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

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JAX: Autograd and XLA Continuous integration

Quickstart| Transformations| Install guide| Change logs| Reference docs| Code search

What is JAX?

JAX is Autograd andXLA, 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 usesXLAto 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 forcomposable function transformations. Both[


]( and [


]( 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 andsharp edges. Please help by trying it out, reporting bugs, and letting us know what you think!

import jax.numpy as np from jax import grad, jit, vmap def predict(params, inputs): for W, b in params: outputs =, W) + b inputs = np.tanh(outputs) return outputs def logprob\_fun(params, inputs, targets): preds = predict(params, inputs) return np.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


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 of primary interest:






, and



Automatic differentiation with


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


]( reverse-mode gradients:

from jax import grad import jax.numpy as np def tanh(x): # Define a function y = np.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



print(grad(grad(grad(tanh)))(1.0)) # 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 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 differentiationand the JAX Autodiff Cookbookfor 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 np from jax import jit def slow\_f(x): # Element-wise ops see a large benefit from fusion return x \* x + x \* 2.0 x = np.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 Notebookfor 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 for W, b in params: output\_vec =, input\_vec) + b # `input_vec` on the right-hand side! input\_vec = np.tanh(output\_vec) return output\_vec

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 = np.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.



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 np # 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 print(pmap(np.mean)(result)) # prints [1.1566595 1.1805978 ... 1.2321935 1.2015157]

In addition to expressing pure maps, you can use fast collective communication operationsbetween 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(np.arange(4.))) # 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 = np.sin(x) @pmap def g(z): return np.cos(z) \* np.tan(y.sum()) \* np.tanh(x).sum() return grad(lambda w: np.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: np.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 Cookbookand the SPMD MNIST classifier from scratch examplefor 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, andto enable double-precision(64-bit, e.g.
    ) one needs to set the
    variable at startup (or set the environment variable
  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
    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[
    parameter](,[structured control flow primitives]([
    ](, 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 via theWindows 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


to a version that supports



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

# install jaxlib PYTHON\_VERSION=cp37 # alternatives: cp36, cp37, cp38 CUDA\_VERSION=cuda100 # alternatives: cuda100, cuda101, cuda102, cuda110 PLATFORM=manylinux2010\_x86\_64 # alternatives: manylinux2010\_x86\_64 BASE\_URL='' pip install --upgrade $BASE\_URL/$CUDA\_VERSION/jaxlib-0.1.51-$PYTHON\_VERSION-none-$PLATFORM.whl pip install --upgrade jax # install jax

The library package name must correspond to the version of the existing CUDA installation you want to use, with


for CUDA 11.0,


for CUDA 10.2,


for CUDA 10.1, and


for CUDA 10.0. To find your CUDA and CUDNN versions, you can run commands like these, depending on your CUDNN install path:

nvcc --version grep CUDNN\_MAJOR -A 2 /usr/local/cuda/include/cudnn.h # might need different path

Note that 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

Or set the following environment variable before importing JAX:


The Python version must match your Python interpreter. There are prebuilt wheels for Python 3.6, 3.7, and 3.8; for anything else, you must build from source. Jax requires Python 3.6 or above. Jax does not support Python 2 any more.

To try automatic detection of the correct version for your system, you can run:

pip install --upgrade`nvidia-smi | sed -En "s/.* CUDA Version: ([0-9]*)\.([0-9]*).*/cuda\1\2/p"`/jaxlib-0.1.51-`python3 -V | sed -En "s/Python ([0-9]*)\.([0-9]*).*/cp\1\2/p"`-none-manylinux2010\_x86\_64.whl jax

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

Building JAX from source

See Building JAX from source.

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 = {}, version = {0.1.55}, 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 thereference documentation.

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

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