When Source-Free Domain Adaptation Meets Learning with Noisy Labels

Deep learning demonstrates strong performance on various tasks across different fields. However, it is limited by the requirement of large-scale labeled and independent, and identically distributed (i.i.d.) data. Unsupervised domain adaptation (UDA) is thus proposed to mitigate the distribution shift between the labeled source and unlabeled target domain. In view of the importance of data privacy, it is crucial to be able to adapt a pre-trained source model to the unlabeled target domain without accessing the private source data, which is known as Source Free Domain Adaptation (SFDA).

The current state-of-the-art SFDA methods (Liang et al., 2020; Yang et al., 2021a; b) mainly focus on learning meaningful cluster structures in the feature space, and the quality of the learned cluster structures hinges on the reliability of pseudo labels generated by the source model. Among these methods, SHOT (Liang et al., 2020) purifies pseudo labels of target data based on nearest centroids, and then the purified pseudo labels are used to guide the self-training. G-SFDA (Yang et al., 2021b) and NRC (Yang et al., 2021a) further refine pseudo labels by encouraging similar predictions to the data point and its neighbors. For a single target data point, when most of its neighbors are correctly predicted, these methods can provide an accurate pseudo label to the data point. However, as we illustrate the problem in Figure 1i(a-b), when the majority of its neighbors are incorrectly predicted to a category, it will be assigned with an incorrect pseudo label, misleading the learning of cluster structures. The experimental result on VisDA (Peng et al., 2017), shown in Figure 1ii, further verifies this phenomenon. By directly applying the pre-trained source model on each target domain instance (central instance), we collect its neighbors and evaluate their quality. We observed that for each class a large proportion of the neighbors are misleading (i.e., the neighbors’ pseudo labels are different from the central instance’s true label), some even with high confidence (e.g., the over-confident misleading neighbors whose prediction score is larger than 0.75). Based on this observation, we can conclude that: (1) the pseudo labels leveraged in current SFDA methods can be heavily noisy; (2) some pseudo-label purification methods utilized in SFDA, which severely rely on the quality of the pseudo label itself, will be affected by such label noise, and the prediction error will accumulate as the training progresses. More details can be found in Appendix A.

In this paper, we address the aforementioned problem by formulating SFDA as learning with label noise (LLN). Unlike existing studies that heuristically rely on cluster structures or neighbors, we investigate the properties of label noise in SFDA and show that there is an intrinsic discrepancy between the SFDA and the LLN problems. Specifically, in conventional LLN scenarios, the label noise is generated by human annotators or image search engines (Patrini et al., 2017; Xiao et al., 2015; Xia et al., 2020a), where the underlying distribution assumption is that the mislabeling rate for a sample is bounded. However, in the SFDA scenarios, the label noise is generated by the source model due to the distribution shift, where we prove that the mislabeling rate for a sample is much higher, and can approach $1$. We term the former label noise in LLN as bounded label noise and the latter label noise in SFDA as unbounded label noise. Moreover, we theoretically show that most existing LLN methods, which rely on bounded label noise assumption, are unable to address the label noise in SFDA due to the fundamental difference (Section 3).

To this end, we leverage early-time training phenomenon (ETP) in LLN to address the unbounded label noise and to improve the efficiency of existing SFDA algorithms. Specifically, ETP indicates that classifiers can predict mislabeled samples with relatively high accuracy during the early learning phase before they start to memorize the mislabeled data (Liu et al., 2020). Although ETP has been previously observed in, it has only been studied in the bounded random label noise in the conventional LLN scenarios. In this work, we theoretically and empirically show that ETP still exists in the unbounded label noise scenario of SFDA. Moreover, we also empirically justify that existing SFDA algorithms can be substantially improved by leveraging ETP, which opens up a new avenue for SFDA. As an instantiation, we incorporate a simple early learning regularization (ELR) term (Liu et al., 2020) with existing SFDA objective functions, achieving consistent improvements on four different SFDA benchmark datasets. As a comparison, we also apply other existing LLN methods, including Generalized Cross Entropy (GCE) (Zhang & Sabuncu, 2018), Symmetric Cross Entropy Learning (SL) (Wang et al., 2019b), Generalized Jensen-Shannon Divergence (GJS) (Englesson & Azizpour, 2021) and Progressive Label Correction (PLC) (Zhang et al., 2021), to SFDA. Our empirical evidence shows that they are inappropriate for addressing the label noise in SFDA. This is also consistent with our theoretical results (Section 4).

Our main contribution can be summarized as: (1) We establish the connection between the SFDA and the LLN. Compared with the conventional LLN problem that assumes bounded label noise, the problem in SFDA can be viewed as the problem of LLN with the unbounded label noise. (2) We theoretically and empirically justify that ETP exists in the unbounded label noise scenario. On the algorithmic side, we instantiate our analysis by simply adding a regularization term into the SFDA objective functions. (3) We conduct extensive experiments to show that ETP can be utilized to improve many existing SFDA algorithms by a large margin across multiple SFDA benchmarks.

Source-free domain adaptation. Recently, SFDA are studied for data privacy. The first branch of research is to leverage the target pseudo labels to conduct self-training to implicitly achieve adaptation (Liang et al., 2021; Tanwisuth et al., 2021; Ahmed et al., 2021; Yang et al., 2021b). SHOT (Liang et al., 2020) introduces k-means clustering and mutual information maximization strategy for self-training. NRC (Yang et al., 2021a) further investigates the neighbors of target clusters to improve the accuracy of pseudo labels. These studies more or less involve pseudo-label purification processes, but they are primarily heuristic algorithms and suffer from the previously mentioned label noise accumulation problem. The other branch is to utilize the generative model to synthesize target-style training data (Qiu et al., 2021; Liu et al., 2021b). Some methods also explore the SFDA algorithms in various settings. USFDA (Kundu et al., 2020a) and FS (Kundu et al., 2020b) design methods for universal and open-set UDA. In this paper, we regard SFDA as the LLN problem. We aim to explore what category of noisy labels exists in SFDA and to ameliorate such label noise to improve the performance of current SFDA algorithms.

Learning with label noise. Existing methods for training neural networks with label noise focus on symmetric, asymmetric, and instance-dependent label noise. For example, a branch of research focuses on leveraging noise-robust loss functions to cope with the symmetric and asymmetric noise, including GCE (Zhang & Sabuncu, 2018), SL (Wang et al., 2019b), NCE (Ma et al., 2020), and GJS (Englesson & Azizpour, 2021), which have been proven effective in bounded label noise. On the other hand, CORES (Cheng et al., 2020) and CAL (Zhu et al., 2021) are shown useful in mitigating instance-dependent label noise. These methods are only tailed to conventional LLN settings. Recently, Liu et al. (2020) has studied early-time training phenomenon (ETP) in conventional label noise scenarios and proposes a regularization term ELR to exploit the benefits of ETP. PCL (Zhang et al., 2021) is another conventional LLN algorithm utilizing ETP, but it cannot maintain the exploit of ETP in SFDA as memorizing noisy labels is much faster in SFDA. Our contributions are: (1) We theoretically and empirically study ETP in the SFDA scenario. (2) Based on an in depth analysis of many existing LLN methods (Zhang & Sabuncu, 2018; Wang et al., 2019b; Englesson & Azizpour, 2021; Zhang et al., 2021), we demonstrate that ELR is useful for many SFDA problems.

The presence of label noise on training datasets has been shown to degrade the model performance (Malach & Shalev-Shwartz, 2017; Han et al., 2018). In SFDA, existing algorithms rely on pseudo-labels produced by the source model, which are inevitably noisy due to the domain shift. The SFDA methods such as Liang et al. (2020); Yang et al. (2021a; b) cannot tackle the situation when some target samples and their neighbors are all incorrectly predicted by the source model. In this section, we formulate the SFDA as the problem of LLN to address this issue. We assume that the source domain $\mathcal{D}_{S}$ and the target domain $\mathcal{D}_{T}$ follow two different underlying distributions over $\mathcal{X}\times\mathcal{Y}$, where $\mathcal{X}$ and $\mathcal{Y}$ are respectively the input and label spaces. In the SFDA setting, we aim to learn a target classifier $f(\mathbf{x};\theta):\mathcal{X}\to\mathcal{Y}$ only with a pre-trained model $f_{S}(\mathbf{x})$ on $\mathcal{D}_{S}$ and a set of unlabeled target domain observations drawn from $\mathcal{D}_{T}$. We regard the incorrectly assigned pseudo-labels as noisy labels. Unlike the “bounded label noise” assumption in the conventional LLN domain, we will show that the label noise in SFDA is unbounded. We further prove that most existing LLN methods that rely on the bounded assumption cannot address the label noise in SFDA due to the difference.

Label noise in conventional LLN settings: In conventional label noise settings, the injected noisy labels are collected by either human annotators or image search engines (Lee et al., 2018; Li et al., 2017; Xiao et al., 2015). The label noise is usually assumed to be either independent of instances (i.e., symmetric label noise or asymmetric label noise) (Patrini et al., 2017; Liu & Tao, 2015; Xu et al., 2019b) or dependent of instances (i.e., instance-dependent label noise) (Berthon et al., 2021; Xia et al., 2020b). The underling assumption for them is that a sample $\mathbf{x}$ has the highest probability of being in the correct class $y$, i.e., $\Pr[\tilde{Y}=i|Y=i,X=x]>\Pr[\tilde{Y}=j|Y=i,X=x],\ \forall x\in\mathcal{X},i\not=j$, where $\tilde{Y}$ is the noisy label and $Y$ is the ground-truth label for input $X$. Equivalently, it assumes a bounded noise rate. For example, given an image to annotate, the mislabeling rate for the image is bounded by a small number, which is realistic in conventional LLN settings (Xia et al., 2020b; Cheng et al., 2020). When the label noise is generated by the source model, the underlying assumption of these types of label noise does not hold.

Label noise in SFDA: As for the label noise generated by the source model, mislabeling rate for an image can approach $1$, that is, $\Pr[\tilde{Y}=j|Y=i,X=x]\to 1,\ \exists\mathcal{S}\subset\mathcal{X},\ \forall x\in\mathcal{S},i\not=j$. To understand that the label noise in SFDA is unbounded, we consider a two-component Multivariate Gaussian mixture distribution with equal priors for both domains. Let the first component ($y=1$) of the source domain distribution $\mathcal{D}_{S}$ be $\mathcal{N}(\bm{\mu_{1}},\sigma^{2}\mathbf{I}_{d})$, and the second component ($y=-1$) of $\mathcal{D}_{S}$ be $\mathcal{N}(\bm{\mu_{2}},\sigma^{2}\mathbf{I}_{d})$, where $\bm{\mu_{1}},\bm{\mu_{2}}\in\mathbb{R}^{d}$ and $\mathbf{I}_{d}\in\mathbb{R}^{d\times d}$. For the target domain distribution $\mathcal{D}_{T}$, let the first component ($y=1$) of $\mathcal{D}_{T}$ be $\mathcal{N}(\bm{\mu_{1}}+\mathbf{\Delta},\sigma^{2}\mathbf{I}_{d})$, and the second component ($y=-1$) of $\mathcal{D}_{T}$ be $\mathcal{N}(\bm{\mu_{2}}+\mathbf{\Delta},\sigma^{2}\mathbf{I}_{d})$, where $\mathbf{\Delta}\in\mathbb{R}^{d}$ is the shift of the two domains. Notice that the domain shift considered is a general shift and it has been studied in Stojanov et al. (2021); Zhao et al. (2019), where we also illustrate the domain shift in Figure 9 in supplementary material.

Let $f_{S}$ be the optimal source classifier. First, we build the relationship between the mislabeling rate for target data and the domain shift:

where $d_{1}=\left\lVert\frac{\bm{\mu_{2}}-\bm{\mu_{1}}}{2}-\mathbf{c}\right\rVert\mathrm{sign}(\left\lVert\frac{\bm{\mu_{2}}-\bm{\mu_{1}}}{2}\right\rVert-\left\lVert\mathbf{c}\right\rVert)$, $d_{2}=\left\lVert\frac{\bm{\mu_{2}}-\bm{\mu_{1}}}{2}+\mathbf{c}\right\rVert$, $\mathbf{c}=\alpha(\bm{\mu_{2}}-\bm{\mu_{1}})$, $\alpha=\frac{\mathbf{\Delta}^{\top}(\bm{\mu_{2}}-\bm{\mu_{1}})}{\left\lVert\bm{\mu_{2}}-\bm{\mu_{1}}\right\rVert^{2}}$ is the magnitude of domain shift, and $\Phi$ is the standard normal cumulative distribution function. Eq. (1) shows that the magnitude of the domain shift inherently controls the mislabeling error for target data. This mislabeling rate increases as the magnitude of the domain shift increases. We defer the proof and details to Appendix B.

More importantly, we characterize that the label noise is unbounded among these mislabeled samples.

Conventional LLN methods assume that the label noise is bounded: $\Pr[f_{H}(\mathbf{x})\not=y]<m,\ \forall(\mathbf{x},y)\sim\mathcal{D}_{T}$, where $f_{H}$ is the labeling function, and $m=0.5$ if the number of clean samples of each component are the same (Cheng et al., 2020). However, Theorem 3.1 indicates that the label noise generated by the source model is unbounded for any $\mathbf{x}\in\mathbf{R}$. In practice, region $\mathbf{R}$ is non-empty as neural networks are usually trained on high dimensional data such that $d\gg 1$, so $\alpha>(\log{\frac{1-\delta}{\delta}})/d\to 0$ is easy to satisfy. The probability measure on $\mathbf{R}=\mathbf{R}_{1}\bigcap\mathbf{R}_{2}$ (i.e., $\Pr_{(\mathbf{x},y)\sim\mathcal{D}_{T}}[\mathbf{x}\in\mathbf{R}]$) increases as the magnitude of the domain shift $\alpha$ increases, meaning more data points contradict the conventional LLN assumption. More details can be found in Appendix C.

Given that the unbounded label noise exists in SFDA, the following Lemma establishes that many existing LLN methods (Wang et al., 2019b; Ghosh et al., 2017; Englesson & Azizpour, 2021; Ma et al., 2020), which rely on the bounded assumption, are not noise tolerant in SFDA.

We denote $\ell_{\text{LLN}}$ by the existing noise-robust loss based LLN methods in Wang et al. (2019b); Ghosh et al. (2017); Englesson & Azizpour (2021); Ma et al. (2020). When the noisy data follows the bounded assumption, these methods are noise tolerant as the minimizer $\tilde{h}^{\star}$ converges to the minimizer $h^{\star}$ with a high probability. We defer the details and proof of the related LLN methods to Appendix D.

Given a fundamental difference between the label noise in SFDA and the label noise in conventional LLN scenarios, existing LLN methods, whose underlying assumption is bounded label noise, cannot be applied to solve the label noise in SFDA. This section focuses on investigating how to address the unbounded label noise in SFDA.

Motivated by the recent studies Liu et al. (2020); Arpit et al. (2017), which observed an early-time training phenomenon (ETP) on noisy datasets with bounded random label noise, we find that ETP does not rely on the bounded random label noise assumption, and it can be generalized to the unbounded label noise in SFDA. ETP describes the training dynamics of the classifier that preferentially fits the clean samples and therefore has higher prediction accuracy for mislabeled samples during the early-training stage. Such training characteristics can be very beneficial for SFDA problems in which we only have access to the source model and the highly noisy target data. To theoretically prove ETP in the presence of unbounded label noise, we first describe the problem setup.

We still consider a two-component Gaussian mixture distribution with equal priors. We denote $y$ by the true label for $\mathbf{x}$, and assume it is a balanced sample from $\{-1,+1\}$. The instance $\mathbf{x}$ is sampled from the distribution $\mathcal{N}(y\bm{\mu},\ \sigma\mathbf{1}_{d})$, where $\left\lVert\bm{\mu}\right\rVert=1$. We denote $\tilde{y}$ by the noisy label for $\mathbf{x}$. We observe that the label noise generated by the source model is close to the decision boundary revealed in Theorem 3.1. So, to assign the noisy labels, we let $\tilde{y}=y\beta(\mathbf{x},y)$, where $\beta(\mathbf{x},y)=\mathrm{sign}(\mathbbm{1}\{y\mathbf{x}^{\top}\bm{\mu}>r\}-0.5)$ is the label flipping function, and $r$ controls the mislabeling rate. If $\beta(\mathbf{x},y)<1$, then the data point $\mathbf{x}$ is mislabeled. Meanwhile, the label noise is unbounded by adopting the label flipping function $\beta(\mathbf{x},y)$: $\Pr[\tilde{y}\not=y|y\mathbf{x}^{\top}\bm{\mu}\leq r]=1$, where $\mathbf{R}=\{\mathbf{x}:y\mathbf{x}^{\top}\bm{\mu}\leq r\}$.

We study the early-time training dynamics of gradient descent on the linear classifier. The parameter $\theta$ is learned over the unbounded label noise data $\{x_{i},\tilde{y}_{i}\}_{i=1}^{n}$ with the following logistic loss function:

where $\theta_{t+1}=\theta_{t}-\eta\nabla_{\theta}\mathcal{L}(\theta_{t})$, and $\eta$ is the learning rate. Then the following theorem builds the connection between the prediction accuracy for mislabeled samples at an early-training time $T$.

The proof is provided in Appendix E. Compared to ETP found in Liu et al. (2020), where the label noise is assumed to be bounded, Theorem 4.1 presents that ETP also exists even though the label noise is unbounded. At a proper time T, the classifier trained by the gradient descent algorithm can provide accurate predictions for mislabeled samples, where its accuracy is lower bounded by a function of the variance of clusters $\sigma$. When $\sigma\to 0$, the predictions of all mislabeled samples equal to their ground-truth labels (i.e., $\kappa(B;\theta_{T})\to 1$). When the classifier is trained for a sufficiently long time, it will gradually memorize mislabeled data. The predictions of mislabeled samples are equivalent to their incorrect labels instead of their ground-truth labels (Liu et al., 2020; Maennel et al., 2020). Based on these insights, the memorization of mislabeled data can be alleviated by leveraging their predicted labels during the early-training time.

To leverage the predictions during the early-training time, we adopt a recently established method, early learning regularization (ELR) (Liu et al., 2020), which encourages model predictions to stick to the early-time predictions for $\mathbf{x}$. Since ETP exists in the scenarios of the unbounded label noise, ELR can be applied to solve the label noise in SFDA. The regularization is given by:

where we overload $f(\mathbf{x};\theta_{t})$ to be the probabilistic output for the sample $\mathbf{x}$, and $\bar{y}_{t}=\beta\bar{y}_{t-1}+(1-\beta)f(\mathbf{x};\theta_{t})$ is the moving average prediction for $\mathbf{x}$, where $\beta$ is a hyperparameter. To see how ELR prevents the model from memorizing the label noise, we calculate the gradient of Eq. (4) with respect to $f(\mathbf{x};\theta_{t})$, which is given by:

Note that minimizing Eq. (4) forces $f(\mathbf{x};\theta_{t})$ to close to $\bar{y}_{t}$. When $\bar{y}_{t}$ is aligned better with $f(\mathbf{x};\theta_{t})$, the magnitude of the gradient becomes larger. It makes the gradient of aligning $f(\mathbf{x};\theta_{t})$ with $\bar{y}_{t}$ overwhelm the gradient of other loss terms that align $f(\mathbf{x};\theta_{t})$ with noisy labels. As the training progresses, the moving averaged predictions $\bar{y}_{t}$ for target samples gradually approach their ground-truth labels till the time $T$. Therefore, Eq. (4) prevents the model from memorizing the label noise by forcing the model predictions to stay close to these moving averaged predictions $\bar{y}_{t}$, which are very likely to be ground-truth labels.

Some existing LLN methods propose to assign pseudo labels to data or require two-stage training for label noise (Cheng et al., 2020; Zhu et al., 2021; Zhang et al., 2021). Unlike these LLN methods, Eq. (4) can be easily embedded into any existing SFDA algorithms without conflict. The overall objective function is given by:

where $\mathcal{L}_{\text{SFDA}}$ is any SFDA objective function, and $\lambda$ is a hyperparameter.

We aim to improve the efficiency of existing SFDA algorithms by using ELR to leverage ETP. We evaluate the performance on four different SFDA benchmark datasets: Office-$31$ (Saenko et al., 2010), Office-Home (Venkateswara et al., 2017), VisDA (Peng et al., 2017) and DomainNet (Peng et al., 2019). Due to the limited space, the results on the dataset Office-$31$ and additional experimental details are provided in Appendix G.

Evaluation. We incorporate ELR into three existing baseline methods: SHOT (Liang et al., 2020), G-SFDA (Zhang & Sabuncu, 2018), and NRC (Yang et al., 2021a). SHOT uses k-means clustering and mutual information maximization strategy to train the representation network while freezing the final linear layer. G-SFDA aims to cluster target data with similar neighbors and attempts to maintain the source domain performance. NRC also explores the neighbors of target data by graph-based methods. ELR can be easily embedded into these methods by simply adding the regularization term into the loss function to optimize without affecting existing SFDA frameworks. We average the results based on three random runs.

Results. Tables 1-4 show the results before/after leveraging the early-time training phenomenon, where Table 4 is shown in Appendix G. Among these tables, the top part shows the results of conventional UDA methods, and the bottom part shows the results of SFDA methods. In the tables, we use SF to indicate whether the method is source free or not. We use Source Only + ELR to indicate ELR with self-training. The results show that ELR itself can boost the performances. As existing SFDA methods are not able to address unbounded label noise, incorporating ELR into these SFDA methods can further boost the performance. The four datasets, including all $31$ pairs (e.g., $A\to D$) of tasks, show better performance after solving the unbounded label noise problem using the early-time training phenomenon. Meanwhile, solving the unbounded label noise on existing SFDA methods achieves state-of-the-art on all benchmark datasets. These SFDA methods also outperform most methods that need to access source data.

Analysis about hyperparameters $\beta$ and $\lambda$. The hyperparameter $\beta$ is chosen from {$0.5$, $0.6$, $0.7$, $0.8$, $0.9$, $0.99$}, and $\lambda$ is chosen from {$1$, $3$, $7$, $12$, $25$}. We conduct the sensitivity study on hyperparameters of ELR on the DomainNet dataset, which is shown in Figure 3(a-b). In each Figure, the study is conducted by fixing the other hyperparameter to the optimal one. The performance is robust to the hyperparameter $\beta$ except $\beta=0.99$. When $\beta=0.99$, classifiers are sensitive to changes in learning curves. Thus, the performance degrades since the learning curves change quickly in the unbounded label noise scenarios. Meanwhile, the performance is also robust to the hyperparameter $\lambda$ except when $\lambda$ becomes too large. The hyperparameter $\lambda$ is to balance the effects of existing SFDA algorithms and the effects of ELR. As we indicated in Tables 1-4, barely using ELR to address the SFDA problem is not comparable to these SFDA methods. Hence, a large value of $\lambda$ makes neural networks neglect the effects of these SFDA methods, leading to degraded performance.

In this paper, we study SFDA from a new perspective of LLN by theoretically showing that SFDA can be viewed as the problem of LLN with the unbounded label noise. Under this assumption, we rigorously justify that robust loss functions are not able to address the memorization issues of unbounded label noise. Meanwhile, based on this assumption, we further theoretically and empirically analyze the learning behavior of models during the early-time training stage and find that ETP can benifit the SFDA problems. Through extensive experiments across multiple datasets, we show that ETP can be exploited by ELR to improve prediction performance, and it can also be used to enhance existing SFDA algorithms.