Systematic generalisation with group invariant predictions
Authors: Faruk Ahmed
Yoshua Bengio
Harm van Seijen
Aaron Courville
Venue: ICLR 2021 Spotlight (2021)
Keywords: Systematic generalisation
invariance penalty
semantic anomaly detection

Abstract: We consider situations where the presence of dominant simpler correlations with the target variable in a training set can cause an SGD-trained neural network to be less reliant on more persistently correlating complex features. When the non-persistent, simpler correlations correspond to non-semantic background factors, a neural network trained on this data can exhibit dramatic failure upon encountering systematic distributional shift, where the correlating background features are recombined with different objects. We perform an empirical study on three synthetic datasets, showing that group invariance methods across inferred partitionings of the training set can lead to significant improvements at such test-time situations. We also suggest a simple invariance penalty, showing with experiments on our setups that it can perform better than alternatives. We find that even without assuming access to any systematically shifted validation sets, one can still find improvements over an ERM-trained reference model.

Published as a conference paper at ICLR 2021
SYSTEMATIC GENERALISATION WITH GROUP INVARI- ANT PREDICTIONS
Faruk Ahmed'; Yoshua Bengio! :2, Harm van Seijen', Aaron Courville 1 Université de Montréal, Mila, 2CIFAR Fellow, Microsoft Research
ABSTRACT
We consider situations where the presence of dominant simpler correlations with the target variable in a training set can cause an SGD-trained neural network to be less reliant more persistently correlating complex features. When the non- persistent, simpler correlations correspond to non-semantic background factors, a neural network trained on this data can exhibit dramatic failure upon encountering systematic distributional shift, where the correlating background features are re- combined with different objects. We perform an empirical study on three synthetic datasets, showing that group invariance methods across inferred partitionings of the training set can lead to significant improvements at such test-time situations. We also suggest simple invariance penalty, showing with experiments on our setups that it can perform better than alternatives. We find that even without assuming access to any systematically shifted validation sets, can still find improvements over an ERM-trained reference model.
INTRODUCTION
If training set is biased such that an easier-to-learn feature correlates with the target variable throughout the training set, a modern neural network trained with SGD will use that factor to perform predictions, ignoring co-occurring harder-to-leamy complex predictive features (Shah et al. 2020). Without any other criteria, this is arguably desirable behaviour, reflecting Occam's razor. We consider the situation where although such simpler correlation is dominant bias in the training set, minority group exists within the dataset where the bias does not manifest. In such cases, relying on more complex predictive features which more pervasively explain the data can be preferable to simpler ones that only explain most of it. For example, if all chairs are red, redness ought to be a predictive rule for chairhood (without any other criteria for predictions). However, if some chairs not red, and all chairs have backs and legs, then one can infer that redness is less relevant.
In this paper, we will study object recognition tasks, where the objects correlate strongly with simpler non-semantic background information for majority of the images, but not for minority group. There is evidence the literature that modern CNNs tend to fixate on simpler features such as texture (Geirhos et al., 2019, Brendel & Bethge 2019), canonical pose (Alcorn et al.) 2019). or contextual background cues (Beery et al.] [2018). We are assuming that semantic features classification context (ones that humans would agree contribute to their labelling of objects) are more likely to persistently correlate with the target variable, while simpler non-semantic background biases more likely to exhibit non-persee non-persistent correlations in real-life data collection processes. Based on this assumption, we will use combinations of objects and backgrounds to compare test-time performances corresponding to particular distributional shifts.
Consider coloured MNIST digits such that there is a dominant, but not universal, correlation between colour and digit identity for majority of the images. In the situation we considering, if the biasing colours in the majority group are not recombined with different digits in the minority group, then there no signal for the model to disregard these biasing factors, which are retained as important predictive rules. This can lead to poor performance at systematic generalisation (Lake & Baroni 2018). where an object occurs with another object's 's biasing factor, and at semantic anomaly detection (Ahmed) & Courville 2020), where a novel object appears with one of the biasing factors. In our example
"Correspondence faruk wwruk.ahmedde. www.tteeddmmomall.....

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Published as a conference paper at ICLR 2021
Table 1: For a coloured MNIST dataset with every digit correlated with a colour 80% of the time, we see poor performance at systematically varying tasks. Performance improves if the minority group combines colours from other biased digits this provides corrective gradients that promote invariance colour. Non-systematic shifts are when unseen colours are used, and anomaly detection measured by decreased predictive confidence for an unseen digit (see Section for more details).
Minority colours
In-distribution
Non-systematii shift
Systematic shift
Anomaly detection
Different
99.60 + 0.02
53.26 1.89
38.72 = 2.27
7.70 = 0.23 46.59 + 6.93
Recombinations
98.67 + 0.39
85.05 1.89
97.56 0.05
with coloured MNIST, if colour the minority group digits with the colours used to bias (different) digits in the majority group, we find a marked improvement at systematically shifted tests over the case when the colours in the minority group are different colours altogether (see Table D).
We investigate the role of encouraging robust predictive behaviour across such groups in terms of improved performance tasks with such distributional shifts. Our experiments suggest that training with cross-group invariance penalties can result in models that have learned to be more reliant on persistent complex correlations without being overwhelmed by simpler, yet less stable features, as indicated by improved performance at systematic generalisation and semantic anomaly detection on our synthetic setups.
We find that recently proposed method (Creager et al. 2020) can be effective at inferring the majority and minority groups along a learned feature- bias, and we use this inferred partition to provide us with groups in the training set in our comparative study. We also suggest a new method for encouraging predictions that rely on persistent correlations across such groups, with the intuition that similar predictive behaviour across the groups should be promoted throughout training. With experiments on three synthetic datasets, we compare the performance of recently proposed invariance penalties and methods, and find that our variant can often perform better at tasks involving such test-time distributional shifts.
SYSTEMATIC AND NON-SYSTEMATIC GENERALISATION
If we assume that data x is generated via a composition C of semantic factors ha and non-semantic factors hn, we can use this decomposition, different scenarios. While ht, is actually independent of y, we shall have the independence property PD (h.,ly) pD(h,,) to not hold when there is bias in the dataset D due to h, correlations.
C(ha, h,). to generate test datasets to capture
We can evaluate, for a particular target y and our system's 's prediction of the target (z)), the average accuracy F. ( h,,)) y}}. as a measure of generalisation for the following different cases.
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Figure 1: COLOURED MNIST training and test sets for evaluating generalisation under non-semantic marginal shift and systematic shift, and anomaly detection. (a) Training set; (b) In-distribution generalisation set Tg. where the test set is coloured following the same scheme as for T,: (c) Systematic-shij gemeralisation set T., where we colour the test set with the biasing colours, but such that no digit is coloured with its own biasing colour; (d) Nor-systematic-ship generalisation sei T,, where the test is coloured with random colours that are different from any of the colours seen in the training set; and (c) Semantic anomaly detection set Ta, where we colour the held-out digits of the test set randomly with the biasing colours.
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In-distribution generalisation h2 p(haly) and hn p(h, ly): The validation and test sets assumed to possess the same biases as the training set, in that the classcconditional distribution of non-semantic features in the test set match that of the training set, p(hn
Generalisation under non-systematic-shitt h2 ~ p(h,lv) y)) and h, o p(h, 1 This estimates a form of generalisation under distributional shift, where the non-semantic factors are sampled from outside the marginal distribution of h, as present in the training set.
Generalisation under systematic-shift h2 p(y) 8.1. y' * y: This estimates another form of generalisation under distributional shift but one where non-semantic factors sampled with intent confuse: non-semantic factors for sampled from the marginal distribution of a randomly picked different target, y * y. Although systematicity, as discussed Fodor & Pylyshyn (1988), and systematic generalisation, as discussed in the NLP literature (Lake & Baroni 2018 Bahdanau et al., 2019) consider recombinations of intra-semantic factors as well, here, in the context of wetgrroundaggnoste object recognition tasks, we only consider h4 h, recombinations.
p(h,ly) and h, p(hnll/") where y'
Semantic anomaly detection ht o p(hbs) and hi, ~ p(h,): Such a datapoint should not be confi- dently categorised as a known y: even if non-semantic features are shared (Ahmed & Courville, 2020). We can use these x to evaluate anomaly detection, as indicated by decreased predictive confidence, and measured by the area under the precision-recall curve (Hendrycks & Gimpel 2017
COLOURED MNIST: Consider an illustrative dataset with coloured MNIST digits. For the training set, T,, MNIST digits are coloured with set of digit-correlated "biasing" colours 80% of the time, and with ten random colours that are different from the biasing colours the remaining 20% of the time. One digit is held out, for testing semantic anomaly detection. See Figure foo examples of the four test sets corresponding to this setting, and also Appendix A for more details on the construction.
Improving performance for such scenarios involving distributional shift might come distribution performance, since more robust features might be harder to learn than simpler dominant correlations that hold in-distribution. In real-world deployments where one is likely encounter unexpected situations, such as in a self-driving car, it can often be preferable to find appropriate trade-offs such that classifiers can indicate reduced confidence upon encountering anomalous objects. or continue to operate in changing environments, while continuing to achieve a desirable degree of in-distribution predictive performance.
for in-
PREDICTIVE GROUP INVARIANCE ACROSS INFERRED SPLITS
In general, do not expect to have direct knowledge of majority and minority groups corresponding to the biasing non-semantic features in a dataset. We will later show how one might infer such groups from the data, but we first describe invariance penalty assuming we have access the groups.
Learning features that are group invariant would require us to match the cclass-conditioned) dis- tribution of features from the majority and minority groups (Ganin et al. 2016 Li et al. (018a). In terms of predictive performance, we can alternatively ask for the class-conditioned distributions of features to match in the sense that they lead to the same softmax distributions on average as training progresses, without modifying the last linear layer. This implementation has the advantage of doing away with an adversarial network, and the issues that tend to accompany the training of such models. We shall refer to this objective as predictive group invariance (PGI). Intuitively, encouraging matched predictive distributions across the groups with fixed last layer pushes for over-emphasis on imnority-group features in the representation, thus acting as implicit re- re-weighting of features in both groups (leading to demoting the relevance of colour in the MNIST case, for example). When persistent feature does exist in both groups, using that feature can lead to equal training rates in regularised networks, satisfying the penalty.
Consider classifier that extracts feature vector (z), where @ are the parameters of a convolutional neural network for example, with a linear layer w on top. The predictive distribution is then
(y] = ow fo(=)),
(1)
where a is softmax, and predictions are made by performing an argmax.
In this paper, we imply sampling from outside the support of p when we say o p(h).
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8, for Published as a conference paper at ICLR 2021
Given partition scheme for splitting the images in our dataset D such that every i-th image xx1) is associated with partition-label (+()), we define distributions Po; Q for the subsets in class ot
p if ( 0,y c,
( O if a = 1,
1, () = =
(3)
We want to minimize empirical risk under the constraint that our feature extractor causes similar predictive distributions on average for pictures of the same object in both partitions. Formally, we want to optimise
min E(0, w w|
(4)
s.t. 0 E argmin d
E z=pe
E [Po(l)=))) Ve, anD
(5)
where l is the standard loss function for ERM training, for example, the categorical cross-entropy. A softened objective for stochastic optimisation can be approximated as
Lww, 8 = DD + Σ d E (pn(())) Lo0te
wa Pa (pa(ll))))
(6)
w=w (fxed)
Since we are comparing distributions, we make the simplest natural choice of be the KL- divergence,
Ex~e Pa (v|z)]
E (mala)) E (v((v = Bnnie
Σ
E Io (v (v) log wnd
En Pa (ylær)
We use this particular ordering of Q because with our grouping, IP consists of examples that "easy" due to a particular bias, and so the mean predictive distribution for IP tends to be correct and low-entropy, while that for Q more high-entropy and inaccurate. We take advantage of the zero-forcing property of this KL divergence, encouraging the mean predictive distribution for Q to closely match that of IP. likely that different choices for d would be better suited for different settings.
Partitioning the dataset Recently, Creager et al. (2020) have considered the question of finding worst-case partitions for invariant learning given a collection of data. The key intuition is that an invariant learning objective, as formulated by IRM (Arjovsky et al. 2019). is maximally violated by splitting along a spurious correlation when predictions rely exclusively on it in reference model (see Theorem 1 in Creager et al. (2020) for details). our case, this would consist of partitioning into the majority and minority groups given ERM-traines model early in training reference.
soft partiiarti predicting network used, g x y), conditioned on the input and the target, to maximise the IRMv1 penalty (Arjovsky et al. 2019). which gives us soft the examples,
Σ eE(0,1)
Ee (e) (OM
max B
Σ
V Valn VIl=110
Σ 300) (e) ((x s
(8)
E. (0(()
Ec (0.1)
where ((x;) = w fo(r) are the logits from the reference model, {0, 1} indexes the partition, (e) [0, signifies the predicted probability for the i-th example being in partition e, such that 1, and 7 is hyper-parameter. We can then compute the partition condition the partition predicting network g on the features fe(x) instead of the input x, and use separate networks for each category, í.e. 3(0 = 9, ()). We find this to perform better in preliminary experiments, improving training and enabling more ligght g networks. This also ensures that the same features as the ones used by our ERM-trained reference model are used to predict partitions, resulting in partitions corresponding to consistent learned-feature biases. We provide more details in Appendix B.3
g0 0) + g
(0) argmax, (e). In implementation,
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Figure 2:
(left) CN-LOURS: left block is the majority group, right block is the "unbiased"
minority group; (right) COCO-ON-PLACES.
RELATED WORK
The dominant perspective towards the issue of unreliable behaviour in novel domains has consisted of treating the problem as that of domain generalisation (Blanchard et al. 2011). One hopes to recover stable features by encouraging invariance across data sampled from different domains, so that performance at test-time out-of- out-of-distribution (OoD) scenarios is less likely to be unstable.
Approaches along such lines typically resemble a cross-domain distribution-matchee to the features being learned, augmenting the usual ERM term (Ganin et al. 20166] Sun & Saenko
penalty applied
2016 Heinze-Deml & Meinshausen 2017 Li et al. 2018
Li et [2018a b), and evaluated on
datasets that consist of data in different modalities (Li et al.
2017
Peng et al. 20199 Venkateswara
al. 2017). or collected through different means (Fang et al. 2013). ). or different contexts (Beery
et al. 2018
Works with the perspective distributionally robust optimisation (DRO) have generally considered using uncertainty sets around training data (Ben-Tal et al.] 2013 Duchi & Namkoong (0188) to minimise worst-case losses, which can often have regularising effect by effectively up-weighting harder examples. More relevant to our discussion, group DRO methods have considered uncertainty sets in terms of different groups of data, for example with different cross-group distributions of labels (Hu al., 2018), or groups collected differently (Oren et al. 2019), similarly to domain generalisation datasets.
More recently, methods promoting the learning of stable features across data from different environ- ments, or sources, have been proposed by using gradient penalties (Arjovsky et al. (001)) risk-based extrapolation (Krueger al. 2020), and masking gradients with opposing signs (Parascandolo et al.) 2020
The typical datasets in such existing works are not curated with testing performance under systematic distributional shift in mind, most often not characterising the specific shift in distribution. In recent times. a commonly adopted synthetic dataset the coloured MNIST variant used in (nrroossyy et al. (2019) since this particular dataset uses flipped colours for the minority group, which is less of a problem with ERM--triniigg the true digit labels were flipped sufficiently high frequency to incapacitate ERM performance by forcing reliance on colour. We believe setups such as ours can be better synthetic testbeds for developing ideas, where not necessary alter ground truth labels to expose failure mode. In general, using better models of dataset bias implies a narrower disconnect with realistic settings, with higher chances of conclusions carrying over.
EXPERIMENTS
We compare performance with our four test sets in-distribution, non-systematically shifted, sys- tematically shifted, semantic anomalies for a range of recently proposed methods for set of three synthesised datasets. Appendix B describes architectural details and training choices.
5.1 METHODS
We compare recent methods aimed at robust predictions across groups, and which do not require changes to network capacity or additional adversaries to impose invariance penalties. We also do not include methods based on advances in self-supervised feature learning, such as Carlucci et al.
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(019)), since such methods developed with prior knowledge of the desired invariances, and are thus limited in their generality.
Baseline: This is reference model, trained via ordinary (regularised) empirical risk minimisation (ERM) without any invariance penalties added. The choices for architecture and regularisers were made to conform the way modern networks are typically trained with in-distribution performance in mind (details in Appendix B
IRMv1, REx, GroupDRO: IRMv1 (Arjovsky et al. 2019) and REx (Krueger et al.) 2020) are two
methods that augment the standard ERM term with invariance penalties across data from different sources. GroupDRO (Sagawa et al.
2020) is algorithm for distributional robustness, which works
by weighting groups of data as a function of their relative losses. See Appendix C for more details about these methods.
cIRMv1, cREx, cGroupDRO: We implement label-conditional variants of the above algorithms, which, to our knowledge, has not been explored. In the context of multi-class classification is reasonable to expect that performances might have multi-modal distributions along different categories earlier training, which suggests stratification by class might improve performance.
We weight the losses in the biased group down. This is a heuristic form of re-balancing the dataset, while choosing hyper-parameter for the weight using the validation set, with the weight serving to downweight the losses for the biased group. In preliminary experiments we found this re-weighting variant (King & Zeng 2001) to significantly outperform oversampling the minority group, as suggested in Buda et al. (018)), or weighting the grouped losses using their population ratios, as performed for imbalanced classes in Cui et al. (2019).
Reweight:
cMMD: Following Li et al. (2018), match the MMD (Gretton et al. 2012) of the distribution of features. preliminary experiments, we find a conditional version (as done with adversarial models in (Li et al. 2018al) perform significantly better, we only report cMMD results here.
5.2 DATASETS
Evaluating performance in an unambiguous manner for the specific kinds of generalisation that we aim study necessitates controlled test-beds. In order to model these tasks, use 3 synthetic datasets of progressively higher complexity, approaching photo-realism.
COLOURED MNIST: This is the simplest setting, where the background information exists as part of the object.
We superimpose 10 segmented COCO (Lin al. 2014) objects on coloured backgrounds. The training set has 800 images per category, with nine in-distribution categories and one held-out category for anomaly detection. Validation and test sets have 100 each images per category. See Figure (left). This is the most extreme dataset in our experiments in terms of the contrast in complexity between the non-semantic correlating factor (background colour) vs. stable features (objects).
COCO-ON-COLOURS:
COCO-ON-PLACES Here we superimpose the same COCO objects on scenes from the PLACES dataset (Zhou et al. 2017), with the place-scenes acting the bias (figure 2. right). See Appendix A for more details about how these datasets are constructed. While the backgrounds this dataset are more complex than colour, they still act as biasing factors, as indicated in the relatively poorer performance at systematic generalisation, and were selected due to visually obvious and distinct colour or texture.
5.3
RESULTS
In all cases, we have used the partition predictor to infer the two groups. The partition accuracies for the three datasets at the end of one epoch of training the base models in the table below. We tested a more naive approach by applying K-Means clustering to the losses, but found it to under-perform, possibly because it cannot account for consistent feature bias learned by reference model.
COLOURED MNIST COCO-ON-COLOURS
COCO-ON-PLACES
97.26 + 0.71
98.22 + 1.05
80.43 + 1.41

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Table 2: Generalisation results on COLOURED MNIST.
Methods
In-distribution
Non-systematic shift
Systematic shift
Anomaly detection
Base (ERM)
99.60 1 0.02
53.26 1: 1.89
38.72 1: 2.27
7.70 + 0.23
IRMv1
99.47 1 0.05
63.24 1: 3.04
55.19 +: 1.07
11.54 1 1.18
REx
98.95 1 0.11
72.12 1: 1.90
71.18 1: 3.27
15.54 1 2.05
GroupDRO
89.47 1 4.52
70.53 + 1.79
79.17 + 1.64
35.15 = 10.83
Reweight
98.51 0.12
75.01 = 1.28
84.85 0.61
28.60 = 1.11
cIRMv1
99.36 + 0.25
65.78 3.53
61.09 + 5.30
14.16 = 2.12
cREx
98.56 + 0.12
74.35 +
80.01 2.11
22.02 +
cGroupDRO
95.65 + 3.23
75.41 + 3.45
81.14 2.41
26.61 = 6.61
cMMD
99.40 + 0.03
97.17 = 0.59
97.86 + 0.16
78.32 = 4.15
PGI
99.05 + 0.08
98.58 + 0.06
98.48 + 0.05
89.42 + 1.95
Table 3: Generalisation performance on COCO-ON-COLOURS.
Methods
In-distribution
Non-systematic shift
Systematic shift
Anomaly detection
Base (ERM)
90.57 + 1.28
26.81 + 4.93
1.10 0.36
5.47 = 0.08
IRMv1
91.61 + 0.38
32.30 +: 4.52
2.11 +: 0.30
5.81 + 0.17
REx
91.69 0.50
36.57 4.03
2.69
5.73 + 0.14
GroupDRO
43.06 + 2.26
41.32 + 4.39
43.24 + 2.89
20.05 + 3.08
Reweight
42.42 + 3.47
47.56 + 2.27
49.12 + 1.63
18.15 + 3.81
cIRMy1
91.53 + 0.31
31.11 4.51
1.74 + 0.40
5.87 + 0.16
cREx
74.75 + 14.14
32.29 + 7.71
29.75 + 5.16
19.77 + 14.98
cGroupDRO
41.10 + 2.37
41.83 + 2.96
42.10 +: 2.15
21.81 + 5.40
cMMD
89.87
1.13
55.02
2.29
27.36
1.57
8.82 + 0.70
PGI
78.23 +
2.01
55.57 1. 4.60
51.62 1. 3.09
18.84 1 2.11
In Tables 23 we find that significant improvements can achieved using group invariance methods. All hyper-parameeee for the results in this set picked on validation set consisting of subset of colours or backgrounds that are different from both the training and test sets, and an equally sized subset of systematically varying colours backgrounds from the biased majority group. In all cases, the split is learned after one epoch of training, and the various penalties dropped in at this point with linearly ramped-in penalty co-efficient. Details about hyper-parameter selection are Appendix C
While conditional variants perform better at systematic generalisation for COLOURED MNIST, perhaps owing our hyper-parameter selection procedure of using mixed-shift validation set, performance at systematic shift appears to be traded off with non-systematic shift in some cases for the more complex datasets. All aggregates are over 5 trials.
5.4 PRACTICAL CONSIDERATIONS FOR HYPER- POPER-PARAMETER SELECTION
While find that with the of group invariance penalties it is possible encourage reliance upon complex persistent correlations in the presence of dominant simple biases, this can sometimes come cost in-distribution performance when picking pprrmpprreeetes using validation sets with specific distributional shift. One might reasonably expect that this can be mis-aligned with real- situations: practice, one typically does not have access to data corresponding exactly to unexpected scenarios, besides not expecting to encounter situations outside the training distribution nearly as often as situations for which a model has been trained and deployed. practitioner might wish to aim for clearer trade-off with such situations, with prior knowledge of how often they might arise compared to in-distribution situations, and with a surrogate validation set to model distributional shift. Here, will simply show that picking parammtersss without assuming access to validation sets consisting of systematic distributional shift can still provide improvements over the baseline reference model. We consider three cases.
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Table Generalisation performance on COCO-ON-PLACES.
Methods
In-distribution
Non-systematic shift
Systematic shift
Anomaly detection
Base (ERM)
81.06 1.01
45.25 + 0.96
29.18 1 1.24
9.21 1: 0.21
IRMv1
80.93 0.71
45.17 1: 0.92
28.78 1 0.73
9.39 0.60
REx
81.55 0.70
45.35 1 0.92
29.56 1 0.77
9,46 1 0.51
GroupDRO
76.05 0.87
43.72 +: 0.43
31.83 + 0.54
9.61 0.55
Reweight
81.14 0.80
45.84 0.70
30.37 = 1.16
9.75 0.69
cIRMv1
80.08 1.90
44.96 2.88
30.06 2.07
9.64 0.94
cREx
81.50 0.76
45.44 + 0.96
29.12 + 0.97
9.17 0.59
cGroupDRO
78.25 0.31
41.69 = 0.08
28.16 + 0.91
9.45 0.22
cMMD
79.64 0.73
49.44 = 0.99
35.86 + 0.66
9.80 + 0.45
PGT
75.00 0.85
46.10 + 0.79
36.25 + 0.42
11.12 + 0.85
cMMD (oracle split)
75.05 0.98
47.88 1.03
37.40 + 1.07
10.76 + 0.61
PGI (oracle split)
70.63 0.48
48.11 = 0.82
42.69 - 0.84
12.56 + 1.20
Table 5:
with different validation sets for COLOURED MNIST
Validation
In-distribution
Non-systematic shift
Systematic shift
Anomaly detection
NS+S (PGI)
99.05 + 0.08
98.58 + 0.06
98.48 = 0.05
89.42 + 1.95
NS (PGI)
99.31 = 0.05
98.21 + 0.26
97.54 = 0.41
76.00 = 4.06
NS+ID (PGI)
99.30 0.07
98.31 + 0.27
97.48 + 0.45
76.07 5.67
ID only (PGI)
99.69 0.03
63.62 + 2.05
58.18 + 2.05
11.81 + 1.89
Base (ERM)
99.60 0.02
53.26 1.89
38.72 : 2.27
7.70 0.23
Hyper-papne papeee-arameters are picked using only the validation set for non-systematic distributional shift (which consists of backgrounds that are different from those in the training set and test sets). This models the situation where we have access to some data that is different from our training data, and also considered somewhat representative of any shifts we might encounter.
NS:
NS + ID: Hyper-parameters are picked using an (equally-weighted) average the NS and in-distribution validation sets. If we have prior knowledge of the likelihood of encountering data from out-distributions the wild, we could use this prior to use an appropriately sampled validation set for hyper-panne parameter optimisation.
ID ONLY:
are picked using only the in-distribution validation set.
We show results for the different schemes for method in Tables 5 6 7
While the accuracies
under distributional shift are, as expected, less strong than in the previous set of results (NS+S tables), still find improvements improved classifier.
the reference model, indicating that one can still achieve
In Appendix D show similar results with all methods, and include only the best performing method for both generalisation under systematic and non-systematic shift corresponding to the different validation strategies in the tables in this section.
Table 6: Hyper-pane parameters with different validation sets for COCO-ON-COLOURS
Validation
In-distribution
Non-systematic shift
Systematic shift
Anomaly detection
NS+S (PGI)
78.23 2.01
55,57 4.60
51.62 = 3.09
18.84 +: 2.11
NS (PGI)
85.78 + 1.45
51,02 + 2.32
38.85 = 2.29
15.71 = 3,25
NS+ID (PGI)
85.78 + 1.45
51.02 + 2.32
38.85 = 2.29
15.71 + 3.25
ID only (cMMD)
92.51 + 0.41
44.59 + 3.28
10.48 = 0.98
6.05 + 0.23
Base (ERM)
90.57 +:: 1.28
26.81 1: 4.93
1.10 + 0.36
5.47 +: 0.08
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Table 7: Hyper-parameters with different validation sets for COCO-ON-PLACES
Validation
In-distribution
Non-systematic shift
Systematic shift
Anomaly detection
NS+S (cMMD)
79.64 1 0.73
49.44 + 0.99
35,86
0.66
9.80 = 0.45
NS (cMMD)
79.64 1 0.73
49.44 0.99
35,86 0.66
9.80 0.45
NS+ID (cMMD)
79.64 0.73
49.44 0.99
35.86 0.66
9.80 0.45
ID only (PGI)
80.99 + 0.52
47.63 0.90
31.91 = 0.89
9.59 = 0.89
Base (ERM)
81.06 1.01
45.25 0.96
29.18 1.24
9.21 = 0.21
CONCLUSION
Our experiments investigate the potential usefulness of invariance penalties and methods at improving performance under distributional shift, such systematic generalisation and semantic anomaly detection.
While our exploratory experiments are conducted in disambiguated synthetic setups, next steps would involve investigating the potential for extending these approaches to real datasets used in the field. Since such methods cannot work when spurious correlations are completely pervasive, is important to include sufficient diversity of data sources and curation in order to be able to reap the advantages such techniques can afford in real world applications. We note that peculiarities in datasets and problems might give rise to different potential failings at robustness, calling for more targeted invariance methods.
We find that our method of learning features that result in matched predictive behaviour throughout training appears to hold promise at handling certain distributional shifts, although does not always perform best across different validation schemes. A practical line of inquiry would be the question of how to make trade-offs in performance between in-distribution and unexpected situations.
ACKNOWLEDGMENTS
We thank Ishaan Guirajani, Tim Cooijmans, David Krueger, and Phong Nguyen for useful discussions, and anonymous reviewers for constructive feedback. We acknowledge the computational resources provided by Compute Canada and Mila, and the financial support of Hitachi, Microsoft Research, CIFAR, and the Natural Sciences and Engineering Research Council of Canada (NSERC Discovery Grant).
REFERENCES
Faruk Ahmed and Aaron Courville. Detecting semantic anomalies. AAAI, 2020.
Michael Alcorn Qi Li, Zhitao Gong, Chengfei Wang, Long Mai, Wei shinn Ku, and Anh Nguyen. Strike (with) a pose: Neural networks are easily fooled by strange poses of familiar objects. CVPR, 2019.
Martin Arjovsky, Léon Bottou, Ishaan Gulrajani, and David Lopez-Paz. Invariant risk minimization. CoRR, 2019.
Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E. Hinton. Layer normalization. CoRR, 2016.
Dzmitry Bahdanau, Shikhar Murty, Michael Noukhovitch, Thien Huu Nguyen, Harm de Vries, and Aaron Courville. Systematic generalization: What required and can be learne?? ICLR, 2019.
Sara Beery, Grant Van Horn, and Pietro Perona. Recognition in terra incognita. CoRR, 2018.
Aharon Ben-Tal, Dick den Hertog, Anja De Waegenaere, Bertrand Melenberg, and Gijs Rennen. Robust solutions of optimization problems affected by uncertain probabilities. 59(2), 2013. ISSN 0025-1909.

---

Published as a conference paper at ICLR 2021
Gilles Blanchard, Gyemin Lee, and Clayton Scott. Generalizing from several related classification tasks to a new unlabeled sample. In J. Shawe- Shawe-Taylor, R. S. Zemel, L. Bartlett, F. Pereira, and K. Q. Weinberger (eds.), Advances in Neural Information Processing Systems 24, pp. 2178-2180 2011.
Wieland Brendel and Matthias Bethge. Approximating cnns with bag-of-local-featuee models works surprisingly well on imagenet. International Conference on Learning Representations, 2019.
Mateusz Buda, Atsuto Maki, and Maciej A Mazurowski. A systematic study of the class imbalance problem convolutional neural networks. Neural Networkx, 106:249- 2299, 2018.
Fabio Maria Carlucci, Antonio D' D'Innocente, Silvia Bucci, Barbara Caputo, and Tatiana Tommasi. Domain generalization by solving jigsaw puzzles. In CVPR. 2019.
Elliot Creager, Jörn-Henrik Jacobson, and Richard Zemel. Environment inference for invariant learning. ICML Workshop on Uncertainty and Robustness, 2020.
Yin Cui, Menglin Jia, Tsung-Y Yi Lin, Yang Song, and Serge Belongie. Class-balanced loss based on effective number of samples. In Proceedings of the IEEE Conference Computer Vision and Pattern Recognition, pp. 9268-9999999 9277, 2019.
Harm de Vries, Florian Strub, Jérémie Mary, Hugo Larochelle, Olivier Pietquin, and Aaron C. Courville. Modulating early visual processing by language. NIPS, 2017.
John Duchi and Hongseok Namkoong. Learning models with uniform performance via distributionally robust optimization. arXiv preprint arXiv:1810.087:0 2018
Chen Fang, Ye Xu, and Daniel N Rockmore. Unbiased metric learning: On the utilization of multiple datasets and web images for softening bias. In Proceedings of the IEEE International Conference on Computer Vision, pp. 1657-1664, 2013.
Jerry A. Fodor and Zenon W. Pylyshyn. Connectionism and cognitive architecture: A critical analysis. Cognition, 28(1):3 71, 1988.
Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario Marchand, and Victor Lempitsky. Domain-adversarial training of neural networks. J. Mach. Learn. Res., 17(1):2096-200 2030, January 2016.
Robert Geirhos, Patricia Rubisch, Claudio Michaelis, Matthias Bethge, Felix A. Wichmann, and Wieland Brendel. Imagenet-trained cnns are biased towards texture; increasing shape bias improves accuracy and robustness. ICLR, 2019.
Arthur Gretton, Karsten M. Borgwardt, Malte J. Rasch, Bernhard Schölkopf, and Alexander Smola. kernel two-sample test. 13(uull), 2012. ISSN 1532-4435.
Christina Heinze-Deml and Nicolai Meinshausen. Conditional variance penalties and domain shift robustness. CoRR, 2017.
Daniel Hendrycks and Kevin Gimpel. A baseline for detecting misclassified and outtof-disttibution examples in neural networks. ICLR, 2017.
Weihua Hu, Gang Niu, Issei Sato, and Masashi Sugiyama. Does distributionally robust supervised learning give robust classifiers? In International Conference on Machine Learning, pp. 2029-2033 PMLR, 2018.
Gary King and Langche Zeng. Logistic regression in rare events data. Political analysis, 9(2): 137-163, 2001.
Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. International Conference on Learning Representations, 12 2014.
David Krueger, Ethan Caballero, and Aaron Courville. dittr---isttibutionn generalization via risk extrapolation (rex). CoRR, 2020.
Jacobsen, Amy Zhang, Jonathan Binas, Remi Le Priol,
10

---

Published as a conference paper at ICLR 2021
Brenden M. Lake and Marco Baroni. Generalization without systematicity: On the compositional skills of to-uueeence recurrent networks. In ICML, 2018.
Da Li, Yongxin Yang, Yi-Zhe Song, and Timothy M Hospedales. Deeper, broader and artier domain generalization. In Proceedings of the IEEE international conference on computer vision, pp. 5542-5550, 2017.
H. S. J. Pan, S. Wang, and A. C. Kot. Domain generalization with adversarial feature learning. pp. 5400-5409, 2018.
Y. Li, X. Tian, M. Gong, Y. Liu, T. Liu, K. Zhang, and D. Tao. Deep domain generalization via conditional invariant adversarial networks. In ECCV, 2a.
Ya Li, Mingming Gong, Xinmei Tian, Tongliang Liu, and Dacheng Tao. Domain generalization via conditional invariant representations. 2018b.
Yuanzhi Li, Colin Wei, and Tengyu Ma. Towards explaining the regularization effect of initial large learning rate in training neural networks. Advances Neural Information Processing Systems 32, pp. 1 11674-1 1685. 2019.
Tsung- Yi Lin, M. Maire, Serge J. Belongie, J. Hays, P. Perona, D. Ramanan, P. Dollár, and C. L. Zitnick. Microsoft coco: Common objects in context. ArXiv, hts/1405.03122 2014.
Takeru Miyato, Toshiki Kataoka, Masanori Koyama, and Yuichi Yoshida. Spectral normalization for generative adversarial networks. CoRR, 2018.
Yonatan Oren, Shiori Sagawa, Tatsunori Hashimoto, and Percy Liang. Distributionally robust language modeling. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP), pp. 4218-4223 2019.
Giambattista Parascandolo, Alexander Neitz, Antonio Orvieto, Luigi Gresele, and Bernhard Schölkopf. Learning explanations that are hard to vary. arXiv preprint wwXiv:2009.00322 2020.
Xingchao Peng, Qinxun Bai, Xide Xia, Zijun Huang, Kate Saenko, and Bo Wang. Moment matching for multi-source domain adaptation. Proceedings of the IEEE International Conference on Computer Vision, pp. 1406-1415, 2019.
Shiori Sagawa, Pang Wei Koh, Tatsunori Hashimoto, and Percy Liang. Distributionally robust neural networks for group shifts: On the importance of regularization for worst-case generalization. ICLR, 2020.
Harshay Shah, Kaustav Tamuly, Aditi Raghunathan, Prateek Jain, and Praneeth Netrapalli. The pitfalls of simplicity bias in neural networks. CoRR, 2020.
Baochen Sun and Kate Saenko. Deep coral: Correlation alignment for deep domain adaptation. pp. 450, 2016.
Hemanth Venkateswara, Jose Eusebio, Shayok Chakraborty, and Sethuraman Panchanathan. Deep hashing network for unsupervised domain adaptation. In Proceedings of the IEEE Conference Computer Vision and Pattern Recognition, pp. 5018-5027, 2017.
Yuichi Yoshida and Takeru Miyato. Spectral norm regularization for improving the generalizability of deep learning. CoRR, 2017.
Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. In BMVC, 2016
Bolei Zhou, Agata Lapedriza, Aditya Khosla, Aude Oliva, and Antonio Torralba. Places: A 10 million image database for scene recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2017.
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A DATASET DETAILS
In this section, we provide more details about how we constructed our synthetic datasets.
A.1 COLOURED MNIST
The training set, T,, is constructed with an 80% colour-digit correlation per digit with nine RGB- colours (with the zero digit held out, for testing semantic anomaly detection).
(0, 10,00),0
(188, 143, 143)
(255, 0,
(255, 215. 0)
(0, 255, 0)
(65, 105, 225)
(0, 225, 225)
(0, 0, 255)
(255, 20, 147)
Table 8: RGB codes used to bias the digits in the majority group.
The colours for the minority group picked such that their L.2 distance is at least 50 units away from the biasing colours. Prior to colouring, the digits were binarised to avoid grayscale tones potentially resulting unintentionally similar colours.
For the non-systematic validation and test sets, ten colours each were chosen such that they were at least 50 units away from all other colours.
A.2 COCO-ON-COLOURS
We use the following nine categories for in-distribution objects: boat, airplane, truck, dog, zebra, horse, bird, train, and bus. We hold out motorcycle for anomaly detection experiments. For background colours, we use the same colours from the coloured MNIST experiments, and also use an 80/20 split for the majority and minority groups
In case of multiple instances of the same object in an image, we pick the largest one, and filter our dataset by mask area, such that only images with objects occupying at least 10K pixels are retained. All images are finally resized to 64 x 64.
The training set uses 800 such pictures per category, and the validation and test sets use 100 each. The colour backgrounds for the minority group. con-systematically shifted validation and test sets are picked using the same strategy as with the COLOURED MNIST datasct.
A.3 COCO-ON-PLACES
This dataset follows the same procedure COCO-ON-COLOUT except using scenes from the Places dataset. In Table we list the backgrounds from the corresponding scenes for the different categories.
B NETWORK ARCHITECTURES AND TRAINING DETAILS
B.1 COLOURED MNIST
We use 4-layer CNN with the first three layers being convolutional and the last layer linear. The convolutional layers have feature dimensions 64 - 128 256. and are all followed by a MAX POOL, BATCH NORM layer, and RELU activation. Before being fed into the final linear layer, there is a spatial mean mean-pooling pooling operation. An L2 weight decay added to all parameters with a co-efficient of le-4.
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Majority group
Minority group
Validation
Test
boat
beach
kasbah
house orchard viaduct
water tower waterfall
airplane
canyon
lighthouse pagoda
truck
building facade
zen garden
dog
staircase
rock arch
zebra
desert (sand)
horse
crevasse
bamboo forest
train broadleaf forest
bus
ball pit
Table 9: Background scenes for the in-distribution majority group, minority group, and the non- systematically shifted validation and test sets. (The mapping to categories only applies to the majority group in the training set.)
Training is conducted for 30 epochs, with SGD + Momentum (0.9), using batch sizes of 512. The learning rate is cut by 10 from its initial value of 0.1 at epochs 9, 18, and 24.
B.2 For both COCO datasets, we use an architecture based off of Wide Resnet 28-10 (Zagoruyko & Komodakis 2016), Since our images are 64 64, append an extra group of 4 residual blocks with the same layer widths as in the previous group, and use a smaller widening factor of instead of 10 to avoid memory overflow (starting base dimension 64). An L2 weight decay regulariser applied on all parameters with a coefficient of 5c-4.
We train for 200 epochs with SGD + Momentum (0.9), using batch sizes of 384, with an initial learning rate of 0.1 which is cut by 10 at the 120th, 160th, I 180th, and 190th epochs. We use the initially large learning rate for longer following prior works such as as et al. (2019) that have suggested annealing schedules with longer periods of higher learning rates can improve generalisation, which we do find help the base network. In both cases, we apply data augmentation of random crops (after symmetric padding) and random horizontal reflections.
B.3 PARTITIONING NETWORK
We use the same MLP with three hidden layers for all our partitioning networks, with dimensions 64 - 32 - 16. We use LAYER NORM (Ba et al. 2016) and RELU activations after each layer. To avoid merely memorising hard examples, it is necessary regularise this network, spectral normalisation (Yoshida & Miyato| 2017); this involves spectrally normalising every linear layer, and excluding the scaling term in the layer normalisation transforms, in Miyato et al. (018).
also apply
We use a separate network for each class, training for 100 iterations each, with the same batch size used for training the rest of the model. We use the Adam optimiser (Kingma & Ba 2014] with learning rate of le-4. In preliminary experiments we found shared network for all categories to also work, using conditional layer normalisation (Ba et al. 2016 de Vries et al. 2017). We didn't investigate it further for all datasets, since in general a larger number of classes in a dataset might require larger capacity in the partition predictor to account for more features, and as the number of classes go up, number of smaller matrices can have a lower footprint than one very large matrix.
Network architecture design for the partitioning network was done only on the COLOURED MNIST dataset, with access to true oracle group labels for smaller set of in-distribution validation images (20 per category). The same network architecture was applied for the two COCO datasets. The 7 hyper-parameter was learned separately for all datasets, using the smaller sets of validation images. We find, in preliminary experiments, that using random partitionings lead to much worse performance. This suggests that, although not performed for present study, in more realistic situations one could potentially tune these aoper-parameter by validating over classification accuracy as for the invariance penalties.
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B.4 INVARIANCE PENALTIES
In all cases, we pause training of the base network after epoch of training, and learn a partitioning of the training set. This learned partition is used to drop in the invariance penalties training proceeds, and as in prior work (Arjovsky et al. 2019 Krueger et al.] 2020), we find ramping in the penalty co-efficient over number of epochs be useful for stable training. For IRM and REx (and conditional variants), we find it helpful to scale the ERM term down by the penalty co-efficient when the optimal validation co-efficient is greater than 1, as implemented by (Arjovsky et al. (2019) and Krueger et al. (2020).
C REVIEW OF BASELINES AND CONDITIONAL VARIANTS
We briefly review the group invariance methods we compared.
C.I IRMv1
In Arjovsky et al. 2019), risk regularisation method is described in order encourage reliance features that ohey stable correlations with the target variable across data from different environments. The regularisation consists of gradient penalty wrt dummy multiplier on the logits, with the intuition that scaling or shrinking the logits in different environments can only result in local improvements within each environment if the classifier uses features that correlate different levels in the different environments. The objective function is
min 4:2
R (++ + V Rs
BEE
comprises the predictor, which in our case TT fx().. l is dummy multiplier, fixed and R is the environment risk, corresponding to the average loss for data in particular environment when using Φ
For our conditional variant (cIRMvl), we stratify the gradient penalty over classes, so that the penalty applied separately per class in each environment.
The syper-parameters we search over for this method include the penalty co-efficient À and the number epochs of training over which to linearly ramp up À to its full value.
C.2 REX
Krueger et al. (2020) proposed risk regularisation method that aims directly match training risks across environments, by imposing a penalty that minimises the variance of risks across environments (V-REx).
min &
ER((B) + +VVat) +."
(10)
3EE
For our conditional variant (cREx), we apply the variance penalty stratified by class.
The hyper-parameten we search over for this method include the penalty co-efficient and the number of epochs of training over which to linearly ramp up À to its full value.
C.3 GROUPDRO
Sagawa et al. (2020) suggest an online algorithm for group-based distributionally robust optimisation, which effectively re- re-weights group losses function of their evolving magnitudes, therefore putting more emphasis on groups that fare worse through training.
For conditional variant (cGroupDRO), losses belonging to the classes separately in each group.
compute the group weights per class, by using the
The dyyer--arameee
search over for this method include the learning rate for the online group- weights and the two group adjustment hyper--paaane paper-parameters. Additionally, we sample equally from both groups for this method, suggested, finding it to improve results in preliminary experiments.
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C.4 REWEIGHT
We learn a hyper-parameter À on validation, such that every example in the majority group is weighted with 1/(A + 1) (because we only want weight the majority group down).
The hyper-parametery we search over for this method include the penalty co-efficient and the number of epochs of training over which to linearly ramp up À to its full value.
C.5 MMD FEATURE MATCHING
Maximum mean discrepancy based distributional matching of features across domains has been shown be effective for domain generalisation (Li et al. 2018), and conditional matching of distributions (usually with adversaries, for example, in Li et al. (2188))) tends to work better. We found in preliminary experiments that conditional MMD significantly outperformed the unconditional variant, we only ran full experiments and reported results using cMMD.
The group invariance penalty looks as follows
[ (o o))] E. (o( grroup 1)]
(11)
where á induces kernel function K, which in implementation is mixture of 3 Gaussians with bandwidths [1, 5, 10), which are the recommended set of bandwidths in Li et al. (2018). Adding sharper or flatter bandwidths appeared to hurt performance in preliminary experiments.
The syper--arameters we search over for this method include the penalty co-efficient and the number of epochs of training over which to linearly ramp up À to its full value.
C.6
GRID SEARCH RANGES
In all cases, À is searched over range of
11--4,1e-3, le- 0.1, 0.5, 1, 2, 10, 20, 50, 100, 200, 500, 1000, 10000, 100000},
and the number of epochs over which to linearly ramp up À is searched over {1, 5, 30) for MNIST search over (0.001.0.01. 0.1, 1.0, 10} for the learning rate of the group-weights, and over {0. 1, 2, 3, 4, 5) for the group-adjustment hyper- parameters, as recommended in Sagawa et al.) (2020). We also average the losses group-wise as already done in IRMv1 and REx for cMMD and PGI, except for 1OCO-ON-PLACES, where we find this choice hurt performance.
and {1, 10, 200} for COCO. For the GroupDRO methods,
D DIFFERENT VALIDATION SETS
In this section, report results for all the methods we compare, when picking hyper-parameters using different validation sets, as discussed in Section 5.4
We note that contrary what one would typically do real-world deployment, we do not augment the training sets with the validation sets for evaluating test time performance. This is because the presence of data with systematic distributional shift at training time improves performance significantly (as observed in Table 1), and our goal here is to perform an illustrative study about the potential effectiveness of invariance methods at learning to generalise systematically.
While we could have augmented the training set with validation data when we are not using validation sets with systematic distributional shift, we follow the same protocol these cases of not augmenting the training set, in order to keep the numbers comparable with each other across different validation schemes.
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Table 10:
Picking
only using
validation set of non-systematic shifts for
COLOURED MNIST.
Methods
In-distribution
Non-systematic shift
Systematic shift
Anomaly detection
Base (ERM)
99.60 + 0.02
53.26 + 1.89
38.72 +: 2.27
7.70 = 0.23
IRMv1
99.61 t 0.05
63.80 + 3.58
55,38 = 1.52
10.35 = 0.43
REx
98.95 + 0.11
72.12 + 1.90
71.18 = 3.27
15.54 2.05
GroupDRO
98.70 + 0.10
71.51 + 2.61
77.95 0.65
18.26 2.11
Reweight
99.06 + 0.06
77.03 + 1.33
83.37 +
17.10 1.11
cIRMvl
99.36 + 0.25
65.78 3.53
61.09 + 5.30
14.16 2.12
cREx
99.20 + 0.10
73.97 + 1.07
76.06 1.71
17.62 2.29
cGroupDRO
97.89 + 0.29
73.71 + 3.21
76.90 + 2.55
20.73 4.63
cMMD
99.40 1 0.07
97.36 1 0.72
97.91 1 0.19
78.14 3.79
PGI
99.31 + 0.05
98.21 +: 0.26
97.54 + 0.41
76.00 = 4.06
Table 11: Picking hyper-parameters using both a validation set of non-systematic shifts and the in-distribution for COLOURED MNIST.
Methods
In-distribution
Non-systematic shift
Systematic shift
Anomaly detection
Base (ERM)
99.60 + 0.02
53.26 + 1.89
38.72 + 2.27
7.70 = 0.23
IRMv1
99.61 + 0.05
63.80 + 3.58
55.38 + 1.52
10.35 0.43
REx
98.95 + 0.11
72.12 + 1.90
71.18 = 3.27
15.54 = 2.05
GroupDRO
98.70 + 0.10
71.51 2.61
77.95 = 0,65
18.26 2.11
Reweight
99.06 + 0.06
77.03 + 1.33
83.37 0.61
17.10 1.11
cIRMv1
99.36 1 0.25
65.78 3.53
61.09 5.30
14.16 2.12
cREx
99.20 1 0.10
73.97 1 1.07
76.06 1.71
17.62 2.29
cGroupDRO
97.89 + 0.29
73.71 + 3.21
76.90 + 2.55
20.73 4.63
cMMD
99.49 + 0.04
96.36 + 0.53
97.68 + 0.17
71.15 2.65
PGI
99.30 1 0.07
98.31 1 0.27
97.48 1 0.45
76.07 5.67
Table 12: Picking hyper-parametest using only in-distribution set for COLOURED MNIST.
Methods
In-distribution
Non-systematic shift
Systematic shift
Anomaly detection
Base (ERM)
99.60 + 0.02
53.26 + 1.89
38.72 = 2.27
7.70 0.23
IRMv1
99.69 1 0.02
60.18 + 1.34
53.20 +: 1.44
9,71 = 0.76
REx
99.71 1 0.04
60.71 + 1.38
50.87 #: 2.79
10.02 0.69
GroupDRO
99.61 1 0.01
52.21 + 2.03
40.27
7.37 0.44
Reweight
99.66 1 0.04
63.36 1 4.60
58.09 1 0.52
11.41 0.49
cIRMv1
99.69 1 0.01
60.43 1 2.71
$2.98 2.14
0.91
cREx
99.70 1 0.02
61.06 1 1.20
50,83 1: 2.33
9,21 0.97
cGroupDRO
99.63 + 0.01
55.53 + 3.63
45.25 + 2.24
8.69 = 1.02
cMMD
99.70 + 0.02
61.10 + 1.66
51.06 + 1.87
9.62 1.09
PGI
99.69 1 0.03
63.62 1: 2.05
58.18 +: 2.05
11.81 1.89
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Table 13: Picking
only using validation set of non-systematic shifts for COCO
ON-COLOURS.
Methods
In-distribution
Non-systematic shift
Systematic shift
Anomaly detection
Base (ERM)
90.57 + 1.28
26.81 +: 4.93
1.10 +: 0.36
5.47 = 0.08
IRMv1
91.61 t 0.38
32.30 + 4.52
2.11 = 0.30
5.81 = 0.17
REx
91.69 + 0.50
36.57 + 4.03
2.69 = 0.81
5.73 0.14
GroupDRO
40.31 + 2.11
38.84 + 3.78
43.24 2.84
17.99 3.68
Reweight
73.17 + 2.48
48.98 + 2.65
39.80
18.20 3.80
cIRMvl
91.53 + 0.31
31.11 + 4.51
1.74 0.40
5.87 0.16
cREx
91.45 + 0.39
32,43 + 2.03
1.98 +
5.75 0.13
cGroupDRO
43.61 + 4.33
39.15 + 4.79
36.63 + 4.81
18.21 3.65
cMMD
89.87 1 1.13
55.02 1 2.29
27.36 1: 1.57
8.82 0.70
PGI
85.78 + 1.45
51.02 + 2.32
38.85 + 2.29
15.71 3.25
Table 14: Picking hyper-parameters using both a validation set of non-systematic shifts and the in-distribution for COCO-ON-COLOOURS,
Methods
In-distribution
Non-systematic shift
Systematic shift
Anomaly detection
Base
90.57 + 1.28
26.81 + 4.93
1.10 + 0.36
5,47 = 0.08
IRMv1
91.61 + 0.38
32.30 + 4.52
2.11 0.30
5,81 = 0.17
REx
91.69 + 0.50
36.57 + 4.03
2.69 + 0.81
5,73 = 0.14
GroupDRO
90.70 1 0.56
3.210
5,66 - 0.95
6.60 = 0.40
Reweight
90.25 + 0.71
40.23 3.32
10.60 + 1.34
7.06 + 0.52
cIRMv1
91.53 1 0.31
31.11 + 4.51
1.74
5,87 + 0.16
cREx
91.45 1 0.39
32.43 2.03
1.98 0.68
5.75 0.13
cGroupDRO
87.68 + 0.59
36.40 + 2.30
14.07 2.47
9,82 - 0.91
cMMD
89.87 + 1.13
55.02 + 2.29
27.36 + 1.57
8.82 - 0.70
PGI
85.78 1 1.45
51.02 1 2.32
38.85 2.29
15.71 3,25
Table 15: Picking hyper-parameters using only the in-distribution set COCO-ON-COLOURS.
Methods
In-distribution
Non-systematic shift
Systematic shift
Anomaly detection
Base
90.57 + 1.28
26.81 + 4.93
1.10 0.36
5.47 = 0.08
IRMv1
91.54 1 0.37
32.40 + 3.62
1.93 + 0.36
5.77 +:2 0.23
REx
91.62 1 0.38
31.89 1 4.08
1.98 0.37
5.74 0.20
GroupDRO
91.44 1 0.27
22.42 1 3.00
0.56 0.15
5,55 0.19
Reweight
91.10 1 0.50
38.63 1 3.23
4.35 1 1.13
6.13 1 0.22
cIRMv1
91.31 1 0.43
30.94 3.73
1.65 0.36
5,83 0.17
cREx
91.70 1 0.50
34.93 1 4.58
2.24 0.48
5.82 1 0.19
cGroupDRO
91.75 + 0.60
24.05 + 3.44
0,94 + 0.27
5.77 + 0.13
cMMD
92.51 + 0.41
44.59 3.28
10.48 + 0.98
6.05 - 0.23
PGI
91.86 1: 0.33
32.46 4: 3.06
2.81 + 0.53
5.88 + 0.19
17

---

Published as a conference paper at ICLR 2021
Table 16: Picking hyper-parameters only using validation of non-systematic shifts for COCO-
ON-PLACES.
Methods
In-distribution
Non-systematic shift
Systematic shift
Anomaly detection
Base (ERM)
81.06 1.01
45.25 = 0.96
29.18 + 1.24
9.21 0.21
IRMv1
80.93 0.71
45.17 0.92
28.78 + 0.73
9.39 0.60
REx
81.25 0.76
45.40 = 0.95
29.20 = 1.28
9.46 0.98
GroupDRO
76.05 0.87
43.72 = 0,43
31.83 + 0.54
9.61 + 0,55
Reweight
80.90 0.50
44.87 = 1.26
29.34 = 0.99
9.59 0.54
cIRMv1
81.48 0.67
45.59 = 1.27
29.28 + 0.96
9.80 + 0.78
cREx
81.50 0.76
45.44 = 0.96
29.12 0.97
9.17 0.59
cGroupDRO
78.25 0.31
41.69 0.08
28.16 = 0.91
9,45 0.22
cMMD
79.64 0.73
49.44 0.99
35.86 + 0.66
9.80 0.45
PG1
80.99 0.52
47.63
0.90
31.91 + 0.89
9.59
cMMD (oracle split)
80.04 1.01
49.02 + 1.18
35.60 + 0.72
10.55 + 0.55
PGT (oracle split)
75.98 0.75
47.50 + 0.87
37.27 + 1.40
11.57 + 0.71
Table 17: Picking hyper-parameters using both a validation set of non-systematic shifts and the in-distribution set for COCO-ON-PLACES.
Methods
In-distribution
Non-systematic shift
Systematic shift
Anomaly detection
(ERM)
81.06 1.01
45.25
0.96
29.18 1
9.21 0.21
IRMv1
80.93 0.71
45.17 1 0.92
28.78 1 0.73
9.39 0.60
REx
81.25 0.76
45.40 0.95
29.20 +::
9.46 0.98
GroupDRO
80.61 0.44
41.96 + 1.00
27.19 + 0.67
9.05 0.06
Reweight
80.90 0.50
44.87 = 1.26
29.34 + 0.99
9.59 + 0.54
cIRMv1
81.48 0.67
45.59 + 1.27
29.28 + 0.96
9.80 + 0.78
cREx
81.50 0.76
45.44 + 0.96
29.12 + 0.97
9.17 + 0.59
cGroupDRO
78.25 0.31
41.69 = 0.08
28.16 = 0.91
9.45 0.22
cMMD
79.64 0.73
49.44 0.99
35.86 + 0.66
9.80 0.45
PG1
80.99 0.52
47.63 + 0.90
31.91 + 0.89
9.59 0.89
cMMD (oracle split)
79.56 0.64
46.74 + 0.83
34.78 + 0.76
9.78 + 0.59
PGI (oracle split)
78.70
0.86
47.28 = 1.05
32.84 + 0.89
11.13 + 0.90
Table 18: Picking hyper- parameters using only the in-distribution set for
Methods
In-distribution
Non-systematic shift
Systematic shift
Anomaly detection
Base (ERM)
81.06 1.01
45.25 1 0.96
29.18 1: 1.24
9.21 0.21
IRMv1
80.93 0.71
45.17 0.92
28.78 + 0.73
9.39 0.60
REx
81.25 0.76
45.40 + 0.95
29.20 + 1.28
9,46 + 0.98
GroupDRO
80.61 0.44
41.96 + 1.00
27.19 + 0.67
9.05 + 0.06
Reweight
81.53 0.66
45.77 = 1.33
29.39 = 0.97
9.55 0.79
cIRMv1
81.48 0.67
45.59 + 1.27
29.28 + 0.96
9.80 + 0.78
cREx
80.68 0.69
44.80 + 1.39
29.76 + 1.05
9.95 0.79
cGroupDRO
80.23 0.13
41.86
0.60
25.88 +
9,43 0.68
cMMD
81.11 0.51
46.57 + 0.97
31.54 + 0.88
9.79 0.79
PGI
80.99 0.52
47.63
0.90
31.91 + 0.89
9,59 0.89
cMMD (oracle split)
81.59 0.65
45.47 - 1.40
29.16 + 0.96
9.15 + 0.36
PGI (oracle split)
81.22 1.09
45.16 0.96
29.24 + 0.64
9.31 0.67
18

---

Published as a conference paper at ICLR 2021
E MEASURING SEMANTIC ANOMALY DETECTION
We use the test set with non-systematic distributional shift as the normal data, and the held-out class data combined systematically with the biasing colours or backgrounds as the anomalous data. For MNIST, this means there is 9 times more normal data than anomalous data, which reflects the typical situation of anomalies being rarer. Our choice of normal data makes this a harder task than usual, since we are assessing for higher (than the anomalies) predictive confidences for non-semantic shift with semantic factors kept the same, and reduced predictive confidence for semantic shift with non-semantic factors from the seen data. For the COCO datasets, we only sample 100 images from the held-out class to resemble the MNIST experimental setup.
Anomaly detection is measured using average precision, treating the anomalous class positive, with the negative the predictive softmax confidence as the score (Hendrycks Gimpel 2017
F ALGORITHM
Algorithm 1: Algorithm for PGI
Initialise all classifier parameters 0, and cerritionn-redittingg networks, 9c, Ve [C] for one epoch do for mini-batches D, 'D do grada == Vel(O, w (D)) gradw (. D[)
0, w = optimiieerggradlee gradd))
end end
for all classes [C] do
Learn partition for images in D with labels c, (Eq. 8)
end
for T epochs do
for mini-batches Db, D do
gradg := (e(( 0, w D2) + X.penally) (Bq, 6,7);
gradw- V (0, D))
w.w: == ogrdmiieerggadde gradw)
end
end
19


Reviews:
[{"review_id": "kvqADlseGnD", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 6, "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: the paper studies a setting where there are simple correlations with the target variables (that are not however robust) and more complex but robust features. The simple correlation is usually such that in most cases the feature is descriptive of the label, but at times it takes values that are part of a “minority group” that is not descriptive of any one class in particular.\nSystematic-shift generalization is tested using the same spurious features that are present in training, in all combinations except the usual pairing of spurious feature and class.\nNon-systematic shifts are tested using novel spurious features.\nAnomaly detection is tested with unseen robust features.\nNeural networks trained with standard ERM notoriously pick up on all features that correlate with the label, and the paper compares several methods (including IRMv1, REx, and GroupDRO) to the proposed Predictive Group Invariance. The latter is shown to work better than the baselines (even when using class conditioning) on systematic shifts and anomaly detection, and often non-systematic shifts.\n\nI found the paper to be mostly very well written (with few exceptions), and especially sections 1 and 2 very easy to read and understand. The experiments setting is clear and not at all trivial.\n\nMy main questions and concern:\n- The description of PGI could be improved, I would suggest adding an algorithm box that sums up what is described.\n- “environments” and “partitions” are the same thing for PGI? How are “environments” chosen for the baseliness that need them? With the same partition networks used for PGI? If not, it might be hard to disentangle the role of using the partition networks from the KL objective of PGI. \n- The part about partitioning is not detailed enough. P13: \"We use a separate network for each object category” what is an object category? From page 3 I understood that there might be 2 categories (easy and hard), is this the case? Is it correct to think of the role of these partition predicting networks as a sort of clustering that focuses on the most easy-to-find features?\n- I haven’t seen any mention of early stopping in the experiments. It sounds like the performance reported is the one at the end of all training epochs. This might explain why for example ERM on COCO-on-Colours only achieves a 1.10% accuracy on Systematic shift. If this is correct, why not using early stopping, given how prone to overfitting these networks might be?\n\nOverall, based on the results, the proposed PGI seems to improve the accuracy over the baselines, even though it seems to be more of an incremental improvement than a substantial and conceptual one.\nI look forward to a constructive discussion with the authors and hope my questions and concerns will be clarified.\n\n\nMinor:\n- I think the blanket statement “it has been reported that highly competitive performances can often be achieved with a baseline model on such domain generalisation benchmarks (Gulrajani & Lopez-Paz, 2020), similarly as in Table 1. “ is a bit too vague, and since it questions the validity of the results in all papers mentioned before, it should be either properly justified or adjusted to make sure that what it says actually applies to all of them.\n- Figure 3 is referenced in the text instead of Figure 2 (perhaps it was not referenced with \\ref )\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "TcOqnndFWrC", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper shows that group invariance methods across inferred partitions show better generalization in (non-)systematic distributional shifts and anomaly detection settings. It also suggests a new invariance penalty and empirically shows that it works better on three synthetic datasets viz. coloured-MNIST, COCO-on-colours, and COCO-on-places.\n\nThe paper is written well and starts off by giving an intuition of why IRM-like methods are important by presenting the results of a simple experiment on coloured-MNIST (table 1). It then goes on to talk about (non-)systematic generalization before introducing the proposed method. The authors use reverse KL divergence between the group distributions as the penalty and use prior work to partition the datasets into groups. They use \n\nThe results look promising across datasets, though it is slightly lower in the 'in-distribution' setting. I am happy to see that they also talk extensively about hyperparameter selection especially in the case where they assume no access to validation sets with a distributional shift.\n\nOverall, I like the work and would like to see it presented at the conference.\nOne minor point: cite work the first time you introduce something, not later on. It can be a little confusing for the readers. I wondered if I missed something. For ex: \"We find that a recently proposed method can be effective at discovering...\", \"IRMv1\", etc.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "LeSYadz6_OM", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Overall, I found the paper well-written, the methods appear reasonable and the math appears correct. I think the case made for the importance/significance of the method could be improved, but think the paper should be accepted regardless.\n\n--- Comments ---\n\n1. I thought the introduction did a good job setting up the high-level problem, but did not really establish why a new method is needed. I got to the end of the intro and wondered why I couldn't use any one of a dozen DRO type methods to get the type of robustness described. The experiments do a reasonable job of demonstrating the utility of the proposed method, but I recommend adding something to the intro like: Methods have been proposed that promote robustness to distributional shift, however, these methods fail to capture XYZ shifts because ABC.\n\n2. I really liked the examples of the different types of shifts that the authors are interested in, but thought the paper could do a better job arguing why these specific shifts are important or relevant. Perhaps swapping the \"red chair\" example for something more compelling might do the trick. In particular, it is worth giving a compelling reason why the drop in average performance might be worth an improvement in, for example, anomaly detection. \n\n3. I found the early parts of Section 3 a bit confusing because it was unclear at that point in the paper where the \"majority\" and \"minority\" groups were coming from. I recommend adding a few sentences to the beginning of that section like: Suppose that we knew the collection of non-semantic features and thus new the relevant majority and minority groups. This will not, in general, be the case and we will show how to derive such groups from the data, but we first establish our method as if such groups were known.\n\n4. I found Equation (10) very hard to follow. First, the notation has *many* problems (superscripts from earlier in the section are changed to subscripts; no domain is given for $\\alpha$; it is not clear what it means to index $\\alpha$ by $e$; $\\mu$ is not defined; $\\gamma$ is not defined). Second, I would give a few sentences explaining what the pieces of this objective are doing and why it achieves the goal of splitting the data into relevant majority and minority groups.\n\n--- Minor comments ---\n\n1. Page 2, par 3, line 1: Not clear what \"modeling bias\" refers to here or why it affects the dataset.\n\n2. Equations (1) and (2) don't really seem necessary for the rest of the work. In particular, $\\mathcal{C}$ isn't reference anywhere else in the paper. I would just define $h_s$ and $h_n$ and move on.\n\n3. I found the $\\not\\sim$ notation a bit confusing. Does it mean \"sampled from any distribution other than $p$? Based on the description, it seems more like it means \"sampled from outside the support of $p$\" which matches the examples given in Fig 1.\n\n4. Equation 6: $\\ell$ is not defined.\n\n5. Equation 9 and surrounding text: I would change reverse-KL to just KL since, in this context, this not an obvious forward/reverse direction.\n\n6. Text under Equation 9: It doesn't really make sense for a categorical distribution to be multi-modal since there is no inherent ordering to values in the support (unless there are literally two or more equivalent modes). I would recommend changing unimodal/multi-modal to peaked/flat or low-entropy/high-entropy.", "extra_scores": {}, "extra_text": {}}, {"review_id": "BTV8FFr0Puo", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 8, "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:\nThis paper studies the behaviour of deep neural networks in situations where simple but irrelevant correlations exist between input and output, and dominate more complex but relevant correlations. The authors conduct experiments on synthetic datasets (like coloured MNIST) and show that an invariance penalty helps the network focus on relevant correlations. \n\nPros:\n- The paper studies neural network behaviour with respect to systemic biases that are likely faced by most neural networks in some form or the other. To make the study tenable, the authors make use of meaningful synthetic datasets, and propose an intuitive regularization to overcome the systemic biases. \n- The analysis done in the paper is very methodical, and the presentation is very clear.\n- The numerical simulations are comprehensive and convincing.\n\nCons:\n- It would be nice to see how this would be applicable to real world datasets. The paper is interesting even without it, and I also appreciate that the authors are honest about it - so I would not hold it against the authors. But it would further strengthen the paper if some basic experiments are done on real world datasets. For instance, will one be able to find a partition on ImageNet?\n\nComments:\n- Section 5.1: Minimization is spelt incorrectly.\n- Equation (6-7): I am not entirely sure what is happening with respect to the constraint on \\theta. What does capital \\theta correspond to? And if \\theta itself is the result of an optimisation (argmin), then why is there another optimisation on the same \\theta in the loss function?\n- In the text that appears before equation (3), it is mentioned that the predicted features f_\\theta will be matched for the two partitions, but equation (7) matches the predicted output post softmax. Could you please clarify?", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: All reviewers seems in favour of accepting this paper, witht he majority voting for marginally above acceptance threshold. 
The authors have taken special heed of the suggestions and improved the clarity of the paper. 
From examination of the reviews, the paper achieves enough to warrant publication. 
My recommendation is therefore to accept the manuscript. 
Decision: Accept (Spotlight)