In previous post, we have mentioned multigraph parsing proccess, how to train a multigraph under decentralized federated learning, and experimental setups for Multigraph. In the post, we will mention the effectiveness and efficiency of multigraph topology design under different configurations.

Our code can be found at: https://github.com/aioz-ai/MultigraphFL

#1. Cycle Time Comparison

Table 1 shows the cycle time of our method in comparison with other recent approaches. This table illustrates that our proposed method significantly reduces the cycle time in all setups with different networks and datasets. In particular, compared to the state-of-the-art RING, our method reduces the cycle time by $2.18$, $1.5$, $1.74$ times in average in the FEMNIST, iNaturalist, Sentiment140 dataset, respectively. Our method also clearly outperforms MACHA, MACHA(+), and MST by a large margin. The results confirm that our multigraph with isolated nodes helps reduce the cycle time in federated learning.

From Table1, our multigraph achieves the minimum improvement under the Amazon network in all three datasets. This can be explained that, under the Amazon network, our proposed topology does not generate many isolated nodes. Hence, the improvement is limited. Intuitively, when there are no isolated nodes, our multigraph will become the overlay, and the cycle time of our multigraph will be equal to the cycle time of the overlay in RING.

#2. Isolated Node Analysis

Isolated Nodes vs. Network Configuration. The numbers of isolated nodes vary based on the network configuration (Amazon, Gaia, Exodus, etc.). The parameter $t$ (maximum number of edges between two nodes), and the delay time which is identified by many factors (geometry distance, model size, computational cost based on tasks, bandwidth, etc also affect the process of generating isolated nodes. Table 2 illustrates the effectiveness of isolated nodes in different network configurations. Specifically, we conduct experiments on the FEMNIST dataset using five network configurations (Gaia, Amazon, Geant, Exodus, Ebone). We can see that our cycle time compared with RING is reduced significantly when more communication rounds or graph states have isolated nodes.

Isolated Nodes vs. RING vs. Random Strategy. Isolated nodes play an important role in our method as we can skip the model aggregation step in the isolated nodes. In practice, we can have a trivial solution to create isolated nodes by randomly removing some nodes from the overlay of RING. Table 3 shows the experiment results in two scenarios on FEMNIST dataset and Exodus Network: i} Randomly remove some silos in the overlay of RING, and ii} Remove the most inefficient silos (i.e., silos with the longest delay) in the overlay of RING. From Table 3, the cycle time reduces significantly when two aforementioned scenarios are applied. However, the accuracy of the model also drops significantly. This experiment shows that although randomly removing some nodes from the overlay of RING is a trivial solution, it can not maintain model accuracy. On the other hand, our multigraph not only reduces the cycle time of the model but also preserves the accuracy. This is because our multigraph can skip the aggregation step of the isolated nodes in a communication round. However, in the next round, the delay time of these isolated nodes will be updated, and they can become normal nodes and contribute to the final model.

Isolated Nodes Illustration. Figure belows shows a detailed illustration of our algorithm with the isolated nodes in a real-world training scenario. The experiment is conducted on Gaia network geometry and their corresponding hardware for supporting link latency computation. The image classification task is chosen for this benchmarking by using FEMNIST dataset and CNN backbone provided by Marfod \etal. Hence, we keep the model transmitted size at $4.62$ Mb, all access links have $10$ Gbps traffic capacity, the number of local updates is set to $1$, and the maximum number of edges $t$ is set to $3$. As shown in this Figure, although there are no isolated nodes in the initialized state, the number of isolated nodes increases in the next consequence states with a vast number (4 nodes). This circumstance leads to a $\sim 4$ times reduction in cycle time compared to the initialized state. The appearance of isolated nodes also greatly reduces the connection between silos by $\sim 3.6$ times, from $11$ down to $3$ connections, and also discarded ones all have high latency.

#3. Multigraph Ablation Study

Accuracy Analysis. In federated learning, improving the model accuracy is not the main focus of topology designing methods. However, preserving the accuracy is also important to ensure model convergence. Table 4 shows the accuracy of different topologies after $6,400$ communication training rounds on the FEMNIST dataset. This table illustrates that our proposed method achieves competitive accuracy with other topology designs. This confirms that our topology can maintain the accuracy of the model, while significantly reducing the training time.

Convergence Analysis. Figure belows shows the training loss versus the number of communication rounds and the wall-clock time under Exodus network using the FEMNIST dataset. This figure illustrates that our proposed topology converges faster than other methods while maintaining the model accuracy. We observe the same results in other datasets and network setups.

Cycle Time and Accuracy Trade-off. In our method, the maximum number of edges between two nodes $t$ mainly affects the number of isolated nodes. This leads to a trade-off between the model accuracy and cycle time. Table 5 illustrates the effectiveness of this parameter. When $t = 1$, we technically consider there are no weak connections and isolated nodes. Therefore, our method uses the original overlay from RING. When $t$ is set higher, we can increase the number of isolated nodes, hence decreasing the cycle time. In practice, too many isolated nodes will limit the model weights to be exchanged between silos. Therefore, models at isolated nodes are biased to their local data and consequently affect the final accuracy.

#4. Conclusion

We proposed a new multigraph topology for cross-silo federated learning. Our method first constructs the multigraph using the overlay. Different graph states are then parsed from the multigraph and used in each communication round. Our method significantly reduces the cycle time by allowing the isolated nodes in the multigraph to do model aggregation without waiting for other nodes. The intensive experiments on three datasets show that our proposed topology achieves new state-of-the-art results in all network and dataset setups.

In previous part, we have investigated that how delay time and cycle time is affected by the modification of multigraph in the design of the topology. Also, we will explore how multigraph can be constructed. In this post, we will mention multigraph parsing proccess, how to train a multigraph under decentralized federated learning, and experimental setups for Multigraph.

Our code can be found at: https://github.com/aioz-ai/MultigraphFL

#1. Multigraph Parsing

In Algorithm~\ref{alg:state_form}, we parse multigraph $\mathcal{G}_m$ into multiple graph states $\mathcal{G}_m^s$. Graph states are essential to identify the connection status of silos in a specific communication round to perform model aggregation. In each graph state, our goal is to identify the isolated nodes. During the training, isolated nodes update their weights internally and ignore all weakly-connected edges that connect to them.

To parse the multigraph into graph states, we first identify the maximum of states in a multigraph $s_{\max}$ by using the least common multiple (LCM). We then parse the multigraph into $s_{\max}$ states. The first state is always the overlay since we want to make sure all silos have a reliable topology at the beginning to ease the training. The reminding states are parsed so there is only one connection between two nodes. Using our algorithm, some states will contain isolated nodes. During the training process, only one graph state is used in a communication round. Figure below illustrates the training process in each communication round using multiple graph states.

#2. Multigraph Training

In each communication round, a state graph $\mathcal{G}_m^s$ is selected in a sequence that identifies the topology design used for training. We then collect all strongly-connected edges in the graph state $\mathcal{G}_m^s$ in such a way that nodes with strongly-connected edges need to wait for neighbors, while the isolated ones can update their models. We train our multigraph with DPASGD algorithm:

where $(k- h)$ is the index of the considered weights; $h$ is initialized to $0$ and $h = h + 1 \forall e_{k-h}(i,j) = 0$. Through Equation above, at each state, if a silo is not an isolated node, it must wait for the model from its neighbor to update its weight. If a silo is an isolated node, it can use the model in its neighbor from the $(k-h)$ round to update its weight immediately. The training procedure is described as below:

#3. Algorithm Complexity

It is trivial to see that the complexity of training procedure is $\mathcal{O}(n^2)$. In practice, since the cross-silo federated learning setting has only a few hundred silos ($n<500$), the time to execute our algorithms is just a tiny fraction of training time. Therefore, our proposed topology still can significantly reduce the overall wall-clock training time.

#4. Experimental Setups

Datasets. We use three datasets in our experiments: Sentiment140, iNaturalist, and FEMNIST. All datasets and the pre-processing process are conducted by following recent works. Table below shows the dataset setups in detail.

Network. We consider five distributed networks in our experiments: Exodus, Ebone, Géant, Amazon and Gaia. The Exodus, Ebone, and Géant are from the Internet Topology Zoo. The Amazon and Gaia network are synthetic and are constructed using the geographical locations of the data centers.

Baselines. We compare our multigraph topology with recent state-of-the-art topology designs for federated learning: STAR, MATCHA, MATCHA(+), MST, and RING.

Hardware Setup. Since measuring the cycle time is crucial to compare the effectiveness of all topologies in practice, we test and report the cycle time of all baselines and our method on the same NVIDIA Tesla P100 16Gb GPUs. No overclocking is used.

Time Simulator. We adapted PyTorch with the MPI backend to run DPASGD and DPASGD++ on a GPU cluster. We take advantage of the network simulator, the Time Simulator, which uses an arbitrary topology and computation times of silos as input to calculate the time instants at which local models are computed. The wall-clock time is reconstructed by this time simulator needs thorough understanding of the topology, including all factors mentioned in Delay Equations in each network configuration. The related configuration information is already provided in GAIA Network, and the simulator is created by Marfod \etal.

#Next

In the next post, we will mention the effectiveness and efficiency of multigraph topology design under different configurations.

Federated learning is an active research topic since it enables several participants to jointly train a model without sharing local data. Currently, cross-silo federated learning is a popular training setting that utilizes a few hundred reliable data silos with high-speed access links to training a model. While this approach has been widely applied in real-world scenarios, designing a robust topology to reduce the training time remains an open problem. In this paper, we present a new multigraph topology for cross-silo federated learning. We first construct the multigraph using the overlay graph. We then parse this multigraph into different simple graphs with isolated nodes. The existence of isolated nodes allows us to perform model aggregation without waiting for other nodes, hence effectively reducing the training time. Intensive experiments on three public datasets show that our proposed method significantly reduces the training time compared with recent state-of-the-art topologies while maintaining the accuracy of the learned model.

Our code can be found at: https://github.com/aioz-ai/MultigraphFL

#1. Introduction

Federated learning involves training models using remote devices or isolated data centers while keeping the data localized to respect user privacy policies. According to available literature, there are two prominent training scenarios: the "cross-device" scenario, which includes numerous unreliable edge devices with limited computational capacity and slow connection speeds, and the "cross-silo" scenario, which features a smaller number of reliable data silos with powerful computing resources and high-speed access links. Recently, the cross-silo scenario has gained traction in various federated learning applications.

In practical terms, federated learning represents a promising research avenue that allows us to harness the capabilities of machine learning techniques while upholding user privacy. Key obstacles in federated learning encompass issues like model convergence, communication bottlenecks, and disparities in data distributions across different silos. A commonly employed federated training approach involves establishing a central node responsible for overseeing the training process and aggregating contributions from all clients. However, a drawback of this client-server approach is the potential for communication bottlenecks, especially when dealing with a large number of clients. To mitigate this limitation, recent research has explored the concept of decentralized or peer-to-peer federated learning, where communication occurs via a peer-to-peer network topology, eliminating the need for a central node. Nevertheless, a major challenge in decentralized federated learning remains achieving rapid training while ensuring model convergence and preserving model accuracy.

In federated learning, the structure of communication networks holds significant importance. Specifically, an efficient network design contributes to quicker convergence, resulting in reduced training duration and energy consumption, as measured by worst-case convergence bounds within the topology's framework. Additionally, the topology's design has direct implications for other training-related challenges, including network congestion, overall model accuracy, and energy efficiency. The development of a resilient network structure capable of minimizing training time while preserving model accuracy remains an ongoing challenge in federated learning. Our paper is dedicated to devising a novel network design tailored for cross-silo federated learning, a prevalent scenario in practical applications.

Lately, various network configurations have emerged for cross-silo federated learning. For instance, the STAR topology involves an orchestrator averaging all models during each communication round. Another approach, known as MATCHA, divides potential communications into pairs of clients, with random selection for model transmission in each round. Additionally, the RING topology employs max-plus linear systems. Despite progress in this field, challenges persist, including access link congestion, straggler effects, and the establishment of diverse topologies across communication rounds.

In this paper, we introduce a novel multigraph topology inspired by recent advancements in federated learning. Our aim is to enhance the efficiency of cross-silo federated learning. Our approach involves constructing a multigraph based on the overlay of existing network topologies. Subsequently, we decompose this multigraph into simpler graphs, each featuring only a single edge connecting two nodes. These individual graphs are referred to as "states" within the multigraph. Importantly, each state can involve isolated nodes that perform model aggregation independently, reducing the cycle time in each communication round significantly. Our intensive experiments demonstrate that our proposed topology outperforms existing state-of-the-art methods by a wide margin in terms of training time for cross-silo federated learning, as illustrated in Figure.1.

#2. Overview

Federated Learning is recognized for its capacity to safeguard data privacy. In its modern incarnation, federated learning adopts a centralized network design, where a central node collects gradients from client nodes to update a global model. Early contributions in federated learning research include pioneering work and seminal papers by various researchers. Subsequent extensions and developments in federated learning and related distributed optimization algorithms have been proposed. Federated Averaging (FedAvg), initially introduced by one group, has inspired variations and other recent state-of-the-art model aggregation techniques, addressing convergence and the non-IID (non-identically and independently distributed) data challenge. Despite its simplicity, the client-server approach faces communication and computational bottlenecks at the central node, particularly when dealing with a large number of clients.

Decentralized Federated Learning flips the traditional federated learning model, enabling direct interactions between siloed data nodes, eliminating the necessity for a central coordinating node. While this approach mitigates communication congestion at a central point, optimizing a fully peer-to-peer network presents substantial challenges. The decentralized periodic averaging stochastic gradient descent method has demonstrated convergence rates comparable to centralized algorithms, making large-scale model training feasible. Furthermore, previous research has conducted systematic analyses of decentralized federated learning. A recent advancement involves leveraging a knowledge distillation mechanism to facilitate collaboration among silos in decentralized federated scenarios while preserving privacy among neighboring nodes.

Communication Topology plays a fundamental role in influencing the complexity and convergence behavior of federated learning. Numerous efforts have been dedicated to improving the efficiency of communication topologies, including star-shaped topologies and optimized-shaped topologies. In particular, a spanning tree topology has been introduced to reduce training time.

The STAR topology is designed for orchestrating the averaging of model updates in each communication round. Meanwhile, the MATCHA approach focuses on accelerating the training process through decomposition sampling. Recognizing the impact of straggler effects on communication round duration, methods for selecting the degree of a regular topology have been explored.

The RING topology is tailored for cross-silo federated learning and leverages the principles of max-plus linear systems. A sample-induced topology has been introduced, capable of effectively recovering the performance of existing SGD-based algorithms and their corresponding convergence rates. In a recent comprehensive survey, various models, frameworks, and algorithms related to network topologies in federated learning have been explored.

Multigraph is a concept that originates from traditional mathematics. In conventional terms, a "graph" typically denotes a simple graph without loops or multiple edges between two nodes. In contrast, a multigraph allows for the presence of multiple edges between two nodes. In the realm of deep learning, multigraphs have found utility across various domains, including clustering, medical image processing, traffic flow prediction, activity recognition, recommendation systems, and cross-domain adaptation. In this research, we employ a multigraph construction to facilitate isolated nodes and expedite training in cross-silo federated learning.

#3. Preliminaries

#3.1 Federated Learning

In federated learning, silos do not share their local data, but still periodically transmit model updates between them. Given $N$ siloed data centers, the objective function for federated learning is:

where $L_{i}(\textbf{w}, \xi_i)$ is the loss of model parameterized by the weight $\textbf{w} \in \mathbb R^d$, $\xi_i$ is an input sample drawn from data at silo $i$, and the coefficient $p_i>0$ specifies the relative importance of each silo. Recently, different distributed algorithms have been proposed to optimize the equation. In this work, DPASGD is used to update the weight of silo $i$ in each training round as follows:

where $b$ is the batch size, $i,j$ denote the silo, $u$ is the number of local updates, $\alpha_k > 0$ is a potentially varying learning rate at $k$-th round, $\textbf{A} \in R^{N \times N}$ is a consensus matrix with non-negative weights, and $\mathcal{N}_i^{+}$ is the in-neighbors set that silo $i$ has the connection to.

#3.2 Multigraph for Federated Learning

Connectivity and Overlay. We consider the \textit{connectivity} $\mathcal{G}_c = (\mathcal{V}, \mathcal{E}_c)$ as a graph that captures possible direct communications among silos. Based on its definition, the connectivity is often a fully connected graph and is also a directed graph. % whenever the upload and download are set during learning. The \textit{overlay} $\mathcal{G}_o$ is a connected subgraph of the connectivity graph, i.e., $\mathcal{G}_o = (\mathcal{V}, \mathcal{E}_o)$, where $\mathcal E_o \subset \mathcal E_c$. Only nodes directly connected in the overlay graph $\mathcal{G}_o$ will exchange the messages during training.

Multigraph. While the connectivity and overlay graph can represent different topologies for federated learning, one of their drawbacks is that there is only one connection between two nodes. In our work, we construct a \textit{multigraph} $\mathcal{G}_m = (\mathcal{V}, \mathcal{E}_m)$ from the overlay $\mathcal{G}_o$. The multigraph can contain multiple edges between two nodes. In practice, we parse this multigraph to different graph states, each state is a simple graph with only one edge between two nodes.

In the multigraph $\mathcal{G}_m$, the connection edge between two nodes has two types: \textit{strongly-connected} edge and \textit{weakly-connected} edge. Under both strong and weak connections, the participated nodes can transmit their trained models to their out-neighbours $\mathcal{N}_i^{-}$ or download models from their in-neighbours $\mathcal{N}_i^{+}$. However, in a strongly-connected edge, two nodes in the graph must wait until all upload and download processes between them are finished to do model aggregation. On the other hand, in a weakly-connected edge, the model aggregation process in each node can be established whenever the previous training process is finished by leveraging up-to-date models which have not been used before from the in-neighbours of that node.

State of Multigraph Given a multigraph $\mathcal{G}_m$, we can parse this multigraph into different simple graphs with only one connection between two nodes (either strongly-connected or weakly-connected). We denote each simple graph as a state $\mathcal{G}_m^s$ of the multigraph.

Isolated Node. A node is called isolated when all of its connections to other nodes are weakly-connected edges. Figure.2 shows the graph concepts and isolated nodes.

#Next

In the next post, we will mention the delay time and cylce time also how multigraph can be constructed.

In previous paart, we how explore decentralized federated learning, how and why multigraph is proposed to improve training process. In this part, we will investigate that how delay time and cycle time is affected by the modification of multigraph in the design of the topology. Also, we will explore how multigraph can be constructed.

Our code can be found at: https://github.com/aioz-ai/MultigraphFL

#1. Delay time in multigraph

A delay to an edge $e(i, j)$ is the time interval when node $j$ receives the weight sending by node $i$, which can be defined by:

where $T_{c}(i)$ denotes the time to compute one local update of the model; $u$ is the number of local updates; $l(i,j)$ is the link latency; $M$ is the model size; $O(i, j)$ is the total network traffic capacity. However, unlike other communication infrastructures, the multigraph only contains connections between silos without other nodes such as routers or amplifiers. Thus, the total network traffic capacity $O(i,j) = \text{min}\left(\frac{C_{\rm{UP}}(i)}{\left|{\mathcal{N}_{i}^{-}}\right|}, \frac{C_{\rm{DN}}(j)}{\left|\mathcal{N}_{i}^{+}\right|}\right)$ where $C_{\rm{UP}}$ and $C_{\rm{DN}}$ denote the upload and download link capacity. Note that the upload and download processes can happen in parallel.

Since multigraph can contain multiple edges between two nodes, we extend the definition of the delay in the previous equation to $d_k(i,j)$, with $k$ is the $k$-th communication round during the training process, as:

where $e(i,j)$$=$$\mathbb{0}$ is weakly-connected edge, $e(i,j)$$=$$\mathbb{1}$ is strongly-connected edge; $ \tau_k(\mathcal{G}_m)$ is the cycle time at the $k$-th computation round during the training process.

In general, using introduced equation, \textit{the delay of the next communication round $d_{k+1}$ is updated based on the delay of the previous rounds} and other factors, depending on the edge type connection.

#2. Cycle time in multigraph

The cycle time per round is the time required to complete a communication round. In this work, we define the cycle time per round is the maximum delay between all silo pairs with strongly-connected edges. Therefore, the average cycle time of the entire training is:

where $\mathcal{N}_{i}^{++}$ is an in-neighbors silo set of $i$ whose edges are strongly-connected.

#3. Multigraph Construction

Algorithm 1 describes our methods to generate the multigraph $\mathcal{G}_m$ with multiple edges between silos. The algorithm takes the overlay $\mathcal{G}_o$ as input. Similar to~\cite{marfoq2020throughput}, we use the Christofides algorithm to obtain the overlay. In Algorithm 1, we establish multiple edges that indicate different statuses (strongly-connected or weakly-connected). To identify the total edges between a silo pair, we divide the delay $d(i,j)$ by the smallest delay $d_{\min}$ overall silo pairs, and compare it with the maximum number of edges parameter $t$ ($t=5$ in our experiments). \textit{We assume that the silo pairs with longer delay will have more weakly-connected edges, hence potentially becoming isolated nodes}. Overall, we aim to increase the number of weakly-connected edges, which generate more isolated nodes to speed up the training process. Note that, from Algorithm 1, each silo pair in the multigraph should have one strongly-connected edge and multiple weakly-connected edges. The role of the strongly-connected edge is to make sure that two silos have a good connection in at least one communication round.

#Next

In the next post, we will mention multigraph parsing proccess and how to train a multigraph under decentralized federated learning.

In previous part, we have discussed about Autonomous driving FADNetwork. In this post, we will verify the effectiveness and efficiency of it.

Our source code can be found at: https://github.com/aioz-ai/FADNet

#1. Experimental Setup

Udacity. We use the popular Udacity dataset to evaluate our results. We only use front-forwarded images in this dataset in our experiment. We use $5$ sequences for training and $1$ for testing. The training sequences are assigned randomly to different silos depending on the federated topology (i.e., Gaia or NWS).

Carla. Since the Udacity dataset is collected in the real-world environment, changing the weather or lighting conditions is not easy. To this end, we collect more simulated data in the Carla simulator. We have applied different lighting (morning, noon, night, sunrise, sunset) and weather conditions (cloudy, rain, heavy rain, wet streets, windy, snowy) when collecting the data. We have generated $73,235$ samples distributed over $11$ sequences of scenes.

Gazebo. Since both the Udacity and Carla datasets are collected in outdoor environments, we also employ Gazebo to collect data for autonomous navigation in indoor scenes. We use a simulated mobile robot and the built-in scenes to collect data. Table.1 shows the statistics of three datasets. We use $80\%$ of the collected data in Gazebo and Carla data for training, and the rest $20\%$ for testing.

Network Topology. We conduct experiments on two topologies: the Internet Topology Zoo (Gaia), and the North America data centers (NWS). We use Gaia topology in our main experiment and provide the comparison of two topologies in our ablation study.

Training. The model in a silo is trained with a batch size of $32$ and a learning rate of $0.001$ using Adam optimizer. We follow the training process to obtain a global weight of all silos. The training process is conducted with $3000$ communication rounds and each silo has one NVIDIA 1080 11 GB GPU for training. Note that, one communication round is counted each time all silos have finished updating their model weights.

Baselines. We compare our results with various recent methods, including Random baseline and Constant baseline, Inception-V3, MobileNet-V2, VGG-16, and Dronet. All these methods use the Centralized Local Learning (CLL) strategy (i.e., the data are collected and trained in one local machine.) For distributed learning, we compare our Deep Federated Learning (DFL) approach with the Server-based Federated Learning (SFL) strategy. As the standard practice, we use the root-mean-square error (RMSE) metric to evaluate the results.

#2. Results

Table 2 summarises the performance of our method and recent state-of-the-art approaches. We notice that our FADNet is trained using the proposed peer-to-peer DFL using the Gaia topology with 11 silos. This table clearly shows our FDANet + DFL outperforms other methods by a fair margin. In particular, our FDANet + DFL significantly reduces the RMSE in Gazebo and Carla datasets, while slightly outperforms DroNet in the Udacity dataset. These results validate the robustness of our FADNet while is being trained in a fully decentralized setting. Table 3 also shows that with a proper deep architecture such as our FADNet, we can achieve state-of-the-art accuracy when training the deep model in FL. Fig. 2 illustrates the spatial support regions when our FADNet making the prediction. Particularly, we can see that FADNet focuses on the ``line-like" patterns in the input frame, which guides the driving direction.

#3. Ablation Studies

#Effectiveness of our DFL.

Table 3 summarises the accuracy of DroNet and our FADNet when we train them using different learning methods: CLL, SFL, and our peer-to-peer DFL. From this table, we can see that training both DroNet and FADNet with our peer-to-peer DFL clearly improves the accuracy compared with the SFL approach. This confirms the robustness of our fully decentralized approach and removes a need of a central server when we train a deep network with FL. Compared with the traditional CLL approach, our DFL also shows a competitive performance. However, we note that training a deep architecture using CLL is less complicated than with SFL or DFL. Furthermore, CLL is not a federated learning approach and does not take into account the privacy of the user data.

#Effectiveness of our FADNet.

Table 3 shows that apart from the learning method, the deep architectures also affect the final results. This table illustrates that our FADNet combined with DFL outperforms DroNet in all configurations. We notice that DroNet achieves competitive results when being trained with CLL. However DroNet is not designed for federated training, hence it does not achieve good accuracy when being trained with SFL or DFL. On the other hand, our introduced FADNet is particularly designed with dedicated layers to handle the data imbalance and model convergence problem in federated training. Therefore, FADNet achieves new state-of-the-art results in all three datasets.

#Network Topology Analysis.

Table 4 illustrates the performance of DroNet and our FADNet when we train them using DFL under two distributed network topologies: Gaia and NWS. This table shows that the results of DroNet and FADNet under DFL are stable in both Gaia and NWS distributed networks. We note that the NWS topology has 22-silos while the Gaia topology has only 11 silos. This result validates that our FADNet and DFL do not depend on the distributed network topology. Therefore, we can potentially use them in practice with more silo data.

#Convergence Analysis.

The effectiveness of federated learning algorithms is identified through the convergence ability, including accuracy and training speed, especially when dealing with the increasing number of silos in practice. Fig.3 shows the convergence ability of our FADNet with DFL using two topologies: Gaia with 11 silos, and NWS with 22 silos. This figure shows that our proposed DFL achieves the best results in Gaia and NWS topology and converges faster than the SFL approach in both Gazebo and Carla datasets. We also notice that the performance of our DFL is stable when there is an increase in the number of silos. Specifically, training our FADNet with DFL reaches the converged point after approximately $150$s, $180$s on the NWS and Gaia topology, respectively. Fig.3 validates the convergence ability of our FADNet and DFL, especially when dealing with the increasing number of silos.

In practice, compared with the traditional CLL approach, federated learning methods such as SFL or DFL can leverage more GPUs remotely. Therefore, we can reduce the total training time significantly. However, the drawback of federated learning is we would need more GPUs in total (ideally one for each silo), and deep architecture also should be carefully designed to ensure model convergence.

#Deployment

To verify the effectiveness of our FADNet in practice, we deploy the model trained on the Gazebo dataset on a mobile robot. The robot is equipped with a RealSense camera to capture the front RGB images. Our FADNet is deployed on a Qualcomm RB5 board to make the prediction of the steering angle for the robot. The processing time of our FADNet on the Qualcomm RB5 board is approximately $12$ frames per second. Overall, we observe that the robot can navigate smoothly in an indoor environment without colliding with obstacles. More qualitative results can be found in our supplementary material.

#Conclusion

We propose a new approach to learn an autonomous driving policy from sensory data without violating the user's privacy. We introduce a peer-to-peer deep federated learning (DFL) method that effectively utilizes the user data in a fully distributed manner. Furthermore, we develop a new deep architecture - FADNet that is well suitable for distributed training. The intensive experimental results on three datasets show that our FADNet with DFL outperforms recent state-of-the-art methods by a fair margin. Currently, our deployment experiment is limited to a mobile robot in an indoor environment. In the future, we would like to test our approach with more silos and deploy the trained model using an autonomous car on man-made roads.

Autonomous driving is an active research topic in both academia and industry. However, most of the existing solutions focus on improving the accuracy by training learnable models with centralized large-scale data. Therefore, these methods do not take into account the user's privacy. In this paper, we present a new approach to learn autonomous driving policy while respecting privacy concerns. We propose a peer-to-peer Deep Federated Learning (DFL) approach to train deep architectures in a fully decentralized manner and remove the need for central orchestration. We design a new Federated Autonomous Driving network (FADNet) that can improve the model stability, ensure convergence, and handle imbalanced data distribution problems while is being trained with federated learning methods. Intensively experimental results on three datasets show that our approach with FADNet and DFL achieves superior accuracy compared with other recent methods. Furthermore, our approach can maintain privacy by not collecting user data to a central server.

Our source code can be found at: https://github.com/aioz-ai/FADNet

#1. Introduction

In this paper, our goal is to develop an end-to-end driving policy from sensory data while maintaining the user's privacy by utilizing FL. We address the key challenges in FL to make sure our deep network can achieve competitive performance when being trained in a fully decentralized manner. Fig.1 shows an overview of different learning approaches for autonomous driving. In Centralized Local Learning (CLL), the data are collected and trained in one local machine. Hence, the CLL approach does not take into account the user's privacy. The Server-based Federated Learning (SFL) strategy requires a central server to orchestrate the training process and receive the contributions of all clients. The main limitation of SFL is communication congestion when the number of clients is large. Therefore, we follow the peer-to-peer federated learningto set up the training. Our peer-to-peer Deep Federated Learning (DFL) is fully decentralized and can reduce communication congestion during training. We also propose a new Federated Autonomous Driving network (FADNet) to address the problem of model convergence and imbalanced data distribution. By training our FADNet using DFL, our approach outperforms recent state-of-the-art methods by a fair margin while maintaining user data privacy.

Our contributions can be summarized as follows:

• We propose a fully decentralized, peer-to-peer Deep Federated Learning framework for training autonomous driving solutions.
• We introduce a Federated Autonomous Driving network that is well suitable for federated training.
• We introduce two new datasets and conduct intensive experiments to validate our results.

#2. Problem Formulation

We consider a federated network with $N$ siloed data centers (e.g., autonomous cars) $\mathcal{D}_{i}$, with $i \in [1,N]$. Our goal is to collaboratively train a global driving policy $\theta$ by aggregating all local learnable weights $\theta_i$ of each silo. Note that, unlike the popular centralized local training setup, in FL training, each silo does not share its local data, but periodically transmits model updates to other silos.

In practice, each silo has the training loss $\mathcal{L}_i(\xi_i, \theta_i)$. $\xi_i$ is the ground-truth in each silo $i$. $\mathcal{L}_i(\xi_i, \theta_i)$ is calculated as the regression loss. This regression loss is modeled by a deep network that takes RGB images as inputs and predicts the associated steering angles.

#3. Deep Federated Learning for Autonomous Driving

A popular training method in FL is to set up a central server that orchestrates the training process and receives the contributions of all clients (Server-based Federated Learning - SFL). The limitation of SFL is the server potentially represents a single point of failure in the system. We also may have communication congestion between the server and clients when the number of clients is massive. Therefore, in this work, we utilize the peer-to-peer FL to set up the training scenario. In peer-to-peer FL, there is no centralized orchestration, and the communication is via peer-to-peer topology. However, the main challenge of peer-to-peer FL is to assure model convergence and maintain accuracy in a fully decentralized training setting.

Fig.2 illustrates our Deep Federated Learning (DFL) method. Our DFL follows the peer-to-peer FL setup with the goal to integrate a deep architecture into a fully decentralized setting that ensures convergence while achieving competitive results compared to the traditional Centralized Local Learning or SFL approach. In practice, we can consider a silo as an autonomous car. Each silo maintains a local learnable model and does not share its data with other silos. We represent the silos as vertices of a communication graph and the FL is performed on an overlay, which is a sub-graph of this communication graph.

#Designing the Overlay

Let $\mathcal{G}_c = (\mathcal{V}, \mathcal{E}_c)$ is the connectivity graph that captures the possible direct communications among $N$ silos. $\mathcal{V}$ is the set of vertices (silos), while $\mathcal{E}_c$ is the set of communication links between vertices. $\mathcal{N}_i^{+}$ and $\mathcal{N}_i^{-}$ are in-neighbors and out-neighbors of a silo $i$, respectively. As in~\cite{marfoq2020throughput}, we note that it is unnecessary to use all the connections of the connectivity graph for FL. Indeed, a sub-graph called an overlay, $\mathcal{G}_o = (\mathcal{V}, \mathcal{E}_o)$ can be generated from $\mathcal{G}_c$. In our work, $\mathcal{G}_o$ is the result of Christofides’ Algorithm~\cite{monnot2003approximation}, which yields a strong spanning sub-graph of $\mathcal{G}_c$ with minimal cycle time. One cycle time or time per communication round, in general, is the time that a vertex waits for messages from the other vertices to do a computational update.

In practice, one block cycle time of an overlay $\mathcal{G}_o$ depends on the delay of each link $(i, j)$, denoted as $d_o(i, j)$, which is the time interval between the beginning of a local computation at node $i$, and the receiving of $i$'s messages by $j$. Furthermore, without concerns about access links delays between vertices, our graph is treated as an edge-capacitated network with:

where $T_c(i)$ is the time to compute one local update of the model; $s$ is the number of local computational steps; $l(i,j)$ is the link latency; $M$ is the model size; $B(i,j)$ is available bandwidth of the path $(i,j)$. As in~\cite{marfoq2020throughput}, we set $s=1$.

#Training Algorithm

At each silo $i$, the optimization problem to be solved is:

We apply the distributed federated learning algorithm, DPASGD, to solve the optimizations of all the silos. In fact, after waiting one cycle time, each silo $i$ will receive parameters $\theta_j$ from its in-neighbor $\mathcal{N}_i^{+}$ and accumulate these parameters multiplied with a non-negative coefficient from the consensus matrix $\mathbf{A}$. It then performs $s$ mini-batch gradient updates before sending $\theta_i$ to its out-neighbors $\mathcal{N}_i^{-}$, and the algorithm keeps repeating. Formally, at each iteration $k$, the updates are described as:

where $m$ is the mini-batch size and $\alpha_k > 0$ is a potentially varying learning rate.

#Federated Averaging

To compute the prediction of models in all silos, we compute the average model $\theta$ using weight aggregation from all the local model $\theta_i$. The federated averaging process is conducted as follow:

where $N$ is the number of silos; $\lambda_i = \{0,1\}$. Note that $\lambda_i = 1$ indicates that silo $i$ joins the inference process and $\lambda_i = 0$ if not. The aggregated weight $\theta$ is then used for evaluation on the testing set $\mathcal{D}_{test}$.

#4. Network Architecture

One of the main challenges when training a deep network in FL is the imbalanced and non-IID (identically and independently distributed) problem in data partitioning across silos. To overcome this problem, the learning architecture should have an appropriate design to balance the trade-off between convergence ability and accuracy performance. In practice, the deep architecture has to deal with the high variance between silo weights when the accumulation process for all silos is conducted. To this end, we design a new Federated Autonomous Driving Network, which is based on ResNet8, as shown in Fig.3.

In particular, our proposed FADNet first comprises an input layer normalization to improve the stability of the abstract layer. This layer aims to handle different distributions of input images in each silo. Then, a convolution layer following by a max-pooling layer is added to encode the input. To handle the vanishing gradient problem, three residual blocks are appended with a following FC layer to extract ResBlock features. However, using residual blocks increases the variance of silo weights during the aggregation process and affects the convergence ability of the model. To address this problem, we add a Global Average Pooling layer (GAP) associated with each residual block. GAP is a non-weight pooling layer which sums out the spatial information from each residual block. Thus, it is not affected by the weighted variance problem. The output of each GAP layer is passed through an Accumulation layer to accrue the Support feature. The ResBlock feature and the Support feature from GAP layers are fed into the Aggregation layer to calculate the model loss in each silo.

In our design, the Accumulation and Aggregation layers aim to reduce the variance of the global model since we need to combine multiple model weights produced by different silos. In particular, the Accumulation layer is a variant of the fully connected (FC) layer. Instead of weighting the contribution of input nodes as in FC, the Accumulation layer weights the contribution of multiple features from input layers. The Accumulation layer has a learnable weight matrix $w \in \mathbb{R}^\text{n}$. Its number of nodes is equal to the \text{n} number of input layers. Note that the support feature from the Accumulation layer has the same size as the input. Let $F = \{f_\text{1}, f_\text{2}, ..., f_\text{n}\}, \forall f_\text{h} \in \mathbb{R}^\text{d}$ be the collection of $\text{n}$ number of the features extracted from $\text{n}$ input GAP layers; $\text{d}$ is the unified dimension. The Accumulation outputs a feature $f_\text{c} \in \mathbb{R}^\text{d}$ in each silo $i$, and is computed as:

The Aggregation layer is a fusion between the ResBlock feature extracted from the backbone and the support feature from the Accumulation layer. For simplicity, we use the Hadamard product to compute the aggregated feature. This feature is then averaged to predict the steering angle. Let $f_\text{s} \in \mathbb{R}^\text{d}$ be the ResBlock features extracted from the backbone. The output driving policy $\theta_i$ of silo $i$ can be calculated as:

where $\odot$ denotes Hadamard product; $\bar{(*)}$ denotes the mean and we set $\text{d} = 6,272$.

#Next

In the next post, we will show the effectiveness and efficiency of FADNet during Federated Learning proccess.

This is the final part of the series group dance choreography, In this part, we will provide detailed analyses of our proposed group dance generation method.

#Experiments

#AIOZ-GDANCE Statistics

In Figure 1, we show the distribution of music genres and dance styles in our dataset. As illustrated in Figure 1 (Left), Pop and Electronic are popular music genres while other music genres nearly share the same distribution. Meanwhile, on the right of Figure 1, Zumba, Aerobic, and Commercial are the dominant dance styles.

Figure 2 (Left) shows the number of dancers in each dance style. Naturally, we see that Zumba, Aerobic, and Commercial have more dancers. On the right of this figure, we illustrate the correlation between music genres and dance styles.

#Evaluation Metrics

Similar to prior works on single-dance generation, we evaluate the generated motion quality by calculating the distribution distance between the generated and the ground-truth motions using Frechet Inception Distance (FID)[1, 2]. To evaluate how well the generated 3D motion correlates to the input music, we use the Motion-Music Consistency metric (MMC) [2,3]. We also evaluate our model's ability to generate diverse dance motions when given various input music by measuring Generation Diversity (GenDiv) [2,3].

To evaluate the group dancing quality, we propose three new metrics: Group Motion Realism (GMR), Group Motion Correlation (GMC), and Trajectory Intersection Frequency (TIF). Detailed calculations of these metrics are described as follows:

Group Motion Realism (GMR). To calculate the realism between generated and ground-truth group motion, we need to find a single unified representation for all dancers' motions in the scene. Based on the kinetic features of a single motion sequence [4], we propose to calculate Group Motion Realism (GMR), smaller is better. Specifically, for each entity, we compute the velocity of each element $j$ of the pose vector: $v^n_t = \frac{y^n_{t+1} - y^n_t}{\Delta t}$ where $\Delta t$ is the time period between two consecutive frames. Note that the pose vector of each entity at each frame consists of the root orientation, root position and joint angles. The group kinetic features of a sequence is approximated by taking the logarithm of the total kinetic energy of all group entities as:

where $m_j$ is the moment of inertia or mass of each joint. We assume that $m_j$ is constant with respect to time and entity. Then, we split the sequence into smaller chunks and calculate the features of these chunks. This process is identical for both the generated and ground-truth sequences. Finally, we utilize these sets of features (from generated and ground-truth group dance) to calculate the GMR using the standard FID formulation as in [1].

Group Motion Correlation (GMC). We also evaluate the synchrony and the correlation between dancers within the generated group. We assume that the correlation of movements between individuals is likely to reflect their interaction in the choreography. For every pair of motions within a group, we first align the two motion sequences using Dynamic Time Warping algorithm based on the Euclidean distance in the joint position space (obtained by SMPL joint regressor). We then calculate the mean cross-correlations between the time-aligned motion pairs using the kinetic features [4]. The generated group motion correlation degree is then calculated as the average of all motion pairs.

Trajectory Intersection Frequency (TIF). For the generated group sequences, the intersection rate is calculated over all $F$ frames as:

where $M$ is the SMPL skinning function [5] which can output a 6890-vertices human mesh from the input pose parameters $y$. $\text{intersect}(x,y)$ is a function that returns 1 if the two meshes are intersect with each other and 0 otherwise. For TIF, smaller value is better and indicates less intersection of the generated group.

#Cross-entity Attention Analysis

We compare our method with FACT [2]. FACT is a recent state-of-the-art method designed for single dance generation, thus giving our method an advantage. However, it is still the closest competing method as we propose a new group dance dataset that is not available for benchmarking before. We also analyse our method with and without using Cross-entity Attention. We train all methods with mini-batch containing all dancers within the group instead of sampling each dancer independently as in FACT’s original implementation.

Table 1 shows the method comparison between the baseline FACT[2] and our proposed GDanceR with and without Cross-entity Attention. The results show that GDanceR, especially with the Cross-entity Attention, outperforms the baseline by a large margin in all metrics. In Figure 3, we also visualize the example outputs of FACT and GDanceR. It is clear that FACT does not handle well the intersection problem. This is understandable as FACT is not designed for group dance generation, while our method with the Cross-entity Attention can deal with this problem better.

#Number of Dancers Analysis.

Table 2 demonstrates the generation results of our method when we want to generate different numbers of dancers. In general, the FID, GMR, and GMC metrics do not show much correlation with the numbers of generated dancers since the results are varied. On the other hand, MMC shows its stability among all setups ($\sim 0.248$), which indicates that our network is robust in generating motion from given music regardless of the changing of initial positions. The generation diversity (GenDiv) decreases while the intersection frequency (TIF) increases when more dancers are generated. These results show that dealing with the collision during the group generation process is worth further investigation.

#Dance Style Analysis

Different dance styles exhibit different challenges in group dance generation. As shown in Table 3, Aerobic and Zumba are quite similar for generating choreography as they usually focus on workout and sporty movements. Besides, while Commercial and Irish are easier for the model to reproduce the motions, Bollywood and Samba contain highly skilled movements that are challenging to capture and represent accurately. In Figure 4, we show the generated results of GDanceR with different dance styles. Our Supplementary Material and Demonstration Video also provide more examples.

#Ablation on Latent Motion Fusion.

We investigate different fusion strategies between the local motion $h^i$ and global-aware motion $g^i$ to obtain the final motion representation $z^i$. Specifically, we experiment with three settings: (i) No Fusion: the final motion is the global-aware motion obtained from our Cross-entity Attention ($z^i = g^i$); (ii) Concatenate: the final motion is the concatenation of the local and global-aware motion ($z^i = [h^i; g^i]$); (iii) Add: the final motion is the addition between local and global ($z^i = h^i + g^i$). Table 4 summarizes the results. We find that fusing the motion by adding both the local and global motion features achieves the best results. In this strategy, the global information between entities is encoded to the local motion in an effective way so that the final motion retain the comprehensive information of their own past motion as well as the motion of every other entity. While in the concatenation, the model is prone to overfitting due to the redundant information of both the local and global representation. On the other hand, No Fusion can degrade the amount of information of the past motion, leading to insufficient input information and the Decoder may fail to generate the temporally-coherent motion aligned with the music.

#Conclusion

In summary, we have introduced AIOZ-GDANCE, the largest dataset for audio-driven group dance generation. Our dataset contains in-the-wild videos and covers different dance styles and music genres. We then propose a strong baseline along with new evaluation metrics for group dance generation task. We also perform extensive experiments to validate our method on this interesting yet unexplored problem, using our new dataset and evaluation protocols. We hope that the release of our dataset will foster more research on audio-driven group choreography.

#References

[1] Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilibrium. NIPS 2017.

[2] Ruilong Li, Shan Yang, David A Ross, and Angjoo Kanazawa. Ai choreographer: Music-conditioned 3d dance generation with aist++. ICCV 2021

[3] Hsin-Ying Lee, Xiaodong Yang, Ming-Yu Liu, Ting-Chun Wang, Yu-Ding Lu, Ming-Hsuan Yang, and Jan Kautz. Dancing to music. NIPS 2019.

[4] Kensuke Onuma, Christos Faloutsos, and Jessica K Hodgins. Fmdistance: A fast and effective distance function for motion capture data. Eurographics 2008.

[5] Matthew Loper, Naureen Mahmood, Javier Romero, Gerard Pons-Moll, and Michael J. Black. SMPL: A skinned multi-person linear model. ACM Trans. Graphics, 2015

In the previous post, we have introduced AIOZ-GDANCE, a new largescale in-the-wild dataset for music-driven group dance generation. On the basis of the new dataset, we introduce the first strong baseline for group dance generation that can jointly generate multiple dancing motions expressively and coherently.

#Music-driven Group Dance Generation Method

#Problem Formulation

Given an input music audio sequence $\{m_1, m_2, ...,m_T\}$ with $t = \{1,..., T\}$ indicates the index of music segments, and the initial 3D positions of $N$ dancers $\{\tau^1_0, \tau^2_0, ..., \tau^N_0 \}$, $\tau^i_0 \in \mathbb{R}^{3}$, our goal is to generate the group motion sequences $\{y^1_1,..., y^1_T; ...;y^n_1,...,y^n_T\}$ where $y^i_t$ is the generated pose of $i$-th dancer at time step $t$. Specifically, we represent the human pose as a 72-dimensional vector $y = [\tau; \theta]$ where $\tau$, $\theta$ represent the root translation and pose parameters of the SMPL model [1], respectively.

In general, the generated group dance motion should meet the two conditions: (i) consistency between the generated dancing motion and the input music in terms of style, rhythm, and beat; (ii) the motions and trajectories of dancers should be coherent without cross-body intersection between dancers. To that end, we propose the first baseline method, for group dance generation that can jointly generate multiple dancing motions expressively and coherently. Figure 1 shows the architecture of our proposed Music-driven 3D Group Dance generatoR (GDanceR), which consists of three main components:

• Transformer Music Encoder.
• Initial Pose Generator.
• Group Motion Generator.

#Transformer Music Encoder

From the raw audio signal of the input music, we first extract music features using the available audio processing library Librosa. Concretely, we extract the mel frequency cepstral coefficients (MFCC), MFCC delta, constant-Q chromagram, tempogram, onset strength and one-hot beat, which results in a 438-dimensional feature vector. We then encode the music sequence $M =\{m_1, m_2, ...,m_T\}$, $m_t \in \mathbb{R}^{438}$ into a sequence of hidden representation $\{a_1, a_2,..., a_T\}$, $a_t \in \mathbb{R}^{d_a}$. In practice, we utilize the self-attention mechanism of transformer [2] to effectively encode the multi-scale information and the long-term dependency between music frames. The hidden audio at each time step is expected to contain meaningful structural information to ensure that the generated dancing motion is coherent across the whole sequence.

Specifically, we first embed the music features $m_t$ using a Linear layer followed by Positional Encoding to encode the time ordering of the sequence.

where $\text{PE}$ denotes the Positional Encoding, and $W^u_a \in \mathbb{R}^{438 \times d_a}$ is the parameters of the linear projection layer. Then, the hidden audio information can be calculated using self-attention mechanism:

where $W^q_a, W^k_a \in \mathbb{R}^{d_a \times d_k}$, and $W^v_a \in \mathbb{R}^{d_a \times d_v}$ are the parameters that transform the linear embedding audio $U$ into a query $U^q$, a key $U^k$, and a value $U^v$ respectively. $d_a$ is the dimension of the hidden audio representation while $d_k$ is the dimension of the query and key, and $d_v$ is the dimension of value. $\text{FF}$ is a feed-forward neural network.

#Initial Pose Generator

Given the initial positions of all dancers, we generate the initial poses by combing the audio feature with the starting positions. We aggregate the audio representation by taking an average over the audio sequence. The aggregated audio is then concatenated with the input position and fed to a multilayer perceptron (MLP) to predict the initial pose for each dancer:

where $[;]$ is the concatenation operator, $\tau^i_0$ is the initial position of the $i$-th dancer.

#Group Motion Generator

To generate the group dance motion, we aim to synthesize the coherent motion of each dancer such that it aligns well with the input music. Furthermore, we also need to maintain global consistency between all dancers. As shown in Figure 3, our Group Generator comprises a Group Encoder to encode the group sequence information and an MLP Decoder to decode the hidden representation back to the human pose space. To effectively extract both the local motion and global information of the group dance sequence through time, we design our Group Encoder based on two factors: Recurrent Neural Network [3] to capture the temporal motion dynamics of each dancer, and Attention mechanism [2] to encode the spatial relationship of all dancers.

Specifically, at each time step, the pose of each dancer in the previous frame $y^i_{t-1}$ is sent to an LSTM unit to encode the hidden local motion representation $h^i_t$:

To ensure the motions of all dancers have global coherency and discourage strange effects such as cross-body intersection, we introduce the Cross-entity Attention mechanism. In particular, each individual motion representation is first linearly projected into a key vector $k^i$, a query vector $q^i$ and a value vector $v^i$ as follows: \begin{equation} k^i = h^i W^{k}, \quad q^i = h^i W^{q}, \quad v^i = h^i W^{v}, \end{equation} where $W^q, W^k \in \mathbb{R}^{d_h \times d_k}$, and $W^v \in \mathbb{R}^{d_h \times d_v}$ are parameters that transform the hidden motion $h$ into a query, a key, and a value, respectively. $d_k$ is the dimension of the query and key while $d_v$ is the dimension of the value vector. To encode the relationship between dancers in the scene, our Cross-entity Attention also utilizes the Scaled Dot-Product Attention as in the Transformer [3].

In practice, we find that people having closer positions to each other tend to have higher correlation in their movement. Therefore, we adopt Spacial Encoding strategy to encode the spacial relationship between each pair of dancers. The Spacial Encoding between two entities based on their distance in the 3D space is defined as follows:

where $d_{\tau}$ is the dimension of the position vector $\tau$. Considering the query $q^i$, which represents the current entity information, and the key $k^j$, which represents other entity information, we inject the spatial relation information between these two entities onto their cross attention coefficient:

To preserve the spatial relative information in the attentive representation, we also embed them into the hidden value vector and obtain the global-aware representation $g^i$ of the $i$-th entity as follows:

where $\gamma \in \mathbb{R}^{d_v}$ is the learnable bias and scaled by the Spacial Encoding. Intuitively, the Spacial Encoding acts as the bias in the attention weight, encouraging the interactivity and awareness to be higher between closer entities. Our attention mechanism can adaptively attend to each dancer and others in both temporal and spatial manner, thanks to the encoded motion as well as the spatial information.

We then fuse both the local and global motion representation by adding $h^i$ and $g^i$ to obtain the final latent motion $z^i$. Our final global-local representation of each entity is expected to carry the comprehensive information of their own past motion as well as the motion of every other entity, enabling the MLP Decoder to generate coherent group dancing sequences. Finally, we generate the next movement ${y}^i_t$ based on the final motion representation $z^i_t$ as well as the hidden audio representation $a_t$, and thus can capture the fine-grained correspondence between music feature sequence and dance movement sequence:

Built upon these components, our model can effectively learn and generate coherent group dance animation given several pieces of music. In the next part, we will go through the experiments and detailed studies of the method.

#References

[1] Matthew Loper, Naureen Mahmood, Javier Romero, Gerard Pons-Moll, and Michael J. Black. SMPL: A skinned multiperson linear model. ACM Trans. Graphics, 2015

[2] Vaswani A, Shazeer N, Parmar N, Uszkoreit J, Jones L, Gomez AN, Kaiser Ł, Polosukhin I. Attention is all you need. NIPS 2017.

[3] Hochreiter S, Schmidhuber J. Long short-term memory. Neural computation. 1997 Nov 15;9(8):1735-80.

Dancing is an important part of human culture and remains one of the most expressive physical art and communication forms. With the rapid development of digital social media platforms, creating dancing videos has gained significant attention from social communities. As a result, millions of dancing videos are created and watched daily on online platforms. Recently, studies of how to create natural dancing motion from music have attracted great attention in the research community.

Nevertheless, generating dance motion for a group of dancers remains an open problem and have not been well-investigated by the community yet. Motivated by these shortcomings and to foster research on group choreography, we establish AIOZ-GDANCE, a new largescale in-the-wild dataset for music-driven group dance generation. Unlike existing datasets that only support single dance, our new dataset contains group dance videos as shown in Figure 1, hence supporting the study of group choreography. On the basis of the new dataset, we propose the first strong baseline for group dance generation that can jointly generate multiple dancing motions expressively and coherently.

#Dataset Construction

In this section, we will elaborate and describe the process to build our dataset from a large variety of videos available on the internet. Because our main goal is to develop a large-scale dataset with in-the-wild videos, setting up a MoCap system as in many classical approaches is not feasible. However, manually creating 3D groundtruth for millions of frames from dancing videos is also an extremely costly and tedious job. To that end, we propose a semi-automatic labeling method with humans in the loop to obtain the 3D ground truth for our dataset. The process to construct the data includes the five following key steps:

1. Video collection
2. Human Tracking
3. Human Pose Estimation
4. Local Fitting for Invidual Motions
5. Global Scene Optimization

#Data Collection and Preprocessing

Video Collection. We collect the in-the-wild, public domain group dancing videos along with the music from Youtube, Tiktok, and Facebook. All group dance videos are processed at 1920 × 1080 resolution and 30FPS.

Human Tracking. We perform tracking for all humans in the videos using the state-of-the-art multi-object tracker [1] to obtain the tracking bounding boxes. Note that although the tracker can produce reasonable results, there are failure cases in some frames. Therefore, we manually correct the bounding box of the incorrect cases. This tracking correction is crucial since we want the trajectory of each person to be accurately tracked in order to reconstruct their motion in latter stages.

Pose Estimation. Given the bounding boxes of each person in the video, we leverage a state-of-the-art 2D pose estimation method [2] to generate the initial 2D poses for each person. In practice, there exist some inaccurately detected keypoints due to motion blur and partial occlusion. We manually fix the incorrect cases to obtain the 2D keypoints of each human bounding box.

#Local Mesh Fitting

To construct 3D group dance motion, we first reconstruct the full body motion for each dancer by fitting the 3D mesh. We then jointly optimize all dancer motions to construct the globally-coherent group motion. Finally, we post-process and remove wrong cases from the optimization results.

We use SMPL model [3] to represent the 3D human. The SMPL model is a differentiable function that maps the pose parameters $\mathbf{\theta}$, the shape parameters $\mathbf{\beta}$, and the root translation $\mathbf{\tau}$ into a set of 3D human body mesh vertices $\mathbf{V}\in \mathbb{R}^{6890\times3}$ and 3D joints $\mathbf{X}\in \mathbb{R}^{J\times3}$, where $J$ is the number of body joints.

Our optimizing motion variables for each individual dancer consist of a sequence of SMPL joint angles $\{\mathbf{\theta}_t\}_{t=1}^T$, a sequence of the root translation $\{\mathbf{\tau}_t\}_{t=1}^T$, and a single SMPL shape parameter $\mathbf{\beta}$. We fit the sequence of SMPL motion variables to the tracked 2D keypoints by extending SMPLify-X framework [4] across the whole video sequence:

where:

• $E_{\rm J}$ is the 2D reprojection term between the 2D keypoints and the 2D projection of the corresponding 3D poses.
• $E_{\theta}$ is the pose prior term from the latent space of the VPoser model [4] to encourage plausible human pose.
• $E_{\beta}$ is the shape prior term to regularize the body shape towards the mean shape of the SMPL body model.
• $E_{\rm S} = \sum_{t=1}^{T-1}\Vert \mathbf{\theta}_{t+1} - \mathbf{\theta}_{t} \Vert^2 + \sum_{j=1}^J\sum_{t=1}^{T-1}\Vert \mathbf{X}_{j,t+1} - \mathbf{X}_{j,t} \Vert^2$ is the smoothness term to encourage the temporal smoothness of the motion.
• $E_{\rm F} = \sum_{t=1}^{T-1} \sum_{j \in \mathcal{F}} c_{j,t}\Vert \mathbf{X}_{j,t+1} - \mathbf{X}_{j,t} \Vert^2$ is to ensure feet joints to stay stationary when in contact (zero velocity). Where $\mathcal{F}$ is the set of feet joint indexes, $c_{j,t}$ is the feet contact of joint $j$ at time $t$.

#Global Optimization

Given the 3D motion sequence of each dancer $p$: $\{\mathbf{\theta}^p_t, \mathbf{\tau}^p_t\}$, we further resolve the motion trajectory problems in group dance by solving the following objective:

$E_{\rm pen}$ is the Signed Distance Function penetration term to prevent the overlapping of reconstructed motions between dancers.

${E_{\rm reg}(p) =\sum_{t=1}^T\Vert \mathbf{\theta}^p_t - \hat{\mathbf{\theta}}^p_t\Vert^2}$ is the regularization term that prevents the motion from deviating too much from the prior optimized individual motion $\{\hat{\mathbf{\theta}}^p_t\}$ obtained by optimizing the local mesh for dancer $p$.

In practice, we find that the relative depth ordering of dancers in the scene can be inconsistent due to the ambiguity of the 2D projection. To ensure the group motion quality, we watch the videos and manually provide the ordinal depth relation information of all dancers in the scene at each frame $t$ as follows:

Given the relative depth information, we derive the depth relation term $E_{\rm dep}$. This term encourages consistent ordinal depth relation between the motion trajectories of multiple dancers, especially when dancers partially occlude each other:

where $z^p_t$ is the depth component of the root translation $\mathbf{\tau}^p_t$ of the person $p$ at frame $t$. Intuitively, for $r(p,p')=1$, $z_p$ should be smaller than $z_{p'}$ and otherwise.

Finally, we apply the global ground contact constraint $E_{\rm gc}$ to further ensure consistency between the motion of every person and the environment based on the ground contact information. This contact term is also needed to reduce the artifacts such as foot-skating, jittering, and penetration under the ground.

where $\mathcal{F}$ is the set of feet joint indexes, $\mathbf{n}^*$ is the estimated plane normal and $\mathbf{f}$ is a 3D fixed point on the ground plane. The first term in Equation~\ref{eq_Egc} is the zero velocity constraint when the feet are in contact with the ground, while the second term encourages the feet position to stay near the ground when in contact. To obtain the ground plane parameters, we initialize the plane point $\mathbf{f}$ as the weighted median of all contact feet positions. The plane normal $\mathbf{n}^*$ is obtained by optimizing a robust Huber objective:

where $\mathcal{H}$ is the Huber loss function, $\mathbf{X}_{\rm feet}$ is the 3D feet positions of all dancers across the whole sequence that are labelled as in contact (i.e., $c^p_{j,t} = 1$) .

#How will AIOZ-GDANCE be useful to the community?

We bring up some interesting research directions that can be benefited from our dataset:

• Group Dance Generation
• Human Pose Tracking
• Dance Education
• Dance style transfer
• Human behavior analysis

While single-person choreography is a hot research topic recently, group dance generation has not yet well investigated. We hope that the release of our dataset will foster more this research direction.

#References

[1] Peize Sun, Jinkun Cao, Yi Jiang, Zehuan Yuan, Song Bai, Kris Kitani, and Ping Luo. Dancetrack: Multi-object tracking in uniform appearance and diverse motion. In CVPR, 2022

[2] Hao-Shu Fang, Shuqin Xie, Yu-Wing Tai, and Cewu Lu. RMPE: Regional multi-person pose estimation. In ICCV, 2017.

[3] Matthew Loper, Naureen Mahmood, Javier Romero, Gerard Pons-Moll, and Michael J. Black. SMPL: A skinned multiperson linear model. ACM Trans. Graphics, 2015

[4] Georgios Pavlakos, Vasileios Choutas, Nima Ghorbani, Timo Bolkart, Ahmed A. A. Osman, Dimitrios Tzionas, and Michael J. Black. Expressive body capture: 3d hands, face, and body from a single image. In CVPR, 2019.

In this part, we will conduct experiements for validating the effectiveness of our proposed Global-Local Attention for Context-aware Emotion Recognition. Here, we only focus on static images with background context as our input. Therefore, we choose the static CAER (CAER-S) dataset [2] to validate our method. However, while experimenting with the CAER-S dataset, we observe that there is a correlation between images in the training and the test sets, which can make the model less robust to changes in data and may not generalize well on unseen samples. More specifically, many images in the training and the test set of the CAER-S dataset are extracted from the same video, hence making them look very similar to each other. To properly evaluate the models, we propose a new way to extract static frames from the CAER video clips to create a new static image dataset called Novel CAER-S (NCAER-S). In particular, for each video in the original CAER dataset, we split the video into multiple parts. Then we randomly select one frame of each part to include in the new NCAER-S dataset. Any original video that provides frames for the training set will be removed from the testing set. This process assures the new dataset is novel while the training frames and testing frames are never from one original input video.

#Context-aware emotion recognition results

Table 1 summarizes the results of our network and other recent state-of-the-art methods on the CAER-S dataset [2]. This table clearly shows that integrating our GLA module can significantly improve the accuracy performance of the recent CAER-Net. In particular, our GLAMOR-Net (original) achieves 77.90% accuracy, which is a +4.38% improvement over the CAER-Net-S. When compared with other recent state-of-the-art approaches, the table clearly demonstrates that our GLAMOR-Net (ResNet-18) outperforms all those methods and achieves a new state-of-the-art performance with an accuracy of 89.88%. This result confirms our global-local attention mechanism can effectively encode both facial information and context information to improve the human emotion classification results.

#Component Analysis

To further analyze the contribution of each component in our proposed method, we experiment with 4 different input settings on the NCAER-S dataset: (i) face only, (ii) context only with the facial region being masked, (iii) context only with the facial region visible, and (iv) both face and context (with masked face). When the context information is used, we compare the performance of the model with different context attention approaches. Note that to compute the saliency map with the proposed GLA in the (ii) and (iii) setting, we extract facial features using the Facial Encoding Module, however, these features are only used as the input of the GLA module to guide the context attention map learning process and not as the input of the Fusion Network to predict the emotion category. The results of these settings are summarized in Table 2.

The results clearly show that our GLA consistently helps improve performance in all settings. Specifically, in setting (ii), using our GLA achieves an improvement of 1.06\% over method without attention. Our GLA also improves the performance of the model when both facial and context information is used to predict emotion. Specifically, our model with GLA achieves the best result with an accuracy of 46.91\%, which is higher than the method with no attention 3.72\%. The results from Table 2 show the effectiveness of our Global-Local Attention module for the task of emotion recognition. They also verify that the use of both the local face region and global context information is essential for improving emotion recognition accuracy.

#Qualitative Analysis

Figure 5 shows the qualitative visualization with learned attention maps obtained by our method GLAMOR-Net in comparison with CAER-Net-S. It can be seen that our Global-Local Attention mechanism produces better saliency maps and helps the model attend to the right discriminative regions in the surrounding background than the attention map produced by CAER-Net-S. As we can see, our model is able to focus on the gesture of the person (Figure 5f) and also the face of surrounding people (Figure 5c and 5d) to infer the emotion accurately.

Figure 6 shows some emotion recognition results of different approaches on the CAER-S dataset. More specifically, the first two rows (i) and (ii) contain predictions of the CAER-Net-S while the last two rows (iii) and (iv) show the results of our GLAMOR-Net. In some cases, our model was able to exploit the context effectively to perform inference accurately. For instance, with the same sad image input (shown on the (i) and (iii) rows), the CAER-Net-S misclassified it as neutral while the GLAMOR-Net correctly recognized the true emotion category. It might be because our model was able to identify that the man was hugging and appeasing the woman and inferred that they were sad. Another example is shown on the (i) and (iii) rows of the fear column. Our model classified the input accurately, while the CAER-Net-S is confused between the facial expression and the wedding surrounding, thus incorrectly predicted the emotion as happy.

#Conclusion

We have presented a novel method to exploit context information more efficiently by using the proposed globallocal attention model. We have shown that our approach can noticeably improve the emotion classification accuracy compared to the current state-of-the-art results in the context-aware emotion recognition task. The results on the context-aware emotion recognition datasets consistently demonstrate the effectiveness and robustness of our method.

#References

[1] Chorowski, J., Bahdanau, D., Serdyuk, D., Cho, K., Bengio, Y.: Attention-based models for speech recognition. In NIPS, 2015.

[2] Lee, J., Kim, S., Kim, S., Park, J., Sohn, K.: Context-aware emotion recognition networks. In ICCV, 2019.

Automatic emotion recognition has been a longstanding problem in both academia and industry. It enables a wide range of applications in various domains, ranging from healthcare, surveillance to robotics and human-computer interaction. Recently, significant progress has been made in the field and many methods have demonstrated promising results. However, recent works mainly focus on facial regions while ignoring the surrounding context, which is shown to play an important role in the understanding of the perceived emotion, especially when the emotions on the face are ambiguous or weakly expressed (see the examples in Figure 1).

We hypothesize that the local information (i.e., facial region) and global information (i.e., context background) have a correlative relationship, and by simultaneously learning the attention using both of them, the accuracy of the network can be improved. This is based on the fact that the emotion of one person can be indicated by not only the face’s emotion (i.e., local information) but also other context information such as the gesture, pose, or emotion/pose of a nearby person. To that end, we propose a new deep network, namely, Global-Local Attention for Emotion Recognition Network (GLAMOR-Net), to effectively recognize human emotions using a novel global-local attention mechanism. Our network is designed to extract features from both facial and context regions independently, then learn them together using the attention module. In this way, both the facial and contextual information is used altogether to infer human emotions.

#Overview

#Face and Context Encoding

Our Encoding Module comprises the Facial Encoding Module to learn the face-specific features, and the Context Encoding Module to learn the context-specific features. Specifically, both the Face Encoding and Context Enconding Module are built on several convolutional layers to extract meaningful features from the corresponding input. Each module is comprised of five convolutional layers followed by a Batch Normalization layer an ReLU activation function. The number of filters starts with 32 in the first layer, increasing by a factor of 2 at each subsequent layer except the last one. Our network ends up with 256-channel feature map, which is the embedded representation with respect to the input image. In practice, we also mask the facial regions in the raw input to prevent the attention module from only focusing on the facial region while omitting the context information in other parts of the image.

#Global-Local Attention

Inspired by the attention mechanism [1], to model the associative relationship of the local information (i.e., the facial region in our work) and global information (i.e., the surrounding context background), we propose the Global-Local Attention Module to guide the network focus on meaningful regions (Figure 3). In particular, our attention mechanism models the hidden correlation between the face and different regions in the context by capturing their similarity.

We first reduce the facial feature map $\mathbf{F}_f$ into vector representation using the Global Pooling operator, denoted as $\mathbf{v}_f$. The context feature map can be viewed as a set of $W_c \times H_c$ vectors with $D_c$ dimensions, each vector in each cell $(i,j)$ represents the embedded features at that location with the corresponding patch in the input image. Therefore, at each region $(i,j)$ in the context feature map, we have $\mathbf{F}_c^{(i,j)} = \mathbf{v}_{i,j}$.

We then concatenate $[\mathbf{v}_f; \mathbf{v}_{i,j}]$ into a holistic vector $\bar{\mathbf{v}}_{i,j}$, which contains both information about the face and some small regions of the scene. We then employ a feed-forward neural network to compute the score corresponding to that region by feeding $\bar{\mathbf{v}}_{i,j}$ into the network. By applying the same process for all regions, each region $(i,j)$ will output a raw score value $s_{i,j}$, we spatially apply the Softmax function to produce the attention map $a_{i,j} = \text{Softmax}(s_{i,j})$. To obtain the final context representation vector, we squish the feature maps by taking the average over all the regions weighted by $a_{i,j}$ as follow:

where $\mathbf{v}_c \in \mathbb{R}^{D_c}$ is the final single vector encoding the context information Intuively, $\mathbf{v}_c$ mainly contains information from regions that have high attention, while other nonessential parts of the context are mostly ignored. With this design, our attention module can guide the network focus on important areas based on both facial information and context information of the image.

#Face and Context Fusion

The Fusion Module takes the face $\mathbf{v}_f$ and the context reprsentation $\mathbf{v}_c$ as inputs, then the face score and context score are produced separately by two neural networks:

where $\phi_f$ and $\phi_c$ are the network parameters of the face branch and context branch, respectively. Next, we normalize these scores by the Softmax function to produce weights for each face and context branch

In this way, we let the two networks competitively determine which branch is more useful than the other. Then we amplify the more useful branch and lower the effect of the other by multiplying the extracted features with the corresponding weight:

Finally, we use these vectors to estimate the emotion category. Specifically, in our experiments, after multiplying both $\mathbf{v}_f$ and $\mathbf{v}_c$ by their corresponding weights, we concatenate them together as the input for a network to make final predictions. Figure 4 shows our fusion procedure in detail.

#References

[1] Chorowski, J., Bahdanau, D., Serdyuk, D., Cho, K., Bengio, Y.: Attention-based models for speech recognition. In NIPS, 2015.

[2] Lee, J., Kim, S., Kim, S., Park, J., Sohn, K.: Context-aware emotion recognition networks. In ICCV, 2019.

Facial expression recognition (FER) plays an important role in understanding people's feelings and interactions between humans. Recently, automatic emotion recognition has gained a lot of attention from the research community due to its tremendous applications in education, healthcare, human analysis, surveillance or human-robot interaction. Recent FER methods are mostly based on deep learning and can achieve impressive results. The success of deep models can be attributed to large-scale FER datasets [1][2]. However, ambiguities of facial expression is still a key challenge in FER. Specifically, people with different backgrounds might perceive and interpret facial expressions differently, which can lead to noisy and inconsistent annotations. In addition, real-life facial expressions usually manifest a mixture of feelings rather than only a single emotion.

#Motivation and Proposed Solution

As an example, Figure 1 shows that people may have different opinions about the expressed emotion, particularly in ambiguous images. Consequently, a distribution over emotion categories is better than a single label because it takes all sentiment classes into account and can cover various interpretations, thus mitigating the effect of ambiguity. However, existing large-scale FER datasets only provide a single label for each sample instead of a label distribution, which means we do not have a comprehensive description for each facial expression. This can lead to insufficient supervision during training and pose a big challenge for many FER systems.

To overcome the ambiguity problem in FER, we proposes a new uncertainty-aware label distribution learning method that constructs emotion distributions for training samples. Specifically, we leverage the neighborhood information of samples that have similar expressions to construct the emotion distributions from single labels and utilize them as training supervision signal.

#Methodology

#Preliminaries

We denote $\mathbf{x} \in \mathcal{X}$ as the instance variable in the input space $\mathcal{X}$ and $\mathbf{x}^{i}$ as the particular $i$-th instance. The label set is denoted as $\mathcal{Y} = \{y_1, y_2,..., y_m\}$ where $m$ is the number of classes and $y_j$ is the label value of the $j$-th class. The logical label vector of $\mathbf{x}^{i}$ is indicated by $\mathbf{l}^{i}$ = $(l^{i}_{y_1}, l^{i}_{y_2}, ..., l^{i}_{y_m})$ with $\mathbf{l}^{i}_{y_j} \in \{0, 1\}$ and $\| \mathbf{l} \| _1 = 1$. We define the label distribution of $\mathbf{x}^{i}$ as $\mathbf{d}^{i}$ = $(d^{i}_{y_1}, d^{i}_{y_2}, ..., d^{i}_{y_m})$ with $\| \mathbf{d} \| _1 = 1$ and $d^{i}_{y_j} \in [0, 1]$ representing the relative degree that $\mathbf{x}^{i}$ belongs to the class $y_j$.

Most existing FER datasets assign only a single class or equivalently, a logical label $\mathbf{l}^{i}$ for each training sample $\mathbf{x}^{i}$. In particular, the given training dataset is a collection of $n$ samples with logical labels $D_l$ = $\{ (\mathbf{x}^{i}, \mathbf{l}^{i}) \vert 1 \le i \le n\}$. However, we find that a label distribution $\mathbf{d}^i$ is a more comprehensive and suitable annotation for the image than a single label.

Inspired by the recent success of label distribution learning (LDL) in addressing label ambiguity [3], we aim to construct an emotion distribution $\mathbf{d}^i$ for each training sample $\mathbf{x}^i$, thus transform the training set $D_l$ into $D_d$ = $\{ (\mathbf{x}^{i}, \mathbf{d}^{i}) \vert 1 \le i \le n\}$, which can provide richer supervision information and help mitigate the ambiguity issue. We use cross-entropy to measure the discrepancy between the model's prediction and the constructed target distribution. Hence, the model can be trained by minimizing the following classification loss:

where $f(\mathbf{x}; \theta)$ is a neural network with parameters $\theta$ followed by a softmax layer to map the input image $\mathbf{x}$ into a emotion distribution.

#Overview

An overview of our method is presented in Figure 2. To construct the label distribution for each training instance $\mathbf{x}^i$, we leverage its neighborhood information in the valence-arousal space. Particularly, we identify $K$ neighbor instances for each training sample $\mathbf{x}^i$ and utilize our adaptive similarity mechanism to determine their contribution degrees to the target distribution $\mathbf{d}^i$. Then, we combine the neighbors' predictions and their corresponding contribution degrees with the provided label $\mathbf{l}^i$ and $\mathbf{l}^i$'s uncertainty factor to obtain the label distribution $\mathbf{d}^i$. The constructed distribution $\mathbf{d}^i$ will be used as supervision information to train the model via label distribution learning.

#Adaptive Similarity

We assume that the label distribution of the main instance $\mathbf{x}^i$ can be computed as a linear combination of its neighbors' distributions. To determine the contribution of each neighbor, we propose an adaptive similarity mechanism that not only leverages the relationships between $\mathbf{x}^i$ and its neighbors in the auxiliary space but also utilizes their feature vectors extracted from the backbone. We choose the valence-arousal [4] as the auxiliary space to construct the target label distribution. We use the $K$-Nearest Neighbor algorithm to identify $K$ closest points for each training sample $\mathbf{x}^i$, denoted as $N(i)$. We calculate the adaptive contribution degrees of neighbor instances as the product of the local similarity $s^i_k$ and the calibration score $\zeta^i_k$ as follows:

where the local similarity $s^i_k$ is defined based on the distance between the instance and its neighbor in the valence-arousal space $\mathbf{a}^i$ and $\mathbf{a}^k$

We utilize a multilayer perceptron (MLP) $g$ with parameter $\phi$ to calculate the adaptive calibration score from the extracted features of the two instances $\mathbf{v}^i$ and $\mathbf{v}^k$ obtained from the backbone.

The proposed adaptive similarity can correct the similarity errors in the valance-arousal space, as the valence-arousal values are not always available in practice and we leverage an existing method to generate pseudo-valence-arousal.

#Uncertainty-aware Label Distribution Construction

After obtaining the contribution degree of each neighbor $\mathbf{x}^k \in N(i)$, we can now generate the target label distribution $\mathbf{d}^i$ for the main instance $\mathbf{x}^i$. The target label distribution is calculated using the logical label $\mathbf{l}^i$ and the aggregated distribution $\tilde{\mathbf{d}}^i$ defined as follows:

where $\lambda^i \in [0,1]$ is the uncertainty factor for the logical label. It controls the balance between the provided label $\mathbf{l}^i$ and the aggregated distribution $\tilde{\mathbf{d}^i}$ from the local neighborhood.

Intuitively, a high value of $\lambda^i$ indicates that the logical label is highly uncertain, which can be caused by ambiguous expression or low-quality input images, thus we should put more weight towards neighborhood information $\tilde{\mathbf{d}^i}$. Conversely, when $\lambda^i$ is small, the label distribution $\mathbf{d}^i$ should be close to $\mathbf{l}^i$ since we are certain about the provided manual label. In our implementation, $\lambda^i$ is a trainable parameter for each instance and will be optimized jointly with the model's parameters using gradient descent.

#Loss Function

To enhance the model's ability to discriminate between ambiguous emotions, we also propose a discriminative loss to reduce the intra-class variations of the learned facial representations. We incorporate the label uncertainty factor $\lambda^i$ to adaptively penalize the distance between the sample and its corresponding class center. For instances with high uncertainty, the network can effectively tolerate their features in the optimization process. Furthermore, we also add pairwise distances between class centers to encourage large margins between different classes, thus enhancing the discriminative power. Our discriminative loss is calculated as follows: