SWAD: Domain Generalization by Seeking Flat Minima

Abstract
Domain generalization (DG) methods aim to achieve generalizability to an unseen target domain by using only training data from the source domains. Although a variety of DG methods have been proposed, a recent study shows that under a fair evaluation protocol, called DomainBed, the simple empirical risk minimization (ERM) approach works comparable to or even outperforms previous methods. Unfortunately, simply solving ERM on a complex, non-convex loss function can easily lead to sub-optimal generalizability by seeking sharp minima. In this paper, we theoretically show that finding flat minima results in a smaller domain generalization gap. We also propose a simple yet effective method, named Stochastic Weight Averaging Densely (SWAD), to find flat minima. SWAD finds flatter minima and suffers less from overfitting than does the vanilla SWA by a dense and overfit-aware stochastic weight sampling strategy. SWAD shows state-of-the-art performances on five DG benchmarks, namely PACS, VLCS, OfficeHome, TerraIncognita, and DomainNet, with consistent and large margins of +1.6% averagely on outof-domain accuracy. We also compare SWAD with conventional generalization methods, such as data augmentation and consistency regularization methods, to verify that the remarkable performance improvements are originated from by seeking flat minima, not from better in-domain generalizability. Last but not least, SWAD is readily adaptable to existing DG methods without modification; the combination of SWAD and an existing DG method further improves DG performances. Source code is available at https://github.com/khanrc/swad.

1 Introduction
Independent and identically distributed (i.i.d.) condition is the underlying assumption of machine learning experiments. However, this assumption may not hold in real-world scenarios, i.e., the training and the test data distribution may differ significantly by distribution shifts. For example, a self-driving car should adapt to adverse weather or day-to-night shifts [1, 2]. Even in a simple image recognition scenario, systems rely on wrong cues for their prediction, e.g., geographic distribution [3], demographic statistics [4], texture [5], or backgrounds [6]. Consequently, a practical system should require generalizability to distribution shift, which is yet often failed by traditional approaches.
Domain generalization (DG) aims to address domain shift simulated by training and evaluating on different domains. DG tasks assume that both task labels and domain labels are accessible. For example, PACS dataset [7] has seven task labels (e.g., “dog”, “horse”) and four domain labels (e.g., “photo”, “sketch”). Previous approaches explicitly reduced domain gaps in the latent space [8–
∗Equal contribution †Part of work done while at NAVER Clova Correspondence to: Junbum Cha <junbum.cha@kakaobrain.com>, Sungrae Park <sungrae.park@upstage.ai>
35th Conference on Neural Information Processing Systems (NeurIPS 2021).
12], obtained well-transferable model parameters by the meta-learning framework [13–16], data augmentation [17–19], or capturing causal relation [20, 21]. Despite numerous previous attempts for a decade, Gulrajani and Lopez-Paz [22] showed that a simple empirical risk minimization (ERM) approach works comparably or even outperforms the previous attempts on diverse DG benchmarks under a fair evaluation protocol, called “DomainBed”.
Unfortunately, although ERM showed surprising empirical success on DomainBed, simply minimizing the empirical loss on a complex and non-convex loss landscape is typically not sufficient to arrive at a good generalization [23–26]. In particular, the connection between the generalization gap and the flatness of loss landscapes has been actively discussed under the i.i.d. condition [23–28]. Izmailov et al. [25] argued that seeking flat minima will lead to robustness against the loss landscape shift between training and test datasets, while a simple ERM converges to the boundary of a wide flat minimum and achieves insufficient generalization. In the DG scenario, because training and test loss landscapes differ more drastically due to the domain shift, we conjecture that the generalization gap between flat and sharp minima is larger than expected in the i.i.d. scenario.
To show that flatter minima generalize better to unseen domains, we formulate a robust risk minimization (RRM) problem defined by the worst-case empirical risks within neighborhoods in parameter space [26, 34]. We theoretically show that the generalization gap of DG, i.e., the error on the target domain, is upper bounded by RRM, i.e., a flat optimal solution. Based on our theoretical observation, we modify stochastic weight averaging (SWA) [25], one of the popular existing flatness-aware solvers, by introducing a dense and overfit-aware stochastic weight sampling strategy. First, we suggest to sample weights densely, i.e., for every iteration. Also, we search the start and end iterations for averaging by considering the validation loss to avoid overfitting. We empirically show that the proposed Stochastic Weight Averaging Densely (SWAD) finds flatter minima than the vanilla SWA does, resulting in better generalization to unseen domains.
Contribution. Our main contribution is introducing flatness into DG, and showing remarkably outperforming performances against existing DG methods. As shown in Table 1, our SWAD improves the average DG performances by 3.6pp against the ERM baseline and 1.6pp against the existing best methods. Furthermore, by combining SWAD and previous SOTA [31], we even achieve 0.4pp improvements against the vanilla SWAD results. We also empirically show that while popular indomain generalization methods without considering flatness, e.g., Mixup [35] or CutMix [36], are not effective to out-of-domain generalization (Table 3), flatness-aware methods, e.g., SWA [25] or SAM [26], are only effective methods to both in-domain and out-of-domain generalization.

2 A Theoretical Relationship between Flatness and Domain Generalization
Let D := {Di}Ii be a set of training domains, where Di is a distribution over input space X , and I is the total number of domains. From each domain, we observe n training data points which consist of input x and target label y, (xij , y i j) n j=1 ∼ Di. We also define a set of target domain T := {Ti} T i similarly, where the number of target domains T is usually set to one. For the sake of simplicity, unlike Ben-David et al. [37], we assume that there exists a global labeling function h(x) that generates target label for multiple domains, i.e., yij = h(x i j) for all i and j. Domain generalization (DG) aims to find a model parameter θ ∈ Θ which generalizes well over both multiple training domains D and unseen target domain T . More specifically, let us consider a bounded instance loss function ` : Y × Y → [0, c], such that `(y1, y2) = 0 holds if and only if y1 = y2 where Y is a set of labels. For simplicity, we set c to one in our proofs, but we note that `(·, ·) can be generalized for any bounded loss function. Then, we can define a population loss over multiple domains by ED(θ) = 1I ∑I i=1 Exi∼Di [`(f(xi; θ), yi))], where f(·; θ) is a model parameterized by θ. Formally,
the goal of DG is to find a model which minimizes both ED(θ) and ET (θ) by only minimizing an empirical risk ÊD(θ) := 1In ∑I i=1 ∑n j=1 `(f(x i; θ), yi)) over training domains D.
In practice, ERM, i.e., arg minθ ÊD(θ), can have multiple solutions that provide similar values of the training losses but significantly different generalizability on ED(θ) and ET (θ). Unfortunately, the typical optimization methods, such as SGD and Adam [38], often lead sub-optimal generalizability as finding sharp and narrow minima even under the i.i.d. assumption [23–28]. In the DG scenario, the generalization gap between empirical loss and target domain loss becomes even worse due to domain shift. Here, we provide a theoretical interpretation of the relationship between finding a flat minimum and minimizing the domain generalization gap, inspired by previous studies [23–28].
We consider a robust empirical loss function defined by the worst-case loss within neighborhoods in the parameter space as ÊγD(θ) := max‖∆‖≤γ ÊD(θ + ∆), where ‖ · ‖ denotes the L2 norm and γ is a radius which defines neighborhoods of θ. Intuitively, if γ is sufficiently larger than the “radius” of a sharp optimum θs of ÊD(θ), θs is no longer an optimum of ÊγD(θ) as well as its neighborhoods within the γ-ball. On the other hand, if an optimum θf has larger “radius” than γ, there exists a local optimum within γ-ball – See Figure 1. Hence, solving the robust risk minimization (RRM), i.e., arg minθ ÊγD(θ), will find
a near solution of a flat optimum showing better generalizability [26, 34]. However, as domain shift worsen the generalization gap by breaking the i.i.d. assumption, it is not trivial that RRM will find an optimum with better DG performance. To answer the question, we first show the generalization bound between ÊγD and ET as follows: Theorem 1. Consider a set of N covers {Θk}Nk=1 such that the parameter space Θ ⊂ ∪Nk Θk where diam(Θ) := supθ,θ′∈Θ ‖θ− θ′‖2, N := ⌈ (diam(Θ)/γ) d ⌉
and d is dimension of Θ. Let vk be a VC dimension of each Θk. Then, for any θ ∈ Θ, the following bound holds with probability at least 1− δ,
ET (θ) < ÊγD(θ) + 1
2I I∑ i=1 Div(Di, T ) + max k∈[1,N ]
√ vk ln (m/vk) + ln(N/δ)
m , (1)
where m = nI is the number of the training samples and Div(Di, T ) := 2 supA |PDi(A)−PT (A)| is a divergence between two distributions.
Proof can be done similarly as [37] and [34]. In Theorem 1, the test loss ET (θ) is bounded by three terms: (1) the robust empirical loss ÊγD(θ), (2) the discrepancy between training distribution and test distribution, i.e., the quantity of domain shift, and (3) a confidence bound related to the radius γ and the number of the training samples m. Our theorem is similar to Ben-David et al. [37], while our theorem does not have the term related to the difference in labeling functions across the domains. It is because we simply assume there is no difference between labeling functions for each domain for simplicity. If one assumes a different labeling function, the dissimilarity term can be derived easily because it is independent and compatible with our main proof. More details of Theorem 1, including proof and discussions on the confidence bound, are in Appendix C.1 and C.2.
From Theorem 1, one can conjure that minimizing the robust empirical loss is directly related to the generalization performances on the target distribution. We show that the domain generalization gap on the target domain T by the optimal solution of RRM, θ̂γ , is upper bounded as follows: Theorem 2. Let θ̂γ denote the optimal solution of the RRM, i.e., θ̂γ := arg minθ ÊγD(θ), and let v be a VC dimension of the parameter space Θ. Then, the gap between the optimal test loss, minθ′ ET (θ′), and the test loss of θ̂γ , ET (θ̂γ), has the following bound with probability at least 1− δ.
ET (θ̂γ)−min θ′ ET (θ′) ≤ ÊγD(θ̂ γ)−min θ′′ ÊD(θ′′) +
1
I I∑ i=1 Div(Di, T )
+ max k∈[1,N ]
√ vk ln (m/vk) + ln (2N/δ)
m +
√ v ln (m/v) + ln (2/δ)
m
(2)
Proof is in Appendix C.3. It implies that if we find the optimal solution of the RRM (i.e., θ̂γ), then the generalization gap in the test domain (i.e., ET (θ̂γ) − minθ′ ET (θ′)) is upper bounded by the gap between the RRM and ERM (i.e., ÊγD(θ̂γ)−minθ′′ ÊD(θ′′)). Other terms in Theorem 2 are the discrepancy between the train domains D and the target domain T , and the confidence bounds caused by sample means. We remark that if we choose a proper γ, the optimal solution of the RRM will find a point near a flat optimum of ERM as shown in Figure 1. Hence, Theorem 2 and the intuition from Figure 1 imply that seeking a flat minimum of ERM will lead to a better domain generalization gap.

3 SWAD: Domain Generalization by Seeking Flat Minima
We have shown that flat minima will bring a better domain generalization. In this section, we propose Stochastic Weight Averaging Densely (SWAD) algorithm, and provide empirical quantitative and qualitative analyses on SWAD and flatness to understand why SWAD works better than ERM.

3.1 A baseline method: stochastic weight averaging
Since the importance of flatness in loss landscapes has emerged [23–28], several methods have been proposed to find flat minima [25, 26, 39]. We select stochastic weight averaging (SWA) [25] as a baseline, which finds flat minima by a weight ensemble approach. More specifically, SWA updates a pretrained model (namely, a model trained with sufficiently enough training epochs, K0) with a cyclical [40] or high constant learning rate scheduling. SWA gathers model parameters for every K epochs during the update and averages them for the model ensemble. SWA finds an ensembled solution of different local optima found by a sufficiently large learning rate to escape a local minimum. Izmailov et al. [25] empirically showed that SWA finds flatter minima than ERM. We also considered sharpness-aware minimization (SAM) [26], which is another popular flatness-aware solver, but SWA finds flatter minima than SAM (See Figure 3). We illustrate an overview of SWA in Figure 2a.

3.2 Dense and overfit-aware stochastic weight sampling strategy
Despite its advantages, directly applying SWA to DG task has two problems. First, SWA averages a few weights (usually less than ten) by sampling weights for every K epochs, results in an inaccurate approximation of flat minima on a high-dimensional parameter space (e.g., 23M for ResNet-50 [41]). Furthermore, a common DG benchmark protocol uses relatively small training epochs (e.g., Gulrajani and Lopez-Paz [22] trained with less than two epochs for DomainNet benchmark), resulting in insufficient stochastic weights for SWA. From this motivation, we propose a “dense” sampling strategy for gathering sufficiently enough stochastic weights.
In addition, widely used DG datasets, such as PACS (≈ 10K images, 7 classes) and VLCS (≈ 11K images, 5 classes), are relatively smaller than large-scale datasets, such as ImageNet [42] (≈ 1.2M images, 1K classes). In this case, we observe that a simple ERM approach is rapidly reached to a local optimum only within a few epochs, and easily suffers from the overfitting issue, i.e., the validation loss is increased after a few training epochs. It implies that directly applying the vanilla SWA will suffer from the overfitting issue by averaging sub-optimal solutions (i.e., overfitted parameters). Hence, we need an “overfit-aware” sampling scheduling to omit the sub-optimal solutions for SWA.
The main idea of Stochastic Weight Averaging Densely (SWAD) is a dense and overfit-aware stochastic weight gathering strategy. First, instead of collecting weights for every K epochs, SWAD collects weights for every iteration. This dense sampling strategy easily collects sufficiently many weights than the sparse one. We also employ overfit-aware sampling scheduling by considering traces of the validation loss. Instead of sampling weights from K0 pretraining epochs to the final epoch, we search the start iteration (when the validation loss achieves a local optimum for the first time) and the end iteration (when the validation loss is no longer decreased, but keep increasing). More specifically, we introduce three parameters: an optimum patient parameter Ns, an overfitting patient parameter Ne, and the tolerance rate r for searching the start iteration ts and the end iteration te. First, we search ts which satisfies mini∈[0,...,Ns−1] E (ts+i) val = E (ts) val , where E (i) val denotes the validation loss at iteration i. Simply, ts is the first iteration where the loss value is no longer decreased during Ns iterations. Then, we find te satisfying mini∈[0,1,...,Ne−1] E (te+i) val > rE (ts) val . In other words, te is the first iteration where the validation loss values exceed the tolerance r during Ne iterations.
We illustrate the overview of SWAD and the comparison of SWAD to SWA in Figure 2. Detailed pseudo code is provided in Appendix B.4. We compare SWAD with other possible SWA strategies in §4.3 and show that our design choice works better for DG tasks.

3.3 Empirical analysis of SWAD and flatness
Here, we analyze solutions found by SWAD in terms of flatness. We first verify that the SWAD solution is flatter than those of ERM, SWA, and SAM. Our loss surface visualization shows that the SWAD solution is located on the center of the flat region, while ERM finds a boundary solution. Finally, we show that the sharp boundary solutions by ERM are not generalized well, resulting in sensitivity to the model selection. All following empirical analyses are conducted on PACS dataset, validating by all four domains (art painting, cartoon, photo, and sketch).
Local flatness anaylsis. To begin with, we quantify the local flatness of a model parameter θ by assuming that flat minima will have smaller changes of loss value within its neighborhoods than sharp minima. For the given model parameter θ, we compute the expected loss value changes between θ and parameters on the sphere surrounding θ with radius γ, i.e., Fγ(θ) = E‖θ′‖=‖θ‖+γ [E(θ′)− E(θ)]. In
practice, Fγ(θ) is approximated by Monte-Carlo sampling with 100 samples. Note that the proposed local flatness Fγ(θ) is computationally efficient than measuring curvature using the Hessian-based quantities. Also, Fγ(θ) has an unbiased finite sample estimator, while the worst-case loss value, i.e., max‖θ′‖=‖θ‖+γ [E(θ′)− E(θ)] has no unbiased finite sample estimator.
In Figure 3, we compare Fγ(θ) of ERM, SAM, SWA with cyclic learning rate, SWA with constant learning rate, and SWAD by varying radius γ. SAM and SWA find the solutions with lower local flatness than ERM on average. SWAD finds the most flat minimum in every experiment.
Loss surface visualization. We visualize the loss landscapes by choosing three model weights on the optimization trajectory (θ1, θ2, θ3)2, and computing the loss values by linear combinations of θ1, θ2, θ33 as [25]. More details are in Appendix B.5. In Figure 4, we observe that for all cases, ERM solutions are located at the boundary of a flat minimum of training loss, resulting in poor generalizability in test domains, that is aligned with our theoretical analysis and empirical flatness analysis. Since ERM solutions are located on the boundary of a flat loss surface, we observe that ERM solutions are very sensitive to model selection. In Figure 5, we illustrate the validation accuracies for each train-test domain combination of PACS by ERM, over training iterations (one epoch is equivalent to 83 iterations). We first observe that ERM rapidly reaches the best accuracy within only a few training epochs, namely less than 6 epochs. Furthermore, the ERM validation accuracies fluctuate a lot, and the final performance is very sensitive to the model selection criterion.
On the other hand, we observe that SWA solutions are located on the center of the training loss surfaces as well as of the test loss surfaces (Figure 4). Also, our overfit-aware stochastic weight gathering strategy (denoted as the vertical dot lines in Figure 5) prevents the ensembled weight from overfitting and makes SWAD model selection-free.
2We choose weights at iteration 2500, 3500, 4500 during the training. 3Each point is defined by two axes u and v computed by u = θ2 − θ1 and v = (θ3−θ1)−〈θ3−θ1,θ2−θ1〉‖θ2−θ1‖2·(θ2−θ1) .

4.1 Evaluation protocols
Dataset and optimization protocol. Following Gulrajani and Lopez-Paz [22], we exhaustively evaluate our method and comparison methods on various benchmarks: PACS [7] (9,991 images, 7 classes, and 4 domains), VLCS [43] (10,729 images, 5 classes, and 4 domains), OfficeHome [44] (15,588 images, 65 classes, and 4 domains), TerraIncognita [45] (24,788 images, 10 classes, and 4 domains), and DomainNet [46] (586,575 images, 345 classes, and 6 domains).
For a fair comparison, we follow training and evaluation protocol by Gulrajani and Lopez-Paz [22], including the dataset splits, hyperparameter (HP) search and model selection (while SWAD does not need it) on the validation set, and optimizer HP, except the HP search space and the number of iterations for DomainNet. We use a reduced HP search space to reduce the computational costs. We also tripled the number of iterations for DomainNet from 5,000 to 15,000 because we observe that 5,000 is not sufficient to convergence. We re-evaluate ERM with 15,000 iterations, and observe 3.1pp average performance improvement (40.9%→ 44.0%) in DomainNet. For training, we choose a domain as the target domain and use the remaining domains as the training domain where 20% samples are used for validation and model selection. ImageNet [42] trained ResNet-50 [41] is employed as the initial weight, and optimized by Adam [38] optimizer with a learning rate of 5e-5. We construct a mini-batch containing all domains where each domain has 32 images. We set SWAD HPs Ns to 3, Ne to 6, and r to 1.2 for VLCS and 1.3 for the others by HP search on the validation sets. Additional implementation details, such as other HPs, are given in Appendix B.
Evaluation metrics. We report out-of-domain accuracies for each domain and their average, i.e., a model is trained and validated on training domains and evaluated on the unseen target domain. Each out-of-domain performance is an average of three different runs with different train-validation splits.
4.2 Main results
Comparison with domain generalization methods. We report the full out-of-domain performances on five DG benchmarks in Table 2. The full tables including outof-domain accuracies for each domain are in Appendix E. In all experiments, our SWAD achieves significant performance gain against ERM as well as the previous best results: +2.6pp in PACS, +0.3pp in VLCS, +1.4pp in TerraIncognita, +1.9pp in OfficeHome, and +2.9pp in DomainNet comparing to the previous best results. We observe that SWAD provides two practical advantages comparing to previous methods. First, SWAD does not need any modification on training objectives or model architecture, i.e., it is universally applicable to any other methods. As an example, we show that SWAD actually improves the performances of other DG methods, such as CORAL [31] in Table 4. Moreover, as we discussed before, SWAD is free to the model selection,
resulting in stable performances (i.e., small standard errors) on various benchmarks. Note that we only compare results with ResNet-50 backbone for a fair comparison. We describe the implementation details of each comparison method and the hyperparameter search protocol in Appendix B.
Comparison with conventional generalization methods. We also compare SWAD with other conventional generalization methods to show that the remarkable domain generalization gaps by SWAD is not achieved by better generalization, but by seeking flat minima. The comparison methods include flatness-aware optimization methods, such as SAM [26], ensemble methods, such as EMA [56], data augmentation methods, such as Mixup [35] and CutMix [36], and consistency regularization methods, such as VAT [57] and Π-model [58]. We also split in-domain datasets into training (60%), validation (20%), and test (20%) splits, while no in-domain test set used for Table 2. Every experiment is repeated three times.
The results are shown in Table 3. We observe that all conventional methods helps in-domain generalization, i.e., performing better than ERM on in-domain test set. However, their out-of-domain performances are similar to or even worse than ERM. For example, CutMix and Π-model improve in-domain performances by 1.0pp and 0.2pp but degrade out-of-domain performances by 1.5pp and 1.8pp. SAM, another method for seeking flat minima, slightly increases both in-domain and out-of-domain performances but the out-of-domain performance is not statistically significant. We will discuss performances of SAM in other benchmarks later. In contrast, the vanilla SWA and our SWAD significantly improve both in-domain and out-of-domain performances. SWAD improves the performances by SWA with statistically significantly gaps: 1.2pp on the out-of-domain and 0.6pp on the in-domain. Further comparison between SWA and SWAD is provided in §4.3.
Combinations with other methods. Since SWAD does not require any modification on training procedures and model architectures, SWAD is universally applicable to any other methods. Here, we combine SWAD with ERM, CORAL [31], and SAM [26]. Results are shown in Table 4. Both CORAL and SAM solely show better performances than ERM with +1.2pp average out-of-domain accuracy gap. Note that SAM is not a DG method but a sharpness-aware optimization method to find flat minima. It supports our theoretical motivation: DG can be achieved by seeking flat minima.
By applying SWAD on the baselines, the performances are consistently improved by 3.6pp on ERM, 2.8pp on CORAL, and 1.0pp on SAM. Interestingly, CORAL + SWAD show the best performances with both incorporating different advantages of utilizing domain labels and seeking flat minima. We also observe that SAM + SWAD shows worse performance than ERM + SWAD, while SAM performs better than ERM. We conjecture that it is because the objective control by SAM restricts the model parameter diversity durinig training, reducing the diversity for SWA ensemble. However, applying SWAD on SAM still leads to better performances than the sole SAM. The results demonstrate that the application of SWAD on other baselines is a simple yet effective method for DG.

4.3 Ablation study
Since SWAD does not rely on domain labels, it can be applied to other robustness tasks not containing domain labels. Table 6 show the generalizability of SWAD on ImageNet [42] and its shifted benchmarks, namely, ImageNet-C [59], ImageNet-R [60], and background challenge (BGC) [61]. SWAD consistently improves robustness performances against the ERM baseline and the SWA baseline. These results support that our method is robustly and widely applicable to improve both in-domain and out-of-domain generalizability. The detailed setup is provided in Appendix B.6.

5 Discussion and Limitations
Despite many benefits from SWAD, such as the significant performance improvements, model selection-free property, working plug-and-play manner for various methods, there are some potential limitations. Here, we discuss the limitations of SWAD for further improvements.
Confidence error in Theorem 1. While the confidence error in Theorem 1 tells the effect of γ on generalization error bound, there exists a limitation in that the confidence error term shows improper behavior with respect to γ if γ is close to zero. The behavior we expect is that the confidence error of RRM converges to the confidence error of ERM as γ decreases to zero, however, the current theorem does not show such tendency since the confidence bound diverges to infinity when γ goes to zero. However, we would like to note that this limitation is not a drawback of RRM, but it is caused by the looseness of the union bound which is a mathematical technique used to derive the confidence error of RRM. Our RRM formulation has a similarity to previous works [26, 34] and we note that the counter-intuitive behavior of the confidence bound and γ also appears in Foret et al. [26].
SWAD is not a perfect flatness-aware optimization method. Note that SWAD is not a perfect and theoretically guaranteed solver for flat minima, but a heuristic approximation with empirical benefits. However, even if a better flatness-aware optimization method is proposed, our theoretical contribution still holds: showing the relationship between flat minima and DG.
SWAD does not strongly utilize domain-specific information. In Theorem 2, the domain generalization gap is bounded by three factors: flat minima, domain discrepancy, and confidence bound. Most of the existing approaches focus on domain discrepancy, reducing the difference between the source domains and the target domain by domain invariant learning [8–12]. SWAD focuses on the first factor, the flat minima. While the domain labels are used to construct a mini-batch, SWAD does not strongly utilize domain-specific information. It implies that if one can consider both flatness and domain discrepancy, better domain generalization can be achievable. Table 4 gives us a clue: the combination of CORAL (utilizing domain-specific information) and SWAD (seeking flat minima) shows the best performance among all comparison methods. As a future research direction, we encourage studying a method that can achieve both flat optima and small domain discrepancy.

6 Concluding Remarks
In this paper, we theoretically and empirically demonstrate that domain generalization (DG) is achievable by seeking flat minima. We propose SWAD that captures flatter minima than the vanilla SWA does. The extensive experiments on five DG benchmarks show superior performances of SWAD compared with existing DG methods. In addition, combinations of SWAD and existing DG methods even show better performances than the vanilla SWAD. We theoretically and empirically observe that seeking flat minima can achieve better generalizability to both in-domain and out-of-domain, while strong in-domain generalization methods without consideration of flatness, e.g., Mixup or CutMix, cannot guarantee to achieve out-of-domain generalizability in both theory and practice. This study first brings the concept of flatness into DG tasks, and shows strong empirical performances not only in DG but also in ImageNet benchmarks. We hope that this study promotes a new research direction of seeking flat minima for domain generalization and other robustness tasks.
Acknowledgments and Disclosure of Funding
NAVER Smart Machine Learning (NSML) [62] and Kakao Brain Cloud platform have been used in experiments. This work was supported by IITP grant funded by the Korea government (MSIT) (No. 2021-0-01341, AI Graduate School Program, CAU).

Reviewer Summary:
Reviewer_3: This paper provides some theoretical justification that so-called flat solutions exhibit better domain generalization along with a strongly performing training methodology, SWAD, for improving domain generalization. They first prove theorems bounding the domain generalization performance in terms of robust risk minimization. Then they propose an approach to finding flat solutions related to previous work on in distribution generalization, and they thoroughly explore this method with regard to the flatness of solutions found, its domain generalization performance, and also perform ablations of the method.

Reviewer_4: This paper studies how to improve the model performance on domain generalization problems from the perspective of loss landscape. Specifically, the authors propose to apply weight averaging (WA) to find flat minima and demonstrate this could lead to improve model generalization on unseen domains. Theoretically, the authors provide generalization bounds for the proposed method. Empirically, the proposed method achieves better performance on benchmark datasets compared with previous methods.

Reviewer_5: This paper studies the domain generalization (DG) problem setting and proposes a modification to the stochastic weight averaging (SWA) method that leads to better performance on DG testbeds. The proposed method, named SWAD, uses the same basic idea as SWA but samples parameters more densely and uses additional techniques for preventing the incorporation of overfit parameters. Some theoretical analysis supports the general thrust of seeking flat minima for generalization. The focus of the paper, however, is empirical, as SWAD demonstrates improved performance compared to many prior methods on several DG testbeds. Furthermore, SWAD can be combined with these prior methods, such as CORAL, for even better performance.

Reviewer_6: This paper extends the idea of Stochastic Weighting Averaging (SWA) to the context of Domain Generalization (DG). Theoretically, the authors argues that finding a flat minima improves domain generalization. Algorithmically, the authors modify SWA by increasing the number of samples being averaged (e.g. per iteration) and by selecting the interval from where the samples are taken. The proposed algorithm is evaluated in the DomainBed benchmark and shows compelling results.

Reviewer Comments:
Reviewer_3: The connection between the theoretical work in the first part of the paper and the method developed and tested is fairly tenuous. I appreciate the motivation for finding flatter minima but the paper would be tighter if there were a more direct connection between the flatness term in eq (1) and the algorithm. In particular, does SWAD encourage flatness w.r.t. the first term in eq. (1) more than other measures of flatness? My apologies if I missed this.
How novel/different is Theorem 1 compared to classic results from e.g. Ben-David (2010)? Of course, flatness doesn’t enter into these results but the rest seems somewhat similar.
The performance of the SWAD algorithm seems quite impressive, and I think this makes the paper a solid submission.
Limitations And Societal Impact:
A thoughtful limitations section was included but societal impacts was not. However, I don’t see particular societal impact concerns specific to this work.
Needs Ethics Review: No
Time Spent Reviewing: 2

Reviewer_4: This paper tries to tackle the domain generalization problem by finding flat minima. It is interesting to investigate the loss landscape of the DNNs on domain generalization problems.
Originality: The method proposed in this paper (i.e., SWAD) is mainly built on previous approaches such as SWA, and the authors mention this in the submission.
Quality: This submission is a complete piece of work and the claims are well supported by the empirical results. The evaluation is reasonable.
Clarity: This submission is well-written and easy to follow.
Significance: This paper provides interesting empirical observations and improves the state-of-the-art by applying their proposed new approach.
Pros:
The proposed method consistently improves the model performance on domain generalization problems, and the proposed method is flexible and can be easily integrated with other training approaches for better performance.
Cons:
The generalization bound does not provide intuition for robust risk minimization (RRM), e.g. the theoretical results do not suggest performing RRM is better than standard ERM.
The proposed method is heavily built on the previous approach SWA, which only differs in terms of how frequently the averaging is performed and which iteration is being selected for averaging.
Suggestions:
It would be better to put the algorithm (Algorithm 1 in appendix B) in the main body for describing the proposed method.
Missing reference:
Reference on generalization bound, [1].
Overall, I think this paper provides a simple approach that can improve model (out-of-domain) performance on domain generalization problems. I would suggest `Marginally above the acceptance threshold'.
[1]. A theory of learning from different domains. Shai Ben-David, John Blitzer, Koby Crammer, Alex Kulesza, Fernando Pereira, Jennifer Wortman Vaughan. Machine learning, 2010.
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Thanks for the authors' response. I have read the authors' responses as well as the other reviews. I will stick with my score.
Limitations And Societal Impact:
N.A.
Ethical Concerns:
N.A.
Needs Ethics Review: No
Time Spent Reviewing: 4

Reviewer_5: Originality
The paper is, perhaps by design, not particularly original. Rather, it makes the observation that an existing set of techniques with a particular goal (seeking flat minima for improved generalization) also has applications to a different goal (DG). This is still, to the best of my knowledge, a novel observation and contribution, and furthermore the modifications made to SWA to arrive at SWAD also appear novel. Nevertheless, these modifications are relatively small, so the method can still be viewed as a version of the general idea of SWA, and thus originality is not the paper's strong suit.
Quality
The paper is of relatively high quality, primarily due to the empirical results. I did not carefully check the theory for correctness, but I am unsure as to the significance of the theorems. They appear rather similar to prior theoretical results on how flat minima can lead to better generalization, but with some additional divergence terms thrown in as we are dealing with the DG setting. So the high level conclusion seems to be: standard (in distribution) generalization also helps with DG, so long as the domain divergence is not too large. This is not an interesting result. It would be nice to have stronger theory that more directly connects flat minima to DG, but the current theory does not do this.
Two more minor comments about the theory. First, the reliance on VC dimension is not desirable, since this could be in the billions for the models used and, in particular for DG testbeds, many orders of magnitude larger than the number of training points. Second, I am unsure about the argument in L114-120. "It implies that..." seems to ignore several terms on the RHS of Eq (2) that could be rather large. The final sentence ("Hence, Theorem 2 and...") also seems to require quite a bit more support for such a general statement.
Nevertheless, as mentioned, I am appreciative of the empirical results that the paper provides, including a detailed study on several testbeds and an ablation study that verifies all of the proposed components of the method. I am inclined to view this paper as an empirical contribution, and a fairly nice one that further supports the notion that strong results can be achieved on existing DG testbeds without paying much attention to the domain labels themselves (instead focusing on standard generalization).
Clarity
The paper is overall well written and easy to follow. See my comments above for some suggestions regarding the clarity of the theory. My other comments are more minor. First, though the technical description of the overfit-aware sampling strategy in L151-160 is nice, it would be made more clear with additional intuition and descriptive text, e.g., what is each parameter doing specifically and how did this particular instantiation come about? Second, there is a minor typo on L157 (remove the "\mathcal{E}_{val}^{(t_s)} ="). Third, it is not immediately obvious what the significance is of Figure 5.
Significance
As noted, I view this paper as a nice (significant) empirical contribution. Perhaps the significance could be further extended by exploring problems beyond DG. Indeed, there seems to be nothing central to the proposed method that ties it to DG, so could it perhaps be applied to other problems? A discussion of this point or, even better, some experiments would further improve the paper.
Edit after reading the author response
Thanks for your additional comments. My scores remains unchanged and I continue to recommend acceptance.
Limitations And Societal Impact:
Yes, the authors have adequately addressed limitations and societal impacts in Section 5 and the Appendix.
Needs Ethics Review: No
Time Spent Reviewing: 3

Reviewer_6: Pros:
The paper tackles an important and relevant problem to the ML community.
The proposed algorithm is relatively simple to implement in practice and perform well experimentally.
Cons:
My major concern is wrt to the novelty of the work and the connection/motivation between the proposed algorithm SWAD, domain generalization and the derived theoretical insights.
Theory. While I appreciate the provided theoretical insights, they directly follow from standard results in domain adaptation [1,2,3] and Norton and Royset [29] . Most importantly, I believe some details have been overseen in the technical proof. Specifically, I believe the proof misses a term (line 86 eq. 20 in suppl material). Lemma 1 is between any two h, but the risk is between h and the labeling function. If I understand correct, this is overseen in line 86. The author should rewrite lemma 1 before directly using it in the proof. Please refer to [1] and [2] for details. The second part of the proof and Lemma 2 follow from Norton and Royset [5]. This should be clearly mentioned and if they are different the authors should highlight main differences. I would appreciate if the authors can clarify on this.
Moreover, in practice, the provided upper bound can be easily dominated by the divergence between source and target. Theorem 1, does not capture either the dissimilarity of the labeling functions (as a result of the missing term), which can also simply dominate the performance on target and being small is necessary for adaptation [2,3,4]. Even if these were all negligible and the result correct, the provided bound only shows that a flat minimal can improve generalization but this is not necessarily novel. It is hard to see a clear connection between these results and the proposed algorithm. In other words, the theoretical results do not necessarily motivate the use of SWAD vs SWA. I would appreciate if the authors can provide further explanation on this aspect.
Algorithm. The proposed algorithm is based on a heuristic on top of SWA that does not necessarily exploit the current scenario (DG). I believe this heuristic improves SWA in general and it might be algorithmically incremental vs vanilla SWA. Moreover, it introduces 3 new hyperparameters (well tuned in the experiments). A better motivation than the one provided in 134-147 can improve the presentation/motivation of the algorithm.
Experiments and Analysis. This is one of the strongest sections of the paper. Just a few questions:
Why SWA is not shown in Table 2?
Table 4 mentioned HP search was not applied but ERM+SWAD includes HP search (This is the same result from Table 2).
Suppl
Lemma 1 is a well know result also from [1,2,3] and should be cited appropriately.
Lemma 2 follows [5] and show be cited appropriately.
Originality: The use of SWA in the context of DG is new. However, algorithmically the proposed method is an extension (based on a heuristic) of a well known technique. Theoretically, the presented results also follow from existing work and it is not clear how they motivate the need of SWA vs SWAD (see theory and algorithm above for more details).
Quality: I believe the proposed algorithm should be better motivated as it is not clear the connection/motivation between SWAD, domain generalization and the derived theoretical insights. Moreover, I believe several small details where overseen in the technical proofs and those should be corrected to make them technically sound. (theory above for more details)
Clarity: The submission is clearly written and well organized. The main issue in this aspect is with respect to the motivation and connection among the two different parts (theoretical results and algorithms). (see algorithm above for more details)
Significance: The empirical section is one of the strongest part of the paper and the results advance SoTA. The method is relatively simple to implement in practice. However, it introduces a few extra hyperparams.
As expressed above, my major concern is wrt to the novelty of the work and the connection between the proposed algorithm SWAD, domain generalization and the derived theoretical insights. I believe the paper could significantly benefit from a major revision.
References
[1] Ben-David et al. A theory of learning from different domains Springer 2010
[2] Impossibility Theorems for Domain Adaptation PMLR,2010.
[3] Acuna et al. f-Domain-Adversarial Learning: Theory and Algorithms. ICML 2021
[4] Zhao et al On Learning Invariant Representations for Domain Adaptation, ICML 2019
[5] Norton et al Diametrical Risk Minimization: Theory and Computations, 2019 (cited)
Limitations And Societal Impact:
Yes.
Needs Ethics Review: No
Time Spent Reviewing: 11