Pytrees#

JAX has built-in support for objects that look like dictionaries (dicts) of arrays, or lists of lists of dicts, or other nested structures — in JAX these are called pytrees. This section will explain how to use them, provide useful code examples, and point out common “gotchas” and patterns.

For an explanation of how to create custom pytrees, see Custom pytree nodes.

What is a pytree?#

A pytree is a container-like structure built out of container-like Python objects — “leaf” pytrees and/or more pytrees. A pytree can include lists, tuples, and dicts. A leaf is anything that’s not a pytree, such as an array, but a single leaf is also a pytree.

In the context of machine learning (ML), a pytree can contain:

  • Model parameters

  • Dataset entries

  • Reinforcement learning agent observations

When working with datasets, you can often come across pytrees (such as lists of lists of dicts).

Below is an example of a simple pytree. In JAX, you can use jax.tree.leaves(), to extract the flattened leaves from the trees, as demonstrated here:

import jax
import jax.numpy as jnp

example_trees = [
    [1, 'a', object()],
    (1, (2, 3), ()),
    [1, {'k1': 2, 'k2': (3, 4)}, 5],
    {'a': 2, 'b': (2, 3)},
    jnp.array([1, 2, 3]),
]

# Print how many leaves the pytrees have.
for pytree in example_trees:
  # This `jax.tree.leaves()` method extracts the flattened leaves from the pytrees.
  leaves = jax.tree.leaves(pytree)
  print(f"{repr(pytree):<45} has {len(leaves)} leaves: {leaves}")

[1, 'a', <object object at 0x7ea80106c460>]   has 3 leaves: [1, 'a', <object object at 0x7ea80106c460>]
(1, (2, 3), ())                               has 3 leaves: [1, 2, 3]
[1, {'k1': 2, 'k2': (3, 4)}, 5]               has 5 leaves: [1, 2, 3, 4, 5]
{'a': 2, 'b': (2, 3)}                         has 3 leaves: [2, 2, 3]
Array([1, 2, 3], dtype=int32)                 has 1 leaves: [Array([1, 2, 3], dtype=int32)]

Any tree-like structure built out of container-like Python objects can be treated as a pytree in JAX. Classes are considered container-like if they are in the pytree registry, which by default includes lists, tuples, and dicts. Any object whose type is not in the pytree container registry will be treated as a leaf node in the tree.

The pytree registry can be extended to include user-defined container classes by registering the class with functions that specify how to flatten the tree; see Custom pytree nodes below.

Common pytree functions#

JAX provides a number of utilities to operate over pytrees. These can be found in the jax.tree_util subpackage; for convenience many of these have aliases in the jax.tree module.

Common function: jax.tree.map#

The most commonly used pytree function is jax.tree.map(). It works analogously to Python’s native map, but transparently operates over entire pytrees.

Here’s an example:

list_of_lists = [
    [1, 2, 3],
    [1, 2],
    [1, 2, 3, 4]
]

jax.tree.map(lambda x: x*2, list_of_lists)

[[2, 4, 6], [2, 4], [2, 4, 6, 8]]

jax.tree.map() also allows mapping a N-ary function over multiple arguments. For example:

another_list_of_lists = list_of_lists
jax.tree.map(lambda x, y: x+y, list_of_lists, another_list_of_lists)

[[2, 4, 6], [2, 4], [2, 4, 6, 8]]

When using multiple arguments with jax.tree.map(), the structure of the inputs must exactly match. That is, lists must have the same number of elements, dicts must have the same keys, etc.

Example of jax.tree.map with ML model parameters#

This example demonstrates how pytree operations can be useful when training a simple multi-layer perceptron (MLP).

Begin with defining the initial model parameters:

import numpy as np

def init_mlp_params(layer_widths):
  params = []
  for n_in, n_out in zip(layer_widths[:-1], layer_widths[1:]):
    params.append(
        dict(weights=np.random.normal(size=(n_in, n_out)) * np.sqrt(2/n_in),
             biases=np.ones(shape=(n_out,))
            )
    )
  return params

params = init_mlp_params([1, 128, 128, 1])

Use jax.tree.map() to check the shapes of the initial parameters:

jax.tree.map(lambda x: x.shape, params)

[{'biases': (128,), 'weights': (1, 128)},
 {'biases': (128,), 'weights': (128, 128)},
 {'biases': (1,), 'weights': (128, 1)}]

Next, define the functions for training the MLP model:

# Define the forward pass.
def forward(params, x):
  *hidden, last = params
  for layer in hidden:
    x = jax.nn.relu(x @ layer['weights'] + layer['biases'])
  return x @ last['weights'] + last['biases']

# Define the loss function.
def loss_fn(params, x, y):
  return jnp.mean((forward(params, x) - y) ** 2)

# Set the learning rate.
LEARNING_RATE = 0.0001

# Using the stochastic gradient descent, define the parameter update function.
# Apply `@jax.jit` for JIT compilation (speed).
@jax.jit
def update(params, x, y):
  # Calculate the gradients with `jax.grad`.
  grads = jax.grad(loss_fn)(params, x, y)
  # Note that `grads` is a pytree with the same structure as `params`.
  # `jax.grad` is one of many JAX functions that has
  # built-in support for pytrees.
  # This is useful - you can apply the SGD update using JAX pytree utilities.
  return jax.tree.map(
      lambda p, g: p - LEARNING_RATE * g, params, grads
  )

Viewing the pytree definition of an object#

To view the pytree definition of an arbitrary object for debugging purposes, you can use:

from jax.tree_util import tree_structure
print(tree_structure(object))

PyTreeDef(*)

Pytrees and JAX transformations#

Many JAX functions, like jax.lax.scan(), operate over pytrees of arrays. In addition, all JAX function transformations can be applied to functions that accept as input and produce as output pytrees of arrays.

Some JAX function transformations take optional parameters that specify how certain input or output values should be treated (such as the in_axes and out_axes arguments to jax.vmap()). These parameters can also be pytrees, and their structure must correspond to the pytree structure of the corresponding arguments. In particular, to be able to “match up” leaves in these parameter pytrees with values in the argument pytrees, the parameter pytrees are often constrained to be tree prefixes of the argument pytrees.

For example, if you pass the following input to jax.vmap() (note that the input arguments to a function are considered a tuple):

vmap(f, in_axes=(a1, {"k1": a2, "k2": a3}))

then you can use the following in_axes pytree to specify that only the k2 argument is mapped (axis=0), and the rest aren’t mapped over (axis=None):

vmap(f, in_axes=(None, {"k1": None, "k2": 0}))

The optional parameter pytree structure must match that of the main input pytree. However, the optional parameters can optionally be specified as a “prefix” pytree, meaning that a single leaf value can be applied to an entire sub-pytree.

For example, if you have the same jax.vmap() input as above, but wish to only map over the dictionary argument, you can use:

vmap(f, in_axes=(None, 0))  # equivalent to (None, {"k1": 0, "k2": 0})

Alternatively, if you want every argument to be mapped, you can write a single leaf value that is applied over the entire argument tuple pytree:

vmap(f, in_axes=0)  # equivalent to (0, {"k1": 0, "k2": 0})

This happens to be the default in_axes value for jax.vmap().

The same logic applies to other optional parameters that refer to specific input or output values of a transformed function, such as out_axes in jax.vmap().

Explicit key paths#

In a pytree each leaf has a key path. A key path for a leaf is a list of keys, where the length of the list is equal to the depth of the leaf in the pytree . Each key is a hashable object that represents an index into the corresponding pytree node type. The type of the key depends on the pytree node type; for example, the type of keys for dicts is different from the type of keys for tuples.

For built-in pytree node types, the set of keys for any pytree node instance is unique. For a pytree comprising nodes with this property, the key path for each leaf is unique.

JAX has the following jax.tree_util.* methods for working with key paths:

For example, one use case is to print debugging information related to a certain leaf value:

import collections

ATuple = collections.namedtuple("ATuple", ('name'))

tree = [1, {'k1': 2, 'k2': (3, 4)}, ATuple('foo')]
flattened, _ = jax.tree_util.tree_flatten_with_path(tree)

for key_path, value in flattened:
  print(f'Value of tree{jax.tree_util.keystr(key_path)}: {value}')

Value of tree[0]: 1
Value of tree[1]['k1']: 2
Value of tree[1]['k2'][0]: 3
Value of tree[1]['k2'][1]: 4
Value of tree[2].name: foo

To express key paths, JAX provides a few default key types for the built-in pytree node types, namely:

  • SequenceKey(idx: int): For lists and tuples.

  • DictKey(key: Hashable): For dictionaries.

  • GetAttrKey(name: str): For namedtuples and preferably custom pytree nodes (more in the next section)

You are free to define your own key types for your custom nodes. They will work with jax.tree_util.keystr() as long as their __str__() method is also overridden with a reader-friendly expression.

for key_path, _ in flattened:
  print(f'Key path of tree{jax.tree_util.keystr(key_path)}: {repr(key_path)}')

Key path of tree[0]: (SequenceKey(idx=0),)
Key path of tree[1]['k1']: (SequenceKey(idx=1), DictKey(key='k1'))
Key path of tree[1]['k2'][0]: (SequenceKey(idx=1), DictKey(key='k2'), SequenceKey(idx=0))
Key path of tree[1]['k2'][1]: (SequenceKey(idx=1), DictKey(key='k2'), SequenceKey(idx=1))
Key path of tree[2].name: (SequenceKey(idx=2), GetAttrKey(name='name'))

Common pytree gotchas#

This section covers some of the most common problems (“gotchas”) encountered when using JAX pytrees.

Mistaking pytree nodes for leaves#

A common gotcha to look out for is accidentally introducing tree nodes instead of leaves:

a_tree = [jnp.zeros((2, 3)), jnp.zeros((3, 4))]

# Try to make another pytree with ones instead of zeros.
shapes = jax.tree.map(lambda x: x.shape, a_tree)
jax.tree.map(jnp.ones, shapes)

[(Array([1., 1.], dtype=float32), Array([1., 1., 1.], dtype=float32)),
 (Array([1., 1., 1.], dtype=float32), Array([1., 1., 1., 1.], dtype=float32))]

What happened here is that the shape of an array is a tuple, which is a pytree node, with its elements as leaves. Thus, in the map, instead of calling jnp.ones on e.g. (2, 3), it’s called on 2 and 3.

The solution will depend on the specifics, but there are two broadly applicable options:

  • Rewrite the code to avoid the intermediate jax.tree.map().

  • Convert the tuple into a NumPy array (np.array) or a JAX NumPy array (jnp.array), which makes the entire sequence a leaf.

Handling of None by jax.tree_util#

jax.tree_util functions treat None as the absence of a pytree node, not as a leaf:

jax.tree.leaves([None, None, None])

[]

To treat None as a leaf, you can use the is_leaf argument:

jax.tree.leaves([None, None, None], is_leaf=lambda x: x is None)

[None, None, None]

Common pytree patterns#

This section covers some of the most common patterns with JAX pytrees.

Transposing pytrees with jax.tree.map and jax.tree.transpose#

To transpose a pytree (turn a list of trees into a tree of lists), JAX has two functions: jax.tree.map() (more basic) and jax.tree.transpose() (more flexible, complex and verbose).

Option 1: Use jax.tree.map(). Here’s an example:

def tree_transpose(list_of_trees):
  """
  Converts a list of trees of identical structure into a single tree of lists.
  """
  return jax.tree.map(lambda *xs: list(xs), *list_of_trees)

# Convert a dataset from row-major to column-major.
episode_steps = [dict(t=1, obs=3), dict(t=2, obs=4)]
tree_transpose(episode_steps)

{'obs': [3, 4], 't': [1, 2]}

Option 2: For more complex transposes, use jax.tree.transpose(), which is more verbose, but allows you specify the structure of the inner and outer pytree for more flexibility. For example:

jax.tree.transpose(
  outer_treedef = jax.tree.structure([0 for e in episode_steps]),
  inner_treedef = jax.tree.structure(episode_steps[0]),
  pytree_to_transpose = episode_steps
)

{'obs': [3, 4], 't': [1, 2]}

Extending pytrees#

Material on extending pytrees has been moved to Custom pytree nodes.