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Composable transformations of Python+NumPy programs: differentiate, vectorize, JIT to GPU/TPU, and more

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JAX: Autograd and XLA Continuous integration

Quickstart | Transformations | Install guide | Neural net libraries | Change logs | Reference docs | Code search

News: JAX tops largest-scale MLPerf Training 0.7 benchmarks!

What is JAX?

JAX is Autograd and XLA, brought together for high-performance machine learning research.

With its updated version of Autograd, JAX can automatically differentiate native Python and NumPy functions. It can differentiate through loops, branches, recursion, and closures, and it can take derivatives of derivatives of derivatives. It supports reverse-mode differentiation (a.k.a. backpropagation) via

grad
as well as forward-mode differentiation, and the two can be composed arbitrarily to any order.

What’s new is that JAX uses XLA to compile and run your NumPy programs on GPUs and TPUs. Compilation happens under the hood by default, with library calls getting just-in-time compiled and executed. But JAX also lets you just-in-time compile your own Python functions into XLA-optimized kernels using a one-function API,

jit
. Compilation and automatic differentiation can be composed arbitrarily, so you can express sophisticated algorithms and get maximal performance without leaving Python. You can even program multiple GPUs or TPU cores at once using
pmap
, and differentiate through the whole thing.

Dig a little deeper, and you'll see that JAX is really an extensible system for composable function transformations. Both

grad
and
jit
are instances of such transformations. Others are
vmap
for automatic vectorization and
pmap
for single-program multiple-data (SPMD) parallel programming of multiple accelerators, with more to come.

This is a research project, not an official Google product. Expect bugs and sharp edges. Please help by trying it out, reporting bugs, and letting us know what you think!

import jax.numpy as jnp
from jax import grad, jit, vmap

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

def logprob_fun(params, inputs, targets): preds = predict(params, inputs) return jnp.sum((preds - targets)**2)

grad_fun = jit(grad(logprob_fun)) # compiled gradient evaluation function perex_grads = jit(vmap(grad_fun, in_axes=(None, 0, 0))) # fast per-example grads

Contents

Quickstart: Colab in the Cloud

Jump right in using a notebook in your browser, connected to a Google Cloud GPU. Here are some starter notebooks: - The basics: NumPy on accelerators,

grad
for differentiation,
jit
for compilation, and
vmap
for vectorization
- Training a Simple Neural Network, with TensorFlow Dataset Data Loading

JAX now runs on Cloud TPUs. To try out the preview, see the Cloud TPU Colabs.

For a deeper dive into JAX: - The Autodiff Cookbook, Part 1: easy and powerful automatic differentiation in JAX - Common gotchas and sharp edges - See the full list of notebooks.

You can also take a look at the mini-libraries in

jax.experimental
, like
stax
for building neural networks
and
optimizers
for first-order stochastic optimization
, or the examples.

Transformations

At its core, JAX is an extensible system for transforming numerical functions. Here are four of primary interest:

grad
,
jit
,
vmap
, and
pmap
.

Automatic differentiation with
grad

JAX has roughly the same API as Autograd. The most popular function is

grad
for reverse-mode gradients:

from jax import grad
import jax.numpy as jnp

def tanh(x): # Define a function y = jnp.exp(-2.0 * x) return (1.0 - y) / (1.0 + y)

grad_tanh = grad(tanh) # Obtain its gradient function print(grad_tanh(1.0)) # Evaluate it at x = 1.0

prints 0.4199743

You can differentiate to any order with

grad
.
print(grad(grad(grad(tanh)))(1.0))
# prints 0.62162673

For more advanced autodiff, you can use

jax.vjp
for reverse-mode vector-Jacobian products and
jax.jvp
for forward-mode Jacobian-vector products. The two can be composed arbitrarily with one another, and with other JAX transformations. Here's one way to compose those to make a function that efficiently computes full Hessian matrices:

from jax import jit, jacfwd, jacrev

def hessian(fun): return jit(jacfwd(jacrev(fun)))

As with Autograd, you're free to use differentiation with Python control structures:

def abs_val(x):
  if x > 0:
    return x
  else:
    return -x

abs_val_grad = grad(abs_val) print(abs_val_grad(1.0)) # prints 1.0 print(abs_val_grad(-1.0)) # prints -1.0 (abs_val is re-evaluated)

See the reference docs on automatic differentiation and the JAX Autodiff Cookbook for more.

Compilation with
jit

You can use XLA to compile your functions end-to-end with

jit
, used either as an

@jit
decorator or as a higher-order function.
import jax.numpy as jnp
from jax import jit

def slow_f(x):

Element-wise ops see a large benefit from fusion

return x * x + x * 2.0

x = jnp.ones((5000, 5000)) fast_f = jit(slow_f) %timeit -n10 -r3 fast_f(x) # ~ 4.5 ms / loop on Titan X %timeit -n10 -r3 slow_f(x) # ~ 14.5 ms / loop (also on GPU via JAX)

You can mix

jit
and
grad
and any other JAX transformation however you like.

Using

jit
puts constraints on the kind of Python control flow the function can use; see the Gotchas Notebook for more.

Auto-vectorization with
vmap

vmap
is the vectorizing map. It has the familiar semantics of mapping a function along array axes, but instead of keeping the loop on the outside, it pushes the loop down into a function’s primitive operations for better performance.

Using

vmap
can save you from having to carry around batch dimensions in your code. For example, consider this simple unbatched neural network prediction function:
def predict(params, input_vec):
  assert input_vec.ndim == 1
  for W, b in params:
    output_vec = jnp.dot(W, input_vec) + b  # `input_vec` on the right-hand side!
    input_vec = jnp.tanh(output_vec)
  return output_vec

We often instead write

jnp.dot(inputs, W)
to allow for a batch dimension on the left side of
inputs
, but we’ve written this particular prediction function to apply only to single input vectors. If we wanted to apply this function to a batch of inputs at once, semantically we could just write
from functools import partial
predictions = jnp.stack(list(map(partial(predict, params), input_batch)))

But pushing one example through the network at a time would be slow! It’s better to vectorize the computation, so that at every layer we’re doing matrix-matrix multiplies rather than matrix-vector multiplies.

The

vmap
function does that transformation for us. That is, if we write
from jax import vmap
predictions = vmap(partial(predict, params))(input_batch)
# or, alternatively
predictions = vmap(predict, in_axes=(None, 0))(params, input_batch)

then the

vmap
function will push the outer loop inside the function, and our machine will end up executing matrix-matrix multiplications exactly as if we’d done the batching by hand.

It’s easy enough to manually batch a simple neural network without

vmap
, but in other cases manual vectorization can be impractical or impossible. Take the problem of efficiently computing per-example gradients: that is, for a fixed set of parameters, we want to compute the gradient of our loss function evaluated separately at each example in a batch. With
vmap
, it’s easy:
per_example_gradients = vmap(partial(grad(loss), params))(inputs, targets)

Of course,

vmap
can be arbitrarily composed with
jit
,
grad
, and any other JAX transformation! We use
vmap
with both forward- and reverse-mode automatic differentiation for fast Jacobian and Hessian matrix calculations in
jax.jacfwd
,
jax.jacrev
, and
jax.hessian
.

SPMD programming with
pmap

For parallel programming of multiple accelerators, like multiple GPUs, use

pmap
. With

pmap
you write single-program multiple-data (SPMD) programs, including fast parallel collective communication operations. Applying
pmap
will mean that the function you write is compiled by XLA (similarly to
jit
), then replicated and executed in parallel across devices.

Here's an example on an 8-GPU machine:

from jax import random, pmap
import jax.numpy as jnp

Create 8 random 5000 x 6000 matrices, one per GPU

keys = random.split(random.PRNGKey(0), 8) mats = pmap(lambda key: random.normal(key, (5000, 6000)))(keys)

Run a local matmul on each device in parallel (no data transfer)

result = pmap(lambda x: jnp.dot(x, x.T))(mats) # result.shape is (8, 5000, 5000)

Compute the mean on each device in parallel and print the result

print(pmap(jnp.mean)(result))

prints [1.1566595 1.1805978 ... 1.2321935 1.2015157]

In addition to expressing pure maps, you can use fast collective communication operations between devices:

from functools import partial
from jax import lax

@partial(pmap, axis_name='i') def normalize(x): return x / lax.psum(x, 'i')

print(normalize(jnp.arange(4.)))

prints [0. 0.16666667 0.33333334 0.5 ]

You can even nest

pmap
functions for more sophisticated communication patterns.

It all composes, so you're free to differentiate through parallel computations:

from jax import grad

@pmap def f(x): y = jnp.sin(x) @pmap def g(z): return jnp.cos(z) * jnp.tan(y.sum()) * jnp.tanh(x).sum() return grad(lambda w: jnp.sum(g(w)))(x)

print(f(x))

[[ 0. , -0.7170853 ],

[-3.1085174 , -0.4824318 ],

[10.366636 , 13.135289 ],

[ 0.22163185, -0.52112055]]

print(grad(lambda x: jnp.sum(f(x)))(x))

[[ -3.2369726, -1.6356447],

[ 4.7572474, 11.606951 ],

[-98.524414 , 42.76499 ],

[ -1.6007166, -1.2568436]]

When reverse-mode differentiating a

pmap
function (e.g. with
grad
), the backward pass of the computation is parallelized just like the forward pass.

See the SPMD Cookbook and the SPMD MNIST classifier from scratch example for more.

Current gotchas

For a more thorough survey of current gotchas, with examples and explanations, we highly recommend reading the Gotchas Notebook. Some standouts:

  1. JAX transformations only work on pure functions, which don't have side-effects and respect referential transparency (i.e. object identity testing with
    is
    isn't preserved). If you use a JAX transformation on an impure Python function, you might see an error like
    Exception: Can't lift Traced...
    or
    Exception: Different traces at same level
    .
  2. In-place mutating updates of arrays, like
    x[i] += y
    , aren't supported, but there are functional alternatives. Under a
    jit
    , those functional alternatives will reuse buffers in-place automatically.
  3. Random numbers are different, but for good reasons.
  4. If you're looking for convolution operators, they're in the
    jax.lax
    package.
  5. JAX enforces single-precision (32-bit, e.g.
    float32
    ) values by default, and to enable double-precision (64-bit, e.g.
    float64
    ) one needs to set the
    jax_enable_x64
    variable at startup (or set the environment variable
    JAX_ENABLE_X64=True
    ).
  6. Some of NumPy's dtype promotion semantics involving a mix of Python scalars and NumPy types aren't preserved, namely
    np.add(1, np.array([2],
    np.float32)).dtype
    is
    float64
    rather than
    float32
    .
  7. Some transformations, like
    jit
    , constrain how you can use Python control flow. You'll always get loud errors if something goes wrong. You might have to use
    jit
    's
    static_argnums
    parameter
    , structured control flow primitives like
    lax.scan
    , or just use
    jit
    on smaller subfunctions.

Installation

JAX is written in pure Python, but it depends on XLA, which needs to be installed as the

jaxlib
package. Use the following instructions to install a binary package with
pip
, or to build JAX from source.

We support installing or building

jaxlib
on Linux (Ubuntu 16.04 or later) and macOS (10.12 or later) platforms. Windows users can use JAX on CPU via the Windows Subsystem for Linux. We're not currently working on native Windows support, but contributions are welcome (see #438).

pip installation

To install a CPU-only version, which might be useful for doing local development on a laptop, you can run

pip install --upgrade pip
pip install --upgrade jax jaxlib  # CPU-only version

On Linux, it is often necessary to first update

pip
to a version that supports
manylinux2010
wheels.

If you want to install JAX with both CPU and GPU support, using existing CUDA and CUDNN7 installations on your machine (for example, preinstalled on your cloud VM), you can run

pip install --upgrade pip
pip install --upgrade jax jaxlib==0.1.57+cuda110 -f https://storage.googleapis.com/jax-releases/jax_releases.html

The jaxlib version must correspond to the version of the existing CUDA installation you want to use, with

cuda110
for CUDA 11.0,
cuda102
for CUDA 10.2, and
cuda101
for CUDA 10.1. You can find your CUDA version with: install path:
nvcc --version

Note that some GPU functionality expects the CUDA installation to be at

/usr/local/cuda-X.X
, where X.X should be replaced with the CUDA version number (e.g.
cuda-10.2
). If CUDA is installed elsewhere on your system, you can either create a symlink:
sudo ln -s /path/to/cuda /usr/local/cuda-X.X

Or set the following environment variable before importing JAX:

XLA_FLAGS=--xla_gpu_cuda_data_dir=/path/to/cuda

Please let us know on the issue tracker if you run into any errors or problems with the prebuilt wheels.

Building JAX from source

See Building JAX from source.

Neural network libraries

Multiple Google research groups develop and share libraries for training neural networks in JAX. If you want a fully featured library for neural network training with examples and how-to guides, try Flax. Another option is Trax, a combinator-based framework focused on ease-of-use and end-to-end single-command examples, especially for sequence models and reinforcement learning. Finally, Objax is a minimalist object-oriented framework with a PyTorch-like interface.

DeepMind has open-sourced an ecosystem of libraries around JAX including Haiku for neural network modules, Optax for gradient processing and optimization, RLax for RL algorithms, and chex for reliable code and testing.

Citing JAX

To cite this repository:

@software{jax2018github,
  author = {James Bradbury and Roy Frostig and Peter Hawkins and Matthew James Johnson and Chris Leary and Dougal Maclaurin and Skye Wanderman-Milne},
  title = {{JAX}: composable transformations of {P}ython+{N}um{P}y programs},
  url = {http://github.com/google/jax},
  version = {0.2.5},
  year = {2018},
}

In the above bibtex entry, names are in alphabetical order, the version number is intended to be that from jax/version.py, and the year corresponds to the project's open-source release.

A nascent version of JAX, supporting only automatic differentiation and compilation to XLA, was described in a paper that appeared at SysML 2018. We're currently working on covering JAX's ideas and capabilities in a more comprehensive and up-to-date paper.

Reference documentation

For details about the JAX API, see the reference documentation.

For getting started as a JAX developer, see the developer documentation.

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