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
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)
/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/4510-2/lib/python3.7/site-packages/jax/lib/xla_bridge.py:130: UserWarning: No GPU/TPU found, falling back to CPU. warnings.warn('No GPU/TPU found, falling back to CPU.')
[-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
474 ms ± 5.63 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.
JAX NumPy functions work on regular NumPy arrays.
import numpy as np x = np.random.normal(size=(size, size)).astype(np.float32) %timeit jnp.dot(x, x.T).block_until_ready()
482 ms ± 4.11 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
from jax import device_put x = np.random.normal(size=(size, size)).astype(np.float32) x = device_put(x) %timeit jnp.dot(x, x.T).block_until_ready()
471 ms ± 3.83 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.
x = np.random.normal(size=(size, size)).astype(np.float32) %timeit np.dot(x, x.T)
523 ms ± 2.44 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
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’s three main ones:
jit, for speeding up your code
grad, for taking derivatives
vmap, for automatic vectorization or batching.
Let’s go over these, one-by-one. We’ll also end up composing these in interesting ways.
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,)) %timeit selu(x).block_until_ready()
4.97 ms ± 88.7 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) %timeit selu_jit(x).block_until_ready()
1.15 ms ± 11.2 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)
Taking derivatives with
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
def sum_logistic(x): return jnp.sum(1.0 / (1.0 + jnp.exp(-x))) x_small = jnp.arange(3.) derivative_fn = grad(sum_logistic) 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
jit compose and can be mixed arbitrarily. In the above example we jitted
sum_logistic and then took its derivative. We can go further:
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)))
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') %timeit naively_batched_apply_matrix(batched_x).block_until_ready()
Naively batched 4.57 ms ± 44.9 µs per loop (mean ± std. dev. of 7 runs, 100 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') %timeit batched_apply_matrix(batched_x).block_until_ready()
Manually batched 188 µs ± 5.49 µs per loop (mean ± std. dev. of 7 runs, 10000 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') %timeit vmap_batched_apply_matrix(batched_x).block_until_ready()
Auto-vectorized with vmap 233 µs ± 5.03 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)
vmap can be arbitrarily composed with
grad, and any other JAX transformation.
This is just a taste of what JAX can do. We’re really excited to see what you do with it!