# JAX Quickstart¶

JAX is NumPy on the CPU, GPU, and TPU, with great automatic differentiation for high-performance machine learning research.

With its updated version of Autograd, JAX can automatically differentiate native Python and NumPy code. It can differentiate through a large subset of Python’s features, including loops, ifs, recursion, and closures, and it can even take derivatives of derivatives of derivatives. It supports reverse-mode 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 code on accelerators, like GPUs and TPUs. Compilation happens under the hood by default, with library calls getting just-in-time compiled and executed. But JAX even 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 having to leave Python.

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


## Multiplying Matrices¶

We’ll be generating random data in the following examples. One big difference between NumPy and JAX is how you generate random numbers. For more details, see Common Gotchas in JAX.

key = random.PRNGKey(0)
x = random.normal(key, (10,))
print(x)

[-0.372111    0.2642311  -0.18252774 -0.7368198  -0.44030386 -0.15214427
-0.6713536  -0.59086424  0.73168874  0.56730247]


Let’s dive right in and multiply two big matrices.

size = 3000
x = random.normal(key, (size, size), dtype=jnp.float32)
%timeit jnp.dot(x, x.T).block_until_ready()  # runs on the GPU

460 ms ± 1.78 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)


We added that block_until_ready because JAX uses asynchronous execution by default (see Asynchronous dispatch).

JAX NumPy functions work on regular NumPy arrays.

import numpy as np
x = np.random.normal(size=(size, size)).astype(np.float32)

412 ms ± 1.41 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)


That’s slower because it has to transfer data to the GPU every time. You can ensure that an NDArray is backed by device memory using device_put().

from jax import device_put

x = np.random.normal(size=(size, size)).astype(np.float32)
x = device_put(x)

403 ms ± 1.05 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)


The output of device_put() still acts like an NDArray, but it only copies values back to the CPU when they’re needed for printing, plotting, saving to disk, branching, etc. The behavior of device_put() is equivalent to the function jit(lambda x: x), but it’s faster.

If you have a GPU (or TPU!) these calls run on the accelerator and have the potential to be much faster than on CPU. See Is JAX faster than NumPy? for more comparison of performance characteristics of NumPy and JAX

JAX is much more than just a GPU-backed NumPy. It also comes with a few program transformations that are useful when writing numerical code. For now, there are three main ones:

Let’s go over these, one-by-one. We’ll also end up composing these in interesting ways.

## Using jit() to speed up functions¶

JAX runs transparently on the GPU (or CPU, if you don’t have one, and TPU coming soon!). However, in the above example, JAX is dispatching kernels to the GPU one operation at a time. If we have a sequence of operations, we can use the @jit decorator to compile multiple operations together using XLA. Let’s try that.

def selu(x, alpha=1.67, lmbda=1.05):
return lmbda * jnp.where(x > 0, x, alpha * jnp.exp(x) - alpha)

x = random.normal(key, (1000000,))

2.8 ms ± 10.4 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)


We can speed it up with @jit, which will jit-compile the first time selu is called and will be cached thereafter.

selu_jit = jit(selu)

690 µs ± 6.51 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)


## Taking derivatives with grad()¶

In addition to evaluating numerical functions, we also want to transform them. One transformation is automatic differentiation. In JAX, just like in Autograd, you can compute gradients with the grad() function.

def sum_logistic(x):
return jnp.sum(1.0 / (1.0 + jnp.exp(-x)))

x_small = jnp.arange(3.)
print(derivative_fn(x_small))

[0.25       0.19661197 0.10499357]


Let’s verify with finite differences that our result is correct.

def first_finite_differences(f, x):
eps = 1e-3
return jnp.array([(f(x + eps * v) - f(x - eps * v)) / (2 * eps)
for v in jnp.eye(len(x))])

print(first_finite_differences(sum_logistic, x_small))

[0.24998187 0.1964569  0.10502338]


Taking derivatives is as easy as calling grad(). grad() and jit() compose and can be mixed arbitrarily. In the above example we jitted sum_logistic and then took its derivative. We can go further:

print(grad(jit(grad(jit(grad(sum_logistic)))))(1.0))

-0.035325598


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 them to make a function that efficiently computes full Hessian matrices:

from jax import jacfwd, jacrev
def hessian(fun):
return jit(jacfwd(jacrev(fun)))


## Auto-vectorization with vmap()¶

JAX has one more transformation in its API that you might find useful: vmap(), 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. When composed with jit(), it can be just as fast as adding the batch dimensions by hand.

We’re going to work with a simple example, and promote matrix-vector products into matrix-matrix products using vmap(). Although this is easy to do by hand in this specific case, the same technique can apply to more complicated functions.

mat = random.normal(key, (150, 100))
batched_x = random.normal(key, (10, 100))

def apply_matrix(v):
return jnp.dot(mat, v)


Given a function such as apply_matrix, we can loop over a batch dimension in Python, but usually the performance of doing so is poor.

def naively_batched_apply_matrix(v_batched):
return jnp.stack([apply_matrix(v) for v in v_batched])

print('Naively batched')

Naively batched

1.32 ms ± 2.92 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)


We know how to batch this operation manually. In this case, jnp.dot handles extra batch dimensions transparently.

@jit
def batched_apply_matrix(v_batched):
return jnp.dot(v_batched, mat.T)

print('Manually batched')

Manually batched

9.56 µs ± 19.4 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)


However, suppose we had a more complicated function without batching support. We can use vmap() to add batching support automatically.

@jit
def vmap_batched_apply_matrix(v_batched):
return vmap(apply_matrix)(v_batched)

print('Auto-vectorized with vmap')

Auto-vectorized with vmap

29.9 µs ± 245 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)

Of course, vmap() can be arbitrarily composed with jit(), grad(), and any other JAX transformation.