Working with 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.

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 0x7fce5df87710>]   has 3 leaves: [1, 'a', <object object at 0x7fce5df87710>]
(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:

The most commonly used pytree function is 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]
] x: x*2, list_of_lists)
[[2, 4, 6], [2, 4], [2, 4, 6, 8]] also allows mapping a N-ary function over multiple arguments. For example:

another_list_of_lists = list_of_lists 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, 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 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:]):
        dict(weights=np.random.normal(size=(n_in, n_out)) * np.sqrt(2/n_in),
  return params

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

Use to check the shapes of the initial parameters: 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.

# Using the stochastic gradient descent, define the parameter update function.
# Apply `@jax.jit` for JIT compilation (speed).
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.
      lambda p, g: p - LEARNING_RATE * g, params, grads

Custom pytree nodes#

This section explains how in JAX you can extend the set of Python types that will be considered internal nodes in pytrees (pytree nodes) by using jax.tree_util.register_pytree_node() with

Why would you need this? In the previous examples, pytrees were shown as lists, tuples, and dicts, with everything else as pytree leaves. This is because if you define your own container class, it will be considered to be a pytree leaf unless you register it with JAX. This is also the case even if your container class has trees inside it. For example:

class Special(object):
  def __init__(self, x, y):
    self.x = x
    self.y = y

    Special(0, 1),
    Special(2, 4),
[<__main__.Special at 0x7fce6c127790>, <__main__.Special at 0x7fce6c126b60>]

Accordingly, if you try to use a expecting the leaves to be elements inside the container, you will get an error: x: x + 1,
    Special(0, 1),
    Special(2, 4)
TypeError: unsupported operand type(s) for +: 'Special' and 'int'

As a solution, JAX allows to extend the set of types to be considered internal pytree nodes through a global registry of types. Additionally, the values of registered types are traversed recursively.

First, register a new type using jax.tree_util.register_pytree_node():

from jax.tree_util import register_pytree_node

class RegisteredSpecial(Special):
  def __repr__(self):
    return "RegisteredSpecial(x={}, y={})".format(self.x, self.y)

def special_flatten(v):
  """Specifies a flattening recipe.

    v: The value of the registered type to flatten.
    A pair of an iterable with the children to be flattened recursively,
    and some opaque auxiliary data to pass back to the unflattening recipe.
    The auxiliary data is stored in the treedef for use during unflattening.
    The auxiliary data could be used, for example, for dictionary keys.
  children = (v.x, v.y)
  aux_data = None
  return (children, aux_data)

def special_unflatten(aux_data, children):
  """Specifies an unflattening recipe.

    aux_data: The opaque data that was specified during flattening of the
      current tree definition.
    children: The unflattened children

    A reconstructed object of the registered type, using the specified
    children and auxiliary data.
  return RegisteredSpecial(*children)

# Global registration
    special_flatten,    # Instruct JAX what are the children nodes.
    special_unflatten   # Instruct JAX how to pack back into a `RegisteredSpecial`.

Now you can traverse the special container structure: x: x + 1,
   RegisteredSpecial(0, 1),
   RegisteredSpecial(2, 4),
[RegisteredSpecial(x=1, y=2), RegisteredSpecial(x=3, y=5)]

Modern Python comes equipped with helpful tools to make defining containers easier. Some will work with JAX out-of-the-box, but others require more care.

For instance, a Python NamedTuple subclass doesn’t need to be registered to be considered a pytree node type:

from typing import NamedTuple, Any

class MyOtherContainer(NamedTuple):
  name: str
  a: Any
  b: Any
  c: Any

# NamedTuple subclasses are handled as pytree nodes, so
# this will work out-of-the-box.
    MyOtherContainer('Alice', 1, 2, 3),
    MyOtherContainer('Bob', 4, 5, 6)
['Alice', 1, 2, 3, 'Bob', 4, 5, 6]

Notice that the name field now appears as a leaf, because all tuple elements are children. This is what happens when you don’t have to register the class the hard way.

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 = x: x.shape, a_tree), 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

  • 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]

Custom pytrees and initialization with unexpected values#

Another common gotcha with user-defined pytree objects is that JAX transformations occasionally initialize them with unexpected values, so that any input validation done at initialization may fail. For example:

class MyTree:
  def __init__(self, a):
    self.a = jnp.asarray(a)

register_pytree_node(MyTree, lambda tree: ((tree.a,), None),
    lambda _, args: MyTree(*args))

tree = MyTree(jnp.arange(5.0))

jax.vmap(lambda x: x)(tree)      # Error because object() is passed to `MyTree`.
TypeError: Cannot interpret '<object object at 0x7fce5df87f80>' as a data type

The above exception was the direct cause of the following exception:

TypeError: Cannot determine dtype of <object object at 0x7fce5df87f80>

During handling of the above exception, another exception occurred:

TypeError: Value '<object object at 0x7fce5df87f80>' with dtype object is not a valid JAX array type. Only arrays of numeric types are supported by JAX.
jax.jacobian(lambda x: x)(tree)  # Error because MyTree(...) is passed to `MyTree`.
/home/docs/checkouts/ FutureWarning: None encountered in jnp.array(); this is currently treated as NaN. In the future this will result in an error.
  return array(a, dtype=dtype, copy=bool(copy), order=order)
TypeError: Cannot interpret '<object object at 0x7fce34e48480>' as a data type

The above exception was the direct cause of the following exception:

TypeError: Cannot determine dtype of <object object at 0x7fce34e48480>

During handling of the above exception, another exception occurred:

TypeError: Value '<object object at 0x7fce34e48480>' with dtype object is not a valid JAX array type. Only arrays of numeric types are supported by JAX.
  • In the first case with jax.vmap(...)(tree), JAX’s internals use arrays of object() values to infer the structure of the tree

  • In the second case with jax.jacobian(...)(tree), the Jacobian of a function mapping a tree to a tree is defined as a tree of trees.

Potential solution 1:

  • The __init__ and __new__ methods of custom pytree classes should generally avoid doing any array conversion or other input validation, or else anticipate and handle these special cases. For example:

class MyTree:
  def __init__(self, a):
    if not (type(a) is object or a is None or isinstance(a, MyTree)):
      a = jnp.asarray(a)
    self.a = a

Potential solution 2:

  • Structure your custom tree_unflatten function so that it avoids calling __init__. If you choose this route, make sure that your tree_unflatten function stays in sync with __init__ if and when the code is updated. Example:

def tree_unflatten(aux_data, children):
  del aux_data  # Unused in this class.
  obj = object.__new__(MyTree)
  obj.a = a
  return obj

Common pytree patterns#

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

Transposing pytrees with and jax.tree.transpose#

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

Option 1: Use 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 *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)]
{'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:

  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]}