rocm_jax/docs/notebooks/jax_Array.md

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Parallelism with JAX

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This tutorial discusses parallelism via jax.Array, the unified array object model available in JAX v0.4.0 and newer.

The features required by jax.Array are not supported by the Colab TPU runtime at this time.

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import os

import functools
from typing import Optional

import numpy as np

import jax
import jax.numpy as jnp

jax.config.update('jax_array', True)

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⚠️ WARNING: The notebook requires multiple devices in order to run correctly.

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Sharding describes how array values are laid out in memory across devices

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Sharding basics, and the PositionalSharding subclass

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To parallelize computation across multiple devices, we first have to describe how the data that computation acts on can be distributed across multiple devices. That means describing distributed layouts conveniently.

In JAX, Sharding objects describe distributed memory layouts. They can be used with jax.device_put to produce a value with distributed layout.

For example, here's a value with a single-device Sharding:

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import jax
x = jax.random.normal(jax.random.PRNGKey(0), (8192, 8192))

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Here, we're using the jax.debug.visualize_array_sharding function to show where the value x is stored in memory. All of x is stored on a single device, so the visualization is pretty boring!

But we can shard x across multiple devices by using jax.device_put and a Sharding object. First, we make a numpy.ndarray of Devices using mesh_utils.create_device_mesh, which takes hardware topology into account for the Device order:

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jax.debug.visualize_array_sharding(x)

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A quick example of what jax.Array can do before we dive into more details:

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from jax.experimental import mesh_utils
from jax.sharding import PositionalSharding

# Let's create a Sharding object which we will use to distribute
# a value across devices.
sharding = PositionalSharding(mesh_utils.create_device_mesh((8,)))

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y = jax.device_put(x, sharding.reshape(4, 2))
jax.debug.visualize_array_sharding(y)

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We distributed x which was on a single device before to 8 devices.

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z = jnp.sin(y)
jax.debug.visualize_array_sharding(z)

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After doing an jnp.sin operation on a distributed jax.Array, the sharding on the output was preserved. Also the operation itself happened on multiple devices. To test that, we can do a timing experiment:

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# `x` is present on a single device
%timeit -n 5 -r 5 jnp.sin(x).block_until_ready()

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:outputId: da362eae-810a-438a-81e3-66c16698a456

# `y` is sharded across 8 devices.
%timeit -n 5 -r 5 jnp.sin(y).block_until_ready()

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We can now continue with the rest of the details of Shardings:

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from jax.experimental import mesh_utils
devices = mesh_utils.create_device_mesh((8,))

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Then, we create a PositionalSharding and use it with device_put:

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from jax.sharding import PositionalSharding

sharding = PositionalSharding(devices)

x = jax.device_put(x, sharding.reshape(8, 1))
jax.debug.visualize_array_sharding(x)

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Here sharding is a PositionalSharding which acts like an array with sets of devices as elements:

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sharding

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By writing PositionalSharding(ndarray_of_devices), we fix the device order and the initial shape. Then we can reshape it:

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sharding.reshape(8, 1)

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sharding.reshape(4, 2)

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To use device_put with a data array x, we can reshape the sharding into a shape that is congruent with x.shape, meaning a shape with the same length as x.shape and where each element evenly divides the corresponding element of x.shape:

def is_congruent(x_shape: Sequence[int], sharding_shape: Sequence[int]) -> bool:
  return (len(x_shape) == len(sharding_shape) and
          all(d1 % d2 == 0 for d1, d2 in zip(x_shape, sharding_shape)))

For example, we can reshape sharding to have shape (4, 2), then use it in a device_put:

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sharding = sharding.reshape(4, 2)
print(sharding)

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y = jax.device_put(x, sharding)
jax.debug.visualize_array_sharding(y)

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Here y represents the same value as x, but its shards (i.e. slices) are stored in different devices' memories.

Different sharding shapes result in different distributed layouts (i.e. shardings) of the result:

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sharding = sharding.reshape(1, 8)
print(sharding)

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y = jax.device_put(x, sharding)
jax.debug.visualize_array_sharding(y)

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In some cases, we don't just want to store each slice of x in a single device's memory; we might want to replicate some slices, meaning storing copies of a slice's values in multiple devices' memories.

With PositionalSharding, we can express replication by calling the reducer method replicate:

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sharding = sharding.reshape(4, 2)
print(sharding.replicate(axis=0, keepdims=True))

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y = jax.device_put(x, sharding.replicate(axis=0, keepdims=True))
jax.debug.visualize_array_sharding(y)

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Here the visualization shows that x is sharded two ways along its second dimension (and not sharded along the first dimension), and each of those shards is replicated four ways (i.e. stored in four device memories).

The replicate method acts similar to the familiar NumPy array reduction methods like .sum() and .prod(). It operates along an axis performing a set union. So if sharding has shape (4, 2), then sharding.replicate(0, keepdims=True) has shape (1, 2), and sharding.replicate(1, keepdims=True) has shape (4, 1). Unlike analogous NumPy methods, keepdims=True is actually the default, so reduced-over axes aren't squeezed:

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print(sharding.replicate(0).shape)
print(sharding.replicate(1).shape)

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y = jax.device_put(x, sharding.replicate(1))
jax.debug.visualize_array_sharding(y)

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NamedSharding gives a way to express shardings with names

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So far we've worked with PositionalSharding, but there are alternative ways to express shardings. In fact, Sharding is an interface, and any class that implements that interface can be used with functions like device_put.

Another convenient way to express sharding is with the NamedSharding:

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from jax.experimental import maps
from jax.experimental import PartitionSpec
from jax.experimental import mesh_utils
from jax.sharding import NamedSharding
P = PartitionSpec

devices = mesh_utils.create_device_mesh((4, 2))
mesh = maps.Mesh(devices, axis_names=('a', 'b'))
y = jax.device_put(x, NamedSharding(mesh, P('a', 'b')))
jax.debug.visualize_array_sharding(y)

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We can define a helper function to make things simpler:

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devices = mesh_utils.create_device_mesh((4, 2))
default_mesh = maps.Mesh(devices, axis_names=('a', 'b'))

def mesh_sharding(
    pspec: PartitionSpec, mesh: Optional[maps.Mesh] = None,
) -> NamedSharding:
  if mesh is None:
    mesh = default_mesh
  return NamedSharding(mesh, pspec)

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y = jax.device_put(x, mesh_sharding(P('a', 'b')))
jax.debug.visualize_array_sharding(y)

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Here, we use P('a', 'b') to express that the first and second axes of x should be sharded over the device mesh axes 'a' and 'b', respectively. We can easily switch to P('b', 'a') to shard the axes of x over different devices:

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y = jax.device_put(x, mesh_sharding(P('b', 'a')))
jax.debug.visualize_array_sharding(y)

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# `None` means that `x` is replicated on the 1st dimension (counting from 0).
y = jax.device_put(x, mesh_sharding(P('a', None)))
jax.debug.visualize_array_sharding(y)

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Here, because P('a', None) doesn't mention the Mesh axis name 'b', we get replication over the axis 'b'. The None here is just acting as a placeholder to line up against the second axis of the value x, without expressing sharding over any mesh axis. (As a shorthand, trailing Nones can be omitted, so that P('a', None) means the same thing as P('a'). But it doesn't hurt to be explicit!)

To shard only over the second axis of x, we can use a None placeholder in the PartitionSpec:

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y = jax.device_put(x, mesh_sharding(P(None, 'b')))
jax.debug.visualize_array_sharding(y)

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y = jax.device_put(x, mesh_sharding(P(None, 'a')))
jax.debug.visualize_array_sharding(y)

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For a fixed mesh, we can even partition one logical axis of x over multiple device mesh axes:

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y = jax.device_put(x, mesh_sharding(P(('a', 'b'), None)))
jax.debug.visualize_array_sharding(y)

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Using NamedSharding makes it easy to define a device mesh once and give its axes names, then just refer to those names in PartitionSpecs for each device_put as needed.

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Computation follows data sharding and is automatically parallelized

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With sharded data, the compiler can give us parallel computation. In particular, functions decorated with jax.jit can operate over sharded arrays without copying data onto a single device. Instead, computation follows sharding: based on the sharding of the input data, the compiler decides shardings for intermediates and output values, and parallelizes their evaluation, even inserting communication operations as necessary.

For example, the simplest computation is an elementwise one:

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from jax.experimental import mesh_utils
from jax.sharding import PositionalSharding
sharding = PositionalSharding(mesh_utils.create_device_mesh((8,)))

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x = jax.device_put(x, sharding.reshape(4, 2))
print('input sharding:')
jax.debug.visualize_array_sharding(x)

y = jnp.sin(x)
print('output sharding:')
jax.debug.visualize_array_sharding(y)

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Here for the elementwise operation jnp.sin the compiler chose the output sharding to be the same as the input. Moreover, the compiler automatically parallelized the computation, so that each device computed its output shard from its input shard in parallel.

In other words, even though we wrote the jnp.sin computation as if a single machine were to execute it, the compiler splits up the computation for us and executes it on multiple devices.

We can do the same for more than just elementwise operations too. Consider a matrix multiplication with sharded inputs:

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y = jax.device_put(x, sharding.reshape(4, 2).replicate(1))
z = jax.device_put(x, sharding.reshape(4, 2).replicate(0))
print('lhs sharding:')
jax.debug.visualize_array_sharding(y)
print('rhs sharding:')
jax.debug.visualize_array_sharding(z)

w = jnp.dot(y, z)
print('out sharding:')
jax.debug.visualize_array_sharding(w)

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Here the compiler chose the output sharding so that it could maximally parallelize the computation: without needing communication, each device already has the input shards it needs to compute its output shard.

How can we be sure it's actually running in parallel? We can do a simple timing experiment:

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x_single = jax.device_put(x, jax.devices()[0])
jax.debug.visualize_array_sharding(x_single)

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np.allclose(jnp.dot(x_single, x_single),
            jnp.dot(y, z))

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%timeit -n 5 -r 5 jnp.dot(x_single, x_single).block_until_ready()

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%timeit -n 5 -r 5 jnp.dot(y, z).block_until_ready()

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Even copying a sharded Array produces a result with the sharding of the input:

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w_copy = jnp.copy(w)
jax.debug.visualize_array_sharding(w_copy)

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So computation follows data placement: when we explicitly shard data with jax.device_put, and apply functions to that data, the compiler attempts to parallelize the computation and decide the output sharding. This policy for sharded data is a generalization of JAX's policy of following explicit device placement.

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When explicit shardings disagree, JAX errors

But what if two arguments to a computation are explicitly placed on different sets of devices, or with incompatible device orders? In these ambiguous cases, an error is raised:

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import textwrap
from termcolor import colored

def print_exception(e):
  name = colored(f'{type(e).__name__}', 'red')
  print(textwrap.fill(f'{name}: {str(e)}'))

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sharding1 = PositionalSharding(jax.devices()[:4])
sharding2 = PositionalSharding(jax.devices()[4:])

y = jax.device_put(x, sharding1.reshape(2, 2))
z = jax.device_put(x, sharding2.reshape(2, 2))
try: y + z
except ValueError as e: print_exception(e)

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devices = jax.devices()
permuted_devices = [devices[i] for i in [0, 1, 2, 3, 6, 7, 4, 5]]

sharding1 = PositionalSharding(devices)
sharding2 = PositionalSharding(permuted_devices)

y = jax.device_put(x, sharding1.reshape(4, 2))
z = jax.device_put(x, sharding2.reshape(4, 2))
try: y + z
except ValueError as e: print_exception(e)

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We say arrays that have been explicitly placed or sharded with jax.device_put are committed to their device(s), and so won't be automatically moved. See the device placement FAQ for more information.

When arrays are not explicitly placed or sharded with jax.device_put, they are placed uncommitted on the default device. Unlike committed arrays, uncommitted arrays can be moved and resharded automatically: that is, uncommitted arrays can be arguments to a computation even if other arguments are explicitly placed on different devices.

For example, the output of jnp.zeros, jnp.arange, and jnp.array are uncommitted:

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y = jax.device_put(x, sharding1.reshape(4, 2))
y + jnp.ones_like(y)
y + jnp.arange(y.size).reshape(y.shape)
print('no error!')

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Constraining shardings of intermediates in jitted code

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While the compiler will attempt to decide how a function's intermediate values and outputs should be sharded, we can also give it hints using jax.lax.with_sharding_constraint. Using jax.lax.with_sharding_constraint is much like jax.device_put, except we use it inside staged-out (i.e. jit-decorated) functions:

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jax.lax.with_sharding_constraint = jax.experimental.pjit.with_sharding_constraint

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sharding = PositionalSharding(mesh_utils.create_device_mesh((8,)))

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x = jax.random.normal(jax.random.PRNGKey(0), (8192, 8192))
x = jax.device_put(x, sharding.reshape(4, 2))

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@jax.jit
def f(x):
  x = x + 1
  y = jax.lax.with_sharding_constraint(x, sharding.reshape(2, 4))
  return y

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jax.debug.visualize_array_sharding(x)
y = f(x)
jax.debug.visualize_array_sharding(y)

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@jax.jit
def f(x):
  x = x + 1
  y = jax.lax.with_sharding_constraint(x, sharding.replicate())
  return y

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jax.debug.visualize_array_sharding(x)
y = f(x)
jax.debug.visualize_array_sharding(y)

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By adding with_sharding_constraint, we've constrained the sharding of the output. In addition to respecting the annotation on a particular intermediate, the compiler will use annotations to decide shardings for other values.

It's often a good practice to annotate the outputs of computations, for example based on how the values are ultimately consumed.

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Examples: neural networks

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⚠️ WARNING: The following is meant to be a demonstration of automatic sharding propagation with jax.Array, but it is not a recommended practice.

For neural network training, it is a good idea to constraint parameter and input sharding to be the same at every training step which can be achieved e.g. using pjit's out_axis_resources parameter and with_sharding_constraint.

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We can use jax.device_put and jax.jit's computation-follows-sharding features to parallelize computation in neural networks. Here are some simple examples, based on this basic neural network:

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import jax
import jax.numpy as jnp

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def predict(params, inputs):
  for W, b in params:
    outputs = jnp.dot(inputs, W) + b
    inputs = jnp.maximum(outputs, 0)
  return outputs

def loss(params, batch):
  inputs, targets = batch
  predictions = predict(params, inputs)
  return jnp.mean(jnp.sum((predictions - targets)**2, axis=-1))

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loss_jit = jax.jit(loss)
gradfun = jax.jit(jax.grad(loss))

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def init_layer(key, n_in, n_out):
    k1, k2 = jax.random.split(key)
    W = jax.random.normal(k1, (n_in, n_out)) / jnp.sqrt(n_in)
    b = jax.random.normal(k2, (n_out,))
    return W, b

def init_model(key, layer_sizes, batch_size):
    key, *keys = jax.random.split(key, len(layer_sizes))
    params = list(map(init_layer, keys, layer_sizes[:-1], layer_sizes[1:]))

    key, *keys = jax.random.split(key, 3)
    inputs = jax.random.normal(keys[0], (batch_size, layer_sizes[0]))
    targets = jax.random.normal(keys[1], (batch_size, layer_sizes[-1]))

    return params, (inputs, targets)

layer_sizes = [784, 8192, 8192, 8192, 10]
batch_size = 8192

params, batch = init_model(jax.random.PRNGKey(0), layer_sizes, batch_size)

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8-way batch data parallelism

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sharding = PositionalSharding(jax.devices()).reshape(8, 1)

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batch = jax.device_put(batch, sharding)
params = jax.device_put(params, sharding.replicate())

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:outputId: 336c1892-a3e1-4a2a-a0ee-a16dd7a24103

loss_jit(params, batch)

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step_size = 1e-5

for _ in range(30):
  grads = gradfun(params, batch)
  params = [(W - step_size * dW, b - step_size * db)
            for (W, b), (dW, db) in zip(params, grads)]

print(loss_jit(params, batch))

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:outputId: 8f0d79b4-9ff3-47a9-d25f-ecc770e72195

%timeit -n 5 -r 5 gradfun(params, batch)[0][0].block_until_ready()

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batch_single = jax.device_put(batch, jax.devices()[0])
params_single = jax.device_put(params, jax.devices()[0])

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:outputId: a96ee2c8-179d-4dfa-aa72-5c0076ff6881

%timeit -n 5 -r 5 gradfun(params_single, batch_single)[0][0].block_until_ready()

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4-way batch data parallelism and 2-way model tensor parallelism

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sharding = sharding.reshape(4, 2)

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batch = jax.device_put(batch, sharding.replicate(1))
jax.debug.visualize_array_sharding(batch[0])
jax.debug.visualize_array_sharding(batch[1])

:id: 7kNJVPBjz5nq

(W1, b1), (W2, b2), (W3, b3), (W4, b4) = params

W1 = jax.device_put(W1, sharding.replicate())
b1 = jax.device_put(b1, sharding.replicate())

W2 = jax.device_put(W2, sharding.replicate(0))
b2 = jax.device_put(b2, sharding.replicate(0))

W3 = jax.device_put(W3, sharding.replicate(0).T)
b3 = jax.device_put(b3, sharding.replicate())

W4 = jax.device_put(W4, sharding.replicate())
b4 = jax.device_put(b4, sharding.replicate())

params = (W1, b1), (W2, b2), (W3, b3), (W4, b4)

:id: I8ZJiiGb0HJk
:outputId: 5c406158-6b71-45ec-bdad-2d6b39b237ca

jax.debug.visualize_array_sharding(W2)

:id: t2fsJ_Ow0LgK
:outputId: 50ad8513-cb50-44d3-e0df-551ddd3c9fcb

jax.debug.visualize_array_sharding(W3)

:id: xnNgGB7-0Nh4
:outputId: 338d8bd4-0e0d-4a4a-99f7-e408b95ab274

print(loss_jit(params, batch))

:id: ygV3-IBV0Qx3

step_size = 1e-5

for _ in range(30):
    grads = gradfun(params, batch)
    params = [(W - step_size * dW, b - step_size * db)
              for (W, b), (dW, db) in zip(params, grads)]

:id: VWXN24Xh0Tkc
:outputId: 13379078-3863-42f3-e5f4-368095c3a35c

print(loss_jit(params, batch))

:id: Cq3TzYU70Vfd
:outputId: 9aae2051-3e4f-4918-c221-81d21edb563a

(W1, b1), (W2, b2), (W3, b3), (W4, b4) = params
jax.debug.visualize_array_sharding(W2)
jax.debug.visualize_array_sharding(W3)

:id: hAeLBs9D0Z8T
:outputId: 3eeacdd1-b44b-46e7-9868-dbdbca4a78e2

%timeit -n 10 -r 10 gradfun(params, batch)[0][0].block_until_ready()

+++ {"id": "3diqi5VRBy6S"}

Sharp bits

+++ {"id": "OTfoXNnxFYDJ"}

Generating random numbers

JAX comes with a functional, deterministic random number generator. It underlies the various sampling functions in the jax.random module, such as jax.random.uniform.

JAX's random numbers are produced by a counter-based PRNG, so in principle, random number generation should be a pure map over counter values. A pure map is a trivially partitionable operation in principle. It should require no cross-device communication, nor any redundant computation across devices.

However, the existing stable RNG implementation is not automatically partitionable, for historical reasons.

+++ {"id": "ht_zYFVXNrjN"}

Consider the following example, where a function draws random uniform numbers and adds them to the input, elementwise:

:id: quUwyGoiHub2

@jax.jit
def f(key, x):
  numbers = jax.random.uniform(key, x.shape)
  return x + numbers

key = jax.random.PRNGKey(42)
x_sharding = jax.sharding.PositionalSharding(jax.devices())
x = jax.device_put(jnp.arange(24), x_sharding)

+++ {"id": "ZgSA9x9NLMaP"}

On a partitioned input, the function f produces output that is also partitioned:

:id: nxPHw0loLgFd
:outputId: 59514362-61c5-4181-c7e9-ad7777218779

jax.debug.visualize_array_sharding(f(key, x))

+++ {"id": "WnjlWDUYLkp6"}

But if we inspect the compiled computation for f on this partitioned input, we see that it does involve some communication:

:id: uf6hTkjvKifH
:outputId: 3af4c656-0753-4dd5-e4be-0097bba43256

f_exe = f.lower(key, x).compile()
print('Communicating?', 'collective-permute' in f_exe.as_text())

+++ {"id": "AXp9i8fbL8DD"}

One way to work around this is to configure JAX with the experimental upgrade flag jax_threefry_partitionable. With the flag on, the "collective permute" operation is now gone from the compiled computation:

:id: G87r_Aq6Ts_F
:outputId: 5c8cd729-5f2a-450d-ee40-fb788f91707a

jax.config.update('jax_threefry_partitionable', True)
f_exe = f.lower(key, x).compile()
print('Communicating?', 'collective-permute' in f_exe.as_text())

+++ {"id": "WV8ZccM5SXOU"}

The output is still partitioned:

:id: 8RplTPyRSTbW
:outputId: 74e2ef51-c5c0-4c25-cc90-2f465961bdae

jax.debug.visualize_array_sharding(f(key, x))

+++ {"id": "kaK--hPmSPpV"}

One caveat, however, is that the random values produced may be different than before, even though they were generated by the same random key:

:id: f_EjYjOpSO18
:outputId: b50278f2-927d-4aea-fb04-cdd5c2fc1a66

jax.config.update('jax_threefry_partitionable', False)
print('Stable:')
print(f(key, x))
print()

jax.config.update('jax_threefry_partitionable', True)
print('Partitionable:')
print(f(key, x))

+++ {"id": "8BDPqgOrTMfK"}

In jax_threefry_partitionable mode, the JAX PRNG remains deterministic, but its implementation is new (and under development). The random values generated for a given key will be the same at a given JAX version (or a given commit on the main branch), but may vary across releases.