Transferability of Compositionality
Authors: Yuanpeng Li
Liang Zhao
Joel Hestness
Ka Yee Lun
Kenneth Church
Mohamed Elhoseiny
Venue:  (2021)
Keywords: Compositionality

Abstract: Compositional generalization is the algebraic capacity to understand and produce large amount of novel combinations from known components. It is a key element of human intelligence for out-of-distribution generalization. To equip neural networks with such ability, many algorithms have been proposed to extract compositional representations from the training distribution. However, it has not been discussed whether the trained model can still extract such representations in the test distribution. In this paper, we argue that the extraction ability does not transfer naturally, because the extraction network suffers from the divergence of distributions. To address this problem, we propose to use an auxiliary reconstruction network with regularized hidden representations as input, and optimize the representations during inference. The proposed approach significantly improves accuracy, showing more than a 20% absolute increase in various experiments compared with baselines. To our best knowledge, this is the first work to focus on the transferability of compositionality, and it is orthogonal to existing efforts of learning compositional representations in training distribution. We hope this work will help to advance compositional generalization and artificial intelligence research.

Under review as a conference paper at ICLR 2021
TRANSFERABILITY OF COMPOSITIONALITY
Anonymous authors Paper under double-blind review
ABSTRACT
Compositional generalization is the algebraic capacity to understand and produce large amount of novel combinations from known components. It is key element of human intelligence for out-of-distribution generalization. To equip neural networks with such ability, many algorithms have been proposed extract compositional representations from the training distribution. However, it has not been discussed whether the trained model can still extract such representations in the distribu- tion. In this paper, we argue that the extraction ability does not transfer naturally, because the extraction network suffers from the divergence of distributions, To address this problem, we propose to use an auxiliary reconstruction network with regularized hidden representations as input, and optimize the representations during inference. The proposed approach significantly improves accuracy, showing more than 20% absolute increase in various experiments compared with baselines. To our best knowledge, this is the first work to focus the transferability of compositionality, and it is orthogonal to existing efforts of learning compositional representations in training distribution. We hope this work will help to advance compositional generalization and artificial intelligence research. The code is in supplementary materials.
INTRODUCTION
Human intelligence (Minsky 1986 Lake et al. 2017) exhibits compositional generalization, the algebraic capacity to understand and produce large amount of novel combinations from known components (Chomsky 1957 Montague 1970). This capacity helps humans to recognize the world efficiently and to be imaginative. is also beneficial to design machine learning algorithms with compositional generalization skills. Current neural network models, however, generally lack such ability. Compositional generalization a type of out-of-distribution generalization (Bengio 2017). where the training and test distributions are different. A sample in such a setting is combination of several components, and the generalization enabled by recombining the seen components of the unseen combination during inference. In the image domain, or properties. In the language domain, a compound word is a combination of multiple words. As an example, consider two digits are overlapped (Figure M Each digit is component, and it appears in training. A test example has new combination of two digits.
object a combination of many parts
The main approach for compositional generalization is to learn compositional representations (Ben- gio 2013), which contain several component representations. Each of them depends only on the underlying generative factor, and does not change when other factors change. We call this the compositionality property, and will formally introduce in Section 3 In the digit example, this means that the representation of one digit does not change when the other digit changes.
Multiple approaches have been proposed to learn compositional representations in the train distribu- tion. However, little discussion has focused on whether the model can still extract the representations in the test distribution. We find that the extraction ability does not transfer naturally, because the extraction network suffers from the divergence of distributions (Bengio) 2017: Pleiss et al. 2020), so that each extracted representation shifts away from the corresponding one in training. Our experiment on the digit example shows that the accuracy drops from 89.6% in training to 49.3% in test (Table ] in Section 5)
To address the problem, we hope each representation is consistent with the training one while reflecting the test sample. We use auxiliary network, which has hidden representations as inputs,

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Figure 1: Examples of compositional generalization with overlapping digits. Each sample is a horizontal block with three images and two digits. The middle image X is input and the right two digits Y = Y1, Y2 are output. The left two images X1, X2 are hidden components. X1 is in its original form, and X2 is flipped over left-top to right-bottom diagonal. The sum of the digits even in train, and odd in test. We hope to learn prediction model in training, and transfer it to test.
and the original input as output. For test sample, we regularize each hidden representation in its training manifold, and optimize them to recover the original input. Then we use the optimized representations for prediction. Experimental results show that the proposed approach has more than 20% absolute increase in various experiments compared to baselines, and even outperforms humans on overlapping digit task. Our contributions can summarized as follows.
We raise and investigate the problem of transferability of compositionality test distribution. This work is orthogonal to many efforts of learning compositionality in training distribution.
We propose to address the problem by using an auxiliary reconstruction network with regu- larized hidden representations as input, and optimize the representations during inference.
We empirically show that the transferability problem exists and the proposed approach has significant improvements over baselines.
2 RELATED WORK
Compositional generalization Chomsky 1957 Montague 1970) is critical human cognition (Min- Johnson & et all 2017 Higgins & et al 2018 Lake et al.) (019)). It helps sky 1986 Lake et al.) Dohnsonn
humans to understand and produce large amount novel combinations from known components. Broadly speaking, compositional generalization is a type of out-of-distribution (o.o.d.) transferring or generalization, which also called domain adaptation (Redko al., 2020)) or concept drift (Gama al. 2014). This is different from traditional i.i.d. setting, where the training and the test distributions are identical. The transferring requires prior knowledge how the distribution is changed, and compositional generalization has particular form of such change, as mentioned in the later section.
Compositional generalization also a desirable property for deep neural networks. Human-level com- positional learning (Marcus 2003 Lake & Baroni 2018) has been an important open challenge (Yang) et al. 2019 Keysers & et all 2020 although there is a long history of studying compositionality in
networks. Classic view (Fodor & Pylyshyn| 1988 Marcus 1998 Fodor & Lepore 2002) con-
siders conventional neural networks lack systematic compositionality. With the breakthroughs in deep neural networks, there are more contemporary attempts to encode compositionality in deep neural networks. Compositionality in neural networks is actively explored for systematic behaviour (Wong Wang 2007 Brakel & Frank 2009), counting ability (Rodriguez & Wiles 1998, Weiss al. 2018) and sensitivity to hierarchical structure (Linzen al. 2016). Researchers have also proposed multiple related tasks (Lake Baroni 2018 Loula et al. 22188
al. 2019 and methods (Lake
al. 2017 Lake & Baroni 2018 Loula et al. 2018; 8: Kliegi & Xu 2018 Li Li al. 2019 Lake 2019
Gorden et al. 2020) for learning compositionality in training distribution. Another line of re related
work is independent disentangled representation learning (Higgins al. [017]] Burgess al.
2018
Kim & Mnih| 2018 Chen et al.) 2018; 8 Kumar et al.) | Hsich et al. 2018:] Locatell et al.
2019
2020). Its main assumption is that the expected components statistically independent in training
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data. This setting does not have transferring problem in test, because all combinations have positive joint probabilities in training (please see Section 3).
More recently, there have been approaches for better compositional learning in NLP tasks by
elaborating RNN models (Bastings et al. 2018), by using pointer networks (Kliegl & Xu 2018). by using representations of sentence (Russin et al.) 2019 Li al. 2019). In these tasks, the input can be divided into words, which have consistent information in different distributions. always available, e.g. the overlapping digits. In this paper, we propose
However, such property to acquire the compositional representations with optimization during inference. We will discuss more in the following sections.
3 COMPOSITIONALITY AND TRANSFERABILITY
In this section, we describe compositionality a model property. We then argue that compositionality may not transfer to test distribution, and discuss ideas of the proposed approach.
Compositionality Compositional generalization has different training and test distributions. Sam- ples in both training and test data are combinations of K components. For example, in Figure [ input X has two digits X1, X2, and output Y has corresponding labels Yi, Y2. While test sample's combination does not appear in training, each component the test sample appears in training.
key for compositional generalization is to have compositional representation, which has multiple component representations. Each component representation corresponds to an underlying input component, and it does not change when other components change. We call this property as compositionality. A compositional representation is computed from an entangled input, and the extraction network needs output correct component representations. If the extraction network transfers to test distribution, the representations can be correctly extracted in test, so that the compositional generalization enabled by recombining them.
Transferability As mentioned above, to enable compositional generalization, there is an assump- tion that the property of compositionality should transfer the test distribution. Since XI....,X$ X are entangled, the model has the entire X as input. However, input X has different distributions in training and test, so the extraction network suffers from the divergence distributions (Bengio) 2017, Pleiss et al. 2020). Hence, even if the model is trained to fit the compositionality property in the training distribution, the property is not guaranteed to transfer to the test distribution.
We propose to obtain compositional representations not from the extractor but reversely from an auxiliary network. We extract compositional representations with optimization, and introduce regularization make each hidden representation in the corresponding training manifold. We then use the optimized hidden representation for prediction. More details are provided in the next section.
APPROACH
In this section, we introduce the proposed approach from model architecture, training and inference perspectives. The architecture contains three modules, trained with routine end-to-end optimization. The inference, different from conventional procedure, includes three steps: extracts initial hidden representations; optimizes hidden representations as module input; predicts output. Figure Saaure/ccome contains overall flowcharts, and Algorithm is a summary for the approach. We describe details here.
4.1 MODEL ARCHITECTURE
The model takes a sample with input X and label Y. We have a representation extractor g with parameter d, which takes X as input, and outputs K hidden representations H = HI...., Hg: H g(;;)) Φ). We also have a prediction network with parameter which takes hidden representations H as input, and outputs Y: Y f(H; 0). These networks can be some existing networks for compositionality learning. In addition to them, we have auxiliary network h with parameters v. which takes hidden representations H input, and combines them to output X: X h(H;; v)).
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((X:6)
F(H:0)
g(X; X
f(H:0) Y
X
H
Y
H X
H H
h(H;0)
(a) Training flowchart. The three mod- ules are trained with end-to-end opti- mization.
(b) Inference flowchart. (Left) initial hidden representations extrac- tion. (Middle) optimization of hidden representations as module input. (Right) output prediction.
Figure 2: Flowcharts of the proposed approach. X is input, Y output, and H is hidden representa- tion. The architecture has three modules: g. f.
Algorithm The proposed approach for training (left) and inference (right). o, 9. 17 are hyper parameters. K is the number of components. M the number of instances in memory.
Training sample: X,Y
Inference sample: X
Hi...., Hx- (x
1: iinit g(X;
2: H' = H + ne, c (0,
2: X h(H;
3: Y ;((l';;), 0), X- h(Hl;;
Lmaan ()) = Σk minm (2 Mem
4: Lrain CEE(Y, 8) aL2(X, X) (H)
4: Linter (X, H) = L2(X, X) 7manr (
5: 0, v arg ming, 0,0. Liain (x 0,0,000 1)
5: H arg.ann min Linter (X, H), Ho = Hinit
6: Mem g(Xm; 5), , M
6: Y f(H:6)
4.2 TRAINING
In training, we sequentially use the extractor g and predictor f. by setting the output of the extractor H input of the predictor. We have a loss Loriginal (containing regularization terms). such cross entropy: CE(Y, Y), train model with compositionality by existing algorithms.
On top of that, we train with as inputs and X as output. We auxiliary loss as the difference (T22 distance) between and X: fauxiliaary = [2(X, X). We also regularize the L2 norm of H, Lhidten = T22 H). and add noise, H' = H + 1Je, E (O,, T), to avoid remembering X. is hyper parameter. The whole train loss Luaio the combination of the original loss, auxiliary loss Lauxiliary: and regularization Chidden, with coefficients a, 3.
Ciaan
Coriginal + aLaxxiliary 3L
We train the model in end-to-end manner. This can be standard training for neural networks.
0, L. の = arg min Cuain (X, 0,
0.0.$
After training, we store hidden representations for M training samples. They are used to restrict test representation manifold during inference.
Mem = =x(X",,) Vm = 1,...,1. M
4.3 INFERENCE
We use optimization to acquire hidden representations during inference. Given a test sample X, the model predicts its output Y. We use the auxiliary network h(H; Φ) to search for hidden represen- tations H so that h can output X that is close to the original intput X. This can be achieved by optimization H with auxiliary loss Lauxilarry The initial value Hinit is the output of extractor g: Hinit (X:: δ).
We also add manifold regularization term Luaat to constrain each hidden representation to lie in the corresponding training manifold. For a test sample, we compute the minimum L2 distance between each of its hidden representation Hg and the corresponding representations in memory Mem We
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then use the sum of the distances as regularization.
mant HT)
min
T22(H, Mem
m=I....M
Inference loss Linfer is the combination of auxiliary loss and the regularization, with y coefficient.
Cinfer (X, H) Lauxiliary (xt (X mant
Then, we obtain the hidden representations by optimization.
A arg min Linter (X, H), Ho Himi
We get prediction from the optimized hidden representations: Y f(i;
EXPERIMENTS
In this section, we show examples that, given a model with compositionality in training distribution, the compositionality does not transfer to test distributions, and how the proposed approach is applied these cases, our focus is orthogonal with learning compositional model training distribution, we obtain this model by directly providing true labels for each compositional component during training. Then we evaluate the transferability to test distribution.
Since this is the first work for the transferability problem of compositionality, we do not have previous baselines, so that we use standard deep neural network models as baselines, and also compare with variations of the proposed approach. The main changes of the proposed approach include using noise, auxiliary network and manifold regularization. A variation removes one of these changes, so is also ablation experiment. The details of hyper parameters and experiment settings can be found in Appendix A For all the experiments, we use accuracy as metric. A prediction is correct if and only if all the components are correctly predicted. We repeat each experiment five times and report mean and variance.
Overlapping digits
Compound words
Colored digits
Method
Test dist.
Train dist.
Test dist.
Train dist.
Test dist.
Train dist.
Human
10.0
Standard DNN
49.3 0.9
89.6 + 0.3
27.0 + 6.7
100.0 + 0.0
48.8 + 4.3
99.0 + 0.2
Variation -noise
44.2 0.9
79.8 + 0.9
11.3 + 6.2
0.0
37.7
4.0
96.9 + 0.6
Variation -auxiliary
51.1 0.7
88.7
46.8
3.7
100,0:000
10.6
98.7 = 0.3
Variation manifold
60.8 = 3.7
2.3
51.9
3.3
99.8 0.5
81.8 = 1.4
92.4 + 1.0
Proposed
69.4 0.3
81.1
51.2 3.0
100.0 0.0
91.2 = 0.6 96.2 + 0.4
Table 1: Evaluation accuracy (%). We see that the proposed method has significant improvement over the standard DNN on three datasets and outperforms human overlapping digit dataset.
5.1 EXPERIMENTS ON OVERLAPPING DIGITS
The first experiment on overlapping hand written digit recognition, as shown in Figure I We construct the dataset from MNIST (LeCun et al. 1998). A sample is made by overlapping and taking the average of two original images, the first one its original form, and the second flipped over up-left to down-right diagonal. The output is a vector of the two labels Y = Yi, Y2 (not exchangeable). Each original label has 10 possible values, so the output has 100 possible values. To evaluate compositional generalization,
different distributions in training and test. In train, the sum of the two labels is even, i.c. (Y1 Y2) mod? 2 = O. In test, the sum is odd.
As baseline, we standard neural network with two sub networks, each for output. Each of the network a three layer convolutional neural network. We train the model with cross entropy on both outputs.
The proposed method uses an auxiliary network that takes hidden layer as input and outputs the reconstruction of the original input. The auxiliary network has two sub-networks each with one
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Input
Outputs
Hidden
2
3
4
januarymarch
0,
january march
februaryoctober
1,
february october
januaryfebruary
0,
january february
2
3
4
augustmay
7.
august may
Table 2: Examples of compound word experi- Figure 3: Examples of input for colored digits ments. The output labels align with corresponding experiments. Output is the digit label and color hidden words. Upper is train, lower is test.
label. Upper is train, lower is test.
hidden representation as input. Each sub-network a three layer trans-convolutional neural network, and we average the outputs recover the original input. We use L2 loss the auxiliary loss.
We also collect human performance data through crowd sourcing. There are 27 participants, and each person works on 20 fixed samples randomly selected from test data. Please refer to Appendix B for more details.
The results in Table T (left) show that the proposed method has significant improvement over the baseline by about 20% absolute increase, and it also outperforms humans. The ablation study shows that performances drop in the experiments, indicating that all the modifications in the proposed approach are necessary to achieve the result.
5.2 EXPERIMENTS ON COMPOUND WORDS
We also conduct experiments for language processing. Language has natural units of words, which have consistent information across different distributions. So we design setting that we cannot use this property. We consider a problem that converts a compound word to two words. We construct compound words from two month names (January October), e.g. "julyfebruary". The output label the zero-based index of the month (0 for January). We have each character as an input unit, and assign one-hot representation to it. Other problem settings are the same as previous one. Please see Table 2 for more examples.
In baseline, We use feed forward neural networks, each for output. For the auxiliary network, we also use two feed forward neural networks, and average their outputs. Each feed forward network has three hidden layers. The training and other settings for baseline, proposed approach and ablations are the same as the previous experiment.
The results listed in Table [ (middle) demonstrate that the proposed method is significantly better than the baseline by around 24% absolute increase. We find removing manifold regularization in ablation study shows slight improvement over the proposed approach. Other ablation experiments have significant reductions. This might be because regularizing manifold is not important when inputs (characters) are discrete.
5.3 EXPERIMENTS ON COLORED DIGITS
We also explore the capability the proposed approach to another hand written digit problem with different types of components: digits and colors. We construct the dataset from MNIST (LeCun et al.) 1998) by changing the color digits. We use digit label Y, as a component (0-9), and color label Y2 as another (0-2). Color label is 0, 1, 2 for red, blue, green, respectively. In training, we use label combinations with: Y1 mod + Yg. In test, we use the rest of the combinations. Please refer to Figure 3 for examples.
We use two compositional neural networks for each output, respectively. Each network has three hidden layers. For the auxiliary network, concatenated the hidden representation input, and use a three layer cans-convolutional neural network. Other settings for the methods and ablation are the same the overlapping digit experiment.
The results shown in Table | (right) demonstrate that the proposed method is significantly better than the baseline by more than 40% absolute increase. It also outperforms the ablations, indicating

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2
3
8
9
90
TER (%)
80
70-
Baseline
50
Proposed
20
40
60
80
100
Inference steps
Figure 4: Out-of-distribution problem has differ- ent distributions in training and test. We hope to learn model in training distribution (blue), and use it in test distribution (orange).
Figure Transfer Error Rate (TER) during in- ference optimization. 100% is upper bound by definition. 50% means errors are balanced for in-distribution and out-of-diittrbbtion
that all the modifications necessary. Among them, auxiliary network contributes the most the performance improvement.
DISCUSSIONS
In this section, we perform visualization and error analysis to better understand the experimental results and the behavior of the proposed approach.
6.1 DISTRIBUTION VISUALIZATION
We visualize hidden representations of both baseline and the proposed approach for overlapping digit experiments (Figure 6). We use t-SNE (Maaten & Hinton 2008) reduce each of hidden representations to one dimension, and jointly plot them along horizontal and vertical coordinates, respectively. Training samples are blue, and test samples are orange.
Our expectation is a chess board like distribution, similar to the true underlying distribution (Figure J. Note that though there are 10 labels, the expected results may not be 10 10 colored blocks, because the labels, along with the representation, may not be in order (e.g. switching label 5, 6 reduces two blocks lines), and the first and the second hidden representations may differ for the same digit.
We find the visualization of the proposed approach (Figure 6b is closer to the expectation than the baseline one (Figure 6a). The proposed approach has less empty areas, indicating that it can recombine the components the out-of-distribution setting. This analysis demonstrates that the proposed approach works the expected way.
6.2 SAMPLE VISUALIZATION
We also hope visualize conerete samples for the proposed approach. Since we have the auxiliary network with two sub-networks for each hidden component representation, we visualize their outputs, and compare with the ground-truth (Figure 7.. The result shows that the original input and hidden components are reasonably recovered for both training and test samples. This means that the proposed approach is able extract information for each component in training distribution, and transfer the ability in test distribution.
6.3 TRANSFER ERROR ANALYSIS
We analyze errors to show that the inference optimization process helps addressing the transfer problem. When the model makes mistakes in test, the predicted output may correspond to in- distribution or -uttt--distriuttion label pairs. We investigate how frequent errors are associated with the distribution transfer. We define metric of Transfer Error Rate (TER) measure this. It is the

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75
75
Second representation
50
Secomd representation
50
25
25
0
-25
-25
-50
-50
-75
-75
-50
50
-50
50
First representation
First representation
(a) Baseline approach
(b) Proposed approach
Figure 6: Visualization hidden representations. Each representation is reduced to one dimension via t-SNE (Maaten & Hinton 2008). We plot them jointly, training in blue, and test in orange. The proposed approach (b) is close to expected chess board like result, similar to Figure
5
6
u
S
5
Train samples
Test samples
Figure 7: Visualization for the proposed approach. The first row is ground truth. The second row is recovered images from auxiliary network. The first column is the overlapping image. The second and third columns are the images for the first and the second components, respectively. The results show that the proposed approach is able to learn and transfer the ability to extract correct components.
number of errors predicted to be in wrong distribution, over all emrors. In this setting, the sum of labels is odd in test, and even in training, so TER is the number of errors with even predicted label sum over all errors. this rate high (100% is upper bound by definition), means that most of errors are associated with transfer. If it is around 50%. it means that the errors balanced in distributions.
TER 09.5 for baseline, and 68.3+2.89 2.8 for the proposed approach, with 500 test samples. Also, before the inference optimization in the proposed approach, the value is 88.8 08.8+2.0%. The results show that the baseline has most of the errors to be transfer error, and the propose approach reduces it significantly. Also, the reduction is during optimization in inference, because the value is close to baseline before optimization. This indicates that the proposed approach is effective to reduce TER, and optimization an important factor for Please refer Figure 5 for more details.
CONCLUSION
In this paper, we discuss our finding that compositionality does not transfer naturally from training to test data distributions. We further propose to address this problem with an auxiliary reconstruction network and a regularized optimization procedure during the inference stage. Experimental results show that the proposed approach has more than 20% absolute increase in various experiments comparing with baselines, and even outperforms human on the overlapping digit recognition task. We hope this work would reshape our thoughts on transferability of compositionality and help to advance compositional generalization and artificial intelligence research.
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Gail Weiss, Yoav Goldberg, and Eran Yahav. On the practical computational power finite precision mos for language recognition. arXiv preprint arXiv: 1805004908 2018.
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Under review as a conference paper at ICLR 2021
Francis CK Wong and William SY Wang. Generalisation towards combinatorial productivity in language acquisition by simple recurrent networks. In 2007 Interaational Conference on Integration of Knowledge Intensive Auett-Agent Systems, pp. 139-144. IEEE, 2007.
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Under review as a conference paper at ICLR 2021
A EXPERIMENT SETTINGS
A.1 EXPERIMENTS ON OVERLAPPING DIGITS
The data of overlapping digits experiment are generated from MNIST dataset (LeCun et al.) 1998), which contains 60,000 training data and 10,000 test data. We use the original training data in training and the original test data in test. In training, use the combinations with even label sum. In test, we use the combinations with odd label sum (evaluation on test distribution). As a reference, we also use the combinations with even label sum (evaluation training distribution).
Each input image has a shape of 28 x 28 x 3 for height, width and depth. Extraction network g(: Φ) has two convolutional neural sub-networks. Each of them has three convolutional layers. The first convolutional layer has 3 x 3 kemel and x strides with 32 depth and ReLU activation. It is followed by a max pooling layer with 2 kernel and 1 strides. The second convolutional layer has 3 x 3 kernel and 1 x strides with 64 depth and ReLU activation. Then the representation is flatten and fed to a fully connected neural network with 32 hidden nodes and linear activation. Prediction network f(:: 0) has a single layer fully connected neural network with Softmax activation for each pair of hidden representation and output. Auxiliary network h(::3/) 3)) has two cons-convolutional neural networks. A hidden representation is converted to 7 x 7 x 32 representation (width, height, depth) by a fully connected neural network with ReLU activation. The first trans-convolutionall layer has 3 x 3 kernel and 2 x 2 strides with 64 depth, SAME padding and ReLU activation. The second wrans-convolutiona layer has x 3 kernel and 2 x strides with 32 depth, SAME padding and ReLU activation. The third trans-convolutional layer has 3 3 kernel and x strides with depth, SAME padding and linear activation.
For the hyper parameters, optimization, we use Adam optimizer (Kingma& & Ba 2014) for 1,000 steps with batch size 500 and learning rate 0.001. In inference optimization, we also use Adam optimizer for 100 steps with batch size 500 and learning rate 0.3. We find the memory size M does not change the result significantly. We have 69. 0.6%% for M 100, 68.2+1.099 1.0% for 300, 09.4 for 500 (reported), 69.6+0.5% for 700 and 69. 0..4% for 900 (Figure 8).
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Figure 8: Evaluation accuracy for different memory sizes in the proposed method.
A.2 EXPERIMENTS ON COMPOUND WORDS
In the compound words experiment, we have inputs with different lengths. We first patch an additional character symbol to make all inputs the same length of 18. Then, we convert each character to one- hot representation with size 27. we use three layer fully connected neural networks as sub-networks for extractor. The input is flatten and fed to each sub-network. The first layer has 96 hidden nodes with ReLU activation. The second layer has 64 hidden nodes with ReLU activation. The third layer has 32 hidden nodes with linear activation. Prediction network is single layer fully connected network with Softmax activation for each pair of hidden representation and output. The auxiliary network also has two three layer fully connected sub-networks. The first layer has 64 hidden nodes
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Under review as a conference paper at ICLR 2021
with ReLU activation. The second layer has 96 hidden nodes with ReLU activation. The third layer has 486 hidden nodes with linear activation. Then the nodes are reshaped to 18 x 27.
For the hyper parameters, optimization, we use Adam optimizer for 1,000 steps with batch size 500 and learning rate 0.001. In inference optimization, we also use Adam optimizer for 100 steps with batch size 500 and learning rate 0.3.
1. 3 0.1,
0.01, " 1, M 500. In training
A.3 EXPERIMENTS ON COLORED DIGITS
The data for colored digits experiment are created from MNIST dataset. Each input image is in shape of 28 x 28 x 3 for height, width and depth. The extractor has two sub-networks, each has three convolutional layers. The first convolutional layer has 3 kernel and 1 x strides with 32 depth and ReLU activation. A max pooling layer follows it with 2 x 2 kernel and 1 x 1 strides. The second convolutional layer has 3 kernel and x 1 strides with 64 depth and ReLU activation. Then the representation is flatten and fed to a fully connected neural network with 32 hidden nodes with linear activation. Prediction network has a fully connected network for each pair of hidden first concatenate the
representation and output with Softmax activation. For auxiliary network, hidden representations to have 64 nodes. It is then converted to x 7 x 32 representation (width, height, depth) by a fully connected neural network with ReLU activation. The first trans-convoluttinall layer has 3 x 3 kernel and x 2 strides with 64 depth, SAME padding and ReLU activation. The second trans-convolutional layer has 3 kernel and 2 strides with 32 depth, SAME padding and ReLU activation. The third trans-convolutional layer has 3 x 3 kernel and 1 x 1 strides with depth, SAME padding and linear activation.
1, M = 500. In training optimization, we use Adam optimizer for 1.000 steps with batch size 500 and learning rate 0.001. In inference optimization, we also use Adam optimizer for 100 steps with batch size 500 and learning rate 0.3.
For the hyper parameters, we use a = 1, B = 0.3, 7 = 0.03,
B CROWD SOURCING FOR HUMAN PERFORMANCE
We collect data human performance through crowd-sourced survey. The survey has training and test sections. We select 20 images uniformly at random from the training distribution as training data. We also select 20 images uniformly at random from the test distribution test data (Figure 9. In the training section we state that each picture is an overlap of two pictures containing two numbers. Then we ask survey takers discover the rule through the training examples before they answer questions. In the test section we put up test images, and two multiple-selection (0-9) list boxes under each image. Then we ask the survey takers identify the correct answers. In this survey we had 27 responses. The mean test accuracy is 10.0% with standard deviation of 7.7%
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Figure 9: Test samples and corresponding component images for crowd sourcing. Each sample is horizontal block with three images. The left two images are hidden components. The right image is the overlapping of them. Only the overlapping image is shown to participants.
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Reviews:
[{"review_id": "xGHcvStyzNG", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 4, "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": "The paper introduces a “transferability of compositionality” problem and proposes an approach to alleviate it. The said problem may arise when one trains neural models to produce “compositional” representations of the input. In the paper “compositional representations” consist of multiple vectors which are supposed to correspond to semantically meaningful aspects of the input, for example different objects in the case of images or different parts of compound words in the case of linguistic inputs. The transferability problem arises when there is a difference between training and test distributions, namely when certain combinations of objects have different probabilities in training & testing. The proposed solution at inference time is to project object representations to the manifold of individual object representations. The manifold is estimated by saving representations of individual object representations from the training time. \n\nThe problem that the paper considers is an interesting one. There have been a lot of papers on learning object-oriented representations recently [1, 2], and an implicit assumption in all these works is that there is no statistical dependency between which objects that occur in the scenes. There is also the literature on disentangled representations that the paper extensively cites, where the independence assumption is also common. \n\nMy concerns regarding the paper are as follows: \n- Positioning with the respect to the prior work. The literature on learning object-oriented representations is not cited. The work on disentangled representation is cited, but still new setups are created from scratch. \n- Related to the previous point, the use of full supervision (in the form of labels) in the proposed tasks strikes me as a deviation from most previously used setups. Previous work aimed to learn compositional representations without supervision, often positioning their efforts as a cog-sci-style inquiry in building human-like models. The use of labels makes this look less like a cog-sci and more like a machine learning paper. Viewing the work as an ML paper, one thing that stands out is the lack of connections to any applied ML problem. \n- I think the negative results in the paper would look stronger if pretrained image- and language- processing models were used in all experiments (e.g. Contrastive Predicting Coding & BERT) \n\nThe proposed method seems appropriate for the tasks that the paper considers. The experiments appear to be technically sound. My main concerns are thus focused on the motivation of the proposed tasks themselves and the positioning with respect to their prior work. I think a great direction to improve the paper would be to add experiments without supervised learning and using 3D-rendered images with multiple objects as it is done e.g. in [1] and [2]. \n\nFew comments on writing: \n- Algorithm 1 is very confusing because sample-level steps 1-4 are mixed with dataset-level steps 5 and 6. \n- A confusing sentence in the intro: “For a test sample, we regularize each hidden representation in its training manifold, and optimize them to recover the original input”\n- for colored digits experiments you might want to compare to and cite [3] \n\n- [1] “Multi-Object Representation Learning with Iterative Variational Inference” by Greff et al, 2020\n- [2] “MONet: Unsupervised Scene Decomposition and Representation” by Burgess et al, 2019\n- [3] “Invariant Risk Minimization” by Arjovsky et al, 2019", "extra_scores": {}, "extra_text": {}}, {"review_id": "kJbEvyHG9U", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "## Summary\n\nThis paper studies \"compositionality\" and in particular the way in which it \"transfers\" on test data. They run simple baselines on three experiments (overlapped MNIST, colored MNIST and concatenated month names) and find that the baselines do not learn compositional representation. They proposed the use of an *auxiliary reconstruction network and a regularized optimization* which improves on these baselines. \n\n## Analysis\n\nThe authors frequently say that *compositionality may not transfer to test distribution* but I have a hard time understanding exactly what they mean by this. As I understand it, \"compositionality\" is a property of a representation. Do authors mean that, on the test data, the representation of an input is able to separate multiple components, yet the same network does not separate the components on the test data? It may be true for the models they trained here, but I would have appreciated a comparison with other methods. As such, I find the claims of this paper difficult to evaluate with respect to previous work. They claim theirs is the *first work for the transferability problem of compositionality* which I find really hard to believe. I would have appreciated a thorough study of the \"compositionaly\" limitations of previous techniques.\n\nI found section 4 particularly hard to understand. A lot of symbols, equations and nomenclatures seem to be used with too little introduction. As a result, I cannot vouch for the correctness of this section.\n\nGiven their claim that this is the *first work for the transferability problem of compositionality* the experiments presented on section 5 are on a new dataset and are not compared to previous work. Moreover, the proposed experiments seem relatively simple (two overlapped MNIST digits, colored MNIST digits, concatenated month names) and their baseline seem trivial (*we use a standard neural network with two sub networks, each for an output*) \n\n## Conclusion\n\nOverall, I find the claim of this paper substantial, while the experiments are relatively simple with trivial baselines and an absence of comparison to related work.\n\n## Typos\n\nI find the text difficult to read, it would benefit from a thorough revision. Ex: *This work is orthogonal to many efforts of learning compositionality in training distribution.*\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "7_3ehygfkjK", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "*Summary*\n\nThis paper proposes an architecture that addresses transferability of compositionality. The proposed architecture consists of three components: a network that transforms the input X into a series of hidden representations {H_1, H_2, ... H_K}, a network that reconstructs the input X from this series of hidden representations, and a prediction network that generates a prediction from the hidden representations. The authors propose several datasets meant to address transferability of compositional generalisation, and show that their architecture significantly improves standard DNN architectures as well as humans on these datasets.\n\n*Motivation for score*\n\nCompositional generalization in neural networks is a relevant and hot topic, with still many open questions. This paper aims to contribute on this topic and proposes some interesting datasets. However, However, despite citing several papers addressing compositionality in neural networks in the related work section, I am not convinced that the authors have properly understood the questions that are asked in this domain and were able to address them properly. Below, I outline my concerns.\n\n1. Definition of compositionality\n\nI do not find the definition of compositionality that the authors propose well motivated.\n- None of the three papers cited in the introduction to motivate the work actually has the word \"compositionality\" in the paper\n- The authors claim that previous work has focused just on whether models can extract compositional representations in the training distribution while ignoring the test distribution, while actually most recent papers they cite in related work test compositionality by considering very specific train/test splits\n- The author's definition of compositional generalisation does not seem to take into account that compositionality is traditionally a property of mapping between input and output, not of a model itself. In addition to that, whether the mapping between input and output is compositional does not depend on what is in the train and what is in the test distribution. A model can understand the compositional structure of a dataset also if it has been trained on *all* examples of the dataset, only it will be impossible to behaviourally evaluate if it has. For this reason, much previous work on compositionality in neural networks has created datasets where the training and testing data were distributionally different (as also the authors of this paper do).\nI would recommend the authors to have a look at the paper _Compositionality decomposed: how do neural networks generalise?_ (Hupkes et al.; 2020), for a detailed account of compositionality in the context of neural networks. In particular, their section on _localism_ is particularly important for the author's definition of compositionality.\n\n2. Architecture\n\nI find the proposed architecture interesting, but it is not completely clear to me how it differs from an auto-encoder setup where the encoding is larger than the input instead of smaller (it is very well possible I misunderstood). Nevertheless, it can be interesting to see if auto-encoding based architectures behave better on datasets proposed to evaluate compositionality. One thing that is not clear to me is how the number of components _K_ is determined.\n\n3. Data\n\nI appreciate the effort of the authors to design new datasets that test out-of-distribution generalisation. I do have a few comments/questions:\n- If the main motivation for wanting compositional generalisation is that this is an important capacity of humans, isn't it a problem that humans perform very poorly on the dataset (much worse than the best deep neural network)?\n- What is the motivation for using a new dataset, rather than one of the previously proposed datasets for out-of-distribution generalisation?\n- It is nice that the authors try to include tests from different domains, but I think that calling a dataset mapping inputs like \"januarymarch\" to (0, 2) cannot really be called \"natural language processing\"\n\nOverall, I do not believe that this paper should be accepted for the conference.\n\n_**Update after author response:** I have read the author response, but do not find that the answers really address my concerns.  I have also not really seen any improvements in the paper itself._ ", "extra_scores": {}, "extra_text": {}}, {"review_id": "2oUn545DV9i", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 2, "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": "##### Impact:\n\nThe submission claims that other works that investigation compositionality in representation learning do not actually test compositional generalization (\"because all combinations have positive joint probabilities in training\"). However, I disagree that this is the case in prior work; here are some examples of prior works that correctly hold out novel combinations (of underlying components) for test time:\n- https://openreview.net/forum?id=HJz05o0qK7\n- http://papers.nips.cc/paper/8825-learning-by-abstraction-the-neural-state-machine\n- https://arxiv.org/abs/1910.09113\n- https://arxiv.org/abs/1912.09713\n- https://arxiv.org/abs/1912.12179\n\nThere are many such examples; they are too numerous to list here.\n\n##### Quality: \n\nThe algorithmic components in Section 4 are not adequately motivated, and the relationship of the algorithm to prior work in compositional representation learning is not discussed.\n\n  The evaluation tasks are extremely simple (overlayed MNIST digits and conjoined word token) and are, as such, far from the complexity of existing work on compositionality (which can deal with, for example, naturalistic image data; see the references above for examples of such works).\n\n##### Clarity:\nThere are many points of ill clarity / inconsistencies; for example:\n- \"The main approach for compositional generalization is to learn compositional representations\" Is this really the \"main approach\"? Compositional generalization has been studied in many contexts outside of representation learning (e.g., see https://semanticsarchive.net/Archive/jcyZDc1Y/Goldberg.Compositionality.RoutledgeHandbook.pdf)\n\n- \"We find that the extraction ability does not transfer naturally, because the extraction network suffers from the divergence of distributions\" Why is it assumed here that there is an extraction network? The \"extraction network\" is referred to several times in the introduction and methods section prior to its introduction/explanation.\n\n- \"compositional generalization is a type of out-of-distribution (o.o.d.) transferring or generalization, which is also called domain adaptation\" This is inconsistent with the previously discussed definition of compositional generalization i.e., that it is not just domain adaptation. I think the submission could do with a better job of dealing with the distinctions between OOD generalization / domain adaptation / concept drift.\n\n- \"We propose to obtain compositional representations not from the extractor but reversely from an auxiliary network.\" At this point, neither the \"extractor\" nor the \"auxiliary network\" are defined.\n\n- \"These networks can be some existing networks for compositionality learning\" If so, what are examples of \"existing networks for compositionality learning\"?", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This work considers an apparent problem with current approaches to compositional generalisation (CG) in neural networks. The problem seems to be roughly:
1. prior work in CG aims to extract 'compositional representations' from the training distribution
2. work on CG, the training set and the test set are drawn from different distributions
therefore
3. we don't know whether these models can also extract compositional representations from the test distribution

All four expert reviewers were, to differing degrees, confused by this problem framing, largely because they consider the premise (1) to be false. 

I am also aware of a large body of recent work on CG in neural networks (see those papers listed by R2) and, as far as i know, none of it involves extracting 'compositional representations' from the training set. Rather, it involves learning something (from the training set) that enables strong performance on a test set that differs from the training set in a way that is informed by ideas of compositionaity. 

As far as I know, there are very few  studies that try to identify compositionality by considering the internal representations of neural networks, so it feels incorrect to claim this is standard practice. Any work that goes down this route ought to have a very thorough treatement of the various thorny philosophical and theoretical treatments of compositionality in the literature. As pointed out by R4, the work in its current form does not do this. 

In summary, this work attempts to solve a problem that none of the four expert reviewers consider to be in need of a solution. 
Decision: Reject