| CIFAR-10, Batch-IID Task with 100 Clients (K=100,F=0.05,H=2) |
| Labels-at-ClientScenario | Labels-at-ServerScenario |
| Methods | Acc.(%) | S2C Cost | C2S Cost | Acc.(%) | S2C Cost | C2S Cost |
| FedAvg-SLFedProx-SL | 58.60 ±0.4259.30 ± 0.31 | 100 %100 % | 100 %100 % | 52.45 ± 0.2349.11 ± 0.38 | 100 %100 % | 100%100 % |
| FedAvg-UDAFedProx-UDAFedAvg-FixMatchFedProx-FixMatch | 46.35 ± 0.2947.45 ± 0.2147.01 ± 0.4347.20 ±0.1252.13 ± 0.34 | 100%100 %100 %100 % | 100%100 %100 %100 % | 24.81±0.7319.91 ± 0.3111.95 ± 0.6025.61 ± 0.3244.95±0.49 | 100%100 %100 %100 %45% | 100%100 %100 %100 %22% |
| FedMatch(Ours) | 79% | 46% |
| CIFAR-10,Batch-NonIID Task with 100 Clients (K=100, F=O.05, H=2) | CIFAR-10,Batch-NonIID Task with 100 Clients (K=100, F=O.05, H=2) |
| FedAvg-SLFedProx-SL | 55.15 ± 0.2157.75 ± 0.15 | 100 %100 % | 100 %100 % | 51.50 ± 0.51 | 100 %100 % | 100 % |
| 49.31 ± 0.18 | | 100 % |
| FedAvg-UDAFedProx-UDAFedAvg-FixMatchFedProx-FixMatch | 44.35 ± 0.3946.31 ± 0.63 | 100%100 % | 100%100 % | 27.61±0.71 | 10 | 100% | 100% |
| 26.01 ± 0.78 | 100 % | 100 % |
| 46.20 ± 0.52 | 100 % | 100 % | 09.45 ± 0.34 | 100 % | 100 % |
| 4 | 45.55 ± 0.63 | 100 % | 100 % | 09.21 ±0.24 | 100 % | 100 %20% |
| FedMatch (Ours) | 52.25 ± 0.81 | 85% | 49% | 44.17 ±0.19 | 42% |
| Batch-lID Task (100 Clients) Batch-NonlID (100 Clients) Batch-lID Task (100 Clients) Batch-NonlID (100 Clients)60 60 60 60Wwy50 50 50 mwwW 50 %) eeeect wwwwy40myyiww% 40eeeeeeeeeeec CM30303030FedProx*SLFedProx*SLFedProx*SL20202020FedAvg*SLFedProx*UDAFedProx*UDA+FedProx*UDAFedProx*UDAFedProx*FixMatch FedProx*FixMatch FedProx*FixMatch FedProx*FixMatch10 10 10 10FedMatch (Ours) FedMatch (Ours) FedMatch (Ours) FedMatch (Ours)100150 150 0 0502005010020050100150200 50 100150 200Communication Round Communication Round Communication Round Communication Round(a)Labels-at-Client Scenario (b)Labels-at-Server Scenario |
+
+Figure 5: Test Accuracy Curves on Batch-IID & NonIID Tasks We visualize test accuracy curves of model performance corresponding to the Table 1. Note that the SL (Supervised Learning) models learn on both $s$ and $\mathcal { U }$ with full labels, and are utilized as the upper bounds for each experiment.
+
+Baselines and Training Details Our baselines are: 1) Local-SL: local supervised learning (SL) with full labels $( S + \mathcal { U } )$ without sharing locally learned knowledge. 2) Local-UDA and 3) LocalFixMatch: local semi-supervised learning, including UDA and FixMatch, without sharing local knowledge. 4) FedAVG-SL and 5) FedProx-SL: supervised learning with full labels $( S + \mathcal { U } )$ while sharing local knowledge via FedAvg and FedProx frameworks. 6) FedAvg-UDA and 7) FedProxUDA: naive combinations of FedAvg/Prox with UDA. 8) FedAvg-FixMatch and 9) FedProxFixMatch: naive combination of with FixMatch with FedAvg/Prox. For training, we use SGD with adaptive-learning rate decay introduced in (Serra et al., 2018) with the initial learning rate $1 \mathrm { e } { - 3 }$ . We use ResNet-9 networks as the backbone architecture for all baselines and our methods. We ensure that all hyper-parameters are set equally for all base models and ours to perform fair evaluation and comparison. Please see the Section A in the Appendix for further details. For all experiments, we report the mean and the standard deviation over 3 runs.
+
+# 6.2 EXPERIMENTAL RESULTS
+
+Results on Batch-IID & NonIID Tasks Table 1 shows performance comparison of our models and naive Fed-SSL algorithms on Batch-IID and NonIID tasks under the two different scenarios. We observe that our model outperforms all naive Fed-SSL baselines for all tasks and scenarios. In particular, under labels-at-server scenario, which is more challenging than labels-at-client scenario, we observe that the naive combination models significantly suffer from the forgetting issue and their performances keeps deteriorating after a certain communication round. This phenomenon is mainly caused by the base models failing to properly perform disjoint learning, in which case the learned knowledge from the labeled and unlabeled data causes inter-task interference. Contrarily, our methods show consistent and robust performance regardless where the labeled data exists, which shows that our decomposition techniques effectively handles the challenging disjoint learning scenario. In addition, when the class-wise distribution is imbalanced for each client (Non-IID task), we observe that the base models’ performance slightly drops by $1 - 3 \% p$ , while our methods show consistent.
+
+
+Figure 6: Ablation Study and Additional Analysis on FedMatch Algorithm We study effectiveness of each components of our method, (a) inter-client consistency loss and (b) parameter decomposition. (c) We effectively tackle the inter-task interference. (d) Performance improvement of our method when labeled data is increased.
+
+This shows that our inter-client consistency effectively enhances consistency with the helper agents selected from server based on model similarity, which is good at, in particular, class-imbalanced tasks. We also visualize the test accuracy curve for our models and naive Fed-SSL in Fig. 5. Our method (Red line) trains faster and consistently outperforms the base models, and is most robustness against inter-task interference in both scenarios. For analysis on the averaged communication costs, please see Section B.1 in the Appendix.
+
+Results on Streaming-NonIID Task Table 2 shows averaged local model performance on Streaming-NonIID tasks with 10 synchronized clients. For the labels-at-client scenario, our proposed method outperforms local-SSL and naive Fed-SSL models with large margins, $4 \mathrm { - } 1 5 \% p$ , except for the SL models. There is no huge difference of performance between local SSL and Fed-SSL models, and this implies that our method effectively utilizes inter-client knowledge in this streaming setting. In the labels-at-server scenario, interestingly, the performance of FedProx-SL decreases by around $5 \% p$ compared to the labels-at-client scenario, while Fed-SSL models obtain improved performance. We conjecture that this is because, for streaming situation, the model may not sufficiently train on the new data, while Fed-SSL models overcomes it by utilizing only the consistent pseudo-labels. Even on this task, FedMatch outperforms all baselines with significantly smaller communication cost on average (see Section B.1 for detailed analysis of the averaged communication costs).
+
+Effectiveness of Inter-Client Consistency To show the effectiveness of our inter-client consistency loss, we eliminate the loss, while learning on Batch-IID task with 100 clients $( F { = } 0 . 0 5 )$ . In Figure 6 (a), when we remove our inter-client consistency loss, we observe that the performance has slightly dropped (Pink line) from one with the loss term (Red line). This gap clearly tells us that our interclient consistency loss improves model consistency across multiple models while keeping reliable knowledge. Interestingly, our model without inter-client consistency loss still outperforms base models. This additionally implies that our another proposed method, parameter decomposition for disjoint learning, also effectively enhances model performance.
+
+Effectiveness of Parameter Decomposition Our model with parameter decomposition alone, without inter-client consistency loss, outperforms base models. For further analysis, we show the effect of each decomposed variables, $\sigma$ and $\psi$ , in Figure 6 (b). Removing either of $\sigma$ and $\psi$ results in substantial drop in the performance, with larger performance degeneration when dropping $\sigma$ , which captures much more essential knowledge from labeled data (Green line). Such decomposition is effective since there exists knowledge interference between supervised learning and unsupervised learning. We show this with an experiment where we perform semi-supervised learning with 5 labeled instances per class and 1, 000 unlabeled instances for 100 rounds. We measure accuracy on the labeled set at each training steps. As shown in Figure 6 (c), our method effectively preserves learned knowledge from labeled set, while other base models suffer from knowledge interference. This effective separation of supervised and unsupervised learning tasks enhances the overall performance of our methods even without inter-client consistency loss as shown in Figure 6 (a) (shown in pink).
+
+Number of Labels per Class We increase the number of labels per class for each client in a range of 1, 5, 10, and 20 on Batch-IID task (CIFAR-10). Our method shows consistent performance improvement as the number of labels increases. Interestingly, we observe that baseline models, FedProx-UDA/FixMatch, show performance degradation even when the labeled data increases $( 5 1 0 )$ ). These results show that our method effectively utilize knowledge from labeled and unlabeled data in federated semi-supervised learning settings, while other naive combinations of FSSL could fail to learn properly from labeled and unlabeled data in federated learning framework.
+
+# 7 RELATED WORK
+
+Federated Learning A variety of approaches for averaging local weights at server have been introduced in the past few years. FedAvg (McMahan et al., 2017) performs weighted-averaging on local weights according to the local train size. FedProx (Li et al., 2018) uniformly averages the local updates while clients perform proximal regularization against the global weights, while FedMA (Wang et al., 2020) matches the hidden elements with similar feature extraction signatures in layer-wise manner when averaging local weights. PFNM (Yurochkin et al., 2019) introduces aggregation policy which leverages Bayesian non-parametric methods. Beyond focusing on averaging local knowledge, there are various efforts to extend FL to the other areas, such as continual learning under federated learning frameworks (Yoon et al., 2020a) inspired by parameter decomposition techniques proposed by (Yoon et al., 2020b). Recently, interests of tackling scarcity of labeled data in FL are emerging and discussed in (Jin et al., 2020; Guha et al., 2019; Albaseer et al., 2020).
+
+Semi-Supervised Learning While there exist numerous work on SSL, we mainly discuss consistency regularization approaches. Consistency regularization (Sajjadi et al., 2016) assumes that the class semantics will not be affected by transformations of the input instances, and enforces the model output to be the same across different input perturbations. Some extensions to this technique perturb inputs adversarially (Miyato et al., 2018), through dropout (Srivastava et al., 2014), or through data augmentation (French et al., 2018). UDA (Xie et al., 2019) and ReMixMatch (Berthelot et al., 2019a) use two sets of augmentations, weak and strong, and enforce consistency between the weakly and strongly augmented examples. Recently, in addition to enforcing consistency between weak-strong augmented pairs, FixMatch (Sohn et al., 2020) performs pseudo-label refinement on model predictions via thresholding. Entropy minimization (Grandvalet & Bengio, 2004) which enforces the classifier to predict low-entropy on unlabeled data, is another popular technique for SSL. Pseudo-Label (Lee, 2013) constructs one-hot labels from highly confident predictions on unlabeled data and uses these as training targets inn a standard cross-entropy loss. MixMatch (Berthelot et al., 2019c) performs sharpening on target distribution on unlabeled data, to further refine the generated pseudo-label.
+
+# 8 CONCLUSION
+
+In this work, we introduced two practical scenarios of Federated Semi-Supervised Learning (FSSL) where each client learns with only partly labeled data (Labels-at-Client scenario), or supervised labels are only available at the server, while clients work with completely unlabeled data (Labels-at-Server scenario). To tackle the problem, we propose a novel method, Federated Matching (FedMatch), which introduces the inter-client consistency loss that aims to maximize the agreement between the models trained at different clients, and the parameter decomposition for disjoint learning which decomposes the parameters into one for labeled data and the other for unlabeled data for preservation of reliable knowledge, reduction of communication costs, and disjoint learning. Through extensive experimental validation, we show that FedMatch significantly outperforms both local semi-supervised learning methods and naive combinations of federated learning algorithms with semi-supervised learning on diverse and realistic scenarios. As future work, we plan to further improve our model to tackle the scenario where pretrained models deployed at each client adapts to a completely unlabeled data stream (e.g. on-device learning of smart speakers).
+
+Acknowledgements This work was supported by Samsung Research Funding Center of Samsung Electronics (No. SRFC-IT1502-51), Samsung Advanced Institute of Technology, Samsung Electronics Co., Ltd., Next-Generation Information Computing Development Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT & Future Plannig (No. 2016M3C4A7952634), the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (2018R1A5A1059921), and Center for Applied Research in Artificial Intelligence (CARAI) grant funded by DAPA and ADD (UDI190031RD). Also, this work was supported by Institute of Information communications Technology Planning Evaluation (IITP) grant funded by the Korea government(MSIT) (No.2019-0-00075, Artificial Intelligence Graduate School Program(KAIST))
+
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+
+Organization We describe detailed experimental setups in Section A, such as our baselines (Section A.1), model architecture (Section A.2), and the training configurations (Section A.3). We also provide additional analysis and experimental results in Section B, including analysis on communication costs (Section B.1) and number of labels per class (Section B.2), experiments on real-world dataset (Section B.3), different backbone architecture (Section B.4), fraction of clients per communication round (Section B.5).
+
+# A EXPERIMENTAL DETAILS
+
+We describe our experimental setups in detail, such as our baseline models, network architecture that is used for all base models and our method, and the detailed training setups.
+
+# A.1 BASELINE MODELS
+
+We consider UDA (Xie et al., 2019) and FixMatch (Sohn et al., 2020) as our baselines, since they are state-of-the-art SSL models and are based on the consistency-based mechanisms that are conceptually similar to our inter-client consistency loss. We reimplement UDA with the Training Signal Annealing (TSA) and exponential scheduling for its best performance as reported in their paper (we use RandAugment (Cubuk et al., 2019) for consistency regularization with random magnitude). We also reimplement FixMatch algorithms with strong augmentation as RandAugment (Cubuk et al., 2019). For weak augmentation (filp-and-shift), however, as the performance has significantly dropped when we apply the weak augmentation, we use original images rather than weakly augmenting the images. We fix confidence threshold $\tau { = } 0 . 8 5$ for all FixMatch and our model experiments. For federated learning frameworks, we use FedAvg (McMahan et al., 2017) and FedProx (Li et al., 2018) algorithms since they are the standard baselines for federated learning and can be easily combined with the SSL baselines. Detailed hyper-parameter settings are described in Table 4.
+
+# A.2 NETWORK ARCHITECTURE
+
+We build ResNet-9 networks as our base architecture for all base model and our method. In the architecture, the first two convolutional neural layers have 64 and 128 filters and the same $3 \times 3$ kernel sizes followed by $2 \times 2$ max-pooling layer. Then we have a skip connection between the subsequent two convolution layers with 128 filters. We then double the filter size from 128 to 256 with the next conv layer and down-sample via the following $2 \times 2$ max-pooling layer. We repeat the previous step, such that we have 512 filter size and $4 \times 4$ kernel size. Then, we perform another skip connec
+
+Table 3: Network Architecture of ResNet-9
+
+| CIFAR-10, Batch-IID Task with 100 Clients (K=100,F=0.05,H=2) |
| Labels-at-ClientScenario | Labels-at-ServerScenario |
| Methods | Acc.(%) | S2C Cost | C2S Cost | Acc.(%) | S2C Cost | C2S Cost |
| FedAvg-SLFedProx-SL | 58.60 ±0.4259.30 ± 0.31 | 100 %100 % | 100 %100 % | 52.45 ± 0.2349.11 ± 0.38 | 100 %100 % | 100%100 % |
| FedAvg-UDAFedProx-UDAFedAvg-FixMatchFedProx-FixMatch | 46.35 ± 0.2947.45 ± 0.2147.01 ± 0.4347.20 ±0.1252.13 ± 0.34 | 100%100 %100 %100 % | 100%100 %100 %100 % | 24.81±0.7319.91 ± 0.3111.95 ± 0.6025.61 ± 0.3244.95±0.49 | 100%100 %100 %100 %45% | 100%100 %100 %100 %22% |
| FedMatch(Ours) | 79% | 46% |
| CIFAR-10,Batch-NonIID Task with 100 Clients (K=100, F=O.05, H=2) | CIFAR-10,Batch-NonIID Task with 100 Clients (K=100, F=O.05, H=2) |
| FedAvg-SLFedProx-SL | 55.15 ± 0.2157.75 ± 0.15 | 100 %100 % | 100 %100 % | 51.50 ± 0.51 | 100 %100 % | 100 % |
| 49.31 ± 0.18 | | 100 % |
| FedAvg-UDAFedProx-UDAFedAvg-FixMatchFedProx-FixMatch | 44.35 ± 0.3946.31 ± 0.63 | 100%100 % | 100%100 % | 27.61±0.71 | 10 | 100% | 100% |
| 26.01 ± 0.78 | 100 % | 100 % |
| 46.20 ± 0.52 | 100 % | 100 % | 09.45 ± 0.34 | 100 % | 100 % |
| 4 | 45.55 ± 0.63 | 100 % | 100 % | 09.21 ±0.24 | 100 % | 100 %20% |
| FedMatch (Ours) | 52.25 ± 0.81 | 85% | 49% | 44.17 ±0.19 | 42% |
| Batch-lID Task (100 Clients) Batch-NonlID (100 Clients) Batch-lID Task (100 Clients) Batch-NonlID (100 Clients)60 60 60 60Wwy50 50 50 mwwW 50 %) eeeect wwwwy40myyiww% 40eeeeeeeeeeec CM30303030FedProx*SLFedProx*SLFedProx*SL20202020FedAvg*SLFedProx*UDAFedProx*UDA+FedProx*UDAFedProx*UDAFedProx*FixMatch FedProx*FixMatch FedProx*FixMatch FedProx*FixMatch10 10 10 10FedMatch (Ours) FedMatch (Ours) FedMatch (Ours) FedMatch (Ours)100150 150 0 0502005010020050100150200 50 100150 200Communication Round Communication Round Communication Round Communication Round(a)Labels-at-Client Scenario (b)Labels-at-Server Scenario |
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+ "Figure 5: Test Accuracy Curves on Batch-IID & NonIID Tasks We visualize test accuracy curves of model performance corresponding to the Table 1. Note that the SL (Supervised Learning) models learn on both $s$ and $\\mathcal { U }$ with full labels, and are utilized as the upper bounds for each experiment. "
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+ "text": "Baselines and Training Details Our baselines are: 1) Local-SL: local supervised learning (SL) with full labels $( S + \\mathcal { U } )$ without sharing locally learned knowledge. 2) Local-UDA and 3) LocalFixMatch: local semi-supervised learning, including UDA and FixMatch, without sharing local knowledge. 4) FedAVG-SL and 5) FedProx-SL: supervised learning with full labels $( S + \\mathcal { U } )$ while sharing local knowledge via FedAvg and FedProx frameworks. 6) FedAvg-UDA and 7) FedProxUDA: naive combinations of FedAvg/Prox with UDA. 8) FedAvg-FixMatch and 9) FedProxFixMatch: naive combination of with FixMatch with FedAvg/Prox. For training, we use SGD with adaptive-learning rate decay introduced in (Serra et al., 2018) with the initial learning rate $1 \\mathrm { e } { - 3 }$ . We use ResNet-9 networks as the backbone architecture for all baselines and our methods. We ensure that all hyper-parameters are set equally for all base models and ours to perform fair evaluation and comparison. Please see the Section A in the Appendix for further details. For all experiments, we report the mean and the standard deviation over 3 runs. ",
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+ "text": "Results on Batch-IID & NonIID Tasks Table 1 shows performance comparison of our models and naive Fed-SSL algorithms on Batch-IID and NonIID tasks under the two different scenarios. We observe that our model outperforms all naive Fed-SSL baselines for all tasks and scenarios. In particular, under labels-at-server scenario, which is more challenging than labels-at-client scenario, we observe that the naive combination models significantly suffer from the forgetting issue and their performances keeps deteriorating after a certain communication round. This phenomenon is mainly caused by the base models failing to properly perform disjoint learning, in which case the learned knowledge from the labeled and unlabeled data causes inter-task interference. Contrarily, our methods show consistent and robust performance regardless where the labeled data exists, which shows that our decomposition techniques effectively handles the challenging disjoint learning scenario. In addition, when the class-wise distribution is imbalanced for each client (Non-IID task), we observe that the base models’ performance slightly drops by $1 - 3 \\% p$ , while our methods show consistent. ",
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+ "text": "This shows that our inter-client consistency effectively enhances consistency with the helper agents selected from server based on model similarity, which is good at, in particular, class-imbalanced tasks. We also visualize the test accuracy curve for our models and naive Fed-SSL in Fig. 5. Our method (Red line) trains faster and consistently outperforms the base models, and is most robustness against inter-task interference in both scenarios. For analysis on the averaged communication costs, please see Section B.1 in the Appendix. ",
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+ "text": "Results on Streaming-NonIID Task Table 2 shows averaged local model performance on Streaming-NonIID tasks with 10 synchronized clients. For the labels-at-client scenario, our proposed method outperforms local-SSL and naive Fed-SSL models with large margins, $4 \\mathrm { - } 1 5 \\% p$ , except for the SL models. There is no huge difference of performance between local SSL and Fed-SSL models, and this implies that our method effectively utilizes inter-client knowledge in this streaming setting. In the labels-at-server scenario, interestingly, the performance of FedProx-SL decreases by around $5 \\% p$ compared to the labels-at-client scenario, while Fed-SSL models obtain improved performance. We conjecture that this is because, for streaming situation, the model may not sufficiently train on the new data, while Fed-SSL models overcomes it by utilizing only the consistent pseudo-labels. Even on this task, FedMatch outperforms all baselines with significantly smaller communication cost on average (see Section B.1 for detailed analysis of the averaged communication costs). ",
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+ "text": "Effectiveness of Inter-Client Consistency To show the effectiveness of our inter-client consistency loss, we eliminate the loss, while learning on Batch-IID task with 100 clients $( F { = } 0 . 0 5 )$ . In Figure 6 (a), when we remove our inter-client consistency loss, we observe that the performance has slightly dropped (Pink line) from one with the loss term (Red line). This gap clearly tells us that our interclient consistency loss improves model consistency across multiple models while keeping reliable knowledge. Interestingly, our model without inter-client consistency loss still outperforms base models. This additionally implies that our another proposed method, parameter decomposition for disjoint learning, also effectively enhances model performance. ",
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+ "text": "Effectiveness of Parameter Decomposition Our model with parameter decomposition alone, without inter-client consistency loss, outperforms base models. For further analysis, we show the effect of each decomposed variables, $\\sigma$ and $\\psi$ , in Figure 6 (b). Removing either of $\\sigma$ and $\\psi$ results in substantial drop in the performance, with larger performance degeneration when dropping $\\sigma$ , which captures much more essential knowledge from labeled data (Green line). Such decomposition is effective since there exists knowledge interference between supervised learning and unsupervised learning. We show this with an experiment where we perform semi-supervised learning with 5 labeled instances per class and 1, 000 unlabeled instances for 100 rounds. We measure accuracy on the labeled set at each training steps. As shown in Figure 6 (c), our method effectively preserves learned knowledge from labeled set, while other base models suffer from knowledge interference. This effective separation of supervised and unsupervised learning tasks enhances the overall performance of our methods even without inter-client consistency loss as shown in Figure 6 (a) (shown in pink). ",
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+ "text": "Number of Labels per Class We increase the number of labels per class for each client in a range of 1, 5, 10, and 20 on Batch-IID task (CIFAR-10). Our method shows consistent performance improvement as the number of labels increases. Interestingly, we observe that baseline models, FedProx-UDA/FixMatch, show performance degradation even when the labeled data increases $( 5 1 0 )$ ). These results show that our method effectively utilize knowledge from labeled and unlabeled data in federated semi-supervised learning settings, while other naive combinations of FSSL could fail to learn properly from labeled and unlabeled data in federated learning framework. ",
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+ "text": "7 RELATED WORK ",
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+ "text": "Federated Learning A variety of approaches for averaging local weights at server have been introduced in the past few years. FedAvg (McMahan et al., 2017) performs weighted-averaging on local weights according to the local train size. FedProx (Li et al., 2018) uniformly averages the local updates while clients perform proximal regularization against the global weights, while FedMA (Wang et al., 2020) matches the hidden elements with similar feature extraction signatures in layer-wise manner when averaging local weights. PFNM (Yurochkin et al., 2019) introduces aggregation policy which leverages Bayesian non-parametric methods. Beyond focusing on averaging local knowledge, there are various efforts to extend FL to the other areas, such as continual learning under federated learning frameworks (Yoon et al., 2020a) inspired by parameter decomposition techniques proposed by (Yoon et al., 2020b). Recently, interests of tackling scarcity of labeled data in FL are emerging and discussed in (Jin et al., 2020; Guha et al., 2019; Albaseer et al., 2020). ",
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+ "text": "Semi-Supervised Learning While there exist numerous work on SSL, we mainly discuss consistency regularization approaches. Consistency regularization (Sajjadi et al., 2016) assumes that the class semantics will not be affected by transformations of the input instances, and enforces the model output to be the same across different input perturbations. Some extensions to this technique perturb inputs adversarially (Miyato et al., 2018), through dropout (Srivastava et al., 2014), or through data augmentation (French et al., 2018). UDA (Xie et al., 2019) and ReMixMatch (Berthelot et al., 2019a) use two sets of augmentations, weak and strong, and enforce consistency between the weakly and strongly augmented examples. Recently, in addition to enforcing consistency between weak-strong augmented pairs, FixMatch (Sohn et al., 2020) performs pseudo-label refinement on model predictions via thresholding. Entropy minimization (Grandvalet & Bengio, 2004) which enforces the classifier to predict low-entropy on unlabeled data, is another popular technique for SSL. Pseudo-Label (Lee, 2013) constructs one-hot labels from highly confident predictions on unlabeled data and uses these as training targets inn a standard cross-entropy loss. MixMatch (Berthelot et al., 2019c) performs sharpening on target distribution on unlabeled data, to further refine the generated pseudo-label. ",
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+ "text": "8 CONCLUSION ",
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+ "text": "In this work, we introduced two practical scenarios of Federated Semi-Supervised Learning (FSSL) where each client learns with only partly labeled data (Labels-at-Client scenario), or supervised labels are only available at the server, while clients work with completely unlabeled data (Labels-at-Server scenario). To tackle the problem, we propose a novel method, Federated Matching (FedMatch), which introduces the inter-client consistency loss that aims to maximize the agreement between the models trained at different clients, and the parameter decomposition for disjoint learning which decomposes the parameters into one for labeled data and the other for unlabeled data for preservation of reliable knowledge, reduction of communication costs, and disjoint learning. Through extensive experimental validation, we show that FedMatch significantly outperforms both local semi-supervised learning methods and naive combinations of federated learning algorithms with semi-supervised learning on diverse and realistic scenarios. As future work, we plan to further improve our model to tackle the scenario where pretrained models deployed at each client adapts to a completely unlabeled data stream (e.g. on-device learning of smart speakers). ",
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+ "text": "Acknowledgements This work was supported by Samsung Research Funding Center of Samsung Electronics (No. SRFC-IT1502-51), Samsung Advanced Institute of Technology, Samsung Electronics Co., Ltd., Next-Generation Information Computing Development Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT & Future Plannig (No. 2016M3C4A7952634), the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (2018R1A5A1059921), and Center for Applied Research in Artificial Intelligence (CARAI) grant funded by DAPA and ADD (UDI190031RD). Also, this work was supported by Institute of Information communications Technology Planning Evaluation (IITP) grant funded by the Korea government(MSIT) (No.2019-0-00075, Artificial Intelligence Graduate School Program(KAIST)) ",
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+ "text": "Organization We describe detailed experimental setups in Section A, such as our baselines (Section A.1), model architecture (Section A.2), and the training configurations (Section A.3). We also provide additional analysis and experimental results in Section B, including analysis on communication costs (Section B.1) and number of labels per class (Section B.2), experiments on real-world dataset (Section B.3), different backbone architecture (Section B.4), fraction of clients per communication round (Section B.5). ",
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+ "text": "A EXPERIMENTAL DETAILS ",
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+ "text": "We describe our experimental setups in detail, such as our baseline models, network architecture that is used for all base models and our method, and the detailed training setups. ",
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+ "text": "A.1 BASELINE MODELS",
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+ "text": "We consider UDA (Xie et al., 2019) and FixMatch (Sohn et al., 2020) as our baselines, since they are state-of-the-art SSL models and are based on the consistency-based mechanisms that are conceptually similar to our inter-client consistency loss. We reimplement UDA with the Training Signal Annealing (TSA) and exponential scheduling for its best performance as reported in their paper (we use RandAugment (Cubuk et al., 2019) for consistency regularization with random magnitude). We also reimplement FixMatch algorithms with strong augmentation as RandAugment (Cubuk et al., 2019). For weak augmentation (filp-and-shift), however, as the performance has significantly dropped when we apply the weak augmentation, we use original images rather than weakly augmenting the images. We fix confidence threshold $\\tau { = } 0 . 8 5$ for all FixMatch and our model experiments. For federated learning frameworks, we use FedAvg (McMahan et al., 2017) and FedProx (Li et al., 2018) algorithms since they are the standard baselines for federated learning and can be easily combined with the SSL baselines. Detailed hyper-parameter settings are described in Table 4. ",
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+ "text": "A.2 NETWORK ARCHITECTURE ",
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+ "text": "We build ResNet-9 networks as our base architecture for all base model and our method. In the architecture, the first two convolutional neural layers have 64 and 128 filters and the same $3 \\times 3$ kernel sizes followed by $2 \\times 2$ max-pooling layer. Then we have a skip connection between the subsequent two convolution layers with 128 filters. We then double the filter size from 128 to 256 with the next conv layer and down-sample via the following $2 \\times 2$ max-pooling layer. We repeat the previous step, such that we have 512 filter size and $4 \\times 4$ kernel size. Then, we perform another skip connec",
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+ "Table 3: Network Architecture of ResNet-9 "
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+ "table_body": "| CIFAR-10, Batch-IID Task with 100 Clients (K=100,F=0.05,H=2) |
| Labels-at-ClientScenario | Labels-at-ServerScenario |
| Methods | Acc.(%) | S2C Cost | C2S Cost | Acc.(%) | S2C Cost | C2S Cost |
| FedAvg-SLFedProx-SL | 58.60 ±0.4259.30 ± 0.31 | 100 %100 % | 100 %100 % | 52.45 ± 0.2349.11 ± 0.38 | 100 %100 % | 100%100 % |
| FedAvg-UDAFedProx-UDAFedAvg-FixMatchFedProx-FixMatch | 46.35 ± 0.2947.45 ± 0.2147.01 ± 0.4347.20 ±0.1252.13 ± 0.34 | 100%100 %100 %100 % | 100%100 %100 %100 % | 24.81±0.7319.91 ± 0.3111.95 ± 0.6025.61 ± 0.3244.95±0.49 | 100%100 %100 %100 %45% | 100%100 %100 %100 %22% |
| FedMatch(Ours) | 79% | 46% |
| CIFAR-10,Batch-NonIID Task with 100 Clients (K=100, F=O.05, H=2) | CIFAR-10,Batch-NonIID Task with 100 Clients (K=100, F=O.05, H=2) |
| FedAvg-SLFedProx-SL | 55.15 ± 0.2157.75 ± 0.15 | 100 %100 % | 100 %100 % | 51.50 ± 0.51 | 100 %100 % | 100 % |
| 49.31 ± 0.18 | | 100 % |
| FedAvg-UDAFedProx-UDAFedAvg-FixMatchFedProx-FixMatch | 44.35 ± 0.3946.31 ± 0.63 | 100%100 % | 100%100 % | 27.61±0.71 | 10 | 100% | 100% |
| 26.01 ± 0.78 | 100 % | 100 % |
| 46.20 ± 0.52 | 100 % | 100 % | 09.45 ± 0.34 | 100 % | 100 % |
| 4 | 45.55 ± 0.63 | 100 % | 100 % | 09.21 ±0.24 | 100 % | 100 %20% |
| FedMatch (Ours) | 52.25 ± 0.81 | 85% | 49% | 44.17 ±0.19 | 42% |
| Batch-lID Task (100 Clients) Batch-NonlID (100 Clients) Batch-lID Task (100 Clients) Batch-NonlID (100 Clients)60 60 60 60Wwy50 50 50 mwwW 50 %) eeeect wwwwy40myyiww% 40eeeeeeeeeeec CM30303030FedProx*SLFedProx*SLFedProx*SL20202020FedAvg*SLFedProx*UDAFedProx*UDA+FedProx*UDAFedProx*UDAFedProx*FixMatch FedProx*FixMatch FedProx*FixMatch FedProx*FixMatch10 10 10 10FedMatch (Ours) FedMatch (Ours) FedMatch (Ours) FedMatch (Ours)100150 150 0 0502005010020050100150200 50 100150 200Communication Round Communication Round Communication Round Communication Round(a)Labels-at-Client Scenario (b)Labels-at-Server Scenario |
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| Labels-at-ClientScenario | Labels-at-ServerScenario |
| Methods | Acc.(%) | S2C Cost | C2S Cost | Acc.(%) | S2C Cost | C2S Cost |
| FedAvg-SLFedProx-SL | 58.60 ±0.4259.30 ± 0.31 | 100 %100 % | 100 %100 % | 52.45 ± 0.2349.11 ± 0.38 | 100 %100 % | 100%100 % |
| FedAvg-UDAFedProx-UDAFedAvg-FixMatchFedProx-FixMatch | 46.35 ± 0.2947.45 ± 0.2147.01 ± 0.4347.20 ±0.1252.13 ± 0.34 | 100%100 %100 %100 % | 100%100 %100 %100 % | 24.81±0.7319.91 ± 0.3111.95 ± 0.6025.61 ± 0.3244.95±0.49 | 100%100 %100 %100 %45% | 100%100 %100 %100 %22% |
| FedMatch(Ours) | 79% | 46% |
| CIFAR-10,Batch-NonIID Task with 100 Clients (K=100, F=O.05, H=2) | CIFAR-10,Batch-NonIID Task with 100 Clients (K=100, F=O.05, H=2) |
| FedAvg-SLFedProx-SL | 55.15 ± 0.2157.75 ± 0.15 | 100 %100 % | 100 %100 % | 51.50 ± 0.51 | 100 %100 % | 100 % |
| 49.31 ± 0.18 | | 100 % |
| FedAvg-UDAFedProx-UDAFedAvg-FixMatchFedProx-FixMatch | 44.35 ± 0.3946.31 ± 0.63 | 100%100 % | 100%100 % | 27.61±0.71 | 10 | 100% | 100% |
| 26.01 ± 0.78 | 100 % | 100 % |
| 46.20 ± 0.52 | 100 % | 100 % | 09.45 ± 0.34 | 100 % | 100 % |
| 4 | 45.55 ± 0.63 | 100 % | 100 % | 09.21 ±0.24 | 100 % | 100 %20% |
| FedMatch (Ours) | 52.25 ± 0.81 | 85% | 49% | 44.17 ±0.19 | 42% |
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| Labels-at-ClientScenario | Labels-at-ServerScenario |
| Methods | Acc.(%) | S2C Cost | C2S Cost | Acc.(%) | S2C Cost | C2S Cost |
| FedAvg-SLFedProx-SL | 58.60 ±0.4259.30 ± 0.31 | 100 %100 % | 100 %100 % | 52.45 ± 0.2349.11 ± 0.38 | 100 %100 % | 100%100 % |
| FedAvg-UDAFedProx-UDAFedAvg-FixMatchFedProx-FixMatch | 46.35 ± 0.2947.45 ± 0.2147.01 ± 0.4347.20 ±0.1252.13 ± 0.34 | 100%100 %100 %100 % | 100%100 %100 %100 % | 24.81±0.7319.91 ± 0.3111.95 ± 0.6025.61 ± 0.3244.95±0.49 | 100%100 %100 %100 %45% | 100%100 %100 %100 %22% |
| FedMatch(Ours) | 79% | 46% |
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| 49.31 ± 0.18 | | 100 % |
| FedAvg-UDAFedProx-UDAFedAvg-FixMatchFedProx-FixMatch | 44.35 ± 0.3946.31 ± 0.63 | 100%100 % | 100%100 % | 27.61±0.71 | 10 | 100% | 100% |
| 26.01 ± 0.78 | 100 % | 100 % |
| 46.20 ± 0.52 | 100 % | 100 % | 09.45 ± 0.34 | 100 % | 100 % |
| 4 | 45.55 ± 0.63 | 100 % | 100 % | 09.21 ±0.24 | 100 % | 100 %20% |
| FedMatch (Ours) | 52.25 ± 0.81 | 85% | 49% | 44.17 ±0.19 | 42% |
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| 1 M←NCM(V,9) //from Def.7 |
| 2 Initialize parameters θmin and θmax |
| 3 fork←1tondo //Estimate from Eq.3 |
| 4 | Pmin ← Estimate(M(0min),V,vk,0,0,m) |
| Pmax ← Estimate(M(0max),V,Vk,0,O,m) |
| 5 6 | ↑0 |
| 7 | qmin qmax←0 |
| 8 | forv ∈Dv do |
| 9 | if Consistent(v,y) then |
| 10 | qmin←qmin+ |
| Estimate(M(0min),V,v,X,x,m) |
| 11 | qmax←qmax+ |
| Estimate(M(0max),V,v,X,x,m) |
| //L from Eq.5 |
| 12 | Lmin ←-log Pmin-λlog(1-qmin) |
| 13 | Lmax←-logPmax -λlogqmax |
| |
| 14 | 0min←0min+nVLmin |
| |
| 0max←0max+nVLmax |
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+# CATEGORICAL REPARAMETERIZATION WITH GUMBEL-SOFTMAX
+
+Eric Jang
+Google Brain
+ejang@google.com Shixiang $\mathbf { G u } ^ { * }$
+University of Cambridge MPI Tubingen ¨
+sg717@cam.ac.uk
+Ben Poole∗
+Stanford University
+poole@cs.stanford.edu
+
+# ABSTRACT
+
+Categorical variables are a natural choice for representing discrete structure in the world. However, stochastic neural networks rarely use categorical latent variables due to the inability to backpropagate through samples. In this work, we present an efficient gradient estimator that replaces the non-differentiable sample from a categorical distribution with a differentiable sample from a novel Gumbel-Softmax distribution. This distribution has the essential property that it can be smoothly annealed into a categorical distribution. We show that our Gumbel-Softmax estimator outperforms state-of-the-art gradient estimators on structured output prediction and unsupervised generative modeling tasks with categorical latent variables, and enables large speedups on semi-supervised classification.
+
+# 1 INTRODUCTION
+
+Stochastic neural networks with discrete random variables are a powerful technique for representing distributions encountered in unsupervised learning, language modeling, attention mechanisms, and reinforcement learning domains. For example, discrete variables have been used to learn probabilistic latent representations that correspond to distinct semantic classes (Kingma et al., 2014), image regions (Xu et al., 2015), and memory locations (Graves et al., 2014; Graves et al., 2016). Discrete representations are often more interpretable (Chen et al., 2016) and more computationally efficient (Rae et al., 2016) than their continuous analogues.
+
+However, stochastic networks with discrete variables are difficult to train because the backpropagation algorithm — while permitting efficient computation of parameter gradients — cannot be applied to non-differentiable layers. Prior work on stochastic gradient estimation has traditionally focused on either score function estimators augmented with Monte Carlo variance reduction techniques (Paisley et al., 2012; Mnih & Gregor, 2014; Gu et al., 2016; Gregor et al., 2013), or biased path derivative estimators for Bernoulli variables (Bengio et al., 2013). However, no existing gradient estimator has been formulated specifically for categorical variables. The contributions of this work are threefold:
+
+1. We introduce Gumbel-Softmax, a continuous distribution on the simplex that can approximate categorical samples, and whose parameter gradients can be easily computed via the reparameterization trick.
+2. We show experimentally that Gumbel-Softmax outperforms all single-sample gradient estimators on both Bernoulli variables and categorical variables.
+3. We show that this estimator can be used to efficiently train semi-supervised models (e.g. Kingma et al. (2014)) without costly marginalization over unobserved categorical latent variables.
+
+The practical outcome of this paper is a simple, differentiable approximate sampling mechanism for categorical variables that can be integrated into neural networks and trained using standard backpropagation.
+
+# 2 THE GUMBEL-SOFTMAX DISTRIBUTION
+
+We begin by defining the Gumbel-Softmax distribution, a continuous distribution over the simplex that can approximate samples from a categorical distribution. Let $z$ be a categorical variable with class probabilities $\pi _ { 1 } , \pi _ { 2 } , . . . \pi _ { k }$ . For the remainder of this paper we assume categorical samples are encoded as $k$ -dimensional one-hot vectors lying on the corners of the $\left( k - 1 \right)$ -dimensional simplex, $\Delta ^ { k - 1 }$ . This allows us to define quantities such as the element-wise mean $\vec { \mathbb { E } } _ { p } [ z ] = [ \pi _ { 1 } , . . . , \bar { \pi _ { k } } ]$ of these vectors.
+
+The Gumbel-Max trick (Gumbel, 1954; Maddison et al., 2014) provides a simple and efficient way to draw samples $z$ from a categorical distribution with class probabilities $\pi$ :
+
+$$
+z = { \mathrm { o n e \_ h o t } } \left( \operatorname { a r g m a x } _ { i } \left[ g _ { i } + \log \pi _ { i } \right] \right)
+$$
+
+where $g _ { 1 } . . . g _ { k }$ are i.i.d samples drawn from Gumbel $( 0 , 1 ) ^ { 1 }$ . We use the softmax function as a continuous, differentiable approximation to arg max, and generate $k$ -dimensional sample vectors $y \in \Delta ^ { k - 1 }$ where
+
+$$
+y _ { i } = { \frac { \exp ( ( \log ( \pi _ { i } ) + g _ { i } ) / \tau ) } { \sum _ { j = 1 } ^ { k } \exp ( ( \log ( \pi _ { j } ) + g _ { j } ) / \tau ) } } \qquad { \mathrm { f o r ~ } } i = 1 , . . . , k .
+$$
+
+The density of the Gumbel-Softmax distribution (derived in Appendix B) is:
+
+$$
+p _ { \pi , \tau } ( y _ { 1 } , . . . , y _ { k } ) = \Gamma ( k ) \tau ^ { k - 1 } \left( \sum _ { i = 1 } ^ { k } \pi _ { i } / y _ { i } ^ { \tau } \right) ^ { - k } \prod _ { i = 1 } ^ { k } \left( \pi _ { i } / y _ { i } ^ { \tau + 1 } \right)
+$$
+
+This distribution was independently discovered by Maddison et al. (2016), where it is referred to as the concrete distribution. As the softmax temperature $\tau$ approaches 0, samples from the GumbelSoftmax distribution become one-hot and the Gumbel-Softmax distribution becomes identical to the categorical distribution $p ( z )$ .
+
+
+Figure 1: The Gumbel-Softmax distribution interpolates between discrete one-hot-encoded categorical distributions and continuous categorical densities. (a) For low temperatures $( \tau = 0 . 1 , \tau = 0 . 5 )$ , the expected value of a Gumbel-Softmax random variable approaches the expected value of a categorical random variable with the same logits. As the temperature increases $\tau = 1 . 0$ , $\tau = 1 0 . 0 $ ), the expected value converges to a uniform distribution over the categories. (b) Samples from GumbelSoftmax distributions are identical to samples from a categorical distribution as $\tau 0$ . At higher temperatures, Gumbel-Softmax samples are no longer one-hot, and become uniform as $\tau \infty$ .
+
+# 2.1 GUMBEL-SOFTMAX ESTIMATOR
+
+The Gumbel-Softmax distribution is smooth for $\tau > 0$ , and therefore has a well-defined gradient $\partial y / \partial \pi$ with respect to the parameters $\pi$ . Thus, by replacing categorical samples with GumbelSoftmax samples we can use backpropagation to compute gradients (see Section 3.1). We denote this procedure of replacing non-differentiable categorical samples with a differentiable approximation during training as the Gumbel-Softmax estimator.
+
+While Gumbel-Softmax samples are differentiable, they are not identical to samples from the corresponding categorical distribution for non-zero temperature. For learning, there is a tradeoff between small temperatures, where samples are close to one-hot but the variance of the gradients is large, and large temperatures, where samples are smooth but the variance of the gradients is small (Figure 1). In practice, we start at a high temperature and anneal to a small but non-zero temperature.
+
+In our experiments, we find that the softmax temperature $\tau$ can be annealed according to a variety of schedules and still perform well. If $\tau$ is a learned parameter (rather than annealed via a fixed schedule), this scheme can be interpreted as entropy regularization (Szegedy et al., 2015; Pereyra et al., 2016), where the Gumbel-Softmax distribution can adaptively adjust the “confidence” of proposed samples during the training process.
+
+# 2.2 STRAIGHT-THROUGH GUMBEL-SOFTMAX ESTIMATOR
+
+Continuous relaxations of one-hot vectors are suitable for problems such as learning hidden representations and sequence modeling. For scenarios in which we are constrained to sampling discrete values (e.g. from a discrete action space for reinforcement learning, or quantized compression), we discretize $y$ using arg max but use our continuous approximation in the backward pass by approximating $\nabla _ { \theta } z \approx \nabla _ { \theta } y$ . We call this the Straight-Through (ST) Gumbel Estimator, as it is reminiscent of the biased path derivative estimator described in Bengio et al. (2013). ST Gumbel-Softmax allows samples to be sparse even when the temperature $\tau$ is high.
+
+# 3 RELATED WORK
+
+In this section we review existing stochastic gradient estimation techniques for discrete variables (illustrated in Figure 2). Consider a stochastic computation graph (Schulman et al., 2015) with discrete random variable $z$ whose distribution depends on parameter $\theta$ , and cost function $f ( z )$ . The objective is to minimize the expected cost $L ( \bar { \theta } ) = \mathbb { E } _ { z \sim p _ { \theta } ( z ) } [ f ( z ) ]$ via gradient descent, which requires us to estimate $\nabla _ { \theta } \mathbb { E } _ { z \sim p _ { \theta } ( z ) } [ f ( z ) ]$ .
+
+# 3.1 PATH DERIVATIVE GRADIENT ESTIMATORS
+
+For distributions that are reparameterizable, we can compute the sample $z$ as a deterministic function $g$ of the parameters $\theta$ and an independent random variable $\epsilon$ , so that $z = g ( \theta , \epsilon )$ . The path-wise gradients from $f$ to $\theta$ can then be computed without encountering any stochastic nodes:
+
+$$
+\frac { \partial } { \partial \theta } \mathbb { E } _ { z \sim p _ { \theta } } \left[ f ( z ) ) \right] = \frac { \partial } { \partial \theta } \mathbb { E } _ { \epsilon } \left[ f ( g ( \theta , \epsilon ) ) \right] = \mathbb { E } _ { \epsilon \sim p _ { \epsilon } } \left[ \frac { \partial f } { \partial g } \frac { \partial g } { \partial \theta } \right]
+$$
+
+For example, the normal distribution $z \sim \mathcal { N } ( \mu , \sigma )$ can be re-written as $\mu + \sigma \cdot \mathcal { N } ( 0 , 1 )$ , making it trivial to compute $\partial z / \partial \mu$ and $\partial z / \partial \sigma$ . This reparameterization trick is commonly applied to training variational autooencoders with continuous latent variables using backpropagation (Kingma $\&$ Welling, 2013; Rezende et al., 2014b). As shown in Figure 2, we exploit such a trick in the construction of the Gumbel-Softmax estimator.
+
+Biased path derivative estimators can be utilized even when $z$ is not reparameterizable. In general, we can approximate $\nabla _ { \theta } z \approx \nabla _ { \theta } m ( \theta )$ , where $m$ is a differentiable proxy for the stochastic sample. For Bernoulli variables with mean parameter $\theta$ , the Straight-Through (ST) estimator (Bengio et al., 2013) approximates $m = \mu _ { \theta } ( z )$ , implying $\nabla _ { \theta } m = 1$ . For $k = 2$ (Bernoulli), ST Gumbel-Softmax is similar to the slope-annealed Straight-Through estimator proposed by Chung et al. (2016), but uses a softmax instead of a hard sigmoid to determine the slope. Rolfe (2016) considers an alternative approach where each binary latent variable parameterizes a continuous mixture model. Reparameterization gradients are obtained by backpropagating through the continuous variables and marginalizing out the binary variables.
+
+One limitation of the ST estimator is that backpropagating with respect to the sample-independent mean may cause discrepancies between the forward and backward pass, leading to higher variance.
+
+
+Figure 2: Gradient estimation in stochastic computation graphs. (1) $\nabla _ { \boldsymbol { \theta } } f ( { \boldsymbol { x } } )$ can be computed via backpropagation if $x ( \theta )$ is deterministic and differentiable. (2) The presence of stochastic node $z$ precludes backpropagation as the sampler function does not have a well-defined gradient. (3) The score function estimator and its variants (NVIL, DARN, MuProp, VIMCO) obtain an unbiased estimate of $\nabla _ { \boldsymbol { \theta } } f ( { \boldsymbol { x } } )$ by backpropagating along a surrogate loss $\hat { f } \log p _ { \theta } ( z )$ , where ${ \hat { f } } = f ( x ) - b$ and $b$ is a baseline for variance reduction. (4) The Straight-Through estimator, developed primarily for Bernoulli variables, approximates $\nabla _ { \theta } z \approx 1$ . (5) Gumbel-Softmax is a path derivative estimator for a continuous distribution $y$ that approximates $z$ . Reparameterization allows gradients to flow from $f ( y )$ to $\theta$ . $y$ can be annealed to one-hot categorical variables over the course of training.
+
+Gumbel-Softmax avoids this problem because each sample $y$ is a differentiable proxy of the corresponding discrete sample $z$ .
+
+# 3.2 SCORE FUNCTION-BASED GRADIENT ESTIMATORS
+
+The score function estimator (SF, also referred to as REINFORCE (Williams, 1992) and likelihood ratio estimator (Glynn, 1990)) uses the identity $\nabla _ { \boldsymbol { \theta } } \log { p _ { \boldsymbol { \theta } } ( z ) } = p _ { \boldsymbol { \theta } } ( z ) \nabla _ { \boldsymbol { \theta } } \log { p _ { \boldsymbol { \theta } } ( z ) }$ to derive the following unbiased estimator:
+
+$$
+\nabla _ { \boldsymbol { \theta } } \mathbb { E } _ { z } \left[ f ( \boldsymbol { z } ) \right] = \mathbb { E } _ { z } \left[ f ( \boldsymbol { z } ) \nabla _ { \boldsymbol { \theta } } \log p _ { \boldsymbol { \theta } } ( \boldsymbol { z } ) \right]
+$$
+
+SF only requires that $p _ { \theta } ( z )$ is continuous in $\theta$ , and does not require backpropagating through $f$ or the sample $z$ . However, SF suffers from high variance and is consequently slow to converge. In particular, the variance of SF scales linearly with the number of dimensions of the sample vector (Rezende et al., 2014a), making it especially challenging to use for categorical distributions.
+
+The variance of a score function estimator can be reduced by subtracting a control variate $b ( z )$ from the learning signal $f$ , and adding back its analytical expectation $\mu _ { b } = \bar { \mathbb { E } _ { z } } \left[ b ( z ) \nabla _ { \theta } \log p _ { \theta } ( z ) \right]$ to keep the estimator unbiased:
+
+$$
+\begin{array} { r l } & { \nabla _ { \theta } \mathbb { E } _ { z } \left[ f ( z ) \right] = \mathbb { E } _ { z } \left[ f ( z ) \nabla _ { \theta } \log p _ { \theta } ( z ) + ( b ( z ) \nabla _ { \theta } \log p _ { \theta } ( z ) - b ( z ) \nabla _ { \theta } \log p _ { \theta } ( z ) ) \right] } \\ & { \qquad = \mathbb { E } _ { z } \left[ ( f ( z ) - b ( z ) ) \nabla _ { \theta } \log p _ { \theta } ( z ) \right] + \mu _ { b } } \end{array}
+$$
+
+We briefly summarize recent stochastic gradient estimators that utilize control variates. We direct the reader to $\mathrm { G u }$ et al. (2016) for further detail on these techniques.
+
+• NVIL (Mnih & Gregor, 2014) uses two baselines: (1) a moving average $\bar { f }$ of $f$ to center the learning signal, and (2) an input-dependent baseline computed by a 1-layer neural network fitted to $f - { \bar { f } }$ (a control variate for the centered learning signal itself). Finally, variance normalization divides the learning signal by $\operatorname* { m a x } ( 1 , \sigma _ { f } )$ , where $\sigma _ { f } ^ { 2 }$ is a moving average of $\mathrm { V a r } [ f ]$ .
+
+• DARN (Gregor et al., 2013) uses $b = f ( \bar { z } ) + f ^ { \prime } ( \bar { z } ) ( \bar { z } - z )$ , where the baseline corresponds to the first-order Taylor approximation of $f ( z )$ from $f ( \bar { z } )$ . $\bar { z }$ is chosen to be $1 / 2$ for Bernoulli variables, which makes the estimator biased for non-quadratic $f$ , since it ignores the correction term $\mu _ { b }$ in the estimator expression.
+MuProp (Gu et al., 2016) also models the baseline as a first-order Taylor expansion: $b =$ $f ( { \bar { z } } ) \stackrel { \_ } { + } f ^ { \prime } ( { \bar { z } } ) ( z - { \bar { z } } )$ and $\mu _ { b } \ = \ f ^ { \prime } ( \bar { z } ) \nabla _ { \theta } \mathbb { E } _ { z } \left[ z \right]$ . To overcome backpropagation through discrete sampling, a mean-field approximation $f _ { M F } ( \mu _ { \theta } ( z ) )$ is used in place of $f ( z )$ to compute the baseline and derive the relevant gradients.
+• VIMCO (Mnih & Rezende, 2016) is a gradient estimator for multi-sample objectives that uses the mean of other samples $\textstyle b = 1 / m \sum _ { j \neq i } f ( z _ { j } )$ to construct a baseline for each sample $z _ { i } \in z _ { 1 : m }$ . We exclude VIMCO from our experiments because we are comparing estimators for single-sample objectives, although Gumbel-Softmax can be easily extended to multisample objectives.
+
+# 3.3 SEMI-SUPERVISED GENERATIVE MODELS
+
+Semi-supervised learning considers the problem of learning from both labeled data $( x , y ) \sim \mathcal { D } _ { L }$ and unlabeled data $x \sim \mathcal { D } _ { U }$ , where $x$ are observations (i.e. images) and $y$ are corresponding labels (e.g. semantic class). For semi-supervised classification, Kingma et al. (2014) propose a variational autoencoder (VAE) whose latent state is the joint distribution over a Gaussian “style” variable $z$ and a categorical “semantic class” variable $y$ (Figure 6, Appendix). The VAE objective trains a discriminative network $q _ { \phi } ( y | x )$ , inference network $q _ { \phi } ( z | x , y )$ , and generative network $p _ { \theta } ( x | y , z )$ end-to-end by maximizing a variational lower bound on the log-likelihood of the observation under the generative model. For labeled data, the class $y$ is observed, so inference is only done on $z \sim$ $q ( \boldsymbol { z } | \bar { \boldsymbol { x } } , \boldsymbol { y } )$ . The variational lower bound on labeled data is given by:
+
+$$
+\log p _ { \theta } ( x , y ) \geq - \mathcal { L } ( x , y ) = \mathbb { E } _ { z \sim q _ { \phi } ( z \mid x , y ) } \left[ \log p _ { \theta } ( x | y , z ) \right] - K L [ q ( z | x , y ) | | p _ { \theta } ( y ) p ( z ) ]
+$$
+
+For unlabeled data, difficulties arise because the categorical distribution is not reparameterizable. Kingma et al. (2014) approach this by marginalizing out $y$ over all classes, so that for unlabeled data, inference is still on $\bar { \boldsymbol { q } } _ { \phi } ( z | x , y )$ for each $y$ . The lower bound on unlabeled data is:
+
+$$
+\begin{array} { l } { \displaystyle \log p _ { \theta } ( x ) \geq - \mathcal { U } ( x ) = \mathbb { E } _ { z \sim q _ { \phi } ( y , z \mid x ) } [ \log p _ { \theta } ( x \mid y , z ) + \log p _ { \theta } ( y ) + \log p ( z ) - q _ { \phi } ( y , z \mid x ) ] } \\ { = \displaystyle \sum _ { y } q _ { \phi } ( y \mid x ) ( - \mathcal { L } ( x , y ) + \mathcal { H } ( q _ { \phi } ( y \mid x ) ) ) } \end{array}
+$$
+
+The full maximization objective is:
+
+$$
+\mathcal { I } = \mathbb { E } _ { ( x , y ) \sim \mathcal { D } _ { L } } \left[ - \mathcal { L } ( x , y ) \right] + \mathbb { E } _ { x \sim \mathcal { D } _ { U } } \left[ - \mathcal { U } ( x ) \right] + \alpha \cdot \mathbb { E } _ { ( x , y ) \sim \mathcal { D } _ { L } } \left[ \log q _ { \phi } ( y | x ) \right]
+$$
+
+ere $\alpha$ is the scalar trade-off between the generative and discriminative objectives.
+
+One limitation of this approach is that marginalization over all $k$ class values becomes prohibitively expensive for models with a large number of classes. If $D , I , G$ are the computational cost of sampling from $q _ { \phi } ( y | x )$ , $q _ { \phi } ( z | x , y )$ , and $p _ { \theta } ( x | y , z )$ respectively, then training the unsupervised objective requires $\mathcal { O } ( D + k ( I + G ) )$ for each forward/backward step. In contrast, Gumbel-Softmax allows us to backpropagate through $y \sim q _ { \phi } ( y | x )$ for single sample gradient estimation, and achieves a cost of $\mathcal { O } ( D + I + G )$ per training step. Experimental comparisons in training speed are shown in Figure 5.
+
+# 4 EXPERIMENTAL RESULTS
+
+In our first set of experiments, we compare Gumbel-Softmax and ST Gumbel-Softmax to other stochastic gradient estimators: Score-Function (SF), DARN, MuProp, Straight-Through (ST), and
+
+Slope-Annealed ST. Each estimator is evaluated on two tasks: (1) structured output prediction and (2) variational training of generative models. We use the MNIST dataset with fixed binarization for training and evaluation, which is common practice for evaluating stochastic gradient estimators (Salakhutdinov & Murray, 2008; Larochelle & Murray, 2011).
+
+Learning rates are chosen from $\{ 3 \mathrm { e } { - } 5 , 1 \mathrm { e } { - } 5 , 3 \mathrm { e } { - } 4 , 1 \mathrm { e } { - } 4 , 3 \mathrm { e } { - } 3 , 1 \mathrm { e } { - } 3 \}$ ; we select the best learning rate for each estimator using the MNIST validation set, and report performance on the test set. Samples drawn from the Gumbel-Softmax distribution are continuous during training, but are discretized to one-hot vectors during evaluation. We also found that variance normalization was necessary to obtain competitive performance for SF, DARN, and MuProp. We used sigmoid activation functions for binary (Bernoulli) neural networks and softmax activations for categorical variables. Models were trained using stochastic gradient descent with momentum 0.9.
+
+# 4.1 STRUCTURED OUTPUT PREDICTION WITH STOCHASTIC BINARY NETWORKS
+
+The objective of structured output prediction is to predict the lower half of a $2 8 \times 2 8$ MNIST digit given the top half of the image $( 1 4 \times 2 8 )$ . This is a common benchmark for training stochastic binary networks (SBN) (Raiko et al., 2014; Gu et al., 2016; Mnih & Rezende, 2016). The minimization objective for this conditional generative model is an importance-sampled estimate of the likelihood objective, Eh∼pθ(hi|xupper) - $\begin{array} { r } { \mathbb E _ { h \sim p _ { \theta } ( h _ { i } | x _ { \mathrm { u p p e r } } ) } \left[ \frac { 1 } { m } et { } { ' } \sum _ { i = 1 } ^ { m } \log p _ { \theta } ( x _ { \mathrm { l o w e r } } | h _ { i } ) \right] } \end{array}$ , where $m = 1$ is used for training and $m =$ 1000 is used for evaluation.
+
+We trained a SBN with two hidden layers of 200 units each. This corresponds to either 200 Bernoulli variables (denoted as 392-200-200-392) or 20 categorical variables (each with 10 classes) with binarized activations (denoted as $3 9 2 - ( 2 0 \times 1 0 ) - ( 2 0 \times 1 0 ) - 3 9 2 )$ .
+
+As shown in Figure 3, ST Gumbel-Softmax is on par with the other estimators for Bernoulli variables and outperforms on categorical variables. Meanwhile, Gumbel-Softmax outperforms other estimators on both Bernoulli and Categorical variables. We found that it was not necessary to anneal the softmax temperature for this task, and used a fixed $\tau = 1$ .
+
+
+Figure 3: Test loss (negative log-likelihood) on the structured output prediction task with binarized MNIST using a stochastic binary network with (a) Bernoulli latent variables (392-200-200-392) and (b) categorical latent variables (392- $( 2 0 \times 1 0 )$ - $( 2 0 \times 1 0 )$ -392).
+
+# 4.2 GENERATIVE MODELING WITH VARIATIONAL AUTOENCODERS
+
+We train variational autoencoders (Kingma & Welling, 2013), where the objective is to learn a generative model of binary MNIST images. In our experiments, we modeled the latent variable as a single hidden layer with 200 Bernoulli variables or 20 categorical variables $( 2 0 \times 1 0 )$ . We use a learned categorical prior rather than a Gumbel-Softmax prior in the training objective. Thus, the minimization objective during training is no longer a variational bound if the samples are not discrete. In practice, we find that optimizing this objective in combination with temperature annealing still minimizes actual variational bounds on validation and test sets. Like the structured output prediction task, we use a multi-sample bound for evaluation with $m = 1 0 0 0$ .
+
+The temperature is annealed using the schedule $\tau = \operatorname* { m a x } ( 0 . 5 , \exp ( - r t ) )$ of the global training step $t$ , where $\tau$ is updated every $N$ steps. $N \in \{ 5 0 0 , 1 0 0 0 \}$ and $r \in \{ 1 \mathrm { e } { - } 5 , 1 \mathrm { e } { - } 4 \}$ are hyperparameters for which we select the best-performing estimator on the validation set and report test performance.
+
+As shown in Figure 4, ST Gumbel-Softmax outperforms other estimators for Categorical variables, and Gumbel-Softmax drastically outperforms other estimators in both Bernoulli and Categorical variables.
+
+
+Figure 4: Test loss (negative variational lower bound) on binarized MNIST VAE with (a) Bernoulli latent variables $( 7 8 4 - 2 0 0 - 7 8 4 )$ and (b) categorical latent variables $( 7 8 4 - ( 2 0 \times 1 0 ) - 2 0 0 )$ .
+
+Table 1: The Gumbel-Softmax estimator outperforms other estimators on Bernoulli and Categorical latent variables. For the structured output prediction (SBN) task, numbers correspond to negative log-likelihoods (nats) of input images (lower is better). For the VAE task, numbers correspond to negative variational lower bounds (nats) on the log-likelihood (lower is better).
+
+