Probable Domain Generalization via Quantile Risk Minimization
Authors: Cian Eastwood
Alexander Robey
Shashank Singh
Julius Von Kügelgen
Hamed Hassani
George J. Pappas
Bernhard Schölkopf
Venue: NeurIPS 2022 Accept (2022)
Keywords: domain generalization
out-of-distribution generalization
invariant prediction
causality
robust optimization
risk-aware optimization

Abstract: Domain generalization (DG) seeks predictors which perform well on unseen test distributions by leveraging data drawn from multiple related training distributions or domains. To achieve this, DG is commonly formulated as an average- or worst-case problem over the set of possible domains. However, predictors that perform well on average lack robustness while predictors that perform well in the worst case tend to be overly-conservative. To address this, we propose a new probabilistic framework for DG where the goal is to learn predictors that perform well with high probability. Our key idea is that distribution shifts seen during training should inform us of probable shifts at test time, which we realize by explicitly relating training and test domains as draws from the same underlying meta-distribution. To achieve probable DG, we propose a new optimization problem called Quantile Risk Minimization (QRM). By minimizing the $\alpha$-quantile of predictor's risk distribution over domains, QRM seeks predictors that perform well with probability $\alpha$. To solve QRM in practice, we propose the Empirical QRM (EQRM) algorithm and provide: (i) a generalization bound for EQRM; and (ii) the conditions under which EQRM recovers the causal predictor as $\alpha \to 1$. In our experiments, we introduce a more holistic quantile-focused evaluation protocol for DG, and demonstrate that EQRM outperforms state-of-the-art baselines on datasets from WILDS and DomainBed.

Probable Domain Generalization via Quantile Risk Minimization
Cian Eastwood 112
Alexander Robey*3
Shashank Singh
Julius von Kügelgen 1A
Hamed Hassani
George J. Pappas Bernhard Schölkopf'
Max Planck Institute for Intelligent Systems, Tübingen
2 University of Edinburgh
University of Pennsylvania
Universityy of Cambridge
Abstract
Domain generalization (DG) seeks predictors which perform well on unseen test distributions by leveraging data drawn from multiple related training distributions domains. To achieve this, DG is commonly formulated as average- worst- case problem over the set of possible domains. However, predictors that perform well average lack robustness while predictors that perform well in the worst case tend to be cerel-coonervvtive. To address this, we propose new probabilistic framework for DG where the goal is to learn predictors that perform well with high probability. Our key idea is that distribution shifts seen during training should inform us of probable shifts at test time, which we realize by explicitly relating training and test domains as draws from the same underlying meta-distribution. To achieve probable DG, we propose a new optimization problem called Quantile Risk Minimization (QRM). By minimizing the a-quantile of predictor's risk distribution over domains, QRM seeks predictors that perform well with probability a. To solve QRM in practice, we propose the Empirical QRM (EQRM) algorithm and provide: (i) generalization bound for EQRM; and (ii) the conditions under which EQRM introduce
recovers the causal predictor + 1. In experiments, holistic crantile-focusse evaluation protocol for DG, and demonstrate that EQRM outperforms oate--of-the-art baselines on datasets from WILDS and DomainBed.
Introduction
Despite remarkable successes in recent years [1-3], machine learning systems often fail calamitously when presented with out-of-distribution (OOD) data [4-7]. Evidence of state-of-the-art systems failing in the face of distribution shift is mounting rapidly-be it due to spurious correlations [8-10], changing sub- sub-populations [1 [1],13], changes in location or time [14-16], or other naturally- variations [17-23]. These OOD failures are particularly concerning in safety-critics applications such as medical imaging [24-28] and autonomous driving [29-31], where they represent one of the most significant barriers to the real-world deployment of machine learning systems [32-35].
Domain generalization (DG) seeks improve system's OOD performance leveraging datasets from multiple environments or domains at training time, each collected under different experimental conditions [36-38] (see Fig. la). The goal is build predictor which exploits invariances across the training domains in the hope that these invariances also hold related but distinct test domains [38- 41J. To realize this goal, DG is commonly formulated as average- [36, 42, 43] or worst-case [9, 44, 45] optimization problem over the set of possible domains. However, optimizing for average perfor- mance provably lacks robustness to OOD data [46], while optimizing for worst-domain performance tends to lead overly-conservatin solutions, with worst-case outcomes unlikely in practice [47, 48).
Equul contribution. Correspondence to eastrood@edd ac. uk or arobey1 1aaaa..e...... upenn. edu.
Code available at:
36th Conference on Neural Information Processing Systems (NeurIPS 2022).

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Training demains
Text domains
Protabiiyy
0.5
Location
mequentlc
(ERMO
Risk
(FQRMM) (DR0)
(c)
Figure I: Overview of Probable Domain Generalization and Quantile Risk Minimization. (a) In domain generalization, training and test data are drawn from multiple related distributions domains. For example, the 1WildCam dataset [50], which contains camera-trap images of animal species, the domains correspond to the different camera-traps which captured the images. (h) We relate training and test domains draws from the same underlying (and often unknown) meta- distriiitiitttoon over domains Q. (c) We consider predictor's 's estimated risk distribution over training domains, iaturally-induced by Q. By minimizing the a q-quantile of this distribution, we learn predictors that perform well with high probability (+S a) rather than on average or in the worst case.
In this work, we argue that DG is neither an average-case nor a worst-case problem, but rather a prob- abilistic one. To this end, we propose probabilistic framework for DG, which we call Probable Do- main Generalization ($ 3), wherein the key idea is that distribution shifts seen during training should inform us of probable shifts at test time. To realize this, we explicitly relate training and test domains as draws from the same underlying deta-distribution (Fig. 1b), and then propose problem called Quantile Risk Minimization (QRM). By minimizing the a-qunntile of predictor's 's risk distribution over domains (Fig. 1c). QRM seeks predictors that perform well with high probability rather than on average or the worst case. In particular, QRM leverages the key insight that this a-quantile is an upper bound on the test-domain risk which holds with probability a, meaning that is corresponding to the worst-case setting.
optimization
an interpretable conservativeness- hawprivveeee--yyypppraeeeeee with
To solve QRM in practice, we introduce the Empirical QRM (FQRM) algorithm ($ 4). Given a predictor's 's empirical risks on the training domains, EQRM forms estimated risk distribution using kernel density estimation (KDE, [49]). Importantly, KDE-smoothing ensures a right tail that extends beyond the largest training risk (see Fig. Ic), with this risk "extrapolation" [41] unlocking invariant prediction for EQRM ($ 4.1). We then provide theory for EQRM ($ 4.2, $ 4.3) and demonstrate empirically that EQRM outperforms state-of-the- baselines on real and synthetic data ($ 6).
Contributions. To summarize our main contributions:
A new probabilistic perspective and objective for DG: We argue that predictors should be trained and tested based on their ability to perform well with high probability. We then propose Quantile Risk Minimization for achieving this probable form of domain generalization ($ 3).
A new algorithm: We propose the EQRM algorithm to solve QRM in practice and ultimately learn predictors that generalize with probability & ($ 4). We then provide several analyses of EQRM:
Learning theory: We prove uniform convergence bound, meaning the empirical a-quantlle risk tends to the population u-quantile risk given sufficiently many domains and samples (Thm. 4.1..
Causality. We prove that EQRM learns predictors with invariant risk as ~ 1 (Prop. 4.3), then provide the conditions under which this is sufficient to recover the causal predictor (Thm. 4.4).
Experiments: We demonstrate that EQRM outperforms state-of-ther oftt---rttt baselines several stan- dard DG benchmarks, including CMNIST [9] and datasets from WILDS [12] and DomainBed [38], and highlight the importance of assessing the tail or quantile performance of DG algorithms ($ 6).
Background: Domain generalization
Setup. In domain generalization (DG), predictors trained on data drawn from multiple related training distributions or domains and then evaluated on related but unseen test domains. For example, in the iWildCam dataset [50], the task is classify animal species in images, and the domains correspond to the different camera-traps which captured the images (see Fig. la). More formally, we consider datasets D {(x,, yf) collected from m different training domains or environments Er := {er, em), with each dataset D containing data pairs (x), sampled i.id.. from

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P )² Then, given a suitable function class F and loss function E, the goal of DG is to learn a predictor f F that generalizes data drawn from a larger set of all possible domains Eall Er-
Average case. Letting R (f) denote the statistical risk of f in domain e, and Q a distribution over the domains in Eall DG was first formulated [36, 37] as the following average-case problem:
min Ee-oR(( fer
where
R == (x (x)))
(2.1)
Worst case. Since predictors that perform well on average provably lack robustness [46], i.e. they can perform quite poorly on large subsets of Eall. subsequent works [9, 22, 41, 44, 45, 51] have sought robustness by formulating DG as the following worst-case problem:
min max R (f)- FEJ eEEill
(2.2)
As we only have access to data from a finite subset of aall during training, solving (2.2) is not just challenging but in fact impossible 52, 53] without restrictions on how the domains may differ.
Causality and invariance in DG. Causal works on DG [9, 41, 53-55] describe domain differences using the language of causality and the notion of interventions [56, 57]. In particular, they assume all domains share the same underlying structural causal model (SCM) [56]. with different domains corresponding to different interventions (see Appendix A. for formal definitions and a simple ex- ample). Assuming the mechanism of Y remains fixed or invariant but all Xs may be intervened upon, recent works have shown that only the causal predictor has invariant: (i) predictive distributions [54], coefficients [9] or risks [41] across domains; and (ii) generalizes to arbitrary interventions on the Xs [9, 54, 55]. These works then leverage some form of invariance across domains to discover causal relationships which, through the invariant mechanism assumption, generalize to new domains.
3 Quantile Risk Minimization
In this section we introduce Quantile Risk Minimization (QRM) for achieving Probable Domain Generalization, The core idea is to replace the worst-case perspective of (2.2) with probabilistic one. This approach is founded on great deal of work in classical fields such as control theory [58, 59] and smoothed analysis [60], wherein approaches that yield nigh-probability guarantees are used in place of worst-case approaches in an effort to mitigate conservatism and computational limitations. This mitigation is of particular interest in domain generalization since generalizing to arbitrary domains is impossible [41, 52, 531. Thus, motivated by this classical literature, our goal is to obtain predictors that are robust with high probability over domains drawn from Eall, rather than in the worst case.
distribution over environments. We start by assuming the existence of probability distribution Q(e) over the set of all environments Eill- For instance, in the context of medical imaging, Q could represent distribution over potential changes to hospital's setup or simply a distribution over candidate hospitals. Given that such a distribution Q exists?, we can think of the risk R (f) random variable for each f E F, where the randomness is engendered by the draw of e ~ Q. This perspective gives rise to the following analogue of the optimization problem in (2.2):
min ess sup Rs (f) where FEF α
ess sus Rf) inf t 2 0: Pr (R*(F) S = =1 (3.1) e
e~Q
Here, ess sup denotes the essential-supremee f E F, ess ess sup Re (f) is the least upper bound on (f) that holds for almost every e e~ Q. In this way, the ess sup in (3.1) is the measure-theoretic analogue of the max operator in (2.2), with the subtle but critical difference being that the ess sup in (3.1) can neglect domains of measure zero under Q. For example, for discrete Q, (3.1) ignores domains which are impossible (i.e. have probability zero) while (2.2) does not, laying the foundation for ignoring domains which are improbable.
operator from measure theory, meaning that for each
High-probabiliity generalization. Although the minimax problem in (3.1) explicitly incorporates the distribution Q over environments, this formulation is no less conservative than (2.2). Indeed, in many cases, 13.1) is equivalent to (2.2); see Appendix B for details. Therefore, rather than considering the worst-case problem in (3.1), we propose the following generalization of (3.1) which requires that predictors generalize with probability rather than in the worst-case:
min fcF,
subject to
(R < 1) 2 e-Q
Pr
(3.2)
2 As Q is often unknown, our analysis does not rely on using explicit expression for Q.
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The optimization problem in (3.2) formally defines what we mean by Probable Domain General- ization. In particular, we say that a predictor f generalizes with risk level a if f has risk at most f with probability at least a over domains sampled from Q. In this way, the conservativeness parameter a controls the strictness of generalizing to unseen domains.
distribution over risks. The optimization problem presented (3.2) offers a principled formulation for generalizing to unseen distributional shifts governed by Q. However, Q is often unknown in practice and its support Eall may be high-dimensional or challenging to define [22]. While many previous works have made progress by limiting the scope of possible shift types over domains [19, 22, 45J, in practice, such structural assumptions are often difficult to justify and impossible test. For this reason, we start our exposition of QRM by offering an alternative view of (3.2) which elucidates how predictor's risk distribution plays central role in achieving probable domain generalization.
To begin, note that for each f E F, the distribution over domains Q naturally induces a distribution T over the risks in each domain (f),. In this way, rather than considering the randomness of Q in the often-unknown and (potentially) high-dimensional space of possible shifts (Fig. 1b), one can consider it in the real-valued space of risks (Fig. Ic). This is analogous to statistical learning theory, where the analysis of convergence of empirical risk minimizers (i.e., of functions) is substituted by that of a weaker form convergence, namely that scalar risk functionals-a a crucial step for VC theory [61]. From this perspective, the statistics of T can be thought of as capturing the sensitivity of f to different environmental shifts, summarizing the effect of different intervention types, strengths, and frequencies. To this end, (3.2) can be equivalently rewritten in terms of the risk distribution y as follows:
min FT. (a) fEF
where (a) inf { ( R: Pr R-T,
2a.
(QRM)
Here, (a) denotes the inverse CDF (or quantile*) function of the risk distribution Ty. By means of this reformulation, we elucidate how solving (QRM) amounts to finding predictor with minimal a-quantile risk. That is, (QRM) requires that a predictor f satisfy the probabilistic constraint for at least a-fraction of the risks R ~ 'I', f. or, equivalently, for an a-fraction of the environments ~ Q. In this way, a can be used to interpolate between typical (a = 0.5, median) and worst-case (a = 1) problems in an interpretable manner. Moreover, if the mean and median of TT coincide, a =.5 gives an average-case problem, with (QRM) recovering several notable objectives for DG special cases.
Proposition 3.1. For 1-1, (QRM) is equivalent the worst-case problem of (3.1). For 0.5. it is equivalent to the average-case problem of (2.1) if the mean and median of coincide Vf
minser ER-I, R minjer E.-QR
(3.3)
Connection to DRO. While fundamentally different in terms of objective and generalization capabilities (see $ 4), we draw connections between QRM and distributionally robust optimization (DRO) in Appendix by considering an alternative problem which optimizes the superquantile.
4 Algorithms for Quantile Risk Minimization
We now introduce the Empirical QRM (EQRM) algorithm for solving (QRM) in practice, akin to Empirical Risk Minimization (ERM) solving the Risk Minimization (RM) problem [63].
4.1 From QRM to Empirical QRM
In practice, given predictor ff and its empirical risks Re (f),.., (f) on the m training domains, we must form an estimated risk distribution Ty.. In general, given no prior knowledge about the form of , (e.g. Gaussian), we use kernel density estimation (KDF, [49, 64]) with Gaussian kernels and either the Gaussian-or -apssian-optimal rule [65] or Silverman's rule-of- ohuuhmbbb [65] for bandwidth selection. Fig. Ic depicts the PDF and CDF for 10 training risks when using Silverman's 's th-umb. Armed
TT can be formally defined as the furs-forward measure of Q through the risk functional RC see App. B. 'In financial optimization, when concerned with a distribution over potential losses, the a-quantile value is known the value risk (VaR) at level a [62].

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with a predictor's 's estimated risk distribution T we can approximately solve (QRM) using the following empirical analogue:
min F feF T
(a)
(4.1)
Note that (4.1) depends only on known quantities so we can compute and minimize it in practice, as detailed in Alg. 1 of Appendix E.I.
Smoothing permits risk extrapolation. Fig. 2 compares the KDE- smoothed CDF (black) to the unsmoothed empirical CDF (gray). As shown, the latter places zero probability mass risks greater than our largest training risk, thus implicitly assuming that test risks cannot be larger than training risks. In contrast, the KmF-shot CDR permits "risk extrapolation" [41] since its right tail extends beyond our largest training risk, with the estimated q-quantile risk going to infinity as i --+1 (when kernels have full support). Note that different Randwidth-selection methods encode different assumptions about right-tail heaviness and thus about projected OOD risk. In 4.3, we discuss how, as a -> 1, this KDE-smoothing allows EQRM to learn predictors with invariant risk over domains. In Appendíx C, we discuss different mondwidth-selectiin methods for EQRM.
Probebiliy
KDB
Empicial
0.
Figure 2: Risk CDFs.
4.2 Theory: Generalization bound
We now give a simplified version of our main generalization bound-Thm. D.1-which states that, given sufficiently many domains and samples, the empirical æ-quaantlee risk is good estimate of the population a qu-qunttle risk. contrast to previous results for DG, we bound the proportion of test do- mains for which a predictor performs well, rather than the average error [36, 42], and make no assump- tions about the shift type, c.g. covariate shift [37]. The full version, stated and proved in Appendix D, provides specific finite-sample bounds é1 and e2 below, depending on the hypothesis class F, the empirical estimator 1-1( (a), and the assumptions on the possible risk profiles of hypotheses f E F.
Theorem 4.1 (Simplified form of Thm. D.I, uniform convergence). Given mn domains and samples in each, then with high probability over the training data,
sup feF
F (a E2) - F (a) Se,
(4.2)
where 6 - as 11 - 00 and 62 + 0 as m - o0.
While many domains are required for this bound be tight, i.c. for to precisely estimate the true quantile, our empirical results in & 6 demonstrate that FQRM performs well in practice given only a few domains, In such settings, & still controls conservativeness, but with a less precise interpretation.
4.3 Theory: Causal recovery
We now prove that EQRM can recover the causal predictor in two parts. First, we prove that, as & + 1. EQRM learns a predictor with minimal, invariant risk over domains. For Gaussian estimators of the risk distribution Ty, some intuition puts increasing weight the sample standard deviation of risks over domains 0,. eventually forcing it to zero. For kernel density estimators, similar intuition applies so long the bandwidth has a certain dependence on &y. as detailed in Appendix A.2.2. Second, we prove that learning such minimal invariant-risk predictor is sufficient to recover the causal predictor under weaker assumptions than Peters et al. [54] and Krueger et al. [41]. Together, these two parts provide the conditions under which EQRM successfully performs "causal recovery". i.e., correctly recovers the true causal coefficients in a linear causal model of the data.
be gained from Eq. (A.3) of Appendix A.2.1, noting
that
Definition 4.2. A predictor f is said to be an invariant-risk predictor if its risk is equal almost surely across domains (i.e., Var.-Q [R)( = 0). A predictor is said to be minimal invariant-risk predictor if it achieves the minimal possible risk across all possible invariant-risk predictors.
Proposition 4.3 (EQRM learns a minimal invariant-risk predictor as a ++ 1, informal version of Props. A.4 and A.5). Assume: (i) F. contains an invariant-Fe risk predictor with finite training risks;
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and (ii) no arbitrarit-negaatvv training risks. Then, as a -++1 1. Gaussian and kernel EQRM predictors (the latter with certain Sondvidth-selectinn methods) converge to minimal invariant-ris predictors.
Props. A.4 and A.5 are stated and proved in Appendices A.2.1 and A.2.2 respectively. In addition, for the special case of Gaussian estimators of f. Appendix A.2.1 relates our a parameter to the parameter of VREx [41, Eq. 8). We next specify the conditions under which learning such minimal invariant-risk predictor is sufficient to recover the causal predictor.
Theorem 4.4 (The causal predictor is the only minimal invariant-risk predictor), Assume that: (i) Y generated from a linear SEM, Y = X + N, with X observed and coefficients & E R2; (ū) FF is the class of linear predictors, indexed by B E I2; (iti) the loss & is squared qquared-error; error; (iv) the risk E[((Y - 5T x of the causal predictor & is invariant across domains; and (v) the system of equations
0 22x 2xTCovx~e (x, X)x + Co, (X,N)
=xTCovx~ew (X, X)x + 2x (X,N)
(4.3)
has the unique solution x = 0. If B is minimal invariant-risk predictor, then B = B.
Assumptions (i-iii). The assumptions that Y is drawn from linear structural equation model (SEM) and that the loss is squared-error, while restrictive, are needed for all comparable causal recovery results [41, 54]. In fact, these assumptions are weaker than both Peters et al. [54, Thm. 2] (assume linear Gaussian SEM for X and Y) and Krueger et al. [41, Thm. 1J (assume a linear SEM for X and Y).
Assumption (iv). The assumption that the risk of the causal predictor is invariant across domains, often called domain homoskedusticity [41], is necessary for any method inferring causality from the invariance of risks across domains. For methods based on the invariance of functions, namely the conditional mean E[Y[Pa(YY) (Y)] [9, 66]. this assumption is not required. Appendix G.1.2 compares methods based on invariant risks and to those based on invariant functions.
Assumption (v). In contrast to both Peters et al. and Krueger et al., we do not require specific types of interventions on the covariates. Instead, we require that a more general condition be satisfied, namely that the system of d-variate quadratic equations in (4.3) has a unique solution. Intuitively, Cov (X, X) captures how correlated the covariates are and ensures they are sufficiently uncorrelated distinguish each of their influences on Y, while Cov( X, N) captures how correlated descendant covariates are with Y (via N). Together, these terms capture the idea that predicting Y from the causal covariates must result in the minimal invariant-riske the first inequality ensures the risk is minimal and the subsequent
equalities that is invariant. While this generality comes the cost of abstraction, Appendix A.2.3 provides several concrete examples with different types of interventions in order to aid understanding and illustrate how this condition generalizes existing causal-recover results based on invariant risks [41, 54]. Appendix A.2.3 also provides a proof of Thm. 4.4 and further discussion.
5 Related work
Robust optimization in DG. Throughout this paper, we follow an established line of work (see e.g., 19, 41, 51J) which formulates the DG problem through the lens of robust optimization [44]. To this end, various algorithms have been proposed for solving constrained [22] and distributionally robust [45] variants of the worst-case problem in (2.2). Indeed, this robust formulation has a firm foundation in the broader machine learning literature, with notable works in adversarial robustness [67- 71] and fair learning [72, 73] employing similar formulations. Unlike these past works, we consider robust but non-ad -oversveraaiaal formulation for DG, where predictors are trained generalize with high probability rather than in the worst case. Moreover, the majority of this literature- both within and outside of DG-relies specific structural assumptions (e.g. covariate shift) on the types of possible interventions or perturbations. In contrast, we make the weaker and more flexible assumption of Li.d.-sampled domains, which ultimately makes use of the observed domain-data to determine the types of shifts that are probable. We further discuss this important difference in $ 7.
Other approaches to DG. Outside of robust optimization, many algorithmss have been proposed
for the DG setting which draw insights from diverse array of fields, including approaches based on tools from meta-learning [40, 43, 74-76), kernel methods [77, 78], and information theory [51]. Also prominent are works that design regularizers generalize OOD [79-81] and works that seek

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comain-invariian representations [82-84]. Many of these works employ hyperparameters which are difficult to interpret, which has no doubt contributed to the well-established model-selection problem in DG [38]. In contrast, in our framework, & can be easily interpreted in terms of quantiles of the risk distribution. In addition, many of these works do not explicitly relate the training and test domains, meaning they lack theoretical results in the non-linear setting (e.g. [9, 41, 43, 85)). For those which do, they bound either average error over test domains [36, 42, 86] or worst-case error under specific shift types (c.g. covariate [22]). As argued above, the former lacks robustness while the latter can be both overly-conservatie and difficult to justify in practice, where shift types are often unknown.
pyopabprbbbbllity generalization. As noted in $ 3, relaxing worst-case problems in favor of prob- abilistic ones has a long history in control theory [58, 59, 87-89], operations research [90], and smoothed analysis [60]. Recently, this paradigm has been applied to several areas of machine learning, including perturbation-
robustness [91, 92], fairness [93], active learning [94], and reinforcement learning [95, 96]. However, it has not yet been applied to domain generalization.
Quantile minimization. In financial optimization, the quantile and superquantile functions [62, 97, 98] are central to the literature surrounding portfolio risk management, with numerous applications spanning banking regulations and insurance policies [99, 100]. In statistical learning theory, several recent papers have derived uniform convergence guarantees in terms of alternative risk functionals besides expected risk [94, 101-103). These results focus on functionals that can be written in terms of expectations over the loss distribution (c.g., the superquantile). In contrast, our uniform convergence guarantee (Theorem D.1) shows uniform convergence of the quantile function, which cannot be written as such an expectation; this necessitates stronger conditions to obtain uniform convergence, which ultimately suggest regularizing the estimated risk distribution (e.g. by kernel smoothing).
Invariant prediction and causality. Early work studied the problem of learning from multiple cause- effect datasets that share a functional mechanism but differ in noise distributions [39]. More generally, given (data from) multiple distributions, one can try to identify components which are stable, robust, or invariant, and find means to transfer them across problems [104-108], As discussed in $ 2. recent works have leveraged different forms of invariance across domains to discover causal relationships which, under the invariant mechanism assumption [57], generalize to new domains [9, 41, 54, 55, 109- 11 111]. In particular, VREx [41] leveraged invariant risks (like EQRM) while IRM [9] leveraged invariant
fuvriiant functions or coefficients-eese see Appendix G.1.2 for a detailed comparison of these approaches.
Experiments
We now evaluate our EQRM algorithm synthetic datasets ( 6.1), real-world datasets from WILDS ($ 6.2), and few-domain datasets from DomainBed ($ 6.3). Appendix G reports further results, while Appendix E reports further experimental details.
6.1 Synthetic datasets
Linear regression. We first consider a linear regression dataset based on the following linear SCM:
X1 + N1,
+ X1 + Ny,
X2 t + N2,
with NJ- ~ (0, 03) Here we have two features: one cause X1 x X and one effect X₂ = Xeffect of Y. By fixing o7=1 and of =2 across domains but sampling 02~~ ~2~Looghe LogNomal 0.5), we create dataset in which X₂ is more predictive of Y than X, but less stable. Importantly, as know the true distribution over domains Q(e) = LogNormal 0.55). we know the true risk quantiles. 1000 domains and n = 200000 samples in each,
Fig. 3 depicts results for different a' with using the mean- seean-squared-erro (MSE) loss. Here we see that: A: for each true quantile (x-axis), the corresponding a has the lowest risk (y-axis), confirming that the empirical a-quantile risk is a good estimate of the population ququantile risk; B: As - 1, the estimated risk distribution of fa approaches an invariant (or Dirac delta) distribution centered on the risk of the causal predictor; C: the regression coefficients approach those of the causal predictor as a 1, trading predictive performance for robustness; and D: reducing the number of domains m reduces the accuracy of the estimated a-quantile risks. In Appendix G.I, we additionally: (i) depict the risk CDFs corresponding to plot B above, and discuss how they depict the predictors' risk-robustness curves 1G.1.1), I); and (ii) discuss the solutions of EQRM on datasets in which of. og and/or of change over domains, compared to existing invariance-secking algorithms like IRM [9] and VREx [41] (G.1.2).
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1 ERR ... Causal
Figure 3: EQRM on a toy linear regression dataset (A-D) and on ColoredMNIST (F-F). A: Test risk at different quantiles or degrees of "OODness". For each quantile (x-axis). the corresponding a has the lowest risk (y-axis). B: Estimated risk distributions (corresponding CDFs in Appendix G.1.1). C: Regression coefficients approach those of the causal predictor (Beause = 1, feffect = 0) a - 1. D: Q-Q plot comparing the "true" risk quantiles (estimated with m 1000) against estimated ones (m 1000), for a 0.9. E: Performance of different a's over increasingly OOD test domains, with dots showing training-dowe domain accuracies. F: KDE-estimated accuracy- depicting accuracy-rohustnes: curves, Larger & ''s make lower accuracies less likely.
CoISTedMNIST. We next consider the ColoredMNIST or CMNIST dataset [9]. Here, the MNIST dataset is used to construct binary classification task (0-4 or 5-9) in which digit color (red green) a highly-informative but spurious feature. particular, the two training domains are constructed such that red digits have an 80% and 90% chance of belonging to class 0, while the single test domain constructed such that they only have 10% chance. The goal is learn an invariant predictor which uses only digit shape-a stable feature having 75% chance of correctly determining the class in all domains. We compare with IRM [9], GroupDRO [45]. SD [112], IGA [113] and VREx [41] using: (i) random initialization (Xavier method [1 14]): and (ii) random initialization followed by several iterations of ERM. The ERM initialization or pretraining directly corresponds to the delicate penalty "annealing" or warm-up periods used by most penalty-based methods [9, 41, 112, 1 113]. For 2-hidden-layer MLP with 390 hidden units, the Adam optimizer, learning rate of 0.0001, and dropout with p= 0.2. We sweep over five penalty weights for the baselines and five a's for EQRM. See Appendix E.2 for more experimental details. Table I shows that: (i) all methods struggle without ERM pretraining, explaining the need for cenalty-nnnealing strategies in previous works and corroborating the results of 15. Table I]: (ii) with ERM pretraining. EQRM matches or outperforms baseline methods, even approaching oracle performance (that of ERM trained on grayscale digits). These results suggest ERM pretraining as an effective strategy for DG methods.
all methods,
In addition, Fig. 3 depicts the behavior of EQRM with different as. Here a leads to more consistent performance across domains, eventually forcing the model to ignore color and focus shape for invariant-risk prediction; and F: predictor's 's (estimated) accuracy-CDF depicts its eccuracy-rousstnee that & = 0.5 gives the best worst-case (i.e. worst-domain) risk over the two training domains the preferred solution of DRO [45] while sacrifices risk for increased invariance or robustness.
that: E: increasing
curve, just as its risk-CDF depicts its risk-robustness curve. Note
6.2 Real-world datasets
We now evaluate our methods on the real-world or in-the-wild distribution shifts of WILDS [12]. We focus our evaluation iWildCam [50] and OGB-Mo1PCBA [116, 117]-two large-scale classification datasets which have numerous test domains and thus facilitate a comparison of the test-domain risk distributions and their quantiles. Additional comparisons (c.g. using average accuracy) can be found in Appendix G.3, Our results demonstrate that, across two distinct data types (images and molccular graphs), EQRM offers superior tail or quantile performance.
8

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Table I: CMNIST test accuracy.
FQRM, WWildCam
FORRM, on OOR-MEIDRR
Initialization
ERM
ERM
Algorithm
EQRM x
EQRME
Rand.
ERM
EQRM, s
EQRM 54
ERM
279 1.5
27.9
Deasity
EQRMM
Demsity
EQRM,
IRM
52.5 :::1:2 2.4
69.7
0.9
EQURM
EQRM x
GrpDRO
0.9
2.10
EQRM
BQRMe 25
SD
1.5
70,3
IGA
50.7
57.7
BQRM
V-REx
55.2+4.0
1.7
71
71.6 0.5
0.00
Risk
0.00
1..02 Risk
0.01
Oracle
72.1 0.7
Figure 4: Test-domain risk distributions.
Table 2: EQRM test risks on 1W1ldCam.
Table 3: EQRM test risks on DGB-Mo1PCBA.
Alg-
Mean
Quantile risk
Alg.
Mean risk
Quantile risk
risk
0.25
0.50
0.90
0.50
0.75
1.0
FRM
0.015
0.42
0.76
2.25
4.99
ERM
0.051
0.0
0.001
0.017
0.060
0.13
0.19
16.04
IRM
1.53
0.098
0.52
1.24
1,86
2.36
6.95
7.46
IRM
0.073
0.098
0.52
1.24
1.86
2.36
6.95
7.46
GrouDRRO
1.73
0.091
0.68
1.65
2.18
3.56
5.29
5.54
GroupDRO
0.21
0.091
0.68
1.85
2.18
3.36
5.29
5.54
CORAL
1.27
0.024
0.45
0.73
2.12
2.66
4.50
4.98
CORAL
0.055
0.0
0.12
0.32
1.23
2.01
5.76
7.44
2ORMa
2.70
3.01
EQRM:
0.016
0.059
15.46
EQRMas0
1.11
0.004
0.24
0.68
1,71
2.15
1.04
1.11
EQRMas0
0.052
0.0
0.003
0.015
0.059
0.13
0:48
11.33
HQRMuus
1.05
0.009
0.21
0.68
1.50
2.35
4.88
5.45
EQRMuss
0.052
0.0
0.005
0.015
0,059
0.13
0.47
12.15
FQRMas0
0.98
0.047
0.28
0.63
1.26
1.81
4.11
4.48
FQR Magei
0.052
0.0
0.003
0.015
0.059
0.12
0.47
10.81
BQ
0.99
0.35
0.64
1.30
2.00
3.58
EQ
0.053
0.003
0,014
0.055
7.16
iWildCam. We first consider the iWildCam image-classification dataset, which has 243 training domains and 48 test domains. Here, the label Y is one of 182 different animal species and the domain is the camera trap which captured the image. In Table 2, we observe that FQRM, does indeed tend to optimize the æ-risk quantile, with larger as during training resulting in lower test-domain risks at the corresponding quantiles. In the left pane of Fig. 4, we plot the (KDE-smoothed) test-domain risk distribution for ERM and EQRM. Here distribution tend to drop below ERM, which corroborates the superior quantile performance reported in Table 2. Note that, in Table 2. EQRM tends to record lower average risks than ERM. This has several plausible explanations. First, the number of testing domains (48) is relatively small, which could result in a biased sample with respect to the training domains. Second, the test domains may not represent i.i.d. draws from Q, as WILDS [12] test domains tend be more challenging.
clear trend:
increases, the tails of the risk
OGB-MoPPBBA. We next consider the DGB-Mo1PCBA (or OGB) dataset, which is a molecular graph-classification benchmark containing 44,930 training domains and 43,793 test domains with an average of 3.6 samples per domain. Table shows that ERM achieves the lowest average test risk on OCB, in contrast to the iWildCam results, while EQRM, still achieves stronger quantile performance. Of particular note is the fact that our methods significantly outperform ERM with respect to worst-case performance (colunns/quantiles labeled 1.0); when QRMa is run with large values of a, we reduce the worst-case risk by more than factor of two. In Fig. 4, we again see that the risk distributions of EQRMa have lighter tails than that of FRM.
A new evaluation protocol for DG. The analysis provided in Tables 2-3 and Fig. 4 diverges from the standard evaluation protocol in DG [12, 38). Rather than evaluating an algorithm's performance on average across test domains, we seek to understand the distribution of its crrformance-parriin in the tails by means of the quantile function. This new evaluation protocol lays bare the importance of multiple test domains in DG benchmarks, allowing predictors' risk distributions to be analyzed and compared. Indeed, shown in Tables 2-3, solely reporting a predictor's ' average worst risk over test domains can be misleading when assessing its ability to generalize OOD, indicating that the performance of DG algorithms was likely never "lost", as reported in [38], but rather invisible through the lens of average performance. This underscores the necessity of incorporating tail- or quantile-risk measures into more holistic evaluation protocol for DG, ultimately providing a more nuanced and complete picture. In practice, which measure is preferred will depend on the application. For example, medical applications could have human buman-specified robustness-leve or quantile-of of-interest.
6.3 DomainBed datasets
Finally, we consider the benchmark datasets of DomainBed [38], in particular VLCS [118], PACS [119], Officefome [120], TerraIncognita [5] and DomainNet [121]. As each of these datasets con- tain just 4 or 6 domains, it is not possible to mcaningfully compare tail or quantile performance. Nonetheless, in line with much recent work, and to compare EQRM to a range of standard basclines on few-domain datasets, Table 4 reports DomainBed results in terms of the average performance

---

Table 4: DomainBed results. Model selection: training-domain validation set.
Algorithm
VLCS
PACS
OfficeHomee
TerraIncognita
DomainNet
Avg
ERM
77.5 + 0.4
85.5 + 0.2
66.5 + 0.3
46.1 + 188
40.9 + 0.1
63.3
IRM
78.5 0.5
83.5 + 0.8
64.3 + 2.2
47.6 + 0.8
33.9 - 2.8
61.6
GroupDRO
7
84.4 = 0.8
66.0 0.7
43.2 = 1.1
33.3 = 0.2
60,9
Mixup
0.6
84.6 0.6
68.1 0.3
47.9 1 0.8
39.2 0.1
63.4
MLDG
0.4
84.9 1.0
66.8 0.6
47.7 - 0.9
41.2 0.1
63.6
CORAL
78.8
86.2 0.3
68.7 - 0.3
47.6 1.0
41.5 0.1
64.6
ARM
77.6 0.3
85.1 0.4
0.3
45,5 + 0.3
35,5 1 0.2
61.7
VREX
78.3
84.9
0.6
66.4 = 0.6
46.4 + 0.6
33.6 - 2.9
61.9
EQRM
77.8 0.6
86.5 : 0.2
67.5 + 0.1
47.8 = 0.6
41.0 0.3
64.1
across each choice of test domain. While EQRM outperforms most baselines, including ERM, we reiterate that comparing algorithms solely in terms of average performance can be mislcading (see final paragraph of ş 6.2). Full implementation details are given in Appendix E.3, with further results in Appendix G.2 (additional baselines, per-dataset results, and test-domain model selection).
Discussion
Interpretable model selection. approximates the probability with which our predictor will generalize with risk below the associated a-quantile value. Thus, a represents an interpretable parameterization of the risk-robustness trade-off. Such interpretability is critical for model selection DG, and for practitioners with application-specifii requirements on performance and/or robustness.
The assumption of i.i.d. domains. For a to approximate the probability of generalizing, training and test domains must be sai.les While this is rarely true in practice e.g. hospitals have shared funders, service providers, etc.-we can better satisfy this assumption by subscribing to a new data collection process in which collect training-domain data which is representative of how the underlying system tends to change. For example: (i) randomly select 100 US hospitals; (ii) gather and label data from these hospitals; (iii) train our system with the desired a; (iv) deploy our system to all US hospitals, where it will be successful with probability = a. While this process may seem expensive, time-consuming and vulnerable (c.g. to new hospitals), it offers promising path to machine learning systems which generalize with high probability. Moreover, it is worth noting the alternative: prior works achieve generalization by assuming that only particular types of shifts can occur, e.g. covariate shifts [22, 122, 123], label shifts [123, 124], concept shifts [125], measurement shifts [19], mean shifts [126]. shifts which leave the mechanism of Y invariant [9, 39, 41, 54), etc. In real-world settings, where the underlying shift mechanisms are often unknown, such assumptions are both difficult to justify and impossible to test. Future work could look to relax the i.i.d.-ommiins assumption by leveraging knowledge of domain dependencies (e.g. time).
has The wider value of risk distributions. As demonstrated in 6, a predictor's risk distribution value beyond guantle-mininizzatoos Thus, regardless of the algorithm used, risk distributions can be used to analyze trained predictors.
estimates the probability associated with each level of risk.
8 Conclusion
We have presented Quantile Risk Minimization for achieving Prohable Domain Generalization, motivated by the argument that the goal of domain generalization should be to learn predictors which perform well with high probability rather than on-average or the worst case. By explicitly relating training and test domains as draws from the same underlying meta-distribution, we proposed to learn predictors with minimal a-quantile risk under the training domains. We then introduced the EQRM algorithm, for which we proved a generalization bound and recovery of the causal predictor as a --> 1, before demonstrating that EQRM outperforms state-of-the-at baselines on real and synthetic data.
Acknowledgments and Disclosure of Funding
The authors thank Chris Williams and Ian Mason for providing feedback earlier draft, well as Lars Lorch, David Krueger, Francesco Locatello and members the MPI Tübingen causality group for helpful discussions and comments. This work was supported by the German Federal Ministry of Education and Research (BMBF): Tübingen AI Center, FKZ: 011918039A, 011S18039B; and by the Machine Learning Cluster of Excellence, EXC number 2064/1
Project number 390727645.
10

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Checklist
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19


Reviews:
[{"review_id": "ea53C3JVSCG", "reviewer": "Reviewer_yw1U", "summary": "This paper introduces a quantile risk minimization (QRM) and its empirical version (EQRM) that aim to learn predictors that can provide minimal risk in a domain generalization with probability $\\alpha$. Interestingly, it has been shown in the paper that EQRM turns to the causal predictor when $\\alpha \\to 1$. The experimental results in the paper show the importance of quantile performance in evaluating domain generalization methods. These results also show that the proposed QRM can produce better results on both real and synthetic data sets compared to some strong baselines. ", "questions": "Please see my comments on the weaknesses part of the paper.", "limitations": "N/A", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Originality: The idea of introducing a probable domain generalization (by using quantile risk minimization) is novel and interesting due to the lack of a theoretical guarantee for a generalization bound or learnability of current domain generalization algorithms.\n\nQuality: It seems that the main theoretical contribution of the paper is Proposition 3.1. However, could the authors provide a detailed definition of the distribution $\\mathbb T_f$ and a representation of each element $R$ of $\\mathbb T_f$?  In the formulation of (QRM), does the condition $Pr_{R \\sim \\mathbb T_f} \\{ R \\leq t \\} \\geq \\alpha$ need to be satisfied $\\forall R$? Otherwise, it would be very hard for me to verify that statement. More importantly, does QRM or EQRM learn an invariant predictor/representation across domains? If so, how does that work, and are there any theoretical guarantees for that?\n\nClarity: The paper is well-written and easy to follow.\n\nSignificance: As mentioned in the paper, the theoretical result in Proposition 4.2 is only true for linear predictors -- which really restricts the application of the proposed method in domain generalization. Note that the result for an invariant optimal predictor across domains has been introduced in a very well-known paper [1].  \n\n[1] @article{arjovsky2019invariant,\n  title={Invariant risk minimization},\n  author={Arjovsky, Martin and Bottou, L{\\'e}on and Gulrajani, Ishaan and Lopez-Paz, David},\n  journal={arXiv preprint arXiv:1907.02893},\n  year={2019}\n}  ", "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": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "aR5ptTja_11", "reviewer": "Reviewer_xBHw", "summary": "This work advocates for achieving high-probability generalization across domains, which is in between of the average-case (domain) generaliaztion for which ERM empirically works well, and the worst-case problem which motivates most of previous work in this area.  It assumes the training and test domains are i.i.d. samples from a shared distribution, and proposes to minimize the $\\alpha$-quantile and superquantile of the risk distribution of predictors. The authors establish consistency guarantees for the proposed objectives, and connect them to other distributionally robust optimization and invariant prediction formulations.  Experiments show that the proposed method outperforms ERM in terms of the (super)quantile risk.\n\n**Post-rebuttal update**: I have read the authors' response to my and other reviewers' questions and find them satisfactory.  I'm happy to see the new DomainBed experiment, which shows the promise of the proposed method in settings with a smaller number of domain, but the reason for such improvement isn't fully clear.  While I don't believe such discussions are in any sense necessary for the present work (and will not change my score in any case), I think this claim would be better supported if there are more experiments in the small-number-of-domain setting, or some theoretical discussion on the probability of improvement over ERM in such cases.", "questions": "- Section 7 claims that we can deploy the trained model to \"all [test domains] where it will be successful with probability ≈ α\".  How can we identify such domains?  Are you assuming we have a labeled validation set in all test domains?\n\n- It may also be helpful to move some of the IRM / GroupDRO results in appendix to the text.", "limitations": "Yes.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The main strength of this work lies in its new formulation; while various relaxiations of worst-case generalization have been proposed before, the new formulation appears easier to implement and analyze, especially for nonlinear models, while still being intuitive and interpretable when its assumptions are adequate.  It thus brings new promise for achieving domain generalization in nonlinear models, as is also evidenced by the experiments.\n\nIt should be noted that the method relies on the assumption of i.i.d. domains, and requires that we have a relatively large number of training domains so that quantiles can be estimated to a suitable precision.  The limitations are understandable, given the challenging nature of the problem, and the authors have been very transparent about them, but they will put the contributions better in context.\n\nAnd while the real-world experiment shows the promise of the method in scenarios where a large number of domains are available, it would strengthen the paper if evaluations are also provided in scenarios where only a handful of domains are available, such as some of the DomainBed benchmarks.\n", "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": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "SKcXkv5tNQ", "reviewer": "Reviewer_MTnN", "summary": "The paper presents a probable domain generalisation framework for finding predictors that generalise with high probability instead of generalisation in the worst or average case. It proposes two algorithms: QRM, SQRM, which are based on minimising $\\alpha$ quantile of training risks with the interpretation that the predictor so trained would generalise well to at least $\\alpha$ fraction of test domains. The last statement holds because the train and test domains are sampled from the same (latent) underlying distribution of domains.\n\n----------\n**Post rebuttal**.   \nI thank the authors for clarifying most of my questions.     \nThe merits of the paper are better understood when there is risk-robustness tradeoff such as in their synthetic setting or the two real-world settings: iWildCam, OGB-MolPCBA. Their synthetic experiment has varying noise levels per environment, and iWildCam likely has varying noise or difficulty level per domain, but I do not see any reason for risk-robustness tradeoff in DomainBed experiments. Therefore, their results on DomainBed are arguable. On CMNIST, they claim to recover causal predictor, as further elaborated in Appendix A, but when the number of training domains is so small as two, $T_F$ estimate is noisy, which is not accounted for in their theorem proofs of Appendix. Besides, in the case of CMNIST, it is unclear why any of the existing methods cannot recover a function with invariant loss across domains. Some of the reviewers (including me) are requesting for a better explanation/intuition on their DomainBed results perhaps because some gains could also arise from implementation differences.    \nAlso, the contribution of SQRM to this paper is unclear. Authors note that it could lead to interesting future work, but I feel it does not contribute much to this work (this is a relatively minor issue though).    \nThis paper has solid contributions, but I believe they could have been expressed more crisply with better experiment design and crisper writing/presentation. For this reason, I am changing my earlier assessment to a weak (but not enthusiastic) accept. ", "questions": "Please respond to questions marked with (Q) in the section above.", "limitations": "Yes.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\nProbable DG is a very sensible and a practical idea. Training data could include domains with varying difficulty levels and imposing invariance like constraints on them could be detrimental. \n\n## Weaknesses/comments \n**Writing**   \nThe writing is unclear and some important information should have been in the main content rather than in the appendix. Let me explain, \n\n* L36-37: (Q) what is “standard DG formulation”, need citations. From the description, it resembles (DG) of L76-77, which inturn resembles DRO. \n* L101-110: I do not see why talking about “esssup” is important. It clearly is just DG as remarked in L113: (3.1) is equivalent to DG.  \nWould help if Preposition 3.1 and eq 3.3 are described more, I could not follow.\n* L152-154: Not a good idea to describe in main content a figure in Appendix. Should have been here if it is important. \n* L163-165: I do not see why the note on finance is interesting here. Could have been in related work.\n* L266-268: “A: each $\\alpha$ is optimal at the corresponding quantile” needs further description. I believe the authors are using “optimal” in a different sense? \n* L261: “Ex. A.3” which shows SCM of linear setting should have been in the main content. \n\n**Evaluation metric**   \nThe paper advocates to go beyond average or worst accuracy measures for DG and suggest looking at quantiles of test risk/accuracy. Tables 2-5 shows 7 quantiles and average risk. From what I understand, authors judge superior method to be the one that minimises risk on all quantiles, but such a method would also have lower average risk on test domains. So, it seems the authors are going for a more nuanced comparison of quantiles. (Q) They should mention explicitly how quantiles are contrasted to reveal the superior method. In my opinion, evaluation should go hand-in-hand with a human specified quantile level or some such. \n\n**Missing comparisons**   \n(Q) I expect to see comparisons with other DG methods in Tables 2-5.\n(Q) And also std. dev. for all numbers.   \n\n**QRM vs SQRM**   \nThe way SQRM is formulated, SQRM is not the same as QRM. Moreover, QRM is never compared with SQRM. In tables 2-5, QRM has consistently lower risk than SQRM. If the contribution of SQRM is efficiency or some such, that should have been discussed. I fail to see the contribution of SQRM.\n\n**ColorMNIST**    \n(Q) What’s the value of $\\alpha$ and how did you pick it?\n(Q) As $\\alpha\\rightarrow 1$, QRM (eq 4.1) is the same as (3.1), which is the same as DRO, correct? (Q) Why does QRM perform so much better than DRO/GrpDRO in Table 1?   \n\n**DG formulation**    \nThe paper misrepresents existing DG work. The related work did some justice by having a section on “Robust DG” and “other DG”, but the main content completely disregarded “other DG” and went on to call min-max formulation of risk objective as the standard. This should be fixed. ", "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": "", "extra_scores": {"ethics_review_area": ["I don’t know"]}, "extra_text": {}}, {"review_id": "LGm5tPl7rT5", "reviewer": "Reviewer_WQN2", "summary": "In this work, the authors proposed the Quantile Risk Minimization to have the *probable* DG, where the authors argue that DG should both seek and evaluate predictors based on their ability to perform well with high probability, rather than in the worst-case or on-average, which are widely adopted in concurrent works. The authors also showed that the empirical QRM can recover the causal predictor for the desired generalization probability. The paper is technically sound and generally well-written.", "questions": "\n1. Theorem 4.1 indicates that the number of samples and domains should be sufficiently large. If so, no matter adopting what criteria ('worst case' or 'on average'), the generalization can succeed since we have enough domains and training samples. Thus, I'm wondering what are the exact benefits opt for quantile risk minimization? \n\n2. Following the first question: If the number of samples and domains should be sufficiently large, will the generalization property rely on the model complexity?\n\n3. I would expect more experimental results comparing recent baselines (e.g. the ones provided in DomainBed) on more standard DG datasets (e.g. the PACS, Office-Home, DomainNet etc.).\n\n4. a minor question:  In line 101, it seems that the difference between the assumption of *a distribution over environments* and the general assumption in many concurrent DG works that there exists a meta distribution between domains is not clear. Some explanations or discussions can help.", "limitations": "The motivation is not quite clear, and empirical evaluations are not sufficient.\n====\nAfter rebuttal comments: I would like to thank the authors for further providing the explanations and experimental results in the rebuttal phase.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Originality: the originality of the paper lies in the proposal of a new objective for the domain generalization problems. I found the contribution fair and somewhat limited.\n\nStrengths:\n\n1. The theoretical analysis was complete, I checked most parts of the appendix, and it seems technically sound.\n\n2. The empirical evaluations were conducted on diverse scenarios including regression, digits and some real-world applications. \n\nWeaknesses:\n\n1. The motivation for adopting quantile risk minimization is not quite clear. The benefits of why opt for Quantile Risk Minimization rather than the 'worst-case' or 'on-average' are not clear.\n\n2. The authors' claim on \"Recovering the causal predictor\" was not formally elaborated.\n\n3. The empirical comparison was only conducted on a very limited number of baselines. For example, the experiments on ColorMNIST missed many recent baselines, and for the experiments on the *real-world* datasets, only the ERM was compared. This makes the effectiveness of the proposed method hardly to be evaluated. \n\n4. The organization of the presentation needs improvements. Many key components of the paper are listed in the appendix rather than in the main text. The paper itself should be self-contented.\n", "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": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept