Alex Minnaar

15 Aug 2020

According to it’s README JAX is “Autograd and XLA, brought together for high-performance machine learning research” from Google. Autograd is a reference to an automatic differentiation library which was originally maintained by the Harvard Intelligent Probabilistic Systems Group (HIPS). XLA is a reference to Tensorflow’s XLA (Accelerated Linear Algebra) compiler. JAX also says “At its core, JAX is an extensible system for transforming numerical functions. Here are four of primary interest: grad, jit, vmap, and pmap”. At this point, these four functions make up the bulk of JAX so this blog post will go through each of them and doing so should provide a good overview of JAX in general.

grad is JAX’s automatic differentiation function. It accepts (nearly) any function and returns a function that is it’s derivative. For example, say you want the derivative of the $$tanh$$ function, you can do this with grad in one line as

import jax



Autograd works in a similar way to the method descibed in my old computational graph post. The basic steps are

1. Tracing the Function: This means parsing the the input function into it's individual ops and building a computational graph where each node is an op.
2. Topological Sort: This means putting the ops in order based on their dependencies in the computational graph.
3. Implement VJPs (vector-Jacobian products) for each Op: One way to effectively compute an op's derivative in the context of computational graphs is to use it's VJP. Check out these slides for a more in-depth explanation.
4. Backpropagate: Using the computational graph and the VJP's for each node/op you can backpropagate through the graph to obtain a derivative value at any point.

Since the above process involves going backwards through the computational graph it is called reverse-mode differentiation.

## jit (XLA)

The jit function exposes the XLA compiler. For example, say you want a function “jited” - all you need to do is pass the function through the jit higher order function and the output is the “jited” version of that function i.e.

import jax

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

fast_selu = jax.jit(selu)


But what does this actually do? In terms of XLA, the jit compiler uses GPU kernel fusion to apply significant performance improvements to the input function. Without XLA, typically a function would be parsed into it’s constituent ops where each op has it’s own kernel that needs to be launched. The problem with this is that each kernel launch requires it’s own memory operations which can be particularly costly in memory-bound computations. XLA side-steps this problem by fusing the ops into one kernel requiring the costly memory opertations to only happen once.

Of course XLA is it’s own project within TensorFlow so you will not find it’s implementation in the JAX project. Behind the scenes JAX interacts with XLA via a downloaded XLA shared object.

## vmap

vmap is short for vectorizing map and it does just that. A map is a particular kind of higher-order function which applies some function argument to each element of a sequential data structure then returns the result (more about maps). vmap does the same thing except it vectorizes the process for better performance.

As a concrete example, consider the example from JAX’s README. Let’s start with a predict function which does a forward pass through a standard MLP neural network taking a vector input.

def predict(params, input_vec):
assert input_vec.ndim == 1
for W, b in params:
output_vec = jnp.dot(W, input_vec) + b
input_vec = jnp.tanh(output_vec)
return output_vec


The important thing to note about the above function is that input_vec is a vector so if you wish to predict a batch of input vectors then you would need to call the predict function on each individual vector sequentially i.e. a map function. vmap can vectorize this map function in the following way.

from jax import vmap

vectorized_predict = vmap(predict, in_axis=(None, 0))


The in_axis argument is a tuple that indicates which axis of the inputs to map over. We don’t want to map over the params argument so that is given a None value. We do want to map over input_vec though. Assume the vectorized input is a sequence of input vectors i.e. a matrix. We would want to vectorize over the rows of that matrix i.e. the 0 dimension. Now we can call the vectorized predict function on an entire batch of of inputs.

fast_predictions = vectorized_predict(params, input_batch)


## pmap

pmap is a higher-order map function that executes functions across multiple GPUs in parallel. The semantics of pmap are relatively simple. pmap’s arugment is the function that is to be parallelized. What is returned is a function that executes across the devices and takes an argument for each devices. For example, from the README, we create a parallelized function that creates random matrices across devices based on some key argument.

from jax import random, pmap

parallel_matrix_create = pmap(lambda key: random.normal(key, (5000, 6000)))


Now we create inputs corresponding to each device (let’s say there are 8) and apply them to the parallelized function.

keys = random.split(random.PRNGKey(0), 8)
mats = parrallel_matrix_create(keys)


That’s it - thanks for reading.