Neural Non-additive Utility Aggregation
Authors: Markus Zopf
Venue:  (2020)

Abstract: Neural architectures for set regression problems aim at learning representations such that good predictions can be made based on the learned representations. This strategy, however, ignores the fact that meaningful intermediate results might be helpful to perform well. We study two new architectures that explicitly model latent intermediate utilities and use non-additive utility aggregation to estimate the set utility based on the latent utilities. We evaluate the new architectures with visual and textual datasets, which have non-additive set utilities due to redundancy and synergy effects. We find that the new architectures perform substantially better in this setup.

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NEURAL NON-ADDITIVE UTILITY AGGREGATION
Anonymous authors
Paper under double-blind review
ABSTRACT
Neural architectures for set regression problems aim learning representations such that good predictions can be made based on the learned representations. This strategy, however, ignores the fact that meaningful intermediate results might be helpful to perform well. We study two new architectures that explicitly model la- tent intermediate utilities and use non-ac additive utility aggregation to estimate the set utility based the latent utilities. We evaluate the new architectures with vi- sual and textual datasets, which have non-additive set utilities due to redundancy and synergy effects. We find that the new architectures perform substantially bet- ter in this setup.
INTRODUCTION
In this paper, study the problem of learning to predict utilities for sets of objects, which denote set regression. Let = {o)...... o be a set of objects. Let C {Ci..... C. be set of sets that consist of objects from set O, i.e., C* = {oi..... 0j} with 0i,-.. Oj E O. During training, models only observe sets from C and their corresponding utilities and have to learn a function e : C - TR that maps from sets to set utilities. At test time, utilities of unseen sets with potentially unseen objects have to be estimated.
The standard idea approach this problem in representation learning is to learn representations for sets in C such that the utilities can be predicted based on the generated representations. More formally, for sets C E C an aggregation function OoeC, p(o) is learned that produces an aggre- gation of the individual object representations ((o). A function v e(Ooecyp(o)) is trained (usually jointly with the aggregation function) to predict the utility of the set. One often used aggregation strategy is to simply add the individual object representations (0() Zaheer et al. (2017). Another strategy is to use recurrent neural networks (RNNs) that learn to aggregate object representations. On a conceptual level, the idea of both approaches is to encode everything that is relevant to predict the utility of a set in its aggregated representation. This approach, however, ignores the fact that meaningful intermediate utilities for the objects contained in the sets can be predicted in many prob- lems. For example, intermediate utilities of sentences in automatic summarization can be predicted and aggregated to obtain the summary utility.
crucial limitation of this idea is the fact that objects in sets do not necessarily have an intrin- sic value that is independent from the other objects in the same set. For example, repeating the same information multiple times in automatic summarization does not improve the summary utility even though the individual sentences might contain crucial information. Furthermore, sentence A might depend on another sentence B (e.g.. to resolve pronouns) such that sentence A can only be understood sentence B is also present in the summary. Even if sentence A contains valuable infor- mation, it does not have an intrinsic value. Its value becomes only effective when B is also included in the summary. Since such redundancy and synergy effects occur many problems, additive utility aggregation is not appropriate.
To resolve this problem, we propose neural non-additive utility aggregation. Contrary to previous approaches, we propose to leam intermediate latent utilities for objects in O and non-additive aggregation function for the intermediate utilities. To this end, we make use of the discrete Choquet integral, which is non-additive generalization of the well-known Lebesgue integral. Instead of using additive measures, the Choquet integral makes use of non-additive capacities. Hence, it is well-suited to deal with phenomena such redundancy and synergies.

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(a) DeepSets (DS)
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(b) Recurrent Neural Network (RNN)
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(c) Recurrent Choquet Network (RCN)
(d) Deep Choquet Regression (DCR)
Figure I: Illustration of prior architectures la and lbj and the newly proposed architectures (Ic and Id.. ip; denote feature representations of the input data and h2 indicate hidden states. are gates that are responsible for handling redundancy and synergy effects in the DCR architecture. Each arrow models (potentially non-linear) function. The sign indicates addition of latent utilities and the sign indicates addition of vectors. and u, indicate the predicted set utility and predicted latent utilities, respectively.
Main Contributions. In this paper, (i) we propose novel network architectures for set utility prediction that explicitly exploit the fact that many problems can be modeled as non-linear aggre- gation of latent element scores, (ii) demonstrate the superiority of the newly proposed architectures compared to traditional representation learning in computer vision and natural language understand- ing datasets, and (iii) make all datasets and code publicly available for further research.
NEURAL NON-ADDITIVE UTILITY AGGREGATION
In this section, we present two neural non-additive utility aggregation architectures based the Choquet integral. The key idea implemented in both architectures is to train models that predict latent intermediate utility for observed objects in given set in each step. The latent utilities are aggregated with a learned non-additive aggregation function. This idea is inspired by the dis- crete Choquet integral Choquet (1954); Grabisch al.) (2009). The non-additivity of the Choquet integral is crucial for our work, because we focus on non non-additive problems such as automatic sum- marization. Both architectures are described below. We illustrate both architectures along with prior approaches in Figure [ Due to space restrictions, we refer to Tehrani et al. (2012a) for solid dis- cussion of the Choquet integral for machine learning and focus in this work on the newly proposed architectures and their evaluation.
2. RECURRENT CHOQUET NETWORKS
The first architecture, Recurrent Choquet Neural Networks (RCNs), uses an RNN-like approach to construct hidden state in each step. The hidden state is used to predict latent intermediate utility that already takes non-additive effects into account. The predicted utilities are directly related to the aggregation of capacities in the Choquet integral, which also take non-additive aggregation effects into account. Consequently, the predicted utilities can simply be summed to obtain non-additive set utility. The architecture is illustrated in Figure Ic
In this work, we model hidden states h4 according to h2 = tanh(w(1) [e]l (he-1 (pe)). where denotes feature vector concatenation, |/| denotes the size of the hidden states and |vpl the size of the input representation. The latent utility at step is computed according to a simple linear layer = w2(2) Al.1 h4. The set utility is then computed by = Σ u, Needless to say, other, more sophisticated functions
also be implemented. In this work, we decided to focus rather simple functions to focus on the essential novel part of the new architectures and to achieve a better comparability to the reference models.
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2.2 DEEP CHOQUET REGRESSION
The second architecture, Deep Choquet Regression (DCR), predicts utility for each object in the set individually. To model non-additive effects, hidden states are learned, which are used predict gates 9t. Note that the gating in DCRs is different to the attention mechanism, since the attention mechanism aggregates representations and not latent intermediate utilities.
Again, the products of gates and predicted intermediate utilities can be viewed as capacities of the Choquet integral. The architecture is illustrated in Figure ld The computation of the hidden states equals to the RCN. The individual utilities are computed directly based on the object representations
and not based on the hidden states. Hence, we obtain u, w 1pl.1 Pr. The gates are computed
according to = (W lb].1 h2). The set utility is computed according to v == E.= u4.
3 EXPERIMENTAL SETUP
In the following experiments, we want to evaluate whether the proposed approach based on the Choquet integral performs better than the standard representation learning approach situations in which aggregated score of multiple elements has to be estimated. Specifically, we are interested in situations which the aggregated score cannot simply be estimated by adding individual scores but tasks in which the aggregation depends previously observed elements. Next, we describe the datasets and the experimental setup that we designed to gain insights on this setup.
3.1 REFERENCE ARCHITECTURES
The goal of the experiments is to investigate whether aggregating intermediate results works better than predicting the utility based single representation. Hence, compare the newly proposed architectures to two standard approaches in representation learning that both generate representa- tions for sets.
3.1.1 1 SUM BASELINE
The Sum Baseline (SB) is inspired by prior works in group rating, in which set utilities usually modeled as the sum of individual instance utilities. This baseline simply learns to predict instance utilities independently and sums all predicted utilities. Similar to the new architectures, we use simple linear model in a first version of the model, namely SB-s (s for small), to predict element utilities: U, Wil.. St. In second version, SB-I, we use slightly more complex function: 44 (! a (W (2)
k (P)).
This baseline also way determine the strength of the non non-additive effects. the set utility is based on less non-additive effects, e.g. only small redundancy effects, the Sum Baseline should be able to yield a good performance.
3.1.2 DEEPSETS
The first architecture computes an non-weighted linear average of intermediate representations. This strategy is also known as DeepSets Zaheer et al. (2017). We visualize the strategy in Figure
Formally, set representation computed according to Pset = Σn= Pi. The newly gained repre- sentation is used to predict the set utility. Again, we experiment with 2 different versions, DS-s and DS-1 which compute
W 1l,1 (Peet and Will) a(W((2) |p].8 (set), respectively.
3.1.3 RECURRENT NEURAL NETWORKS
The third reference architecture are recurrent neural networks, which consume object representations iteratively and generate a new representation that is supposed to represent all consumed object in each step. Contrary DeepSets, recurrent neural networks have more sophisticated representation aggregation function. Formally, models based on RNNs compute in each step a hidden state based previously computed hidden state. To increase comparability of the difference architectures,
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the same method to compute hidden states: h tanh(W MIJAa (h4-1 (1)). The final
hidden state is used as set representation to predict the utility of the set according to v W [A],1 hn- RNNs are visualized in Figure lb.
3.2 DATASETS
To evaluate the different architectures, we conduct experiments based on visual and textual data. The datasets are described below.
3.2.1 MNIST REDUNDANCY AND MNIST SYNERGY
Task Description. The first datasets that use are based on the MNIST dataset LeCun et al. (1998. Similar to Zaheer et al. (207)) : lsse et al. 2018), we use the images in the MNIST dataset to generate sets of number images. We assign each set a utility according to different aggregation functions. The aggregation functions are unknown to the machine learning models, whose task is to learn to estimate the correct utility from training data.
Set Utility. We generate two datasets based on MNIST. The first dataset, MNIST-R, models re- dundancy effects. In MNIST-R, the set utility equals to the of all uniquely appearing dig- its. Hence, each element utility is only considered once and each further appearance of an al- ready counted element utility is ignored. For example, the {or, 02, 03) with element utilities 01 = 3, 02 5, 03 = 3 has a utility of 8 in this dataset. The second dataset, MNIST-S, models synergy effects. In MNIST-S, the utility of the set equals the sum of all elements that appear at least twice. Elements that appear only once are ignored. The previously discussed example set {01, 02. 03) has a utility of in MNIST-S.
Image Representation. To represent the images, we implement and train simple neural network model according to publicly available digit classification model The model compresses the 784- dimensional input images in several layers to 128-dimensional and 64-dimensional representations before softmax is applied to 10-dimensional layer. We feed MNIST images into the trained model and extract the activations of the 64-dimensional layer, which yields 64-dimensional image representations.
3.2.2 SUMMARY UTILITY PREDICTION
Task Description. The second task used in the experiments estimating summary utilitics in au- tomatic summarization Mani (2001); Nenkova & McKeown (2011). In automatic summarization, set of source documents is given and the task is to generate a summary which contains the most im- portant information from the source documents. The more important information the automatically generated summary contains, the higher the utility of the summary. Automatic summarization is often modeled as sentence selection problem in which the order of the selected sentence is not irrel- evant for the evaluation. In this case, the summary is simply a set of sentences, which makes the task appropriate for the scope of this paper. Non-additee aggregation effects occur automatic summa- rization naturally. Repeating the same information multiple times, for example, does not increase the information content of the summary and, hence, does not increase the summary quality. Simi- larly to the previous dataset, we make use of the well-known TAC2009 dataset Dang & Owczarzak) 22099) for automatic summarization. The TAC2009 contains several topics, each of which contains multiple source documents. The dataset contains automatically generated summaries from over 55 systems.
Set Utility. Each automatically generated summary in TAC2009 annotated with so called Pyra- mid SCUs (Summary Content Units) Nenkova al. (2007). Each Pyramid SCU refers to in- formation nugget that is contained in the text. The SCUs have been weighted with the Pyramid method. More important SCUs have higher weight than less important SCUs. The weight of each SCU is an integer between and 4. The utility of a summary is calculated by adding the weights of all contained SCUs. However, including SCU multiple times summary does not improve
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Table 1: Experiment results for visual and textual data
MNIST-R
MNIST-S
Sum-64
Sum- num-768
Model
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
SB-s
5.19
40.60
7.60
88.71
1.78
5.03
0.65
1.02
SB-1
5.07
39.50
6.99
75.08
0.75
1.23
DS-s
5.20
40.73
7.62
89.26
1.78
5.04
0.49
0.55
DS-1
4.19
25.77
7.69
88.72
1.53
4.24
RNN
3.88
22.09
7.36
82.50
0.88
1.42
0.18
0.34
RCN
3.43
17.91
7.00
75.01
0.91
1.49
0.12
0.16
DCR
3.23
15.57
6.91
74.60
0.90
1.02
0.55
1.00
the utility of the summary, because repeating an SCU means that the same information is contained multiple times in the summary, which is not desired. Synergy effects not modeled by SCUs.
Sentence Representation. For this task, we need a feature representation for sentences. We use two different representations in the experiments. The first representation are derived from the BERT language model Devlin et al. (2019). We use the publicly available Sentence Transformers library convert the original sentences into 768 188-dimensional sentence embeddings. Since the representation is high-dimensional, also train simple auto-encoder to generate smaller sentence embeddings of size 64.
3.3 IMPLEMENTATION AND TRAINING DETAILS
We implemented all reference and newly proposed architectures PyTorch 1.2. The source code and the used datasets will be publicly available and can be used to reproduce experiments. All experiments have been performed on NVIDIA Tesla K80 GPUs. All datasets contain 10,000 sets. We use random splits of size 8,000, 1,000, and 1,000 for training, validation, and testing, respectively. Experiment-specific details can be found in the corresponding experiment description below. For each experiment, limit the maximum number of epochs 1,000 and perform early stopping based on the validation set with patience of 50 epochs. For all experiments, we use the Adam optimizer and start with learning rate 0.001. We use batch size of 100. We use the mean squared eтror (MSE) as well as the mean absolute error (MAE) as loss functions and evaluation metries. Initial hidden states are initialized with zero-vectors. We use default initialization for all model weights.
EXPERIMENTS
In this section, describe experiment-specifee details, report the outcomes of the experiments, and discuss the results.
4.1 EXPERIMENTS WITH VISUAL AND TEXTUAL DATA
Description. In the first experiment, we evaluate the reference architectures and the newly pro- posed architectures both visual and textual data figure out how well the resulting models can deal with redundancy and synergy effects in both modalities. Based on the MNIST data, we gener- ate sets with length 10 images and compute the set utility according to the previously described redundancy and synergy functions. For summarization, we create sets of size 4 because the average length of the summaries contained in the TAC2009 is very close 4. Hence, set sizes of 4 rea- sonable and realistic. We perform experiments with the compressed 64-dimensional and the original f68-dimensional BERT embeddings.
Results. To improve the quality of the experiment, performed each experiment 5 times with different random seeds. Table [ contains the mean of the 5 runs. We optimize for mean average

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200
SB-s
SB-s
SB-1
square root of frequency
20
SB-1
DS-s
150
DS-s
DS-1
DS-1
frequency
RNN
RNN
RCN
100
RCN
DCR
10
DCR
50
0
10
15
10
squared error
absolut error
Figure 2: Distribution of absolute errors for MNIST-R experiments.
Figure 3: Distribution of squared errors for MNIST-R experiments.
emror and mean squared error as described before and report results for both experiments. Overall, the newly proposed Deep Choquet Regression (DCR) architecture performs best. It achieves the lowest error out of 8 experiments. The only weakness the architecture occurs the sum- marization experiment with fo8-dimensional BERT embeddings. The Recurrent Choquet Network (RCN) achieves best results in the last experiments and is placed second in all MNIST experiments. In Sum-64, it is close the best results. With some distance, the Recurrent Neural Network (RNN) performs mediocre in the MNIST experiments. In summarization, it works better. This result might size the summarization experiments compared to the MNIST exper-
from the smaller iments (set size of vs. 10). The DeepSets (DS) and the Sum Baseline (SB) perform poorly in MNIST, SB achieves good results Sum-64. This might be explained by the smaller impact of the redundancy effects compared to the MNIST data. In general, optimizing according to MAE and MSE correlates well. The best architecture according to MAE and MSE only differs Sum-64. In general, conclude that recurrent models perform better than the non-recurrent models although the input data is permutation invariant. For SB-I and DS-I, no experiments have been conducted with the large 768- 1--dimmeniooal BERT embeddings.
Error Distribution. In addition to the averages reported Table we plot the distribution the Figure errors in Figure 3 for for the MNIST redundancy experiment for the absolute error and the mean squared error, respectively. For both mean average errors and mean squared errors, it can be that the pared to the reference models (orange/blue/red). The mean squared error curves are much steeper, which results from the fact that larger errors are penalized much more than small errors.
models (black) make much more small mistakes and fewer large mistakes com-
4.2 VARYING SET LENGTH
Description. In the second experiment, we are interested in how well the architectures can deal with varying set sizes. To shed light on this question, we generate sets based the MNIST data with redundancy and synergy aggregation functions. In contrast to the previous MNIST experiments, which have fixed set length of 10, randomly generate sets with lengths of 6, 8, 10, 12, and 14 MNIST images. The results of the experiments can be found in Table This time, we only averaged 2 runs for each experiment due computational limitations.
Results. In general, the errors tend to increase, which is reasonable due to the more difficult setup. However, the increase is rather moderate. Similar the previous experiment, DCR and RCN per- form best. This time, no clear winner can be distinguished between DCR and RCN. RNN is again placed third and the other architectures perform poorly.

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Table 2: Results for varying set lengths
Table Extrapolation results
MNIST-R
MNIST-S
MNIST-R
MNIST-S
Model
MAE
MSE
MAE
MSE
Model
MAE
MSE
MAE
MSE
SB-s
4.93
38.56
7.77
91.13
SB-s
4.92
57.17
25.57
1292.23
SB-1
4.87
37.37
7.12
74.70
SB-1
7.25
80.18
32.95
1350.70
DS-s
4.93
38.08
7.76
91.07
DS-s
6.55
66.38
16.22
406.44
DS-1
4.06
24.00
7.83
86.73
DS-1
6.58
53.21
37.51
1702.10
RNN
3.85
21.35
7.46
80.90
RNN
9.37
128.19
35.84
1687.71
RCN
3.24
17.34
7.16
76.87
RCN
4.23
29.75
21.74
866.62
DCR
3.68
20.20
7.02
76.28
DCR
4.25
37.79
16.69
450.81
4.3 EXTRAPOLATION TO LONGER/SHORTED SEQUENCES
Description. In the third experiment, we are interested in the extrapolation abilities of the different architectures, i.c. how well they are able to extrapolate beyond the observed training instances. This provides a good insight whether the models were able to identify and learn the underlying nature of the problem or if they fit to tightly to the observed distribution. To this end, we train the models on sets with length 10. Furthermore, the validation set, according to which the best model is selected, also contains only sets with length 10. The test set, however, only contains sets of size 5 and 20. Hence, the model has to extrapolate from set sizes of 10 to set sizes and 20.
Results. The results can be found in Table 3 Again, we report the average of 2 runs for each experiment. For MNIST-R, the RCN architecture performs best and DCR is second. The RNN architecture performs worst in the extrapolation experiment in MNIST-R. In MNIST-S, DS-s is best. This can be explained by the nature of the synergy effect. The larger the set, the becomes that images representing the same numbers appear multiple times in the set. Hence, it not important anymore count the appearing digits and simply identifying the appearance of a digit can lead to reasonable results. The same effect can be observed when DS-s and RNN are compared. In prior experiments, RNN performed much better than DS-s. In this experiment, however, DS-s performs better. DCR is close to DS-s and performs second best while RCN placed third. The other architectures, namely SB-s, SB-1, and DS-I perform poorly on MNIST-S.
likely
5 RELATED WORK
Set Representation Learning. Most relevant to our work are works that learn representations for set regression problems. We have already discussed recurrent neural networks and DeepSets Zaheer al. (2017). Zhang al. (2019) Iearn how to order the elements of set to feed it into an LSTM such that the resulting representation can be used to predict the desired output. Hence, is smart representation aggregation function. In this work, focus simple RNN-like hidden state aggregations in the RNN, DCN, and DCR architectures because we want to focus on whether modeling intermediate utilities can be beneficial. We leave learning of better hidden states for future work. Improving the learning hidden stated and maybe also reorder objects Zhang et al.) 2019 could be interesting for future work and might improve the performance of the newly proposed DCN and DCR architectures further.
Multiple Instance Learning. The problem considered in this work related to multiple instance learning Dietterich al. (199)); Maron Lozano- Perez (1998), and in particular multiple instance regression Ray & Page (000). In multiple instance learning, only labels for sets of instances are observed. Usually, the problem have binary labels and the set is labeled positive if at least one instance is positive. This is known as the standard multiple instance learning assumption Carbonneau et al. (2018). Similarly, Ray & Page (001)) assume that sets contain primary instances, which are solely responsible for the set utility. In both settings, it is sufficient to learn labels/values the underlying instances. Furthermore, our main contributions which is not considered by most prior works. Only very recently, deep learning has been applied multiple instance classification under the standard assumption llse al. (2018). In this work,
framed as binary classification problem in which all instances
two new neural architectures, i.e. a contribution representation learning,

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however, we focus consider problems in which more difficult non addaiiviivve aggregation function has to be learned. Hence, learning instance utilities is not sufficient and the problem at hand is substantially different to multiple instance learning.
Group Rating Problems. Group rating problems relax the standard multiple instance learning assumption and do not assume the presence of a primary instance. Instead, all individuals group contribute to its utility. However, as already mentioned previously, most prior works in group rating assume that the utility can be modeled as sum of the individual utilities. Most notably are the Bradely-Terry model Bradley & Terry (1922) and TrueSkill (merrricHHrrbrich Herbrich et al. (206). Recently, (eanntlyy.iii al. (2017) proposed model which learns parameters of different parametrized aggregation functions. However, their model is provided with supervision for element and group ratings. Compared to our work, we do not provide the model element scores. Furthermore, we do not provide group features. In our setting, groups are just sets of elements without any additional features.
Sequence-to-Sequence Learning The proposed architectures look similar to sequence-to sequence problems such architectures for machine translation (017). There are, however, notable differences. First and most importantly. the predicted values in the proposed architectures are latent intermediate utilities, which are not part of the output of the prediction. Instead. the output is an aggregation of the latent utilities. In step such as individual words are visible the output such as a translated sentence. Furthermore, like in multiple instance learning, no feedback is available for intermediate predictions. Further- more, our models predict intermediate utilities, which are numbers. Sequence-to-sequenee models usually predict tokens.
-o-euueecee problems, the outputs of each time
Choquet Integral for Machine Learning The Choquet integral Choquet (1954); Grabisch et al. 2009) has been used for non-neural works before. [Beliakov] (2008) fit values of the discrete Choquet integral with linear programming techniques. Tehrani et al. (2012a) use the Choquet integral to model monotone nonlinear aggregations for binary classification. Tehrani et al. (2012b) use the Choquet in pairwise preference learning scenario. To the best of our knowledge, this is the first work which combines the Choquet integral with deep learning.
CONCLUSIONS & FUTURE WORK
We found that standard methods for set regression such architectures that belong the DeepSets family and recurrent neural networks do not explicitly make use of the fact that meaningful inter- mediate utilities be predicted in many problems. Instead, they aim learning representation for sets such that the set utility can be predicted solely on the learned representation. We proposed two architectures, which have been inspired the non-additive Choquet integral. The resulting models learn to predict and aggregate intermediate utilities. Non-additivity is essential to deal with redundancy and synergy effects.
We validated the superiority of neural non-additive utility aggregation on computer vision and sum- marization datasets. Besides the superior performance, the new models also enjoys better inter- pretability than prior models because each prediction step is meaningful and can be inspected by laypeople that do not have any machine learning knowledge. Misconceptions of the model can be identified easily.
We focused in this paper on comparing neural non-linear utility aggregation with standard repre- sentation learning approaches for set regression problems and did not focus on achieving very high overall results. Multiple improvements of the proposed architectures can be investigated future work. For example, more sophisticated representation aggregation functions such LSTM and GRU cells can be used, attention mechanisms can be used to learn better hidden state representa- tions, and different object utility functions and gate functions be investigated. Furthermore, the proposed architectures are not limited to set regression problems, but can also be applied to sequence regression problems.
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Under review as a conference paper at ICLR 2020
Ali Fallah Tehrani, Weiwei Cheng, Krzysztof Dembezytiski, and Eyke Hüllermeier. Learning mono- tone nonlinear models using the Choquet integral. Machine Learning, 89(1-2):18321) 2012a.
ISSN 15730565, doi: 10.1007/s100944-12-53311
Ali Fallah Tehrani, Weiwei Cheng, and Eyke Hullermeier. Preference learning using the choquet integral: The case of multipartite ranking. IEEE Transactions on Fuzzy Systems, 20(6): 102- 1113, 2012b. ISSN 10636706. doi: 10. 10.109//FUUZ2220222100055
Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poezos, Ruslan Salakhutdinov, and Alexander Smola. Deep Sets. In Proceeding of the 31st Conference Neural Infor- mation Processing Systems, pp. 1-11, 2017. URL /paps::/ppppraaaam mips. ccpppprr/
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Yan Zhang, Adam Prügel-Bennett and Jonathon Hare. Learning Representations of Sets through Optimized Permutations. In Proceeding of the 7th International Conference on Learning Repre- sentations, pp. 1-15, 2019.
10


Reviews:
[{"review_id": "ryxs4ypg9r", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": null, "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 proposes two new architectures for processing set-structured data: An RNN with an accumulator on its output, and an RNN with gating followed by an accumulator on its output. While sensible, this seems to me to be too minor a contribution to stand alone as a paper.\n\nAdditionally, I believe the experimental tasks are new, and as a result all implementations of competing techniques are by the paper authors. This makes it difficult to have confidence in the higher reported performance of the proposed techniques.", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I made a quick assessment of this paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "N/A"}, "extra_text": {}}, {"review_id": "BJl-jwU6tB", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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 studies non-additive utility aggregation for sets. The problem is very interesting. Choquet Integral is used to deal with set input. The authors propose two architectures. The two architectures, though not novel enough, are towards representing “non-additive utility”.\nHowever, the experimental comparison is not fair, the description of the model (e.g. how Choquet is integrated into the model and help to learn “intermediate meaningful results”) is not clear, some claims are not true.\n\nFirst, the authors claim that they are the first to combine Choquet integral with deep learning. However, there are a few, though not many, works in the literature trying to combine Choquet integral with deep learning. For example, “Fuzzy Choquet Integration of Deep Convolutional Neural Networks for Remote Sensing” by Derek T. Anderson et al. \n\nSecond, the authors claim they are using/motivated by Choquet integral, but do not have any (appendix) sections to explain how this mathematical tool is really integrated into their models. How do you guarantee that the representation learned by the neural network still obeys the property of Choquet integral? What is your loss or your algorithm? These need to be further clarified.\n\nThird, the comparison to baseline and “DeepSet” is not fair. According to the illustration, it seems that you first obtain “features/representations”. Then the representations are fed to the four architectures you listed in figure one. RNN-based approaches are with better “complexity” comparing to your sum baseline and “Deepset” approach. So, I have some doubts about the experimental results.\n", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "N/A"}, "extra_text": {}}, {"review_id": "BJeididLqS", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": null, "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": "In this paper, the authors propose two RNN-based models to learn non-additive utility functions for sets of objects. The two architectures are inspired by the discrete Choquet integral. The proposed models are evaluated on visual and textual data against an MLP baseline and deep sets. \n\nOverall, the paper is clearly written and easily understandable. However, the novelty of the paper is limited and the empirical support of the proposed models is insufficient. The motivation of using \"Choquet integral\" seems obscure to me. The author might want to provide a short introduction to Choquet integral and elaborate on the connection with the proposed models. The proposed models seem very basic and do not have much novelty. The tasks in the experimental study seem overly simple. The authors might want to consider other tasks, for example, Point Cloud Classification in [1]. \n\nQuestions:\n* For RCN and DCR, how to decide the ordering of phi_i, given that they are the objects of an unordered set?\n* It would be helpful it the authors can also provide the number of parameters of the baseline models in Tables 1, 2, and 3.\n* To model the interaction among objects in a set, GNN might be a better choice than RNN.\n\n[1] Zaheer, Manzil, et al. \"Deep sets.\" Advances in neural information processing systems. 2017.", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I made a quick assessment of this paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "N/A"}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper presents two new architectures that model latent intermediate utilities and use non-additive utility aggregation to estimate the set utility based on the computed latent utilities. These two extensions are easy to understand and seem like a simple extension to the existing RNN model architectures, so that they can be implemented easily. However, the connection to Choquet integral is not clear and no theory has been provided to make that connection. Hence, it is hard for the reader to understand why the integral is useful here.  The reviewers have also raised objection about the evaluation which does not seem to be fair to existing methods. These comments can be incorporated to make the paper more accessible and the results more appreciable. 
Decision: Reject