Distributed arrays and automatic parallelization#
This tutorial discusses parallelism via jax.Array, the unified array object model available in JAX v0.4.1 and newer.
import os
import functools
from typing import Optional
import numpy as np
import jax
import jax.numpy as jnp
⚠️ WARNING: The notebook requires 8 devices to run.
if len(jax.local_devices()) < 8:
raise Exception("Notebook requires 8 devices to run")
Intro and a quick example#
By reading this tutorial notebook, you’ll learn about jax.Array, a unified
datatype for representing arrays, even with physical storage spanning multiple
devices. You’ll also learn about how using jax.Arrays together with jax.jit
can provide automatic compiler-based parallelization.
Before we think step by step, here’s a quick example.
First, we’ll create a jax.Array sharded across multiple devices:
from jax.experimental import mesh_utils
from jax.sharding import PositionalSharding
# Create a Sharding object to distribute a value across devices:
sharding = PositionalSharding(mesh_utils.create_device_mesh((8,)))
# Create an array of random values:
x = jax.random.normal(jax.random.key(0), (8192, 8192))
# and use jax.device_put to distribute it across devices:
y = jax.device_put(x, sharding.reshape(4, 2))
jax.debug.visualize_array_sharding(y)
┌──────────┬──────────┐
│ TPU 0 │ TPU 1 │
├──────────┼──────────┤
│ TPU 2 │ TPU 3 │
├──────────┼──────────┤
│ TPU 6 │ TPU 7 │
├──────────┼──────────┤
│ TPU 4 │ TPU 5 │
└──────────┴──────────┘
Next, we’ll apply a computation to it and visualize how the result values are stored across multiple devices too:
z = jnp.sin(y)
jax.debug.visualize_array_sharding(z)
┌──────────┬──────────┐
│ TPU 0 │ TPU 1 │
├──────────┼──────────┤
│ TPU 2 │ TPU 3 │
├──────────┼──────────┤
│ TPU 6 │ TPU 7 │
├──────────┼──────────┤
│ TPU 4 │ TPU 5 │
└──────────┴──────────┘
The evaluation of the jnp.sin application was automatically parallelized
across the devices on which the input values (and output values) are stored:
# `x` is present on a single device
%timeit -n 5 -r 5 jnp.sin(x).block_until_ready()
The slowest run took 13.32 times longer than the fastest. This could mean that an intermediate result is being cached
5 loops, best of 5: 9.69 ms per loop
# `y` is sharded across 8 devices.
%timeit -n 5 -r 5 jnp.sin(y).block_until_ready()
5 loops, best of 5: 1.86 ms per loop
Now let’s look at each of these pieces in more detail!
Computation follows data sharding and is automatically parallelized#
With sharded input 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:
from jax.experimental import mesh_utils
from jax.sharding import PositionalSharding
sharding = PositionalSharding(mesh_utils.create_device_mesh((8,)))
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)
input sharding:
┌──────────┬──────────┐
│ TPU 0 │ TPU 1 │
├──────────┼──────────┤
│ TPU 2 │ TPU 3 │
├──────────┼──────────┤
│ TPU 6 │ TPU 7 │
├──────────┼──────────┤
│ TPU 4 │ TPU 5 │
└──────────┴──────────┘
output sharding:
┌──────────┬──────────┐
│ TPU 0 │ TPU 1 │
├──────────┼──────────┤
│ TPU 2 │ TPU 3 │
├──────────┼──────────┤
│ TPU 6 │ TPU 7 │
├──────────┼──────────┤
│ TPU 4 │ TPU 5 │
└──────────┴──────────┘
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:
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)
lhs sharding:
┌───────────────────────┐
│ TPU 0,1 │
├───────────────────────┤
│ TPU 2,3 │
├───────────────────────┤
│ TPU 6,7 │
├───────────────────────┤
│ TPU 4,5 │
└───────────────────────┘
rhs sharding:
┌───────────┬───────────┐
│ │ │
│ │ │
│ │ │
│ │ │
│TPU 0,2,4,6│TPU 1,3,5,7│
│ │ │
│ │ │
│ │ │
│ │ │
└───────────┴───────────┘
out sharding:
┌──────────┬──────────┐
│ TPU 0 │ TPU 1 │
├──────────┼──────────┤
│ TPU 2 │ TPU 3 │
├──────────┼──────────┤
│ TPU 6 │ TPU 7 │
├──────────┼──────────┤
│ TPU 4 │ TPU 5 │
└──────────┴──────────┘
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:
x_single = jax.device_put(x, jax.devices()[0])
jax.debug.visualize_array_sharding(x_single)
┌───────────────────────┐
│ │
│ │
│ │
│ │
│ TPU 0 │
│ │
│ │
│ │
│ │
└───────────────────────┘
np.allclose(jnp.dot(x_single, x_single),
jnp.dot(y, z))
True
%timeit -n 5 -r 5 jnp.dot(x_single, x_single).block_until_ready()
5 loops, best of 5: 19.3 ms per loop
%timeit -n 5 -r 5 jnp.dot(y, z).block_until_ready()
5 loops, best of 5: 3.25 ms per loop
Even copying a sharded Array produces a result with the sharding of the input:
w_copy = jnp.copy(w)
jax.debug.visualize_array_sharding(w_copy)
┌──────────┬──────────┐
│ TPU 0 │ TPU 1 │
├──────────┼──────────┤
│ TPU 2 │ TPU 3 │
├──────────┼──────────┤
│ TPU 6 │ TPU 7 │
├──────────┼──────────┤
│ TPU 4 │ TPU 5 │
└──────────┴──────────┘
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.
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:
import textwrap
from termcolor import colored
def print_exception(e):
name = colored(f'{type(e).__name__}', 'red')
print(textwrap.fill(f'{name}: {str(e)}'))
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)
ValueError: Devices of all `Array` inputs and outputs should
be the same. Got array device ids [0, 1, 2, 3] on platform TPU and
another array's device ids [4, 5, 6, 7] on platform TPU
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)
ValueError: Devices of all `Array` inputs and outputs should
be the same. Got array device ids [0, 1, 2, 3, 4, 5, 6, 7] on platform
TPU and another array's device ids [0, 1, 2, 3, 6, 7, 4, 5] on
platform TPU
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:
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!')
no error!
Constraining shardings of intermediates in jitted code#
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:
sharding = PositionalSharding(mesh_utils.create_device_mesh((8,)))
x = jax.random.normal(jax.random.key(0), (8192, 8192))
x = jax.device_put(x, sharding.reshape(4, 2))
@jax.jit
def f(x):
x = x + 1
y = jax.lax.with_sharding_constraint(x, sharding.reshape(2, 4))
return y
jax.debug.visualize_array_sharding(x)
y = f(x)
jax.debug.visualize_array_sharding(y)
┌──────────┬──────────┐
│ TPU 0 │ TPU 1 │
├──────────┼──────────┤
│ TPU 2 │ TPU 3 │
├──────────┼──────────┤
│ TPU 6 │ TPU 7 │
├──────────┼──────────┤
│ TPU 4 │ TPU 5 │
└──────────┴──────────┘
┌───────┬───────┬───────┬───────┐
│ │ │ │ │
│ TPU 0 │ TPU 1 │ TPU 2 │ TPU 3 │
│ │ │ │ │
│ │ │ │ │
├───────┼───────┼───────┼───────┤
│ │ │ │ │
│ TPU 6 │ TPU 7 │ TPU 4 │ TPU 5 │
│ │ │ │ │
│ │ │ │ │
└───────┴───────┴───────┴───────┘
@jax.jit
def f(x):
x = x + 1
y = jax.lax.with_sharding_constraint(x, sharding.replicate())
return y
jax.debug.visualize_array_sharding(x)
y = f(x)
jax.debug.visualize_array_sharding(y)
┌──────────┬──────────┐
│ TPU 0 │ TPU 1 │
├──────────┼──────────┤
│ TPU 2 │ TPU 3 │
├──────────┼──────────┤
│ TPU 6 │ TPU 7 │
├──────────┼──────────┤
│ TPU 4 │ TPU 5 │
└──────────┴──────────┘
┌───────────────────────┐
│ │
│ │
│ │
│ │
│ TPU 0,1,2,3,4,5,6,7 │
│ │
│ │
│ │
│ │
└───────────────────────┘
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.
Examples: neural networks#
⚠️ WARNING: The following is meant to be a simple demonstration of automatic sharding propagation with jax.Array, but it may not reflect best practices for real examples. For instance, real examples may require more use of with_sharding_constraint.
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:
import jax
import jax.numpy as jnp
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))
loss_jit = jax.jit(loss)
gradfun = jax.jit(jax.grad(loss))
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.key(0), layer_sizes, batch_size)
8-way batch data parallelism#
sharding = PositionalSharding(jax.devices()).reshape(8, 1)
batch = jax.device_put(batch, sharding)
params = jax.device_put(params, sharding.replicate())
loss_jit(params, batch)
Array(23.469475, dtype=float32)
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))
10.760101
%timeit -n 5 -r 5 gradfun(params, batch)[0][0].block_until_ready()
5 loops, best of 5: 26.3 ms per loop
batch_single = jax.device_put(batch, jax.devices()[0])
params_single = jax.device_put(params, jax.devices()[0])
%timeit -n 5 -r 5 gradfun(params_single, batch_single)[0][0].block_until_ready()
5 loops, best of 5: 122 ms per loop
4-way batch data parallelism and 2-way model tensor parallelism#
sharding = sharding.reshape(4, 2)
batch = jax.device_put(batch, sharding.replicate(1))
jax.debug.visualize_array_sharding(batch[0])
jax.debug.visualize_array_sharding(batch[1])
┌───────┐
│TPU 0,1│
├───────┤
│TPU 2,3│
├───────┤
│TPU 4,5│
├───────┤
│TPU 6,7│
└───────┘
┌───────┐
│TPU 0,1│
├───────┤
│TPU 2,3│
├───────┤
│TPU 4,5│
├───────┤
│TPU 6,7│
└───────┘
(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)
jax.debug.visualize_array_sharding(W2)
┌───────────┬───────────┐
│ │ │
│ │ │
│ │ │
│ │ │
│TPU 0,2,4,6│TPU 1,3,5,7│
│ │ │
│ │ │
│ │ │
│ │ │
└───────────┴───────────┘
jax.debug.visualize_array_sharding(W3)
┌───────────────────────┐
│ │
│ TPU 0,2,4,6 │
│ │
│ │
├───────────────────────┤
│ │
│ TPU 1,3,5,7 │
│ │
│ │
└───────────────────────┘
print(loss_jit(params, batch))
10.760103
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))
10.752466
(W1, b1), (W2, b2), (W3, b3), (W4, b4) = params
jax.debug.visualize_array_sharding(W2)
jax.debug.visualize_array_sharding(W3)
┌───────────┬───────────┐
│ │ │
│ │ │
│ │ │
│ │ │
│TPU 0,2,4,6│TPU 1,3,5,7│
│ │ │
│ │ │
│ │ │
│ │ │
└───────────┴───────────┘
┌───────────────────────┐
│ │
│ TPU 0,2,4,6 │
│ │
│ │
├───────────────────────┤
│ │
│ TPU 1,3,5,7 │
│ │
│ │
└───────────────────────┘
%timeit -n 10 -r 10 gradfun(params, batch)[0][0].block_until_ready()
10 loops, best of 10: 30.5 ms per loop