“If batch size could be made arbitrarily large […], then training is amenable to standard weak scaling approaches. However, if the training […] is restricted to small batch sizes, then we will need to find other algorithmic and architectural approaches to their acceleration.”
– J. Dean et al. , March 2018
Deep learning has revolutionised many application fields, including computer vision[32, 17], speech recognition [18, 64]28]
. The training of deep learning models is expensive: it takes roughly half a month to reproduce the state-of-the-art accuracy for the ImageNet challenge on a single NVIDIA Titan X GPU. To reduce training time, systems exploit data parallelism across many GPUs to speed up training [35, 19, 13]. Consequently multi-GPU servers have become widely available: a new 10-GPU server with NVIDIA Pascal GPUs costs less than $40,000 , and public cloud providers offer GPU server instances with up to 16 GPUs .
Users expect training time to go down with the number of GPUs in a server. Scaling the training process is challenging though because it requires a system to fully utilise the parallelism of all GPUs without introducing bottlenecks. Existing systems, including TensorFlow , MXNet , CNTK , and Caffe2 , use parallel synchronous stochastic gradient descent (S-SGD)  for training: input batches of training data are partitioned across GPUs. Each GPU then updates its local model replica before a synchronisation operation calculates a new global model for the training of the next input batch.
To utilise many GPUs effectively, S-SGD must therefore use a large batch size. The batch size typically grows linearly with (1) the number of GPUs, and (2) the performance of each GPU. In practice, batch sizes of 64,000 are now not uncommon . With large batch sizes though, statistical efficiency  of the training process reduces [30, 38]. As the per-GPU model replicas synchronise less frequently in relation to the processed training data, the converge rate decreases, which in turn increases the time-to-accuracy until the trained model reaches a target accuracy. Users try to compensate for this reduction in statistical efficiency by increasing the learning rate , or adjusting the batch size adaptively . These techniques, however, require model-specific tuning and do not fundamentally solve the problem but eventually fail for very large batch sizes [14, 24, 15]. Given these implications of large batches, users prefer to use small batches when possible .
The goal of our work is to explore how to design a deep learning system that effectively trains with small batch sizes, between 2 and 32 , while still scaling to many GPUs. The starting point for our design is that, on each GPU, we simply train a model replica with a small batch size. This introduces two challenges, which we address in the paper: (i) how to synchronise this potentially large number of model replicas without adversely affecting statistical efficiency; and (ii) how to ensure that the hardware resources of each GPU are fully utilised, thus achieving high hardware efficiency?
We describe the design and implementation of Crossbow, a new single-server
multi-GPU deep learning system that decreases time-to-accuracy when increasing
the number of GPUs, irrespective of the batch size.111The open source
release of Crossbow is available at:
https://github.com/lsds/Crossbow. The design of Crossbow makes the following new contributions:
(1) Synchronous model averaging (SMA) Crossbow uses SMA, a new synchronisation approach that synchronises model replicas in a scalable fashion with a low reduction in statistical efficiency. In SMA, multiple parallel learners each train their own model replica independently. Learners access a global average model to coordinate: they adjust their trajectories with an update proportional to their divergence from the average model. The average model thus reaches better minima faster than individual learners. All replicas, including the average model, are updated after each learner processes a single batch, and all accesses to the average model are strongly consistent.
(2) Auto-tuning the number of learners With a small batch size, a single learner may not fully utilise the resources of a GPU. Crossbow therefore places multiple concurrent learners on the same GPU. The number of learners per GPU is tuned automatically. During training, Crossbow increases the number of learner until there is no increase in training throughput, the maximum hardware efficiency has been reached. It then uses the number of learners that resulted in peak throughput.
(3) Concurrent task engine Crossbow has a task scheduler that dynamically schedules learners that process the next batch on the first available GPU. The scheduler issues learning and synchronisation tasks concurrently in order to prevent the synchronisation performed by SMA from becoming a bottleneck. Crossbow achieves this by breaking the global synchronisation barrier of SMA into a hierarchical tree: each learner synchronises using a local copy of the average model that resides on its GPU; and local models synchronise across GPUs. The local and global synchronisation operations have optimised implementations according to their communication scopes (using all-reduce ). They also overlap with the forward and backwards error propagation of learners.
In our experimental evaluation, we show that, when training ResNet-50 with 2 model replicas per GPU and a batch size of 16, Crossbow reaches a given target accuracy 1.5 faster than TensorFlow. Training with multiple model replicas per GPU reduces time-to-accuracy by 1.9 for ResNet-32, by 4.2 for VGG-16, and by 2.7 for LeNet, respectively. SMA improves statistical efficiency with multiple model replicas by up to 1.6; with multiple model replicas per GPU, the task engine of Crossbow improves hardware efficiency by up to 1.8.
The rest of the paper is organised as follows: §2 discusses the challenges when scaling the training by increasing the batch size; §3 introduces Crossbow’s synchronous model averaging approach with independent learners; §4 describes the design and implementation of the Crossbow task engine; §5 presents out experimental results; §6 surveys related work; and §7 concludes.
2 Scaling Deep Learning
Deep learning models, multi-layer convolutional neural networks, have been shown to achieve high accuracy for many image or speech classification problems [17, 3]. Since increasing the amount of training data and the number of model parameters improves their accuracy [17, 13], deep learning models require training approaches that exploit the parallelism of modern hardware.
2.1 Mini-batch gradient descent
Supervised training of deep learning models uses labelled samples, split into training and test data. A model gradually “learns” to predict the labels of training data by adjusting its model parameters based on the error. It usually takes several passes (or epochs) over the training data to minimise the prediction error. The test data is used to measure the model accuracy on previously unseen data. The most important metric is test accuracy, which measures the ability of the model to make predictions when deployed “in the wild”.
More formally, let
be a vector of model parameters (weights), and
be a loss function that, given, measures the difference between the predicted label of a sample and the ground truth . The training problem is to find a that minimises the average loss over all training data. In today’s systems, this is achieved using stochastic gradient descent (SGD) [52, 4, 5], an iterative training algorithm that adjusts based on a few samples at a time:
where is the learning rate in the -th iteration of the algorithm, is a mini-batch of training samples, and is the gradient of the loss function, averaged over the batch samples:
It is common to augment Eq. 1 with momentum, a technique known to accelerate the convergence of deep learning models . Using momentum, the training process favours gradients that descent in directions known to improve accuracy from previous iterations. The iterative training algorithm with momentum becomes:
where is the momentum parameter and denotes the algorithm’s previous search direction.
Gradient back-propagation  is used to compute the model gradients when weights are spread across multiple layers. This is done in two steps: (i) an input batch propagates forward through the layers to compute the predicted label. This is compared with the ground-truth label associated with each sample in the batch, measuring the error; and (ii) the error propagates backwards from layer to layer in reverse order. The error is used to compute the gradient for the weights in each layer, and the weights can then be updated incrementally by applying Eq. 3.
When training a deep learning model, the goal is to reduce the time to reach a target level of test accuracy (time-to-accuracy). Two factors affect the time-to-accuracy: (i) the number of iterations that a training algorithm such as SGD requires to find a solution with a given test accuracy (statistical efficiency); and (ii) the execution time of each iteration (hardware efficiency).
2.2 Training with GPUs
GPU architectures are well suited for increasing the hardware efficiency of the training process. A GPU is a many-core processor that is designed for high processing throughput. It features thousands of cores, which are simple floating-point arithmetic units. Arranged in groups, cores form tens of streaming multi-processors (SMs). Multiple threads can execute the same instruction per cycle on different data, such as a training sample or a weight.
When training a deep learning model on a GPU, a batch of training data (or, during the backwards phase, its error) is transformed via a series of matrix or vector floating-point operations () as it propagates from one layer to the next. GPUs can perform more floating-point operations per weight read than a CPU, thus achieving more model updates per second .
Programs for the GPU are kernels, which can be executed in a blocking or non-blocking fashion. Complex multi-layer deep learning models may comprise of hundreds of kernels. Kernels are executed in-order as part of a GPU stream, which is a queue of device work. A GPU can have more than one stream, which allows kernels to execute concurrently. Modern GPUs support events, which are a publish/subscribe mechanism to synchronise across different streams, without having to stall the entire GPU pipeline.
Copying input data from CPU to GPU memory over the PCIe bus is typically assisted by a copy engine on the GPU, which runs independently from the GPU’s compute engine that schedules kernels. Systems therefore hide the latency of communication by overlapping communication with computation tasks (using NVIDIA’s Collective Communication Library (NCCL) ).
A server can have multiple GPUs, and GPU-to-GPU data transfers use the PCIe bus or exploit a fast direct interconnect such as NVIDIA’s NVLink bridge . The GPUs in a server are interconnected in a topology with varying bandwidth: in a two-socket multi-GPU server, the GPUs may form a binary tree in which each GPU pair is connected to a PCI switch, and two pairs are connected with a PCI host bridge, attached to a CPU socket.
2.3 Parallel synchronous gradient descent
Current parallel training approaches distribute the gradient computation across multiple GPUs, but differ in how they synchronise the gradients. The prevailing training algorithm is parallel synchronous SGD (S-SGD) . It requires all GPUs to have a consistent view of the -th version of the model before the -th iteration starts: (i) at each iteration, S-SGD partitions a batch equally across GPUs; (ii) each GPU computes a partial gradient from a batch partition and the latest model version; (iii) GPUs then coordinate to merge partial gradients into an aggregate gradient (according to Eq. 2); and (iv) the aggregate gradient is used to update the models (according to Eq. 3) before the next iteration.
Figure 1 shows the execution of S-SGD on a two-GPU server. Each GPU has a local model replica in its memory, which is used to compute the partial gradients. The GPUs coordinate so that the same aggregate gradient is applied to all local replicas ensuring consistency: a GPU collects partial gradients, averages them, and then disseminates the result.
To address stragglers during gradient computation or synchronisation, researchers have proposed an asynchronous variant of SGD (A-SGD) . In A-SGD, a GPU progresses to the next iteration immediately after its partial gradient was added to the aggregate gradient, and uses the value accumulated thus far to update its model replica. This leads to stale gradients and hard-to-understand asynchrony, making it difficult to train complex neural network models such as ResNet effectively. In contrast, S-SGD has better convergence properties, which is why it has become the de-facto standard for the training of deep neural networks [15, 24]. We therefore also focus on synchronous training.
2.4 Challenges in scaling training
“Training with large mini-batches is bad for your health. More importantly, it’s bad for your test error. Friends don’t let friends use mini-batches larger than 32.”
–Y. LeCun (@ylecun), April 2018
The batch size is a critical parameter when training with parallel S-SGD: if the batch is too small, the GPU is not fully utilised as the communication overhead to move data to and from the GPU dominates. In parallel training, the (aggregate) batch size must therefore increase linearly with the number of GPUs, resulting in a constant batch size per GPU; otherwise, the overall throughput scales poorly. This effect can be seen in the plot of hardware efficiency in Figure 2. The figure shows the relative throughput speed-up when training a deep learning model with TensorFlow as we increase the number of GPUs. If the aggregate batch size remains constant (64), the throughput does not increase linearly because the batch size per GPU decreases (with 8 GPUs the batch per GPU is just 8). In contrast, if we increase the aggregate batch size (to 512 or 1,024 for 8 GPUs), thus maintaining a constant batch size per GPU, we observe a linear speed-up.
While the above shows that large batch sizes are ideal to ensure high hardware efficiency, they exhibit poor statistical efficiency , which is expressed as the number of epochs required to converge to a given accuracy (ETA). This is shown in the plot of statistical efficiency in Figure 3: as the batch size increases, TensorFlow requires more epochs to converge. The reasons are twofold: (1) with large and redundant training datasets (as it is often the case), small batches ensure faster training because only few batches are sufficient to capture the dimensionality of the problem space and converge quickly to good solutions [34, 5]; (2) a small batch size leads to “noisier” gradient updates, which widen the exploration of the loss landscape, making it more likely to find better solutions with a higher test accuracy [21, 20, 30, 23]
This trade-off between hardware and statistical efficiency is particularly detrimental in parallel training. While increasing the batch size enables a linear throughput speed-up with the number of GPUs, beyond a certain threshold (256 in the example in Figure 3), the number of epochs increases super-linearly, thus preventing a linear reduction of training time.
A typical solution to mitigate this issue and compensate for the loss of statistical efficiency with larger batch sizes is hyper-parameter tuning, dynamically adjusting the batch size as well as other hyper-parameters, such as the learning rate and the momentum, during the training progresses. In particular, it has been observed that, as long as the ratio between the learning rate and the batch size remains constant, training may be improved by varying the batch size [23, 58, 15, 21]. This only holds when the learning rate remains relatively small though .
While hyper-parameter tuning can achieve quasi-linear scaling of the training time for large networks such as ResNet-50 on up to 1,024 GPUs [15, 58, 24], it requires a time-consuming model-specific methodology, which is often beyond the reach of non-experts and cannot be applied easily to new models or hardware architectures. In some cases, even with hyper-parameter tuning, it is hard to scale training time on multiple GPUs: a recent study from Google Brain 
shows that convolutional networks exhibit only limited scaling with batch sizes larger than 64 and, for recurrent neural networks, long short-term memory (LSTM), the threshold seems to be even lower (16).
A general approach for scaling S-SGD on multiple GPUs therefore remains an open challenge due to the conflicting impact of large batch sizes on hardware and statistical efficiency. In the next section, we show how Crossbow addresses this problem by leveraging a new synchronisation approach among fully-utilised GPUs, thus achieving high GPU throughput without sacrificing converge speed.
3 Synchronous Model Averaging with Learners
Our approach relies on the concept of a learner, which independently trains a model replica for a given input batch (Section 3.1). Having many independent learners requires careful synchronisation in order to achieve high statistical efficiency. We introduce a new algorithm, named synchronous model averaging (SMA), that consolidates the model updates computed by many learners (Section 3.2). After that, we discuss how to train multiple learners per GPU (Section 3.3) and how to determine the number of learners to use (Section 3.4).
3.1 Independent learners
Parallel S-SGD imposes tight synchronisation when processing partitioned batches. The gradients computed based on all model replicas are aggregated, and the obtained result is incorporated by all replicas. After each iteration, before the computation of gradients for the next batch begins, all replicas are therefore the same.
Our idea is to introduce more diversity into the learning process based on the notion of a learner. A learner is an entity that trains a single model replica independently with a given batch size. Figure 4 shows two learners, each being assigned a different complete batch. A learner computes a gradient and immediately updates its replica based on the gradient. It then continues with the gradient computation for the next batch. As we describe in the next section, to prevent learners from diverging, each learner also applies a correction to its model, which is incorporated synchronously as part of the next update of the replica.
In contrast with parallel S-SGD, model replicas with learners evolve independently because they are not reset to a single global model after each batch. Unlike asynchronous learning approaches [51, 42, 66], each replica is corrected in each iteration to maintain the convergence of the learning process. Learners enable us to achieve both high statistical efficiency and hardware efficiency, avoiding the trade-off between the two faced by other parallel training approaches.
3.2 Sma algorithm
To synchronise the local models of learners, we propose synchronous model averaging (SMA), a new algorithm based on model averaging [48, 47, 54]. SMA consolidates the model updates of learners by maintaining a central average model. We illustrate the idea behind SMA in Figure 5. Starting with the initial model, two learners train replicas, and , with distinct batches. Once the learners have computed the gradients and updated their local replicas, the updates are applied to a central average model. The average model is used to compute corrections for each local replica, ensuring that they follow the trajectory of the central average model.
Corrections are defined as the difference of a replica with the current central average model. A correction represents a penalty of the replica disagreeing with the consensus in the previous iteration , and pulls a replica toward the average value over all replicas. The expectation is that all replicas gradually agree on the optimality of the central average model, and eventually reach the same minima.
In general, a desirable property of model averaging is that the asymptotic variance of the central average model is lower than the variance of individual model replicas. This difference prevents the accumulation of training errors on each learner. As the number of learners increases, however, the difference in variance tends to decrease, preventing the central averaging model from converging faster than the learners. To compensate for this effect, SMA uses momentum, as introduced in Section 2.1. Unlike past work that applies momentum for individual model replicas only [69, 36], we incorporate the directions of model weights into the updates to the central average model: updates in directions of persistent descent are kept; those in other directions are cancelled or diminished. We use Polyak’s momentum method  because, compared to Nesterov’s accelerated gradient 
, it fits model averaging better: the update to the central average model is computed by all learners based on their current position and not an estimated one.
We formalise the SMA algorithm in Alg. 1. It takes as input a model, initialised as , a set of model replicas that are managed by learners, a set of batches , along with two hyper-parameters: the learning rate and the momentum . Upon termination, the algorithm returns the trained model.
First SMA initialises the central average model and a reference to a previous version of it, (lines 1–1). It defines an iterative learning process (lines 1–1) that terminates when the target accuracy has been reached by the central average model or there are insufficient batches available.
In each iteration, a learner proceeds as follows: (i) it selects a batch (Alg. 1) and, using the batch and its replica , the learner computes a gradient (as in Eq. 1) under the given learning rate (Alg. 1); (ii) it computes a correction as the difference between the replica and the central average model where is a constant (Alg. 1); and (iii) the model replica is then updated by applying the gradient and the correction (Alg. 1).
The iteration completes with an update to the central average model (Alg. 1). This update is twofold: (i) it includes the corrections derived for all the model replicas assigned to the independent learners, which represent the current differences of the replicas with respect to ; and (ii) a second component exploits the configured momentum and the previous version of the central average model to accelerate convergence by maintaining the direction of gradients . In the -th iteration of SMA, is the model at the beginning of the -th iteration.
Similar to most parallel training approaches [31, 69], SMA benefits from an online adaptation of hyper-parameters. Based on the accuracy observed after each iteration, the learning rate (parameter in Alg. 1) may be reduced step-wise to overcome oscillation of the trained model and improve accuracy . Such adaptation can be realised by updating directly in each iteration in SMA. For example, when training ResNet-50, it is common to reduce the learning rate twice, at the 30th and 60th epochs , which are chosen empirically based on the accuracy observed after each iteration.
With SMA, however, oscillatory behaviour is observed on the central average model, whereas a change in the learning rate affects the training of each model separately. Since SMA does not reset all models in each iteration, the accuracy may not necessarily improve each time the learning rate changes. When detecting such a situation, we therefore restart SMA: Alg. 1 is executed again with the latest version of the central average model as the new initial model .
3.3 Training multiple learners per GPU
When selecting a small batch size for training in order to achieve high statistical efficiency, training a single model replica may not saturate all resources of a GPU. Learners enable us to decouple the processing of a batch from the available hardware though. To achieve high hardware utilisation, we can execute multiple learners per GPU. If a batch size as low as 2 is used, dozens of learners can run concurrently on a multi-GPU server.
Given a potentially large number of learners, we take advantage of the fact that some learners reside on the same GPU and access shared memory, which is at least an order of magnitude faster than PCIe for inter-GPU communication. Rather than aggregating all differences in a single step, we therefore separate synchronisation on the intra-GPU and inter-GPU level.
We organise synchronisation as illustrated in Figure 6. To synchronise the learners executing on a single GPU, one learner is chosen to manage a reference model. Each learner then computes the difference between its model replica and the local reference model. This difference is then applied to the respective replica. At the global level, the SMA algorithm is executed. It uses one of the local reference models as the central average model ( in Alg. 1); all other reference models (one per GPU) are the replicas incorporated into the model averaging process ( in Alg. 1).
3.4 Choosing the number of learners
The number of learners per GPU or, put differently, the number of model replicas to be trained in parallel, is an important parameter. It must be chosen carefully for a given batch size: when training too few replicas, a GPU is under-utilised, wasting resources; when training too many, the execution of otherwise independent learners is partially sequentialised on a GPU, leading to a slow-down.
We propose to tune the number of learners per GPU based on the training throughput at runtime. By observing the number of processed batches per second, we can detect under- and over-utilisation of a GPU during training. As described in Alg. 2, we initially use a single learner per GPU (Alg. 2). For each GPU, we then consider the learning throughput (lines 2–2): if a significant increase in throughput is observed, the increase is a above a predefined tolerance threshold , a new learner is added to the respective GPU (Alg. 2). Upon observing a decrease in throughput, we reduce the number of learners again (Alg. 2).
Changing the number of learners is also beneficial in terms of statistical efficiency. Having initially few learners reduces the noise of stochastic gradients, fostering convergence of the reference models and thus the central average model. Eventually, though, this hampers the optimality of convergence as a smaller part of the loss space is explored.222Similar observations have been made regarding dynamic changes of the batch size  and learning rates . By increasing the parallelism gradually, we avoid this issue. Intuitively, a small initial number of learners allows the central average model to reach the neighbourhood of the solution quickly, which is then comprehensively explored with increased training parallelism.
4 Crossbow System Design
To support the training of deep learning models using SMA, the design of Crossbow has several unique features:
(1) Since we train multiple learners per GPU, Crossbow must share GPUs efficiently. Crossbow executes learners concurrently on a GPU by scheduling each to run on a separate GPU stream.
(2) We decide on the number of learners per GPU at runtime. The design of Crossbow must support changing the number of learners per GPU dynamically based on the available GPU resources.
(3) SMA synchronises all learners when they access the central average model. The design of Crossbow must implement this global synchronisation operation efficiently, and exploit concurrency during synchronisation to avoid bottlenecks.
Next we introduce the main component of Crossbow’s design in Section 4.1. Based on the above requirements, we then describe hows tasks execute (Section 4.2) and are scheduled (Section 4.3), and how the number of learners is tuned dynamically (Section 4.4). We finish with an explanation of memory management in Section 4.5.
4.1 System overview
Figure 7 shows the main components of Crossbow:
(1) The data pre-processors read the training dataset into memory and arrange samples into batches, possibly after some transformations such as image decoding and cropping.
(2) The task manager controls the pools of model replicas, input batches and learner streams. It handles task completion events originating from the GPUs.
(3) The task scheduler assigns learning tasks to GPUs based on the available resources. It also triggers synchronisation operations at the end of each iteration.
(4) The auto-tuner monitors the training throughput and creates new learners on a GPU on-demand.
Figure 7 shows how Crossbow executes an iteration of SMA: the data pre-processors populate the input batch pool with pointers to data, one complete batch at a time (step 1). The task scheduler checks if a model replica and a co-located learner stream are available (step 2) and then schedules a learning task (step 3). Upon completion, the task manager handles the event and returns the learner stream and the model replica to the pool (step 4). It also frees up a slot in the input batch pool, to be populated by one of the data pre-processors (step 1). The auto-tuner monitors the rate at which learning tasks complete and interrupts the training by adding a new learner (step 5).
4.2 Task execution
Crossbow trains a deep learning model by executing a dataflow graph, as shown in Figure 8. The dataflow graph consists of a set of learning tasks interleaved with synchronisation tasks. Crossbow represents the processing layers of a deep learning model as a graph of operators (a convolution or a matrix multiplication). A learning task encapsules multiple operators (see a). It takes as input a batch of training samples and a model replica and outputs a gradient. The task scheduler executes a learning task on any of the learner streams available on the GPU on which the replica used by that learning task resides.
Synchronisation tasks can be local or global: (i) a local synchronisation task computes the difference between a model replica and the central average model (see b). It uses the difference, together with the gradient of the corresponding learning task, to update the model replica. There is one local synchronisation tasks per replica on the same learner stream as the corresponding learning task; (ii) a global synchronisation task aggregates all local differences to update the central average model (see c). Crossbow allocates one average model replica per GPU, and uses a separate stream, the synchronisation stream, to execute global synchronisation tasks.
Placing a global execution barrier between the synchronisation and all preceding learning tasks would be expensive: it would block the task scheduler and delay the GPUs already waiting for tasks to complete. Instead, the task scheduler overlaps the synchronisation tasks from one iteration with the learning tasks of the next.
When overlapping tasks, the task scheduler considers data dependencies between tasks (see d). The horizontal direction in the figure represents time, and the vertical direction a spatial GPU partitioning based on streams. A line connecting two tasks denotes that the task on the right cannot start until the one on the left has completed.
Within an iteration, all local synchronisation tasks can execute concurrently (see e). They only require read-only access to the central average model in order to compute the difference from each model replica. The local synchronisation tasks, however, depend on the average model being updated consistently on each GPU by the global synchronisation tasks of the previous iteration.
Global synchronisation tasks can also execute concurrently within an iteration (see Section 3.3). The intra-GPU operations of a global synchronisation task execute as soon as their dependency to a local synchronisation task is satisfied (see d). The inter-GPU operations are implemented as a collective all-reduce primitive . All-reduce creates a ring topology in which each GPU exchanges data partitions with its peers. A GPU reduces the partition data that it receives by combining it with its own, and eventually every GPU holds the final aggregated data. As a result, all-reduce evenly distributes the computation of the update for the average model among the GPUs.
The global synchronisation tasks in one iteration can execute concurrently with the learning tasks in the next (see f). Once the local synchronisation tasks have completed, each replica is updated independently. At this point, the task scheduler can issue the next learning task to the learner stream without waiting for other tasks to complete (see g).
4.3 Task scheduling
In each iteration of SMA
, the task scheduler schedules one learning task for each model replica in the pool, followed by synchronisation tasks. As the task manager returns newly-updated model replicas to the pool, the task scheduler schedules further learning tasks and associates the next batch with a model replica on a first-come, first-served basis. Compared to round-robin scheduling, as used by PyTorch or TensorFlow , this improves hardware efficiency because the task scheduler does not need to wait for a given replica to become available. After an assignment, the task scheduler hands the learning task over to one of its worker threads, in particular, one that runs on the same socket as the task’s designated GPU. The worker thread issues the task to one of the GPU’s streams as a sequence of kernels. All kernel calls to the GPU are non-blocking, and the thread returns immediately to schedule the next task.
A challenge for the task scheduler is to ensure that tasks that can run concurrently are executed concurrently by a GPU. As there are multiple streaming multi-processors (SMs) per GPU with no shared resources among them, a GPU can execute multiple tasks concurrently by assigning them to different sets of SMs. This favours our approach of training with small batch sizes because the kernels of a learning task usually require few SMs. As a solution, the task scheduler assigns concurrent tasks to different GPU streams, which enables the GPU to make efficient internal scheduling decisions: tasks submitted to the same stream execute in issue order; tasks on different streams may execute concurrently. A worker thread issues the task’s kernels to its assigned streams, along with any event generators or handlers.
The task scheduler uses GPU events to respect any data dependencies between submitted tasks. If there is a dependency between tasks and , the task scheduler submits an event generator after completes and an event handler before begins. When the event generator on ’s stream executes, it signals ’s handler that all preceding tasks on that stream, included, have completed; when the event handler on ’s stream executes, all subsequent tasks on that stream block until it receives a signal from ’s event generator. Analogous to the task scheduler, the task manager also uses multiple threads to handle task completion events in parallel, returning model replicas return to the pool in a timely manner.
4.4 Tuning learners
The auto-tuner changes the number of learners per GPU at runtime without prior knowledge of the trained model (the number and computational complexity of each model layer) or the training environment (the number and capability of each GPU). To this end, it implements the adaptation procedure introduced in Section 3.4 and formalised in Alg. 2.
The auto-tuner measures the training throughput by considering the rate at which learning tasks complete, as recorded by the task manager. As defined in Alg. 2, the number of learners per GPU is then increased or decreased based on the observed change in training throughput. Note that, on a server with homogeneous GPUs, the auto-tuner may measure only the throughput of a single GPU to adapt the number of learners for all GPUs.
The auto-tuner initiates the creation of a new learner after the learning and synchronisation tasks of the current iteration have been scheduled. Adding a learner to a GPU requires allocating a new model replica and a corresponding learner stream. The auto-tuner places temporarily a global execution barrier between two successive iterations (step f in Figure 8), avoiding overlap with other tasks. The new model replica is initialised with the latest value of the average model. The auto-tuner also locks the resources pools, preventing access by the task scheduler or manager during resizing.
Even for large models, such as ResNet-50, auto-tuning completes within milliseconds. The main overhead comes from the memory allocation and the initialisation of the model weights. Since model weights and their gradients are kept in contiguous memory, a single allocation call suffices.
|Model||Dataset||Input size (MB)||# Ops||Model size (MB)|
4.5 Memory management
Data pre-processors transfer the input data from CPU to GPU memory using direct memory access (DMA). They write the pre-processed training samples to a page-aligned, page-locked circular buffer whose memory range is registered to the GPU’s address space. This memory can be read directly by the GPU with higher bandwidth than unregistered, pageable memory. The buffer size must accommodate at least one input batch per learner (enough for a complete iteration of SMA). Crossbow uses double buffering to create a pipeline between data pre-processors and the task scheduler. When the pre-processors stall the pipeline because it takes more time to prepare the data on the CPU than to process it on a GPU, some or all of the input data transformations are scheduled on the GPUs as a preamble to each learning task.
Deep learning models require more memory to store the output of their dataflow operators than the model itself. For example, the ResNet-50 model is 97.5MB in size but consumes 7.5GB of memory to store the outputs from 384 operators. The memory requirement scales linearly with the batch size, as well as the number of learners. Crossbow must therefore reduce the memory footprint of each learner when training multiple of them per GPU.
and Super-Neurons, devises an offline memory plan to reuse the output buffers of operators using reference counters. During initialisation, Crossbow traverses the operators of a learning task. When visiting an operator, it considers preceding operators and reuses an output buffer if the reference count is zero; otherwise it assumes that a new output buffer must be created. To account for data dependencies, it then decrements the reference counter of the operator’s inputs and increments the counters of the operator’s output. Such an offline plan reduces the memory footprint of a learner by up to 50% because outputs are mostly reused during the backwards phase.
When executing multiple learners per GPU, however, replicating the offline plan for each learner would over-allocate memory. Crossbow exploits that, in practice, not all instances of the same operator would execute concurrently. This enables the sharing of some of the output buffers among learners on the same GPU using an online memory plan. For each operator, the task scheduler maintains a pool of output buffer pointers to GPU memory. Pools are shared by all learners on the same GPU. At runtime, when the task scheduler considers an operator for execution, it reuses the first available buffer in the output buffer pool; if none are available, it allocates a new one. The task scheduler increments the reference counter of the operator’s output according to its data dependencies and issues the kernel on a stream. When the operator completes, the task manager decrements the reference counter of the operator’s input and output buffers. A buffer with zero references returns to the pool, and it can be reused by other learning tasks.
In this section, we evaluate the performance of our Crossbow prototype when training on a multi-GPU server. We begin by comparing its behaviour against TensorFlow  using four deep learning macro-benchmarks, as we vary the number of GPUs and the number of learners per GPU (Section 5.2). Next, we assess the impact of Crossbow’s core features through a set of micro-benchmarks: we explore the benefits of executing multiple learners per GPU in terms of statistical and hardware efficiency (Section 5.3); and we measure the ability of the auto-tuning mechanism to identify the best number of learners per GPU (Section 5.4) and the impact of the model averaging used in SMA (Section 5.5). Finally, we quantify the efficiency of the synchronisation task implementation (Section 5.6).
5.1 Experimental set-up
All experiments are conducted on a server with two Intel Xeon E5-2650 v3 2.3GHz CPUs (20 CPU cores in total) and 256GB of RAM. The server has 8 GeForce GTX Titan X (Pascal) GPUs, each with 3,072 cores and 12GB of RAM, connected via PCIe 3.0 (16). It runs Linux kernel 4.4 with the NVIDIA driver 367.57 and CUDA 8.0 with the cuDNN 6.0 library.
As a baseline, we use TensorFlow version 1.4, and we compare its performance against Crossbow using TensorFlow’s suite of benchmarks  shown in Table 1. We select a mix of models, representing different sizes and shapes of networks. This includes both small (LeNet) and large networks (ResNet-50) as well as deep and low-dimension networks (ResNet-32 and ResNet-50) and shallow and high-dimension ones (VGG). To enable a fair comparison, we configure both systems with the same data augmentation, model variable initialisation and hyper-parameter settings. Following common practices for training, in ResNet-32, the learning rate is multiplied by 0.1 at epochs 80 and 120 ; in VGG, the learning rate is halved every 20 epochs .
In our experiments, we vary two main parameters: the batch size per learner () and the number of model replicas (). Our main metric is the time-to-accuracy , defined as the time taken for the median test accuracy of the last 5 epochs to be equal or above a given threshold . For each of the four deep learning models, we choose different thresholds based on the highest accuracy reached by TensorFlow in our set-up. According to the results in Figure 9, we select the following thresholds: 99% (LeNet), 88% (ResNet-32), 69% (VGG-16) and 53% (ResNet-50). Higher accuracies can be achieved by leveraging dynamic hyper-parameter tuning. As discussed in §2.4, these techniques, however, are model- and architecture-specific and lack generality. In contrast, the goal of our evaluation is to compare the different approaches underlying Crossbow and TensorFlow under uniform settings.
Figure 10 shows the performance of Crossbow and TensorFlow when training the four deep learning models. For ResNet-32 and VGG (Figures 9(b) and 9(a)), we scale the number of GPUs from 1 to 8; for ResNet-50 (Figure 9(c)), we show the results only for 8 GPUs due to the long time required to train with fewer GPUs (with 1 GPU, it would take more than 5 days with TensorFlow); for LeNet (Figure 9(d)), we only run the experiment with 1 GPU because, given the small model size, training on multiple GPUs leads to an increase in training time due to the synchronisation overhead.
First we consider the performance of Crossbow when using only one learner per GPU, (black bar in Figure 10). In this configuration, for ResNet-32 and VGG, Crossbow achieves a performance comparable or slightly worse than TensorFlow when training on a small number of GPUs (1 or 2). The reason is that both ResNet and VGG are relatively compute-intensive models. With few learners, the synchronisation overhead is limited and, hence, the benefits of SMA (§3) and the fast task scheduling (§4.3) are less relevant. As we increase the number of GPUs to 4 or 8, the number of learners and the amount of synchronisation among them increases proportionally. The results now highlight the higher performance of SMA compared to TensorFlow’s S-SGD scheme, with up to a 72% reduction in for VGG with 8 GPUs (and 7% for ResNet-32, respectively). A similar improvement can be observed for ResNet-50 in Figure 9(c), with Crossbow achieving a 18% reduction in with 8 GPUs.
Crossbow, however, can offer a benefit even when training on a single GPU (one learner in total) if the model is not compute-intensive. For LeNet, each learning task takes 1ms or less (compared to 220ms for ResNet-50) and, hence, the scheduling overhead becomes more critical. By leveraging its efficient task scheduler, Crossbow yields a significant reduction (43%) compared to TensorFlow with one learner (see Figure 9(d)).
Thus far, we have focused on a Crossbow configuration with . A key advantage of Crossbow is its ability to increase hardware efficiency without affecting statistical efficiency by adding more learners per GPU (). In this case, Crossbow significantly improves performance even when training with few GPUs. For ResNet-32, Crossbow with achieves a 46% reduction with 1 GPU and a 24% reduction with and 8 GPUs; for VGG, the reduction is 10% for 1 GPU and 77% for 8 GPUs. Similar benefits with also exist for ResNet-50 (33% reduction, corresponding to 5hours) and LeNet (63% reduction). As explained in §3, the reason is that increases the GPU throughput without necessitating a larger batch size that would affect statistical efficiency. For ResNet-50, Crossbow with uses a small batch size of compared to an aggregate batch size of for TensorFlow (64 and 1,024 for ResNet-32, respectively).
While we choose to use the highest accuracy reached by TensorFlow as the threshold in the metric, similar improvements hold for other accuracy thresholds. In Figure 11, we plot over time for both ResNet-32 and VGG with 1 and 8 GPUs, respectively. Crossbow achieves high accuracy within a few minutes: with 8 GPUs, Crossbow takes 92seconds to exceed a 80% accuracy for ResNet-32 compared to 252seconds for TensorFlow—a 63% reduction. Similarly, Crossbow achieves a 74% reduction for VGG. This indicates that SMA converges quickly to a region containing good minima.
5.3 Statistical and hardware efficiency
Crossbow with outperforms TensorFlow because it can increase hardware efficiency without negatively impacting statistical efficiency. As discussed in §2.4, the only way to improve hardware efficiency in TensorFlow is to increase the batch size but this comes at the cost of reduced statistical efficiency. In contrast, Crossbow can use the number of learners as an additional control parameter to increase hardware efficiency without having to resort to larger batch sizes. We investigate this in more detail in Figures 13 and 12, in which we show how hardware and statistical efficiency, and the resulting , are affected by when using 1 and 8 GPUs respectively. We only report the results for ResNet-32 () but similar trends hold for the other models. Compared to the experiments in Figure 9(a), we lower the target accuracy for
to 80% as otherwise the results would be skewed by the change in the learning rate at epoch 80 (see §5.1).
When training with 1 GPU, using increases the throughput by a factor of 1.4 compared to the case with a single learner (Figure 11(a)) because the multiple learners fully utilise a GPU. Interestingly, this improves statistical efficiency as well—the number of epochs required to converge drop from 30 () to 14 () (see Figure 11(b)). This is because multiple model replicas can explore a larger portion of the space in parallel while the average model can reduce the variance among them, thus requiring fewer epochs to find good minima. As a result of the higher hardware and statistical efficiencies, the is also reduced by 3.2 (Figure 11(c)).
In contrast, the behaviour with 8 GPUs is somewhat different. While yields higher throughput (1.3), increasing the number of learners to does not further improve the throughput (Figure 12(a)). The reason is that, with 8 GPUs and 4 learners per GPU, the overall number of learners is 32, which introduces a significant amount of synchronisation overhead. In terms of statistical efficiency (Figure 12(b)), increasing the number of learners to does not significantly affect the number of epochs to converge but increasing it further leads to reduced statistical efficiency—with 32 learners in total, there is not enough stochastic noise in the training process, which makes it harder for the average model to escape bad minima. In this case, represents the best trade-off because it allows for higher hardware efficiency without noticeably worsening statistical efficiency. Indeed, this configuration reduces training time by 1.3 (Figure 12(c)).
These results show that increasing the number of learners per GPU is beneficial to reduce the training time. However, identifying the correct number of learners is crucial to achieving the best performance, as we show next.
5.4 Selecting the number of learners
To select the number of learners per GPU , Crossbow progressively increases until the throughput (expressed as images processed per second) stops improving (see Section 3.4). To validate this approach, Figure 14 shows the and throughput achieved for increasing values of when training ResNet-32 and VGG with 1 and 8 GPUs, respectively.
For ResNet-32, we observe that, with 1 GPU, the throughput grows until and then decreases; with 8 GPU, the throughput improves until and then remains relatively constant for . As predicted by our auto-tuning technique, the lowest is achieved with for 1 GPU and for 8 GPUs, respectively. Similarly, for VGG, the throughput plateaus at for 1 GPU and for 8 GPUs, respectively, which correspond to the lowest values of . This demonstrates the ability of Crossbow’s auto-tuning technique to identify quickly the correct number of learners in order to minimise .
5.5 Synchronisation model
Now we turn our attention to SMA, the synchronisation approach used by Crossbow, by comparing it to elastic averaging SGD (EA-SGD) . To quantify the impact of momentum in SMA, we compare the performance of Crossbow when training ResNet-32 using SMA against the performance using EA-SGD for an increasing number of GPUs.
Figure 15 shows that SMA reduces compared to EA-SGD by 9% with 1 GPU and and by 61% with 8 GPUs and (16 learners). The reason why the gap increases with the number of GPUs is that the more learners are used, the lower the (asymptotic) variance of the average model becomes, which makes it hard to escape from local minima. Therefore, without including momentum, the average model converges more slowly.
Crossbow synchronises the different replicas with the average model in each iteration. The authors of EA-SGD propose to synchronise every iterations to reduce the communication overhead. We study the impact of this optimisation on Crossbow in Figure 16. While less frequent synchronisation () increases the overall throughput (up to 31% for compared to ), it negatively affects convergence speed, resulting in a higher (53% longer with compared to ). Therefore, we always use .
5.6 Synchronisation efficiency
In the previous set of experiments, we focused on the algorithmic benefits of the SMA synchronisation approach. Now, instead, we want to understand the performance of our synchronisation implementation. Similar to Figure 16, we conduct an experiment in which we measure the throughput achieved by Crossbow for increasing values of , including also the case with no synchronisation at all. Since we are only interested in the performance aspect, we do not report the (we have already shown that yields the shortest ). The goal of this experiment is rather to observe the increase in throughput as we reduce the synchronisation frequency. The rationale is that, if synchronisation incurred a high cost, the throughput would drastically increase as we reduce the amount of synchronisation.
Contrary to this expectation, the results in Figure 17 show that throughput is only marginally higher, growing from 15,500images/s with to 18,500images/s with no synchronisation at all (20%) with (27% for , respectively). This indicates that Crossbow’s synchronisation implementation is well-optimised and introduces only a modest overhead.
6 Related Work
Training with multiple GPUs Machine learning systems use data parallelism, model parallelism, or a mix or both (see DistBelief and, more recently, FlexFlow ) to decrease training time. TensorFlow , PyTorch , MXNet , CNTK  and Caffe2  exploit data parallelism by default and use S-SGD as their de-facto training algorithm. S-SGD, however, couples the batch size and the number of GPUs. To compensate for the loss of statistical efficiency incurred by large batch sizes, users tune other hyper-parameters using techniques such as auto-tuning [67, 24], scaling  and warming up [15, 31] the learning rate, auto-tuning the batch size , auto-tuning the momentum  and others. The effectiveness of these tuning techniques is problem-specific , and users invest substantial time to find a scalable set-up . Crossbow explores a different direction by decoupling the batch size and the number of GPUs. It provides a design for a task engine that can fully utilise a multi-GPU server even when the batch size is small.
Increasing GPU utilisation There are proposals to improve the hardware utilisation of machine learning systems using cooperative scheduling. ModelBatch  and NVIDIA’s Multi-Process Service train multiple deep learning models on a GPU in a cooperative manner, but the problem of model synchronisation remains unresolved. Litz  explores the scheduling of training and synchronisation tasks on CPUs, but its lack of GPU support makes it ineffective for deep learning. Ray  trains deep learning models using cooperative GPU tasks but only shares GPUs using time-sharing. In contrast, Crossbow provides efficient concurrent execution of learning and synchronisation tasks on GPUs, which is the key to achieve high hardware efficiency when training deep learning models with small batch sizes.
Asynchronous training Prior work also attempts to improve hardware utilisation using asynchronous SGD , often at the expense of statistical efficiency . Hogwild!  and Dogwild!  do not specify data dependencies between learning and synchronisation tasks: all workers access a central model concurrently, leading to higher training throughput. To compensate for the loss in statistical efficiency, DimmWitted  coordinates parallel replicas in a CPU-oriented NUMA architecture. Each NUMA node has its own model replica shared by its cores. Within a node, cores update a shared replica asynchronously. In contrast to these efforts, Crossbow follows a synchronous training approach and therefore does not compromise statistical efficiency with stale updates.
Model averaging was originally proposed as an asynchronous method to distribute training. Polyak-Ruppert’s averaged SGD [48, 47, 54] first demonstrated that an average model can asymptotically converge faster to a solution than the individual model replicas used to compute it. In practice, it is difficult to find this asymptotic region , especially with models that have complex loss spaces . To improve the statistical efficiency of model averaging, recent studies [6, 36] propose to use the average model to correct the trajectory of model replicas, but the effectiveness of this approach was shown only for non-deep-learning problems.
In deep learning, elastic averaging SGD (EA-SGD)  uses the average model to correct model replicas occasionally, keeping the communication cost low. Asynchronous decentralised SGD (AD-SGD)  further reduces server communication traffic by requiring each replica to perform model averaging with only one worker per iteration. Compared to these techniques, SMA is a synchronous algorithm that shares and maintains a consistent view of the central average model across all learners in each iteration. SMA further improves the statistical efficiency of model averaging by adopting momentum (see Section 3.2) to correct the average model. When there are changes to hyper-parameters during training, SMA also restarts the averaging process as an effective way to preserve statistical efficiency.
Distributed training When scaling the training of deep learning models in distributed clusters, a parameter server (PS)  design is the de-facto approach. In contrast to Crossbow, which improves the training performance with small batch sizes on a single multi-GPU server, PS-based systems address the challenges of using a cluster for distributed learning, including the handling of elastic and heterogeneous resources [22, 27], the mitigation of stragglers [13, 11, 16], the acceleration of synchronisation using hybrid hardware , and the avoidance of resource fragmentation using collective communication [63, 56, 24]. Similar to prior model averaging systems , Crossbow could adopt a PS design to manage its average model in a distributed deployment. We view the distribution of Crossbow as future work.
Crossbow improves hardware efficiency when training with the preferred batch size, however small, with a low loss of statistical efficiency. It trains multiple model replicas on the same GPU, tuning their number automatically as to maximise training throughput. Despite training many more model replicas compared to existing approaches, Crossbow avoids reduced statistical efficiency using SMA. The latter is a new training algorithm in which replicas independently explore the solution space with gradient descent, but adjust their search synchronously based on the trajectory of a globally-consistent central average model. Our experimental results with a set of deep learning models show that Crossbow shortens the time-to-accuracy during training by up to 4 compared to TensorFlow.
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