Public API: jax package

Subpackages

• jax.numpy package
• jax.scipy package
• jax.experimental package
• jax.image package
• jax.lax package
• jax.nn package
• jax.ops package
• jax.random package
• jax.tree_util package
• jax.dlpack module
• jax.profiler module

Just-in-time compilation (jit)

jit(fun[, static_argnums, device, backend, …])	Sets up fun for just-in-time compilation with XLA.
disable_jit()	Context manager that disables jit() behavior under its dynamic context.
xla_computation(fun[, static_argnums, …])	Creates a function that produces its XLA computation given example args.
make_jaxpr(fun[, static_argnums, return_shape])	Creates a function that produces its jaxpr given example args.
eval_shape(fun, *args, **kwargs)	Compute the shape/dtype of fun without any FLOPs.
device_put(x[, device])	Transfers x to device.
named_call(fun, *[, name])	Adds a user specified name to a function when staging out JAX computations.

Automatic differentiation

grad(fun[, argnums, has_aux, holomorphic, …])	Creates a function which evaluates the gradient of fun.
value_and_grad(fun[, argnums, has_aux, …])	Create a function which evaluates both fun and the gradient of fun.
jacfwd(fun[, argnums, holomorphic])	Jacobian of fun evaluated column-by-column using forward-mode AD.
jacrev(fun[, argnums, holomorphic, allow_int])	Jacobian of fun evaluated row-by-row using reverse-mode AD.
hessian(fun[, argnums, holomorphic])	Hessian of fun as a dense array.
jvp(fun, primals, tangents)	Computes a (forward-mode) Jacobian-vector product of fun.
linearize(fun, *primals)	Produces a linear approximation to fun using jvp() and partial eval.
linear_transpose(fun, *primals)	Transpose a function that is promised to be linear.
vjp(fun, *primals[, has_aux])	Compute a (reverse-mode) vector-Jacobian product of fun.
custom_jvp(fun[, nondiff_argnums])	Set up a JAX-transformable function for a custom JVP rule definition.
custom_vjp(fun[, nondiff_argnums])	Set up a JAX-transformable function for a custom VJP rule definition.
checkpoint(fun[, concrete])	Make fun recompute internal linearization points when differentiated.

Vectorization (vmap)

vmap(fun[, in_axes, out_axes, axis_name])	Vectorizing map.
jax.numpy.vectorize(pyfunc, *[, excluded, …])	Define a vectorized function with broadcasting.

Parallelization (pmap)

pmap(fun[, axis_name, in_axes, out_axes, …])	Parallel map with support for collective operations.
devices([backend])	Returns a list of all devices for a given backend.
local_devices([host_id, backend])	Like jax.devices(), but only returns devices local to a given host.
host_id([backend])	Returns the integer host ID of this host.
host_ids([backend])	Returns a sorted list of all host IDs.
device_count([backend])	Returns the total number of devices.
local_device_count([backend])	Returns the number of devices on this host.
host_count([backend])	Returns the number of hosts.

jax.jit(fun, static_argnums=(), device=None, backend=None, donate_argnums=())

Sets up fun for just-in-time compilation with XLA.

Parameters

• fun (~F) – Function to be jitted. Should be a pure function, as side-effects may only be executed once. Its arguments and return value should be arrays, scalars, or (nested) standard Python containers (tuple/list/dict) thereof. Positional arguments indicated by static_argnums can be anything at all, provided they are hashable and have an equality operation defined. Static arguments are included as part of a compilation cache key, which is why hash and equality operators must be defined.

• static_argnums (Union[int, Iterable[int]]) – An int or collection of ints specifying which positional arguments to treat as static (compile-time constant). Operations that only depend on static arguments will be constant-folded in Python (during tracing), and so the corresponding argument values can be any Python object. Static arguments should be hashable, meaning both __hash__ and __eq__ are implemented, and immutable. Calling the jitted function with different values for these constants will trigger recompilation. If the jitted function is called with fewer positional arguments than indicated by static_argnums then an error is raised. Arguments that are not arrays or containers thereof must be marked as static. Defaults to ().

• device (Optional[Device]) – This is an experimental feature and the API is likely to change. Optional, the Device the jitted function will run on. (Available devices can be retrieved via jax.devices().) The default is inherited from XLA’s DeviceAssignment logic and is usually to use jax.devices()[0].

• backend (Optional[str]) – This is an experimental feature and the API is likely to change. Optional, a string representing the XLA backend: 'cpu', 'gpu', or 'tpu'.

• donate_argnums (Union[int, Iterable[int]]) – Specify which arguments are “donated” to the computation. It is safe to donate arguments if you no longer need them once the computation has finished. In some cases XLA can make use of donated buffers to reduce the amount of memory needed to perform a computation, for example recycling one of your input buffers to store a result. You should not reuse buffers that you donate to a computation, JAX will raise an error if you try to.

Return type

~F

Returns

A wrapped version of fun, set up for just-in-time compilation.

In the following example, selu can be compiled into a single fused kernel by XLA:

>>> import jax
>>>
>>> @jax.jit
... def selu(x, alpha=1.67, lmbda=1.05):
...   return lmbda * jax.numpy.where(x > 0, x, alpha * jax.numpy.exp(x) - alpha)
>>>
>>> key = jax.random.PRNGKey(0)
>>> x = jax.random.normal(key, (10,))
>>> print(selu(x))  
[-0.54485  0.27744 -0.29255 -0.91421 -0.62452 -0.24748
 -0.85743 -0.78232  0.76827  0.59566 ]

jax.disable_jit()

Context manager that disables jit() behavior under its dynamic context.

For debugging it is useful to have a mechanism that disables jit() everywhere in a dynamic context.

Values that have a data dependence on the arguments to a jitted function are traced and abstracted. For example, an abstract value may be a ShapedArray instance, representing the set of all possible arrays with a given shape and dtype, but not representing one concrete array with specific values. You might notice those if you use a benign side-effecting operation in a jitted function, like a print:

>>> import jax
>>>
>>> @jax.jit
... def f(x):
...   y = x * 2
...   print("Value of y is", y)
...   return y + 3
...
>>> print(f(jax.numpy.array([1, 2, 3])))
Value of y is Traced<ShapedArray(int32[3])>with<DynamicJaxprTrace(level=0/1)>
[5 7 9]

Here y has been abstracted by jit() to a ShapedArray, which represents an array with a fixed shape and type but an arbitrary value. The value of y is also traced. If we want to see a concrete value while debugging, and avoid the tracer too, we can use the disable_jit() context manager:

>>> import jax
>>>
>>> with jax.disable_jit():
...   print(f(jax.numpy.array([1, 2, 3])))
...
Value of y is [2 4 6]
[5 7 9]

jax.xla_computation(fun, static_argnums=(), axis_env=None, in_parts=None, out_parts=None, backend=None, tuple_args=False, instantiate_const_outputs=None, return_shape=False, donate_argnums=())

Creates a function that produces its XLA computation given example args.

Parameters

• fun (Callable) – Function from which to form XLA computations.

• static_argnums (Union[int, Iterable[int]]) – See the jax.jit() docstring.

• axis_env (Optional[Sequence[Tuple[Any, int]]]) – Optional, a sequence of pairs where the first element is an axis name and the second element is a positive integer representing the size of the mapped axis with that name. This parameter is useful when lowering functions that involve parallel communication collectives, and it specifies the axis name/size environment that would be set up by applications of jax.pmap(). See the examples below.

• in_parts – Optional, how each argument to fun should be partitioned or replicated. This is used to specify partitioned XLA computations, see sharded_jit for more info.

• out_parts – Optional, how each output of fun should be partitioned or replicated. This is used to specify partitioned XLA computations, see sharded_jit for more info.

• backend (Optional[str]) – This is an experimental feature and the API is likely to change. Optional, a string representing the XLA backend: 'cpu', 'gpu', or 'tpu'.

• tuple_args (bool) – Optional bool, defaults to False. If True, the resulting XLA computation will have a single tuple argument that is unpacked into the specified function arguments. If None, tupling will be enabled when there are more than 100 arguments, since some platforms have limits on argument arity.

• instantiate_const_outputs (Optional[bool]) – Deprecated argument, does nothing.

• return_shape (bool) – Optional boolean, defaults to False. If True, the wrapped function returns a pair where the first element is the XLA computation and the second element is a pytree with the same structure as the output of fun and where the leaves are objects with shape and dtype attributes representing the corresponding types of the output leaves.

• donate_argnums (Union[int, Iterable[int]]) – Specify which arguments are “donated” to the computation. It is safe to donate arguments if you no longer need them once the computation has finished. In some cases XLA can make use of donated buffers to reduce the amount of memory needed to perform a computation, for example recycling one of your input buffers to store a result. You should not reuse buffers that you donate to a computation, JAX will raise an error if you try to.

Return type

Callable

Returns

A wrapped version of fun that when applied to example arguments returns a built XLA Computation (see xla_client.py), from which representations of the unoptimized XLA HLO computation can be extracted using methods like as_hlo_text, as_serialized_hlo_module_proto, and as_hlo_dot_graph. If the argument return_shape is True, then the wrapped function returns a pair where the first element is the XLA Computation and the second element is a pytree representing the structure, shapes, and dtypes of the output of fun.

Concrete example arguments are not always necessary. For those arguments not indicated by static_argnums, any object with shape and dtype attributes is acceptable (excepting namedtuples, which are treated as Python containers).

For example:

>>> import jax
>>>
>>> def f(x): return jax.numpy.sin(jax.numpy.cos(x))
>>> c = jax.xla_computation(f)(3.)
>>> print(c.as_hlo_text())  
HloModule xla_computation_f.6

ENTRY xla_computation_f.6 {
  constant.2 = pred[] constant(false)
  parameter.1 = f32[] parameter(0)
  cosine.3 = f32[] cosine(parameter.1)
  sine.4 = f32[] sine(cosine.3)
  ROOT tuple.5 = (f32[]) tuple(sine.4)
}

Alternatively, the assignment to c above could be written:

>>> import types
>>> scalar = types.SimpleNamespace(shape=(), dtype=np.float32)
>>> c = jax.xla_computation(f)(scalar)

Here’s an example that involves a parallel collective and axis name:

>>> def f(x): return x - jax.lax.psum(x, 'i')
>>> c = jax.xla_computation(f, axis_env=[('i', 4)])(2)
>>> print(c.as_hlo_text())  
HloModule jaxpr_computation.9
primitive_computation.3 {
  parameter.4 = s32[] parameter(0)
  parameter.5 = s32[] parameter(1)
  ROOT add.6 = s32[] add(parameter.4, parameter.5)
}
ENTRY jaxpr_computation.9 {
  tuple.1 = () tuple()
  parameter.2 = s32[] parameter(0)
  all-reduce.7 = s32[] all-reduce(parameter.2), replica_groups={{0,1,2,3}}, to_apply=primitive_computation.3
  ROOT subtract.8 = s32[] subtract(parameter.2, all-reduce.7)
}

Notice the replica_groups that were generated. Here’s an example that generates more interesting replica_groups:

>>> from jax import lax
>>> def g(x):
...   rowsum = lax.psum(x, 'i')
...   colsum = lax.psum(x, 'j')
...   allsum = lax.psum(x, ('i', 'j'))
...   return rowsum, colsum, allsum
...
>>> axis_env = [('i', 4), ('j', 2)]
>>> c = xla_computation(g, axis_env=axis_env)(5.)
>>> print(c.as_hlo_text())  
HloModule jaxpr_computation__1.19
[removed uninteresting text here]
ENTRY jaxpr_computation__1.19 {
  tuple.1 = () tuple()
  parameter.2 = f32[] parameter(0)
  all-reduce.7 = f32[] all-reduce(parameter.2), replica_groups={{0,2,4,6},{1,3,5,7}}, to_apply=primitive_computation__1.3
  all-reduce.12 = f32[] all-reduce(parameter.2), replica_groups={{0,1},{2,3},{4,5},{6,7}}, to_apply=primitive_computation__1.8
  all-reduce.17 = f32[] all-reduce(parameter.2), replica_groups={{0,1,2,3,4,5,6,7}}, to_apply=primitive_computation__1.13
  ROOT tuple.18 = (f32[], f32[], f32[]) tuple(all-reduce.7, all-reduce.12, all-reduce.17)
}

jax.make_jaxpr(fun, static_argnums=(), return_shape=False)

Creates a function that produces its jaxpr given example args.

Parameters

• fun (Callable) – The function whose jaxpr is to be computed. Its positional arguments and return value should be arrays, scalars, or standard Python containers (tuple/list/dict) thereof.

• static_argnums (Union[int, Iterable[int]]) – See the jax.jit() docstring.

• return_shape (bool) – Optional boolean, defaults to False. If True, the wrapped function returns a pair where the first element is the jaxpr and the second element is a pytree with the same structure as the output of fun and where the leaves are objects with shape and dtype attributes representing the corresponding types of the output leaves.

Return type

Callable[…, ClosedJaxpr]

Returns

A wrapped version of fun that when applied to example arguments returns a ClosedJaxpr representation of fun on those arguments. If the argument return_shape is True, then the returned function instead returns a pair where the first element is the ClosedJaxpr representation of fun and the second element is a pytree representing the structure, shape, and dtypes of the output of fun.

A jaxpr is JAX’s intermediate representation for program traces. The jaxpr language is based on the simply-typed first-order lambda calculus with let-bindings. make_jaxpr() adapts a function to return its jaxpr, which we can inspect to understand what JAX is doing internally. The jaxpr returned is a trace of fun abstracted to ShapedArray level. Other levels of abstraction exist internally.

We do not describe the semantics of the jaxpr language in detail here, but instead give a few examples.

>>> import jax
>>>
>>> def f(x): return jax.numpy.sin(jax.numpy.cos(x))
>>> print(f(3.0))
-0.83602
>>> jax.make_jaxpr(f)(3.0)
{ lambda  ; a.
  let b = cos a
      c = sin b
  in (c,) }
>>> jax.make_jaxpr(jax.grad(f))(3.0)
{ lambda  ; a.
  let b = cos a
      c = sin a
      _ = sin b
      d = cos b
      e = mul 1.0 d
      f = neg e
      g = mul f c
  in (g,) }

jax.eval_shape(fun, *args, **kwargs)

Compute the shape/dtype of fun without any FLOPs.

This utility function is useful for performing shape inference. Its input/output behavior is defined by:

def eval_shape(fun, *args, **kwargs):
  out = fun(*args, **kwargs)
  return jax.tree_util.tree_map(shape_dtype_struct, out)

def shape_dtype_struct(x):
  return ShapeDtypeStruct(x.shape, x.dtype)

class ShapeDtypeStruct:
  __slots__ = ["shape", "dtype"]
  def __init__(self, shape, dtype):
    self.shape = shape
    self.dtype = dtype

In particular, the output is a pytree of objects that have shape and dtype attributes, but nothing else about them is guaranteed by the API.

But instead of applying fun directly, which might be expensive, it uses JAX’s abstract interpretation machinery to evaluate the shapes without doing any FLOPs.

Using eval_shape() can also catch shape errors, and will raise same shape errors as evaluating fun(*args, **kwargs).

Parameters

• fun (Callable) – The function whose output shape should be evaluated.

• *args – a positional argument tuple of arrays, scalars, or (nested) standard Python containers (tuples, lists, dicts, namedtuples, i.e. pytrees) of those types. Since only the shape and dtype attributes are accessed, only values that duck-type arrays are required, rather than real ndarrays. The duck-typed objects cannot be namedtuples because those are treated as standard Python containers. See the example below.

• **kwargs – a keyword argument dict of arrays, scalars, or (nested) standard Python containers (pytrees) of those types. As in args, array values need only be duck-typed to have shape and dtype attributes.

For example:

>>> import jax
>>> import jax.numpy as jnp
>>>
>>> f = lambda A, x: jnp.tanh(jnp.dot(A, x))
>>> class MyArgArray(object):
...   def __init__(self, shape, dtype):
...     self.shape = shape
...     self.dtype = dtype
...
>>> A = MyArgArray((2000, 3000), jnp.float32)
>>> x = MyArgArray((3000, 1000), jnp.float32)
>>> out = jax.eval_shape(f, A, x)  # no FLOPs performed
>>> print(out.shape)
(2000, 1000)
>>> print(out.dtype)
float32

jax.device_put(x, device=None)

Transfers x to device.

Parameters

• x – An array, scalar, or (nested) standard Python container thereof.

• device (Optional[Device]) – The (optional) Device to which x should be transferred. If given, then the result is committed to the device.

If the device parameter is None, then this operation behaves like the identity function if the operand is on any device already, otherwise it transfers the data to the default device, uncommitted.

For more details on data placement see the FAQ on data placement.

Returns

A copy of x that resides on device.

jax.named_call(fun, *, name=None)

Adds a user specified name to a function when staging out JAX computations.

When staging out computations for just-in-time compilation to XLA (or other backends such as TensorFlow) JAX runs your Python program but by default does not preserve any of the function names or other metadata associated with it. This can make debugging the staged out (and/or compiled) representation of your program complicated because there is limited context information for each operation being executed.

named_call tells JAX to stage the given function out as a subcomputation with a specific name. When the staged out program is compiled with XLA these named subcomputations are preserved and show up in debugging utilities like the TensorFlow Profiler in TensorBoard. Names are also preserved when staging out JAX programs to TensorFlow using experimental.jax2tf.convert().

Parameters

• fun (Callable[…, Any]) – Function to be wrapped. This can be any Callable.

• name (Optional[str]) – Optional. The prefix to use to name all sub computations created within the name scope. Use the fun.__name__ if not specified.

Return type

Callable[…, Any]

Returns

A version of fun that is wrapped in a name_scope.

jax.grad(fun, argnums=0, has_aux=False, holomorphic=False, allow_int=False)

Creates a function which evaluates the gradient of fun.

Parameters

• fun (Callable) – Function to be differentiated. Its arguments at positions specified by argnums should be arrays, scalars, or standard Python containers. Argument arrays in the positions specified by argnums must be of inexact (i.e., floating-point or complex) type. It should return a scalar (which includes arrays with shape () but not arrays with shape (1,) etc.)

• argnums (Union[int, Sequence[int]]) – Optional, integer or sequence of integers. Specifies which positional argument(s) to differentiate with respect to (default 0).

• has_aux (bool) – Optional, bool. Indicates whether fun returns a pair where the first element is considered the output of the mathematical function to be differentiated and the second element is auxiliary data. Default False.

• holomorphic (bool) – Optional, bool. Indicates whether fun is promised to be holomorphic. If True, inputs and outputs must be complex. Default False.

• allow_int (bool) – Optional, bool. Whether to allow differentiating with respect to integer valued inputs. The gradient of an integer input will have a trivial vector-space dtype (float0). Default False.

Return type

Callable

Returns

A function with the same arguments as fun, that evaluates the gradient of fun. If argnums is an integer then the gradient has the same shape and type as the positional argument indicated by that integer. If argnums is a tuple of integers, the gradient is a tuple of values with the same shapes and types as the corresponding arguments. If has_aux is True then a pair of (gradient, auxiliary_data) is returned.

For example:

>>> import jax
>>>
>>> grad_tanh = jax.grad(jax.numpy.tanh)
>>> print(grad_tanh(0.2))
0.961043

jax.value_and_grad(fun, argnums=0, has_aux=False, holomorphic=False, allow_int=False)

Create a function which evaluates both fun and the gradient of fun.

Parameters

• fun (Callable) – Function to be differentiated. Its arguments at positions specified by argnums should be arrays, scalars, or standard Python containers. It should return a scalar (which includes arrays with shape () but not arrays with shape (1,) etc.)

• argnums (Union[int, Sequence[int]]) – Optional, integer or sequence of integers. Specifies which positional argument(s) to differentiate with respect to (default 0).

• has_aux (bool) – Optional, bool. Indicates whether fun returns a pair where the first element is considered the output of the mathematical function to be differentiated and the second element is auxiliary data. Default False.

• holomorphic (bool) – Optional, bool. Indicates whether fun is promised to be holomorphic. If True, inputs and outputs must be complex. Default False.

• allow_int (bool) – Optional, bool. Whether to allow differentiating with respect to integer valued inputs. The gradient of an integer input will have a trivial vector-space dtype (float0). Default False.

Return type

Callable[…, Tuple[Any, Any]]

Returns

A function with the same arguments as fun that evaluates both fun and the gradient of fun and returns them as a pair (a two-element tuple). If argnums is an integer then the gradient has the same shape and type as the positional argument indicated by that integer. If argnums is a sequence of integers, the gradient is a tuple of values with the same shapes and types as the corresponding arguments.

jax.jacfwd(fun, argnums=0, holomorphic=False)

Jacobian of fun evaluated column-by-column using forward-mode AD.

Parameters

• fun (Callable) – Function whose Jacobian is to be computed.

• argnums (Union[int, Sequence[int]]) – Optional, integer or sequence of integers. Specifies which positional argument(s) to differentiate with respect to (default 0).

• holomorphic (bool) – Optional, bool. Indicates whether fun is promised to be holomorphic. Default False.

Return type

Callable

Returns

A function with the same arguments as fun, that evaluates the Jacobian of fun using forward-mode automatic differentiation.

>>> import jax
>>> import jax.numpy as jnp
>>>
>>> def f(x):
...   return jnp.asarray(
...     [x[0], 5*x[2], 4*x[1]**2 - 2*x[2], x[2] * jnp.sin(x[0])])
...
>>> print(jax.jacfwd(f)(jnp.array([1., 2., 3.])))
[[ 1.       0.       0.     ]
 [ 0.       0.       5.     ]
 [ 0.      16.      -2.     ]
 [ 1.6209   0.       0.84147]]

jax.jacrev(fun, argnums=0, holomorphic=False, allow_int=False)

Jacobian of fun evaluated row-by-row using reverse-mode AD.

Parameters

• fun (Callable) – Function whose Jacobian is to be computed.

• argnums (Union[int, Sequence[int]]) – Optional, integer or sequence of integers. Specifies which positional argument(s) to differentiate with respect to (default 0).

• holomorphic (bool) – Optional, bool. Indicates whether fun is promised to be holomorphic. Default False.

• allow_int (bool) – Optional, bool. Whether to allow differentiating with respect to integer valued inputs. The gradient of an integer input will have a trivial vector-space dtype (float0). Default False.

Return type

Callable

Returns

A function with the same arguments as fun, that evaluates the Jacobian of fun using reverse-mode automatic differentiation.

>>> import jax
>>> import jax.numpy as jnp
>>>
>>> def f(x):
...   return jnp.asarray(
...     [x[0], 5*x[2], 4*x[1]**2 - 2*x[2], x[2] * jnp.sin(x[0])])
...
>>> print(jax.jacrev(f)(jnp.array([1., 2., 3.])))
[[ 1.       0.       0.     ]
 [ 0.       0.       5.     ]
 [ 0.      16.      -2.     ]
 [ 1.6209   0.       0.84147]]

jax.hessian(fun, argnums=0, holomorphic=False)

Hessian of fun as a dense array.

Parameters

• fun (Callable) – Function whose Hessian is to be computed. Its arguments at positions specified by argnums should be arrays, scalars, or standard Python containers thereof. It should return arrays, scalars, or standard Python containers thereof.

• argnums (Union[int, Sequence[int]]) – Optional, integer or sequence of integers. Specifies which positional argument(s) to differentiate with respect to (default 0).

• holomorphic (bool) – Optional, bool. Indicates whether fun is promised to be holomorphic. Default False.

Return type

Callable

Returns

A function with the same arguments as fun, that evaluates the Hessian of fun.

>>> import jax
>>>
>>> g = lambda x: x[0]**3 - 2*x[0]*x[1] - x[1]**6
>>> print(jax.hessian(g)(jax.numpy.array([1., 2.])))
[[   6.   -2.]
 [  -2. -480.]]

hessian() is a generalization of the usual definition of the Hessian that supports nested Python containers (i.e. pytrees) as inputs and outputs. The tree structure of jax.hessian(fun)(x) is given by forming a tree product of the structure of fun(x) with a tree product of two copies of the structure of x. A tree product of two tree structures is formed by replacing each leaf of the first tree with a copy of the second. For example:

>>> import jax.numpy as jnp
>>> f = lambda dct: {"c": jnp.power(dct["a"], dct["b"])}
>>> print(jax.hessian(f)({"a": jnp.arange(2.) + 1., "b": jnp.arange(2.) + 2.}))
{'c': {'a': {'a': DeviceArray([[[ 2.,  0.], [ 0.,  0.]],
                               [[ 0.,  0.], [ 0., 12.]]], dtype=float32),
             'b': DeviceArray([[[ 1.      ,  0.      ], [ 0.      ,  0.      ]],
                               [[ 0.      ,  0.      ], [ 0.      , 12.317766]]], dtype=float32)},
       'b': {'a': DeviceArray([[[ 1.      ,  0.      ], [ 0.      ,  0.      ]],
                               [[ 0.      ,  0.      ], [ 0.      , 12.317766]]], dtype=float32),
             'b': DeviceArray([[[0.      , 0.      ], [0.      , 0.      ]],
                              [[0.      , 0.      ], [0.      , 3.843624]]], dtype=float32)}}}

Thus each leaf in the tree structure of jax.hessian(fun)(x) corresponds to a leaf of fun(x) and a pair of leaves of x. For each leaf in jax.hessian(fun)(x), if the corresponding array leaf of fun(x) has shape (out_1, out_2, ...) and the corresponding array leaves of x have shape (in_1_1, in_1_2, ...) and (in_2_1, in_2_2, ...) respectively, then the Hessian leaf has shape (out_1, out_2, ..., in_1_1, in_1_2, ..., in_2_1, in_2_2, ...). In other words, the Python tree structure represents the block structure of the Hessian, with blocks determined by the input and output pytrees.

In particular, an array is produced (with no pytrees involved) when the function input x and output fun(x) are each a single array, as in the g example above. If fun(x) has shape (out1, out2, ...) and x has shape (in1, in2, ...) then jax.hessian(fun)(x) has shape (out1, out2, ..., in1, in2, ..., in1, in2, ...). To flatten pytrees into 1D vectors, consider using jax.flatten_util.flatten_pytree().

jax.jvp(fun, primals, tangents)

Computes a (forward-mode) Jacobian-vector product of fun.

Parameters

• fun (Callable) – Function to be differentiated. Its arguments should be arrays, scalars, or standard Python containers of arrays or scalars. It should return an array, scalar, or standard Python container of arrays or scalars.

• primals – The primal values at which the Jacobian of fun should be evaluated. Should be either a tuple or a list of arguments, and its length should equal to the number of positional parameters of fun.

• tangents – The tangent vector for which the Jacobian-vector product should be evaluated. Should be either a tuple or a list of tangents, with the same tree structure and array shapes as primals.

Return type

Tuple[Any, Any]

Returns

A (primals_out, tangents_out) pair, where primals_out is fun(*primals), and tangents_out is the Jacobian-vector product of function evaluated at primals with tangents. The tangents_out value has the same Python tree structure and shapes as primals_out.

For example:

>>> import jax
>>>
>>> y, v = jax.jvp(jax.numpy.sin, (0.1,), (0.2,))
>>> print(y)
0.09983342
>>> print(v)
0.19900084

jax.linearize(fun, *primals)

Produces a linear approximation to fun using jvp() and partial eval.

Parameters

• fun (Callable) – Function to be differentiated. Its arguments should be arrays, scalars, or standard Python containers of arrays or scalars. It should return an array, scalar, or standard python container of arrays or scalars.

• primals – The primal values at which the Jacobian of fun should be evaluated. Should be a tuple of arrays, scalar, or standard Python container thereof. The length of the tuple is equal to the number of positional parameters of fun.

Return type

Tuple[Any, Callable]

Returns

A pair where the first element is the value of f(*primals) and the second element is a function that evaluates the (forward-mode) Jacobian-vector product of fun evaluated at primals without re-doing the linearization work.

In terms of values computed, linearize() behaves much like a curried jvp(), where these two code blocks compute the same values:

y, out_tangent = jax.jvp(f, (x,), (in_tangent,))

y, f_jvp = jax.linearize(f, x)
out_tangent = f_jvp(in_tangent)

However, the difference is that linearize() uses partial evaluation so that the function f is not re-linearized on calls to f_jvp. In general that means the memory usage scales with the size of the computation, much like in reverse-mode. (Indeed, linearize() has a similar signature to vjp()!)

This function is mainly useful if you want to apply f_jvp multiple times, i.e. to evaluate a pushforward for many different input tangent vectors at the same linearization point. Moreover if all the input tangent vectors are known at once, it can be more efficient to vectorize using vmap(), as in:

pushfwd = partial(jvp, f, (x,))
y, out_tangents = vmap(pushfwd, out_axes=(None, 0))((in_tangents,))

By using vmap() and jvp() together like this we avoid the stored-linearization memory cost that scales with the depth of the computation, which is incurred by both linearize() and vjp().

Here’s a more complete example of using linearize():

>>> import jax
>>> import jax.numpy as jnp
>>>
>>> def f(x): return 3. * jnp.sin(x) + jnp.cos(x / 2.)
...
>>> jax.jvp(f, (2.,), (3.,))
(DeviceArray(3.26819, dtype=float32), DeviceArray(-5.00753, dtype=float32))
>>> y, f_jvp = jax.linearize(f, 2.)
>>> print(y)
3.2681944
>>> print(f_jvp(3.))
-5.007528
>>> print(f_jvp(4.))
-6.676704

jax.linear_transpose(fun, *primals)

Transpose a function that is promised to be linear.

For linear functions, this transformation is equivalent to vjp, but avoids the overhead of computing the forward pass.

The outputs of the transposed function will always have the exact same dtypes as primals, even if some values are truncated (e.g., from complex to float, or from float64 to float32). To avoid truncation, use dtypes in primals that match the full range of desired outputs from the transposed function. Integer dtypes are not supported.

Parameters

• fun (Callable) – the linear function to be transposed.

• *primals – a positional argument tuple of arrays, scalars, or (nested) standard Python containers (tuples, lists, dicts, namedtuples, i.e., pytrees) of those types used for evaluating the shape/dtype of fun(*primals). These arguments may be real scalars/ndarrays, but that is not required: only the shape and dtype attributes are accessed. See below for an example. (Note that the duck-typed objects cannot be namedtuples because those are treated as standard Python containers.)

Return type

Callable

Returns

A callable that calculates the transpose of fun. Valid input into this function must have the same shape/dtypes/structure as the result of fun(*primals). Output will be a tuple, with the same shape/dtypes/structure as primals.

>>> import jax
>>> import types
>>>
>>> f = lambda x, y: 0.5 * x - 0.5 * y
>>> scalar = types.SimpleNamespace(shape=(), dtype=np.float32)
>>> f_transpose = jax.linear_transpose(f, scalar, scalar)
>>> f_transpose(1.0)
(DeviceArray(0.5, dtype=float32), DeviceArray(-0.5, dtype=float32))

jax.vjp(fun, *primals, has_aux=False)

Compute a (reverse-mode) vector-Jacobian product of fun.

grad() is implemented as a special case of vjp().

Parameters

• fun (Callable) – Function to be differentiated. Its arguments should be arrays, scalars, or standard Python containers of arrays or scalars. It should return an array, scalar, or standard Python container of arrays or scalars.

• primals – A sequence of primal values at which the Jacobian of fun should be evaluated. The length of primals should be equal to the number of positional parameters to fun. Each primal value should be a tuple of arrays, scalar, or standard Python containers thereof.

• has_aux (bool) – Optional, bool. Indicates whether fun returns a pair where the first element is considered the output of the mathematical function to be differentiated and the second element is auxiliary data. Default False.

Return type

Union[Tuple[Any, Callable], Tuple[Any, Callable, Any]]

Returns

If has_aux is False, returns a (primals_out, vjpfun) pair, where primals_out is fun(*primals). vjpfun is a function from a cotangent vector with the same shape as primals_out to a tuple of cotangent vectors with the same shape as primals, representing the vector-Jacobian product of fun evaluated at primals. If has_aux is True, returns a (primals_out, vjpfun, aux) tuple where aux is the auxiliary data returned by fun.

>>> import jax
>>>
>>> def f(x, y):
...   return jax.numpy.sin(x), jax.numpy.cos(y)
...
>>> primals, f_vjp = jax.vjp(f, 0.5, 1.0)
>>> xbar, ybar = f_vjp((-0.7, 0.3))
>>> print(xbar)
-0.61430776
>>> print(ybar)
-0.2524413

class jax.custom_jvp(fun, nondiff_argnums=())

Set up a JAX-transformable function for a custom JVP rule definition.

This class is meant to be used as a function decorator. Instances are callables that behave similarly to the underlying function to which the decorator was applied, except when a differentiation transformation (like jax.jvp() or jax.grad()) is applied, in which case a custom user-supplied JVP rule function is used instead of tracing into and performing automatic differentiation of the underlying function’s implementation.

There are two instance methods available for defining the custom JVP rule: defjvp() for defining a single custom JVP rule for all the function’s inputs, and for convenience defjvps(), which wraps defjvp(), and allows you to provide separate definitions for the partial derivatives of the function w.r.t. each of its arguments.

For example:

import jax.numpy as jnp

@jax.custom_jvp
def f(x, y):
  return jnp.sin(x) * y

@f.defjvp
def f_jvp(primals, tangents):
  x, y = primals
  x_dot, y_dot = tangents
  primal_out = f(x, y)
  tangent_out = jnp.cos(x) * x_dot * y + jnp.sin(x) * y_dot
  return primal_out, tangent_out

For a more detailed introduction, see the tutorial.

defjvp(jvp)

Define a custom JVP rule for the function represented by this instance.

Parameters

jvp – a Python callable representing the custom JVP rule. When there are no nondiff_argnums, the jvp function should accept two arguments, where the first is a tuple of primal inputs and the second is a tuple of tangent inputs. The lengths of both tuples is equal to the number of parameters of the custom_jvp function. The jvp function should produce as output a pair where the first element is the primal output and the second element is the tangent output. Elements of the input and output tuples may be arrays or any nested tuples/lists/dicts thereof.

Returns

None.

Example:

import jax.numpy as jnp

@jax.custom_jvp
def f(x, y):
  return jnp.sin(x) * y

@f.defjvp
def f_jvp(primals, tangents):
  x, y = primals
  x_dot, y_dot = tangents
  primal_out = f(x, y)
  tangent_out = jnp.cos(x) * x_dot * y + jnp.sin(x) * y_dot
  return primal_out, tangent_out

defjvps(*jvps)

Convenience wrapper for defining JVPs for each argument separately.

This convenience wrapper cannot be used together with nondiff_argnums.

Parameters

*jvps – a sequence of functions, one for each positional argument of the custom_jvp function. Each function takes as arguments the tangent value for the corresponding primal input, the primal output, and the primal inputs. See the example below.

Returns

None.

Example:

@jax.custom_jvp
def f(x, y):
  return jnp.sin(x) * y

f.defjvps(lambda x_dot, primal_out, x, y: jnp.cos(x) * x_dot * y,
          lambda y_dot, primal_out, x, y: jnp.sin(x) * y_dot)

class jax.custom_vjp(fun, nondiff_argnums=())

Set up a JAX-transformable function for a custom VJP rule definition.

This class is meant to be used as a function decorator. Instances are callables that behave similarly to the underlying function to which the decorator was applied, except when a reverse-mode differentiation transformation (like jax.grad()) is applied, in which case a custom user-supplied VJP rule function is used instead of tracing into and performing automatic differentiation of the underlying function’s implementation. There is a single instance method, defvjp(), which may be used to define the custom VJP rule.

This decorator precludes the use of forward-mode automatic differentiation.

For example:

import jax.numpy as jnp

@jax.custom_vjp
def f(x, y):
  return jnp.sin(x) * y

def f_fwd(x, y):
  return f(x, y), (jnp.cos(x), jnp.sin(x), y)

def f_bwd(res, g):
  cos_x, sin_x, y = res
  return (cos_x * g * y, sin_x * g)

f.defvjp(f_fwd, f_bwd)

For a more detailed introduction, see the tutorial.

defvjp(fwd, bwd)

Define a custom VJP rule for the function represented by this instance.

Parameters

• fwd – a Python callable representing the forward pass of the custom VJP rule. When there are no nondiff_argnums, the fwd function has the same input signature as the underlying primal function. It should return as output a pair, where the first element represents the primal output and the second element represents any “residual” values to store from the forward pass for use on the backward pass by the function bwd. Input arguments and elements of the output pair may be arrays or nested tuples/lists/dicts thereof.

• bwd – a Python callable representing the backward pass of the custom VJP rule. When there are no nondiff_argnums, the bwd function takes two arguments, where the first is the “residual” values produced on the forward pass by fwd, and the second is the output cotangent with the same structure as the primal function output. The output of bwd must be a tuple of length equal to the number of arguments of the primal function, and the tuple elements may be arrays or nested tuples/lists/dicts thereof so as to match the structure of the primal input arguments.

Returns

None.

Example:

import jax.numpy as jnp

@jax.custom_vjp
def f(x, y):
  return jnp.sin(x) * y

def f_fwd(x, y):
  return f(x, y), (jnp.cos(x), jnp.sin(x), y)

def f_bwd(res, g):
  cos_x, sin_x, y = res
  return (cos_x * g * y, sin_x * g)

f.defvjp(f_fwd, f_bwd)

jax.checkpoint(fun, concrete=False)

Make fun recompute internal linearization points when differentiated.

The jax.checkpoint() decorator, aliased to jax.remat, provides a way to trade off computation time and memory cost in the context of automatic differentiation, especially with reverse-mode autodiff like jax.grad() and jax.vjp() but also with jax.linearize().

When differentiating a function in reverse-mode, by default all the linearization points (e.g. inputs to elementwise nonlinear primitive operations) are stored when evaluating the forward pass so that they can be reused on the backward pass. This evaluation strategy can lead to a high memory cost, or even to poor performance on hardware accelerators where memory access is much more expensive than FLOPs.

An alternative evaluation strategy is for some of the linearization points to be recomputed (i.e. rematerialized) rather than stored. This approach can reduce memory usage at the cost of increased computation.

This function decorator produces a new version of fun which follows the rematerialization strategy rather than the default store-everything strategy. That is, it returns a new version of fun which, when differentiated, doesn’t store any of its intermediate linearization points. Instead, these linearization points are recomputed from the function’s saved inputs.

See the examples below.

Parameters

• fun (Callable) – Function for which the autodiff evaluation strategy is to be changed from the default of storing all intermediate linearization points to recomputing them. Its arguments and return value should be arrays, scalars, or (nested) standard Python containers (tuple/list/dict) thereof.

• concrete (bool) – Optional, boolean indicating whether fun may involve value-dependent Python control flow (default False). Support for such control flow is optional, and disabled by default, because in some edge-case compositions with jax.jit() it can lead to some extra computation.

Return type

Callable

Returns

A function (callable) with the same input/output behavior as fun but which, when differentiated using e.g. jax.grad(), jax.vjp(), or jax.linearize(), recomputes rather than stores intermediate linearization points, thus potentially saving memory at the cost of extra computation.

Here is a simple example:

>>> import jax
>>> import jax.numpy as jnp

>>> @jax.checkpoint
... def g(x):
...   y = jnp.sin(x)
...   z = jnp.sin(y)
...   return z
...
>>> jax.grad(g)(2.0)
DeviceArray(-0.25563914, dtype=float32)

Here, the same value is produced whether or not the jax.checkpoint() decorator is present. But when using jax.checkpoint(), the value jnp.sin(2.0) is computed twice: once on the forward pass, and once on the backward pass. The values jnp.cos(2.0) and jnp.cos(jnp.sin(2.0)) are also computed twice. Without using the decorator, both jnp.cos(2.0) and jnp.cos(jnp.sin(2.0)) would be stored and reused.

The jax.checkpoint() decorator can be applied recursively to express sophisticated autodiff rematerialization strategies. For example:

>>> def recursive_checkpoint(funs):
...   if len(funs) == 1:
...     return funs[0]
...   elif len(funs) == 2:
...     f1, f2 = funs
...     return lambda x: f1(f2(x))
...   else:
...     f1 = recursive_checkpoint(funs[:len(funs)//2])
...     f2 = recursive_checkpoint(funs[len(funs)//2:])
...     return lambda x: f1(jax.checkpoint(f2)(x))
...

jax.vmap(fun, in_axes=0, out_axes=0, axis_name=None)

Vectorizing map. Creates a function which maps fun over argument axes.

Parameters

• fun (~F) – Function to be mapped over additional axes.

• in_axes –

  An integer, None, or (nested) standard Python container (tuple/list/dict) thereof specifying which input array axes to map over.

  If each positional argument to fun is an array, then in_axes can be an integer, a None, or a tuple of integers and Nones with length equal to the number of positional arguments to fun. An integer or None indicates which array axis to map over for all arguments (with None indicating not to map any axis), and a tuple indicates which axis to map for each corresponding positional argument. Axis integers must be in the range [-ndim, ndim) for each array, where ndim is the number of dimensions (axes) of the corresponding input array.

  If the positional arguments to fun are container types, the corresponding element of in_axes can itself be a matching container, so that distinct array axes can be mapped for different container elements. in_axes must be a container tree prefix of the positional argument tuple passed to fun.

  At least one positional argument must have in_axes not None. The sizes of the mapped input axes for all mapped positional arguments must all be equal.

  Arguments passed as keywords are always mapped over their leading axis (i.e. axis index 0).

  See below for examples.

• out_axes – An integer, None, or (nested) standard Python container (tuple/list/dict) thereof indicating where the mapped axis should appear in the output. All outputs with a mapped axis must have a non-None out_axes specification. Axis integers must be in the range [-ndim, ndim) for each output array, where ndim is the number of dimensions (axes) of the array returned by the vmap()-ed function, which is one more than the number of dimensions (axes) of the corresponding array returned by fun.

Return type

~F

Returns

Batched/vectorized version of fun with arguments that correspond to those of fun, but with extra array axes at positions indicated by in_axes, and a return value that corresponds to that of fun, but with extra array axes at positions indicated by out_axes.

For example, we can implement a matrix-matrix product using a vector dot product:

>>> import jax.numpy as jnp
>>>
>>> vv = lambda x, y: jnp.vdot(x, y)  #  ([a], [a]) -> []
>>> mv = vmap(vv, (0, None), 0)      #  ([b,a], [a]) -> [b]      (b is the mapped axis)
>>> mm = vmap(mv, (None, 1), 1)      #  ([b,a], [a,c]) -> [b,c]  (c is the mapped axis)

Here we use [a,b] to indicate an array with shape (a,b). Here are some variants:

>>> mv1 = vmap(vv, (0, 0), 0)   #  ([b,a], [b,a]) -> [b]        (b is the mapped axis)
>>> mv2 = vmap(vv, (0, 1), 0)   #  ([b,a], [a,b]) -> [b]        (b is the mapped axis)
>>> mm2 = vmap(mv2, (1, 1), 0)  #  ([b,c,a], [a,c,b]) -> [c,b]  (c is the mapped axis)

Here’s an example of using container types in in_axes to specify which axes of the container elements to map over:

>>> A, B, C, D = 2, 3, 4, 5
>>> x = jnp.ones((A, B))
>>> y = jnp.ones((B, C))
>>> z = jnp.ones((C, D))
>>> def foo(tree_arg):
...   x, (y, z) = tree_arg
...   return jnp.dot(x, jnp.dot(y, z))
>>> tree = (x, (y, z))
>>> print(foo(tree))
[[12. 12. 12. 12. 12.]
 [12. 12. 12. 12. 12.]]
>>> from jax import vmap
>>> K = 6  # batch size
>>> x = jnp.ones((K, A, B))  # batch axis in different locations
>>> y = jnp.ones((B, K, C))
>>> z = jnp.ones((C, D, K))
>>> tree = (x, (y, z))
>>> vfoo = vmap(foo, in_axes=((0, (1, 2)),))
>>> print(vfoo(tree).shape)
(6, 2, 5)

Here’s another example using container types in in_axes, this time a dictionary, to specify the elements of the container to map over:

>>> dct = {'a': 0., 'b': jnp.arange(5.)}
>>> x = 1.
>>> def foo(dct, x):
...  return dct['a'] + dct['b'] + x
>>> out = vmap(foo, in_axes=({'a': None, 'b': 0}, None))(dct, x)
>>> print(out)
[1. 2. 3. 4. 5.]

The results of a vectorized function can be mapped or unmapped. For example, the function below returns a pair with the first element mapped and the second unmapped. Only for unmapped results we can specify out_axes to be None (to keep it unmapped).

>>> print(vmap(lambda x, y: (x + y, y * 2.), in_axes=(0, None), out_axes=(0, None))(jnp.arange(2.), 4.))
(DeviceArray([4., 5.], dtype=float32), 8.0)

If the out_axes is specified for an unmapped result, the result is broadcast across the mapped axis:

>>> print(vmap(lambda x, y: (x + y, y * 2.), in_axes=(0, None), out_axes=0)(jnp.arange(2.), 4.))
(DeviceArray([4., 5.], dtype=float32), DeviceArray([8., 8.], dtype=float32))

If the out_axes is specified for a mapped result, the result is transposed accordingly.

jax.numpy.vectorize(pyfunc, *, excluded=frozenset({}), signature=None)

Define a vectorized function with broadcasting.

vectorize() is a convenience wrapper for defining vectorized functions with broadcasting, in the style of NumPy’s generalized universal functions. It allows for defining functions that are automatically repeated across any leading dimensions, without the implementation of the function needing to be concerned about how to handle higher dimensional inputs.

jax.numpy.vectorize() has the same interface as numpy.vectorize, but it is syntactic sugar for an auto-batching transformation (vmap()) rather than a Python loop. This should be considerably more efficient, but the implementation must be written in terms of functions that act on JAX arrays.

Parameters

• pyfunc – function to vectorize.

• excluded – optional set of integers representing positional arguments for which the function will not be vectorized. These will be passed directly to pyfunc unmodified.

• signature – optional generalized universal function signature, e.g., (m,n),(n)->(m) for vectorized matrix-vector multiplication. If provided, pyfunc will be called with (and expected to return) arrays with shapes given by the size of corresponding core dimensions. By default, pyfunc is assumed to take scalars arrays as input and output.

Returns

Vectorized version of the given function.

Here a few examples of how one could write vectorized linear algebra routines using vectorize():

import jax.numpy as jnp
from functools import partial

@partial(jnp.vectorize, signature='(k),(k)->(k)')
def cross_product(a, b):
  assert a.shape == b.shape and a.ndim == b.ndim == 1
  return jnp.array([a[1] * b[2] - a[2] * b[1],
                    a[2] * b[0] - a[0] * b[2],
                    a[0] * b[1] - a[1] * b[0]])

@partial(jnp.vectorize, signature='(n,m),(m)->(n)')
def matrix_vector_product(matrix, vector):
  assert matrix.ndim == 2 and matrix.shape[1:] == vector.shape
  return matrix @ vector

These functions are only written to handle 1D or 2D arrays (the assert statements will never be violated), but with vectorize they support arbitrary dimensional inputs with NumPy style broadcasting, e.g.,

>>> cross_product(jnp.ones(3), jnp.ones(3)).shape
(3,)
>>> cross_product(jnp.ones((2, 3)), jnp.ones(3)).shape
(2, 3)
>>> cross_product(jnp.ones((1, 2, 3)), jnp.ones((2, 1, 3))).shape
(2, 2, 3)
>>> matrix_vector_product(jnp.ones(3), jnp.ones(3))
ValueError: input with shape (3,) does not have enough dimensions for all
core dimensions ('n', 'k') on vectorized function with excluded=frozenset()
and signature='(n,k),(k)->(k)'
>>> matrix_vector_product(jnp.ones((2, 3)), jnp.ones(3)).shape
(2,)
>>> matrix_vector_product(jnp.ones((2, 3)), jnp.ones((4, 3))).shape
(4, 2)  # not the same as jnp.matmul

jax.pmap(fun, axis_name=None, *, in_axes=0, out_axes=0, static_broadcasted_argnums=(), devices=None, backend=None, axis_size=None, donate_argnums=(), global_arg_shapes=None)

Parallel map with support for collective operations.

The purpose of pmap() is to express single-program multiple-data (SPMD) programs. Applying pmap() to a function will compile the function with XLA (similarly to jit()), then execute it in parallel on XLA devices, such as multiple GPUs or multiple TPU cores. Semantically it is comparable to vmap() because both transformations map a function over array axes, but where vmap() vectorizes functions by pushing the mapped axis down into primitive operations, pmap() instead replicates the function and executes each replica on its own XLA device in parallel.

The mapped axis size must be less than or equal to the number of local XLA devices available, as returned by jax.local_device_count() (unless devices is specified, see below). For nested pmap() calls, the product of the mapped axis sizes must be less than or equal to the number of XLA devices.

Note

pmap() compiles fun, so while it can be combined with jit(), it’s usually unnecessary.

Multi-host platforms: On multi-host platforms such as TPU pods, pmap() is designed to be used in SPMD Python programs, where every host is running the same Python code such that all hosts run the same pmapped function in the same order. Each host should still call the pmapped function with mapped axis size equal to the number of local devices (unless devices is specified, see below), and an array of the same leading axis size will be returned as usual. However, any collective operations in fun will be computed over all participating devices, including those on other hosts, via device-to-device communication. Conceptually, this can be thought of as running a pmap over a single array sharded across hosts, where each host “sees” only its local shard of the input and output. The SPMD model requires that the same multi-host pmaps must be run in the same order on all devices, but they can be interspersed with arbitrary operations running on a single host.

Parameters

• fun (~F) – Function to be mapped over argument axes. Its arguments and return value should be arrays, scalars, or (nested) standard Python containers (tuple/list/dict) thereof. Positional arguments indicated by static_broadcasted_argnums can be anything at all, provided they are hashable and have an equality operation defined.

• axis_name (Optional[Any]) – Optional, a hashable Python object used to identify the mapped axis so that parallel collectives can be applied.

• in_axes – A non-negative integer, None, or nested Python container thereof that specifies which axes of positional arguments to map over. Arguments passed as keywords are always mapped over their leading axis (i.e. axis index 0). See vmap() for details.

• out_axes – A non-negative integer, None, or nested Python container thereof indicating where the mapped axis should appear in the output. All outputs with a mapped axis must have a non-None out_axes specification (see vmap()).

• static_broadcasted_argnums (Union[int, Iterable[int]]) – An int or collection of ints specifying which positional arguments to treat as static (compile-time constant). Operations that only depend on static arguments will be constant-folded. Calling the pmapped function with different values for these constants will trigger recompilation. If the pmapped function is called with fewer positional arguments than indicated by static_argnums then an error is raised. Each of the static arguments will be broadcasted to all devices. Arguments that are not arrays or containers thereof must be marked as static. Defaults to ().

• devices (Optional[Sequence[Device]]) – This is an experimental feature and the API is likely to change. Optional, a sequence of Devices to map over. (Available devices can be retrieved via jax.devices()). If specified, the size of the mapped axis must be equal to the number of local devices in the sequence. Nested pmap() s with devices specified in either the inner or outer pmap() are not yet supported.

• backend (Optional[str]) – This is an experimental feature and the API is likely to change. Optional, a string representing the XLA backend. ‘cpu’, ‘gpu’, or ‘tpu’.

• axis_size (Optional[int]) – Optional; the size of the mapped axis.

• donate_argnums (Union[int, Iterable[int]]) – Specify which arguments are “donated” to the computation. It is safe to donate arguments if you no longer need them once the computation has finished. In some cases XLA can make use of donated buffers to reduce the amount of memory needed to perform a computation, for example recycling one of your input buffers to store a result. You should not reuse buffers that you donate to a computation, JAX will raise an error if you try to.

• global_arg_shapes (Optional[Tuple[Tuple[int, …], …]]) – Optional, must be set when using pmap(sharded_jit) and the partitioned values span multiple processes. The global cross-process per-replica shape of each argument, i.e. does not include the leading pmapped dimension. Can be None for replicated arguments. This API is likely to change in the future.

Return type

~F

Returns

A parallelized version of fun with arguments that correspond to those of fun but with extra array axes at positions indicated by in_axes and with output that has an additional leading array axis (with the same size).

For example, assuming 8 XLA devices are available, pmap() can be used as a map along a leading array axis:

>>> import jax.numpy as jnp
>>>
>>> out = pmap(lambda x: x ** 2)(jnp.arange(8))  
>>> print(out)  
[0, 1, 4, 9, 16, 25, 36, 49]

When the leading dimension is smaller than the number of available devices JAX will simply run on a subset of devices:

>>> x = jnp.arange(3 * 2 * 2.).reshape((3, 2, 2))
>>> y = jnp.arange(3 * 2 * 2.).reshape((3, 2, 2)) ** 2
>>> out = pmap(jnp.dot)(x, y)  
>>> print(out)  
[[[    4.     9.]
  [   12.    29.]]
 [[  244.   345.]
  [  348.   493.]]
 [[ 1412.  1737.]
  [ 1740.  2141.]]]

If your leading dimension is larger than the number of available devices you will get an error:

>>> pmap(lambda x: x ** 2)(jnp.arange(9))  
ValueError: ... requires 9 replicas, but only 8 XLA devices are available

As with vmap(), using None in in_axes indicates that an argument doesn’t have an extra axis and should be broadcasted, rather than mapped, across the replicas:

>>> x, y = jnp.arange(2.), 4.
>>> out = pmap(lambda x, y: (x + y, y * 2.), in_axes=(0, None))(x, y)  
>>> print(out)  
([4., 5.], [8., 8.])

Note that pmap() always returns values mapped over their leading axis, equivalent to using out_axes=0 in vmap().

In addition to expressing pure maps, pmap() can also be used to express parallel single-program multiple-data (SPMD) programs that communicate via collective operations. For example:

>>> f = lambda x: x / jax.lax.psum(x, axis_name='i')
>>> out = pmap(f, axis_name='i')(jnp.arange(4.))  
>>> print(out)  
[ 0.          0.16666667  0.33333334  0.5       ]
>>> print(out.sum())  
1.0

In this example, axis_name is a string, but it can be any Python object with __hash__ and __eq__ defined.

The argument axis_name to pmap() names the mapped axis so that collective operations, like jax.lax.psum(), can refer to it. Axis names are important particularly in the case of nested pmap() functions, where collective operations can operate over distinct axes:

>>> from functools import partial
>>> import jax
>>>
>>> @partial(pmap, axis_name='rows')
... @partial(pmap, axis_name='cols')
... def normalize(x):
...   row_normed = x / jax.lax.psum(x, 'rows')
...   col_normed = x / jax.lax.psum(x, 'cols')
...   doubly_normed = x / jax.lax.psum(x, ('rows', 'cols'))
...   return row_normed, col_normed, doubly_normed
>>>
>>> x = jnp.arange(8.).reshape((4, 2))
>>> row_normed, col_normed, doubly_normed = normalize(x)  
>>> print(row_normed.sum(0))  
[ 1.  1.]
>>> print(col_normed.sum(1))  
[ 1.  1.  1.  1.]
>>> print(doubly_normed.sum((0, 1)))  
1.0

On multi-host platforms, collective operations operate over all devices, including those on other hosts. For example, assuming the following code runs on two hosts with 4 XLA devices each:

>>> f = lambda x: x + jax.lax.psum(x, axis_name='i')
>>> data = jnp.arange(4) if jax.host_id() == 0 else jnp.arange(4, 8)
>>> out = pmap(f, axis_name='i')(data)  
>>> print(out)  
[28 29 30 31] # on host 0
[32 33 34 35] # on host 1

Each host passes in a different length-4 array, corresponding to its 4 local devices, and the psum operates over all 8 values. Conceptually, the two length-4 arrays can be thought of as a sharded length-8 array (in this example equivalent to jnp.arange(8)) that is mapped over, with the length-8 mapped axis given name ‘i’. The pmap call on each host then returns the corresponding length-4 output shard.

The devices argument can be used to specify exactly which devices are used to run the parallel computation. For example, again assuming a single host with 8 devices, the following code defines two parallel computations, one which runs on the first six devices and one on the remaining two:

>>> from functools import partial
>>> @partial(pmap, axis_name='i', devices=jax.devices()[:6])
... def f1(x):
...   return x / jax.lax.psum(x, axis_name='i')
>>>
>>> @partial(pmap, axis_name='i', devices=jax.devices()[-2:])
... def f2(x):
...   return jax.lax.psum(x ** 2, axis_name='i')
>>>
>>> print(f1(jnp.arange(6.)))  
[0.         0.06666667 0.13333333 0.2        0.26666667 0.33333333]
>>> print(f2(jnp.array([2., 3.])))  
[ 13.  13.]

jax.devices(backend=None)

Returns a list of all devices for a given backend.

Each device is represented by a subclass of Device (e.g. CpuDevice, GpuDevice). The length of the returned list is equal to device_count(backend). Local devices can be identified by comparing Device.host_id() to the value returned by jax.host_id().

If backend is None, returns all the devices from the default backend. The default backend is generally 'gpu' or 'tpu' if available, otherwise 'cpu'.

Parameters

backend (Optional[str]) – This is an experimental feature and the API is likely to change. Optional, a string representing the xla backend: 'cpu', 'gpu', or 'tpu'.

Return type

List[Device]

Returns

List of Device subclasses.

jax.local_devices(host_id=None, backend=None)

Like jax.devices(), but only returns devices local to a given host.

If host_id is None, returns devices local to this host.

Parameters

• host_id (Optional[int]) – the integer ID of the host. Host IDs can be retrieved via jax.host_ids().

• backend (Optional[str]) – This is an experimental feature and the API is likely to change. Optional, a string representing the xla backend: 'cpu', 'gpu', or 'tpu'.

Return type

List[Device]

Returns

List of Device subclasses.

jax.host_id(backend=None)

Returns the integer host ID of this host.

On most platforms, this will always be 0. This will vary on multi-host platforms though.

Parameters

backend (Optional[str]) – This is an experimental feature and the API is likely to change. Optional, a string representing the xla backend: 'cpu', 'gpu', or 'tpu'.

Return type

int

Returns

Integer host ID.

jax.host_ids(backend=None)

Returns a sorted list of all host IDs.

Parameters

backend (Optional[str]) –

Return type

List[int]

jax.device_count(backend=None)

Returns the total number of devices.

On most platforms, this is the same as jax.local_device_count(). However, on multi-host platforms, this will return the total number of devices across all hosts.

Parameters

backend (Optional[str]) – This is an experimental feature and the API is likely to change. Optional, a string representing the xla backend: 'cpu', 'gpu', or 'tpu'.

Return type

int

Returns

Number of devices.

jax.local_device_count(backend=None)

Returns the number of devices on this host.

Parameters

backend (Optional[str]) –

Return type

int

jax.host_count(backend=None)

Returns the number of hosts.

Parameters

backend (Optional[str]) –

Return type

int