Distributionally Robust Models with Parametric Likelihood Ratios
Authors: Paul Michel
Tatsunori Hashimoto
Graham Neubig
Venue: ICLR 2022 Poster (2022)
Keywords: distributionally robust optimization
fairness
deep learning
robustness
adversarial learning

Abstract: As machine learning models are deployed ever more broadly, it becomes increasingly important that they are not only able to perform well on their training distribution, but also yield accurate predictions when confronted with distribution shift. The Distributionally Robust Optimization (DRO) framework proposes to address this issue by training models to minimize their expected risk under a collection of distributions, to imitate test-time shifts. This is most commonly achieved by instance-level re-weighting of the training objective to emulate the likelihood ratio with possible test distributions, which allows for estimating their empirical risk via importance sampling (assuming that they are subpopulations of the training distribution). However, re-weighting schemes in the literature are usually limited due to the difficulty of keeping the optimization problem tractable and the complexity of enforcing normalization constraints. In this paper, we show that three simple ideas -- mini-batch level normalization, a KL penalty and simultaneous gradient updates -- allow us to train models with DRO using a broader class of parametric likelihood ratios. In a series of experiments on both image and text classification benchmarks, we find that models trained with the resulting parametric adversaries are consistently more robust to subpopulation shifts when compared to other DRO approaches, and that the method performs reliably well with little hyper-parameter tuning.

Published as a conference paper at ICLR 2022
DISTRIBUTIONALLY ROBUST MODELS WITH PARAMETRIC LIKELIHOOD RATIOS
Paul Michel
Tatsunori Hashimoto
Graham Neubig
Centre Sciences des Données
Computer Science Department Stanford University Stanford, CA 94305, USA
School of Computer Science Carnegie Mellon University Pittburgh, PA 15213, USA gneubig@css cmu edu
École normale supérieure PSL Paris, 75005, France
pmiche 1hh413311555 56 com
ABSTRACT
As machine learning models are deployed ever more broadly, it becomes increas- ingly important that they are not only able to perform well on their training distri- bution, but also yield accurate predictions when confronted with distribution shift. The Distributionally Robust Optimization (DRO) framework proposes to address this issue by training models to minimize their expected risk under a collection of distributions, to imitate test-time shifts. This is most commonly achieved by instance-level re- re-weighting of the training objective to emulate the likelihood ratio with possible test distributions, which allows for estimating their empirical risk via importance sampling (assuming that they are subpopulations of the training distri- bution). However, re-weighting schemes in the literature are usually limited due to the difficulty of keeping the optimization problem tractable and the complexity enforcing normalization constraints. In this paper, we show that three simple ideas mini-batch level normalization, a KL penalty and simultancous gradient updates allow us to train models with DRO using a broader class of parametric likelihood ratios. In series of experiments on both image and text classification benchmarks, we find that models trained with the resulting parametric adversaries are consistently more robust to subpopulation shifts when compared to other DRO approaches, and that the method performs reliably well with little tuning
INTRODUCTION
is well acknowledged that modern neural network based machine learning models tend to under- perform when they are evaluated on data distributions that differ from the one they were trained on. For example, machine learning model performance has been observed to degrade under train-test mismatch in topics (Gururangan et al. 2020), demographics (Blodgett et al., 2016 Ammdei et al.) 2016; Hovy & Søgaard 2015 Grother et al. 2019), geographic regions (Koh el al. 20020), and even data collection processes (Beery et al. [018 Zech al. 2018 Michel & Neubig 2018). In partieular, these models often perform poorly when evaluated on subpopulations. domains that are present but underrepresented in their training data (Sagawa al., 2020). and they can latch on to spurious correlations (McCoy et al. 2019). This has problematic real-world consequences; when such models are deployed at large, this representation disparity can, for example, unfairly affect minority groups (Buolamwini & Gebru 2018 Hashimoto et al. 2018).
This behaviour can largely be attributed to the the empirical risk minimization (ERM))rinciphl) principle which underlies the majority of learning algorithms used in practice. In |RM models are trained to minimize the average loss over finite sample from fixed training distribution (Vapnik) 1992). proxy for the expected loss random example drawn from the fixed, albeit unknown, data distribution. This favors models which perform well on average on a fixed training set, as opposed to models which would perform equally well variety of subpopulations that better reflects the diverse set of distributions that can be encountered at test time. On the other hand, Distributionally robust optimization (DRO))rrposss an appealing alternative to ERM In DRO models trained to Code reproduce our experiments can be found wttpt:tp////m ///p:////iii thub. wwt.c//.m/.c.cc.. che 1 PRO

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minimize their worst case risk (or empirical estimate thereof computed on finite sample, via e.g. importance weighting) under a pre-determined family of distributions Q, called the "uncertainty set" (or "ambiguity set"):
LRO (0) = max ((c.))> (o(2, y)- 9E
(1)
In the absence of explicit information about the subpopulations of interest (which would naturally define Q), it is up to the practitioner to carefully define this uncertainty set. This has been the subject of much work in the literature (see Rahimian & Mehrotra (2019) for a survey). Recently, Miichel et al.) (2021) proposed P-DRO, a promising approach where the uncertainty set is defined by a parametric family of generative models, which allows for more flexibility in defining the uncertainty set. P-DRO shows significant improvement over comparable baselines, but it suffers from several drawbacks. First, it presupposes the availability of generative models capable of outputting exact densities, which limits its field of application to modalities where such models are readily available (such as language models in NLP). Second, is challenging to use in practice due to its reliance on a number of hyper-parameters and approximations to the objective function.
In this paper, we propose a new approach for [RO] called RP-DRO, based on a key modification of the P-DRO algorithm: instead of modeling the worst-case distributions directly, parametrize the likelihood ratio between the training distribution and the worst-case distribution. This removes the dependency on an unwieldy generative model, making the method useful for more applications. While likelihood ratio formulations of DRO have been tried in prior work (Sagawa et al. 2020), show that they are particularly effective for parametrie, neural network based adversaries. Our approach relies on three simple ideas: a mini-batch level normalization strategy to enforce likelihood ratio constraints, penalty-form of the KL divergence uncertainty set, and the use of simultancous gradient updates for training. RP-DRO consistently achieves equal or better robust subpopulation accuracy compared to P-DRO and other baselines on variety of standard benchmarks in image and text classification. In addition, we find it is both faster than P-DRO and depends fewer hyper- parameters. Additional ablation experiments demonstrate that both our mini-batch normalization strategy and simultaneous gradient updates are necessary for high performance. Finally, perform experimental analyses to shed light on the advantages brought by parametric adversaries compared to their nonparametric counterparts.
2 BACKGROUND
In the following, we consider a model parametrized by E R.daas Our goal is to find model which minimizes the loss function &a (z, y) pairs of inputs and outputs (z, y) X x V. For instance, might represent images and y a categorical label. Parameters 0 are estimated on a training dataset D (x y:) which assume be drawn from training distribution p.
The DRO optimization problem with uncertainty set Q is
min max B.y (o y) qEQ
Note that the DRO loss in Eq. is the inner maximum of the DRO problem, and it provides an upper bound on the expected loss of the model under any distribution in the uncertainty set Q. This motivates the use the minimizer of the min-max game in Eq. 2 as a robust model. We refer to the solution of the inner maximum as the "adversary" from now on
However this objective is only useful insofar that (1) Q covers test distributions of interest (corre- sponding to different domains, demographics, etc.) and (2) Q is not overly pessimistic. To fulfil this second condition, there should exist some model 0 that achieves low loss simultancously on the test distribution well as Q. This often requires that Q only contain distributions that are covariate shifts of the test distribution, i.e. that are such that the conditional distribution g(v) x)) coincides with that of training distribution ply x)).
2.1 NONPARAMETRIC DRO
There substantial existing work on nomparametric formulations of DRO. where Q is expressed as a divergence ball centered at the training distribution. This includes f-divergences (Ben- Tal et al. 2013 Hu & Hong, 2013; Faury et al. 2020), Wasserstein/IPM (Sinha et al.) 2018 Husain 2020), moment
constraints (Delage & Ye 2010 Nguyen al. 2020), and CVaR (Fan et al. 2017 Curi al. 2020
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Levy al. 2020) based uncertainty sets. These nonparametric approaches appealing they require very little domain- -spuciffc knowledge, have well-understood theory (Duchi & Namkoong 2018). and optimization procedures (e.g. Huu & Hong 2013) for constraints and Levy al. (2020)
for X and CVaR constraints).
Unfortunately, nonparametrie DRO algorithms suffer from being overly pessimistic. Their uncertainty sets tend to include distributions that are exceedingly difficult to learn, or not representative of real- world distribution shifts. Furthermore, they often cannot enforce even basic constraints such as covariate shift structures (Duchi et al. 2020 Hu et al. 2018). Group-structured DRO uncertainty sets (Sagawa et al. 2020) overcome some of these challenges, but require significant domain expertise to pre-specify target subpopulations that model should be robust to.
2.2 PARAMETRIC DRO
Parametric DRO Michel et al. 2021 is a method for DRO in which the uncertainty set Q is defined family of parametric generative models, which avoids the extreme pessimism of nonparametric DRO without the explicit specification of subpopulations. Specifically, given generative model q parameterized by va E ded the KL-constrained parametric DRO objective can be written as follows:
min
max
(2 (xx, y, 0)
0
KL(ge Spp)sk
As demonstrated by Michel et al. metric counterpart. However, the difficulty of optimizing Eq. 3 directly results a number of approximations and additional peram-rrrseeerrss that are hard to tune.
(02T), P-DRO yields significant improvements over its nonpara-
addition, central drawback of P-DRO is that it necessitates training an auxiliary generative model of the data. This be difficult for several reasons. First, this limits the applicability of the method to domains with generative models that allow for exact probability computations. Moreover, even when such generative models are available, they are often more computationally demanding than their discriminative counterparts. In language models for instance, probabilities for sequences of text obtained by iteratively producing conditional probabilities over all tokens in the vocabulary. This additional step results in considerable computational overhead compared to discriminative models.
3 PARAMETRIC LIKELIHOOD RATIO
3.1 DRO AS LIKELIHOOD RATIO OPTIMIZATION PROBLEM
In the situation that all distributions in Q are absolutely continuous with respect to p (i.e. for all measurable subset A C X x V, g Q, ((A) > 0 only p(A) 0) the inner maximum in Eq. can be rewritten purely as a function of the likelihood ratio P
g 4) p(x, y)
Bi (a v) = ((
lo (0( y).
(4)
Such absolute continuity assumptions standard in f-divergence and group DRO methods, which both rely upon re- re-weighting the training distributions. In fact, the KL divergence constraint in P-DRO presupposes absolute continuity.
This suggests that the inner maximum can be re-written as an optimization problem on functions r: X T44 within the uncertainty set R E (r pr E ②}
min max (x, HER
This reparametrization of the problem will allow us to replace a parametric family of generative models with a parametric family over probability ratios.
3.2 RATIO-BASED P-DRO
The likelihood ratio formulation described above is appealing for P-DRO because it enables the use of discriminative style neural architectures for parametrizing the ratio r, which opens up many more
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consists of using the normalized ratio at the mini-batch level by setting Published as a conference paper at ICLR 2022
options for defining the parametric uncertainty set. Specifically, we can set the adversary to be any parametric function T : : x Y + R* verifying Ex,y (2 y) = 1. The key insight that we use to realize our proposed method is that do not need to restrict the choice of adversaries to those that implicitly satisfy this normalization condition (i.e. generative models). Instead, we can pick any adversary and treat normalization
additional constraint (the "normalization constraint").
Note that in this case, the KL constraint takes the simple form (L llp) = Epry log Pee
Ep To log rg. The final min-met max problem, which we dub ratio- P-DRO (RP-DRO), is:
min
max * Ep rog Epra=1
B(z:))~pre(a ()
(6)
Carsen
As in P-DRO, we can look for equilibria of this differentiable min max game by performing simulta- neous gradient updates (Singh et al. (2000) to & and si in directions - -RORRODOO and +V ei CR respectively. Although inding global equilibria is not guaranteed in high dimensional non-convex settings (Balduzzi et al. 20188) empirical evidence suggests that models trained in this manner still reach useful solutions (Michel al. 2021).
In experiments, we adopt an exponential parametrization (a y) x e (m.a)) where fe is the output of any parametric model with values in R. Similarly to P-DRO, we do not explicitly enforce the KL constraint (due to the difficulty of projecting onto the KL ball), and instead we relax it in the form of a term T Ep Tw logry, added to the loss function. The regularization strength is treated as a hyper-parameter
3.3 ENFORCING THE NORMALIZATION CONSTRAINT
In addition to the KL constraint, RP-DRO necessitates that Tw satisfies a normalization constraint Ep Tv = 1 to ensure that pre is a proper probability distribution over Duain- If that were not the case, the adversary To could artificially increase the weighted expectation Ep To by assigning a total weight greater than 1 to the entire dataset.
Existing methods for ratio based DRO such Sagawa al. (2000) achieve this by either projecting Tp onto the set fr Ep = 1} after each gradient update on t/ or by directly parametrizing the ratio as rwe (Ir, y) = ( / EEp e/v. Unfortunately, these solutions are computationally infeasible in practical scenarios with large datasets. Indeed, they necessitate computing the entire expectation over p. which can be costly when each fo (z, y) is the output of neural model.
We propose two simple, yet effective solutions for addressing this issue in the context of mini-batch training where we can only compute f for small number of samples (x1, yI),..., (Ia, Un) at each step of training.
nell-normalization is inspired by the idea of "ssel--normalizationn developed for globally normal- ized structured prediction models (Andreas et al. 2015)) Goyal et al. (019). It consists in adding a relaxation of the normalization constraint to the objective. Specifically, following (Goyal et al. (2019) we add a squared penalty on the log normalizer at the mini-batch level. Ignoring the KL penalty, this regularized objective takes the following empirical form:
2
n
ro (Z vz)
Eselt-norm (9, v)
Σ (z Eo 9:) - Bog
Σ
(7)
The hyper-paramedt 3 controls the regularization strength. Intuitively, this penalizes adversaries that assign too much (or too little) total weight to the mini-batch. However, the choice of an optimal B adds an additional degree of freedom to RP-DRO, which suggests our second option as a simpler alternative.
Batch-level normalization
(f(())))))
(a yi) = ΣE= 36))
for each sample (zx, y1) in the mini-batch. An obvious downside of this approach is that the weight of each sample now depends the mini-batch it was sampled from. This can be problematic for
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small batch sizes: as extreme example, for batch size of 1, this normalization scheme assigns the same weight of to every sample, making the objective equivalent to ERM.
However, mini-batch approximations have proven effective for other forms of DRO (Hu et al. 2018 Levy et al. 2020) and there is some evidence that they can yield accurate estimates for higher batch sizes (Cortes et al. 20T0). In practice we find that this approach yields good results for commonly used batch sizes, is generally more stable than the self-normalization penalty, and does not introduce the additional nyper-parameter B. In most of our experiments, we adopt this approach unless specified otherwise. In that case, the empirical RP-DRO objective on mini-batch becomes
72
Lratch level norm (0, 1b)
Σ To (2:4, (o)
To (24, Ui) To
expected loss under
KL. penalty
The KL term serves to penalize ratios that deviate too much from (Note that the penalty subtracted because are maximizing with respect to vb). The only hyper-parameter that needs to be tuned is the KL regularization strength T.
4 EXPERIMENTS
4.1 DATASETS
We perform experiments on four datasets: two text classification datasets used in Michel et al. (2021)
BiasedSST and FDCL18, and two image classification datasets (rommSagame et al. (2020). Waterbirds and CelebA. Specifie details for each dataset follow these previous works, as described below:
BiasedSST is based on the SST-2 sentiment classification dataset (Radford et al. 2018). but modified to introduce spurious correlation between distractor token ("So,") and positive labels in around 95% of the dataset. In this setting models trained with ERM can very easily learn this spurious correlation, which hurts performance on the small subpopulation that doesn not suffer from this bias.
FDCL18 A toxicity detection dataset of tweets labeled as hateful (5%), abusive (27%), normal (54%) and spam (14%). The group-DRO problem is formulated by partitioning the evaluation data along labels as dialectal annotation obtained automatically with an off-the shelf classifier (Blodgett) et al. 2016 Sap et al. 2019). In particular these dialects align closely with self-reported race, and Sap al. (2019) found that machine learning models trained on such toxicity detection datasets tend to exhibit bias towards certain labels depending the demographics of the tweets' authors, particularly with minorities. In order to report more reliable accuracy numbers, all groups containing less than 100 samples are aggregated when computing test accuracies.
Waterbirds An image classification dataset where the task is to predict "land bird" or "water bird" with the confounding factor of the background; most water (resp. land) bird pictures have water (resp. land) on the background.
CelebA A popular face recognition dataset originally published by Liu ct al. (2015). The group- DRO problem is formulated task of predicting the hair color ("Blond" or "Dark") across groups formed by the combination of the label and the (binary) gender of the subject. Due to the spurious correlation between blond hair/female and dark hair/male, models trained with ERM tend to achieve lower accuracies on underrepresented groups such as "blond-haired male" which totals only 0.85% of the training data.
4.2 EXPERIMENTAL SETTING
On BiasedSST and FDCL18 follow Michel et al. (2021) and train BiLSTM and BERT-base
model Devlin et al. (2018) respectively. On the image classification datasets we train Resnet-50
architectures (He et al. 2016) praiteeined on ImageNet (Deng et al. 2009) as in Sagawa et al. (202))
Since the adversary in RP-DRO can be any discriminative architecture, we opt for the natural solution of using similar architecture for this model. For instance BiasedSST, we take ( v) as the
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Table 1: Robust and average test accuracies on the Biased SST and FDCL18 datasets. Underlined
numbers indicates statistically significant difference compared to ERM (p-value<< 0.05). Bold numbers indicates the best number in each column (barring Oracle DRO).
Biased SST
FDCL18
Robust
Average
Robust
Average
ERM
2.15 i:
95.09 i
19.57
7.00
81.56 0.26
Topic DRO
5.18 +: 1.46
95.00 1 0.10
16.48 5.46
80.49
1: 0.49
NonParam X2
4.65 +: 1.61
95.10 + 0.22
15.70
2.56
81.55
+: 0.23
NonParam CVRR
24.42
6.82
89.03 = 3.73
11.19 7.64
69.19
+ 8.28
NonParam-Ke
28.11
2.16
92.45 + 1.55
17.54 = 6.41
81.20
+ 0.11
P-DRO
34.98
9.39
84.21
1 2.11
30.25
10.13
79.91
1: 1.41
RP-DRO
50.70
+ 7.33
86.60
+ 2.96
1.66 53.52
76.62 # 1.43
Oracle DRO
67.71
3.03
77.91 + 4.49
55.23
3.97
72.43 + 2.61
Table 2:
Robust and average test accuracies on the Waterbirds and CelebA datasets. Underlined numbers indicates statistically significant difference compared ERM (p-value< 0.05). Bold numbers indicates the best number in each column (barring Oracle DRO).
Waterbirds
CelebA
Robust
Average
Robust
Average
ERM
68.32 i: 2.02
89.23 1 0.36
40.33
1 2.29
95.89 1: 0.05
NonParam- X²
68.54
+: 0.65
89.56
0.61
41.78
+ 2.03
95.87
0.07
NonParam CVaa
44.71
+ 14.27
71.94
10.30
36.00
+ 7.50
94.63 0.58
-I.
72.21
+: 0.95
90.54
+ 0.72
43.33
+ 3.58
95.72
+: 0.10
RP-DRO
73.49
+: 3.01
90.15
0.74
55.78
9.15
93.10
3.87
Oracle DRO
85.60 +:
89.12
1.20
89.22
92.59 0.40
raw logit output by a BiLSTM architecture identical to that of the elassifier (without the final softmax layer). We do the same for the other datasets, with the exception of FDCL18 where we use smaller DistilIBETT model (Sanh et al. 2019) for efficiency. We use the same learning rate and optimizer for both model and adversary and only vary the KL penalty weight use batch size of for BiasedSST and FDCL 18 and 32 for Waterbirds and CelebA. We perform optimal stopping using the Minmax criterion proposed in Michel et al. (2021): every epoch T, we min-max game between
{0.001, 0.01, 0.1, 1.0). We
determine the best model by explicitly solving a greedy approximation all T previously checkpointed adversaries and models the validation dataset Dalld
0" arg min max
Σ
Psy (2, (x, (x,y)
(10)
6,-1...T 1j-1..7 Dvalia
Z.VE Dims
similar strategy applied for for la in Eq. (only for validation) as we found in preliminary experiments on BiasedSST that it consistently produced better results.
selection. Importantly, we substitute the 0-1 loss
We compare our results to 5 different methods experimented with by Michel et al. (2021):
ERM: standard training to minimize the average loss.
NonParam: nonparametric DRO with various uncertainty sets We report results for uncertainty set constrained by KL divergence (Hu & Hong 2013 Hu et al. 2018). x² divergence and CVaR (Levy et al. 2020). respectively referred to NonParam-Ke KL. NonParam-x" and NonParam-CVas all of these variants, the inner maximization problem has an analytical solution where q y) depends some function of Co y). Consequently, examples with high loss se-ieighted. Topic-DRO: a variation on Topic CVaR (Oren et al.) 2019) using the online algorithm from Sagawa et al. (2020) to minimize the worst case loss collection of pseudo domains automatically generated via topic modeling FDCLI8).
We use this baseline for the text datasets only (BiasedSST and
This baseline was inaccurately referred as "Topic CVaR" in Michel al. 2021

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P-DRO: the parametric DRO approach proposed by Michel al. (2021). For image datasets, preliminary experiments using auto-regressive models for image modeling (ann Oord et al. 2016)
proved to be prohibitively slow. Therefore, only report P-DRO text datasets in Michel
et al. 202T).
Oracle DRO: online algorithm for minimizing the worst-case loss on the true uncertainty set (Sagawa et al. 2020). Contrary to all other methods, Oracle DRO presupposes that we know the groups of interest at training time. As such, it is not directly comparable and serves provide an upper bound of the robust accuracy that can be achieved.
Across all experiments, we report results with mean and standard deviation across reruns with different seeds. Note that the model selection criterion can have large effect on final performance (Guraaaaii Lopez-Paz 2020). In contrast to some related work (e.g. Koh et al. (2020); [drissi et al. (202T)) except for Oracle DRO we only use validation criteria that do not require group annotation on the validation set. We provide more details on the various hhperr-paamme parameters used in Appendix
4.3 RESULTS
For all methods, we report the worst accuracy over all groups in the test set (the "robust accuracy"), Models that are robust to subpopulation shift will have higher robust accuracies. Since there is often a trade-off between being robust to distribution shift and performing well on the training distribution, we also report the standard accuracy the original test set (the "average accuracy")
As shown in Table RP-DRO works particularly well on the two text classification tasks, beating all baselines by large margin on both BiasedSST and FDCL18. In fact, its robust accuracy almost matches that of Oracle DRO on FDCL18, 18, despite the fact that the former does not use an any group information at training time. Compared to P-DRO, we find that results are not only better, but also more consistent as evidenced by the lower standard deviation.
Results are also generally good on image datasets (see Table (2) On Waterbirds both the NonParam baseline and RP-DRO perform much better than ERM. with slight edge for RP-DRO (although the latter exhibits a higher variance across reruns). On CelebA, RP-DRO largely outperforms the baselines.
ANALYSIS AND ABLATIONS
We perform a series of analysis experiments and ablation studies to better (1) identify how the parametric representation of the ratio provides improvements nonparametric alternatives and (2) understand the importance of the various choices made with regards to the renormalization scheme described in Section
Throughout this section, experiments are performed the BiasedSST dataset.
5.1 WHY ARE PARAMETRIC ADVERSARIES BETTER? THE CASE OF LABEL NOISE
Our experimental results in Section 4 shows that parametric approaches such as P-DRO and RP- DRO consistently outperform their nonparametric counterparts. A possible explanation for this phenomenon is that for nonparametric approaches, optimal weights generally only depends on the loss of the model. This can be problematic because the nonparametric worst-case distribution will indiscriminately up- up-weight noisy samples that have high loss. On the other hand, we hypothesize that it is more difficult for the parametric adversary to "fit the noise" and that it tends to focus more on systematic patterns of failures of the model.
To corroborate this hypothesis, we perform experiments by adding increasing amounts of noise to the BiasedSST. Specifically, for each example in the training set we replace the original label with random label with probability Pnoise O, 0.1. o....... 0.5. We then train models on these increasingly noisy datasets using both parametric (RP-DRO) and nonparametric (NonParam- ( [KL, CVaR.x²)) configuration for each (T 0.1
approach. To simplify experiments we only run one hyper-parament and 1 for RP-DRO and NonParam respectively) and report the test accuracies of the model with the highest robust accuracy on the validation set. Results are averaged over 5 runs with different random seeds.
As showcased in Figure I we find that nonparametric approaches are very sensitive to label noise, losing around 20 points when adding the smallest amount of noise (proise 0.1 (85 65), whereas
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90
100
R-PDRO
Adversary
NonParam-
90
Linear
80
NonFarom
nom--
Robust accuracy
80
MUP-2
Acciracy
NonParam-CVaR
MLP-4
70
70
60
50
50
40
30
0.0
0.1
0.2
0.3
0.4
0.5
10
20
50
100
Label noise probability
of adversary staps classitler stops
Figure 1: Effect of label noise performance (average test accuracy on BiasedSST) in para- curacy as the adversary takes more gradient metric and nonparametric DRO.
Figure 2: Evolution of RP-DRO's robust ac-
steps than the classifier in toy setting.
RP-DRO is comparatively more robust with only a loss of around points for the same amount of noise. The trend holds all the way to Pnoise 0.5 where all models collapse to chance accuracies. This further supports our hypothesis that nonparametric adversaries tend to fit these noisy examples. which decreases the overall quality of the resulting classifier. We show a finer-grained analysis of evolution accuracies broken down by groups in Appendix B. Appendix [..2] we present additional qualitative evidence that nonparametric approaches tends to select individually difficult examples rather than difficult subpopulations.
5.2
OPTIMIZATION WITH SIMULTANEOUS GRADIENT UPDATES PLAYS A CRUCIAL ROLE
Despite the aforemeentonned results, it remains unclear why RP-DRO learns re-weightings that are less sensitive label noise or difficult examples compared to NonParam- methods. Indeed, since the nonparametric adversary is the optimal solution of the inner minimization problem in Eq. [ it stands to reason that (large, sometimes over-paranee peer-parameterieed parametric adversaries from RP-DRO would converge solution close to NonParam. Our hypothesis is that the simultancous updates to both model and adversary parameters prevent the parametric adversary from converging towards such solutions, and provides some implicit regularization against up- up-weighting examples that noisy too difficult.
To verify this hypothesis, conduct toy experiment where allow the adversary to take additional gradient steps in-between each update to the classifier. At the limit, this would allow the adversary to find optimum of the inner maximization problem at each step of training the classifier (which for large enough adversaries, might come close to the nonparametric solution).
For computational efficiency, these experiments are performed on toy setting similar that of Michel et al. (202T): a linear model is trained on a binary classification problem with two domains, one of which is severely under-represented. For our adversary, we experiment with linear adversary, as well as larger multilayer perceptrons with one hidden layer and 2 (MLP-2) and 4 (MLP-4) hidden units. In Figure 2 we report the average robust accuracy (across reruns) for classifiers trained with RP-DRO when the adversary is allowed to take more steps than the classifier.
We observe that RP-DRO's robust accuracy suffers from giving the adversary too much time to catch up with the classifier: as the number of updates to the adversary increases, robust accuracy decreases. This effect is amplified in larger adversaries (e.g. MLP-4). We find that a key effect simultaneous updates is to dramatically improve stability (see Appendix DD). This experiment underlines the importance of the simultaneous gradient updates, which prevent large, over peee-paammterizee adversaries from converging to the sub-optimal nonparametric solution. We also find that simultaneous updates lead to dramatic
5.3 BATCH-LEVEL NORMALIZATION VS SELF-NORMALIZATION
we discussed two alternatives for enforcing the normalization constraint on the ratios (E,rs 1): a regularization-bun based approach ("self- normalization") and batch level renormalization. In Figure
In Section 3.3
show the effect of self- normaaoaaaiiaionn with different values of the regularization weight for a fixed value of the KL penalty т = 0.1. We find that batch-level normalization achieves a slightly better robust accuracy than the best nelf-normalization configuration.
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Robust
Average
100
80
Accuracy
60
Self-normalization
ERM
20
Batch-level normalization
10-3
10-?
10-1
10 10a
10-3
10
10-1
10c
10¹
B
B
Figure 3: Effect of self-normalization coefficient 3 on robust and average accuracy. We report results of ERM (which corresponds to B 00) and batch level renormalization for comparison.
In Michel et al. 2021), ). the Minmax stopping criterion was identified as a major contributing factor to the performance of parametrie DRO approaches. To understand how it affects each normalization strategy, we perform the same experiment as above but this time without the Minmax criterion, selecting instead the model with the highest robust accuracy on the validation set ("Oracle stopping"), which provides an upper bound to the robust accuracy that can be achieved. We find that although accuracies are generally closer, batch-level normalization matches the hest self-normalization penalty (the full figure can be found in Appendix C). This indicates that batch-level normalization not only performs as well as nofmaaarmliztioon (without the need for tuning the additional but also that it interacts better with the Minmax stopping criterion, making it preferable alternative.
3),
5.4 EFFECT OF BATCH SIZE ON RE-NORMALIZATION
As pointed out in Section 3.3 a potential downside of the mini-batch-leve normalization approach is that the effec- tive weight of each sample then depends on the weights of the other samples within the mini-batch. For example, consider an adversary that assigns high weight to only 5% of the training data. With small enough batch size, it is likely that some batches may not contain any example of the high weight subpopulation, in which case mini-batch level renormalization will overestimate the weight of the sample in the mini-batch.
Table 3: Effect of batch size on RP-DRO performance.
Batch size
Robust
Average
64 (ERM)
32.97 1: 2.34
92.42 :: 0.38
4
37.50 1: 2.37
92.32 1: 0.75
38.08 +
91.61 + 0.48
41.67 + 3.53
91.24 0.34
32
42.32 + 0.72
89.77 : 1.42
64
44.15
+ 6.83
88.30 + 2.33
To assess the severity of this issue, we run RP-DRO on BiasedSST with т 0.1 and vary the batch size in {4, 8, 16, 32, 64, 128]. Each configuration is run times, and we report average and standard devia- tion of the robust and average test accuracies in Table Results suggest that while robust accuracy indeed deteriorates for lower batch sizes (4 and 8), results are consistently good for batch sizes upwards of 16, a reasonable number considering that larger batch sizes are often preferred in the literature (Popel & Bojar 2018 Goyal et al. 2017)
128
42.25
+ 7.90
88.21 2.02
6 CONCLUSION
In this paper we have proposed a parametric, likelihood ratio based approach to distributionally robust optimization of machine learning models. With the proposed method, we can use any type of parametric function estimator to define the uncertainty set of the DRO min-max game. We showed that with a careful renormalization strategy, the proposed method (RP-DRO) can be used to train robust models. It depends on very few hyper-parameters and consistently performs well on number of benchmarks, making it an appealing "off-the-shelf" option. Finally we have shown that such parametric approaches are more resilient to the presence of noise in the training data when compared to their nonparametric alternatives, and that simultaneous gradient descent is a key component of RP-DRO's 's success.
The main downside of RP-DRO is the computational overhead of jointly training second neu- ral model. An interesting direction for future work is to improve its efficiency through parallel computation or by sharing parameters between the classifier and the adversary.

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Published as a conference paper at ICLR 2022
ACKNOWLEDGMENTS
The authors would like to thank the anonymous reviewers whose feedback helped improve the paper to its current form. Moreover, this project benefited from fruitful discussions with Chunting Zhou, Daniel Levy, Zachary Lipton and Zico Kolter. The first author was supported by the ENS-CFM Data Science Chair.
ETHICS STATEMENT
The work presented in this paper places itself in the continuity of series of papers dedicated training models that are more robust to certain types of distribution shifts (Sagawa et al. (2020); Levy et al. (2020); Zhou et al. (202T) cite only few). This line of work allows for training models that perform more equitably across subpopulations of the training set, with positive implications in terms of fairness and equal representations of marginalized groups.
A potential negative impact of RP-DRO is in the increased computational overhead that results from training the adversary, which leads to a larger carbon footprint, and ultimately negatively affects the environment. This is shortcoming which we hope to address in future work.
REPRODUCIBILITY STATEMENT
To facilitate reproducibility of our work. our experiments are conducted on datasets that are openly available: BiasedSST FDCL18 (Founta et al., 2018). Waterbirds (Sagawa al.) (020) and CelebA (Liu et al. 2015). We describe experimental settings in Section 4.2 and additional hyper-
parameters are reported in Appendix A Code to reproduce our experiments available at https 7/gi Sigithubb... com/ pmi che Shee314115.. 5/P-DRO
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A ADDITIONAL EXPERIMENTAL DETAILS
We describe additional hyper-parameter choices specific to each dataset or method facilitate reproduction of our results.
A.1 DATASET-SPECINC HYPER-PARAMETERS
All hyper-parameters listed below are constant across all methods:
Text Datasets The imput data is tokenized using the bert - base-aaaeaas- unsseed sub-word tokenizer from Devlin al. (2018). We train both classifier and adversary with Adam (Kingma & Ba 2014) using a learning rate of x 10 linearly decay the learning rate to O at each step. We train with batches of size 64 (or containing up 2500 tokens, whichever is lower) for 50 and 20 epochs for
BiasedSST and FDCL18 88 respectively, evaluating model on the validation data every epoch.
Image Datasets On both datasets, images are rescaled to 224 x 224 pixels and pixel values are normalized to have mean and variance across all color channels on the training data. At training time, augment the data by randomly cropping flipping the images horizontally. We train using regular stochastic gradient descent using a constant learning rate of 10 and a batch size of 32. We train for 75 and 13 epochs match the number of steps trained to Sagawa et al. (2020) despite the smaller batch size), and validate every 100 (for Waterbirds) and 1000 (for CelebA) training steps.
Waterbirds and CelebA respectively (those numbers were chosen to
A.2 METIOO SPECIFIC HYPER-PARAMETERS
For NonParam we follow the adaptation of Hu et al. (2018) used in Michel et al. (2021) and choose the optimal temperature r* based on mini-batch level estimates of the KI. divergence. We treat the KI. bound к as a hyper-parameter We adapt the Minmax stopping criterion of P-DRO to the nonparametric adversaries as we found it yiclded more robust models than those selected with average validation accuracy. We sweep over E {0.01, 0.1, 1.0, 10.0}
For RP-DRO we perform min-mas max stopping using the Minmax criterion with a KL threshold of log 10 in all experiments, to match the value recommended for P-DRO. Specifically, estimate the KL divergence of checkpointed adversaries 4/ on the validation data as follows:
(x v) log (x,
(11)
D valid
K.yE Dutia
and reject adversaries for which this quantity exceeds log 10.
B ADDITIONAL COMPARISONS BETWEEN PARAMETRIC AND NONPARAMETRIC DRO
B.I EVOLUTION OF GROUP ACCURACIES UNDER INCREASING LABEL NOISE
In Figure 4 we report the evolution of individual group accuracies under increasing amounts of label noise (as described in Section 5.T). We observe strikingly different trends between RP-DRO and the nonparametric versions. Indeed, for all nonparametrie methods the accuracy within each group converges to the chance level (50%) rapidly. We interpret this to mean that nonparametric adversaries assign disproportionately more weight the noisy examples. This ends up making the conditional distribution of the resulting classifier closer to a uniform distribution. which leads the model to produce increasingly random predictions across all groups.
In RP-DRO however, we find that the model is less affected by the uniform label noise: the accuracy decreases for all groups, but the same (much slower) rate than NonParam
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R-PDRO
NonParam-KL
NonParam-CVaR
NonParam-2
Group accuracy
80
60
40
20
Label noise probability
Figure Effect of label noise on model performance on each individual group in parametric (R-PDRO) and non parametric DRO (NonParam- (a..
B.2 QUALITATIVE COMPARISON OF PARAMETRIC AND NONPARAMETRIC RATIOS
We take a qualitative look at the top-10 up-weighted samples by the RP-DRO adversary and the top-10 10 up- weightet examples by NonParam adversary (which is equivalent to the top- examples with the highest loss).
We report these examples for model trained with RP-DRO on the small BiasedSST dataset. Results are shown for checkpoint early training (after epoch), on the validation data. We observe that the RP-DRO adversary assigns high weight to examples of one of the under--epresented groups (examples containing the distractor token "so, but labeled as positive, representing only 2.5% the training data).
On the other hand, the examples with the highest loss (which would be up-weighted the most under NonParam adversary) do not exhibit this pattern. To us, these seem to represent more difficult examples. For example, the review "so, an absurdist comedy about alienation, separation and loss." " does exhibit clear negative sentiment. Overall find that among the top 10% examples in the validation data with the highest loss, only 9.19% belong to the high error minority group (distractor token + positive label), versus 26% (or almost 3x more) for the top 10% most up- up-weighted examples by the RP-DRO adversary.
While this only qualitative evidence, it meshes with our intuition that even in the absence of label noise, nonparametrie adversaries would tend to focus on difficult examples, rather than consistent patterns of failures exhibited by the model.
C EFFECT OF ORACLE STOPPING ON RENORMALIZATION STRATEGIES
In Figure we report the evolution of robust and average accuracy for model trained with RP-DRO using a self normalization penalty with a coefficient 3 varying from 10-3 3 10
Robust
Average
100
80
Accuracy
60
Self-normalization
40
ERM
20
Batch-level normalization
10-3
10-2
10
100
10
10
10
10
10a
10a
B
Effect of normaaiiaaiinno coefficient 3 on robust and average accuracy using Oracle stopping. We report results of ERM (which corresponds oc) and batch level renormalization for comparison.
Figure 5:
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Label
Text
RP-DRO
positive
so, not only is undercover brother as funny, if not more so, than both austin powers films, but it's also one of the smarter, savvier spoofs to come along in some time.
positive positive
so, too, is this comedy about mild culture clashing in today's 's new delhi. so, we root for clara and paul ), even like them, though perhaps it's emotion closer to pity.
positive
so, thanks to scott's charismatic roger and eisenberg's 's sweet nephew, roger dodger is one of the most compelling variations on in the company of men. " so, the sort of film that makes me miss hitchcock, but also feel optimistic that there's hope for popular cinema yet.
positive
positive
so, visually imaginative, thematically instructive and thoroughly delightful, it takes us on roller- coaster ride from innocence to experience without even a hint of that typical kiddie flick sentimentality.
positíve
so, looking aristocratic, luminous yet careworn in jane hamilton's exemplary costumes, rampling gives a performance that could not improved upon. so, the far future may be awesome to consider, but from period detail to matters of the heart, this film is most transporting when stays put in the past. "
positive
positive
so, whether you like rap music or loathe it, you can't 't deny either the tragic loss of two young men in the prime of their talent or the power of this movie.
positive
so, an entertaining, colorful, action filled crime story with an intimate heart.
NonParam
negative negative
that's a cheat. its well of thorn and vinegar and simple humanity) has long been plundered by similar works featuring the insight and punch this picture so conspicuously lacks."
negative
so, an absurdist comedy about alienation, separation and loss.
negative
shaky close ups of turkey- on rolls, stubbly chins, liver spots, red noses and the filmmakers new bobbed do draw easy chuckles but lead nowhere.
negative
paid full is so stale, in fact, that its most vibrant scene one that uses clips from brian de palma's scarface."
negative
so, and the lesson, in the end, nothing new.
negative
may reawaken discussion of the kennedy assassination but this fictional film looks made for cable rather than for the big screen.
positive
so, atom egoyan has conjured up multilayered work that tackles any number of fascinating issues" "
negative positive
so, dull, lifeless, and amateurishly assembled.
( oesn't 't bother being as cloying or preachy as equivalent evangelical christian movies maybe the filmmakers know that the likely audience will already be among the faithful."
Table 4: Top-10 most up- up-weighted examples in the BiasedSST validation set with parametric (R-DRO) and non-parametric adversary.
D STABILITY OF SIMULTANEOUS GRADIENT DESCENT VS. EXACT MINMAX
In the toy setting of Section when both the model and the adversary are linear, the resulting min- max problem becomes convex-concave This means that any stationary point 0*, o- of simultancous gradient descent will be a global aadle--pint of the CRP-DRO(G (0, e)) objective. Why then is simultaneous gradient descent systematically achieving higher robust accuracy than the more "exact case" where we take gradient steps on the max. LRP- (8, +)
We find that the benefit of simultaneous updates lies in increasing the stability of training. In Figure 6 we report training curves in terms of accuracies on each the two domains (for 5 random seeds). We vary the number of steps that the adversary is allowed take in between each classifier update
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100
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1000
1500
2000
Classitter stops
Classiffer stops
Classitier steps
(a) aadversary step (simultane- ous updates).
(b) adversary steps.
(c) 100 adversary steps.
Figure 6: Evolution of training trajectories for in toy setting the linear adversary is able take more steps than the classifier. We report accuracies on the two domains for multiple restarts. The bold curves correspond to the average trajectory across all seeds.
Accuracy
75
Accuracy
75
Accuracy
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stops
stops
steps
(a) adversary step (simultane- updates).
(b) 10 adversary steps.
(c) 100 adversary steps.
Evolution of training trajectories for in our toy setting as the MI.P2 adversary is able to take more steps than the classifier. We report accuracies on the domains for multiple restarts. The bold curves correspond to the average trajectory across all seeds.
Figure 7:
from (simultaneous updates) to 100 (which closer to "exact" case of taking descent steps on max. CRP-DRO (0, v)).
We observe that models trained with simultancous gradient updates consistently converge to the global optimum where both domains achieve the same accuracy. On the other hand the more "exact" variants (where the adversary is allowed to take more steps) is much less stable. In the most extreme case (100 adversary steps for each classifier step) the model fails to converge.
When models are bigger, and we lose the convex convex-concavits concavity of the problem, we find that taking steps max; (n v) is also unstable, and sometimes converges to worse local optima compared to simultaneous updates (see Figure
17


Reviews:
[{"review_id": "qToVC3JQoGX", "reviewer": "Reviewer_hEDd", "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": "This paper tackles an important problem of data set shift. Most methods in industrial application will need to have some robustness to data set shift since there is always the potential for temporal drifts in the data distribution from when the training data was collected to when a system is deployed.\n\nThe setup to reparameterize from a constraint on the shift distribution q to a reweighting function r in (5) is clever and useful.\n\nComments:\n- It would be interesting to see this method benchmarked against simpler baselines like randomly reweighting the training points in a non-adversarial way. If that weighting was done in a way that was tuned across many problems it might be quite competitive and simpler.\n\n- In order to find more \"natural\" examples of data set shift, it would be good to find benchmark data sets that are in principle iid but have a timestamp associated with them. If the train test split were done temporally instead of randomly it would give some natural example of more subtle data set shift.\n\n- I think the paper should make a clearer distinction between the case that p(x) shifts and p(y|x) remains the same, and the much more difficult case where p(y|x) changes as well. Which methods work best will likely be a function of which scenario one is in.\n\nQuestions:\n- The paper considers bounding the data set shift in terms of KL. However, would it make more sense to consider limiting the data set shift in terms of JSD? That might be more interpretable because it could phrased in terms of the number of data points required to distinguish the new distribution from the original training distribution.\n\n- The paper considers a batch level normalization constraint to make sure the adversarial distribution is actually normalized. It would be interesting to see a study into how effective that actually is at enforcing normalization. Are those distributions actually normalized if we use heavier duty methods (e.g., HMC) to assess it?\n\n- The study of label noise is interesting in that proposes a theory on why the nonparametric robust optimization methods don't work as well. It would be interesting to further follow up on this ablation using a semi-synthetic data set that is homoscedastic by construction. Because in that case, the nonparametric method should not be able to hyper focus on the high noise regions of the space.", "summary_of_the_paper": "This paper presents a method to robustify any risk minimization learning to data set shift in the test set. It sets up an adversarial min-max framework where the learner needs to minimize a loss function but under a max for perturbations of reweighting the training distribution.", "summary_of_the_review": "Overall good paper with potential impact. Some things should have been evaluated better.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "fJ9r1lJm7SQ", "reviewer": "Reviewer_5Xhf", "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": "PROS\n\nThe paper deals with an interesting problem of timely relevance.\n\nThe proposed practical method for DRO looks promising and may deserve the attention of the community.\n\nThe paper is relatively well written (some editorial feedback is given below).\n\n\nCONS\n\nSome unclarity in the presentation of ERM and DRO methods (distinction between \"ideal\" and \"empirical\" objectives).\n\nMissed some pointers to work on \"DRO with guarantees\" e.g. what's been done on the theory literature for DRO. \n\nThe previous is suggesting that some pointers could be helpful just to inform the reader (maybe in the section on related literature).\n\nA number of editorial changes (minor mostly) need to be made for the paper to be in publishable form.\n\n\nEDITORIAL FEEDBACK\n\nSection 1 (Introduction):\n\nReplace \"neural-network\" with \"neural network\" (no hyphen)\n\nFirst line of second paragraph: replace \"explained by\" with \"attributed to\"\n\nRegarding this second paragraph: The ERM principle consists of using the empirical risk (average loss over the finite sample) as optimization objective. Ideally the optimization objective should be the risk, i.e. the expected loss on a random example, but the latter is inaccessible because the distribution is unknown. This passage as currently written might a bit misleading.\n\nSuggesting a rewrite:\n\n\"In ERM, models are trained to minimize the average loss over a finite sample from a fixed training distribution (Vapnik, 1992). This is because the empirical risk is taken as a proxy for the risk $\\mathbb{E}_{(x,y) \\sim p}[\\ell_\\theta(x,y)]$, namely the expected loss on a random example chosen from the training distribution $p$, which is assumed to be a fixed distribution, albeit unknown.\"\n\nNext sentence: \"This favors models which perform well \\emph{on average} on samples from a fixed distribution, as opposed to models which would perform equally well on a variety of distributions that may better reflect the diverse set of subpopulations that can be encountered at test time, including those that may not have been represented adequately by the training distribution.\"\n\nNext sentence: \"On the other hand, distributionally robust optimization (DRO) proposes an appealing alternative to risk minimization.\"\n\nMight be worth mentioning that the \"uncertainty set\" ($\\mathcal{Q}$) is also called \"ambiguity set\" in some literature.\n\nThe objective $\\mathcal{L}_\\mathrm{DRO}(\\theta)$ defined in Eq. (1) corresponds to the \"ideal objective\" I think, while in practice the implemented objective will be an empirical version of this objective? Say $\\hat{\\mathcal{L}}_\\mathrm{DRO}(\\theta)$, where the hat indicates the empirical version where the expectation is replaced with average over the finite sample? \n\nAlternatively, clarify how the expectations $\\mathbb{E}_{(x,y) \\sim q}[\\ell_\\theta(x,y)]$ are to be implemented.\n\nNext page: Maybe use the name RP-DRO for your approach? \nTo indicate the ratio-based (R) parametric (P) distributionally robust optimization (DRO).\n\n\"In this paper, we propose a new approach for DRO, called RP-DRO, based on a key modification\nof the P-DRO algorithm: instead of modeling the worst-case distributions directly, we reparametrize \nthe likelihood ratio between the training distribution and the worst-case distribution. This [...]\"\n \nReplace \"likelihood-ratio\" with \"likelihood ratio\" (no hyphen)\n\nReplace \"neural-network\" with \"neural network\" (no hyphen)\n\nThis looks like a broken sentence: \"a penalty-form to the KL divergence uncertainty set\" -- rectify as per the intended meaning.\n\nChoose one of \"mini-batch\" or \"minibatch\" to use consistently throughout the paper.\n\nSame comment for \"sub-population(s)\" versus \"subpopulation(s)\"\n\nSection 2:\n\nSuggesting to rewrite what comes after Eq. (2):\n\n\"Notice that the DRO loss in Eq. (1) is the inner maximum in Eq. (2), it provides an upper bound on the expected loss of the model under any distribution in the uncertainty set Q, which motivates the use the minimizer of the min-max game in Eq. (2) as a robust model. We refer to the solution of the inner maximum as the “adversary” from now on.\"\n\nNext sentence:\n\n\"However this objective is only useful insofar that (1) $\\mathcal{Q}$ covers test distributions of interest (corresponding to different domains, demographics, etc.) and (2) $\\mathcal{Q}$ is not overly pessimistic. To fulfil [...]\"\n\nSection 2.1:\n\nLast line of first paragraph: replace $\\chi_2$ with $\\chi^2$ ?\n\nSection 2.2:\n\nWrite \"Equation (3)\" or \"Eq. (3)\" for cross-references to equation numbers.\n\nChoose one of \"non-parametric\" or \"nonparametric\" and use consistently throughout the paper.\n\nSection 3:\n\nInsert some text describing the section: \"In this section we [...]\"\n\nSection 3.1:\n\n\"In the situation that all distributions in $\\mathcal{Q}$ are absolutely continuous with respect to the training distribution $p$\"\n\nThis \"definition\" of absolute continuity is restricted to discrete distributions. In case you consider more general distributions, the corresponding general definition of absolute continuity should be given instead.\n\nSection 3.2:\n\nReplace \"likelihood-ratio\" with \"likelihood ratio\" (no hyphen)\n\nReplace \"limit the choice\" with \"restrict the choice\"\n\nTop of next page: make the expression for the KL into a displayed math expression (no number, just display)\n\nTwo lines after Eq. (6): the method name in the subscript needs to be corrected (twice)\n\nNext paragraph, the right hand side of the equation for $r-\\psi$ is missing the normalizing constant.\n\n\"and instead we relax it\" (insert \"we\")\n\n\"in the form of a term $\\tau\\mathbb{E}_p[..]$\" (delete \"KL\" here)\n\nSection 3.3:\n\nIn the second paragraph, I did not understand why this set is called a \"simplex\" -- can this be justified?\n\nOr else replace \"simplex\" with \"set\"\n\nLine after Eq. (7): \"this penalizes adversaries that [..]\"\n\nSection 4:\n\nInsert some text describing the section: \"In this section [...]\"\n\nSection 4.1:\n\nSecond line of first paragraph: insert a comma after FDCL18\n\nNext paragraph: replace \"doesn't\" with \"does not\"\n\nNext paragraph: \"off-the-shelf\"\n\nSection 4.2:\n\n\"BERT-based model (Devlin et al., 2018)\"\n\nIn the captions of Table 1 and Table 2: \"($p$-value < 0.05)\"\n\nParagraph after Eq. (10): Using the zero-one loss makes complete sense for classification problems. Please write this part to make it very clear for what purpose the 0-1 loss is used (e.g. only for hyperparameter selection, or also for optimization?)\n\nReplace $\\chi_2$ with $\\chi^2$ (twice)\n\nTop of page 7: \"We will release code for our experiments in a future revision\" -- does this mean that the code will be released upon de-anonymization? (i.e. will the code be made available with the camera ready paper if accepted?)\n\nSection 4.3:\n\nShould \"higher robust accuracies\" be replaced with \"more robust accuracies\"? -- rectify as per intended meaning.\n\nNext paragraph: replace \"doesn't\" with \"does not\"\n\nSection 5:\n\nChoose one of \"Biased SST\" or \"BiasedSST\" and use consistently throughout the paper.\n\nSection 5.1:\n\nLine 3 of first paragraph: \"optimal weights generally only depend on\"\n\nNext paragraph: $\\chi^2$\n\nSection 5.2:\n\nLine 2 of first paragraph: Replace \"R-PDRO\" with the method to which the comparison is made.\n\n\"Eq. (5)\"\n\nSection 5.3:\n\n\"matches still matches\" has one too many matches:)\n\nSection 5.4:\n\n\"the effective weight of each sample then depends on the weights of the other samples within the minibatch\" ?\n\nChoose one of \"hyper-parameter(s)\" or \"hyperparameter(s)\" and use consistently throughout the paper.\n\n\n\n\n", "summary_of_the_paper": "As far as I can see, this work is about practical methods for distributionally robust optimization (DRO) with a special focus on trying to overcome the limitations of previous methods that have been proposed for this kind of problem. This work builds on the idea of instance-reweighting of the loss function via a re-weighting function that plays the role of the likelihood ratios (between the distributions in the class used for enforcing robustness and the training distribution). The re-weighting functions considered here are the well-known \"exponential weights\" and this work proposes and explores (i) using mini-batch level renormalizations and (ii) ensuring the KL-constraint by adding a KL penalty term to the training objective. The framework and corresponding objectives are described, and they are tested in experiments on benchmark image and text classification data sets, and compared to other existing DRO methods and to plain ERM.", "summary_of_the_review": "Overall I enjoyed reading this paper. I think the main strength of this paper is in the experiments sections, demonstrating the advantages of their proposed DRO method with respect to previously proposed methods. I am being conservative with my evaluation mainly because I am not familiar with the whole literature in this area (notice my confidence of 3) and I look forward to see other reviewers' reactions to this work. I am willing to reconsider my evaluation if my feedback is addressed adequately and provided that no other flaws are found.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "f-LX2RszFMZ", "reviewer": "Reviewer_iYtW", "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": "Strengths:\n- The idea is simple and yields better results than other methods. \n- The empirical evaluation is thorough. R-PDRO is studied from quite a few perspectives. In particular, the ablation study does a good job at demonstrating the effect and importance of each part of the algorithm. \n- The paper is well written and quite pleasant to read. Ideas are presented clearly.\n\nWeaknesses:\n- The likelihood ratio is known to be easy to estimate numerically when the denominator distribution $p$ covers the numerator distribution $q$ well enough but is otherwise difficult to estimate accurately. What happens when the training distribution $p$ is distributionally far from the worst-case distribution $q$?\n- In Section 5.2, it is observed that simultaneous updates to both model and adversary parameters yield better results than when the adversary is allowed to take more gradient steps. I believe this is a major observation that the authors should investigate further since it shows that if Eq. 6 is solved exactly, as initially proposed in this work, then this leads to suboptimal results. As a matter of act, shouldn't this call for a new formulation of the objective? [A study of the training dynamics, similar to 1705.10461, is certainly worth carrying out, albeit well outside the scope of this paper.]\n\nQuestions:\n- Batch-level normalization is said to \"not introduce any additional hyper-parameters\", but Eq. 9 includes the hyper-parameter $\\tau$. Can you clarify the first statement?\n- The evaluation is carried out on four benchmark datasets: Biased SST, FDCL18, Waterbirds, and CelabA. Sagawa et al (2020) evaluate on MultiNLI while Michel et al (2021) evaluate on DWMW17. Is there a reason why these two benchmarks were not considered in the experimental evaluation of R-PDRO?", "summary_of_the_paper": "This work introduces R-PDRO, a distributionally robust optimization framework based on parametric likelihood ratios. Experimental results show that training models with R-PDRO are consistently more robust to distribution shifts when compared to other DRO approaches. \n", "summary_of_the_review": "This paper proposes a simple and effective algorithm for training models with DRO. Experimental results are convincing. My main issue concerns the training objective, which leads to weaker results when solved more accurately. The preliminary observations discussed in Section 5.2 are a first step in this direction, but the contribution could be much stronger if the authors had investigated this more deeply. Nevertheless, I believe the overall contribution to be worthy of acceptance.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "JqkLZeFwdfw", "reviewer": "Reviewer_UdKV", "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": "The paper does a good job of outlining the problem of interest and steps through the rationale for the design choices. Parts of the method seem a bit heuristic but the paper acknowledges this and the choices are largely justified by previous literature and empirical validation. The experiments support the claim that D-PDRO improves the performance for under-represented subpopulations but does so at the expense of the better-represented populations. This seems like a reasonable trade-off and is consistent with robust/standard accuracies in the adversarial setting. The problem of reducing bias against under-represented populations is a relevant issue and this method takes a positive step in that direction.\nOverall, the paper is well-organized and the development makes sense but the method has many parts that build on one another which makes things seem somewhat convoluted. Intermediate figures/cartoons or an algorithm block may help to clarify/streamline the method. Additionally, the front-matter focuses heavily on the general distribution shift problem but the experiments focus entirely on subpopulation under-representation within a larger dataset. The issues are clearly related and I do not have any misgivings with this focus in the experiment section, however, it would feel less disjoint if the paper alluded to this focus earlier.\n\nMinor issues:\nSome of your figures have gridlines and some do not. I would recommend adding the to all the figures.", "summary_of_the_paper": "The paper proposes a modficiation to the DRO/PDRO method based on a parameterization of the various distributions that allows the problem to be recast in an optimization directly over likelihood ratios which belong to the exponential family. The paper additionally proposes two methods for controlling the normalizations and then tests the method against several datasets with under-represented subpopulations.", "summary_of_the_review": "The paper addresses an important, modern issue (subpopulation bias in large datasets) and does a good job explaining, developing, and exercising the idea on appropriate datasets with relevant baselines. I recommend accepting the paper.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper builds upon parametric distributionally robust optimization (PDRO) and proposes ratio PDRO (R-PDRO) where the ratio of the worst case distribution and training distribution is parameterized by a discriminative network. This has a benefit over PDRO which needs to do generative modeling of worst case distribution. The paper empirically demonstrates R-PDRO improves over existing methods on group robustness problems. Reviewer are overall positive about the paper, and have appreciated the significance of the problem, writing clarity, and thorough empirical evaluation. There were some minor questions which have been adequately addressed by the authors.
Decision: Accept (Poster)