# domain_generalisation_via_imprecise_learning__e33bc383.pdf Domain Generalisation via Imprecise Learning Anurag Singh 1 Siu Lun Chau 1 Shahine Bouabid 2 Krikamol Muandet 1 Out-of-distribution (OOD) generalisation is challenging because it involves not only learning from empirical data, but also deciding among various notions of generalisation, e.g., optimising the average-case risk, worst-case risk, or interpolations thereof. While this choice should in principle be made by the model operator like medical doctors, this information might not always be available at training time. The institutional separation between machine learners and model operators leads to arbitrary commitments to specific generalisation strategies by machine learners due to these deployment uncertainties. We introduce the Imprecise Domain Generalisation framework to mitigate this, featuring an imprecise risk optimisation that allows learners to stay imprecise by optimising against a continuous spectrum of generalisation strategies during training, and a model framework that allows operators to specify their generalisation preference at deployment. Supported by both theoretical and empirical evidence, our work showcases the benefits of integrating imprecision into domain generalisation. 1. Introduction The capability to generalise knowledge, a hallmark of both biological and artificial intelligence (AI), has seen remarkable progress in recent years. Developments in generalpurpose learning algorithms (Vapnik, 1991; Hofmann et al., 2008; Le Cun et al., 2015; Goodfellow et al., 2016), model architectures (Krizhevsky et al., 2012; Cohen and Welling, 2016; Vaswani et al., 2017), and training infrastructures (Ratner et al., 2019) have given rise to AI systems such as generative models (Gen AI) and large language models Anurag Singh is part of the Graduate School of Computer Science at Saarland University, Saarbr ucken, Germany. 1CISPA Helmholtz Center for Information Security, Saarbr ucken, Germany 2Department of Statistics, University of Oxford, UK. Correspondence to: Anurag Singh . Proceedings of the 41 st International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s). (LLM) that surpass human-level generalisation capabilities in specific domains. Despite notable achievements, these systems may catastrophically fail when operated on out-of-domain (OOD) data because theoretical guarantees for their generalisation hinge on the assumption of independent and identically distributed (IID) training and deployment data, with empirical risk minimisation (ERM) being the dominant learning algorithm (Vapnik, 1991; 1995). Emerging challenges like distribution shifts (Quionero-Candela et al., 2009; Beery et al., 2018; 2020; Koh et al., 2021), adversarial attacks (Szegedy et al., 2013; Goodfellow et al., 2014), and strategic manipulations (Hardt et al., 2016; Perdomo et al., 2020; Vo et al., 2023) have prompted researchers to question the validity of algorithms developed under this assumption. This gap has fueled interest in OOD generalisation, prompting the exploration of novel learning algorithms and resulting in rapid developments in domain adaptation (Wilson and Cook, 2020; Zhao et al., 2022), domain generalisation (Wang et al., 2021b; Zhou et al., 2023; Shen et al., 2021), and test-time adaptation (Sun et al., 2020; Wang et al., 2021a; Chen et al., 2023a), among others. In IID generalisation, where test loss aligns with training loss, the learner s goal of minimising the training loss aligns with the operator s expectation of small test loss. Bounded data uncertainty, within finite data, enables the learner to assess model generalisation during deployment. Historically, the IID assumption is accompanied by another critical, but often overlooked assumption: the overlap between the learner and the operator, who employs the model in realworld contexts. Conversely, OOD generalisation still lacks a precise definition, leading to additional ambiguity termed generalisation uncertainty . Unlike data uncertainty, generalisation uncertainty arises from a lack of knowledge about deployment environments, whether due to natural shifts (across hospitals, experimental conditions, and time) or artificial ones (adversarial attacks, strategic manipulation), and cannot be mitigated by additional data collection. Prior research has addressed generalisation uncertainty independently by introducing various concepts of OOD generalisation including worst-case generalisation (Arjovsky et al., 2019; Ben-Tal et al., 2009; Sagawa et al., 2020; Krueger et al., 2021), average-case generalisation (Blan- Domain Generalisation via Imprecise Learning Figure 1: An illustration of our proposed imprecise learning framework. We allow learners to stay imprecise to avoid over-commit in light of generalisation uncertainty. Instead, we defer this choice of precise generalisation to the operator. chard et al., 2011; 2021; Muandet et al., 2013; Zhang et al., 2021), and their interpolations (Eastwood et al., 2022a). Learning algorithms like distributional robust optimisation (DRO) (Rahimian and Mehrotra, 2022), invariant risk minimisation (IRM) (Arjovsky et al., 2019), and quantile risk minimisation (QRM) (Eastwood et al., 2022a) have been tailored for these OOD generalisation notions. This line of research relaxes the IID assumption, but still assumes alignment between the learner s objective and the operator s goal to tackle generalisation uncertainty. Due to the need for precise concept of generalisation in these scenarios, we collectively term them precise generalisation . Precise generalisation hinges on the assumption that the learner s objective aligns with the operator s goal, necessitating close collaboration during model development. However, this approach presents two primary drawbacks. Firstly, institutional separation between the learner (e.g., machine learning engineers) and model operators (e.g., doctors) can make collaboration costly, time-consuming, or even impractical. Secondly, tailoring the model to a specific operator may restrict its deployment usability, as the operator s beliefs can change or conflicts may emerge when the model is operated by a different individual. Consider an example depicted in Figure 1. Using data obtained from hospitals across Europe, an engineer is developing a machine learning model that will be embedded into a medical software that will be used by the doctors. Here, the engineer confronts uncertainty regarding where the model will ultimately be deployed it could be within Europe (IID) or outside it (OOD). The engineer might anticipate the doctor s generalisation strategy during the model s training phase. For instance, if the doctor is perceived to be risk-averse, the engineer might prioritise training a model robust to worst-case scenarios. However, ideally, it should be the doctor, often equipped with domain-specific expertise, who decides the generalisation strategy, drawing upon their in-depth knowledge of the field, at deployment time. Customising models effectively to the clinical settings where they operate can significantly impact healthcare outcomes (Beede et al., 2020). In this work, we extend the relaxation of the IID assumption further by loosening the requirement for overlap between the learner and the operator. Since there is no need of specific concept of generalisation at training time, we term this scenario imprecise generalisation (see Figure 1). We operationalise imprecise learning in the context of domain generalisation (Blanchard et al., 2011; 2021; Muandet et al., 2013), aiming to answer the question: How to take knowledge acquired from an arbitrary number of related domains and apply it to previously unseen domains? This concept comprises two main components: (1) An optimisation process enabling learners to remain imprecise during learning, thus not committing to a specific generalisation notion during training, and (2) a model framework allowing operators to define their preferred generalisation strategy at deployment. We delve into the formulation and existing work on OOD generalisation in Section 2. Our primary contribution, the framework of Imprecise Domain Generalisation, is detailed in Section 3, along with its optimisation strategy, termed Imprecise Risk Optimisation (IRO), in Section 4. Experimental results are presented in Section 5, and we conclude our paper in Section 6. All proofs are in the appendix and we open-source our code at https://github.com/muandet-lab/dgil. 2. Preliminaries Consider X Rd as our instance space and Y as our target space, where Y R is used for regression and Y = 1, . . . , C for C-class classification. In supervised learning, the process of learning a function mapping from X to Y involves the learner specifying their inductive biases. These inductive biases include: (1) selecting a hypothesis class H, consisting of functions f : X Y, (2) defining a suitable loss function ℓ: Y Y R+ based on the problem, (3) assuming the presence of a joint probability distribution P over the variables (X, Y ) X Y from which the data are sampled. Most critical to our work are (4) the assumptions regarding the deployment environment where the model f is expected to generalise. 2.1. Precise Learning In the following, we briefly review various generalisation assumptions commonly adopted in the literature and unify Domain Generalisation via Imprecise Learning them under the setting of precise learning. IID assumption. Perhaps the most fundamental generalisation assumption in supervised learning is that the training and deployment environments are independent and identically distributed (IID). Under this assumption, a model that performs well in training is expected to generalize effectively in deployment. This concept is formalized by finding the function f H that minimizes the population risk for P, known as the Bayes optimal model: R(f) EP[Zf] = E(X,Y ) P ℓ(f(X), Y ) . (1) For simplicity, we have denoted Z (X, Y ), Z X Y, and Zf ℓ(f(X), Y ) as the random loss associated with f H. In practice, since the true distribution P is unknown, we focus on minimizing an empirical estimate of this risk based on IID samples (xi, yi)n i=1 from P, expressed as: i=1 ℓ(f(xi), yi) + η f 2 H, f H, (2) where the second term is a regularization term to prevent overfitting, following the empirical risk minimization (ERM) principle (Vapnik, 1991; 1995). This scenario introduces data uncertainty, stemming from the finite nature of data when approximating Bayes optimal models. The IID assumption also enjoys favourable guarantees, e.g., as the sample size n increases, the uniform convergence of b R( ) over H ensures that the gap between the empirical and the population risk becomes negligible with high probability; see, e.g., Vapnik (1998); Cucker and Smale (2002). Beyond IID assumptions. The IID assumption is often not viable in real-world scenarios due to various factors such as distribution shifts (Quionero-Candela et al., 2009; Beery et al., 2018; 2020; Koh et al., 2021), sub-population shifts (Santurkar et al., 2021; Yang et al., 2023), adversarial attacks (Szegedy et al., 2013; Goodfellow et al., 2014), strategic manipulation (Hardt et al., 2016; Perdomo et al., 2020; Vo et al., 2023), and time shifts (Gagnon-Audet et al., 2022). In response to these challenges, learners must consider generalisation uncertainty when designing their learning algorithm. This uncertainty is typically represented by a credal set K(Z) (Walley, 1991), a closed set of potential probability distributions that reflect the learner s ignorance, or partial knowledge about the deployment environments. For example, in distributionally robust optimisation (Rahimian and Mehrotra, 2022), the credal set comprises distributions within an ϵ distance from the empirical distribution, and the goal is to optimise f for the worst-case empirical risk within it. Another approach, involves learning across multiple domains P1, . . . , Pd, and assumes the deployment distribution lies within their convex hull (Mansour et al., 2012; Krueger et al., 2021; F oll et al., 2023). In invariant causal prediction (Peters et al., 2016; Heinze Deml et al., 2018), hypothetical interventional distributions associated with a structural causal model (SCM) constitute the credal set. Learning algorithms here aim to optimize for worst-case empirical risk (Arjovsky et al., 2019; Ben Tal et al., 2009; Sagawa et al., 2020; Krueger et al., 2021), average-case empirical risk (Blanchard et al., 2011; 2021; Muandet et al., 2013; Zhang et al., 2021), and interpolations thereof (Eastwood et al., 2022a). The choice of risk corresponds to selecting a particular distribution within the credal set, such as the centroid of the convex hull referring to the average case. Notably, the credal set in the IID case reduces to a single distribution, K(Z) = {P}. 2.2. Previous Work Limitation of precise learning. A majority of previous work in both IID and OOD generalisation falls into the precise learning setting. A fundamental requirement is for the learner to commit to a specific notion of generalisation. This involves precisely selecting a particular distribution P K(Z) during training and performing statistical learning to develop the model f. Although widely used, this might not always be optimal in modern machine learning settings, especially when there is a clear institutional separation between those who build and those who operate the model (cf. Section 3). This separation presents two significant challenges. First, it assumes that the learners either fully understand the specific generalisation needs of the operators, or that the operators have comprehensive access to the datasets and a thorough understanding of statistical inference, effectively making them the learners. Second, the choice of generalisation strategy is inherently subjective, involving normative decisions by the operators. For instance, a risk-averse operator might lean towards a worst-case empirical risk optimiser, while an operator with in-depth knowledge of the deployment environment might prefer an average-case empirical risk optimiser. Domain generalisation strategies. The core of domain generalisation is the invariance principle (Muandet et al., 2013; Arjovsky, 2019), which asserts that certain properties remain constant across different environments and thus are expected to generalise to unseen settings. This principle is reflected in approaches focusing on feature representation (Muandet et al., 2013; Ghifary et al., 2015; Arjovsky et al., 2019), causal mechanism (Peters et al., 2016; Rojas-Carulla et al., 2018; Heinze-Deml and Meinshausen, 2021), and risk functional (Krueger et al., 2021), all aimed at identifying and leveraging these invariant properties. While necessary, this principle faces two challenges: it abstracts away the inherent heterogeneity across environments (Heckman, 2001), which might give rise to non-invariant yet generalisable properties (Eastwood et al., 2023; Nastl and Hardt, 2024). Furthermore, identifying and utilising invariant properties Domain Generalisation via Imprecise Learning faces practical difficulties due to their need for large sample sizes (Rosenfeld et al., 2021; Kamath et al., 2021). Addressing these challenges, recent work suggests combining domain-invariant with domain-specific properties (Liu et al., 2021a; Mahajan et al., 2021). While these approaches have been shown to improve in-domain generalisation performance, how domain-specific properties affect OOD generalisation in unseen environments remains unclear. To overcome this, it is popular to utilise various forms of test-time adaptation via auxiliary tasks (Sun et al., 2020; Wang et al., 2021a; Chen et al., 2023a). However, Liu et al. (2021b) has shown that this strategy can improve the pre-trained model only when the auxiliary loss aligns with the main loss. This suggests that a certain degree of precision in aligning losses is essential for effective domain generalisation. Generalisation uncertainty representation. As opposed to the credal set K(Z), some authors have instead adopted a second-order probability (aka meta distributions) as a belief over the true or ideal probabilities P(Z) (Blanchard et al., 2011; 2021; Muandet et al., 2013; Eastwood et al., 2022a). However, Walley (1991, Sec. 5.10) pointed out that if probability distributions entail behavioural dispositions, it is necessary that the credal set must collapse to a singleton to avoid incoherent behaviour. Paradoxically, this implies that assuming the existence of meta-distributions is equivalent to making the IID assumption in the first place, emphasising that one must clearly differentiate generalisation uncertainty from data uncertainty. Learning under imprecision. Machine learning inherently grapples with imprecision due to its inductive nature. One common approach to mitigate this is to create precision at various stages of model development. Techniques like data up/downsampling (He and Garcia, 2009) and fusion (Chau et al., 2021a;b) address issues of granularity and missing data by drawing information from a precise empirical distribution. During algorithm selection, approaches like the Bayesian paradigm, ensembling, and Auto ML (He et al., 2021) are used to handle potential model misspecification by selecting a precise model from a set of alternatives. Furthermore, model deployment requires a precise definition of generalisation, such as optimising for average-case or worst-case risks, to be determined before training. When the introduced precision is not warranted, imprecise probabilists advocate for learning along with imprecision. For instance, Walley s Imprecise Dirichlet Model effectively handles incomplete and missing data (Utkin et al., 2021). Dempster-Shafer Theory (Shafer, 1992) enables the fusion of multiple information sources, considering all available evidence. Credal learners, including credal decision trees (Abellan and Masegosa, 2010), credal networks (Cozman, 2000), and imprecise Bayesian neural networks (Caprio et al., 2023), propagate imprecision to pre- diction, resulting in models that capture the full range of possible outcomes. Central to these methods is the concept of a set of permissible solutions. This approach leads to indeterminate yet credible models, particularly in domains where uncertainty is prevalent. Our research aligns with this line of work, focusing on developing domain generalisation strategies that acknowledge and adapt to imprecision. By embracing imprecision, we aim to create models that offer a range of permissible solutions, empowering model operators to make informed choices at test time. The use of a credal set to model epistemic uncertainty has been concurrently explored by Caprio et al. (2024) to derive generalization bounds under credal uncertainty. 3. Imprecise Domain Generalisation In this work, we advocate for an imprecise learning, where learners do not commit to any particular P K(Z) at training time, but express their uncertainty through a credal set K(Z), where we discuss our choice in Section 3.2. We operationalise this idea in the context of domain generalisation (DG) problems. To this end, consider data coming from d distinct domains, each with its own distribution P1, . . . , Pd, and corresponding risk profiles (R1, . . . , Rd) R. The learner s objective is to select an optimal hypothesis from H considering both the risk profiles and K(Z), based on a certain optimality criterion defined below. While we mainly focus on multi-domain environments, this framework is also relevant and adaptable to single-domain scenarios (see Appendix C for further discussion). Credal set and partial preference. A crucial distinction between precise and imprecise learning lies in their approach to learner s preferences (Chau et al., 2022a;b). Precise learners commit to a specific distribution P K(Z) during training, creating a complete1 and transitive preference order based on empirical risk in H. That is, for any f, g H, f g if and only if b R(f) b R(g). Conversely, imprecise learning based on the credal set K(Z) results in a partial order over H (Giron and Rios, 1980; Walley, 1991): Lemma 3.1. The binary relation represented by K(Z) is such that for f, g H, f g, if and only if EP[Zf] EP[Zg] for every P K(Z). This leads to an incomplete preference ordering. Lemma 3.1 highlights the challenge of learning with imprecision, implying that unless the learners are willing to exert their judgement over the distributions in K(Z), as was previously done in precise learning, it is no longer possible to unanimously identify the best hypothesis in H from the observed data alone. In the following, we describe how the learners can implement imprecise learning at training time such that the operators can make prediction efficiently at test time. 1For every f, g H, either f g, g f, or both hold. Domain Generalisation via Imprecise Learning 3.1. Aggregation Functions and Optimality Criteria To facilitate learning, we need a certain notion of optimality taking into account K(Z). We formalise this by considering an aggregated learning algorithm P : R 7 h that takes a risk profile and returns a hypothesis h H. In particular, we focus on a specific type of aggregation function called an aggregated risk minimizers: P : R 7 arg min θ Θ ρλ[R](hθ), λ Λ, (3) where ρλ : Ld 2(H) L2(H) is a risk aggregation function indexed by λ Λ, which yields the non-negative realvalued statistical functional ρλ[R] : H R+. Here, we assume that our model class H is parametrized by a parameter space Θ Rp, e.g., a weight vector in a neural network. We call Λ a choice space which arises exclusively due to the imprecision of the learning problem (cf. Lemma 3.1) and serves merely as an index set. In practice, we only have access to the empirical risks ( b R1, . . . , b Rd) b R which we can substitute directly into (3). In Section 3.2, we consider Conditional Value-at-Risk (CVa R) as a concrete example of the risk aggregator ρλ. Our formulation (3) is not only pertinent to financial risk measures but has also gained traction for creating interpretable, risk-aware machine learning algorithms (Williamson and Menon, 2019). For each λ Λ, we denote the Bayes optimal models by h λ arg minθ Θ ρλ[R](hθ) and the associated parameter by θ λ. Unfortunately, for continuous choice space, it is unrealistic for the learner to find the Bayes optimal models in H simultaneously for all λ Λ. For this reason, we generalise the notion of Pareto optimality (Pareto, 1897) from multi-objective optimisation to its continuous counterpart and propose an alternative optimality criterion with respect to all λ Λ : C-Pareto optimality. Definition 3.2 (C-Pareto optimality). The hypothesis hθ dominates hθ , denoted by hθ hθ , if ρλ[R](hθ) ρλ[R](hθ ) for all λ Λ and ρ λ[R](hθ) < ρ λ[R](hθ ) for some λ Λ. Then, hθ is C-Pareto optimal if there exists no hθ such that hθ hθ. When λ takes values on a finite set Λ with m elements, i.e., Λ = {λ1, . . . , λm}, Definition 3.2 coincides with the Pareto optimality in standard multi-objective optimization (MOO); see, e.g., Sener and Koltun (2018); Lin et al. (2019); Zhang and Golovin (2020); Ma et al. (2020) and references therein. Chen et al. (2023b) have recently studies trade-offs between ERM and existing OOD objectives using MOO. It is not hard to show that, like introduced in Lemma 3.1, can be incomplete and that any Bayes optimal models h λ are also C-Pareto optimal. Intuitively, instead of obtaining the Bayes optimal model for all λ Λ, the learner can at best find the non-dominating models, i.e., the models upon which an improvement is only possible at a cost of deterioration of another non-dominating model. Next, we introduce the notion of C-Pareto stationary used to check if a model is C-Pareto optimal. Definition 3.3 (C-Pareto stationary). Suppose ρλ[R](hθ) is a smooth function of hθ and define the local gradient at h as vλ := ρλ[R](h ). The point h is called C-Pareto stationary if and only if there exists a probability density q such that R vλ dq(λ) = 0. 3.2. Conditional Value-at-Risk (CVa R) In theory, all aggregation functions ρλ[R] can be expressed as a type of weighted average of R, as detailed in Proposition B.1. A high level of generality could be achieved by formulating K(Z) as the convex hull of P1, . . . , Pd. This corresponds to treating the choice parameter λ Rd as all possible averaging weight, thus defining ρλ[R] = λ R. However, this approach has its serious drawbacks, since λ might be difficult for the operators to interpret, potentially leading to irrational decisions. For instance, operators may inappropriately assign more weight to domains that are easier to train, resulting in atypical risk-seeking behaviour. To select an appropriate aggregation function (equivalent to formulating an appropriate credal set) that is both interpretable and aligned with typical behaviour such as risk aversion, we opt for ρλ from the class of risk measures. This corresponds to formulating credal set as distributions that are mixtures of P1, . . . Pd with weights determined by the aggregation function. Notably, we choose the Conditional Value-at-Risk (CVa R): Definition 3.4 (Conditional Value-at-Risk (Rockafellar and Uryasev, 2002)). Let R = (R1, . . . , Rd) represent our risk profile, and FR(r) = 1 d Pd i=1 I[Ri r] as the cumulative distribution function (CDF) for R. Define rλ = minr{r | FR(r) λ} as the λ-level quantile. Then, the Conditional Value-at-Risk for R at level λ is given by: ηλI[Ri = rλ] + (1 ηλ)I[Ri rλ] Pd i=1I[Ri rλ] where ηλ = (FR(rλ) λ)(1 λ) 1, indicating the discontinuity level of the CDF at λ. CVa R effectively enables operators to express their level of risk aversion through λ, which in turn influences the selection of riskier domains for optimization. Additionally, this approach provides a means to transition smoothly between two prevalent notions of generalisation (Robey et al., 2022; Eastwood et al., 2022a; Li et al., 2023), namely optimising average risks (λ = 0) and worst-case risks (λ = 1). Furthermore, CVa R belongs to a class of coherent risk measures, which possess desirable properties (Artzner et al., 1999) and Domain Generalisation via Imprecise Learning have been studied in the robust optimisation literature; see, e.g., Ben-Tal et al. (2010). 3.3. Augmented Hypothesis To further institutionalize the separation of statistical decision-making, i.e., choosing appropriate notion of generalisation (performed by the operator) from statistical learning (performed by the learner), we propose to shift the problem view to an imprecise setting where the learner does not assume a priori which λ Λ is relevant to the operator, but instead designs a model that allows the operator to choose their own λ at deployment time.2 To this end, we extend the hypothesis space to an augmented hypothesis space HΛ of functions of both x and λ, and propose to learn an augmented hypothesis hξ : X Λ Y parametrized by a parameter ξ Ξ Rq. In contrast with hθ H which is fixed across λ Λ, an augmented hypothesis hξ HΛ describes a range of possible hypothesis hξ( , λ) for each λ Λ, such that the user can choose the one that best fits their needs. By abuse of notation, we consider ρλ[R]( hξ( , λ)) as a point-wise aggregated risk for the augmented hypothesis hξ HΛ. The subtle difference here is that we evaluate the objective at hξ( , λ) for the same λ used in ρλ. While the idea of augmented hypothesis with loss-conditional learning has previously been considered (Brault et al., 2019; Dosovitskiy and Djolonga, 2020), existing work still fall into the setting of precise learning, as we describe subsequently in Section 4. The function h : (x, λ) 7 h λ(x) that maps onto a Bayes optimal model for each λ Λ is an example of such augmented hypothesis. However, we may again prefer to consider a more amenable optimality criterion that seeks optimality jointly across Λ. To this end, we extend Definition 3.2 to an augmented hypothesis. Definition 3.5 (C-Pareto optimal augmented hypothesis). The augmented hypothesis hξ dominates hξ , denoted hξ hξ , if ρλ[R] hξ( , λ) ρλ[R] hξ ( , λ) for all λ Λ and ρ λ[R] hξ( , λ) < ρ λ[R] hξ ( , λ) for some λ Λ. Then hξ is C-Pareto optimal if there exists no hξ such that hξ hξ. We can again verify that a function h : (x, λ) 7 h λ(x) that maps onto a Bayes optimal model for each λ is in fact C-Pareto optimal. The following result shows that, under the assumption of existence of a Bayes optimal model, C-Pareto optimality is in fact equivalent to Bayes optimality. Proposition 3.6. Suppose there exists h HΛ such that h ( , λ) is Bayes optimal for all λ Λ. Then an augmented 2While we focus primarily on the learning aspect and assume throughout that the operator knows how to specify λ, we acknowledge the challenge of eliciting operators preferences at test time; see Appendix F.2 for further discussion on test-time elicitation. hypothesis g HΛ is C-Pareto optimal if and only if g ( , λ) is a Bayes optimal model for all λ Λ. Proposition 3.6 illustrates that all C-Pareto optimal augmented hypotheses can simultaneously learn all the Bayes optimal models. While this provides a strong guarantee, finding a C-Pareto optimal solution may still in practice be challenging and, when possible, one will prefer optimising against a scalar objective. Let (Λ) be the space of probability density functions over Λ. In our imprecise learning setting, a learner can scalarise the objective by choosing a distribution Q (Λ), and taking an expectation over all objectives. This substitutes the learning problem over all of Λ with the scalarised objective JQ( hξ) = Eλ Q ρλ[R]( hξ( , λ)) , (5) where the choice of distribution Q corresponds to a choice of scalarisation from the learner. The following proposition shows that all choices of Q lead to Bayes optimal models on their support. Proposition 3.7. Let Q (Λ). If g HΛ solves the scalarised optimisation problem, i.e., g arg ming HΛ JQ(g), then g ( , λ) is a Bayes optimal model for all λ Λ such that Q(λ) > 0. A similar result has previously been shown in Dosovitskiy and Djolonga (2020, Proposition 1) under the continuity and infinite model capacity assumptions. This result implies in particular that for any choice of distribution Q with full support, the scalarised objective can in theory yields a Bayes optimal model for every λ Λ. 4. Imprecise Risk Optimisation Unfortunately, Proposition 3.7 does not inform specific choices of Q for the learner, leaving them in a state of ignorance. Under this scenario, the most popular narrative in the literature is to leave the choice of Q to the operators or to adopt non-informative priors such as Jeffreys prior and uniform priors (Brault et al., 2019; Dosovitskiy and Djolonga, 2020). However, both approaches would defeat the purpose of this work as they render the learning problem precise again (see the discussions in Section 2). In particular, it has been argued that complete or partial ignorance cannot be fully represented by a single precise probability (Walley, 1991, Sec. 5.5). For example, uniform distribution is not an appropriate way of representing ignorance because it coincides with a precise judgement of uniform belief. C-Pareto improvement. To overcome this challenge, we adopt the concept of C-Pareto improvement which allows us to develope a learning algorithm that respects not only the limitation of evidence and resource, but also the complete ignorance of the learner. Specifically, we focus on the Domain Generalisation via Imprecise Learning gradient-based method: ξt ξt 1 η ξJQt( hξ), Qt (Λ). (6) We say that the update (6) makes a C-Pareto improvement if hξt dominates hξt 1 according to the aggregation ρλ[R]. The central concept involves the adaptive selection of Qt at each step, ensuring that the parameter update remains consistently non-dominant. This approach bears resemblance to the multiple-gradient descent algorithm (MGDA) utilised in multi-objective optimisation (D esid eri, 2012). The subsequent result demonstrates the specific selection of Qt that results in C-Pareto improvement. Theorem 4.1. For λ Λ, suppose ξ 7 ρλ[R]( hξ( , λ)) is locally continuously differentiable in a neighbourhood of ξ. Define Q t arg min Q (Λ) ξt 1JQ(ξt 1) 2 (7) and vt(ξt) = ξt JQ t (ξt). Then the update ξt ξt 1 η vt(ξt) for an appropriate choice of η > 0 always makes C-Pareto improvement. 4.1. Practical Algorithm with Theoretical Justification In practice, we have access to data from d distinct domains. The empirical risk for the augmented hypothesis hξ HΛ for the ith domain can be computed for each λ Λ as b Ri( hξ( , λ)) = 1 j=1 ℓ( hξ(x(i) j , λ), y(i) j ), (8) where (x(i) j , y(i) j ) Pi. In principle, the choice of λ determines how to aggregate the risk profile. However, in practice once λ is known, only then the corresponding hξ( , λ) HΛ can be used to compute the empirical risk profile. For a particular objective ρλ[ b R]( hξ( , λ)), we can compute the corresponding empirical risk profile as b R( hξ( , λ)) = { b Ri( hξ( , λ)), . . . , b Rd( hξ( , λ))}. If Q is known to the learner, they can sample {λj}m j=1 Q and compute the corresponding empirical risk profiles for each λj. However, for an imprecise learner, the right choice of distribution Q is unknown a priori. Therefore, we defer the computation of the empirical risk profile until the corresponding λ is known. That is, given a candidate distribution Q (Λ) we compute the risk profile and aggregate it with {λj}m j=1 Q. We can then estimate Q t with b Qt using Monte Carlo estimate of (7), i.e., b Qt = arg min Q (Λ) j=1 ρλj[ b R]( hξ( , λj)) where {λj}m j=1 Q. The direction of C-Pareto improvement is obtained by ˆvt(ξt) = ξt J b Qt(ξt). Algorithm 1 summarises the proposed algorithm. Algorithm 1 Imprecise Risk Optimisation (IRO) 1: Input: Data from d distinct domains {x(d) i , y(d) i }n i=1 Pd(X, Y ), a loss function ℓ: Y Y R+, a probability space (Λ), a (augmented) hypothesis class HΛ, risk aggregator ρλ : Ld(H) L(H), number of Monte Carlo samples m. 2: Initialise the parameter ξ Ξ. 3: repeat 4: Estimate Q t with b Qt by solving (9) by computing b Qt = arg min Q (Λ) 1 m Pm j=1 ρλj[ b R]( hξ( , λj)) 2 where λ1, . . . , λm Q. 5: Compute ˆvt(ξ) = 1 m Pm k=1 ρλk[ b R]( hξ( , λk)) where λ1, . . . , λm b Qt. 6: Update ξ = ξ ηˆvt(ξ). 7: until ˆvt(ξ) 2 > ϵ Proposition 4.2. Let Q (Λ) and let λop Λ such that Q(λop) > 0. Assume that ρλ is a linear, idempotent aggregation operator and that the loss ℓis upper bounded by M 0. Let n 1 be the number of samples we observe from each environment, assumed equal across environments. Then, there exists q (0, 1) such that if ˆg arg min g HΛ i=1 ρλi[ b R]( g( , λi)) (10) where λ1, . . . , λm Q, then for any δ > qm, the following inequality holds with probability 1 δ: ρλop[R](ˆg( , λop)) ρλop[R](h ( , λop)) log(6/ηδ) 2m(1 q)(1 qm) where ηδ = (δ qm)/(1 qm). This proposition shows that even when the learner does not know the operator s true preference λop, the operator excess risk on the solution of the empirical scalarised IRO problem ˆg is bounded with high probability in O(n 1/2 + m 1/2), provided Q has full support. This means in particular that, provided an unlimited budget on the number of samples (the λis) that can be drawn from Q, the operator excess risk has a bound that matches standard learning rates for ERM. The constant q (0, 1) depends on the choice of distribution Q and the operator s true preference λop. If Q has a high density around λop, then q can be chosen closer to zero. Conversely, if Q has a lower density around λop, the values of q will be closer to one, requiring a larger number of samples λ1, . . . , λm to achieve a comparable bound. Domain Generalisation via Imprecise Learning 0.0 0.2 0.4 0.6 0.8 1.0 λop CVa Rλop(R) PL-f (λlr = 0) PL-f (λlr = 1) PL-f (λlr U(0, 1)) PL-f (λlr = λop) (a) (Synthetic data) Comparing IL with PL-f trained using different λlr. 0.0 0.2 0.4 0.6 0.8 1.0 λop CVa Rλop(R) PL-h(λlr Beta(5, 5)) PL-h(λlr Beta(5, 1)) PL-f (λlr = λop) (b) (Synthetic data) Comparing IL with PLh trained using different priors over λlr. 0.0 0.2 0.4 0.6 0.8 1.0 λop CVa Rλop(R) PL-f (λlr = λop) PL-f (λlr = 0) PL-f (λlr = 1) (c) (UCI Bike Rentals) Comparing IL with various PLh and PL-f. Figure 2: Experiments comparing imprecise learning (IL) with various precise learners with precise hypothesis (PL-f) and with augmented hypothesis (PLh). 1 standard deviation is included and experiments are repeated 5 times. 5. Experiments Our framework features a learner, who trains the model, and an operator, who employs it, with their preferences denoted as λlr and λop. Due to the institutional separation, the operator s preferred generalisation strategy cannot be communicated to the learner. We assess our Imprecise Learning (IL) framework, allowing learners to train an augmented hypothesis h using our IRO algorithm (see Algorithm 1), enabling operators to provide λop at deployment. This contrasts with Precise Learners (PL-f) who commit to a fixed generalisation (λlr) during training, producing a precise hypothesis f : X Y, and PLh, who create an augmented hypothesis but with a pre-determined prior over λlr. We evaluate using the objective ρλop[R], comparing IL, PL-f with fixed (0 or 1) or uniform λlr, and PLh with prior of λlr as Beta distributions (5,5), (5,1), and (1,1). The strategy aligning with Beta(1, 1) corresponds to the approach in Brault et al. (2019), thus is termed INF-TASKh. We benchmark against an ideal scenario where λlr equals λop and also calculate the maximum regret, i.e., for any model h (or f), max-regret( h) supλop Λ ρλop[R]( h) ρλop[R]( h λop), to gauge the models deviation from optimality across all λop. Synthetic data: Following Eastwood et al. (2022b), we construct a simulated experiment to compare learners. We consider a linear model for each domain d: Yd = θd X + ϵ with X N(1, 0.5) and ϵ N(0, 0.1). We simulate different domains by drawing θd with probability p = 0.5 from Uniform distributions U(1,1.1) and U( 1.1, 1). This allows data to exhibit multi-modality, thus creating a discontinuous risk profile which becomes harder for a single augmented hypothesis to capture. We consider 250 train and 250 test domains with 100 samples from each domain. CMNIST dataset. We also experiment on the CMNIST dataset (Arjovski, 2021), which is a modified version of the MNIST dataset. The task is to classify digits {0, 1, 2, 3, 4} and {5, 6, 7, 8, 9} into negative and positive classes, respectively. A color is introduced as an additional domainspecific predictive feature that varies across domains, e.g., Table 1: Reporting the maximum regret averaged over 5 repetitions for each experiment with one standard error included. Top: Comparing IL with PL-f (Synthetic). Middle: Comparing IL with PLh (Synthetic). Bottom: Comparing IL with PL-f and PLh (Bike Rentals). PL-f (U(0, 1)) PL-f (λlr = 0) PL-f (λlr = 1) IL (ours) 1.971 (0.0098) 6.177 (0.0617) 2.010 (0.0564) 0.867 (0.0058) PLh (Beta(5,5)) PLh (Beta(5,1)) INF-TASKh IL (ours) 1.79 (0.12) 1.57 (0.03) 0.935 (0.04) 0.56 (0.00) PL-f(λlr = 0) PL-f(λlr = 1) INF-TASKh IL (ours) 4.81 0.27 0.66 0.01 0.72 0.13 0.42 0.08 P(Y = 1 | color = red) = 0.9 for domains in which the true label is highly correlated with the color feature. As a result, the mechanism by which color influences the label changes across domains, but the shape has a stable mechanism across domains (see Figure 4a). We sample 10 training environments from a long-tailed Beta(0.9, 1) distribution, resulting in over-represented (majority) and under-represented (minority) subgroups (see Figure 4b). Note that we do not make the IID assumption over environments since we evaluate all subgroups at test time. We further discuss the dataset and experiment setup in Appendix E. Real-world data: Following Rothenh ausler et al. (2021) and Subbaswamy et al. (2019), we use the UCI Bike Sharing dataset (Fanaee-T and Gama, 2014) to predict the number of hourly bike rentals R from various weather-related features. Here, R is transformed from count to continuous with normalization. The data contains 17, 379 observations with temporal information such as season and year. The data is partitioned by season (1-4) and year (1-2) to create 8 different domains. Domains from the first year are used for training and the subsequent year as test domains. 5.1. Insights from Experiments Comparing IL with PL-f. Our initial experiment on synthetic data contrasts Imprecise Learning (IL) with Precise Learners (PL-f) across different λlr settings, including Domain Generalisation via Imprecise Learning average-case (λlr = 0) and worst-case (λlr = 1) scenarios. Results, shown in Figure 2a, indicate that PL-f models achieve the lowest aggregated risk compared to other learners when λlr = λop. However, when λlr = λop, PL-f then deviates from this ideal scenario, which is expected since PL-f models are finely tuned to their specific λlr. Conversely, IL achieves aggregated risks that remain close to the ideal scenario across the spectrum of λop, matching or exceeding the worst-case PL-f in risk-averse settings (λop > 0.6) and surpassing both average-case and worstcase PL-f as λop increases. Notably, IL achieved the lowest maximum regret (see Table 1), underscoring the advantage of imprecise learning in handling generalization uncertainty. Comparing IL with PLh. In our second experiment using synthetic data, we evaluate IL s augmented hypothesis trained using imprecise risk optimisation (IRO) against precise learners (PLh) employing various optimization strategies influenced by their subjective beliefs about λop. Results in Figure 2b indicate IL s performance is close to the ideal baseline across most λop values, except at higher risk levels where INF-TASK and PLh trained under Beta(5,1) excel. This outcome aligns with expectations, as INF-TASK uniformly aggregates objectives, favoring higher-risk scenarios, similar to Beta(5,1) s weighting towards higher λ. Despite this, IL outperforms these methods across other λ and achieves the lowest maximum regret (see Table 1), demonstrating the efficacy of the proposed method. Comparing DG algorithms on CMNIST. In Table 2, we compare IL to other DG methods on three representative domains from minority and majority subgroups (see Figure 4). The domains e {0.0, 1.0} demonstrate opposite mechanisms, i.e., in domain e = 0.0, the color red is fully predictive of the negative class, whereas for e = 1.0, it is fully predictive of the positive class. In domain e = 0.5, color is uncorrelated with the target. We can see that IL can learn relevant features in context with appropriate λ and generalises in all scenarios. By setting λ = 0, the model operator can be less risk averse and generalise better to domains from the majority subgroup, as noted in the performance of IL for e = 0.0. With λ 1, the model operator can be risk averse and generalise better to the minority subgroup and is also reflected in the performance of IL for e {0.5, 1.0}. Furthermore, with λ 1, it performs similarly to the invariant learners. We discuss the results on all test domains in Table 3 in Appendix E. Real-world experiment. Figure 2c demonstrates similar comparisons between IL and various PL-f and PLh models as in previous experiments. Notably, IL surpassed the ideal scenario at higher risk levels. This can be attributed to the fact that CVAR as an objective discards data from lower-risk environments (see Section 3.2), thus the optimisation has lower statistical efficiency as risk level increases. Table 2: Accuracy and maximal regret of different domain generalisation algorithms on the CMNIST test environments from P(Y = 1 | color = red) = e with e {0.0, 0.5, 1.0}, respectively. The hypothetical best invariant and Bayes classifier are listed in bold. Domain-wise best acc & regret are highlighted in green. Bayes classifier is defined w.r.t. the IID learner trained for a particular environment Objective Algorithm e = 0.0 e = 0.5 e = 1.0 Regret Average ERM 96.1 59.2 28.3 72.7 Worst Grp DRO 54.1 64.5 75.5 46.9 SD 52.1 63.7 73.3 47.9 IGA 71.8 65.2 50.3 49.7 IRM 72.0 69.7 67.7 32.3 VREx 72.7 69.5 68.5 31.5 EQRM 67.8 69.1 72.1 32.2 Oracle 73.5 27.9 PLh Inf-Task 96.0 63.1 68.3 31.7 IL (Ours) IRO 95.8 69.5 70.3 29.7 Bayes ERM (IID) 100.0 75.0 100.0 Augmented hypothesis mitigates this downside because it is smooth in the λ parameter by design, thus can borrow information from nearby risk regions. At last, IL consistently achieved the lowest maximum regret as shown in Table 1. 6. Conclusion In out-of-distribution (OOD) generalisation, a clear institutional separation between machine learners and model operators creates generalisation uncertainty that prevents consensus on a specific generalisation approach during training. To overcome this, we presented imprecise domain generalisation. Our approach incorporates imprecise risk optimisation, allowing learners to maintain imprecision during training, coupled with a model framework that lets operators specify their generalisation strategy at deployment. Both theoretical analysis and experimental evaluations demonstrate the effectiveness of our proposed framework. Our work faces two main limitations. First, it assumes that model operators are aware of their level of risk aversion. In practice, they may however struggle to precisely articulate their preferences. Consequently, this necessitates preference elicitation at test time, which may result in a probability distribution over λ rather than a single value. Second, imprecise learning is more computationally intensive compared to precise counterparts as it involves optimising for a continuum of objectives. In our future work, we aim to broaden the scope of imprecise learning by implementing methods to elicit user preferences more effectively, improving computational efficiency, and exploring alternative aggregation functions. This approach would empower operators to weigh various criteria such as fairness, privacy, and algorithmic performance effectively. Domain Generalisation via Imprecise Learning Impact Statement This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here. Acknowledgement We thank Kevin Murphy, Chris Holmes, Uri Shalit, Emtiyaz Khan, Hugo Monz on, Shai Ben-David, and Amartya Sanyal for fruitful discussion, and anonymous reviewers for their insightful feedback. 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IEEE Transactions on Neural Networks and Learning Systems, 33(2): 473 493, 2022. K. Zhou, Z. Liu, Y. Qiao, T. Xiang, and C. C. Loy. Domain generalization: A survey. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(4):4396 4415, apr 2023. Domain Generalisation via Imprecise Learning This section provides the detailed proofs of our main results presented in the paper. A.1. Proof of Proposition 3.6 We can easily verify that if g ( , λ) is Bayes optimal for all λ Λ, then it is C-Pareto optimal. Suppose g HΛ is C-Pareto optimal. We have g C-Pareto optimal h HΛ, h g h HΛ, (h g ) h HΛ, λ Λ, ρλ[R](h( , λ)) > ρλ[R](g ( , λ)) λ Λ, ρ λ[R](h( , λ)) ρ λ[R](g ( , λ)) . This implies in particular that λ Λ, ρλ[R](h ( , λ)) > ρλ[R](g ( , λ)) λ Λ, ρ λ[R](h ( , λ)) ρ λ[R](g ( , λ)) . Since h ( , λ) is Bayes optimal for all λ Λ, the first statement cannot be true. Therefore, the second statement must hold and we have g C-Pareto optimal λ Λ, ρ λ[R](h ( , λ)) ρ λ[R](g ( , λ)) λ Λ, ρ λ[R](h ( , λ)) = ρ λ[R](g ( , λ)) (h ( , λ) Bayes optimal) λ Λ, g ( , λ) Bayes optimal . This concludes the proof. A.2. Proof of Proposition 3.7 Proof. Since h ( , λ) is a Bayes optimal model for all λ Λ, we have ρλ[R](g ( , λ)) ρλ[R](h ( , λ)) 0, λ Λ Eλ Q[ρλ[R](g ( , λ)) ρλ[R](h ( , λ))] 0. But by definition of g we also have JQ(g ) JQ(h ) Eλ Q[ρλ[R](g ( , λ)) ρλ[R](h ( , λ))] 0. Therefore, Z ρλ[R](g ( , λ)) ρλ[R](h ( , λ)) Q(λ) dλ = 0. Since the integrand is positive, it implies that for all λ Λ such that Q(λ) > 0, ρλ[R](g ( , λ)) = ρλ[R](h ( , λ)) which concludes the proof. A.3. Proof of Theorem 4.1 and C-Pareto Improvement When the choice of scalarisation, i.e., Q improves some objectives at the cost of degrading other objectives, it induces a preference. Therefore, the problem becomes multi-objective again as there will be a trade-off among these objectives. The imprecise choice of scalarization will be the distribution Q such that it improves at least one of the objectives without degrading any other objective, i.e., it ensures C-Pareto improvement. Formally, Domain Generalisation via Imprecise Learning Proposition A.1. Suppose a learning algorithm P learns an augmented hypothesis g HΛ using aggregated objective JQ(g) (ref. Eq 5). Then, the learner does not induce an additional preference over HΛ if it makes Pareto improvement. Proof. Consider a learner P which learns an augmented hypothesis g HΛ which aggregates the objectives ρλ[R](g( , λ)) for all λ Λ with respect to Q to obtain the aggregated objective Eλ Q[ρλ[R](g( , λ))]. Assume that G = {gi}n i=0 denotes the sequence of models that the learner obtains at every update while minimizing the aggregated objective. We know that i, j such that i < j and gi, gj G Eλ Q[ρλ[R](gi( , λ))] > Eλ Q[ρλ[R](gj( , λ))] which defines a preference on G HΛ with aggregated objective as the utility function u Q(g) := Eλ Q[ρλ[R](g( , λ))]. This additional preference relation agrees with the original binary preference relation ( ) on HΛ which defines dominance and C-Pareto optimality if there does not exist gi, gj G such that i < j, gi gj and gj gi with u Q(gi) > Q(gj). This implies that aggregated objective u Q must be such that for all gi, gj G and i < j, gj gi. That is, u Q should make C-Pareto improvement to not induce any additional preference. Therefore, we propose an alternate characterization of C-Pareto optimality based on the concept of C-Pareto improvement with the idea of local gradients. Proposition A.2. An augmented hypothesis hξ is C-Pareto optimal if and only if there exists no w Ξ such that for an ϵ > 0, ρλ[R]( hξ ϵw( , λ)) ρλ[R]( hξ( , λ)) for all λ Λ and ρ λ[R]( hξ ϵw( , λ)) < ρ λ[R]( hξ( , λ)) for some λ Λ. Proof. We prove the forward direction using contradiction. Assume h is C-Pareto optimal and there exists w Ξ such that for an ϵ > 0, ρλ[R](hξ ϵw( , λ)) ρλ[R](hξ( , λ)) for all λ Λ and ρ λ[R](hξ ϵw( , λ)) < ρ λ[R](hξ( , λ)) for some λ Λ. Then hξ ϵw strictly dominates hξ according to our definition of C-Pareto optimality for augmented hypothesis Def ?? which contradicts that hξ is C-Pareto optimal. We prove the reverse direction using the contraposition. Assume hξ is not C-Pareto optimal. Then there exists hξ that strictly dominates hξ, i.e., ρλ[R](hξ ( , λ)) ρλ[R](hξ( , λ)) for all λ Λ and ρ λ[R](hξ ( , λ)) < ρ λ[R](hξ( , λ)) for some λ Λ. Then there exists w = ξ ξ and ϵ = 1 such that hξ ϵw strictly dominates hξ. Proposition A.2 shows that for C-Pareto optimality there must not be any direction w Ξ for Pareto improvement. The non-existence of a direction for Pareto improvement is an if and only-if condition for C-Pareto optimality. Proposition A.3. In an ϵ-neighbourhood of ξ let ρξ[R](hξ( , λ)) be a smooth function of ξ and the local gradient is defined as vλ(hξ) := ξρλ[R](hξ( , λ)). If hξ is not pareto optimal then there exists a local pareto improvement direction w Ξ such that for all λ Λ w vλ(hξ) 0 and for some λ Λ w v λ(hξ) > 0. Proof. From Proposition A.2, when hξ is not Pareto optimal, there exists a w Ξ such that for an ϵ > 0, hξ ϵw hξ. Then for all λ Λ, ρλ[R](hξ ϵw) ρλ[R](hξ) ρλ[R](hξ) ϵw vλ(hξ) + ϵ2R ρλ[R](hξ) ϵw vλ(hξ) + ϵ2R 0 (R : Remainder Higher order terms) ϵR w vλ(hξ) Since ρλ[R](hξ ϵw) ρλ[R](hξ), then w vλ(hξ) 0 as ϵ 0 otherwise a contradiction would arise for sufficiently small ϵ. Similarly for an λ Λ, since ρ λ[R](hξ ϵw) < ρ λ[R](hξ), then w v λ(hξ) > 0. Proposition A.3 extends the argument from Proposition A.2 that when hξ is not C-Pareto optimal, a direction for Pareto improvement must exist. Remark explains that a local Pareto improvement direction must align with the gradient of all objectives. Since the direction opposite to the local gradient of an objective shows us the direction of the improvement for the objective, then the direction opposite to local Pareto improvement w Ξ must align with the local gradient if w Ξ improves the corresponding objective. Note that the local gradient of aggregated objective (5) is ξJQ(ξ) := Eλ Q[vλ(hξ)] (12) Domain Generalisation via Imprecise Learning Where vλ(hξ) := ξρλ[R](hξ( , λ)) denotes the local gradient of ρλ[R](hξ( , λ)). Then the choice of Q such that ξJQ(ξ) is the the direction of local Pareto improvement is given by Proposition A.4. For λ Λ, suppose ξ 7 ρλ[R]( hξ( , λ)) is locally continuously differentiable in a neighbourhood of ξ. Define Q t arg min Q (Λ) ξt 1JQ(ξt 1) 2 (13) and vt(ξt) = ξt JQ t (ξt). Then the update ξt ξt 1 η vt(ξt) for an appropriate choice of η > 0 always makes C-Pareto improvement. i.e., vt(ξt) for all objectives ρλ[R](hβ( , λ)), λ Λ such that vt(ξt)T vλ(hξ) ||vt(ξt)||2 2. Proof. We start by assuming that a given Q t exists then the update ξt ξt 1 η vt(ξt) performs local C-Pareto improvement. First we show that λ Λ the vt(ξt)T vλ(hξ) ||vt(ξt)||2 2. For any distribution Q (Λ), v = Eλ Q[vλ(hξ)] vt(ξt). We can say that ϵ [0, 1] vt(ξt) + ϵv is essentially vt(ξt) + ϵv = vt(ξt) + ϵ(Eλ Q[vλ(hξ)] vt(ξt)) = (1 ϵ)vt(ξt) + ϵEλ Q[vλ(hξ)] = Eλ ϵQ+(1 ϵ)Q t [vλ(hξ)] Where ϵQ + (1 ϵ)Q t is some other valid probability distribution. Therefore the norm of vt(ξt) + ϵv must be larger than or equal to the minimum norm obtained from Equation (13). (vt(ξt) + ϵv)T (vt(ξt) + ϵv) vt(ξt)T vt(ξt) 2ϵvt(ξt)T v + ϵ2v T v 0 ϵ 2vt(ξt)T v Since the above statement must be true for all ϵ (0, 1]. For ϵ = 0 equality must hold that vt(ξt)T vt(ξt) = vt(ξt)T vt(ξt). Therefore, the lower bound from above must be less than or equal to 0. vt(ξt)T v 0 Replacing v by Eλ Q[vλ(hξ)] vt(ξt) then gives us that vt(ξt)T (Eλ Q[vλ(hξ)] vt(ξt)) 0 vt(ξt)T Eλ Q[vλ(hξ)] vt(ξt)T vt(ξt) Thus we obtain that λ Λ the vλ(hξ)T vt(ξt) ||vt(ξt)||2 2 by setting Q to be dirac delta function at λ. Therefore from Proposition A.3 we can say that hξt 1 ηvt(ξt) hξt 1. This makes w Ξ the common direction for local C-Pareto improvement. Analogous to the definition 3.3 we define C-Pareto stationarity for augmented hypothesis as Definition A.5. Let ρλ[R]( h( , λ)) be a smooth function of augmented hypothesis h and vλ( hξ) := ρλ[R]( hξ( , λ)) be the local gradient then the augmented hypothesis is said to be C-Pareto Stationary if and only if there exists a probability density q such that R vλ( hξ) dq(λ) = 0. Intuitively, C-Pareto Stationarity corresponds to local C-Pareto Optimality. For a single objective, C-Pareto stationarity is equivalent to the first-order derivative being zero. Therefore, If an augmented hypothesis h is C-Pareto optimal, it is C-Pareto stationary. This means that C-Pareto stationarity is a necessary condition for C-Pareto optimality. From Proposition A.2 we know that for a C-Pareto optimal point, no direction for Pareto improvement must exist, which implies that no direction for local Pareto improvement must also not exist. From theorem 4.1 we know that a local direction for pareto improvement is vt(ξt) = R vλ(hξ)d Q t (λ) where Q t = arg min Q (Λ) ||Eλ Q[vλ(hξ)]||. Given that no direction for local C-Pareto improvement must exist implies that vt(ξt) = 0. This means that there exists a distribution Q such that R vλ(hξ)d Q(λ) = 0. This illustrates that C-Pareto stationarity is a necessary condition for C-Pareto optimality which intuitively illustrates that local C-Pareto optimality is necessary for C-Pareto optimality. Domain Generalisation via Imprecise Learning A.4. Proof of Proposition 4.2 A.4.1. USEFUL RESULTS Proposition A.6. Let X be a random variable taking values in X and let f : X R+ and g : X R+ be non-negative functions. Define Z = (f(X), g(X)) and suppose that it admits a continuous density p Z with respect to the Lebesgue measure on R2. Let α, β > 0 such that p Z(α, β) > 0 and let Z1, . . . , Zn be independent copies of Z. Then there exists r 1 and a random subsampling operator π such that π([n]) 2{1,...,n}, |π([n])| Binomial(n, 1/r), and for any index i π([n]) E[Zi] = (α, β), (14) where the expectation is taken against both the variable and the index. Proof. The proof consists in showing that the assumptions made are sufficient to construct a rejection sampling procedure where the proposal density is the density of Z and the target density is a uniform centered over (α, β). Since p Z(α, β) > 0 and p Z is continuous, there exists an open neighbourhood of (α, β) where p Z is strictly positive. Therefore, there exists η > 0 such that if we define the closed rectangle Aα = [α η/2, α + η/2] Aβ = [β η/2, β + η/2] then p Z(x, x ) > 0 for any (x, x ) A and admits a positive lower bound on A. Further, we can define the uniform random variable U Uniform(A) with probability density p U(x, x ) = 1 η2 , (x, x ) A. Then, by upper boundedness of p U over A and lower-boundedness of p Z over A, there exists r 1 such that for any (x, x ) A, p U(x, x ) p Z(x, x ) r. As a result, we can formally construct a rejection sampling procedure to sample from U using samples from Z with acceptance rate 1/r. It is important to note this is only a formal construction to show the existence of an appropriate subsampling procedure. In practice, we may not be able to evaluate p Z and therefore may be unable to effectively implement the procedure. Algorithm 2 Algorithmic definition of the random subsampling operator π 1: Input: p U, p Z, r, Z1, . . . , Zn 2: Initialise subsampled = {} 3: for i {1, . . . , n} do 4: Let Ui Uniform([0, 1]) 5: if Ui p U(Zi)/rp Z(Zi) then 6: Append i to subsampled 7: end if 8: end for 9: Return subsampled Algorithm 2 outlines an algorithmic definition of a random subsampling operator π : 2[n] 2[n] based on rejection sampling. We emphasise the random nature of the operator π as Z1, . . . , Zn are treated throughout as random variables. By property of rejection sampling, the number of accepted samples |π([n])| or |subsampled| follows a Binomial distribution with n trials and probability of success 1/r. Finally, we have by construction that for any i π([n]) E[Zi] = E[Uniform(A)] = (α, β) which concludes the proof. Domain Generalisation via Imprecise Learning A.4.2. PROOF OF THE MAIN RESULT We begin by introducing notations which will be used in this proof. Suppose we observe n N of IID observations from each environment, i.e., we observe (x(i) 1 , y(i) 1 ), . . . , (x(i) n , y(i) n ) Pi for every i {1, . . . , d}. Furthermore, let Q (Λ) be the scalarisation density the learner chooses and let λ1, . . . , λm Q be independent samples from this distribution. For each environment i {1, . . . , d}, we define an empirical risk j=1 ℓ(y(i) j , f(x(i) j )), f H, which we concatenate into an empirical risk profile ˆR = ( ˆR1, . . . , ˆRd). We can easily verify that for any i {1, . . . , d}, E[ ˆRi] = Ri where the expectation is taken against Pi, thus E[ ˆR] = R. Therefore, if we take the empirical aggregated risk to be ρλ[ ˆR] for λ Λ and assume that ρλ : Ld 2(H) L2(H) is a linear risk aggregation function, it follows that E h ρλ[ ˆR] i = ρλ h E[ ˆR] i = ρλ[R]. Finally, define the empirical scalarised risk using the values λ1, . . . , λm sampled above, for g HΛ as i=1 ρλi[ ˆR](g( , λi)). In what follows, we will always assume there exists a function ˆh HΛ such that for any λ Λ, ˆh( , λ) is a minimiser of the empirical aggregated risk ρλ[ ˆR], i.e., ˆh( , λ) arg min f H ρλ[ ˆR](f) , λ Λ, and that the empirical scalarised risk also admits a minimiser which we denote ˆg HΛ, i.e., ˆg arg min g HΛ ˆJQ(g). The following lemma shows that when such minimisers exists, then ˆg( , λi) is automatically a minimiser of the empirical aggregated risk ρλi[ ˆR]. Lemma A.7. Suppose there exists ˆh, ˆg defined as above. Then ˆg( , λi) minimises ρλi[ ˆR] for all i {1, . . . , m}. Proof. Let ˆh HΛ such that ˆh( , λ) arg min f H ρλ[ ˆR](f) for any λ Λ. Then, we have ρλ[ ˆR](ˆg( , λ)) ρλ[ ˆR](ˆh( , λ)) , λ Λ ρλi[ ˆR](ˆg( , λi)) ρλi[ ˆR](ˆh( , λi)) , i {1, . . . , m} ˆJQ(ˆg) ˆJQ(ˆh) ˆJQ(ˆg) = ˆJQ(ˆh) (ˆg arg min ˆJQ) i=1 ρλi[ ˆR](ˆg( , λi)) ρλi[ ˆR](ˆh( , λi)) = 0 ρλi[ ˆR](ˆg( , λi)) = ρλi[ ˆR](ˆh( , λi)) , i {1, . . . , m} (sum of positives) ˆg( , λi) arg min f H ρλi[ ˆR](f) , i {1, . . . , m}. This concludes the proof. Domain Generalisation via Imprecise Learning Finally, before we turn to the main result, recall that we assume there exists h HΛ such that h ( , λ) H is a Bayes optimal model for any λ Λ, i.e., h ( , λ) arg min f H ρλ[R](f). For any λ Λ, we denote the resulting Bayes risk as ρλ[R] = ρλ[R](h ( , λ)). Let λop Λ be the choice of λ which reflects the operator s preference, but is unknown to the learner. The following result provides a bound on the excess risk at λop when using ˆg as a hypothesis. Proposition A.8. Let Q (Λ) and let λop Λ such that Q(λop) > 0. Suppose that ρλ is a linear, idempotent aggregation operator and that the loss ℓis upper bounded by M 0. Then there exists q (0, 1) such that for any δ > qm, the following inequality holds with probability 1 δ: ρλop[R](ˆg( , λop)) ρλop[R] 2M log(2/ηδ) 2m(1 q)(1 qm) where ηδ = (δ qm)/(1 qm). Proof. The proof consists in (1) constructing a subsequence from λ1, . . . , λm such that the empirical scalarised risks converge to appropriate limits, (2) using these subsequences to apply concentration inequalities to the excess risk when the subsequence exists and (3) combining the results together in the general case. (1) Constructing an appropriate subsampling procedure Let λ be a random variable with probability density function Q over Λ. It induces a real-valued distribution over the risks ρλ[ ˆR](ˆg( , λ)) and ρλ[ ˆR](ˆh( , λ)). We assume that ρλ[ ˆR](ˆg( , λ)), ρλ[ ˆR](ˆh( , λ)) admits a continuous density with respect to the Lebesgue measure in R2 we denote p. Further, define αop = ρλop[ ˆR](ˆg( , λop)) βop = ρλop[ ˆR](ˆh( , λop)). Since Q(λop) > 0, we have p(αop, βop) > 0. Then by Proposition A.6, given λ1, . . . , λm Q(λ) IID, there exists r 1 and a random subsampling π([m]) 2[m] such that for any index i π([m]) we have E h ρλi[ ˆR](ˆg( , λi)), ρλi[ ˆR](ˆh( , λi)) i = (αop, βop). In particular, let p = |π([m])| Binomial(m, 1/r) denote the number of subsampled elements and assume without loss of generality these are the first p ones. Then, conditionally on p 1, we have that 1 p Pp i=1 ρλi[ ˆR](ˆg( , λi)) and 1 p Pp i=1 ρλi[ ˆR](ˆh( , λi)) are respectively unbiased estimators of αop = ρλop[ ˆR](ˆg( , λop)) and βop = ρλop[ ˆR](ˆh( , λop)). (2.1) Bounding the regret when p 1 is fixed Domain Generalisation via Imprecise Learning Suppose we are in a fixed setting where p 1. Then we can decompose and upper bound the regret following ρλop[R](ˆg( , λop)) ρλop[R] = ρλop[R](ˆg( , λop)) ρλop[ ˆR](ˆg( , λop)) + ρλop[ ˆR](ˆg( , λop)) 1 i=1 ρλi[ ˆR](ˆg( , λi)) i=1 ρλi[ ˆR](ˆg( , λi)) ρλop[ ˆR](ˆh( , λop)) + ρλop[ ˆR](ˆh( , λop)) ρλop[R] ρλop[R](ˆg( , λop)) ρλop[ ˆR](ˆg( , λop)) + ρλop[ ˆR](ˆg( , λop)) 1 i=1 ρλi[ ˆR](ˆg( , λi)) i=1 ρλi[ ˆR](ˆh( , λi)) ρλop[ ˆR](ˆh( , λop)) (Lemma A.7) + ρλop[ ˆR](h ( , λop)) ρλop[R] ˆh( , λop) arg min ρλop[ ˆR] ρλop[R](f) ρλop[ ˆR](f) ρλop[ ˆR](ˆg( , λop)) 1 i=1 ρλi[ ˆR](ˆg( , λi)) i=1 ρλi[ ˆR](ˆh( , λi)) ρλop[ ˆR](ˆh( , λop)) Let η (0, 1) fixed. By linearity of ρλ, we have shown that ρλ[ ˆR](f) is an unbiased estimator of ρλ[R](f). Therefore, Mc Diarmid s inequality gives us that we have with probability at least 1 η/3 ρλop[R](f) ρλop[ ˆR](f) M If we denote Zλi = ρλi[ ˆR](ˆg( , λi)) for the p accepted samples from the rejection sampling procedure, then we have by construction that 1 p Pp i=1 Zλi is an unbiased estimator of ρλop[ ˆR](ˆg( , λop)). Therefore, we can also apply Mc Diarmid s inequality to obtain that with probability at least 1 η/3 ρλop[ ˆR](ˆg( , λop)) 1 i=1 ρλi[ ˆR](ˆg( , λi)) Applying the same reasoning to the last line and combining the bounds together using the union bound we get that with probability at least 1 η ρλop[R](ˆg( , λop)) ρλop[R] 2M (2.2) Integrating the upper bound against p given p 1 Domain Generalisation via Imprecise Learning We now consider the random setting, conditional on p 1. Recall that the number of accepted samples from the rejection sampling procedure p follows a Binomial(n, 1/r) distribution. We want to take the expectation of the established probabilistic upper bound with respect to p given that p 1. Let q = 1 1/r denote the rejection rate, then we have for any k 1 P(p = k | p 1) = 1 1 qm (1 q)kqm k. This corresponds to a positive Bernoulli distribution (Grab and Savage, 1954), and in particular if we denote Bm,r(k) = P(Bernoulli(m, 1/r) k), we have from (Grab and Savage, 1954) Eq. (12) that p | p 1 1 (m + 1)(1 q)(1 qm) [1 Bm+1,r(1)] + 3 (1 q)(m + 2) [1 Bm+2,r(2)] 1 m(1 q)(1 qm). Therefore, it follows that log(6/η)1/p log(6/η)E[1/p | p 1] log(6/η) 2m(1 q)(1 qm), and by applying this to the probabilistic upper bound on the excess risk we have obtained earlier, we get that with probability at least 1 η ρλop[R](ˆg( , λop)) ρλop[R] 2M log(6/η) 2m(1 q)(1 qm). (3) Combining things together Now that we have established a probabilistic upper-bound on the excess risk when at least one sample is accepted by π, we set out to obtain a general probabilistic bound on the excess risk. Let q = 1 1/r be the rejection rate of the rejection sampling procedure and fix δ (qm, 1). Take ηδ = (δ qm)/(1 qm) and εδ = 2M q log(6/ηδ) 2m(1 q)(1 qm), then we have P ρλop[R](ˆg( , λop) ρλop[R] > εδ) = P ρλop[R](ˆg( , λop)) ρλop[R] > εδ | p = 0 P (p = 0) | {z } =qm + P ρλop[R](ˆg( , λop)) ρλop[R] > εδ | p 1 P (p 1) qm + (1 qm)P ρλop[R](ˆg( , λop)) ρλop[R] > εδ | p 1 qm + (1 qm)ηδ = qm + (1 qm)δ qm where the last derivations follow from the construction of εδ and ηδ. This shows that for any δ (qm, 1), the following inequality holds with probability 1 δ ρλop[R](ˆg( , λop)) ρλop[R] 2M log(6/ηδ) 2m(1 q)(1 qm), Domain Generalisation via Imprecise Learning where ηδ = (δ qm)/(1 qm). This concludes the proof. B. Conditional Value-at-Risk (CVa R) Proposition B.1. Let I = {1, . . . , m} be an index set and R : I R+ such that R(i) = ˆRi for i I. Denote C(I) as the space of real-valued, continuous function on I and C(I) its dual, i.e., {T : C(I) R}. Then there is a finite measure µ on I such that for any T C(I) and R C(I), we have Sketch Proof. The key is to notice I is a compact metric space because it is bounded. Furthermore, all functions on discrete space are automatically continuous. This allows us to directly apply the Riesz-Markov-Kakuani representation theorem. The proposition implies that no matter how we aggregate a risk profile, it will always correspond to some kind of weighted average. From the perspective of optimisation, since these weights are always convex (noramlising the weights does not change the optimisation), it can then be understood that whenever we aggregate the risk profile, we are picking a particular weighted distribution to perform the standard ERM. C. Single-Domain Scenario In a single-domain setting, we envision two possible approaches to imprecise learning. The first approach treats each training data point as an individual domain, estimating the risk profile through point-wise loss functions, denoted as R(f) = (ℓ(f(x1), y1), . . . , ℓ(f(xn), yn)). The second approach delineates a credal set by an ϵ-ball around the empirical distribution of the training data, akin to Distributionally Robust Optimisation (DRO). Subsequently, it extracts a finite number of extreme points from this credal set, which then represent the risk profile. While the first approach can be directly implemented within the current framework, the second approach entails a non-trivial extension of the existing setup. D. Risk Profiles of Simulation Simulation of Risk Profile: In economic theory, risk aversion explains the inclination to accept a situation with a more predictable but possibly lower payoff than another situation with a very unpredictable but possibly higher payoff. In OOD research, the term risk averseness has been conceptually used to describe the operator s risk perception for the model s risk profile (i.e., the distribution of ˆR). A risk-averse operator prefers models whose risk is more predictable but possibly higher than models whose risk is less predictable but possibly lower. Given that the operator at test time have a risk averseness between less risk averse and risk averse and by having h(x, λ) we can cover this spectrum of the operator s risk averseness. Given that we use CVa R, the entire spectrum of an ML Operators potential risk averseness is encoded in the interval of λ being between 0 and 1. By construction, h(x, λ) can cover the spectrum of risk averseness because it corresponds to the prediction function we obtain at CV a R(λ). Hence, we verify this hypothesis. Experiment 1A: Assume a linear model Ye = θe X + ϵ, where X N(2, 0.2) and ϵ N(0, 0.1). We simulate different environments by drawing θ from a Beta distribution Beta(0.1, 0.2). In total, we generate for 250 train and test domains 100 observations each. Each data line corresponds to a domain in Figure 3a. Hence the domains differ in their slope. Since we take θ from the bimodal distribution Beta(0.1, 0.2), we observe that the domains form two clusters. The more dominant cluster includes the domains with smaller θ. Subsequently, we aim to find the optimal ˆθ for all λ {0.05, . . . , 0.95} by solving the corresponding CVa R objective. As we can see from this plot, the optimal lines for small values of λ cluster around the dominant cluster of the environments. We consider the dark blue line (λ=0.05) as the average case . When increasing λ, the lines get closer to the second cluster of domains, which could be considered as the worst-case . Hence, the dark red line (λ=0.95) could be somewhat considered to be the estimated θ that works well in the worst cases. Domain Generalisation via Imprecise Learning Figure 3: Figure 3a illustrated the data and the ideal learner fλ(ˆθ) H for λ {0.05, . . . , 0.95}. Figure 3b describes the landscape of the objective function ρ (CVa R) for the ideal learner. We plot ˆθ as circles.Figure 3c describes the Risk profile for λ {0.05, . . . , 0.95} for the ideal learner. Figure 3d describes the Risk profile for λ {0.05, . . . , 0.95} Imprecise Learner. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 X Estimated regression functions h( , x) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 Landscape for CVa R ( ) and circles represent CVa R value 0.2 0.0 0.2 0.4 0.6 0.8 0 Risk Profiles wrt for Ideal Learner 0.0 0.2 0.4 0.6 0.8 0 Risk Profiles wrt for IRO In Figure 3b, we observe that for higher values of λ, the optimal solutions for θ vary a lot, while for smaller values of λ, the optimal solutions for θ do not vary significantly. As an interpretation, we can say that it is likely that the problem becomes harder for small λ. This interpretation is supported by the fact that when we choose λ to be high, we condition on the tail of the R, thus considering only a subset of the domains (i.e., lesser data for optimization). When looking at the optimal ˆθ, they form a smooth curve across all λ {0.05, . . . , 0.95}. Lastly, in Figure 3c, we see how the distribution of the risk changes across all λ. As expected, when choosing higher λ we consider higher risks from the risk profile R and minimize these parts in the optimization. We observe that for higher values of λ the risk profile does not transition smoothly contrary to the case of IRO in Figure 3d. We postulate that this is because an ideal learner essentially throws away the data from low-risk domains when focusing on high-risk domains due to the formulation of CVa R as an aggregator. However, since IRO learns all the objectives simultaneously it can implicitly address this issue of a finite number of domains for training in λ corresponding to higher risks. This observation is consistent with our observation from real-world experiments on UCI bike rentals in Figure 2c. E. Experiments on CMNIST E.1. Dataset Setup We conduct a large-scale experiment using an extension of the CMNIST dataset (Arjovski, 2021). The CMNIST comprises data from the MNIST dataset modified to the task of binary classification. For the standard task in CMNIST, the digits (0-4) and (5-9) have to be classified into two labels 0 and 1. Another feature as color is introduced in the training domain where digits are colored red or green such that the color is predictive of the true label e.g. domain 0.3 i.e. P(Y = 1 | color = red) = 0.3 and P(Y = 0 | color = red) = 0.7. Whereas for domain 0.9 it would mean P(Y = 1 | color = red) = 0.9 and P(Y = 0 | color = red) = 0.1. That is the mechanism by which color influences the label changes across domains. However, shape has a stable mechanism of prediction across domains i.e. P(Y = 0 | shape {0, 1, . . . , 4}) = 0.75 and P(Y = 1 | shape {5, 6, . . . , 9}) = 0.75. E.2. Experimental Setup and Baselines We consider a scenario where we sample environments from a long-tail distribution at training time to model data collection in the real world, such as low-resource languages. We sample 10 training environments from a Beta(0.9,1) distribution exactly {0.01, 0.02, 0.05, 0.07, 0.09, 0.12, 0.14, 0.58, 0.7, 0.99}. However, we do not assume IID distribution on environments, i.e. at test time we evaluate all the environments {0.0, 0.1, . . . , 0.9, 1.0}. Each environment is assumed to be influenced by both color and shape where the mechanism of color s influence changes but shape affects the target stably. This forces all the precise learners with a fixed hypothesis, i.e., PL-f to learn the invariant risk minimizer across domains that rely only on shape as a predictor to generalize to minority domains. We compare performance to baselines (precise learners with fixed hypothesis PL-f) based on different assumptions like ERM (average-case risk), Grp DRO (Sagawa et al., 2020), V-REx (Krueger et al., 2021) (worst-case risk) and IRM (Arjovski, 2021), IGA (Koyama and Yamaguchi, 2020) (Invariant Predictors), EQRM (Eastwood et al., 2022a) (probable domain generalizer) and SD (Pezeshki et al., 2021) which avoids Domain Generalisation via Imprecise Learning Color Shape (a) DAG of features and target in CMNIST 0.0 0.2 0.4 0.6 0.8 1.0 P(Y = 1|Color = Red) = e e Beta(0.9,1.0) Majority Minority (b) Long tail distribution of train environments Figure 4: In Figure 4a we describe the features that affect the target. The mechanism by which color affects target changes across environments. However, shape has a stable mechanism across environments. In Figure 4b we consider a long tail distribution of environments from which we sample training environments. This is often realistic that many subpopulations are underrepresented in training data, eg low resource languages for translation tasks. implicit regularization from Gradient starvation by decoupling features. We also consider Inf-Task which is a baseline for comparing how an Imprecise Learner (IL) performs against precise learners with an augmented hypothesis (PLh). Based on the initialization setup for CMNIST described by Eastwood et al. (2022a), all baseline methods perform poorly without ERM pretraining. Therefore, to ensure a fair comparison, we consider the ERM pretraining for PL-f learners for the initial 400 steps out of a 600-step training. All other hyper-parameters remain consistent with the established setup. For the learners with augmented hypotheses, it does not make sense to initialize with ERM because it may predispose the imprecise learner towards specific outcomes. Therefore, we assess the best-case performance across all learners across types of initialization. To implement the augmented hypothesis, we append FILM layers (Perez et al., 2018) to MLP architecture used in Eastwood et al. (2022a). E.3. Imprecise Learner can learn relevant features in context Table 3: Maximal regret and test accuracy across all CMNIST test environments.Bold denotes the hypothetical best invariant and Bayes classifier performance. Highlighted Green denotes the best performance amongst all algorithms for each domain and best regret. Bayes classifier is defined w.r.t the IID learner trained for a particular environment Objective Algorithm Test Environments based on P(Y = 1 | color = red) = e Regret 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Average Case ERM 96.1 87.1 78.0 72.1 65.8 59.2 51.8 47.1 39.9 33.6 28.3 72.7 Worse Case Grp DRO 54.1 55.6 58.1 595 61.5 64.5 66.3 69.1 70.5 73.9 75.5 46.9 SD 52.1 54.1 56.6 58.6 59.7 63.7 65.8 67.0 68.5 70.3 73.3 47.9 IRM 72.0 72.0 72.0 72.0 72.1 69.7 69.3 69.9 69.2 69.7 67.7 32.3 IGA 71.8 72.0 72.0 72.1 69.8 65.2 62.4 60.5 57.2 57.7 50.3 49.7 EQRM (λ 1) 67.8 67.7 68.3 68.8 70.5 69.1 70.3 72.0 72.1 71.4 72.1 32.2 VREx 72.7 71.3 71.8 71.4 71.7 69.5 69.5 70.2 69.5 71.6 68.5 31.5 Oracle 73.5 27.9 PLh Inf-Task 96.0 86.3 78.6 68.0 62.1 61.3 63.2 65.0 66.6 68.4 68.3 31.7 IL (Ours) IRO 95.8 87.2 78.8 68.9 69.4 69.5 70.8 70.1 70.0 70.4 70.3 29.7 Bayes Classifier ERM (IID) 100.0 90.0 80.0 75.0 75.0 75.0 75.0 75.0 80.0 90.0 100.0 In Table 3 we compare IL to other methods, showing that IL can learn relevant features in context. This also allows us to guide model operators on selecting appropriate λ. Suppose the user expects data at test time to come from the majority environments of their training. In that case, they can be less risk averse and use λ = 0 whereas if the user is unsure and anticipates test environments to look like unlike training, i.e. more minority environments they can choose λ 1. This is also reflected in the performance of IL such that for the majority domains e {0.0, . . . , 0.4} it performs similar to average case learner and for relatively less seen i.e. minority domains e {0.5, . . . , 1.0} it performs similar to the invariant learner. Domain Generalisation via Imprecise Learning F. Limitations of Imprecise Learner F.1. Computational Complexity The additional computation costs result from solving (9) compared to solving for a single notion of generalization which grows by the O(m) where m is the number of estimates needed. Since the convergence rate for Monte Carlo estimates is O( 1 m) the quality of estimates of the gradient improves slowly w.r.t. the number of samples. The generalization to the user s choice of risk λop with high probability is also given by O( 1 n + 1 m) in Proposition 4.2, where n is the number of data samples from each environment. In practice, there is room to obtain a better approximation of (9) with possibly quasi-Monte Carlo sampling methods. F.2. Challenges in Specifying User Preferences One of the main challenges in the Imprecise Learning (IL) framework is to specify user preference in terms of risk level i.e. a choice of λop. In practical scenarios, model operators may encounter challenges in precisely articulating their level of risk aversion. Additionally, bridging the operator s concept of generalization to a specific domain with an appropriate risk level remains ambiguous. In our experiments on modified CMNIST, we address this by allowing the model operator to be more risk-averse to generalize to minority environments. In contrast, for generalizing to a domain from majority environments users can be more risk-seeking. F.3. Generalization with no access to minority environments In the context of the standard CMNIST setup where the learner has access to no minority environments, CVa R as a risk measure does not allow to generalize beyond the credal set which can be constructed from the convex combination of majority environments alone. For standard CMNIST setup training envs are {0.1, 0.2} and test env is {0.9}. This means that the mechanism by which color affects the target is anti-correlated at test time, such situations can arise in adversarial settings. Since for λ 1, CVa R only minimizes the higher risks in a profile to achieve invariance it cannot recover the invariant mechanism without access to at least one environment from a subgroup. However, we argue that by using additional assumptions i.e. a different risk measure Imprecise learners can still learn to generalize to novel unseen domains outside of the credal set. We can extend the risk measure to enforce invariance by using VREx as an additional regularizer. ρλ[R] := CVa Rλ[R] + λVariance(R) (15) In Table 4, we observe that IL for λ = 1 obtains poor performance on a novel test domain however with an additional risk measure it obtains a closer performance to ERM on grayscale (Oracle) and outperforms several baselines. Note that with random initialization IL+VREx significantly outperforms other baselines. Table 4: CMNIST Test Accuracy. Training Environments are {0.1, 0.2} & Test Environment {0.9} Objective Algorithm Initialization Rand. ERM Best Case ERM 27.9 1.5 27.9 1.5 27.9 1.5 IRM 52.5 2.4 69.7 0.9 69.7 0.9 Grp DRO 27.3 0.9 29.0 1.1 29.0 1.1 SD 49.4 1.5 70.3 0.6 70.3 0.6 IGA 50.7 1.4 57.7 3.3 57.7 3.3 V-REx 55.2 4.0 71.6 0.5 71.6 0.5 EQRM 53.4 1.7 71.4 0.4 71.4 0.4 IL IRO 28.4 0.7 27.4 0.1 28.4 0.7 PLh+VREX Inf-Task 68.4 0.1 64.6 0.0 68.4 0.1 IL+VREX IRO 71.4 0.2 65.4 0.1 71.4 0.2 Invariant Pred. Oracle 73.5 0.2 G. Implementation Details This section provides the details of specific implementations used in our experiments. Domain Generalisation via Imprecise Learning G.1. Augmented Hypothesis For implementing the augmented hypothesis, we use hypernetworks (Ha et al., 2016) to realize the dependence of h on model operator s preference, i.e., λ. In this scenario, the weights of the augmented model are dependent on λ, i.e., hξ(x, λ) := fgw(λ)(x) where gw(λ) is the hypernetwork and ξ := {w, gw(λ)}. For neural networks with multiple layers, we use FILM layers (Perez et al., 2018) to augment the network such that it can be conditioned upon λ. G.2. Imprecise Risk Optimisation To operationalise the imprecise risk optimization, we need to minimise (9) with respect to the family of probability distributions (Λ). Since for our case Λ = [0, 1], we parameterise the family of distributions with Beta(α, β). We sample λ from Q via uniform sampling from the inverse CDF of Q, which we denote as F 1. We approximate the gradient of F 1 by first-order difference as described in Algorithm 3. Algorithm 3 Sampling from a Beta Distribution using ICDF with Gradient Computation 1: class ICDFBeta: 2: def forward(u): # Compute ICDF 3: return F 1(α,β)(u) 4: def backward(u): # Compute Gradient 5: δ := 1e 6 6: θF 1(α, β)(u) := (F 1(α+δ,β)(u) F 1(α,β)(u)) δ 7: ϕF 1(α, β)(u) := (F 1(α,β+δ)(u) F 1(α,β)(u)) δ 8: return αF 1(α, β)(u), βF 1(α, β)(u) 9: Initialize: α, β 1.0, 1.0 10: icdfbeta = ICDFBeta(α, β) 11: for epoch = 1 to k do 12: for i = 1 to m do 13: ui Uniform([0, 1]) 14: λi = icdfbeta.forward(ui) 15: end for 16: end for 17: Return Set of samples {λ1, λ2, . . . , λm} and gradients for each epoch