Robust Federated Learning with Noisy Labels

We use feature vectors extracted from $F_{k}$ to compute local class-wise centroids ${\bf f}_{k}$. If we calculate ${\bf f}_{k}$ by using all local samples with given labels, noise labels have a negative effect on the correct formulation of centroids. Therefore, we introduce loss-based local centroids, which are motivated by (Han et al. 2018; Arpit et al. 2017). We only use features of samples with relatively small-losses to create accurate feature centroids. At first, we refine the dataset $D_{k}$ by selecting $R(t)$ percentage of small-loss instances on each client as follows:

where $D^{\prime}_{k}$ is the optimization variable, $\left|\cdot\right|$ stands for the cardinality of a set the number of samples, and $R(t)$ controls how many small-loss samples should be selected in each round. Then, the $k$-th local model calculates naive average features of each class ${\bf\hat{f}}_{k}^{c}$ depending on the small-loss samples as follows:

where $\tilde{n}_{k}^{c}$ is the number of samples corresponding to the label $c$ in $\hat{{\it D}}_{k}$, and $\mathds{1}(\cdot)$ is the indicator function returning 1 for true statements and 0 otherwise.

However, these average features may differ from the other clients’. To avoid these undesirable deviations, we derive local centroids ${\bf f}_{k}^{c}$ by weighted average depending on the similarity between global centroids ${\bf f}_{G}^{c}$ and the average features ${\bf\hat{f}}_{k}^{c}$ as follows:

where ${sim}(\cdot,\cdot)$ can be any similarity function, but we choose cosine similarity for our experiments. ${\bf\hat{f}}_{k}^{c}$ is calculated by Eq. 3, and global centroids ${\bf f}_{G}^{c}$ are transmitted from the server reflecting entire clients’ centroids, which is described in the next section.

We expect that class-wise centroids are the central features of clean samples. At the beginning of training, deep networks tend to prioritize learning simple patterns first (Arpit et al. 2017), and we exploit this property to form global centroids less susceptible to noisy samples. After that, we update local centroids to reflect similarity with these global centroids. This similarity-based update can keep centroids less corrupted by noisy data even after a large number of training epochs.

We exploit these local centroids to reduce weight divergence of clients’ models. In detail, we design a loss function to map the features of the confident sample onto the centroids corresponding to the class as follows:

where $m_{k}^{i}$ denotes a binary mask of the k-th client that returns 1 for confident samples and 0 otherwise.

We introduce a sampling approach to select confident samples for training each client’s model without a detrimental influence from noisy labels. To this end, we introduce the feature similarity-based labels as follows:

Note that we should not fully trust given labels because they may not be annotated accurately. Also they should not depend on feature similarity-based labels inducing the wrong labels for the hard samples. The complementary use of feature similarity-based and ground truth labels can help to find accurate confident samples. Therefore, we consider the similarity-based labels with local centroids and the ground truth labels at the same time. By adopting the ground truth label and the similarity-based label together, $m_{k}^{i}$ for masking a confident sample is obtained as follows:

We exploit this mask for $L_{c}$ and $L_{cen}$ to reduce the impact of noise samples. Since the number of confident samples is not fixed for each class, this mask can choose confident samples well regardless of different noise ratio for each class.

To fully utilize the local data information, we exploit the well-known label correction method (Tanaka et al. 2018). Although this self-learning strategy with pseudo-labeling is powerful for label correction in the centralized setting, it leads local models to be self-biased (Arazo et al. 2019). Therefore, we propose global-guided pseudo-labeling, which corrects labels of local data by employing the server model. Our technique for the label estimation prevents local models from being self-biased. Each client receives the global model at the broadcast time and uses the model to generate global-guided pseudo-labels ${\bf\hat{y}}_{k}$ as follows:

where $C_{G}$ and $F_{G}$ are the client’s networks with global parameters. After that, each client trains its network with these global-guided pseudo-labels by Eq. 1.

Finally, the k-th local model is trained to minimize the sum of three losses:

where $L_{e}^{k}$ is the entropy regularization of prediction results. Note that this term forces probability distribution of each softmax output to a single class. ${\bf p}^{i}$ indicates softmax outputs $C_{k}(F_{k}(x_{k}^{i}))$, and the loss $L_{e}^{k}$ is calculated by $-\sum_{i}{{\bf p}^{i}}\mbox{log}{{\bf p}^{i}}$. $\lambda_{cen}$ and $\lambda_{e}$ indicate trade-off parameters. Our complete algorithm is illustrated in Fig. 3.

We execute FedAvg (McMahan et al. 2017) for weight aggregation, which is suitable for an i.i.d. dataset. Since only noisy labels are added in i.i.d. data, we expect that the FedAvg algorithm works well enough in our experimental settings. FedAvg takes a weighted average of local parameters $\theta_{L}$ as follows:

where $\theta_{G}$ is global parameters, and $n$ and $n_{k}$ indicate the total number of data and the number of the k-th client’s data, respectively.

To tackle the different noise distributions in clients explicitly, we adjust global centroids to employ the similarity-based summation of local centroids. In every round, local centroids of selected clients update global centroids by using the similarity to previous global centroids in the server. Let $K$ be the set of indices of clients selected in the current round, then global centroids are updated as follows:

where $w_{k}^{c}$ indicates similarity between stored global centroids ${\bf\hat{f}}_{G}^{c}$ and the uploaded the k-th client centroids of class $c$, and it is obtained by:

Therefore, this weight update rule allows global centroids to reflect the similarity with local centroids, which depends on the client and class. The complete pseudo-code is shown in supplementary material.

CIFAR-10 (Krizhevsky and Hinton 2009) is a benchmark dataset of 10 categories, which contains 50,000 images for training and 10,000 images for testing. We replace the ground truth labels of CIFAR-10 with two types of noisy labels: symmetric flipping (Van Rooyen, Menon, and Williamson 2015) and pair flipping (Han et al. 2018), which are described in supplementary material. Since our paper mainly focuses on the robustness in federated learning with various noisy ratios, the noise ratio $\epsilon$ is chosen from $\{0.1,0.2,0.3,0.4,0.5,0.6\}$ for symmetric flipping and $\{0.1,0.2,0.3,0.4,0.45\}$ for pair flipping. In detail, we give noisy labels to the entire dataset with the designated noise ratio, and then randomly distribute it to 100 clients. This process induces different noise distributions in clients, and we fix the seed for a fair comparison. In ablation studies, we experiment with extremely different noise ratios.

Clothing1M (Xiao et al. 2015) is a large real-world dataset of 14 categories, which contains 1 million images of clothing with noisy labels since it is obtained from several online shopping websites. In (Xiao et al. 2015), it is reported that the overall noise ratio is approximately 38.46%. The Clothing1M dataset also contains 50k, 14k, and 10k of clean data for training, validation, and testing, respectively, but we do not use the clean training data. For federated learning, we randomly divide the clothing1M dataset into 100 groups, which indicates the number of clients, and we set them as local datasets.

We report the results of CIFAR-10 with symmetric flipping and pair flipping in Table 1. As shown in Table 1, our method achieves better overall test accuracy at various noise ratios.

Co-teaching selects a fixed number of loss-based samples, which is vulnerable to different noise distributions in clients. It causes serious performance degradation of the server model since each client model is affected by noisy data. Joint Optimization follows a self-training scheme, which is a naive pseudo-labeling method. In federated learning, this self-training method may lead the network to be self-biased due to a large number of local epochs, and the weights of self-biased local models can be diverged severely. Notably in extremely noisy cases, clients cannot be trained properly due to high noise ratios, and when these local parameters are aggregated in the server, performance is further deteriorated. Our proposed method is also dependent on a loss-based algorithm but robust to different noise distributions in clients because of the similarity-based centroids update. Moreover, we exploit a global-guided pseudo-labeling method, which mitigates self-bias of each client, and we validate the effectiveness of this algorithm in ablation studies.

Furthermore, neither of the two previous methods can guarantee that all local models are trained to form similar decision boundaries, which makes aggregation of local models unsuccessful. Our proposed method constrains class representations of all client models not to diverge from global class representations. This induces all local models to be trained to have similar boundaries, and we demonstrate the efficacy of the algorithm in noisy federated learning.

Different from the artificial noise in CIFAR-10, Clothing1M is a real-world noisy label dataset, including lots of unknown structure noise. By comparing the results in Table 2, we can see that our proposed method outperforms the others by a large margin in the federated setting. In the case of Deep self-learning (Han, Luo, and Wang 2019), which corrects the labels of the data comparing the similarity with the several prototypes of the features, it achieves great performance improvement in the centralized setting. However, in the federated setting, this algorithm suffers from the significant performance degradation since it does not constrain local models from having similar decision boundaries. We get the centroids with relatively small-losses and update them to be similar with global centroids. It can keep centroids less corrupted by noise data as well as achieve local decision boundaries similar to others. Our algorithm is effective for the real-world noisy label dataset corrupted by unknown structure noise.

In the federated setting, clients may have different amounts of noise because of the discrepancy between clients’ labeling systems. In detail, we split clients into five groups and assign different noise ratios. We set noise variance $\eta$, then divide noise range $[\epsilon-\eta,\epsilon+\eta]$ equally into five noise ratios and assign the noise ratio to each group. For example, if we set the noise ratio $\epsilon$ and noise variance $\eta$ to $0.4$ and $0.2$, respectively, the noise ratio in each group is assigned one of $\{0.2,0.3,0.4,0.5,0.6\}$. In Table 3, we experiment by fixing the noise ratio to $0.4$ and changing noise variance. Our approach achieves consistent performance regardless of the different noise ratios in clients.

Due to background knowledge of the client user, the local data can have different noise ratios in classes. We assume an extreme situation where each client has totally erroneous samples for a single class. In detail, we force clients to have wrong labels for a single random class entirely and replace the labels of other classes with noisy labels of the designated noise ratio $\epsilon$. We show that the proposed algorithm is robust to different noise ratios in classes in Table 3.

We evaluate the effectiveness of our sampling approach in Table 4. Note that we set the noise ratio to $0.4$ by using symmetric flipping. As shown in Table 4, the mask for confident samples and the pseudo-labeling method complement each other. The network trained only with the selected samples by removing unconfident samples has better performance than the one trained with all samples, and the performance increases considerably when the unconfident samples’ labels are replaced by pseudo-labels. Moreover, we show the precision and recall of noisy label detection of our sampling approach in Table 5. The precision means the number of correctly detected noisy labels over the entire number of detected noisy labels and the recall means the number of correctly detected noisy labels over the entire number of noisy labels in data. Our sampling approach selects confident samples by the complementary use of feature similarity-based and ground truth labels. It leads to high accuracy for the precision of noise labels.

We have conducted the experiments by removing global-guided pseudo-labeling or replacing the proposed method with naive pseudo-labeling in Table 4. Although the self-learning strategy with pseudo-labeling is powerful for label correction in centralized learning, it leads local models to be self-biased to their own datasets. Our global-guided pseudo-labeling outperforms a naive approach to prevent local models from being self-biased.

To show the effectiveness of interchanging centroids, we experiment our algorithm while local models are updated without using global centroids. In detail, we use the loss function Eq. 5 by calculating local centroids without using Eq. 4, which cannot explicitly constrain local models to have similar boundaries. Figure 4 shows that global centroids help all clients to have similar feature representations, which leads to reducing weight divergence.