martinetz (2020)
OURS
Data Split
Barrett et al. (2018)
Wang et al. (2020)
Neural Interpolation 67.4%
76.9%
89.6%
98.0%
90.1%
84.6%
57.8%
85.4%
N MAXIMUM OF SET: ADDITIONAL BASELINE
For the "Maximum of a set" experiment, we additionally tested a Relation- based model that explicitly takes the comparison results input. The Relation Network first encoded pairwise between all pairs of element (I,, æy) using MLP (32-32) taking the concatenation of element values as input. For each element x; we sum all pairwise embeddings between this element and others elements v, as Σj MLP((s:: Zz). The embeddings e; are then passed through max-pooling layer, and then another MLP to produce the final output. This model achieves 16.2 + 2.45 M.S.E, which is very large and thus are not included in the main paper.
O ADDITIONAL PLOTS OF COMPARATOR FUNCTION LANDSCAPE
Figure 11 shows plots of the "equal' comparator function landscape for independent runs of x-coord comparison task.
19

---

Under review as a conference paper at ICLR 2021
equal
equal
20
50
20
-25
-50
-40
-75
150
150
1000
100
100
-60
50
-125
-150
-100
-100
-150
150
50
0
-100
-150
-150
150
50
-50
-100 -150
equal
equal
150
50
0
-10
150
-20
-50
100
50
-100
-50
-0
50
-100
-150
150
50
-50
-100 -150
-150
150
100
0
-50
-100
-150
Figure 11: Plots of the 'equal' comparator function landscape for independent runs of x-coord comparison task
20


Reviews:
[{"review_id": "wVQRxIkH6h", "reviewer": "AnonReviewer5", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "This paper showcases three studies on relational reasoning where objects need to be compared on a certain attribute (like size). The experiments show large improvements in generalization performance over previous work. The authors attempt to formulate a general method from their experiments.\n\nMy biggest criticism is that the method needs to be described more clearly. There are lots of variations and add-ons from one experiment to the next. When you introduce notation, it is not clear what it maps to in your architecture descriptions (e.g. the choice of g and c). Is it fair to say that the main insight of the paper is the following: it is possible and desirable (in terms of generalization) to learn comparators between objects by feeding the difference between low-dimensional (or specifically one-dimensional) projections of the object representations, taken pairwise, into small MLPs?\n\nThe hyperparameters/architectural choices need to be summarized better. Could you perhaps make a table specifying the following for each experiment (I’ll refer to maximum of a set, dSprites, and RPM as experiments 1, 2, and 3 respectively):\n- the value of K (this is clear for experiment 1 but not others). Appendix E seems to suggest you use K = 512 for experiment 3. This is much larger than what I would expect.\n- the size of the low-dimensional manifold for each projection p. Is this always 1? Figure 10 seems to show 1 works best, which is a nice result. \n- the comparison summarizer g \n- the exact form of the comparator function c. Are you feeding in p(o_i) - p(o_j) into an MLP or |p(o_i) - p(o_j)| or something else altogether? This isn't clear especially for experiment 1.\n- how many objects are you ultimately comparing/learning a relation over. If I understand correctly, this is N, 2, and 8 for the three experiments respectively. Please do specify N.\n- what is the final output. This is clear for experiment 2 and 3. For experiment 1, I assume the output is a categorical probability vector with N dimensions? Or is it in fact the value of the maximum element?\n- what type of loss you are using. You defined a fairly non-intuitive loss L(d(a_t(o_i), a_t(o_j)), f(o_i, o_j)) in section 2.1. It is unclear where you’re using this.\n\nThe strengths of the paper include:\n- Solid results showing better strictly better generalization\n- Good attempts to analyze the method\n\nMinor: I do also take exception when you refer to your proposed comparator as “an inductive bias module.” (I realize there is no consensus on the definition of the term “inductive bias.” If we do define it–as something which changes the preferred solutions of a learning algorithm–then most deep learning architecture choices are inductive biases. So it is rather vacuous to refer to your architectures as inductively biased.) Moreover, an extra input (e.g. p(o_i) - p(o_j) which is fed to the comparator in your case) cannot be termed as an inductive bias, because it changes the solution space completely (consider learning the XOR function with a single linear layer when the difference is provided as an additional input versus when it is not).\n\nLastly, the inspiration from neuroscience is quite loose. The fact that human brains have high-level neurons which disentangle object position and size does not lead to your method directly; you could argue that all of deep learning attempts to learn higher-level features progressively with depth in the network. I would suggest revising the paper to focus on the insights, along with a clear description of the method, rather than the loose inspiration.\n\nQuestions and request for clarification:\n1) Could you please provide error bars for the RPM experiment? \n2) On your observations (section 3.4) answering why comparators on low-dimensional manifolds improve generalization:\n- I would suggest moving Figure 10 from the appendix to this section to answer why low-dimensional projects are important to learn good comparators.\n- I see that observation 1 holds for 2D projection spaces, but could you illustrate the linearity of the projection in 3+ dimensions? Alternatively, please change the phrasing to refer to 2D specifically to avoid overclaiming.\n- Under observation 3, your claim that the 2D comparator is “random [...] outside of the training points’ sub-manifold” in Figure 4b: in what sense do you mean that? Could you add similar plots over independent runs so the “randomness” is evident?\n3) I do appreciate your efforts to show systematic generalization. But your o.o.d algorithmic alignment metric is sloppy and arbitrarily defined. For instance:\n- What is V in the definition? I don’t see it defined or used anywhere in section 2.5.\n- What are the requirements on \\Tau, the truncation function? It seems to be constrained by \\beta and \\textbf{u} but this isn’t fully specified. I suppose you want to ensure the truncation function is not vacuous (e.g. the identity function)?\n- Are all dimensions in \\textbf{u} truncated by the same \\beta parameter? Why choose a subset of dimensions for truncation at all (rather than truncate all dimensions)? This needs justification. \n- How would the truncation work when the training data are images?\n- Given than D_s is only a truncation of D, it seems fair to say that the sample {x^s_i, y^s_i} has non-zero likelihood of coming from D instead of D_s. So strictly speaking, your definition of (M, \\epsilon, …)-learnability is not measuring o.o.d performance; rather it seems to be measuring performance on a “superset” of the training sample. Is this a correct reading?\n4) Your algorithmic alignment results (section 3.5) show that your comparator generalizes favorably over the baseline for various values of \\beta and the sample size. But what dataset are the scores based on? What is the truncation function? What is the baseline?\n5) Notation: could you please avoid overloading the use of \"a\" as the ground-truth mapping function (from an object's representation to its t'th attribute) as well as the attentional weights for pooling? Ideally, avoid defining the former unless you're actually using it.\n\nIn conclusion, I do consider this work substantial and important. I would be open to revising my score if the presentation is sharpened.", "extra_scores": {}, "extra_text": {}}, {"review_id": "jQT2trbUiwc", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "The paper proposes a simple low-dimensional projection and comparison module for extrapolatable relational reasoning. The proposed module first projects object embeddings into lower-dimensional manifolds, compare and concatenate them to form a vector, before finally making a prediction. The paper shows that on the maximum-of-a-set task, the visual-object-comparison task, and the IQ test, the proposed module, when combined with deep models, outperforms existing works in extrapolation.\n\nThe paper empirically investigates how the proposed module can improve relational reasoning in ood extrapolation tasks, which is considered an important aspect for deep neural networks. The paper is well-presented, providing intuitive sketches to ease understanding.\n\n------------------------------------\n\na. It looks like the major claim of the work is that low-dimension manifolds can help learning. But I wonder, for most of tasks, especially the representation used for classification, when will low-dimension manifolds hurt? Aren't the final layers of deep networks always low-dimensional? Low dimensions ease understanding, interpretation, visualization, and learning. When the true hidden representation is low-dimensional, why do we need to make it high-dimensional? I'm confused what the authors are trying to argue in this point.  \nb. From a technical point of view, the true lower-dimensional representation, if well-learned, will improve generalization without doubt. However, the work does not demonstrate the representation learned has this property. For example, if the authors are trying to say they learn the attribute, say size, can it be shown that learned low-dim representation is correlated with the actual size of it, independent of its position, during extrapolatory tests? And can the comparator be shown to act like a sign function. Especially in the IQ test problem? The first two tests look particularly simple, as one can even design transformations that show that. E.g., for numbers you can use identity transformation and do a subtraction and a sign function. For the black-and-white images, you can use a simple identity kernel to compute the number of foreground pixels.  \nc. The module design looks to be adopted directly from existing works like relational module, PrediNet, and CoPINet. Relational module and PrediNet have this object-pair and comparison idea, and the CoPINet mentions the contrast implementation. The low-dim comparator is a simple modification of the existing works.  \nd. While the argument is on extrapolation, I wonder what makes the module infeasible for interpolation. If the module learns well the true representation and knows what it means to compare, interpolation shouldn't be a problem either. Can it be shown that the representation is good for interpolation as well?  \ne. The authors, during experiments, use pre-trained models rather than training from scratch. It is possible that it's the pre-trained representation that helps the learning and extrapolation rather than the proposed low-dim projection? What if the models are not pre-trained?  \nf.  Finally, for Section 3.4. It looks like the low-dim representation is not well-learned. Test data representation is much clustered that the training data. Different ranges of training data are well-separated, but test data points are distributed across the test range. Also, for the comparator plot, response for training data is sure to be better than that for unseen data, but what kind of insight is there?  ", "extra_scores": {}, "extra_text": {}}, {"review_id": "eivJj68qKQ7", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "The paper presents two neural network design ideas: low-dimensional projection and arithmetic comparator. By integrating these two ideas into CNNs, the model can solve a set of tasks that require recognizing 1-dimensional properties of objects (such as the size, the color, etc), and making a comparison of these properties. The authors show that their method has strong out-of-distribution generalization: such as generalization to larger objects.\n\nThe presentation of the paper is relatively clear. However, I believe the authors could make the paper stronger by strengthening their experiments with more results and ablation studies. Details follow.\n\n1. The numbers in Table 4 and Table 5 are suspicious. Can the authors verify that these two tables have exactly the same results?\n2. The last sentence of the first paragraph of Page3 says \"the vector distance p(o_i) - p(o_j) or absolute distance improves the generalization performance\". Are there any ablation studies for those?\n3. The sec 3.1 \"maximum of a set\" experiment is toy-ish: in some sense, only the proposed model takes comparison results as the model input, while all other models take numbers. The authors should consider other baselines that also take comparison results as their inputs.\n4. I suggest the authors discuss more the results in Table 2: what are the failure cases of the model? Also in Table 2, the author should make ablation studies to discuss the importance of 1) low-dim embeddings and 2) comparison operations. For example, what will the baseline perform if it also projects the input image into a very low-dim space?\n5. Table 3. I suggest the authors supply the performance on standard train-test splits for RPM as well. This will give us an idea of how the model performs on \"i.i.d.\" test splits. Also, what are the failure cases for Table 3?\n6. I have some concerns about the applicability of this proposed model. Learning meaningful low-dimensional embeddings can be hard. For example, how to extract the number of people in an image? Or in an even simpler task, how to count the number of vertices of an arbitrary polygon in an image? The authors should discuss the limitation of their method.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "VIxzF413Ybh", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "\"Projecting into a lower dimensional space, so that irrelevant dimensions are projected out, will help in extrapolation\" is an interesting hypothesis that should be tested. However, I was not convinced that the experiments gave convincing evidence for the hypothesis. It is standard in science to test a hypothesis by keeping as many of the extraneous variables as invariant as possible. However, in this case a different architecture (which projects onto a lower dimensional space) was used for each problem, and there was an impressive improvement. It wasn't clear to me whether it was the lower dimensionality or other architecture choices that made the difference.\n\nDetailed question. What is g in equation (2)? I can make sense if it if the output of f is a K-dimensional vector.  But then you say \"g is a function that combines....to make a prediction\" so. I interpret that as producing a value. So then f gives a number, but then I can't iWork out what is the input to the MLP in equation (2); I was trying to look for a, say,  d-dimensional input, but I can't work out what d is.\n\nThe results of finding the maximum (Table 1) are dramatic. Is it really just the lower dimensional embedding? I'm guessing that you can learn the correct hypothesis, but the others can't (or there are lots of hypotheses consistent with the data).", "extra_scores": {}, "extra_text": {}}, {"review_id": "B7GXl9Asl-0", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "Review: \nThis paper addresses an inductive bias for relational reasoning tasks to improve generalization performance on out-of-distribution scenarios (so called extrapolation), which the value ranges (continuous variables) or value itself (discrete variables) of the training/test dataset do not overlap. The proposed idea is to project high-dimensional representations onto low-dimensional space and compare those of the targets of relational reasoning. To validate effectiveness of the idea, the authors provide the improved results on three problems: (1) finding the maximum of a set of real numbers, (2) comparing the attributes of visual objects with dSprites dataset, and (3) visual reasoning with Raven progressive matrices (RPM) based on the PGM dataset. The experimental results of each  show better performance than other baseline methods. This research deals with one of important topics in AI/ML research. Out-of-distribution generalization is potentially extended to world models, common sense reasoning, learning and reasoning with small number of instances, and abstraction capability of human intelligence. As Section of related work in this paper also mentions these points, it is understandable to figure out the position of this work. \nOn the other hand, I think that the idea of projection and comparison is similar to the approaches of zero-shot learning and deep metric learning. So, the novelty of this idea is not so big and it would be better to explain them as related work. Considering the experimental results in Section 3.4, it is not sure whether the proposed method still shows similar efficacy regardless to the degree of difference between the training datasets and the out-of-distribution ones.  \nI recommend as ‘Marginally below acceptance threshold’\n\nPros:\n- In Section 3.4, it is interesting that the analyses of the projection space and low-dimensional embedding and why the dimension should be small. I like it.\n- The authors propose a generalized improvement strategy for relational reasoning tasks on the out-of-distribution scenarios, and they show that the key idea effectively improves the performance.\n\nConcerns:\n- The proposed idea is not so concrete to apply as practical solutions. I think it seems close to a guideline. If there is someone to use this approach, it needs lots of effort and time for the actual problems.\n- It seems that the distribution difference between training datasets and the out-of-distribution ones influences explicitly on their performance. Does the main claim still valid in that situation? What is the relationship between the difference and the validness?\n- How does the dataset for visual object comparison split? Overall, there are not much explanation on details of the configurations of comparative methods. It seems difficult to reproduce the results. \n- What is the baseline method of Figure 5 (b)?\n- Overall presentation is not so clear. That makes difficult to follow.\n\nMinors:\n- I think ‘Less defined’ is not so good expression.\n- Figure 4 (b) has no information on each axis.\n- The right end of Figure 5 (a) is covered by Figure 5 (b).\n- Figure 5 (b) shows poor readability.\n", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper is attempting to improve the OOD generalization performance of neural networks on relational reasoning tasks. This is an important failure point of general neural network architectures and important research topic. The results of the paper shows impressive improvements on a set of subject.

* The paper is improved during the rebuttal, however, I do agree with the R5 and the paper is still lacking a lot in terms o clarity. The writing of this paper still requires some work. 

* As R2 also has written, the proposed idea is not so concrete to apply as practical solutions, and the presentation of the paper still requires some more work.

* R3 pointed out some inaccuracies and it seems like authors have added some ablations in the direction that R3 has suggested.

I am suggesting to reject this paper given that the majority of the reviewers are also leaning towards rejection as well. I would recommend the authors to improve the clarity of the paper, do more ablations for their models and resubmit to a different conference.
Decision: Reject