google / jax

GPU- and TPU-backed NumPy with differentiation and JIT compilation.


JAX: Autograd and XLA

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) 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.

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 np
from jax import grad, jit, vmap
from functools import partial

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(lambda params, inputs, targets:  # fast per-example gradients
                  vmap(partial(grad_fun, params), inputs, targets))

JAX started as a research project by Matt Johnson, Roy Frostig, Dougal Maclaurin, and Chris Leary, and is now developed in the open by a growing number of contributors.


Quickstart: Colab in the Cloud

Jump right in using a notebook in your browser connected to a Google Cloud GPU.


JAX is written in pure Python, but it depends on XLA, which needs to be compiled and installed as the jaxlib package. Use the following instructions to build JAX from source or install a binary package with pip.

Building JAX from source

First, obtain the JAX source code:

git clone
cd jax

To build XLA with CUDA support, you can run

python build/ --enable_cuda
pip install -e build  # install jaxlib (includes XLA)
pip install -e .      # install jax (pure Python)

See python build/ --help for configuration options, including ways to specify the paths to CUDA and CUDNN, which you must have installed. The build also depends on NumPy, and a compiler toolchain corresponding to that of Ubuntu 16.04 or newer.

To build XLA without CUDA GPU support (CPU only), drop the --enable_cuda:

python build/
pip install -e build  # install jaxlib (includes XLA)
pip install -e .      # install jax

To upgrade to the latest version from GitHub, just run git pull from the JAX repository root, and rebuild by running if necessary. You shouldn't have to reinstall because pip install -e sets up symbolic links from site-packages into the repository.

pip installation

Installing XLA with prebuilt binaries via pip is still experimental, especially with GPU support. Let us know on the issue tracker if you run into any errors.

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

pip install jax jaxlib  # CPU-only version

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=py2  # alternatives: py2, py3
CUDA_VERSION=cuda92  # alternatives: cuda90, cuda92, cuda100
PLATFORM=linux_x86_64  # alternatives: linux_x86_64
pip install$CUDA_VERSION/jaxlib-0.1-$PYTHON_VERSION-none-$PLATFORM.whl

pip install jax  # install jax

The library package name must correspond to the version of the existing CUDA installation you want to use, with cuda100 for CUDA 10.0, cuda92 for CUDA 9.2, and cuda90 for CUDA 9.0. To find your CUDA and CUDNN versions, you can run command 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

A brief tour

In [1]: import jax.numpy as np

In [2]: from jax import random

In [3]: key = random.PRNGKey(0)

In [4]: x = random.normal(key, (5000, 5000))

In [5]: print(, x.T) / 2)  # fast!
[[  2.52727051e+03   8.15895557e+00  -8.53276134e-01 ...,  # ...

In [6]: print(, x.T) / 2)  # even faster!
[[  2.52727051e+03   8.15895557e+00  -8.53276134e-01 ...,  # ...

What’s happening behind-the-scenes is that JAX is using XLA to just-in-time (JIT) compile and execute these individual operations on the GPU. First the random.normal call is compiled and the array referred to by x is generated on the GPU. Next, each function called on x (namely transpose, dot, and divide) is individually JIT-compiled and executed, each keeping its results on the device. It’s only when a value needs to be printed, plotted, saved, or passed into a raw NumPy function that a read-only copy of the value is brought back to the host as an ndarray and cached. The second call to dot is faster because the JIT-compiled code is cached and reused, saving the compilation time.

The fun really starts when you use grad for automatic differentiation and jit to compile your own functions end-to-end. Here’s a more complete toy example:

from jax import grad, jit
import jax.numpy as np

def sigmoid(x):
    return 0.5 * (np.tanh(x / 2.) + 1)

# Outputs probability of a label being true according to logistic model.
def logistic_predictions(weights, inputs):
    return sigmoid(, weights))

# Training loss is the negative log-likelihood of the training labels.
def loss(weights, inputs, targets):
    preds = logistic_predictions(weights, inputs)
    label_probs = preds * targets + (1 - preds) * (1 - targets)
    return -np.sum(np.log(label_probs))

# Build a toy dataset.
inputs = np.array([[0.52, 1.12,  0.77],
                   [0.88, -1.08, 0.15],
                   [0.52, 0.06, -1.30],
                   [0.74, -2.49, 1.39]])
targets = np.array([True, True, False, True])

# Define a compiled function that returns gradients of the training loss
training_gradient_fun = jit(grad(loss))

# Optimize weights using gradient descent.
weights = np.array([0.0, 0.0, 0.0])
print("Initial loss: {:0.2f}".format(loss(weights, inputs, targets)))
for i in range(100):
    weights -= 0.1 * training_gradient_fun(weights, inputs, targets)

print("Trained loss: {:0.2f}".format(loss(weights, inputs, targets)))

To see more, check out the quickstart notebook, a simple MNIST classifier example and the rest of the JAX examples.

What's supported

If you’re using JAX just as an accelerator-backed NumPy, without using grad or jit in your code, then in principle there are no constraints, though some NumPy functions haven’t been implemented yet. Generally using, B) is better than because the former gives us more opportunities to run the computation on the device. NumPy also does a lot of work to cast any array-like function arguments to arrays, as in np.sum([x, y]), while jax.numpy typically requires explicit casting of array arguments, like np.sum(np.array([x, y])).

For automatic differentiation with grad, JAX has the same restrictions as Autograd. Specifically, differentiation works with indexing (x = A[i, j, :]) but not indexed assignment (A[i, j] = x) or indexed in-place updating (A[i] += b). You can use lists, tuples, and dicts freely: jax doesn't even see them. Using, B) rather than is required for automatic differentiation when A is a raw ndarray.

For compiling your own functions with jit there are a few more requirements. Because jit aims to specialize Python functions only on shapes and dtypes during tracing, rather than on concrete values, Python control flow that depends on concrete values won’t be able to execute and will instead raise an error. If you want compiled control flow, use structured control flow primitives like lax.cond and lax.while. Some indexing features, like slice-based indexing A[i:i+5] for argument-dependent i, or boolean-based indexing A[bool_ind] for argument-dependent bool_ind, produce abstract values of unknown shape and are thus unsupported in jit functions.

In general, JAX is intended to be used with a functional style of Python programming. Functions passed to transformations like grad and jit are expected to be free of side-effects. You can write print statements for debugging but they may only be executed once if they're under a jit decorator.

TLDR Do use

  • Functional programming
  • Many of NumPy’s functions (help us add more!)
  • Some SciPy functions
  • Indexing and slicing of arrays like x = A[[5, 1, 7], :, 2:4]
  • Explicit array creation from lists like A = np.array([x, y])

Don’t use

  • Assignment into arrays like A[0, 0] = x
  • Implicit casting to arrays like np.sum([x, y]) (use np.sum(np.array([x, y]) instead)
  • method syntax for functions of more than one argument (use, B) instead)
  • Side-effects like mutation of arguments or mutation of global variables
  • The out argument of NumPy functions

For jit functions, also don’t use

  • Control flow based on dynamic values if x > 0: .... Control flow based on shapes is fine: if x.shape[0] > 2: ... and for subarr in array.
  • Slicing A[i:i+5] for dynamic index i (use lax.dynamic_slice instead) or boolean indexing A[bool_ind] for traced values bool_ind.

You should get loud errors if your code violates any of these.


At its core, JAX is an extensible system for transforming numerical functions. We currently expose three important transformations: grad, jit, and vmap.

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 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 0.41997434161402603

You can differentiate to any order with grad.

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
    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)

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

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 =, 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 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 = 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.

The vmap function does that transformation for us. That is, if we write

from jax import vmap
predictions = vmap(partial(predict, 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.

Random numbers are different

JAX needs a functional pseudo-random number generator (PRNG) system to provide reproducible results invariant to compilation boundaries and backends, while also maximizing performance by enabling vectorized generation and parallelization across random calls. The numpy.random library doesn’t have those properties. The jax.random library meets those needs: it’s functionally pure, but it doesn’t require you to pass stateful random objects back out of every function.

The jax.random library uses count-based PRNGs and a functional array-oriented splitting model. To generate random values, you call a function like jax.random.normal and give it a PRNG key:

import jax.random as random

key = random.PRNGKey(0)
print(random.normal(key, shape=(3,)))  # [ 1.81608593 -0.48262325  0.33988902]

If we make the same call again with the same key, we get the same values:

print(random.normal(key, shape=(3,)))  # [ 1.81608593 -0.48262325  0.33988902]

The key never gets updated. So how do we get fresh random values? We use jax.random.split to create new keys from existing ones. A common pattern is to split off a new key for every function call that needs random values:

key = random.PRNGKey(0)

key, subkey = random.split(key)
print(random.normal(subkey, shape=(3,)))  # [ 1.1378783  -1.22095478 -0.59153646]

key, subkey = random.split(key)
print(random.normal(subkey, shape=(3,)))  # [-0.06607265  0.16676566  1.17800343]

By splitting the PRNG key, not only do we avoid having to thread random states back out of every function call, but also we can generate multiple random arrays in parallel because we can avoid unnecessary sequential dependencies.

There's a gotcha here, which is that it's easy to unintentionally reuse a key without splitting. We intend to add a check for this (a sort of dynamic linear typing) but for now it's something to be careful about.


JAX provides some small, experimental libraries for machine learning. These libraries are in part about providing tools and in part about serving as examples for how to build such libraries using JAX. Each one is only a few hundred lines of code, so take a look inside and adapt them as you need!

Neural-net building with Stax

Stax is a functional neural network building library. The basic idea is that a single layer or an entire network can be modeled as an (init_fun, apply_fun) pair. The init_fun is used to initialize network parameters and the apply_fun takes parameters and inputs to produce outputs. There are constructor functions for common basic pairs, like Conv and Relu, and these pairs can be composed in series using stax.serial or in parallel using stax.parallel.

Here’s an example:

from jax.experimental import stax
from jax.experimental.stax import Conv
from jax.experimental.stax import Dense
from jax.experimental.stax import MaxPool
from jax.experimental.stax import Relu
from jax.experimental.stax import LogSoftmax

# Set up network initialization and evaluation functions
net_init, net_apply = stax.serial(
    Conv(32, (3, 3), padding='SAME'), Relu,
    Conv(64, (3, 3), padding='SAME'), Relu,
    MaxPool((2, 2)), Flatten,
    Dense(128), Relu,
    Dense(10), SoftMax,

# Initialize parameters, not committing to a batch shape
in_shape = (-1, 28 * 28)
out_shape, net_params = net_init(in_shape)

# Apply network
predictions = net_apply(net_params, inputs)

First-order optimization with Minmax

Minmax is an optimization library focused on stochastic first-order optimizers. Every optimizer is modeled as an (init_fun, update_fun) pair. The init_fun is used to initialize the optimizer state, which could include things like momentum variables, and the update_fun accepts a gradient and an optimizer state to produce a new optimizer state. The parameters being optimized can be ndarrays or arbitrarily-nested list/tuple/dict structures, so you can store your parameters however you’d like.

Here’s an example, using jit to compile the whole update end-to-end:

from jax.experimental import minmax
from jax import jit

# Set up an optimizer
opt_init, opt_update = minmax.momentum(step_size=1e-3, mass=0.9)

# Define a compiled update step
def step(i, opt_state, batch):
  params = minmax.get_params(opt_state)
  g = grad(loss)(params, batch)
  return opt_update(i, g, opt_state)

# Optimize parameters in a loop
opt_state = opt_init(net_params)
for i in range(num_steps):
  opt_state = step(i, opt_state, next(data_generator))
net_params = minmax.get_params(opt_state)

How it works

Programming in machine learning is about expressing and transforming functions. Transformations include automatic differentiation, compilation for accelerators, and automatic batching. High-level languages like Python are great for expressing functions, but usually all we can do with them is apply them. We lose access to their internal structure which would let us perform transformations.

JAX is a tool for specializing and translating high-level Python+NumPy functions into a representation that can be transformed and then lifted back into a Python function.


JAX specializes Python functions by tracing. Tracing a function means monitoring all the basic operations that are applied to its input to produce its output, and recording these operations and the data-flow between them in a directed acyclic graph (DAG). To perform tracing, JAX wraps primitive operations, like basic numerical kernels, so that when they’re called they add themselves to a list of operations performed along with their inputs and outputs. To keep track of how data flows between these primitives, values being tracked are wrapped in instances of the Tracer class.

When a Python function is provided to grad or jit, it’s wrapped for tracing and returned. When the wrapped function is called, we abstract the concrete arguments provided into instances of the AbstractValue class, box them for tracing in instances of the Tracer class, and call the function on them. Abstract arguments represent sets of possible values rather than specific values: for example, jit abstracts ndarray arguments to abstract values that represent all ndarrays with the same shape and dtype. In contrast, grad abstracts ndarray arguments to represent an infinitesimal neighborhood of the underlying value. By tracing the Python function on these abstract values, we ensure that it’s specialized enough so that it’s tractable to transform, and that it’s still general enough so that the transformed result is useful, and possibly reusable. These transformed functions are then lifted back into Python callables in a way that allows them to be traced and transformed again as needed.

The primitive functions that JAX traces are mostly in 1:1 correspondence with XLA HLO and are defined in This 1:1 correspondence makes most of the translations to XLA essentially trivial, and ensures we only have a small set of primitives to cover for other transformations like automatic differentiation. The jax.numpy layer is written in pure Python simply by expressing NumPy functions in terms of the LAX functions (and other NumPy functions we’ve already written). That makes jax.numpy easy to extend.

When you use jax.numpy, the underlying LAX primitives are jit-compiled behind the scenes, allowing you to write unrestricted Python+Numpy code while still executing each primitive operation on an accelerator.

But JAX can do more: instead of just compiling and dispatching to a fixed set of individual primitives, you can use jit on larger and larger functions to be end-to-end compiled and optimized. For example, instead of just compiling and dispatching a convolution op, you can compile a whole network, or a whole gradient evaluation and optimizer update step.

The tradeoff is that jit functions have to satisfy some additional specialization requirements: since we want to compile traces that are specialized on shapes and dtypes, but not specialized all the way to concrete values, the Python code under a jit decorator must be applicable to abstract values. If we try to evaluate x > 0 on an abstract x, the result is an abstract value representing the set {True, False}, and so a Python branch like if x > 0 will raise an error: it doesn’t know which way to go! See What’s supported for more information about jit requirements.

The good news about this tradeoff is that jit is opt-in: JAX libraries use jit on individual operations and functions behind the scenes, allowing you to write unrestricted Python+Numpy and still make use of a hardware accelerator. But when you want to maximize performance, you can often use jit in your own code to compile and end-to-end optimize much bigger functions.

What we're working on

  1. Documentation!
  2. Cloud TPU support
  3. Multi-GPU and multi-TPU support
  4. Full NumPy coverage and some SciPy coverage
  5. Full coverage for vmap
  6. Make everything faster
    • Lowering the XLA function dispatch overhead
    • Linear algebra routines (MKL on CPU, MAGMA on GPU)
  7. cond and while primitives with efficient automatic differentiation

Current gotchas

Some things we don't handle that might surprise NumPy users:

  1. No in-place mutation syntax. Functional code. Can use lax.dynamic_update_slice.
  2. PRNG can be awkward, and linearity is not checked with a warning.


So far, JAX includes lots of help and contributions from Jamie Townsend, Peter Hawkins, Alex Wiltschko, George Dahl, Eli Bendersky, Zak Stone, Alexey Radul, Michael Isard, Skye Wanderman-Milne, and many others.

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