1 Introduction
Deep nets LeCunNature2015 ; BengioPAMI2013
are omnipresent across fields from computer vision and natural language processing to computational biology and robotics. Across domains and tasks they have demonstrated impressive results by automatically extracting hierarchical abstractions of representations from many different datasets. The surge in popularity pivoted in the 2010s, with impressive results being demonstrated on the ImageNet dataset
KrizhevskyNIPS2012 ; RussakovskyIJCV2015 . Since then, deep nets have been applied to many more tasks. Prominent examples include recognition of places ZhouNIPS2014 , playing of Atari games MnihNIPSWS2013 ; MnihNature2015 , and the game of Go SilverNature2016 . Common to all those methods is the use of large datasets to fuel the many layers of deep nets.Importantly, in the last few years, the number of layers, or more generally the depth of the computation tree has increased significantly from a few layers for LeNet LeCunIEEE1998 to several 100s or 1000s HeCVPR2016 ; LarssonARXIV2016 . Inherent to the increasing complexity of the computation graph is an increase in training time and often also an increase in the amount of data that is processed. Traditionally, computational performance increases do not keep up with the desired processing needs despite the use of accelerators like GPUs.
Beyond accelerators, parallelization of computation on multiple computers is therefore popular. However, it requires frequent communication to exchange a large amount of data among compute nodes while the bandwidth of network interfaces is limited. This in turn significantly diminishes the benefit of parallelization, as a substantial fraction of training time is spent to communicate data. The fraction of time spent on communication is further increased when applying accelerators IandolaCVPR2016 ; CMU_EuroSys16 ; Nvidia15 ; Cong_FPGA15 ; Li_ISCAS16 ; Li_NeuroCom17 ; dnnweaver , as they decrease computation time while leaving communication time untouched.
To take advantage of parallelization across machines, a variety of approaches have been developed starting from the popular MapReduce paradigm DeanACM2008 ; ZahariaUSENIX2010 ; Isard2007 ; Murray2013 . Despite their benefits, communication heavy training of deep nets is often based on custom implementations Dean2012 ; Chilimbi2014 ; MoritzICLR2016 ; Kim2016 relying on the parameter server architecture MuLiNIPS2014 ; MuLiOSDI2014 ; SSP , where the centralized server aggregates the gradients from workers and distributes the updated weights, either in a synchronous or asynchronous manner. Recent research proposed to use a decentralized architecture with global synchronization among nodes Facebook1Hour ; deep_g_compression . However, in common to all the aforementioned techniques, little is reported regarding the timing analysis of distributed deep net training.
In this paper, we analyze the wallclock time tradeoffs between communication and computation. To this end we develop a model to assess the training time based on a set of parameters such as latency, cluster size, network bandwith, model size, etc. Based on the results of our model we develop PipeSGD, a framework with pipelined training and balanced communication, and show its convergence properties by adjusting proofs of langford2009slow ; SSP . We also show what types of compression can be efficiently included in an AllReduce based framework. Finally, we assess the speedups of our proposed approach on a GPU cluster of four nodes with 10GbE network, showing wallclock time training improvements by a factor of compared to conventional centralized and decentralized approaches without degradation in accuracy.
2 Background
General Training of Deep Nets: Training of deep nets involves finding the parameters of a predictor given input data
. To this end we minimize a loss function
which compares the predictor output for given data and the current to the groundtruth annotation . Given a dataset , finding is formally summarized via:(1) 
Optimization of the objective given in Eq. (1) w.r.t. the parameters , e.g., via gradient descent using , can be challenging due to not only the complexity of evaluating the predictor and its derivative, but also the size of the dataset
. Consequently, stochastic gradient descent (SGD) emerged as a popular technique. We randomly sample a subset
of the dataset, often also referred to as a minibatch. Instead of computing the gradient on the entire dataset , we approximate it using the samples in the minibatch, i.e., we assume . However, for present day datasets and predictors, computation of the gradient on a single machine is still challenging. Minibatch sizes of less than 20 samples are common, e.g., when training for semantic image segmentation ChenICLR2015 .Distributed Training of Deep Nets: To train larger models or to increase the minibatch size, distributed training on multiple compute nodes is used Dean2012 ; SSP ; Chilimbi2014 ; MuLiOSDI2014 ; MuLiNIPS2014 ; MoritzICLR2016 ; IandolaCVPR2016 . A popular architecture to facilitate distributed training is the parameter server framework SSP ; MuLiOSDI2014 ; MuLiNIPS2014 . The parameter server maintains a copy of the current parameters, and communicates with a group of worker nodes, each of which operates on a small minibatch to compute local gradients based on the retrieved parameters . Upon having completed its task, the worker shares the gradients with the parameter server. Once the parameter server has obtained all or some of the gradients it updates the parameters using the negative gradient direction and afterwards shares the latest values with the workers.
Asynchronous updates where each worker independently pulls from the server, computes its own local gradient, and pushes results back are available and illustrated in Fig. 1 (a). Due to the asynchrony, minimal synchronization overhead is traded with staleness of gradients. Methods for staleness control exist, which bound the number of delay steps SSP . However, note that stale gradients may slow down training significantly.
Importantly, all those frameworks are based on a centralized compute topology which forms a communication bottleneck, increasing the training time as the cluster size scales. The time taken by pushing gradient, update, and pulling can be linear in the cluster size due to network congestion.
Therefore, most recently, decentralized training frameworks gained popularity in both the synchronous and asynchronous setting DPSGD ; ADPSGD . However, those approaches assume decentralized workers are either completely synchronous (as in Fig. 1 (b)) or completely asynchronous, which requires to either deal with long execution time every iteration or pay for uncontrolled gradient staleness.
Compression in Distributed Training: As the model size increases and cluster size scales, communication overhead in distributed learning system dominates the training time, e.g., up to even in a highspeed network environment INCEPTIONN ; Aluminum . To reduce the communication time, various compression algorithms have been proposed recently Microsoft_1BitSGD ; Amazon_FixThresh ; Lawrence_Adaptive ; IBM_AdaComp ; TernGradNIPS2017 ; deep_g_compression ; QSGDNIPS2017 , some of which focus on reducing the precision of communicated gradients through scalar quantization into 1 bit, while others focus on reducing the quantity of gradients to be transferred. Most compression works, however, only emphasize on achieving high compression ratio or low loss in accuracy without reporting the wallclock training time.
In practice, compression without knowledge of the communication process is usually counterproductive INCEPTIONN , i.e., the total training time often increases. This is due to the fact that AllReduce is a multistep algorithm which requires transferred gradients to be compressed and decompressed repeatedly with a worstcase complexity linear in the cluster size, as we discuss below in Sec. 3.2.
3 Decentralized Pipelined Stochastic Gradient Descent
Overview: To address the aforementioned issues (network congestion for a central server, long execution time for synchronous training, and stale gradients in asynchronous training) we propose a new decentralized learning framework, PipeSGD, shown in Fig. 1 (c). It balances communication among nodes via AllReduce and pipelines the local training iterations to hide communication time.
We developed PipeSGD by analyzing a timing model for wallclock train time under different resource conditions using various communication approaches. We find that the proposed PipeSGD is optimal when gradient updates are delayed by only one iteration and the time taken by each iteration is dominated by local computation on workers. Moreover, we found lossy compression to further reduce communication time without impacting accuracy.
Due to local pipelined training, balanced communication, and compression, the communication time is no longer part of the critical path, i.e., it is completely masked by computation, leading to linear speedup of endtoend training time as the cluster size scales. Finally, we prove the convergence of PipeSGD for convex and strongly convex objectives by adjusting the proof of langford2009slow ; SSP .
3.1 Timing Models and Decentralized PipeSGD
Timing Model: We propose timing models based on decentralized synchronous SGD to analyze the wallclock runtime of training. Each training iteration consists of three major stages: model update, gradient computation, and gradient communication. Classical synchronous SGD (Fig. 1 (b)) runs local iterations on workers sequentially, i.e., each update depends on the gradient from the previous iteration, i.e., the iteration dependency is . Therefore the total runtime of synchronous SGD can be formulated easily as:
(2) 
where denotes the total number of training iterations and refer to the time taken by update, compute, and communication, respectively. It is apparent that synchronous SGD depends on the sum of execution time taken by all stages, which leads to long endtoend training time.
On the contrary, PipeSGD relaxes the iteration dependency to , i.e., each update depends only on the gradients of the th last iteration. This enables interleaving between neighboring iterations while maintaining globally synchronized communication, as shown in Fig. 1 (c). If we assume ideal conditions where both computation resources (CPU, GPU, other accelerators) and communication resources (communication links) are unlimited or abundant in counts/bandwidth, then the total runtime of PipeSGD is:
(3) 
where denotes the iteration dependency or the gradient staleness. We observe that the endtoend training time in PipeSGD can be shortened by a factor of . However, the ideal resource assumption doesn’t hold in practice, because both computation and communication resources are strictly limited on each worker node in today’s distributed systems. As a result, the timing model for distributed learning is resource bound, either communication or computation bound, as shown in Fig. 2 (a), i.e., the total runtime is:
(4) 
where the total runtime is solely determined by either computation or communication resources, regardless of (when ). Also, since gradient updates are always delayed by iterations, increasing only harms, i.e., the optimal value of for PipeSGD with limited resources. Hence, the staleness of gradients is limited to iteration, i.e., the minimal staleness achievable in asynchronous updates. Besides, we generally prefer a computationbound setting for distributed training system, i.e., . To achieve this we discuss compression techniques in Sec. 3.2.
In addition to pipelined execution of iterations, we also analyze pipelined gradient communication within each iteration to reduce train time. Computation of gradients, i.e., the backwardpass, and communication of gradients are often executed in a strictly sequential manner (see Fig. 2 (b)). However, pipelined gradient communication, i.e., communicating gradients immediately after they are computed, is feasible. Again, we assume limited resources and compare the sequential and pipelined gradient communication in Fig. 2 (b).
To analyze the detailed timing of the those two approaches, we use the timing models for communication ThakurRG05
. Communication of gradients is an AllReduce operation which aggregates the gradient vector from all workers, performs the sum reduction elementwise, and then sends the result back to all. In practice, the underlying algorithms are much more involved
ThakurRG05 . For example, RingAllReduce, one of the fastest AllReduce algorithms, performs gradient aggregation collectively among workers through balanced communication. As shown in Fig. 2 (c), each worker transmits only a block of the entire gradient vector to its neighbor and performs the sum reduction on the received block. This “transmitandreduce” runs in parallel on all workers, until the gradient blocks are fully reduced on a worker (different for each block). Afterwards those fully reduced blocks are sent back to the remaining workers along the virtual ring. This approach optimally utilizes the network bandwidth of all nodes.Adopting the RingAllReduce model of ThakurRG05 , we obtain the total runtime of PipeSGD with sequential gradient communication under the limited resource assumption via:
(5) 
where and denote forwardpass and backwardpass time, denotes the number of workers, the network latency, the model size in bytes, the byte transfer time, the byte sum reduction time, and the global synchronization time.
Similarly, we obtain the total runtime of PipeSGD with pipelined gradient communication via:
(6) 
where denotes the number of gradient segments, and denotes the backwardpass time taken by the first segment.
Based on Eq. (5) and Eq. (6) we note: if a pipelined system remains communication bound, then sequential gradient communication is preferred over the pipelined gradient communication (Eq. (5) is smaller than Eq. (6) due to positive ). In practice, distributed training of large models is often communication bound, making sequential exchange the best option.
To sum up, based on our timing models, we find: PipeSGD is optimal for , system is compute bound (after compression), and sequential gradient communication is used. Note that although our model is derived based on the RingAllReduce, this conclusion also applies to other AllReduce algorithms, such as recursive doubling, recursive halving and doubling, pairwise exchange, etc. ThakurRG05 .
Decentralized Pipeline SGD:
Guided by the timing models, we develop the decentralized PipeSGD framework illustrated in Fig. 1 (c) where neighboring training iterations on workers are interleaved with a width of while the execution within each iteration remains strictly sequential. Decentralized workers perform pipelined training in parallel with synchronization on gradient communication after every iteration. Due to the synchronous nature of our framework, the gradient update is always delayed by iterations, which enforces a deterministic rather than an uncontrolled staleness. In our optimal setting, the number of iterations for a delayed update is 1, as compared to where is the cluster size in the conventional asynchronous parameter server training SSP ; ADPSGD ; TensorFlowOSDI . Importantly, our framework still enjoys the advantage of an asynchronous approach – interleaving of training iterations to reduce endtoend runtime. Also, different from the parameter server architecture, we don’t congest the head node. Instead, in our case, every worker is only responsible for aggregating part of the gradients in a balanced manner such that communication and aggregate operation time are much more scalable.
More formally, we outline the algorithmic structure of our implementation for each worker in Alg. 1. To be specific, each worker has two threads: one for computation and one for communication, where the former thread consumes the aggregated gradient of the th last iteration and generates the local gradient to be communicated, and the latter thread exchanges the local gradient and buffers the aggregated results to be consumed by the former thread.
3.2 Compression in PipeSGD
To further reduce the communication time we integrate lossy compression into our decentralized PipeSGD framework. Unlike the conventional parameter server or recent decentralized framework transferring parameters over the network Dean2012 ; Chilimbi2014 ; MuLiOSDI2014 ; MuLiNIPS2014 ; MoritzICLR2016 ; IandolaCVPR2016 ; SSP ; ADPSGD ; DPSGD , our approach communicates only gradients and we justified empirically that gradients are much more tolerant to lossy compression than the model parameters. This seems intuitive since reducing the precision of parameters in every iteration harms the final precision of the trained model directly.
Importantly, as mentioned in Sec. 3.1, compressing the communication overhead contributes to the optimal setting of PipeSGD. Once PipeSGD is completely computation bound, linear speedups of endtoend training time can be realized as the cluster size increases. Analytically, we show this observation by deriving the scaling efficiency using the timing model given in Eq. (4). Assume that: 1) the singenode training takes iterations to complete with an execution time of taken by each iteration; 2) given a PipeSGD cluster with workers we use the same batch size on each worker as the singlenode Facebook1Hour ; 3) the single node and PipeSGD
train the same epochs on the dataset. From 2) and 3), we find that the total number of iterations required for
PipeSGD is , because PipeSGD has a times larger batch size while still training the same number of samples. From this we obtain the scaling efficiency of PipeSGD via(7) 
Thus, we showed that once our system becomes compute bound with compressed communication, PipeSGD can achieve linear speedup as the cluster scales, i.e., .
To maintain applicability of RingAllReduce, we choose two simple compression approaches: truncation and scalar quantization. Truncation drops the less significant mantissa bits of floatingpoint values for each gradient. The scalar quantization discretizes each gradient value into an integer of limited bits, with a quantization range determined by the maximal element of a gradient vector. Due to their simplicity, we easily parallelize those compression approaches to minimize overhead.
Note that compression itself can be computeheavy and the introduced computation overhead can outweigh the benefit of compressed communication. Particularly when considering that AllReduce based communication performs multiple steps to transfer and reduce the data (see Fig. 2 (c)), requiring repeated invocation of compression and decompression, i.e., for each “transmitandreduce” step, with an invocation complexity linear in cluster size. Therefore, many proposed complex compression techniques Microsoft_1BitSGD ; Amazon_FixThresh ; Lawrence_Adaptive ; IBM_AdaComp ; TernGradNIPS2017 ; deep_g_compression often fail in the communicationoptimal AllReduce setting, resulting in longer wallclock time. For these reasons, compression embedded inside AllReduce must be light, fast and easy to parallelize, such as a floatingpoint truncation or our elementwise quantization.
Indeed, pipelining within AllReduce can help alleviate the heavy overhead of complex compression. However, its benefit might still be limited. Instead of pipelining of training iterations as in PipeSGD, pipelining within AllReduce interleaves the gradient communication and reduction within each AllReduce process, as illustrated in Fig. 3 (a). Since the communication time is often larger than the reduction time, the latter can be hidden by the former. Once compression is used (as in Fig. 3 (b)), the two stage pipeline becomes (decompression, sum, compression) and (compressed communication) such that light compression overhead can be masked completely. Although complex compression may also benefit from the pipelined AllReduce, the improvement is limited because the time spent by complex compression often outweighs the communication time. For example, we implemented TernGradNIPS2017 within the pipelined AllReduce and found that the compression overhead is the uncompressed communication time and the compressed communication time for the benchmarks in Sec. 4, in which case the heavy overhead cannot be masked. Complete masking requires the compression overhead to be smaller than the compressed communication. In the remainder, we only consider light compressions (truncation/quantization) with native AllReduce.
3.3 Convergence
To prove the convergence of PipeSGD we adapt the derivation from parameterserver based asynchronous training SSP ; langford2009slow . We can show that the convergence rate of PipeSGD for convex objectives via SGD is , where , and are constants for gradient distance and Lipschitz continuity, respectively. We can also show the convergence of PipeSGD for strongly convex functions, and find a rate of for gradient descent. These rates are consistent with SSP ; langford2009slow . Due to the page limit we defer details to the supplementary material.
4 Experimental Evaluation
In this section, we demonstrate the efficacy of our approach on four benchmarks using three datasets: MNIST LeCunIEEE1998 , CIFAR100 Krizhevsky2009 and ImageNet RussakovskyIJCV2015 . We briefly review characteristics of those datasets before discussing metrics and setup, and finally presenting experimental results and analysis.
Datasets and Deep Net Architecture

[leftmargin=*]

MNIST: The MNIST dataset consists of 60,000 training and 10,000 test images, each showing one of ten possible digits. The images are of size
pixels with digits located at the center of the images. We use a classical 3layer perceptron, MNISTMLP, with both hidden layers being 500dimensional and with a global batch size of
. 
ImageNet: For our experiments we use 1,281,167 training and 50,000 validation examples from the ImageNet challenge. Each example comprises a color image of 256256 pixels and belongs to one of 1000 classes. We use the classical AlexNet KrizhevskyNIPS2012 and ResNet HeCVPR2016 , both with a global batch size of .

CIFAR100: The CIFAR100 dataset is composed of 50,000 training and 10,000 test examples with 100 classes. The simple AlexNetstyle CIFAR100 architecture in RenjieCNN
is used for benchmarking this datasets. It consists of 3 convolutional layers and 2 fully connected layers followed by a softmax layer. The detailed parameters are available in
RenjieCNN . Importantly, we adapt this 5 layer CIFAR100CNN into a convex optimization benchmark, CIFAR100Convex, to match our proof of convergence. The convexity is achieved by training only the last fully connected layer while fixing the parameters of all previous layers.
Metrics and Setup
We measure the wallclock time of endtoend training, i.e., the same number of iterations for different settings. For each benchmark, we evaluate the timing model we proposed using endtoend train time and detailed timing breakdowns. We plot the test/validation accuracy over training time to evaluate the actual convergence. Also, final top1 accuracies on the test/validation set are reported. For the setup, we use a cluster of four nodes, each of which consists of a Titan XP GPU TitanXP and a Xeon CPU E52640 IntelXeon . We employ an additional node as the parameter server to support the conventional centralized design. All nodes are connected by 10Gb Ethernet. We implement a distributed training framework in C++ using CUDA 8.0 CUDA , MKL 2018 MKL , and OpenMPI 2.0 OpenMPI , which supports the parameterserver and PipeSGD approach.
Results and Analysis
We evaluate the performance of three different frameworks: parameter server with synchronous SGD (PSSync), decentralized synchronous SGD (DSync), and PipeSGD. Our compression schemes, i.e., 16bit truncation (T) and 8bit quantization (Q), are also applied to AllReduce communication in DSync and PipeSGD. Evaluation results are summarized in Fig. 4 where the first two columns show the convergence performances and the third column shows detailed timing breakdowns with final accuracies labeled.
Convergence: From Fig. 4, we observe: decentralized approaches, i.e., DSync and PipeSGD, converge much faster than the parameter server even without compression, and PipeSGD shows the fastest convergence among these frameworks, especially when compression is applied. For example, the convergence curve of the CIFAR100Convex shows that DSync is around faster than PSSync and PipeSGD is another faster than DSync. The advantage of PipeSGD is further boosted by compression, i.e., truncation in this case, and demonstrates an additional faster convergence than the DSync with the same compression scheme. Therefore PipeSGD prevails with a great margin.
Timing Breakdown: From Fig. 4, the comparison between centralized and decentralized designs shows reduction in uncompressed communication time, thus justifying the efficacy of balanced communication. Once compression is applied, further reduction is observed. However, the actual improvement in DSync is not ideal considering compression factors of for truncation and for quantization, because the compression overhead is paid at the critical path of DSync. In contrast, our PipeSGD can hide this overhead together with computation due to the pipelined nature, as shown in “DSync+T” vs. “PipeSGD +T” in the MNIST benchmark. As communication is further reduced by quantization, the system becomes compute bound and PipeSGD switches to hide the communication instead, thus reaching the optimal setting of PipeSGD. This optimum can also be achieved via the simplest truncation for models with less dominant communication time, e.g., ResNet18 and CIFAR100Convex. As a result, our approach achieves a speedup of compared to DSync and compared to PSSync for these benchmarks. Note that these speedups are based on the comparison between different approaches in the same cluster without scaling the cluster size.
Accuracy: Considering the potential drawback of the 1iteration staled update and lossy compression in PipeSGD, we also evaluate the final test/validation accuracies after endtoend training, as shown in Fig. 4. Interestingly, in our optimal settings “PipeSGD +T/Q,” we find that only AlexNet drops top1 accuracy by compared to baseline DSync while all other benchmarks show slightly improved accuracies. To obtain the best accuracies for the two large nonconvex models such as AlexNet and ResNet, we employ a similiar warmup scheme as in deep_g_compression , i.e., we don’t turn on the pipelined training until the th epoch, before which we still stick to DSync training to avoid the undesirable gradient change in the initial stage. Since the warmup period is marginal compared to total number of epochs, the system performance benefits from PipeSGD most of the time. Note that for smaller models, especially convex ones (e.g., CIFAR100Convex), no warmup is required.
5 Related Work
Li et al. MuLiOSDI2014 ; MuLiNIPS2014 proposed a parameter server framework for distributed learning and a few approaches to reduce the cost of communication among compute nodes, such as exchanging only nonzero parameter values, local caching of index list, and random skip of messages to be transmitted. Abadi et al. TensorFlowOSDI
also proposed a centralized framework, TensorFlow, which incorporates model and data parallelism for training deep nets. Both works support the asynchronous setting to improve communication efficiency but without controlling the staleness of the gradient update. Ho
et al. SSP proposed SSP, another centralized asynchronous framework but with bounded staleness for gradients. The key idea of SSP: 1) each worker has its own iteration index, 2) the slowest and fastest worker must be within iterations, otherwise, the fastest worker is forced to wait until the slowest worker catches up. However, this bound applies to the iteration drift among workers instead of directly on the stale updates of the parameter server. As a result, each worker within the bound can still commit their updates to the server asynchronously, making the last gradient update staled heavily. In the worst case, the staleness is linear in the cluster size.Lin et al. deep_g_compression employed AllReduce as the gradient aggregation method in their synchronous framework, but little is reported regarding wallclock time benefits, especially considering that the full synchronous design suffers from the longest execution time among all workers. Besides, Lian et al. proposed ADPSGD ADPSGD which parallelizes the SGD process over decentralized workers in a completely asynchronous fashion. Workers run completely independently, and only communicate with a set of neighboring nodes to exchange trained weights, i.e., neighboring models are averaged to replace each worker’s local model in each iteration. However, this approach suffers from uncontrolled staleness, which in practice increases with cluster size and the time taken by each iteration. In addition, such a communication method requires each worker to act as the center node of a local graph, which results in a local communication bottleneck. As a result, each worker suffers from long iteration time which further increases the staleness of weight updates. Although Lian et al. ADPSGD compared their framework with the full synchronous design in wallclock time, the performance turns out to be similar when network speeds are roughly equal.
Recently, independent work PipeDream also proposed a distributed pipelined system for DNN training. Different from PipeSGD, PipeDream focuses on pipelining with model parallelism, partitioning the DNN layers onto different machines and pipelining the execution of the machines by injecting consecutive minibatches into the first one. This approach reduces communication load since only activations and gradients of a subset of layers are communicated between machines. However, complex mechanisms (such as profiling, partitioning algorithm, and replicated stages) are necessary to balance the workload among different machines, otherwise compute resources turn idle. Furthermore, PipeDream may suffer from staleness of the weight update, which is linear in the number of stages. This limits the effectiveness of model pipelining and throttles speedups.
6 Conclusion
We developed a rigorous timing model for distributed deep net training which takes into account network latency, model size, byte transfer time, etc. Based on our timing model and realistic resource assumptions, e.g., limited network bandwidth, we assessed scalability and developed PipeSGD, a pipelined training framework which is able to mask the faster of computation or communication time. We showed efficacy of the proposed method on a fournode GPU cluster connected with 10Gb links. Rigorously assessing wallclock time for PipeSGD, we are able to achieve improvements of up to compared to conventional approaches.
Acknowledgement
This work is supported in part by grants from NSF (IIS 1718221, CNS 1705047, CNS 1557244, CCF1763673 and CCF1703575). This work is also supported by 3M and the IBMILLINOIS Center for Cognitive Computing Systems Research (C3SR). Besides, this material is based in part upon work supported by Defense Advanced Research Projects Agency (DARPA) under Contract No. HR001117C0053. The views, opinions, and/or findings expressed are those of the author(s) and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government.
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