# domaininspired_sharpnessaware_minimization_under_domain_shifts__c4ed1cd3.pdf

Published as a conference paper at ICLR 2024

DOMAIN-INSPIRED SHARPNESS-AWARE MINIMIZATION UNDER DOMAIN SHIFTS

Ruipeng Zhang , , Ziqing Fan , , Jiangchao Yao , ,B, Ya Zhang , , Yanfeng Wang , ,B

Cooperative Medianet Innovation Center, Shanghai Jiao Tong University Shanghai Artificial Intelligence Laboratory {zhangrp, zqfan_knight, Sunarker, ya_zhang, wangyanfeng}@sjtu.edu.cn

This paper presents a Domain-Inspired Sharpness-Aware Minimization (DISAM) algorithm for optimization under domain shifts. It is motivated by the inconsistent convergence degree of SAM across different domains, which induces optimization bias towards certain domains and thus impairs the overall convergence. To address this issue, we consider the domain-level convergence consistency in the sharpness estimation to prevent the overwhelming (deficient) perturbations for less (well) optimized domains. Specifically, DISAM introduces the constraint of minimizing variance in the domain loss, which allows the elastic gradient calibration in perturbation generation: when one domain is optimized above the averaging level w.r.t. loss, the gradient perturbation towards that domain will be weakened automatically, and vice versa. Under this mechanism, we theoretically show that DISAM can achieve faster overall convergence and improved generalization in principle when inconsistent convergence emerges. Extensive experiments on various domain generalization benchmarks show the superiority of DISAM over a range of stateof-the-art methods. Furthermore, we show the superior efficiency of DISAM in parameter-efficient fine-tuning combined with the pretraining models. The source code is released at https://github.com/Media Brain-SJTU/DISAM.

1 INTRODUCTION

Although deep learning has achieved remarkable advances in various areas (He et al., 2016; Dosovitskiy et al., 2020), it remains a challenge for optimization in pursuit of strong generalization. Especially, a lower training loss does not necessarily guarantee a better generalization, as there exist numerous local minima in the complex and non-convex hypothesis space. Recent empirical and theoretical investigations (Dziugaite & Roy, 2017; Chaudhari et al., 2019; Jiang et al., 2020; 2023; Dinh et al., 2017b; Keskar et al., 2017b) have identified a significant correlation between generalization and the sharpness of the loss landscape. This correlation suggests that generalizability can be interpreted as flatness in the loss surface, leading to a wide range of explorations that have contributed to the rapid development of Sharpness-Aware Minimization (SAM) (Foret et al., 2021).

Existing SAM-based methods predominantly focus on the narrowly defined generalizability between training and test data under the Independent and Identically Distributed (i.i.d) assumption, which can be summarized as two categories. The first strives to improve the performance by creating a more effective estimation of sharpness like the enhanced minimization in GSAM (Zhuang et al., 2022), PGN (Zhao et al., 2022), SAGM (Wang et al., 2023b) and Va SSO (Li & Giannakis, 2023), as vanilla perturbation in SAM fails to accurately capture the geometric flatness of the loss landscape. The other category targets to improve computational efficiency by reducing perturbation directions (Liu et al., 2022) or using a more efficient perturbation surrogate (Du et al., 2022a;b), as the original SAM incurs double the computational overhead compared to Empirical Risk Minimization (ERM). Nonetheless, these methods cannot solve generalizability scenarios that involve training data of multiple domains with domain shifts like Domain Generalization (DG) (Ben-David et al., 2010; Li et al., 2017).

In this study, we observed that sometimes SAM even has a detrimental impact in situations where there exist domain shifts across multiple domains as shown in Figure 1(a). While a few studies have incorporated SAM-based methods in domain generalization tasks (Wang et al., 2023b; Foret et al.,

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(a) Performance under domain shifts.

(b) Convergence curves under domain shifts.

Figure 1: Illustration of SAM s degradation of the training process under domain shifts. (a) Performance comparison between ERM and SAM, where SAM consistently performs worse than ERM across all hyperparameters ρ. (b) Convergence curves of SAM and ERM for each domain during training, with the convergence degree normalized to [0,1]. SAM exacerbates the disparity in convergence degree among different domains in domain shift scenarios, resulting in inferior generalization performance. The dataset used here is Terra Inc from the Domain Bed benchmark, and the backbone is Res Net50. Further experimental details are provided in Section 4.1 and Appendix C.5.

2021), they cannot ensure consistent improvements in generalizability during domain shifts due to their reliance on the i.i.d assumption. Upon a thorough analysis of the behavior of SAM under domain shifts, we discovered that the degradation of the training process caused by SAM from the disparity in convergence degree among different domains as shown in Figure 1(a). Given the inconsistency in the degree and direction of convergence among different domains during training (Arjovsky et al., 2019; Krueger et al., 2021), the straightforward application of SAM for perturbations may not only disrupt convergence but also generate perturbation directions that are not adequately coherent to the geometric characteristics of the entire loss landscape.

To solve the aforementioned problem, we propose a Domain-Inspired Sharpness-Aware Minimization (DISAM) algorithm. As the degradation origins from the inconsistency in convergences of these domains, DISAM incorporates domain-level convergence information to intervene in the perturbation of the vanilla SAM: the perturbation direction should focus more on domains with higher convergence degree while being mild to domains with lower convergence degree. Under such balancing in perturbation, the gradient update actually implements the domain-level personalization, thus mitigates the impact of domain shifts and enhances the generalization performance. Technically, we ingeniously accomplish the adaptive adjustment of the perturbation direction in accordance with the degree of convergence through the domain loss variance minimization constraint. The perturbation of DISAM directs towards a location with a more consistent convergence degree, enabling a better global view of the loss landscape for gradient update. We summarize our contributions as follows:

We identify that the use of SAM has a detrimental impact on training under domain shifts, thereby compromising generalizability, and further analyze that the reason is the inconsistent convergence of training domains that deviates from the underlying i.i.d assumption of SAM.

We introduce a novel approach called Domain-Inspired Sharpness-Aware Minimization to mitigate the problem above. DISAM incorporates domain-level convergence consistency by imposing a variance minimization constraint on domain loss during the sharpness estimation process, thereby enabling a more representative perturbation location and enhancing generalization.

Extensive experiments show the superiority of DISAM in improving the current state-of-the-art methods on several benchmarks. We also provide a comprehensive analysis of its merit of faster convergence compared to SAM, and show its persistent generalization capabilities under parameter-efficient fine-tuning with large models like CLIP.

2 PRELIMINARIES

2.1 BASIC NOTATIONS

S = {D1, D2, , DM}: Overall training set of a M-source domain generalization task. We denote each domain by Di and the number of samples in Di by ni = |Di|.

ξ, ξi j: A specific sample and the j-th sample in i-th domain Di, respectively.

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L, L(w), L(w; ξ): A loss function, expected loss under w and the specific loss of ξ, respectively.

Li(w) = Eξ Di L(w; ξ): Expected loss under w for each domain Di.

Var{ }M i=1: The variance among M training domains, which holds: Var{Li(w)}M i=1 = 1 2M 2 PM i=1 PM j=1(Li(w) Lj(w))2.

LDI(w) = L(w) λVar{Li(w)}M i=1: A loss function with domain-inspired regularizer Var{Li(w)}M i=1 on L. λ is a constant value that controls the strength of the constraint.

Lp(w) = max ϵ 2 ρ L(w + ϵ): The perturbed loss and the objective of SAM.

wasc t = wt + ρ LDI(wt)

LDI(wt) : The sharpness estimation of DISAM with gradient ascend at step t.

ηt: Learning rate at step t.

w: Parameters of a neural network Rk, where k is the dimension.

ϵ Rk: A perturbation on the parameters w with scale ρ R.

2.2 SHARPNESS-AWARE MINIMIZATION

In general, simply minimizing ERM tends to overfit training data and extensive studies show the correlation between generalizability and the sharpness of minima (Dinh et al., 2017b; Hochreiter & Schmidhuber, 1994b; Mc Allester, 1999; Chaudhari et al., 2019). We clarify the concepts as below.

Sharpness. The sharpness on parameter w with a dataset D and loss function L is:

s(w, D) max ϵ 2 ρ Eξ D[L(w + ϵ; ξ) L(w; ξ)]. (1)

Sharpness-Aware Minimization (SAM). Foret et al. (2021) proposed SAM to improve the generalization by simultaneously minimizing the loss and the sharpness of the overall loss surface. The objective is defined as: min w max ϵ 2 ρ Eξ D[L(w + ϵ; ξ)]. (2)

From the above equation, we can see SAM minimizes a perturbed loss max ϵ 2 ρ Eξ D[L(w + ϵ; ξ)] , which aims to maximize the loss L within radius ρ centered at the parameter w.

3.1 MOTIVATION

Although existing SAM-based methods that minimize the sharpness have achieved good generalization, in the case of multiple domains with shifts, the inherent heterogeneity in quantity and task difficulty among domains can considerably distort their sharpness estimation, yielding a degradation in the performance. Concretely, with a collection S of M domains, each of which contains a set of ni samples, i.e., {ξi j = (xi j, yi j)}ni j=1, the training objective can be then formulated as follows:

min w Eξ S[L(w; ξ)] = 1

j=1 L(w; ξi j) =

i=1 αi Li(w), (3)

where N = PM i=1 ni, αi = ni

N and Li(w) = 1 ni Pni j=1 L(w; ξi j). Note that, we clarify here that we will ignore the notations of data properly in some subsequent equations to avoid clutter. Then, on the basis of Eq. (3), the corresponding objective of SAM under domain shifts is defined as:

min w Eξ S[LSAM(w; ξ)] = min w max ϵ 2 ρ Eξ S [L(w + ϵ; ξ)] ? min w max ϵ 2 ρ

i=1 αi Li(w + ϵ). (4)

The core that we should point out is whether the approximation from max ϵ 2 ρ Eξ S [L(w + ϵ; ξ)] to max ϵ 2 ρ PM i=1 αi Li(w + ϵ) in Eq. (4) is reasonable. There is no harm when samples in S

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Aware Minimization Under Domain Shift

The neighborhood

of radius 𝝆

Loss Surface for Training Domain

Flat minima (generalizable)

Sharp minima (Ungeneralizable)

Overall loss surface for all domains

Gradient vector

Loss surface

Perturbation 𝒘𝒕

𝒂𝒔𝒄after gradient ascend on 𝒘𝒕

DISAM will rescale

each domain s

gradient by convergence degree

Actual update direction

Convergence Degree

SAM and DISAM will get different

𝒂𝒔𝒄, thus have different update

directions.

Visualization of Loss Landscape

Domain shifts

Varied convergence degrees across domains

Figure 2: Toy example illustrating the problem of SAM under domain shifts. Left: Domain shifts on the loss surface of training domains, which causes the inconsistency of convergence degree. Middle: Differences between SAM and DISAM in the perturbation generation and convergence. Specifically, SAM is affected by the inconsistent degree of convergence. Right: Visualization of loss landscape for ERM, SAM, and DISAM on unseen test domain. DISAM is flatter than SAM and ERM.

are intrinsically independently and identically distributed. However, this is actually ill-posed under domain shifts. Differences in the amount of data or inconsistency in task difficulty can result in biased sharpness estimation towards specific domains, hindering the overall convergence. As shown in Figure 2, neglecting the domain shifts and the consequent convergence inconsistency issue, using SAM directly at this point, significantly misdirects the perturbation direction towards the domain with the largest gradient vectors (implying a lower degree of convergence). Consequently, it does not help find a better convergence path and conversely leads to a suboptimal sharp minima.

3.2 DOMAIN-INSPIRED SAM

To address the problem described in Eq. (4) and Figure 2, we need to design an adjustment mechanism that takes into account the convergence degree of each domain during the perturbation generation. Specifically, we should make the perturbation direction Lp(w) to efficiently pull domains that are close to convergence out of sharp minima while minimizing the negative impact on domains that have not yet converged. Here, we define the convergence degree of domain i with model parameter w as Ci(w) = L i Li(w), where L i represents the optimal minimum of domain i (L i 0). The design principle is to prioritize the contribution of domains with larger Ci(w) to the overall perturbation direction. To achieve this, a simple approach involves directly adding Ci(w) to the weight αi of SAM with a controlling coefficient term β.

i=1 αi Li(w)

αi + βCi(w) PM j=1(αj + βCj(w)) Li(w) =

β(Ci(w) αi PM j=1 Cj(w))

1 + β PM j=1 Cj(w) Li(w) (5)

However, we observe that the weight adjustment in Eq. (5) that is affected by the convergence degree, is constrained by the magnitude of αi. That is, domains with higher αi values can tolerate lower convergence degrees, which may not accurately satisfy our goals. To refine this, we propose to use an adaptive way to ensure fairness by calculating the average convergence at the domain level, namely, Ci(w) αi PM i=1 Ci(w) Li(w) 1 M PM i=1 Li(w). With this intuition, we introduce a method called Domain-Inspired Sharpness-Aware Minimization (DISAM) that incorporates a variance constraint between domain losses to estimate sharpness. It enables the adaptive adjustment of the perturbation direction similar to our spirit, which we will provide a detailed analysis in the following Eq. (8). First of all, we give the definition of the variance between different domain losses as:

Var{Li(w + ϵ)}M i=1 = 1 2M 2

j=1 (Li(w + ϵ) Lj(w + ϵ))2. (6)

Then, putting the above variance term into the loss, the new training objective can be defined as:

min w Eξ S[LDISAM(w; ξ)] min w max ϵ 2 ρ

i=1 αi Li(w + ϵ) λVar{Li( ˆw + ϵ)}M i=1

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Here ˆw is w without derivative taken during backpropagation, and it only makes effect in the max ϵ 2 ρ loop without affecting the optimization of the first term w.r.t. w. Following the computing way of perturbation ϵ in SAM, namely, using first-order Taylor expansion (Foret et al., 2021), we will have ϵ ρ LDISAM

LDISAM , where LDISAM w.r.t. w has the form:

| {z } Adaptive adjustment: increase weights for smaller losses, reduce for larger ones.

The first term in the RHS of Eq. (8) recovers the gradient term for perturbation generation in SAM, and the second term characterizes the working mechanism for the adaptive adjustment. As can be seen, when the loss Li(w) of one certain domain is above the averaging level, the second term will generate a residual gradient for this domain to cancel out the gradient contribution in LSAM, and vice versa. It means to have a mild perturbation for the domain that is not well optimized, and have an aggressive perturbation for the domain that is well optimized. In total, the variance constraint ensures that the perturbation location is at a more consistent point, enabling a better global view of the loss landscape for gradient update. The complete algorithm is described in Appendix B. Regarding λ, a default value of 0.1 is relatively stable, and we provide more discussion about λ in Appendix B.3.

Difference and Compatibility. Similarly, the current SAM variants will meet the same challenge, if they are directly applied to this scenario. Different from existing state-of-the-art methods like GSAM (Zhuang et al., 2022) and SAGM (Wang et al., 2023b) that modify the optimization objective based on the second derivative of SAM, DISAM rectifies the domain shift issue by the domain-level adjustment in the perturbation generation, which actually alleviates the negative impacts on the training objective. In Appendix A.2.5, we present a table to comprehensively characterize the difference between DISAM and other domain-invariant robust optimization methods. Besides, DISAM can be easily extended into other SAM-based methods to improve the generalization performance. We have provided a comparison of the similarities and differences between DISAM and general convergence consistency methods (such as V-REx(Krueger et al., 2021) and Fishr(Rame et al., 2022)) in Appendix B.1.

3.3 UNDERSTANDING DOMAIN-INSPIRED SAM

Complexity. Compared to SAM-based methods, our algorithm only additionally computes the loss variance between different domains as Eq. (6) and requires no extra storing space. Therefore it has the same space complexity and the time complexity can be represented as ODISAM = OSAM + OVar. Since it only needs to additionally count the domain loss and the corresponding variance according to the domain label when calculating the empirical loss, the overall cost on OVar is negligible.

Convergence. In the following, we provide the convergence analysis of SAM and DISAM. Similar to (Zhuang et al., 2022; Jiang et al., 2023), our theorem is established on assumptions that a nonconvex function L(w) is L Lipschitz-smooth, the lower bound of the empirical loss is bounded by Lmin, and the norm of noisy stochastic gradients is bounded ( Lp(wt) 2 G) at the t-step. Theorem 1. Consider a non-convex function L(w) with Lipschitz-smooth constant L and lower bound Lmin. With the bounded norm assumption of noisy stochastic gradients ( Lp(w) 2 G) at the t-step, the learning rate ηt = η0/

t and a fixed perturbation amplitude ρ, we have:

t=1 E Lp(wt) 2 2 Lp(w0) Lmin

T + (LG2 + ρ2LΓ2)η0 log(T)

where in SAM, Γ = L and in our DISAM, Γ L.

The complete proof is presented in Appendix B. As can be seen in Theorem 1, the critical convergence difference between SAM and DISAM is on Γ, and especially the Γ in our DISAM is smaller than that in SAM due to the canonically correlated perturbations during training (see the proof for the details),

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0 1000 2000 3000 4000 5000

Convergence curve on test domain A

0 1000 2000 3000 4000 5000

Convergence curve on test domain S

0.21 0.2 0.1 0.09 0.05 ρ

Max ρ = 0.09 Max ρ = 0.2

Max ρ in PACS

0.14 0.13 0.07 0.06 0.05 ρ

Max ρ = 0.06 Max ρ = 0.13

Max ρ in Office Home

Figure 3: Convergence curves and Max ρ search for SAM and DISAM. (a) & (b) show the trend of L(w) during the training process on PACS dataset, while (c) & (d) search for the maximum perturbation amplitude ρ of SAM and DISAM on PACS and Office Home datasets.

which thus leads to a faster convergence rate. Note that, the overall ρ2LΓ2 in Theorem 1 indicates that a larger perturbation amplitude ρ will result in larger upper bound of convergence. However, as analyzed in SAM (Foret et al., 2021), a larger perturbation amplitude ρ has the merit of reaching a smaller upper bound on generalization error. This means that ρ actually has a trade-off between accelerating the convergence and improving the generalization. Fortunately, when in the same overall value of ρ2LΓ2, as DISAM enjoys a smaller Γ than SAM, DISAM can permit potential larger ρ than that in SAM, thus yielding a better generalization (Please refer to Appendix B.4 for more discussion).

To verify the theoretical analysis, we present our empirical results of DISAM and SAM in Figure 3. As shown in Figures 3(a) and 3(b), the training curves on the PACS dataset show that DISAM achieves faster and steeper convergence than SAM. In addition, as DISAM has a smaller Γ, it is able to utilize a larger perturbation amplitude ρ. In Figures 3(c) and 3(d), we show the experimental support that DISAM allows larger ρ values (0.2 and 0.13) than SAM (0.09 and 0.06) on PACS and Office Home datasets, while achieving a better performance. In total, these experiments confirm the advantage of DISAM that allows larger ρ for better generalization.

4 EXPERIMENTS

4.1 EXPERIMENT SETUPS

Datasets. We evaluate DISAM on five datasets PACS (Li et al., 2017), VLCS (Fang et al., 2013) Office Home (Venkateswara et al., 2017), Terra Incognita (Beery et al., 2018) (abbreviated as Terra Inc), and Domain Net (Peng et al., 2019), following the Domain Bed benchmark (Gulrajani & Lopez-Paz, 2021). For fair comparison, we adhere to the training and evaluation protocol outlined in Domain Bed.

Evaluation. The standard leave-one-domain-out strategy is used in evaluation. Specially, the unseen domain is used to evaluate the out-of-domain generalization, and the validation sets of source domains are used to measure the in-domain generalization, while the others are used for training. Final accuracy is averaged across all settings, and the performance is the averaging over three trials with distinct random seeds. Detailed statistics for each case of all datasets are provided in Appendix C.

Implementation details. Our backbones are Res Net50 pretrained on Image Net (He et al., 2016) and a pretrained CLIP (Radford et al., 2021) with Vi T-B/16 structure (Dosovitskiy et al., 2020). For model hyperparameters, we adopt settings in (Wang et al., 2023b) for experiments using Res Net50 and in (Shu et al., 2023) for experiments using CLIP. As the default, we set the perturbation hyperparameter ρ to 0.05 (Wang et al., 2023b) (Fixed value during training), and the weight of the variance constraint λ to 0.1. For a detailed description of the hyparameter settings, please see Appendix C.

4.2 PERFORMANCE UNDER RESNET50 BACKBONE

We propose incorporating our domain-inspired adaptive adjustment into three SAM-based methods: SAM (Foret et al., 2021), GSAM (Zhuang et al., 2022), and SAGM (Wang et al., 2023b) on five datasets of Domain Bed with Res Net50 backbone. Table 1 shows that our Domain-Inspired SAM can mitigate issues arising from SAM s training under domain shifts, by comparing averaged in-domain and out-of-domain performance of leading SAM methods, with and without DISAM. In-domain results show domain-inspired perturbations enhance convergence, especially on the Terra Inc dataset with substantial domain gaps. In Out-of-domain results, DISAM consistently improves generalization,

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Table 1: Comparison with state-of-the-art domain generalization methods based on Res Net50. In-domain and Out-of-domain accuracies on five datasets from Domain Bed.

Algorithm PACS VLCS Office Home Terra Inc Domain Net Avg.

In-domain results

ERM 96.6 0.2 84.6 0.4 84.2 0.3 93.6 0.3 67.1 1.6 85.2

SAM 97.3 0.1 84.8 0.3 85.8 0.2 88.9 0.2 68.5 0.1 85.1 Domain-Inspired 97.8 0.1 84.4 0.3 86.3 0.2 94.8 0.2 70.2 0.1 86.7

GSAM 97.8 0.2 83.9 0.2 85.9 0.2 92.1 0.2 69.1 0.1 85.8 Domain-Inspired 97.9 0.1 85.1 0.4 86.2 0.2 94.8 0.3 70.0 0.1 86.8

SAGM 97.6 0.1 84.6 0.3 86.1 0.2 92.0 0.2 69.2 0.1 85.9 Domain-Inspired 97.9 0.1 85.0 0.2 86.5 0.3 94.9 0.2 70.5 0.1 87.0

Out-of-domain results

ERM 85.5 0.2 77.3 0.4 66.5 0.3 46.1 1.8 43.8 0.1 63.9 CORAL (SOTA) 86.2 0.3 78.8 0.3 68.7 0.3 47.6 1.0 41.5 0.1 64.5

SAM 85.8 0.2 79.4 0.1 69.6 0.1 43.3 0.7 44.3 0.0 64.5 Domain-Inspired 87.3 0.2 80.1 0.5 70.7 0.2 47.9 0.8 45.8 0.2 66.4

GSAM 85.9 0.1 79.1 0.2 69.3 0.0 47.0 0.8 44.6 0.2 65.1 Domain-Inspired 87.2 0.3 80.0 0.3 70.8 0.3 50.6 1.2 45.6 0.1 66.8

SAGM 86.6 0.2 80.0 0.3 70.1 0.2 48.8 0.9 45.0 0.2 66.1 Domain-Inspired 87.5 0.3 80.7 0.2 71.0 0.2 50.0 1.2 46.0 0.1 67.0 + CORAL 88.4 0.3 81.2 0.4 71.7 0.2 51.7 0.3 46.3 0.2 67.9

Table 2: Comparison with state-of-the-art domain generalization methods based on CLIP with Vi T-B/16. Out-of-domain accuracies on five datasets from Domain Bed.

Algorithm PACS VLCS Office Home Terra Inc Domain Net Avg.

Zero-shot 96.2 81.7 82.0 33.4 57.5 70.2

Co Op 96.8 81.2 84.2 44.9 59.9 73.4 + SAM 97.1 0.1 81.3 0.8 84.6 0.2 47.7 1.3 60.3 0.2 74.2 + DISAM 97.2 0.1 81.8 0.4 84.8 0.2 49.5 1.2 60.6 0.2 74.8

ERM 96.1 0.5 83.0 0.2 83.3 0.3 60.9 0.2 59.9 0.1 76.7 CLIPOOD1 97.3 0.1 85.0 0.4 87.0 0.2 60.4 0.7 63.5 0.1 78.6 CLIPOOD 2 96.6 0.4 84.1 0.3 86.1 0.2 59.7 0.8 63.1 0.1 77.9 + SAM 96.9 0.2 84.3 0.6 84.4 0.4 60.0 1.4 58.6 0.2 76.9 + DISAM 97.1 0.1 85.6 0.2 86.6 0.0 61.1 0.7 63.6 0.1 78.8

with average improvements of 1.9% for SAM, 1.7% for GSAM, and 1.9% for SAGM. Notably, SAM performs well when the performance gap between in-domain and out-of-domain is small but worse than ERM on datasets like Terra Inc with large gaps, which proves our analysis of SAM s shortcomings under domain shifts. This shows SAM s inconsistent convergence for large domain shifts, which DISAM addresses by incorporating domain-inspired adaptive adjustments based on domain-level convergence degree. Incorporating CORAL constraints, a recognized effective traditional DG method on Domain Bed improves SAGM with DISAM and sets new state-of-the-art results on all settings.

4.3 PERFORMANCE UNDER CLIP-BASED PRETRAINED LARGE MODEL

The CLIP-based large pretrained models (Radford et al., 2021) show great zero-shot performance but struggle with domain shifts in downstream tasks. We assess DISAM s out-of-domain results on CLIP using the experimental setup of CLIPOOD (Shu et al., 2023). We test two downstream adaptation methods: Co Op (Zhou et al., 2022a), an effective prompt learning approach, and CLIPOOD, an image encoder finetuning approach for DG problems. For Co Op, we use a 16-length learnable generic prompt and 5000 training steps, and For CLIPOOD settings, we follow Shu et al. (2023). As

1The original results reported in Shu et al. (2023). 2Our reproduced results are based on their official code at https://github.com/thuml/CLIPood.

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Table 3: Accuracy on Office Home and Domain Net with both domain shifts and open classes.

Split Algorithm Office Home Domain Net Avg. A C P R C I P Q R S

Zero-shot 86.7 75.9 89.6 92.2 72.6 51.8 65.4 13.6 83.5 67.2 72.6 Co Op 87.3 76.7 92.2 92.5 74.6 58.2 67.9 15.0 83.7 69.9 74.4 +SAM 89.2 79.6 93.0 93.7 73.5 58.4 67.8 14.8 83.6 69.5 75.1 +DISAM 88.0 80.5 92.7 92.4 75.0 59.9 68.7 14.9 84.4 70.5 75.3 CLIPOOD 88.9 79.5 92.2 94.0 76.3 58.7 69.9 17.5 85.6 72.4 76.0 +SAM 89.1 78.9 92.3 94.1 78.7 62.1 72.0 19.9 86.5 73.5 77.0 +DISAM 89.7 79.4 92.7 94.2 77.1 61.8 71.5 20.0 86.0 73.1 77.0

Zero-shot 76.8 59.7 88.7 86.4 69.7 45.0 67.0 14.3 83.9 60.8 67.4 Co Op 73.7 56.4 86.6 85.0 69.7 47.4 67.0 15.2 82.5 61.5 66.3 +SAM 75.2 59.1 89.6 86.0 71.1 49.2 69.3 15.4 82.2 62.9 67.9 +DISAM 79.3 61.5 90.9 88.4 72.1 49.0 69.6 15.5 85.5 62.9 69.6 CLIPOOD 75.2 58.6 87.5 85.8 69.3 46.4 67.2 15.2 83.2 60.6 66.9 +SAM 77.2 60.0 89.8 87.6 66.8 45.4 64.9 14.8 82.0 57.1 66.9 +DISAM 79.7 62.0 90.5 89.0 71.8 48.7 68.7 17.5 84.7 63.0 69.7

Zero-shot 82.6 67.3 88.8 89.5 71.4 47.1 66.2 13.8 83.4 63.4 69.8 Co Op 81.4 65.7 88.9 88.8 71.9 51.3 67.4 15.1 83.1 65.1 70.1 +SAM 83.5 69.1 91.3 90.1 72.3 52.8 68.6 15.1 82.9 65.7 71.5 +DISAM 84.2 70.1 91.7 90.4 73.4 53.2 69.1 15.2 85.0 65.8 72.2 CLIPOOD 83.3 68.8 89.9 90.1 72.5 51.2 68.5 16.4 84.4 65.6 71.4 +SAM 84.2 69.2 91.0 91.0 72.3 52.0 68.4 17.3 84.3 64.0 71.8 + DISAM 84.6 69.5 91.3 91.2 73.5 53.2 69.4 18.3 84.9 66.3 72.6

shown in Table 2, DISAM effectively mitigates the impact of domain shifts on model generalization during downstream task adaptation. In addition, as Co Op and CLIPOOD primarily focus on rapid adaptation with limited parameters, the overfitting risk can be alleviated through early stopping, resulting in the relatively marginal improvements for DISAM in Table 2. Despite this, when handling challenging tasks like Terra Inc and Domain Net, our approach still exhibits substantial enhancements.

4.4 PERFORMANCE UNDER OPEN-CLASS GENERALIZATION

0 1 2 3 4 5 6 7 8 9 Epoch

In-domain results on test domain C of Office Home

ERM SAM DISAM

0 1 2 3 4 5 6 7 8 9 Epoch

Out-of-domain results on test domain C of Office Home

ERM SAM DISAM

Figure 4: Comparison of Co Op based ERM, SAM and DISAM on accuracy curves for base/new classes. (Top: In Domain; Bottom: Out-of-Domain)

In this part, we evaluate the performance of our DISAM in a more realistic in-the-wild setting, where both domain shifts and open-class scenarios may arise in the test domain. This setting was first proposed by Shu et al. (2023). Office Home and Domain Net are selected to conduct related experiments because they offer an ample number of classes suitable for evaluating open-class situations. To delineate, we segregate the classes within each dataset into two categories, based on the class ID. The initial half denotes the base classes, and the latter half signifies the new classes. Based on Section 4.1, we eliminate the data corresponding to new classes in the training domains. Due to CLIP s open vocabulary property, we can evaluate the new classes on the unseen test domain.

As presented in Table 3, we evaluated the classification accuracy of "Base" classes, "New" classes, and "Total" classes in the test domain, where total classes represent the overall test domain. It revealed that the existing adaptation approach while performing well on base classes, lacks generalization on new classes during the fitting process. Our DISAM mitigates open-class overfitting using domain-level convergence constraints, improving performance by 3.3% over Co Op and 3.1% over CLIPOOD. Figure 4 provides a detailed analysis of open classes and domain shifts dimensions. ERM tends to overfit to both in-domain and base class. While SAM outperforms ERM, it struggles with sharp minima perturbations, failing to effectively escape from them. This difficulty hampers its generalization capabilities in larger models. Please refer to Appendix C.6 for more discussion about DISAM and other methods for open-class generalization.

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0.2 0.1 .02 .05 .01 5e-31e-3 0.0 0.84

Different on PACS

(a) ρ on PACS.

0.1 .07 .05 .02 .01 5e-31e-3 0.0 0.66

Different on Office Home

(b) ρ on Office Home.

0.0 0.1 0.3 0.5 0.7 Weight

Different weights on PACS

(c) λ on PACS.

0.0 0.1 0.3 0.5 0.7 Weight

0.69 0.69 0.69 0.70 0.70 0.70 0.70 0.71 0.71

Different weights on Office Home

(d) λ on Office Home.

Figure 5: Ablation study investigating the sensitivity of hyperparameters, namely perturbation amplitude ρ and variance constraint weight λ in DISAM.

0 10 20 30 40 50 Epoch

100 150 200 250 300 350 400

Var L(w; B)

Sharpness on test domain A of PACS

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SAM GSAM SAGM Algorithms

Speed per step (s)

Speed Comparison w/wo Domain-Inspired

/ w/wo DI / Res Net50/Vi T

Figure 6: (a) & (b): Sharpness curves for SAM and DISAM trained on PACS dataset, which show the trend of the estimated sharpness of the model on the test domain. (c): Computation cost with and without Domain-Inspired SAM used on Res Net50 and Vi T-B/16 backbone.

4.5 ABLATION STUDIES

Hyperparameter sensitivity. We performed a sensitivity analysis of the perturbation amplitude ρ, and the variance constraint weight λ, on the PACS and Office Home datasets. The default ρ and λ is set to 0.05 (Zhuang et al., 2022) and 0.1, respectively. As illustrated in Figure 5(a) and 5(b) within a wide range of ρ, DISAM consistently achieves stable and superior results compared to SAM. However, when ρ is too large or small, experimental results worsen. Large ρ hiders convergence, while small ρ weakens sharpness constraint, both affecting generalization. As for λ, Figure 5(c) and 5(d) show stable results when λ [0.1, 0.7]. However, larger λ values increase the variance due to excessive over-conditioning weight, which can also influence the convergence.

Estimated sharpness on unseen test domain. Estimating sharpness has a high computational cost. Early methods (Dinh et al., 2017b; Hochreiter & Schmidhuber, 1994b) relied on Monte Carlo sampling, but recent advancements (Jiang et al., 2023; 2020) use gradient-based approximations for efficiency. We assess model sharpness on unseen test domains at each epoch s end, based on the work of Jiang et al. (2023). As depicted in Figure 6(a) and 6(b), our DISAM achieves much smaller gradient variance Var{ L(wt; Bt)} than SAM during the whole training, indicating the incorporation of domain-inspired information can further reduce the sharpness of the loss surface.

Computation cost of DISAM. In Figure 6(c), we show the extra computational cost from adding domain-inspired perturbation direction generation versus the original algorithm (time cost/step, batch size 32, RTX 3090 GPU). Empirical findings show DISAM integration incurs minimal overhead ( 0.01s) across algorithms/backbones, linked solely to domain number and batch size, not model size, via strategic domain loss variance constraints for domain-level convergence consistency.

5 CONCLUSION

This paper presents Domain-Inspired Sharpness-Aware Minimization (DISAM), an algorithm that incorporates domain-level convergence consistency into the generation of SAM s perturbations, to address the dilemma under multiple domains. DISAM mitigates SAM s bias in domain shifts that can detrimentally impact the convergence during training, yielding perturbations towards highly converged domains and limiting those in less optimized ones. This is achieved by minimizing the variance of domain loss during perturbation generation, enabling an adaptive weight adjustment for each domain based on its convergence degree, thereby enhancing the convergence across training domains and generalization on unseen domains. Extensive experiments on the domain generalization benchmarks prove DISAM s superiority over existing methods. In addition, DISAM persistents generalization capabilities under parameter-efficient fine-tuning with large models like CLIP.

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ETHICS STATEMENT

This paper does not raise any ethics concerns. This study does not involve any human subjects, practices to data set releases, potentially harmful insights, methodologies and applications, potential conflicts of interest and sponsorship, discrimination/bias/fairness concerns, privacy and security issues, legal compliance, and research integrity issues.

REPRODUCIBILITY STATEMENT

All experiments were conducted using NVIDIA Ge Force RTX 3090 GPU, Python 3.9.15, Pytorch 1.12.1, and clip 1.0. Further details regarding experimental setups and implementation can be found in Section 4.1 and Appendix C, while theoretical proofs are provided in Appendix B. The principal code for implementing Domain-Inspired SAM is presented in Appendix D.

ACKNOWLEDGMENTS

This work is supported by the National Key R&D Program of China (No. 2022ZD0160702), STCSM (No. 22511106101, No. 18DZ2270700, No. 21DZ1100-100), 111 plan (No. BP0719010), and State Key Laboratory of UHD Video and Audio Production and Presentation. Ruipeng Zhang and Ziqing Fan are partially supported by Wu Wen Jun Honorary Doctoral Scholarship, AI Institute, Shanghai Jiao Tong University.

Martin Arjovsky, Léon Bottou, Ishaan Gulrajani, and David Lopez-Paz. Invariant risk minimization. ar Xiv preprint ar Xiv:1907.02893, 2019. 2, 17

Yogesh Balaji et al. Metareg: Towards domain generalization using meta-regularization. In Neur IPS, pp. 998 1008, 2018. 17

Sara Beery, Grant Van Horn, and Pietro Perona. Recognition in terra incognita. In Proceedings of the European conference on computer vision (ECCV), pp. 456 473, 2018. 6, 22

Shai Ben-David, John Blitzer, Koby Crammer, Alex Kulesza, Fernando Pereira, and Jennifer Wortman Vaughan. A theory of learning from different domains. Machine learning, 79:151 175, 2010. 1

Fabio Maria Carlucci, Paolo Russo, Tatiana Tommasi, and Barbara Caputo. Hallucinating agnostic images to generalize across domains. In 2019 IEEE/CVF International Conference on Computer Vision Workshop (ICCVW), pp. 3227 3234. IEEE, 2019. 17

Junbum Cha, Sanghyuk Chun, Kyungjae Lee, Han-Cheol Cho, Seunghyun Park, Yunsung Lee, and Sungrae Park. Swad: Domain generalization by seeking flat minima. Advances in Neural Information Processing Systems, 34:22405 22418, 2021. 17, 18

Woong-Gi Chang, Tackgeun You, Seonguk Seo, Suha Kwak, and Bohyung Han. Domain-specific batch normalization for unsupervised domain adaptation. In Proceedings of the IEEE/CVF conference on Computer Vision and Pattern Recognition, pp. 7354 7362, 2019. 17

Pratik Chaudhari, Anna Choromanska, Stefano Soatto, Yann Le Cun, Carlo Baldassi, Christian Borgs, Jennifer Chayes, Levent Sagun, and Riccardo Zecchina. Entropy-sgd: Biasing gradient descent into wide valleys. Journal of Statistical Mechanics: Theory and Experiment, 2019(12):124018, 2019. 1, 3

Laurent Dinh, Razvan Pascanu, Samy Bengio, and Yoshua Bengio. Sharp minima can generalize for deep nets. In International Conference on Machine Learning, pp. 1019 1028. PMLR, 2017a. 24

Laurent Dinh, Razvan Pascanu, Samy Bengio, and Yoshua Bengio. Sharp minima can generalize for deep nets. In International Conference on Machine Learning, pp. 1019 1028. PMLR, 2017b. 1, 3, 9, 16, 24

Published as a conference paper at ICLR 2024

Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. ar Xiv preprint ar Xiv:2010.11929, 2020. 1, 6

Qi Dou, Daniel Coelho de Castro, Konstantinos Kamnitsas, and Ben Glocker. Domain generalization via model-agnostic learning of semantic features. In Neur IPS, pp. 6450 6461, 2019. 17

Jiawei Du, Hanshu Yan, Jiashi Feng, Joey Tianyi Zhou, Liangli Zhen, Rick Siow Mong Goh, and Vincent Tan. Efficient sharpness-aware minimization for improved training of neural networks. In International Conference on Learning Representations, 2022a. 1, 16

Jiawei Du, Daquan Zhou, Jiashi Feng, Vincent Tan, and Joey Tianyi Zhou. Sharpness-aware training for free. Advances in Neural Information Processing Systems, 35:23439 23451, 2022b. 1, 16

Gintare Karolina Dziugaite and Daniel M. Roy. Computing nonvacuous generalization bounds for deep (stochastic) neural networks with many more parameters than training data. In Proceedings of the 33rd Annual Conference on Uncertainty in Artificial Intelligence (UAI), 2017. 1

Chen Fang, Ye Xu, and Daniel N Rockmore. Unbiased metric learning: On the utilization of multiple datasets and web images for softening bias. In Proceedings of the IEEE International Conference on Computer Vision, pp. 1657 1664, 2013. 6, 22

Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adaptation of deep networks. In International conference on machine learning, pp. 1126 1135. PMLR, 2017. 17

Pierre Foret, Ariel Kleiner, Hossein Mobahi, and Behnam Neyshabur. Sharpness-aware minimization for efficiently improving generalization. In International Conference on Learning Representations, 2021. 1, 3, 5, 6, 16, 17, 18, 21

Ishaan Gulrajani and David Lopez-Paz. In search of lost domain generalization. In International Conference on Learning Representations, 2021. URL https://openreview.net/forum? id=l Qd Xe XDo Wt I. 6, 22

Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770 778, 2016. 1, 6

Sepp Hochreiter and Jürgen Schmidhuber. Simplifying neural nets by discovering flat minima. Advances in neural information processing systems, 7, 1994a. 16

Sepp Hochreiter and Jürgen Schmidhuber. Simplifying neural nets by discovering flat minima. Advances in neural information processing systems, 7, 1994b. 3, 9, 24

Pavel Izmailov, Dmitrii Podoprikhin, Timur Garipov, Dmitry Vetrov, and Andrew Gordon Wilson. Averaging weights leads to wider optima and better generalization. ar Xiv preprint ar Xiv:1803.05407, 2018. 17, 18

Weisen Jiang, Hansi Yang, Yu Zhang, and James Kwok. An adaptive policy to employ sharpnessaware minimization. In The Eleventh International Conference on Learning Representations, 2023. URL https://openreview.net/forum?id=6Wl7-M2BC-. 1, 5, 9, 24

Yiding Jiang, Behnam Neyshabur, Hossein Mobahi, Dilip Krishnan, and Samy Bengio. Fantastic generalization measures and where to find them. In International Conference on Learning Representations, 2020. URL https://openreview.net/forum?id=SJg IPJBFv H. 1, 9, 24

Xin Jin, Cuiling Lan, Wenjun Zeng, and Zhibo Chen. Style normalization and restitution for domain generalization and adaptation. IEEE Transactions on Multimedia, 24:3636 3651, 2022. doi: 10.1109/TMM.2021.3104379. 17

Nitish Shirish Keskar, Dheevatsa Mudigere, Jorge Nocedal, Mikhail Smelyanskiy, and Ping Tak Peter Tang. On large-batch training for deep learning: Generalization gap and sharp minima. In International Conference on Learning Representations, 2017a. 16

Published as a conference paper at ICLR 2024

Nitish Shirish Keskar, Dheevatsa Mudigere, Jorge Nocedal, Mikhail Smelyanskiy, and Ping Tak Peter Tang. On large-batch training for deep learning: Generalization gap and sharp minima. In International Conference on Learning Representations, 2017b. URL https://openreview. net/forum?id=H1oy Rl Ygg. 1, 24

Daehee Kim, Youngjun Yoo, Seunghyun Park, Jinkyu Kim, and Jaekoo Lee. Selfreg: Self-supervised contrastive regularization for domain generalization. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 9619 9628, 2021. 17

Minyoung Kim, Da Li, Shell X Hu, and Timothy Hospedales. Fisher sam: Information geometry and sharpness aware minimisation. In International Conference on Machine Learning, pp. 11148 11161. PMLR, 2022. 16

David Krueger, Ethan Caballero, Joern-Henrik Jacobsen, Amy Zhang, Jonathan Binas, Dinghuai Zhang, Remi Le Priol, and Aaron Courville. Out-of-distribution generalization via risk extrapolation (rex). In International Conference on Machine Learning, pp. 5815 5826. PMLR, 2021. 2, 5, 17, 18

Jungmin Kwon, Jeongseop Kim, Hyunseo Park, and In Kwon Choi. Asam: Adaptive sharpness-aware minimization for scale-invariant learning of deep neural networks. In International Conference on Machine Learning, pp. 5905 5914. PMLR, 2021. 16

Bingcong Li and Georgios B Giannakis. Enhancing sharpness-aware optimization through variance suppression. ar Xiv preprint ar Xiv:2309.15639, 2023. 1, 16, 17

Da Li, Yongxin Yang, Yi-Zhe Song, and Timothy M Hospedales. Deeper, broader and artier domain generalization. In Proceedings of the IEEE international conference on computer vision, pp. 5542 5550, 2017. 1, 6, 22

Da Li, Yongxin Yang, Yi-Zhe Song, and Timothy M. Hospedales. Learning to generalize: Metalearning for domain generalization. In AAAI, 2018a. 17

Hao Li, Zheng Xu, Gavin Taylor, Christoph Studer, and Tom Goldstein. Visualizing the loss landscape of neural nets. Advances in neural information processing systems, 31, 2018b. 16

Haoliang Li, Sinno Jialin Pan, Shiqi Wang, and Alex C Kot. Domain generalization with adversarial feature learning. In CVPR, pp. 5400 5409, 2018c. 16

Ya Li, Xinmei Tian, Mingming Gong, Yajing Liu, Tongliang Liu, Kun Zhang, and Dacheng Tao. Deep domain generalization via conditional invariant adversarial networks. In Proceedings of the European conference on computer vision (ECCV), pp. 624 639, 2018d. 16

Yiying Li, Yongxin Yang, Wei Zhou, and Timothy Hospedales. Feature-critic networks for heterogeneous domain generalization. In International Conference on Machine Learning, pp. 3915 3924. PMLR, 2019. 17

Yajing Liu, Zhiwei Xiong, Ya Li, Xinmei Tian, and Zheng-Jun Zha. Domain generalization via encoding and resampling in a unified latent space. IEEE Transactions on Multimedia, 25:126 139, 2023. doi: 10.1109/TMM.2021.3121564. 17

Yong Liu, Siqi Mai, Xiangning Chen, Cho-Jui Hsieh, and Yang You. Towards efficient and scalable sharpness-aware minimization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 12360 12370, 2022. 1, 16

David A Mc Allester. Pac-bayesian model averaging. In Proceedings of the twelfth annual conference on Computational learning theory, pp. 164 170, 1999. 3

Peng Mi, Li Shen, Tianhe Ren, Yiyi Zhou, Xiaoshuai Sun, Rongrong Ji, and Dacheng Tao. Make sharpness-aware minimization stronger: A sparsified perturbation approach. In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho (eds.), Advances in Neural Information Processing Systems, 2022. 16

Published as a conference paper at ICLR 2024

Saeid Motiian, Marco Piccirilli, Donald A Adjeroh, and Gianfranco Doretto. Unified deep supervised domain adaptation and generalization. In Proceedings of the IEEE international conference on computer vision, pp. 5715 5725, 2017. 17

Ziwei Niu, Junkun Yuan, Xu Ma, Yingying Xu, Jing Liu, Yen-Wei Chen, Ruofeng Tong, and Lanfen Lin. Knowledge distillation-based domain-invariant representation learning for domain generalization. IEEE Transactions on Multimedia, pp. 1 11, 2023. doi: 10.1109/TMM.2023. 3263549. 17

Matthew D Norton and Johannes O Royset. Diametrical risk minimization: Theory and computations. Machine Learning, pp. 1 19, 2021. 17

Xingchao Peng, Qinxun Bai, Xide Xia, Zijun Huang, Kate Saenko, and Bo Wang. Moment matching for multi-source domain adaptation. In Proceedings of the IEEE/CVF international conference on computer vision, pp. 1406 1415, 2019. 6, 22

Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual models from natural language supervision. In International conference on machine learning, pp. 8748 8763. PMLR, 2021. 6, 7

Alexandre Rame, Corentin Dancette, and Matthieu Cord. Fishr: Invariant gradient variances for outof-distribution generalization. In International Conference on Machine Learning, pp. 18347 18377. PMLR, 2022. 5, 17, 18

Seonguk Seo, Yumin Suh, Dongwan Kim, Geeho Kim, Jongwoo Han, and Bohyung Han. Learning to optimize domain specific normalization for domain generalization. In Computer Vision ECCV 2020: 16th European Conference, Glasgow, UK, August 23 28, 2020, Proceedings, Part XXII 16, pp. 68 83. Springer, 2020. 17

Rui Shao, Xiangyuan Lan, Jiawei Li, and Pong C Yuen. Multi-adversarial discriminative deep domain generalization for face presentation attack detection. In CVPR, pp. 10023 10031, 2019. 16

Yuge Shi, Jeffrey Seely, Philip HS Torr, N Siddharth, Awni Hannun, Nicolas Usunier, and Gabriel Synnaeve. Gradient matching for domain generalization. ar Xiv preprint ar Xiv:2104.09937, 2021. 17, 18

Yang Shu, Xingzhuo Guo, Jialong Wu, Ximei Wang, Jianmin Wang, and Mingsheng Long. Clipood: Generalizing clip to out-of-distributions. In International Conference on Machine Learning, 2023. 6, 7, 8, 22

Baochen Sun and Kate Saenko. Deep coral: Correlation alignment for deep domain adaptation. In ECCV, pp. 443 450. Springer, 2016. 16

Hemanth Venkateswara, Jose Eusebio, Shayok Chakraborty, and Sethuraman Panchanathan. Deep hashing network for unsupervised domain adaptation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 5018 5027, 2017. 6, 22

Jindong Wang, Cuiling Lan, Chang Liu, Yidong Ouyang, Tao Qin, Wang Lu, Yiqiang Chen, Wenjun Zeng, and Philip S. Yu. Generalizing to unseen domains: A survey on domain generalization. IEEE Transactions on Knowledge and Data Engineering, 35(8):8052 8072, 2023a. doi: 10.1109/ TKDE.2022.3178128. 16

Pengfei Wang, Zhaoxiang Zhang, Zhen Lei, and Lei Zhang. Sharpness-aware gradient matching for domain generalization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 3769 3778, 2023b. 1, 5, 6, 16, 17, 22

Shujun Wang, Lequan Yu, Caizi Li, Chi-Wing Fu, and Pheng-Ann Heng. Learning from extrinsic and intrinsic supervisions for domain generalization. In ECCV, 2020. 17

Qinwei Xu, Ruipeng Zhang, Ya Zhang, Yanfeng Wang, and Qi Tian. A fourier-based framework for domain generalization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 14383 14392, 2021. 17

Published as a conference paper at ICLR 2024

Qinwei Xu, Ruipeng Zhang, Ziqing Fan, Yanfeng Wang, Yi-Yan Wu, and Ya Zhang. Fourier-based augmentation with applications to domain generalization. Pattern Recognition, 139:109474, 2023a. 17

Qinwei Xu, Ruipeng Zhang, Yi-Yan Wu, Ya Zhang, Ning Liu, and Yanfeng Wang. Simde: A simple domain expansion approach for single-source domain generalization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 4797 4807, 2023b. 17

Qinwei Xu, Ruipeng Zhang, Ya Zhang, Yi-Yan Wu, and Yanfeng Wang. Federated adversarial domain hallucination for privacy-preserving domain generalization. IEEE Transactions on Multimedia, pp. 1 13, 2023c. doi: 10.1109/TMM.2023.3257566. 17

Shen Yan, Huan Song, Nanxiang Li, Lincan Zou, and Liu Ren. Improve unsupervised domain adaptation with mixup training. ar Xiv preprint ar Xiv:2001.00677, 2020. 17

Lei Zhang, Yingjun Du, Jiayi Shen, and Xiantong Zhen. Learning to learn with variational inference for cross-domain image classification. IEEE Transactions on Multimedia, 25:3319 3328, 2023a. doi: 10.1109/TMM.2022.3158072. 17

Ruipeng Zhang, Qinwei Xu, Chaoqin Huang, Ya Zhang, and Yanfeng Wang. Semi-supervised domain generalization for medical image analysis. In 2022 IEEE 19th International Symposium on Biomedical Imaging (ISBI), pp. 1 5. IEEE, 2022. 17

Ruipeng Zhang, Qinwei Xu, Jiangchao Yao, Ya Zhang, Qi Tian, and Yanfeng Wang. Federated domain generalization with generalization adjustment. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 3954 3963, 2023b. 17

Xingxuan Zhang, Renzhe Xu, Han Yu, Hao Zou, and Peng Cui. Gradient norm aware minimization seeks first-order flatness and improves generalization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 20247 20257, 2023c. 16

Yang Zhao, Hao Zhang, and Xiuyuan Hu. Penalizing gradient norm for efficiently improving generalization in deep learning. In International Conference on Machine Learning, pp. 26982 26992. PMLR, 2022. 1, 16

Kaiyang Zhou, Yongxin Yang, Timothy Hospedales, and Tao Xiang. Deep domain-adversarial image generation for domain generalisation. In Proceedings of the AAAI conference on artificial intelligence, volume 34, pp. 13025 13032, 2020a. 17

Kaiyang Zhou, Yongxin Yang, Timothy Hospedales, and Tao Xiang. Learning to generate novel domains for domain generalization. In Computer Vision ECCV 2020: 16th European Conference, Glasgow, UK, August 23 28, 2020, Proceedings, Part XVI 16, pp. 561 578. Springer, 2020b. 17

Kaiyang Zhou, Yongxin Yang, Yu Qiao, and Tao Xiang. Domain generalization with mixstyle. ar Xiv preprint ar Xiv:2104.02008, 2021. 17

Kaiyang Zhou, Jingkang Yang, Chen Change Loy, and Ziwei Liu. Learning to prompt for visionlanguage models. International Journal of Computer Vision, 130(9):2337 2348, 2022a. 7

Zhihan Zhou, Jiangchao Yao, Yan-Feng Wang, Bo Han, and Ya Zhang. Contrastive learning with boosted memorization. In International Conference on Machine Learning, pp. 27367 27377. PMLR, 2022b. 17

Zhihan Zhou, Jiangchao Yao, Feng Hong, Ya Zhang, Bo Han, and Yanfeng Wang. Combating representation learning disparity with geometric harmonization. In Thirty-seventh Conference on Neural Information Processing Systems, 2023. 17

Juntang Zhuang, Boqing Gong, Liangzhe Yuan, Yin Cui, Hartwig Adam, Nicha C Dvornek, sekhar tatikonda, James s Duncan, and Ting Liu. Surrogate gap minimization improves sharpness-aware training. In International Conference on Learning Representations, 2022. 1, 5, 6, 9, 16

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Table of Contents

A Related Work 16 A.1 Sharpness-Aware Minimization (SAM) . . . . . . . . . . . . . . . . . . . . . . 16 A.2 Domain Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

B Details of DISAM 18 B.1 Comparative Analysis of DISAM Versus General Convergence Consistency . . . 18 B.2 Algorithm of DISAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 B.3 Proof of DISAM s Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 19 B.4 Discussion of the role of ρ in DISAM . . . . . . . . . . . . . . . . . . . . . . . 21

C Detailed Experiments 22 C.1 Detailed Experiment Setups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 C.2 Detailed Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 C.3 Details about Estimated Sharpness on Unseen Test Domain . . . . . . . . . . . 24 C.4 Details about Comparison of Computation Cost . . . . . . . . . . . . . . . . . . 24 C.5 Details about Convergence Curves of SAM and ERM . . . . . . . . . . . . . . 24 C.6 Detailed Analysis about Open-Class Generalization . . . . . . . . . . . . . . . 25 C.7 Detailed Analysis of Convergence Speed Comparison . . . . . . . . . . . . . . 25

D Pseudo code of DISAM 26

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A RELATED WORK

A.1 SHARPNESS-AWARE MINIMIZATION (SAM)

Numerous studies (Hochreiter & Schmidhuber, 1994a; Li et al., 2018b; Dinh et al., 2017b) have been conducted to enhance our understanding of the generalization capabilities of deep learning models through an exploration of the geometric properties of the loss landscape. These investigations have consistently demonstrated that deep neural networks exhibiting a flat minimum tend to exhibit superior generalization performance. In order to obtain a flat minimum, the Sharpness-Aware Minimization (SAM) approach (Foret et al., 2021) was proposed, which utilizes a base optimizer to simultaneously minimize both the vanilla training loss and the sharpness metric. The sharpness metric, as defined by (Keskar et al., 2017a), quantifies the flatness of a minimum through the eigenvalues of the Hessian matrix. In practice, SAM involves obtaining a fixed-length perturbation through gradient ascent on the initial parameter, followed by updating the gradient based on this perturbed parameter with respect to the initial parameter. Although SAM can result in a flat minimum and substantially enhance the generalization capability, it incurs a twofold increase in computational overhead. The variants of SAM have been extensively investigated from two perspectives: the first pertains to the enhancement of SAM s generalizability (Kwon et al., 2021; Zhuang et al., 2022; Zhang et al., 2023c; Zhao et al., 2022; Wang et al., 2023b), while the second focuses on improving its efficiency (Liu et al., 2022; Du et al., 2022a;b; Mi et al., 2022).

A.1.1 GENERALIZABILITY IMPROVEMENT OF SAM

One key problem of SAM is that the perturbation obtained by gradient ascent might disagree with sharpness since gradient ascent is only a first-order approximation of the sharpness calculation. Zhuang et al. (2022) introduced a surrogate gap to enhance the evaluation of sharpness, while (Wang et al., 2023b) integrated the perturbed loss and the surrogate gap from (Zhuang et al., 2022) into a unified objective. Additionally, (Zhao et al., 2022) revealed that SAM inherently optimizes both the empirical risk loss and the corresponding gradient norm. Besides, Fisher SAM (Kim et al., 2022) and ASAM (Kwon et al., 2021) achieved improved perturbations by adjusting the scales of the perturbation magnitudes. (Zhang et al., 2023c) further proposed Gradient norm Aware Minimization (GAM), which regularized the Hessian of the gradient norm. Va SSO (Li & Giannakis, 2023) focuses on addressing the issue of SAM s subpar performance in perturbation direction generation due to the noise introduced by mini-batch sampling.

A.1.2 EFFICIENCY IMPROVEMENT OF SAM

Due to the doubled overhead of SAM in comparison to a base optimizer like SGD (Stochastic Gradient Descent), considerable efforts have been devoted to mitigating this overhead. (Liu et al., 2022) introduced Look SAM as a means to reduce the number of perturbations. Meanwhile, (Mi et al., 2022) achieved sparse perturbations through the use of a binary mask. Furthermore, Du et al.explored various proxy methods (ESAM (Du et al., 2022a), SAF (Du et al., 2022b)) for computing perturbations, thereby replacing the gradient ascent derivation process employed in SAM.

A.2 DOMAIN GENERALIZATION

Domain generalization is a vital research direction that focuses on training models capable of generalizing well to unseen domains by leveraging knowledge from multiple source domains (Wang et al., 2023a). Over the past decade, several methods have been proposed to address the challenges of domain generalization. These methods can be broadly categorized into five main approaches: domain alignment, meta-learning, domain hallucination, domain disentanglement, and robustness training. In this section, we provide a brief overview of each of these categories.

A.2.1 DOMAIN ALIGNMENT-BASED METHOD

The goal of domain alignment is to mitigate discrepancies among distinct source domains by aligning the marginal feature distributions to extract domain-invariant representations. This objective can be accomplished using various strategies, including adversarial training (Li et al., 2018d; Shao et al., 2019), maximum mean discrepancy (Li et al., 2018c), moment matching (Sun & Saenko, 2016),

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self-supervised learning (Wang et al., 2020), or contrastive learning (Kim et al., 2021; Motiian et al., 2017; Zhou et al., 2022b; 2023). All of these methods improve generalization across unseen domains by either directly or indirectly reducing the discrepancy between different feature distributions and imposing domain-invariant constraints on these discriminative features.

A.2.2 META LEARNING-BASED METHODS

These approaches aim to address unforeseen domain shifts and enhance the generalizability of models to such shifts through meta-optimization, achieved by partitioning the training domains into distinct meta-train and meta-test domains. (Li et al., 2018a) first introduced meta learning into DG, following the concept of Modal-Agnostic Meta-Learning (MAML) (Finn et al., 2017). Subsequently, (Balaji et al., 2018) designed a weight regularizer based on the meta-learning framework, while (Li et al., 2019) chose to meta-learn a feature critic loss. (Dou et al., 2019) constrained the invariance of learned semantic relations between the meta-train and meta-test domains. Additionally, (Zhang et al., 2023a) integrated meta learning into a Bayesian framework and enforced the model to learn a meta-variational distribution to enhance knowledge transfer.

A.2.3 DOMAIN HALLUCINATION-BASED METHODS

Domain hallucination, also known as data augmentation in the presence of domain shifts, aims to encompass a wider range of domain variations by generating additional training samples from fictional domains while preserving their semantic integrity. Early approaches such as (Xu et al., 2021; 2023a; Zhang et al., 2022; Xu et al., 2023b; Zhou et al., 2020a; Yan et al., 2020; Zhou et al., 2020b; Carlucci et al., 2019; Xu et al., 2023c) involve cross-domain data augmentation in the input space and can be categorized into non-parametric and adversarial sample-based approaches. Non-parametric methods (Xu et al., 2021; 2023a; Yan et al., 2020; Zhang et al., 2022) employ traditional image transformations to achieve enhancement, while adversarial sample-based methods (Xu et al., 2023b; Zhou et al., 2020a;b; Carlucci et al., 2019; Xu et al., 2023c) generate samples from a new domain through adversarial training. Adversarial training ensures the quality of generation by enforcing consistency in terms of category among the samples from the generated fictional domain. Some recent work focuses on augmentation in the latent space (Liu et al., 2023; Zhou et al., 2021), which achieves more efficient augmentation perturbations by applying perturbations to the latent features to improve the generalization of the model.

A.2.4 DOMAIN DISENTANGLEMENT-BASED METHODS

In contrast to the majority of domain generalization approaches that aim for domain invariance, disentanglement-based approaches focus on separating the domain-invariant and domain-specific components. To achieve this, Seo et al. (2020) introduced domain-specific batch normalization (Chang et al., 2019) for each training domain, effectively balancing feature discrimination and invariance. In a similar vein,Jin et al. (2022) designed a style restitution module that encourages the separation of taskrelevant and task-irrelevant features. Furthermore, Niu et al. (2023) proposed a two-stage distillation approach, aimed at learning a domain-invariant representation while preserving domain-specific features.

A.2.5 ROBUSTNESS TRAINING-BASED METHODS

The objective of robustness training-based methods is to incorporate constraints that enhance the model s robustness or flatness during the training process. Robustness-related methods aim to learn domain-invariant representations by employing a technique known as Invariant Risk Minimization (IRM) (Arjovsky et al., 2019). By minimizing the risk across different domains, these methods (Arjovsky et al., 2019; Krueger et al., 2021; Norton & Royset, 2021; Shi et al., 2021; Rame et al., 2022; Li & Giannakis, 2023) seek to learn features that are insensitive to domain variations, thereby improving the model s ability to generalize. On the other hand, a separate class of flatness-related methods (Izmailov et al., 2018; Cha et al., 2021; Zhang et al., 2023b; Foret et al., 2021; Wang et al., 2023b) aims to address the effects of domain shifts by identifying flat minima. These methods strive to find regions in the loss landscape where small perturbations in the input have minimal impact on the model s predictions. By leveraging flat minima, these methods enhance the model s robustness to domain variations.

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Table 4: Comparison of SAM-based methods and other robustness training-based methods on the optimization objective.

Method Total Optimization Function Optimization on w Optimization on ϵ

i=1 αi Li(w) Same to left

V-REx min w

i=1 αi Li(w) + βVar{Li(w)}M i=1 Same to left

i=1 αi Li(w) γ 2 M(M 1)

i,j [1,M] Li(w) Lj(w) Same to left

Fishr min w

i=1 αi Li(w) λ 1

i=1 Li(w) L(w) 2 Same to left

SAM min w max ϵ 2 ρ

i=1 αi Li(w + ϵ) min w

i=1 αi Li(w + ϵ) max ϵ 2 ρ

i=1 αi Li(w + ϵ)

DISAM min w max ϵ 2 ρ

i=1 αi Li(w + ϵ) λVar{Li( ˆw + ϵ)}M i=1

i=1 αi Li(w + ϵ) max ϵ 2 ρ

i=1 αi Li(w + ϵ) λVar{Li(w + ϵ)}M i=1

In Table 4, we provide the comparison of optimization objectives between representative algorithms in the two categories. Domain-invariant methods solely concentrate on optimizing the parameter w. For instance, V-REx (Krueger et al., 2021) directly minimizes the variance of the domain loss, which can have a detrimental effect on convergence. Similarly, Fish (Shi et al., 2021) and Fishr (Rame et al., 2022) impose constraints on gradient updates. SAM-based methods require the estimation of sharpness, so in addition to optimizing the parameters w, they also need to optimize the perturbation directions ϵ.

This paper primarily concentrates on the flatness-based method, which encompasses two main approaches for enhancing the flatness of the model. The first approach involves leveraging the self-ensemble of multiple minima attained during the training process to passively acquire a result that favors flatness minima. Notable examples of this approach include Stochastic Weight Averaging (SWA) (Izmailov et al., 2018) and Stochastic Weight Averaging Densely (SWAD) (Cha et al., 2021). The second approach involves directly optimizing for flatness and is referred to as Sharpness Aware Minimization (SAM) (Foret et al., 2021). In the subsequent section, we will delve into a comprehensive review of the relevant literature pertaining to these approaches.

B DETAILS OF DISAM

B.1 COMPARATIVE ANALYSIS OF DISAM VERSUS GENERAL CONVERGENCE CONSISTENCY

Here, we present a thorough examination of the distinctions between our proposed DISAM framework and the broader, conventional convergence consistency issue like V-REx(Krueger et al., 2021) and Fishr(Rame et al., 2022). Specifically, we address the following aspects:

Distinct Focus: DISAM focuses on the issue where SAM-based methods are unable to accurately estimate sharpness in domain shift scenarios, leading to the ineffective sharpness minimization and reduction in generalization performance.

Enhancing on Top of General Methods: While traditional solutions(Krueger et al., 2021;

Rame et al., 2022; Shi et al., 2021) aim at convergence consistency in parameter optimization, DISAM s methodology is distinct and orthogonal. It builds upon methods like V-REx(Krueger et al., 2021) and Fishr(Rame et al., 2022), but goes further in enhancing out-of-domain generalization through better sharpness minimization. This is evident in our experiments, where combining DISAM with Fishr results in significant performance gains (shown in Table 5).

We also provide extensive experimental results to validate DISAM s effectiveness and its practical implications in various domain-shift scenarios.

It is imperative to reiterate the contributions of our DISAM. We provide a detailed exposition of how simplistic applications of SAM compromise training robustness, especially when dealing with domain shifts. DISAM strategically mitigates these issues by finely tuning the perturbation vectors and their

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Table 5: Comparison with other general convergence consistency methods.

Algorithm PACS VLCS Office Home Terra Inc Domain Net Avg.

V-REx 84.9 78.3 66.4 46.4 33.6 61.9 V-REx + DISAM 85.8 78.4 70.5 45.9 42.3 64.6

Fishr 86.9 78.2 68.2 53.6 41.8 65.7 Fishr + DISAM 87.5 79.2 70.7 54.8 43.9 67.2

Algorithm 1 Domain-Inspired Sharpness-Aware Minimization (DISAM).

Input: Source Domains S = {D1, , DM}, initial model w1, perturbation ratio ρ, variance constraint weight λ, learning rate ηt, training iterations T. Output: Generalization model w T .

1: for t in 1 T do 2: Sample mini-batch B = {B1, , BM} S, where Bi Di and |Bi| 0. 3: Compute the domain-inspired loss gradient: LDI(wt; B) = L(wt; B) λ Var{Li(wt); Bi}M i=1.

4: Get the perturbation weight: wasc t = wt + ρ LDI(wt;B)

LDI(wt;B) .

5: Update weights: wt+1 = wt ηt L(wasc t ; B)}. 6: end for

location points, thus significantly enhancing model generalization. Furthermore, we underscore the notable enhancements achieved with DISAM, as corroborated by comprehensive experimental analyses and the ensuing performance metrics.

B.2 ALGORITHM OF DISAM

We give specific algorithmic details for DISAM in Algorithm 1, and the python code implementation is in Appendix D.

B.3 PROOF OF DISAM S CONVERGENCE

Theorem 1. (Convergence During Training). Consider a non-convex function L(w) with Lipschitzsmooth constant L and lower bound Lmin. With the bounded norm assumption of noisy stochastic gradients ( Lp(w) 2 G) at the t-step, the learning rate ηt = η0/

t and a fixed perturbation amplitude ρ, we have:

t=1 E Lp(wt) 2 2 Lp(w0) Lmin

T + (LG2 + ρ2LΓ2)η0 log(T)

where in SAM, Γ = L and when use DISAM Γ L.

Proof. For simplicity of notation, we denote the update at step t as dt = ηtg(t) p , where ηt is the decayed learning rate and gt p is the expected gradient of perturbation loss Lp. By L-smoothness of the loss function L and the definition of Lp(wt) = L(wasc t ), where wasc t represents the parameters after the perturbation of gradient ascent, we have:

Lp(wt+1) = L(wasc t+1) L(wasc t ) + L(wasc t ), wasc t+1 wasc t + L

2 wasc t+1 wasc t 2 (10)

where L is the Lipschitz constant of loss L and with the definition of dt = wt+1 wt and wasc t = wt + ϵt, we have:

Lp(wt+1) L(wasc t ) + L(wasc t ), wt+1 + ϵt+1 wt ϵt + L

2 wt+1 + ϵt+1 wt ϵt 2

L(wasc t ) + L(wasc t ), dt + L dt 2 + L(wasc t ), ϵt+1 ϵt + L ϵt+1 ϵt 2

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Let us take the expectation conditioned on observations up to step t. For the sake of simplicity, we use the symbol E to denote the expectation over all possible data points on the training data distribution. Moreover, given the observations up to step t, we can use the definition of dt to obtain:

E[Lp(wt+1)] L(wasc t ) ηt L(wasc t ), E[g(t) p ] + η2 t LE g(t) p 2

+ E L(wasc t ), ϵt+1 ϵt + LE ϵt+1 ϵt 2

L(wasc t ) ηt E L(wasc t ) 2 2 + η2 t LG2

+ E L(wasc t ), ϵt+1 ϵt + LE ϵt+1 ϵt 2

By the definition of ϵt, we have:

ϵt = ρ g(t)

g(t) , ϵt+1 = ρ g(t+1)

g(t+1) (13)

where g(t) is the gradient of L at wt with the domain-inspired gradient in Eq.( 8). We denote ϵt = Ld(wt) = PM i=1 βi t Li(wt), where βi t = αi 2λ

M Li(wt) 1 M PM j=1 Lj(wt) . Since both ϵt and ϵt+1 are unit length vectors, ϵt+1 ϵt can be bounded by the arc length ϕt between them. Here the difference vector between ϵt+1 and ϵt can be regarded as a random noise in the gradient direction and in SAM ρ ηt, so the expectation of the inner product with the gradient direction L(wasc t ) can be approximated as 0 (wasc t is updated from wt with a larger step size ρ, and its gradient direction can be considered approximately independent of the gradient direction in the neighborhood of wt, so its difference with the inner product between ϵt+1 and ϵt is negligible). Therefore, we have:

E[Lp(wt+1)] L(wasc t ) ηt E L(wasc t ) 2 2 + η2 t LG2

+ E[ L(wasc t ), ϵt+1 L(wasc t ), ϵt ] + Lρ2E g(t+1)

g(t+1) g(t)

L(wasc t ) ηt E L(wasc t ) 2 2 + η2 t LG2 + Lρ2ϕ2 t

Because of the continuity of the optimization, the angle between the gradient perturbations before and after is small. Therefore, we can get:

ϕt tan ϕt = ϵt+1 ϵt

ϵt + O(ϕ2 t) = Ld(wt+1) Ld(wt)

Ld(wt) + O(ϕ2 t)

= PM i=1 βi t+1 Li(wt+1) βi t Li(wt) Ld(wt) + O(ϕ2 t)

= PM i=1 (βi t+1 βi t) Li(wt+1) + βi t( Li(wt+1) Li(wt)) Ld(wt) + O(ϕ2 t)

Here we consider the effect of the weight coefficients generated by DISAM in the perturbation of Ld, for the part of Li(wt) that is large, βi t is smaller, we assume that the larger Li(wt) is, the larger the corresponding gradient Li(wt) is also, and after one optimization process, the variability between the domains will be reduced, so βi t+1 is a little bit smaller than the weight of βi t, in the place where the gradient is large, and by the rearranging inequality, we can obtained:

i=1 βi t+1 Li(wt+1)

i=1 βi t+1 Li(wt) (16)

So bring Eq.( 16) to Eq.( 15), and with L(wt+1) = L(wt + dt) = L(wt) + Hdt + O( dt 2) we can get:

ϕt PM i=1(βi t+1 βi t) Li(wt+1) + Hdt + O( dt 2)

Ld(wt) + O(ϕ2 t) ηtΓ (17)

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Here since PM i=1(βi t+1 βi t) Li(wt+1) 0, we use Γ to denote an upper bound that is smaller than L.

Plug Eq.( 17) into Eq.( 14), we have:

E[Lp(wt+1)] L(wasc t ) ηt E L(wasc t ) 2 2 + η2 t LG2 + Lρ2η2 t Γ2 (18)

Perform telescope sum and note that ηT = η0

T , we have:

ELp(w T ) Lp(w0)

t=1 ηt E L(wasc t ) 2 2 + (LG2 + ρ2LΓ2)η2 0

t=1 ηt E L(wasc t ) 2 2 + (LG2 + ρ2LΓ2)η2 0 log(T)

t=1 E L(wasc t ) 2 2

t=1 ηt E L(wasc t ) 2 2 Lp(w0) Lmin + (LG2 + ρ2LΓ2)η2 0 log(T)

Note that ηT = η0

T , we have:

t=1 E L(wasc t ) 2 2 Lp(w0) Lmin

T + (LG2 + ρ2LΓ2)η0 log(T)

The influence of λ: In the proof of Theorem 1, specifically in Eq. (15), λ is integrated into β, serving as a hyperparameter that regulates the weight adjustment in DISAM. It functions by modulating the degree of correction for domain shifts:

βi t = αi 2λ

The choice of λ influences how aggressively DISAM responds to variance or domain shifts, with a higher λ leading to more pronounced adjustments in β. Our experimental analysis in Figure 5(c) and 5(d), reveals that DISAM s performance remains relatively stable across a wide range of λ values. However, choosing too large λ can result in overly aggressive early training adjustments, yielding the increased variance in repeated experiments. Consequently, we adopted a default λ value of 0.1 in all experiments.

B.4 DISCUSSION OF THE ROLE OF ρ IN DISAM

Here, we provide a detailed discussion on how ρ affects both generalization and convergence. First, we introduce the generalization theorem of the upper bound on generalization error, which is only related to the magnitude of ρ, and DISAM follows the same upper bound on generalization error as SAM. In the SAM framework, the parameter ρ plays a crucial role in determining generalizability. As established in SAM (Foret et al., 2021), there exists an upper bound on the generalization error for SAM, suggesting that a larger ρ could potentially enhance generalization, provided that convergence is not impeded. Here is the relevant generalization bound from SAM (Foret et al., 2021): Theorem 2. (Generalization Bound of SAM). For any ρ > 0 and any distribution D, with probability 1 δ over the choice of the training set S D,

LD(w) max ϵ 2 ρ LS(w + ϵ) +

v u u u tk log 1 + w 2 2 ρ2 (1 + q

k )2 + 4 log n

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where n = |S|, k is the number of parameters and we assumed LD(w) Eϵi N(0,ρ)[LD(w + ϵ)]. This theorem s proof focuses solely on the magnitude of ρ, thus affirming the applicability of this theoretical framework to DISAM.

When considering domain shift, the upper bound on generalization error for the test domain is: Theorem 3. (PAC-Bayesian Generalization Bound). For any ρ > 0 and the unseen domain T, suppose we have multi-source domains S = {S1, S2, } with a total of N samples. Let H be the hypothesis space and Ωbe the corresponding parameter space, where the VC dimension of H is d. We denote the domain divergence between two domains Di and Dj on the hypothesis space H as d H H(Di, Dj). Then, for any δ (0, 1), with probability at least 1 δ, for all w Ω, we have:

LT (w) max ϵ 2 ρ LS(w + ϵ) + 1

2d H H(S, T) +

log d + log 1

v u u u u t

1 + w 2 2 ρ2

2 + 4 log N

where λ is the optimal combined risk on T and S that can be achieved by the parameters in Ω.

Combining this with the convergence theorem (Theorem 1), there is a trade-off with respect to ρ. A larger ρ might theoretically enhance generalization but poses greater challenges for convergence. This reflects the intuitive notion that searching for flatter minima across a broader range is inherently more complex, which can potentially affect training efficiency. However, if LS(w + ϵ) can be converged with a sufficiently small value, a larger ρ corresponds to better generalization. DISAM, compared to SAM, converges faster, which means that under the same convergence speed, a larger ρ can be used to achieve better generalization. This advantage is empirically showcased in Figure 3(c) and (d), where we demonstrate that DISAM effectively employs a larger ρ compared to traditional SAM. This ensures both convergence and enhanced generalization. Such a capability to balance between convergence efficiency and generalization is a distinguishing feature of DISAM over conventional SAM methods.

C DETAILED EXPERIMENTS

C.1 DETAILED EXPERIMENT SETUPS

We present the detailed results obtained from five datasets, namely PACS (Li et al., 2017) (9,991 images, 7 classes, 4 domains), VLCS (Fang et al., 2013) (10,729 images, 5 classes, 4 domains), Office Home (Venkateswara et al., 2017) (15,588 images, 65 classes, 4 domains), Terra Incognita (Beery et al., 2018) (abbreviated as Terra Inc, 24,788 images, 10 classes, 4 domains), and Domain Net (Peng et al., 2019) (586,575 images, 345 classes, 6 domains), following the Domain Bed benchmark (Gulrajani & Lopez-Paz, 2021) with the Res Net50 backbone architecture. We set the hyperparameters for the Domain-Inspired + SAM method as follows: ρ = 0.5 and λ = 0.1 for PACS, VLCS, Office Home, and Domain Net; for Terra Inc, we use ρ = 0.01 and λ = 0.2. Both Domain-Inspired + GSAM and Domain-Inspired + SAGM employ the strategy described in the supplementary material of SAGM (Wang et al., 2023b). As for the Co Op with CLIP, we set the batch size as 32 and the default learning rate as 2e-3. Given the detailed experimental hyperparameter settings provided in the SAGM supplement (Wang et al., 2023b) and the official open-source CLIPOOD code (Shu et al., 2023), we directly applied these official settings. The results, replicated using the official open-source CLIPOOD code, are presented in Table 2 of the main text.

As for the experiments on open class, we found that CLIPOOD requires a lower learning rate and correspondingly lower ρ, and therefore used learning rate 1e-07 and ρ 1e-05 as default settings.

C.2 DETAILED EXPERIMENTAL RESULTS

We present the specific out-of-domain experimental results for each dataset in Table 1, corresponding to each leave-one-domain-out setting.

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Table 6: Comparison with state-of-the-art domain generalization methods. Out-of-domain accuracies on the PACS dataset with Res Net50 backbone.

Algorithm Art Cartoon Photo Sketch Avg.

ERM 84.7 0.4 80.0 0.6 97.2 0.3 79.3 1.0 85.5

SAM 85.6 2.1 80.9 1.2 97.0 0.4 79.6 1.6 85.8 Domain-Inspired 87.1 0.4 81.9 0.5 96.2 0.3 83.1 0.7 87.1

GSAM 86.9 0.1 80.4 0.2 97.5 0.0 78.7 0.8 85.9 Domain-Inspired 88.4 0.2 81.1 0.3 97.0 0.0 82.3 0.6 87.2

SAGM 87.4 0.2 80.2 0.3 98.0 0.2 80.8 0.6 86.6 Domain-Inspired 89.7 0.6 81.5 0.0 97.0 0.1 81.8 0.6 87.5 +CORAL 89.8 0.5 82.9 0.2 97.4 0.2 83.4 0.2 88.4

Table 7: Comparison with state-of-the-art domain generalization methods. Out-of-domain accuracies on the VLCS dataset with Res Net50 backbone.

Algorithm Caltech Label Me Pascal Sun Avg.

ERM 98.0 0.3 64.7 1.2 71.4 1.2 75.2 1.6 77.3

SAM 99.1 0.2 65.0 1.0 73.7 1.0 79.8 0.1 79.4 Domain-Inspired 99.3 0.0 66.3 0.5 81.0 0.1 73.2 0.1 79.9

GSAM 98.7 0.3 64.9 0.2 74.3 0.0 78.5 0.8 79.1 Domain-Inspired 99.8 0.0 66.6 0.1 74.2 0.9 79.3 0.1 80.0

SAGM 99.0 0.2 65.2 0.4 75.1 0.3 80.7 0.8 80.0 Domain-Inspired 99.9 0.1 66.1 0.6 75.1 0.3 81.8 0.0 80.7 +CORAL 99.7 0.1 67.8 0.7 75.5 0.8 81.6 0.2 81.2

Table 8: Comparison with state-of-the-art domain generalization methods. Out-of-domain accuracies on the Office Home dataset with Res Net50 backbone.

Algorithm Art Clipart Product Real World Avg.

ERM 61.3 0.7 52.4 0.3 75.8 0.1 76.6 0.3 66.5

SAM 64.5 0.3 56.5 0.2 77.4 0.1 79.8 0.4 69.6 Domain-Inspired 65.8 0.2 55.6 0.2 79.2 0.2 80.6 0.1 70.3

GSAM 64.9 0.1 55.2 0.2 77.8 0.0 79.2 0.2 69.3 Domain-Inspired 65.7 0.3 57.4 0.3 79.4 0.1 80.7 0.3 70.8

SAGM 65.4 0.4 57.0 0.3 78.0 0.3 80.0 0.2 70.1 Domain-Inspired 67.2 0.0 56.3 0.3 79.6 0.2 81.0 0.3 71.0 +CORAL 68.5 0.1 57.6 0.1 79.3 0.4 81.3 0.2 71.7

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Table 9: Comparison with state-of-the-art domain generalization methods. Out-of-domain accuracies on the Terra Inc dataset with Res Net50 backbone.

Algorithm L100 L38 L43 L46 Avg.

ERM 49.8 4.4 42.1 1.4 56.9 1.8 35.7 3.9 46.1

SAM 46.3 1.0 38.4 2.4 54.0 1.0 34.5 0.8 43.3 Domain-Inspired 46.2 2.9 41.6 0.1 58.0 0.5 40.5 2.2 46.6

GSAM 50.8 0.1 39.3 0.2 59.6 0.0 38.2 0.8 47.0 Domain-Inspired 56.7 1.5 46.7 1.0 59.2 0.7 39.9 1.5 50.6

SAGM 54.8 1.3 41.4 0.8 57.7 0.6 41.3 0.4 48.8 Domain-Inspired 57.6 1.6 44.8 1.5 58.6 1.2 38.9 0.6 50.0 + CORAL 57.9 0.3 46.6 0.6 59.9 0.3 42.5 0.1 51.7

Table 10: Comparison with state-of-the-art domain generalization methods. Out-of-domain accuracies on the Domain Net dataset with Res Net50 backbone.

Algorithm Clipart Infograph Painting Quickdraw Real Sketch Avg.

ERM 62.8 0.4 20.2 0.3 50.3 0.3 13.7 0.5 63.7 0.2 52.1 0.5 43.8

SAM 64.5 0.3 20.7 0.2 50.2 0.1 15.1 0.3 62.6 0.2 52.7 0.3 44.3 Domain-Inspired 65.9 0.2 20.7 0.2 51.7 0.3 16.6 0.3 62.8 0.5 54.8 0.4 45.4

GSAM 64.2 0.3 20.8 0.2 50.9 0.0 14.4 0.8 63.5 0.2 53.9 0.2 44.6 Domain-Inspired 65.7 0.1 21.3 0.1 52.2 0.1 15.6 0.0 64.5 0.2 54.1 0.2 45.6

SAGM 64.9 0.2 21.1 0.3 51.5 0.2 14.8 0.2 64.1 0.2 53.6 0.2 45.0 Domain-Inspired 65.9 0.2 21.4 0.0 52.6 0.1 15.8 0.0 65.3 0.0 54.8 0.2 46.0 +CORAL 66.4 0.3 21.9 0.2 53.1 0.1 16.1 0.0 65.3 0.0 55.0 0.0 46.3

C.3 DETAILS ABOUT ESTIMATED SHARPNESS ON UNSEEN TEST DOMAIN

Estimating sharpness involves a significant computational overhead. In the earliest methods, Monte Carlo random sampling was the only viable approach (Dinh et al., 2017b; Hochreiter & Schmidhuber, 1994b). However, recent advancements have introduced efficient approximation techniques based on gradients to estimate sharpness (Jiang et al., 2023; 2020). Based on the work of Jiang et al. (2023), we assess the sharpness of the training model on the unseen test domain at the end of each epoch. Sharpness is commonly characterized by the eigenvalues of the Hessian matrix (Keskar et al., 2017b; Dinh et al., 2017a), but direct computation incurs substantial overhead. To address this, a computationally efficient measurement of sharpness is proposed by Jiang et al. (2020), which utilizes the gradient variance Var{ L(wt; Bt)} as an estimate (Bt represent the batch data sampled at step t).

C.4 DETAILS ABOUT COMPARISON OF COMPUTATION COST

We selected the PACS dataset for experimentation, using a platform with a 16-core CPU, a single RTX3090 GPU, and 64GB RAM. The time overhead for one training step was calculated and averaged over 500 iterations. Due to the lack of optimization for parallel acceleration in the variance calculation code, which employs a simple for loop approach, the actual overhead is larger than theoretically expected. Nonetheless, DISAM s advantage lies in its overhead being unrelated to gradient size, but only to batch size and domain number. This drawback can be addressed through parallel code optimization, and no additional memory overhead is present.

C.5 DETAILS ABOUT CONVERGENCE CURVES OF SAM AND ERM

In this section, we provide a detailed analysis of the convergence curves depicted in Figure 1(b). Figure 7(a) presents the same as Figure 1(b), with a normalized representation of the loss curves,

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(a) Convergence curves under domain shifts

(b) Loss curves under domain shifts

Figure 7: Illustration of SAM s degradation of the training process under domain shifts. (a) Convergence curves of SAM and ERM for each domain during training, with the convergence degree normalized to [0,1]. (b) Loss curves of SAM and ERM for each domain during training.

0 1000 2000 3000 4000 5000

Convergence curve on test domain A

ERM SAM DISAM

0 1000 2000 3000 4000 5000

Convergence curve on test domain S

ERM SAM DISAM

Figure 8: Convergence curves for ERM, SAM and DISAM. (a) & (b) show the trend of L(w) during the training process on PACS dataset.

ranging from 0 to 1, achieved by subtracting the minimum loss value and dividing by the maximum loss value. Our intention is to emphasize the inconsistency in convergence trends across different SAM domains, as illustrated by the optimization overshoot observed in Figure 7(a). Figure 7(b) showcases the actual loss change curve. It is apparent that due to the consistency issue encountered during the early phase of convergence, the in-domain convergence is compromised, resulting in poor generalization performance in the out-of-domain scenario.

C.6 DETAILED ANALYSIS ABOUT OPEN-CLASS GENERALIZATION

In the experiments of open-class generalization, as presented in Table 3 and Figure 4 of section 4.4, we specifically explore the effectiveness of DISAM for parameter-efficient fine-tuning (PEFT). Our quantitative analysis compares the performance of ERM, SAM, and DISAM in fine-tuning scenarios. As shown in Table 3, although Co Op and CLIPOOD perform better on base classes than zero-shot, their performance on new classes is worse than zero-shot. This suggests that the fine-tuned parameters overfit to the existing training data distribution from both the domain and class perspectives. This overfitting is particularly detrimental to the generalization of large VLM models, which often have feature representations too rich for the downstream task, especially when only a small number of parameters are fine-tuned. Figure 4 visualizes the change in performance trends during the training process, and we observe a trend where ERM initially performs well on base classes but then exhibits a decline on new classes, suggesting a collapse of the feature space onto the training data classes. Although SAM offers some relief from overfitting, its performance on new classes does not match zero-shot levels. In contrast, DISAM, by minimizing sharpness more effectively, shows improved performance on new classes, especially in domain shift scenarios.

C.7 DETAILED ANALYSIS OF CONVERGENCE SPEED COMPARISON

We presented a comparison of the convergence speed with the inclusion of ERM in Figure 8. It can be observed that although DISAM converges much faster compared to SAM, the overall convergence speed is still slower than ERM due to the introduction of ρ.

Published as a conference paper at ICLR 2024

D PSEUDO CODE OF DISAM

We present pseudo-code for DISAM using Python syntax. Py Torch is utilized as the deep learning experimental framework. The code for the optimizer in the SAM-based method can be referenced from the provided open source links in the relevant papers.

Listing 1: Training Code for DISAM

def train_epoch_disam(dataloader, model, optimizer):

""" Train the DISAM model for one epoch.

dataloader (Data Loader): The training dataloader. model (nn.Module): The training model. optimizer (Optimizer): The SAM-based optimizer, such as SAM,

GSAM, and SAGM. """ model.train() for i, data_list in tqdm(enumerate(dataloader)): imgs, labels = data_list imgs, labels = imgs.cuda(), labels.cuda() preds = model(imgs) # Calculate domain losses and total loss using the

cross-entropy loss function domain_loss_list, total_loss = get_domain_loss(preds, labels, domain_labels, loss_func) loss_variance = compute_variance(domain_loss_list) loss = total_loss - lamda * loss_variance optimizer.zero_grad() loss.backward() # Perform the first step of SAM: gradient ascent with a fixed

length rho optimizer.first_step(zero_grad=True)

output = model(imgs) loss = loss_func(output, labels) loss.backward() # Obtain the actual gradient from the perturbation location of

DISAM optimizer.second_step(zero_grad=True)

def get_domain_loss(preds, labels, domain_labels, loss_func):

""" The function to compute the loss for each domain.

preds (Tensor): The predictions of the training model in one

batch. labels (Tensor): The labels of batch data. domain_labels (Tensor): The domain labels of batch data. loss_func: (Function): The loss function. """ # Get a list of all domains domain_list = list(set(domain_labels)) domain_loss_list = [] total_loss = 0.

for domain_name in domain_list:

# Get the mask for the current domain domain_mask = domain_labels == domain_name

labels_per_domain = labels[domain_mask]

Published as a conference paper at ICLR 2024

preds_pre_domain = preds[domain_mask]

# Compute the loss for the current domain single_domain_loss = loss_func(preds_pre_domain, labels_per_domain)

domain_loss_list.append(single_domain_loss)

# Add the loss for the current domain to the total loss, taking

into account the number of samples in the domain total_loss += len(labels_per_domain) * single_domain_loss

total_loss /= len(labels)

return domain_loss_list, total_loss

def compute_variance(domain_loss_list):

""" The function to compute the variance of the list of domain losses

domain_loss_list (List): the list of each domain s loss. """ loss_variance = 0. for domain_i_loss in domain_loss_list:

for domain_j_loss in domain_loss_list:

# Compute the square of the difference in loss between each

pair of elements and add it to the loss variance loss_variance += (domain_i_loss - domain_j_loss)**2

loss_variance /= (2*len(domain_loss_list)**2)

return loss_variance