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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 PyPI version

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

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

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,

. 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
, 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

are instances of such transformations. Others are
for automatic vectorization and
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 =, W) + b inputs = jnp.tanh(outputs) # inputs to the next layer return outputs # no activation on last layer

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

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


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,

for differentiation,
for compilation, and
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

, like
for building neural networks
for first-order stochastic optimization
, or the examples.


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

, and

Automatic differentiation with

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

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

# prints 0.62162673

For more advanced autodiff, you can use

for reverse-mode vector-Jacobian products and
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
    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

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

, used either as an

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

and any other JAX transformation however you like.


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

Auto-vectorization with

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.


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
  activations = input_vec
  for W, b in params:
    outputs =, activations) + b  # `activations` on the right-hand side!
    activations = jnp.tanh(outputs)        # inputs to the next layer
  return outputs                           # no activation on last layer

We often instead write, W)
to allow for a batch dimension on the left side of
, 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 multiplication rather than matrix-vector multiplication.


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

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

, 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
, it’s easy:
per_example_gradients = vmap(partial(grad(loss), params))(inputs, targets)

Of course,

can be arbitrarily composed with
, and any other JAX transformation! We use
with both forward- and reverse-mode automatic differentiation for fast Jacobian and Hessian matrix calculations in
, and

SPMD programming with

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

. With

you write single-program multiple-data (SPMD) programs, including fast parallel collective communication operations. Applying
will mean that the function you write is compiled by XLA (similarly to
), 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:, x.T))(mats) # result.shape is (8, 5000, 5000)

Compute the mean on each device in parallel and print the 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')


prints [0. 0.16666667 0.33333334 0.5 ]

You can even nest

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)


[[ 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

function (e.g. with
), 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
    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...
    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
    , 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
  5. JAX enforces single-precision (32-bit, e.g.
    ) values by default, and to enable double-precision (64-bit, e.g.
    ) one needs to set the
    variable at startup (or set the environment variable
    ). On TPU, JAX uses 32-bit values by default for everything except internal temporary variables in 'matmul-like' operations, such as
    . Those ops have a
    parameter which can be used to simulate true 32-bit, with a cost of possibly slower runtime.
  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],
    rather than
  7. Some transformations, like
    , constrain how you can use Python control flow. You'll always get loud errors if something goes wrong. You might have to use
    , structured control flow primitives like
    , or just use
    on smaller subfunctions.


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

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

We support installing or building

on Linux (Ubuntu 16.04 or later) and macOS (10.12 or later) platforms. Windows users can use JAX on CPU and GPU via the Windows Subsystem for Linux. There is some initial native Windows support, but since it is still somewhat immature, there are no binary releases and it must be built from source.

pip installation: CPU

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

pip install --upgrade pip
pip install --upgrade "jax[cpu]"

On Linux, it is often necessary to first update

to a version that supports

pip installation: GPU (CUDA)

If you want to install JAX with both CPU and NVidia GPU support, you must first install CUDA and CuDNN, if they have not already been installed. Unlike some other popular deep learning systems, JAX does not bundle CUDA or CuDNN as part of the

package. JAX provides pre-built CUDA-compatible wheels for linux only; the CUDA 10 JAX wheels require CuDNN 7, whereas the CUDA 11 wheels of JAX require CuDNN 8. Other combinations of CUDA and CuDNN are possible but require building from source.

Next, run

pip install --upgrade pip
pip install --upgrade "jax[cuda111]" -f  # Note: wheels only available on linux.

The jaxlib version must correspond to the version of the existing CUDA installation you want to use: * For CUDA 11.1, 11.2, or 11.3, use

. The same wheel should work for CUDA 11.x releases from 11.1 onwards. * Older CUDA versions are not supported.

You can find your CUDA version with the command:

nvcc --version

Some GPU functionality expects the CUDA installation to be at

, where X.X should be replaced with the CUDA version number (e.g.
). 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

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

pip installation: Google Cloud TPU

JAX also provides pre-built wheels for Google Cloud TPU. To install JAX along with appropriate versions of

, you can run the following in your cloud TPU VM:
pip install --upgrade pip
pip install "jax[tpu]>=0.2.16" -f
Colab TPU runtimes come with JAX pre-installed, but before importing JAX you must run the following code to initialize the TPU:

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. (Watch the NeurIPS 2020 JAX Ecosystem at DeepMind talk here)

Citing JAX

To cite this repository:

  author = {James Bradbury and Roy Frostig and Peter Hawkins and Matthew James Johnson and Chris Leary and Dougal Maclaurin and George Necula and Adam Paszke and Jake Vander{P}las and Skye Wanderman-{M}ilne and Qiao Zhang},
  title = {{JAX}: composable transformations of {P}ython+{N}um{P}y programs},
  url = {},
  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/, 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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