| Writing Distributed Applications with PyTorch |
| ============================================= |
| **Author**: `Séb Arnold <https://seba1511.com>`_ |
|
|
| Prerequisites: |
|
|
| - `PyTorch Distributed Overview <../beginner/dist_overview.html>`__ |
|
|
| In this short tutorial, we will be going over the distributed package |
| of PyTorch. We'll see how to set up the distributed setting, use the |
| different communication strategies, and go over some the internals of |
| the package. |
|
|
| Setup |
| ----- |
|
|
| .. raw:: html |
|
|
| <!-- |
| * Processes & machines |
| * variables and init_process_group |
| --> |
|
|
| The distributed package included in PyTorch (i.e., |
| ``torch.distributed``) enables researchers and practitioners to easily |
| parallelize their computations across processes and clusters of |
| machines. To do so, it leverages messaging passing semantics |
| allowing each process to communicate data to any of the other processes. |
| As opposed to the multiprocessing (``torch.multiprocessing``) package, |
| processes can use different communication backends and are not |
| restricted to being executed on the same machine. |
|
|
| In order to get started we need the ability to run multiple processes |
| simultaneously. If you have access to compute cluster you should check |
| with your local sysadmin or use your favorite coordination tool. (e.g., |
| `pdsh <https://linux.die.net/man/1/pdsh>`__, |
| `clustershell <https://cea-hpc.github.io/clustershell/>`__, or |
| `others <https://slurm.schedmd.com/>`__) For the purpose of this |
| tutorial, we will use a single machine and fork multiple processes using |
| the following template. |
|
|
| .. code:: python |
|
|
| """run.py:""" |
| |
| import os |
| import torch |
| import torch.distributed as dist |
| from torch.multiprocessing import Process |
|
|
| def run(rank, size): |
| """ Distributed function to be implemented later. """ |
| pass |
|
|
| def init_process(rank, size, fn, backend='gloo'): |
| """ Initialize the distributed environment. """ |
| os.environ['MASTER_ADDR'] = '127.0.0.1' |
| os.environ['MASTER_PORT'] = '29500' |
| dist.init_process_group(backend, rank=rank, world_size=size) |
| fn(rank, size) |
|
|
|
|
| if __name__ == "__main__": |
| size = 2 |
| processes = [] |
| for rank in range(size): |
| p = Process(target=init_process, args=(rank, size, run)) |
| p.start() |
| processes.append(p) |
|
|
| for p in processes: |
| p.join() |
|
|
| The above script spawns two processes who will each setup the |
| distributed environment, initialize the process group |
| (``dist.init_process_group``), and finally execute the given ``run`` |
| function. |
|
|
| Let's have a look at the ``init_process`` function. It ensures that |
| every process will be able to coordinate through a master, using the |
| same ip address and port. Note that we used the ``gloo`` backend but |
| other backends are available. (c.f. |
| `Section 5.1 <#communication-backends>`__) We will go over the magic |
| happening in ``dist.init_process_group`` at the end of this tutorial, |
| but it essentially allows processes to communicate with each other by |
| sharing their locations. |
|
|
| Point-to-Point Communication |
| ---------------------------- |
|
|
| .. figure:: /_static/img/distributed/send_recv.png |
| :width: 100% |
| :align: center |
| :alt: Send and Recv |
|
|
| Send and Recv |
|
|
|
|
| A transfer of data from one process to another is called a |
| point-to-point communication. These are achieved through the ``send`` |
| and ``recv`` functions or their *immediate* counter-parts, ``isend`` and |
| ``irecv``. |
|
|
| .. code:: python |
|
|
| """Blocking point-to-point communication.""" |
|
|
| def run(rank, size): |
| tensor = torch.zeros(1) |
| if rank == 0: |
| tensor += 1 |
| |
| dist.send(tensor=tensor, dst=1) |
| else: |
| |
| dist.recv(tensor=tensor, src=0) |
| print('Rank ', rank, ' has data ', tensor[0]) |
|
|
| In the above example, both processes start with a zero tensor, then |
| process 0 increments the tensor and sends it to process 1 so that they |
| both end up with 1.0. Notice that process 1 needs to allocate memory in |
| order to store the data it will receive. |
|
|
| Also notice that ``send``/``recv`` are **blocking**: both processes stop |
| until the communication is completed. On the other hand immediates are |
| **non-blocking**; the script continues its execution and the methods |
| return a ``Work`` object upon which we can choose to |
| ``wait()``. |
|
|
| .. code:: python |
|
|
| """Non-blocking point-to-point communication.""" |
|
|
| def run(rank, size): |
| tensor = torch.zeros(1) |
| req = None |
| if rank == 0: |
| tensor += 1 |
| |
| req = dist.isend(tensor=tensor, dst=1) |
| print('Rank 0 started sending') |
| else: |
| |
| req = dist.irecv(tensor=tensor, src=0) |
| print('Rank 1 started receiving') |
| req.wait() |
| print('Rank ', rank, ' has data ', tensor[0]) |
|
|
| When using immediates we have to be careful about with our usage of the sent and received tensors. |
| Since we do not know when the data will be communicated to the other process, |
| we should not modify the sent tensor nor access the received tensor before ``req.wait()`` has completed. |
| In other words, |
|
|
| - writing to ``tensor`` after ``dist.isend()`` will result in undefined behaviour. |
| - reading from ``tensor`` after ``dist.irecv()`` will result in undefined behaviour. |
|
|
| However, after ``req.wait()`` |
| has been executed we are guaranteed that the communication took place, |
| and that the value stored in ``tensor[0]`` is 1.0. |
|
|
| Point-to-point communication is useful when we want a fine-grained |
| control over the communication of our processes. They can be used to |
| implement fancy algorithms, such as the one used in `Baidu's |
| DeepSpeech <https://github.com/baidu-research/baidu-allreduce>`__ or |
| `Facebook's large-scale |
| experiments <https://research.fb.com/publications/imagenet1kin1h/>`__.(c.f. |
| `Section 4.1 <#our-own-ring-allreduce>`__) |
|
|
| Collective Communication |
| ------------------------ |
|
|
| +----------------------------------------------------+-----------------------------------------------------+ |
| | .. figure:: /_static/img/distributed/scatter.png | .. figure:: /_static/img/distributed/gather.png | |
| | :alt: Scatter | :alt: Gather | |
| | :width: 100% | :width: 100% | |
| | :align: center | :align: center | |
| | | | |
| | Scatter | Gather | |
| +----------------------------------------------------+-----------------------------------------------------+ |
| | .. figure:: /_static/img/distributed/reduce.png | .. figure:: /_static/img/distributed/all_reduce.png | |
| | :alt: Reduce | :alt: All-Reduce | |
| | :width: 100% | :width: 100% | |
| | :align: center | :align: center | |
| | | | |
| | Reduce | All-Reduce | |
| +----------------------------------------------------+-----------------------------------------------------+ |
| | .. figure:: /_static/img/distributed/broadcast.png | .. figure:: /_static/img/distributed/all_gather.png | |
| | :alt: Broadcast | :alt: All-Gather | |
| | :width: 100% | :width: 100% | |
| | :align: center | :align: center | |
| | | | |
| | Broadcast | All-Gather | |
| +----------------------------------------------------+-----------------------------------------------------+ |
|
|
|
|
|
|
| As opposed to point-to-point communcation, collectives allow for |
| communication patterns across all processes in a **group**. A group is a |
| subset of all our processes. To create a group, we can pass a list of |
| ranks to ``dist.new_group(group)``. By default, collectives are executed |
| on the all processes, also known as the **world**. For example, in order |
| to obtain the sum of all tensors at all processes, we can use the |
| ``dist.all_reduce(tensor, op, group)`` collective. |
|
|
| .. code:: python |
|
|
| """ All-Reduce example.""" |
| def run(rank, size): |
| """ Simple point-to-point communication. """ |
| group = dist.new_group([0, 1]) |
| tensor = torch.ones(1) |
| dist.all_reduce(tensor, op=dist.reduce_op.SUM, group=group) |
| print('Rank ', rank, ' has data ', tensor[0]) |
|
|
| Since we want the sum of all tensors in the group, we use |
| ``dist.reduce_op.SUM`` as the reduce operator. Generally speaking, any |
| commutative mathematical operation can be used as an operator. |
| Out-of-the-box, PyTorch comes with 4 such operators, all working at the |
| element-wise level: |
|
|
| - ``dist.reduce_op.SUM``, |
| - ``dist.reduce_op.PRODUCT``, |
| - ``dist.reduce_op.MAX``, |
| - ``dist.reduce_op.MIN``. |
|
|
| In addition to ``dist.all_reduce(tensor, op, group)``, there are a total |
| of 6 collectives currently implemented in PyTorch. |
|
|
| - ``dist.broadcast(tensor, src, group)``: Copies ``tensor`` from |
| ``src`` to all other processes. |
| - ``dist.reduce(tensor, dst, op, group)``: Applies ``op`` to all |
| ``tensor`` and stores the result in ``dst``. |
| - ``dist.all_reduce(tensor, op, group)``: Same as reduce, but the |
| result is stored in all processes. |
| - ``dist.scatter(tensor, src, scatter_list, group)``: Copies the |
| :math:`i^{\text{th}}` tensor ``scatter_list[i]`` to the |
| :math:`i^{\text{th}}` process. |
| - ``dist.gather(tensor, dst, gather_list, group)``: Copies ``tensor`` |
| from all processes in ``dst``. |
| - ``dist.all_gather(tensor_list, tensor, group)``: Copies ``tensor`` |
| from all processes to ``tensor_list``, on all processes. |
| - ``dist.barrier(group)``: block all processes in `group` until each one has entered this function. |
|
|
| Distributed Training |
| -------------------- |
|
|
| .. raw:: html |
|
|
| <!-- |
| * Gloo Backend |
| * Simple all_reduce on the gradients |
| * Point to optimized DistributedDataParallel |
|
|
| TODO: Custom ring-allreduce |
| --> |
|
|
| **Note:** You can find the example script of this section in `this |
| GitHub repository <https://github.com/seba-1511/dist_tuto.pth/>`__. |
|
|
| Now that we understand how the distributed module works, let us write |
| something useful with it. Our goal will be to replicate the |
| functionality of |
| `DistributedDataParallel <https://pytorch.org/docs/stable/nn.html#torch.nn.parallel.DistributedDataParallel>`__. |
| Of course, this will be a didactic example and in a real-world |
| situation you should use the official, well-tested and well-optimized |
| version linked above. |
|
|
| Quite simply we want to implement a distributed version of stochastic |
| gradient descent. Our script will let all processes compute the |
| gradients of their model on their batch of data and then average their |
| gradients. In order to ensure similar convergence results when changing |
| the number of processes, we will first have to partition our dataset. |
| (You could also use |
| `tnt.dataset.SplitDataset <https://github.com/pytorch/tnt/blob/master/torchnet/dataset/splitdataset.py#L4>`__, |
| instead of the snippet below.) |
|
|
| .. code:: python |
|
|
| """ Dataset partitioning helper """ |
| class Partition(object): |
|
|
| def __init__(self, data, index): |
| self.data = data |
| self.index = index |
|
|
| def __len__(self): |
| return len(self.index) |
|
|
| def __getitem__(self, index): |
| data_idx = self.index[index] |
| return self.data[data_idx] |
|
|
|
|
| class DataPartitioner(object): |
|
|
| def __init__(self, data, sizes=[0.7, 0.2, 0.1], seed=1234): |
| self.data = data |
| self.partitions = [] |
| rng = Random() |
| rng.seed(seed) |
| data_len = len(data) |
| indexes = [x for x in range(0, data_len)] |
| rng.shuffle(indexes) |
|
|
| for frac in sizes: |
| part_len = int(frac * data_len) |
| self.partitions.append(indexes[0:part_len]) |
| indexes = indexes[part_len:] |
|
|
| def use(self, partition): |
| return Partition(self.data, self.partitions[partition]) |
|
|
| With the above snippet, we can now simply partition any dataset using |
| the following few lines: |
|
|
| .. code:: python |
|
|
| """ Partitioning MNIST """ |
| def partition_dataset(): |
| dataset = datasets.MNIST('./data', train=True, download=True, |
| transform=transforms.Compose([ |
| transforms.ToTensor(), |
| transforms.Normalize((0.1307,), (0.3081,)) |
| ])) |
| size = dist.get_world_size() |
| bsz = 128 / float(size) |
| partition_sizes = [1.0 / size for _ in range(size)] |
| partition = DataPartitioner(dataset, partition_sizes) |
| partition = partition.use(dist.get_rank()) |
| train_set = torch.utils.data.DataLoader(partition, |
| batch_size=bsz, |
| shuffle=True) |
| return train_set, bsz |
|
|
| Assuming we have 2 replicas, then each process will have a ``train_set`` |
| of 60000 / 2 = 30000 samples. We also divide the batch size by the |
| number of replicas in order to maintain the *overall* batch size of 128. |
|
|
| We can now write our usual forward-backward-optimize training code, and |
| add a function call to average the gradients of our models. (The |
| following is largely inspired from the official `PyTorch MNIST |
| example <https://github.com/pytorch/examples/blob/master/mnist/main.py>`__.) |
|
|
| .. code:: python |
|
|
| """ Distributed Synchronous SGD Example """ |
| def run(rank, size): |
| torch.manual_seed(1234) |
| train_set, bsz = partition_dataset() |
| model = Net() |
| optimizer = optim.SGD(model.parameters(), |
| lr=0.01, momentum=0.5) |
|
|
| num_batches = ceil(len(train_set.dataset) / float(bsz)) |
| for epoch in range(10): |
| epoch_loss = 0.0 |
| for data, target in train_set: |
| optimizer.zero_grad() |
| output = model(data) |
| loss = F.nll_loss(output, target) |
| epoch_loss += loss.item() |
| loss.backward() |
| average_gradients(model) |
| optimizer.step() |
| print('Rank ', dist.get_rank(), ', epoch ', |
| epoch, ': ', epoch_loss / num_batches) |
|
|
| It remains to implement the ``average_gradients(model)`` function, which |
| simply takes in a model and averages its gradients across the whole |
| world. |
|
|
| .. code:: python |
|
|
| """ Gradient averaging. """ |
| def average_gradients(model): |
| size = float(dist.get_world_size()) |
| for param in model.parameters(): |
| dist.all_reduce(param.grad.data, op=dist.reduce_op.SUM) |
| param.grad.data /= size |
|
|
| *Et voilà*! We successfully implemented distributed synchronous SGD and |
| could train any model on a large computer cluster. |
|
|
| **Note:** While the last sentence is *technically* true, there are `a |
| lot more tricks <https://seba-1511.github.io/dist_blog>`__ required to |
| implement a production-level implementation of synchronous SGD. Again, |
| use what `has been tested and |
| optimized <https://pytorch.org/docs/stable/nn.html#torch.nn.parallel.DistributedDataParallel>`__. |
|
|
| Our Own Ring-Allreduce |
| ~~~~~~~~~~~~~~~~~~~~~~ |
|
|
| As an additional challenge, imagine that we wanted to implement |
| DeepSpeech's efficient ring allreduce. This is fairly easily implemented |
| using point-to-point collectives. |
|
|
| .. code:: python |
|
|
| """ Implementation of a ring-reduce with addition. """ |
| def allreduce(send, recv): |
| rank = dist.get_rank() |
| size = dist.get_world_size() |
| send_buff = send.clone() |
| recv_buff = send.clone() |
| accum = send.clone() |
|
|
| left = ((rank - 1) + size) % size |
| right = (rank + 1) % size |
|
|
| for i in range(size - 1): |
| if i % 2 == 0: |
| |
| send_req = dist.isend(send_buff, right) |
| dist.recv(recv_buff, left) |
| accum[:] += recv_buff[:] |
| else: |
| |
| send_req = dist.isend(recv_buff, right) |
| dist.recv(send_buff, left) |
| accum[:] += send_buff[:] |
| send_req.wait() |
| recv[:] = accum[:] |
|
|
| In the above script, the ``allreduce(send, recv)`` function has a |
| slightly different signature than the ones in PyTorch. It takes a |
| ``recv`` tensor and will store the sum of all ``send`` tensors in it. As |
| an exercise left to the reader, there is still one difference between |
| our version and the one in DeepSpeech: their implementation divide the |
| gradient tensor into *chunks*, so as to optimally utilize the |
| communication bandwidth. (Hint: |
| `torch.chunk <https://pytorch.org/docs/stable/torch.html#torch.chunk>`__) |
|
|
| Advanced Topics |
| --------------- |
|
|
| We are now ready to discover some of the more advanced functionalities |
| of ``torch.distributed``. Since there is a lot to cover, this section is |
| divided into two subsections: |
|
|
| 1. Communication Backends: where we learn how to use MPI and Gloo for |
| GPU-GPU communication. |
| 2. Initialization Methods: where we understand how to best setup the |
| initial coordination phase in ``dist.init_process_group()``. |
|
|
| Communication Backends |
| ~~~~~~~~~~~~~~~~~~~~~~ |
|
|
| One of the most elegant aspects of ``torch.distributed`` is its ability |
| to abstract and build on top of different backends. As mentioned before, |
| there are currently three backends implemented in PyTorch: Gloo, NCCL, and |
| MPI. They each have different specifications and tradeoffs, depending |
| on the desired use case. A comparative table of supported functions can |
| be found |
| `here <https://pytorch.org/docs/stable/distributed.html#module-torch.distributed>`__. |
|
|
| **Gloo Backend** |
|
|
| So far we have made extensive usage of the `Gloo backend <https://github.com/facebookincubator/gloo>`__. |
| It is quite handy as a development platform, as it is included in |
| the pre-compiled PyTorch binaries and works on both Linux (since 0.2) |
| and macOS (since 1.3). It supports all point-to-point and collective |
| operations on CPU, and all collective operations on GPU. The |
| implementation of the collective operations for CUDA tensors is not as |
| optimized as the ones provided by the NCCL backend. |
|
|
| As you have surely noticed, our |
| distributed SGD example does not work if you put ``model`` on the GPU. |
| In order to use multiple GPUs, let us also do the following |
| modifications: |
|
|
| 1. Use ``device = torch.device("cuda:{}".format(rank))`` |
| 2. ``model = Net()`` :math:`\rightarrow` ``model = Net().to(device)`` |
| 3. Use ``data, target = data.to(device), target.to(device)`` |
|
|
| With the above modifications, our model is now training on two GPUs and |
| you can monitor their utilization with ``watch nvidia-smi``. |
|
|
| **MPI Backend** |
|
|
| The Message Passing Interface (MPI) is a standardized tool from the |
| field of high-performance computing. It allows to do point-to-point and |
| collective communications and was the main inspiration for the API of |
| ``torch.distributed``. Several implementations of MPI exist (e.g. |
| `Open-MPI <https://www.open-mpi.org/>`__, |
| `MVAPICH2 <http://mvapich.cse.ohio-state.edu/>`__, `Intel |
| MPI <https://software.intel.com/en-us/intel-mpi-library>`__) each |
| optimized for different purposes. The advantage of using the MPI backend |
| lies in MPI's wide availability - and high-level of optimization - on |
| large computer clusters. `Some <https://developer.nvidia.com/mvapich>`__ |
| `recent <https://developer.nvidia.com/ibm-spectrum-mpi>`__ |
| `implementations <https://www.open-mpi.org/>`__ are also able to take |
| advantage of CUDA IPC and GPU Direct technologies in order to avoid |
| memory copies through the CPU. |
|
|
| Unfortunately, PyTorch's binaries can not include an MPI implementation |
| and we'll have to recompile it by hand. Fortunately, this process is |
| fairly simple given that upon compilation, PyTorch will look *by itself* |
| for an available MPI implementation. The following steps install the MPI |
| backend, by installing PyTorch `from |
| source <https://github.com/pytorch/pytorch#from-source>`__. |
|
|
| 1. Create and activate your Anaconda environment, install all the |
| pre-requisites following `the |
| guide <https://github.com/pytorch/pytorch#from-source>`__, but do |
| **not** run ``python setup.py install`` yet. |
| 2. Choose and install your favorite MPI implementation. Note that |
| enabling CUDA-aware MPI might require some additional steps. In our |
| case, we'll stick to Open-MPI *without* GPU support: |
| ``conda install -c conda-forge openmpi`` |
| 3. Now, go to your cloned PyTorch repo and execute |
| ``python setup.py install``. |
|
|
| In order to test our newly installed backend, a few modifications are |
| required. |
|
|
| 1. Replace the content under ``if __name__ == '__main__':`` with |
| ``init_process(0, 0, run, backend='mpi')``. |
| 2. Run ``mpirun -n 4 python myscript.py``. |
|
|
| The reason for these changes is that MPI needs to create its own |
| environment before spawning the processes. MPI will also spawn its own |
| processes and perform the handshake described in `Initialization |
| Methods <#initialization-methods>`__, making the ``rank``\ and ``size`` |
| arguments of ``init_process_group`` superfluous. This is actually quite |
| powerful as you can pass additional arguments to ``mpirun`` in order to |
| tailor computational resources for each process. (Things like number of |
| cores per process, hand-assigning machines to specific ranks, and `some |
| more <https://www.open-mpi.org/faq/?category=running#mpirun-hostfile>`__) |
| Doing so, you should obtain the same familiar output as with the other |
| communication backends. |
|
|
| **NCCL Backend** |
|
|
| The `NCCL backend <https://github.com/nvidia/nccl>`__ provides an |
| optimized implementation of collective operations against CUDA |
| tensors. If you only use CUDA tensors for your collective operations, |
| consider using this backend for the best in class performance. The |
| NCCL backend is included in the pre-built binaries with CUDA support. |
|
|
| Initialization Methods |
| ~~~~~~~~~~~~~~~~~~~~~~ |
|
|
| To finish this tutorial, let's talk about the very first function we |
| called: ``dist.init_process_group(backend, init_method)``. In |
| particular, we will go over the different initialization methods which |
| are responsible for the initial coordination step between each process. |
| Those methods allow you to define how this coordination is done. |
| Depending on your hardware setup, one of these methods should be |
| naturally more suitable than the others. In addition to the following |
| sections, you should also have a look at the `official |
| documentation <https://pytorch.org/docs/stable/distributed.html#initialization>`__. |
|
|
| **Environment Variable** |
|
|
| We have been using the environment variable initialization method |
| throughout this tutorial. By setting the following four environment |
| variables on all machines, all processes will be able to properly |
| connect to the master, obtain information about the other processes, and |
| finally handshake with them. |
|
|
| - ``MASTER_PORT``: A free port on the machine that will host the |
| process with rank 0. |
| - ``MASTER_ADDR``: IP address of the machine that will host the process |
| with rank 0. |
| - ``WORLD_SIZE``: The total number of processes, so that the master |
| knows how many workers to wait for. |
| - ``RANK``: Rank of each process, so they will know whether it is the |
| master of a worker. |
|
|
| **Shared File System** |
|
|
| The shared filesystem requires all processes to have access to a shared |
| file system, and will coordinate them through a shared file. This means |
| that each process will open the file, write its information, and wait |
| until everybody did so. After what all required information will be |
| readily available to all processes. In order to avoid race conditions, |
| the file system must support locking through |
| `fcntl <http://man7.org/linux/man-pages/man2/fcntl.2.html>`__. |
|
|
| .. code:: python |
|
|
| dist.init_process_group( |
| init_method='file:///mnt/nfs/sharedfile', |
| rank=args.rank, |
| world_size=4) |
|
|
| **TCP** |
|
|
| Initializing via TCP can be achieved by providing the IP address of the process with rank 0 and a reachable port number. |
| Here, all workers will be able to connect to the process |
| with rank 0 and exchange information on how to reach each other. |
|
|
| .. code:: python |
|
|
| dist.init_process_group( |
| init_method='tcp://10.1.1.20:23456', |
| rank=args.rank, |
| world_size=4) |
|
|
| .. raw:: html |
|
|
| <!-- |
| |
| * The magic behind init_process_group: |
|
|
| 1. validate and parse the arguments |
| 2. resolve the backend: name2channel.at() |
| 3. Drop GIL & THDProcessGroupInit: instantiate the channel and add address of master from config |
| 4. rank 0 inits master, others workers |
| 5. master: create sockets for all workers -> wait for all workers to connect -> send them each the info about location of other processes |
| 6. worker: create socket to master, send own info, receive info about each worker, and then handshake with each of them |
| 7. By this time everyone has handshake with everyone. |
| --> |
|
|
| .. raw:: html |
|
|
| <center> |
|
|
| **Acknowledgements** |
|
|
| .. raw:: html |
|
|
| </center> |
|
|
| I'd like to thank the PyTorch developers for doing such a good job on |
| their implementation, documentation, and tests. When the code was |
| unclear, I could always count on the |
| `docs <https://pytorch.org/docs/stable/distributed.html>`__ or the |
| `tests <https://github.com/pytorch/pytorch/blob/master/test/test_distributed.py>`__ |
| to find an answer. In particular, I'd like to thank Soumith Chintala, |
| Adam Paszke, and Natalia Gimelshein for providing insightful comments |
| and answering questions on early drafts. |
|
|