JAX 内部原理：原语

JAX 原语简介

JAX 原语是 JAX 程序的基本计算单元。本文档解释了 JAX 原语必须支持的接口，以允许 JAX 执行其所有转换（这不是操作指南）。

例如，乘加运算可以使用底层的 jax.lax.* 原语（类似于 XLA 运算符包装器）或 jax.core.Primitive("multiply_add") 来实现，如下文进一步所示。

JAX 能够获取这些原始操作的序列，并通过其 Python 函数的可组合变换（例如 jax.jit()、 jax.grad() 和 jax.vmap()）来转换它们。JAX 以JAX 可追踪的方式实现这些转换。这意味着当执行 Python 函数时，它对数据应用的唯一操作是：

• 数据属性的检查： 数据信息，例如形状或类型；或

• JAX 原语： 这些是本教程中涵盖的 JAX 特殊操作。

JAX 原语知道如何操作具体的数据值和抽象的 JAX 值。一个 JAX 可追踪函数可以由 JAX 使用抽象参数调用。例如，一个 JAX 抽象值 — ShapedArray(float32[2,2]) — 捕获值的类型和形状，但不捕获具体的数据值。

JAX 转换的函数本身必须是 JAX 可追踪的函数，以确保这些转换是可组合的，例如像 jax.jit(jax.jacfwd(jax.grad(f))) 这样。

JAX 提供了对应于大多数 XLA 操作的预定义原语，包括加法、矩阵乘法、正弦、余弦和索引。

此外，JAX 还提供了使用 JAX 原语实现的 NumPy 函数。这意味着使用 JAX 实现的 NumPy 的 Python 程序是 JAX 可追踪的，因此是可转换的。通过使用 JAX 原语实现其他库，可以使其成为 JAX 可追踪的。

此外，JAX 原语的集合是可扩展的，因此您可以定义一个新的原语来封装函数行为，而不是使用预定义的 JAX 原语重新实现函数。

考虑以下示例：您希望为 JAX 添加对具有三个参数的乘加函数的支持，该函数在数学上定义为 multiply_add(x, y, z) = x * y + z。此函数对 3 个形状相同的浮点值张量进行操作，并逐点执行操作。您可以通过以下方式执行此操作：

• 使用现有的 JAX 原语；或

• 定义新的 JAX 原语

使用现有的 JAX 原语

定义新函数的最简单方法是使用 JAX 原语编写它们，或者使用本身使用 JAX 原语编写的其他函数，例如在 jax.lax() 模块中定义的那些。

from jax import lax
from jax._src import api

def multiply_add_lax(x, y, z):
  """Implementation of multiply-add using the `jax.lax` primitives."""
  return lax.add(lax.mul(x, y), z)


def square_add_lax(a, b):
  """A square-add function using the newly defined multiply-add."""
  return multiply_add_lax(a, a, b)

print("square_add_lax = ", square_add_lax(2., 10.))
# Differentiate w.r.t. the first argument
print("grad(square_add_lax) = ", api.grad(square_add_lax, argnums=0)(2.0, 10.))

square_add_lax =  14.0
grad(square_add_lax) =  4.0

要理解 JAX 在内部如何使用原语，请为跟踪函数调用添加一些辅助函数。

#@title Helper functions (execute this cell)
import functools
import traceback

_indentation = 0
def _trace(msg=None):
    """Print a message at current indentation."""
    if msg is not None:
        print("  " * _indentation + msg)

def _trace_indent(msg=None):
    """Print a message and then indent the rest."""
    global _indentation
    _trace(msg)
    _indentation = 1 + _indentation

def _trace_unindent(msg=None):
    """Unindent then print a message."""
    global _indentation
    _indentation = _indentation - 1
    _trace(msg)

def trace(name):
  """A decorator for functions to trace arguments and results."""

  def trace_func(func):  # pylint: disable=missing-docstring
    def pp(v):
        """Print certain values more succinctly"""
        vtype = str(type(v))
        if "jax._src.xla_bridge._JaxComputationBuilder" in vtype:
            return "<JaxComputationBuilder>"
        elif "jaxlib.xla_extension.XlaOp" in vtype:
            return "<XlaOp at 0x{:x}>".format(id(v))
        elif ("partial_eval.JaxprTracer" in vtype or
              "batching.BatchTracer" in vtype or
              "ad.JVPTracer" in vtype):
            return "Traced<{}>".format(v.aval)
        elif isinstance(v, tuple):
            return "({})".format(pp_values(v))
        else:
            return str(v)
    def pp_values(args):
        return ", ".join([pp(arg) for arg in args])
    
    @functools.wraps(func)
    def func_wrapper(*args):
      _trace_indent("call {}({})".format(name, pp_values(args)))
      res = func(*args)
      _trace_unindent("|<- {} = {}".format(name, pp(res)))
      return res

    return func_wrapper

  return trace_func

class expectNotImplementedError(object):
  """Context manager to check for NotImplementedError."""
  def __enter__(self): pass
  def __exit__(self, type, value, tb):
    global _indentation
    _indentation = 0
    if type is NotImplementedError:
      print("\nFound expected exception:")
      traceback.print_exc(limit=3)
      return True
    elif type is None:  # No exception
      assert False, "Expected NotImplementedError"
    else:
      return False

您可以直接使用 jax.lax() 原语，也可以使用其他使用这些原语编写的函数，例如 jax.numpy 中的那些。

import jax.numpy as jnp
import numpy as np

@trace("multiply_add_numpy")
def multiply_add_numpy(x, y, z):
    return jnp.add(jnp.multiply(x, y), z)

@trace("square_add_numpy")
def square_add_numpy(a, b):
    return multiply_add_numpy(a, a, b)

print("\nNormal evaluation:")  
print("square_add_numpy = ", square_add_numpy(2., 10.))
print("\nGradient evaluation:")
print("grad(square_add_numpy) = ", api.grad(square_add_numpy)(2.0, 10.))

Normal evaluation:
call square_add_numpy(2.0, 10.0)
  call multiply_add_numpy(2.0, 2.0, 10.0)
  |<- multiply_add_numpy = 14.0
|<- square_add_numpy = 14.0
square_add_numpy =  14.0

Gradient evaluation:
call square_add_numpy(Traced<ShapedArray(float32[], weak_type=True)>, 10.0)
  call multiply_add_numpy(Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>, 10.0)
  |<- multiply_add_numpy = Traced<ShapedArray(float32[], weak_type=True)>
|<- square_add_numpy = Traced<ShapedArray(float32[], weak_type=True)>
grad(square_add_numpy) =  4.0

请注意，在计算 jax.grad() 的过程中，JAX 使用特殊参数 ConcreteArray(...)（在下面的 colab 中进一步描述）调用 square_add_numpy 和 multiply_add_numpy。请务必记住，JAX 可追踪函数不仅必须能够对具体参数进行操作，还必须能够对 JAX 可能用来抽象函数执行的特殊抽象参数进行操作。

只要该函数是使用 JAX 原语编写的，JAX 可追踪性属性就会得到满足。

定义新的 JAX 原语

为乘加添加支持的正确方法是使用如上所示的现有 JAX 原语。但是，为了演示 JAX 原语的工作原理，假设您要为 JAX 添加一个新的乘加功能原语。

from jax import core

multiply_add_p = core.Primitive("multiply_add")  # Create the primitive

@trace("multiply_add_prim")
def multiply_add_prim(x, y, z):
  """The JAX-traceable way to use the JAX primitive.
  
  Note that the traced arguments must be passed as positional arguments
  to `bind`. 
  """
  return multiply_add_p.bind(x, y, z)

@trace("square_add_prim")
def square_add_prim(a, b):
  """A square-add function implemented using the new JAX-primitive."""
  return multiply_add_prim(a, a, b)

/tmp/ipykernel_1057/1751132419.py:3: DeprecationWarning: jax.core.Primitive is deprecated. Use jax.extend.core.Primitive instead, and see https://jax.ac.cn/en/latest/jax.extend.html for details.
  multiply_add_p = core.Primitive("multiply_add")  # Create the primitive

如果您尝试调用新定义的函数，您将收到一个错误，因为您尚未告知 JAX 有关新原语的任何语义。

with expectNotImplementedError():
  square_add_prim(2., 10.)

call square_add_prim(2.0, 10.0)
  call multiply_add_prim(2.0, 2.0, 10.0)

Found expected exception:

Traceback (most recent call last):
  File "/tmp/ipykernel_1057/2844449444.py", line 2, in <module>
    square_add_prim(2., 10.)
  File "/tmp/ipykernel_1057/1393342955.py", line 48, in func_wrapper
    res = func(*args)
  File "/tmp/ipykernel_1057/1751132419.py", line 17, in square_add_prim
    return multiply_add_prim(a, a, b)
NotImplementedError: Evaluation rule for 'multiply_add' not implemented

原始评估规则

@trace("multiply_add_impl")
def multiply_add_impl(x, y, z):
  """Concrete implementation of the primitive.

  This function does not need to be JAX traceable.

  Args:
    x, y, z: The concrete arguments of the primitive. Will only be called with 
      concrete values.

  Returns:
    the concrete result of the primitive.
  """
  # Note: you can use the ordinary (non-JAX) NumPy, which is not JAX-traceable.
  return np.add(np.multiply(x, y), z)

# Now, register the primal implementation with JAX:
multiply_add_p.def_impl(multiply_add_impl)

<function __main__.multiply_add_impl(x, y, z)>

assert square_add_prim(2., 10.) == 14.

call square_add_prim(2.0, 10.0)
  call multiply_add_prim(2.0, 2.0, 10.0)
    call multiply_add_impl(2.0, 2.0, 10.0)
    |<- multiply_add_impl = 14.0
  |<- multiply_add_prim = 14.0
|<- square_add_prim = 14.0

当您使用 jit 时会发生什么

现在，如果您尝试使用 jit，您将收到一个 NotImplementedError 错误

with expectNotImplementedError():
  api.jit(square_add_prim)(2., 10.)

call square_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>)
  call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>)

Found expected exception:

Traceback (most recent call last):
  File "/tmp/ipykernel_1057/1813425700.py", line 2, in <module>
    api.jit(square_add_prim)(2., 10.)
  File "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/traceback_util.py", line 180, in reraise_with_filtered_traceback
    return fun(*args, **kwargs)
  File "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/pjit.py", line 340, in cache_miss
    pgle_profiler) = _python_pjit_helper(fun, jit_info, *args, **kwargs)
NotImplementedError: Abstract evaluation for 'multiply_add' not implemented

抽象评估规则

为了 JIT 函数以及进行其他转换，JAX 首先仅使用参数的形状和类型以抽象方式对其进行评估。此抽象评估有多个用途：

• 获取计算中使用的 JAX 原语序列。此序列将被编译。

• 计算计算中使用的所有向量和操作的形状和类型。

例如，具有 3 个元素的向量的抽象可能是 ShapedArray(float32[3]) 或 ConcreteArray([1., 2., 3.])。在后一种情况下，JAX 使用作为抽象值包装的实际具体值。

from jax import core

@trace("multiply_add_abstract_eval")
def multiply_add_abstract_eval(xs, ys, zs):
  """Abstract evaluation of the primitive.

  This function does not need to be JAX traceable. It will be invoked with
  abstractions of the actual arguments

  Args:
    xs, ys, zs: Abstractions of the arguments.

  Result:
    a ShapedArray for the result of the primitive.
  """
  assert xs.shape == ys.shape
  assert xs.shape == zs.shape
  return core.ShapedArray(xs.shape, xs.dtype)

# Now, register the abstract evaluation with JAX:
multiply_add_p.def_abstract_eval(multiply_add_abstract_eval)

<function __main__.multiply_add_abstract_eval(xs, ys, zs)>

如果您重新尝试应用 jit，您可以检查抽象评估的进行方式，但您会收到另一个有关缺少实际 XLA 编译规则的错误。

with expectNotImplementedError():
  api.jit(square_add_prim)(2., 10.)

call square_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>)
  call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>)
    call multiply_add_abstract_eval(ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True))
    |<- multiply_add_abstract_eval = ShapedArray(float32[])
  |<- multiply_add_prim = Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>
|<- square_add_prim = Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>

Found expected exception:

Traceback (most recent call last):
  File "/home/docs/.asdf/installs/python/3.10.15/lib/python3.10/runpy.py", line 196, in _run_module_as_main
    return _run_code(code, main_globals, None,
  File "/home/docs/.asdf/installs/python/3.10.15/lib/python3.10/runpy.py", line 86, in _run_code
    exec(code, run_globals)
  File "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/ipykernel_launcher.py", line 18, in <module>
    app.launch_new_instance()
jax._src.source_info_util.JaxStackTraceBeforeTransformation: NotImplementedError: MLIR translation rule for primitive 'multiply_add' not found for platform cpu

The preceding stack trace is the source of the JAX operation that, once transformed by JAX, triggered the following exception.

--------------------

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

Traceback (most recent call last):
  File "/tmp/ipykernel_1057/1813425700.py", line 2, in <module>
    api.jit(square_add_prim)(2., 10.)
  File "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/traceback_util.py", line 180, in reraise_with_filtered_traceback
    return fun(*args, **kwargs)
  File "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/pjit.py", line 340, in cache_miss
    pgle_profiler) = _python_pjit_helper(fun, jit_info, *args, **kwargs)
NotImplementedError: MLIR translation rule for primitive 'multiply_add' not found for platform cpu

XLA 编译规则

JAX 编译的工作原理是将每个原语编译为 XLA 操作图。

这是向 JAX 添加新功能的最大障碍，因为 XLA 操作集是有限的，并且 JAX 已经为其中的大多数操作预定义了原语。但是，XLA 包括一个 CustomCall 操作，该操作可用于封装使用 C++ 定义的任意功能。

from jax._src.lib.mlir.dialects import hlo

@trace("multiply_add_lowering")
def multiply_add_lowering(ctx, xc, yc, zc):
  """The compilation to XLA of the primitive.

  Given an mlir.ir.Value for each argument, return the mlir.ir.Values for
  the results of the function.

  Does not need to be a JAX-traceable function.
  """
  return [hlo.AddOp(hlo.MulOp(xc, yc), zc).result]

# Now, register the lowering rule with JAX.
# For GPU, refer to the https://jax.ac.cn/en/latest/Custom_Operation_for_GPUs.html
from jax.interpreters import mlir

mlir.register_lowering(multiply_add_p, multiply_add_lowering, platform='cpu')

<function __main__.multiply_add_lowering(ctx, xc, yc, zc)>

现在，您将成功应用 jax.jit。请注意，JAX 首先以抽象方式评估该函数，这会触发 multiply_add_abstract_eval 函数，然后编译它遇到的原语集，包括 multiply_add。此时，JAX 调用 multiply_add_lowering。

assert api.jit(lambda x, y: square_add_prim(x, y))(2., 10.) == 14.

call square_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>)
  call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>)
    call multiply_add_abstract_eval(ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True))
    |<- multiply_add_abstract_eval = ShapedArray(float32[])
  |<- multiply_add_prim = Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>
|<- square_add_prim = Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>
call multiply_add_lowering(LoweringRuleContext(module_context=ModuleContext(context=<jax._src.interpreters.mlir.JaxIrContext object at 0x7fb25450d1c0>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7fb255765580>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7fb2544ff910>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7fb2557a1c50>, platforms=('cpu',), backend=<jaxlib.xla_extension.Client object at 0x7fb258145080>, axis_context=ShardingContext(num_devices=1, device_assignment=None, abstract_mesh=None), keepalives=[], channel_iterator=count(1), host_callbacks=[], shape_poly_state=<jax._src.interpreters.mlir.ShapePolyLoweringState object at 0x7fb2557a1ba0>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x56102ed26c00>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at 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"/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py", line 3541>, 76): loc("run_code"("/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py":3577:0)), (<code object run_ast_nodes at 0x7fb29538aef0, file "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py", line 3418>, 500): loc("run_ast_nodes"("/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py":3517:0)), (<code object run_cell_async at 0x7fb29538ab80, file "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py", line 3183>, 828): loc("run_cell_async"("/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py":3334:0)), (<code object _pseudo_sync_runner at 0x7fb295255790, file "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/async_helpers.py", line 119>, 8): loc("_pseudo_sync_runner"("/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/async_helpers.py":128:0))}, canonical_name_cache={'/tmp/ipykernel_1057/1751132419.py': '/tmp/ipykernel_1057/1751132419.py', '/tmp/ipykernel_1057/1393342955.py': '/tmp/ipykernel_1057/1393342955.py', '/tmp/ipykernel_1057/1570919344.py': '/tmp/ipykernel_1057/1570919344.py', '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py': '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py', '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/async_helpers.py': '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/async_helpers.py'}, is_user_file_cache={'/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/source_info_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/interpreters/partial_eval.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/core.py': False, '/tmp/ipykernel_1057/1751132419.py': True, '/tmp/ipykernel_1057/1393342955.py': True, '/tmp/ipykernel_1057/1570919344.py': True, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/api_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/linear_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/profiler.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/pjit.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/traceback_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py': True, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/async_helpers.py': True}), lowering_parameters=LoweringParameters(override_lowering_rules=None, global_constant_computation=False, for_export=False, export_ignore_forward_compatibility=False)), name_stack=NameStack(stack=(Scope(name='jit(<lambda>)'), Scope(name='jit(main)'))), primitive=multiply_add, avals_in=[ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True)], avals_out=[ShapedArray(float32[])], tokens_in=<jax._src.interpreters.mlir.TokenSet object at 0x7fb2557a18a0>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None, threefry_partitionable=True, cur_abstract_mesh=None, xla_metadata={}), platforms=None), Value(<block argument> of type 'tensor<f32>' at index: 0), Value(<block argument> of type 'tensor<f32>' at index: 0), Value(<block argument> of type 'tensor<f32>' at index: 1))
|<- multiply_add_lowering = [<jaxlib.mlir._mlir_libs._mlir.ir.OpResult object at 0x7fb25574e870>]

下面是 jit 的另一种用法，您仅针对第一个参数进行编译。请注意，square_add_prim 的第二个参数是具体的，这导致 multiply_add_abstract_eval 的第三个参数为 ConcreteArray。请注意，multiply_add_abstract_eval 可以与 ShapedArray 和 ConcreteArray 一起使用。

assert api.jit(lambda x, y: square_add_prim(x, y), 
               static_argnums=1)(2., 10.) == 14.

call square_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, 10.0)
  call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, 10.0)
    call multiply_add_abstract_eval(ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True))
    |<- multiply_add_abstract_eval = ShapedArray(float32[])
  |<- multiply_add_prim = Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>
|<- square_add_prim = Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>
call multiply_add_lowering(LoweringRuleContext(module_context=ModuleContext(context=<jax._src.interpreters.mlir.JaxIrContext object at 0x7fb25450d8c0>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7fb2545402c0>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7fb2545403f0>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7fb2557a32d0>, platforms=('cpu',), backend=<jaxlib.xla_extension.Client object at 0x7fb258145080>, axis_context=ShardingContext(num_devices=1, device_assignment=None, abstract_mesh=None), keepalives=[], channel_iterator=count(1), host_callbacks=[], shape_poly_state=<jax._src.interpreters.mlir.ShapePolyLoweringState object at 0x7fb2557a3370>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x56102ed26c00>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1057/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_1057/4165789807.py":1:0) at callsite("<module>"("/tmp/ipykernel_1057/4165789807.py":1:0) at callsite("run_code"("/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py":3577:0) at callsite("run_ast_nodes"("/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py":3517:0) at callsite("run_cell_async"("/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py":3334:0) at "_pseudo_sync_runner"("/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/async_helpers.py":128:0)))))))))))}, location_cache={(<code object multiply_add_prim at 0x7fb2580fbaa0, file "/tmp/ipykernel_1057/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0)), (<code object func_wrapper at 0x7fb2580face0, file "/tmp/ipykernel_1057/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0)), (<code object square_add_prim at 0x7fb2580fb730, file "/tmp/ipykernel_1057/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_1057/1751132419.py":17:0)), (<code object <lambda> at 0x7fb2557ae550, file "/tmp/ipykernel_1057/4165789807.py", line 1>, 6): loc("<lambda>"("/tmp/ipykernel_1057/4165789807.py":1:0)), (<code object <module> at 0x7fb2557ae8c0, file "/tmp/ipykernel_1057/4165789807.py", line 1>, 20): loc("<module>"("/tmp/ipykernel_1057/4165789807.py":1:0)), (<code object run_code at 0x7fb29538b050, file "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py", line 3541>, 76): loc("run_code"("/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py":3577:0)), (<code object run_ast_nodes at 0x7fb29538aef0, file "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py", line 3418>, 500): loc("run_ast_nodes"("/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py":3517:0)), (<code object run_cell_async at 0x7fb29538ab80, file "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py", line 3183>, 828): loc("run_cell_async"("/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py":3334:0)), (<code object _pseudo_sync_runner at 0x7fb295255790, file "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/async_helpers.py", line 119>, 8): loc("_pseudo_sync_runner"("/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/async_helpers.py":128:0))}, canonical_name_cache={'/tmp/ipykernel_1057/1751132419.py': '/tmp/ipykernel_1057/1751132419.py', '/tmp/ipykernel_1057/1393342955.py': '/tmp/ipykernel_1057/1393342955.py', '/tmp/ipykernel_1057/4165789807.py': '/tmp/ipykernel_1057/4165789807.py', '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py': '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py', '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/async_helpers.py': '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/async_helpers.py'}, is_user_file_cache={'/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/source_info_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/interpreters/partial_eval.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/core.py': False, '/tmp/ipykernel_1057/1751132419.py': True, '/tmp/ipykernel_1057/1393342955.py': True, '/tmp/ipykernel_1057/4165789807.py': True, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/api_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/linear_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/profiler.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/pjit.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/traceback_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py': True, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/async_helpers.py': True}), lowering_parameters=LoweringParameters(override_lowering_rules=None, global_constant_computation=False, for_export=False, export_ignore_forward_compatibility=False)), name_stack=NameStack(stack=(Scope(name='jit(<lambda>)'), Scope(name='jit(main)'))), primitive=multiply_add, avals_in=[ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True)], avals_out=[ShapedArray(float32[])], tokens_in=<jax._src.interpreters.mlir.TokenSet object at 0x7fb2557a3be0>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None, threefry_partitionable=True, cur_abstract_mesh=None, xla_metadata={}), platforms=None), Value(<block argument> of type 'tensor<f32>' at index: 0), Value(<block argument> of type 'tensor<f32>' at index: 0), Value(%0 = "stablehlo.constant"() <{value = dense<1.000000e+01> : tensor<f32>}> : () -> tensor<f32>))
|<- multiply_add_lowering = [<jaxlib.mlir._mlir_libs._mlir.ir.OpResult object at 0x7fb254548b70>]

前向微分

JAX 以雅可比矩阵-向量积 (JVP) 的形式实现前向微分（您可以在 高级自动微分 中了解更多信息）。

如果您尝试计算 jvp 函数，您将收到一个错误，因为您尚未告知 JAX 如何区分 multiply_add 原语。

# The second argument is set to `(2., 10.)` values where you
# evaluate the Jacobian, and the third argument `(1., 1.)`
# contains the values of the tangents for the arguments.
with expectNotImplementedError():
  api.jvp(square_add_prim, (2., 10.), (1., 1.))

call square_add_prim(Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>)
  call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>)

Found expected exception:

Traceback (most recent call last):
  File "/tmp/ipykernel_1057/459539105.py", line 5, in <module>
    api.jvp(square_add_prim, (2., 10.), (1., 1.))
  File "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/api.py", line 1700, in jvp
    return _jvp(lu.wrap_init(fun), primals, tangents, has_aux=has_aux)
  File "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/api.py", line 1729, in _jvp
    out_primals, out_tangents = ad.jvp(flat_fun).call_wrapped(ps_flat, ts_flat)
NotImplementedError: Differentiation rule for 'multiply_add' not implemented

from jax.interpreters import ad

@trace("multiply_add_value_and_jvp")
def multiply_add_value_and_jvp(arg_values, arg_tangents):
  """Evaluates the primal output and the tangents (Jacobian-vector product).

  Given values of the arguments and perturbation of the arguments (tangents), 
  compute the output of the primitive and the perturbation of the output.

  This method must be JAX-traceable. JAX may invoke it with abstract values 
  for the arguments and tangents.

  Args:
    arg_values: A tuple of arguments
    arg_tangents: A tuple with the tangents of the arguments. The tuple has 
      the same length as the arg_values. Some of the tangents may also be the 
      special value `ad.Zero` to specify a zero tangent

  Returns:
     A pair of the primal output and the tangent.
  """
  x, y, z = arg_values
  xt, yt, zt = arg_tangents
  _trace("Primal evaluation:")
  # Now, you have a JAX-traceable computation of the output. 
  # Normally, you can use the multiply add (`ma`) primitive itself to compute the primal output. 
  primal_out = multiply_add_prim(x, y, z)

  _trace("Tangent evaluation:")
  # You must use a JAX-traceable way to compute the tangent. It turns out that 
  # the output tangent can be computed as (xt * y + x * yt + zt),
  # which you can implement in a JAX-traceable way using the same "multiply_add_prim" primitive.

  # You do need to deal specially with `Zero`. Here, you just turn it into a 
  # proper tensor of 0s (of the same shape as 'x'). 
  # An alternative would be to check for `Zero` and perform algebraic 
  # simplification of the output tangent computation.
  def make_zero(tan):
    return lax.zeros_like_array(x) if type(tan) is ad.Zero else tan  

  output_tangent = multiply_add_prim(make_zero(xt), y, multiply_add_prim(x, make_zero(yt), make_zero(zt)))
  return (primal_out, output_tangent)

# Register the forward differentiation rule with JAX:
ad.primitive_jvps[multiply_add_p] = multiply_add_value_and_jvp

# Tangent is: xt*y + x*yt + zt = 1.*2. + 2.*1. + 1. = 5.
assert api.jvp(square_add_prim, (2., 10.), (1., 1.)) == (14., 5.)

call square_add_prim(Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>)
  call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>)
    call multiply_add_value_and_jvp((2.0, 2.0, 10.0), (1.0, 1.0, 1.0))
      Primal evaluation:
      call multiply_add_prim(2.0, 2.0, 10.0)
        call multiply_add_impl(2.0, 2.0, 10.0)
        |<- multiply_add_impl = 14.0
      |<- multiply_add_prim = 14.0
      Tangent evaluation:
      call multiply_add_prim(2.0, 1.0, 1.0)
        call multiply_add_impl(2.0, 1.0, 1.0)
        |<- multiply_add_impl = 3.0
      |<- multiply_add_prim = 3.0
      call multiply_add_prim(1.0, 2.0, 3.0)
        call multiply_add_impl(1.0, 2.0, 3.0)
        |<- multiply_add_impl = 5.0
      |<- multiply_add_prim = 5.0
    |<- multiply_add_value_and_jvp = (14.0, 5.0)
  |<- multiply_add_prim = Traced<ShapedArray(float32[])>
|<- square_add_prim = Traced<ShapedArray(float32[])>

前向微分的 JIT

您可以将 jit 应用于前向微分函数。

assert api.jit(lambda arg_values, arg_tangents: 
                   api.jvp(square_add_prim, arg_values, arg_tangents))(
         (2., 10.), (1., 1.)) == (14., 5.)

call square_add_prim(Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>)
  call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>)
    call multiply_add_value_and_jvp((Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>), (Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>))
      Primal evaluation:
      call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>)
        call multiply_add_abstract_eval(ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True))
        |<- multiply_add_abstract_eval = ShapedArray(float32[])
      |<- multiply_add_prim = Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>
      Tangent evaluation:
      call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>)
        call multiply_add_abstract_eval(ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True))
        |<- multiply_add_abstract_eval = ShapedArray(float32[])
      |<- multiply_add_prim = Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>
      call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>)
        call multiply_add_abstract_eval(ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True), ShapedArray(float32[]))
        |<- multiply_add_abstract_eval = ShapedArray(float32[])
      |<- multiply_add_prim = Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>
    |<- multiply_add_value_and_jvp = (Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>)
  |<- multiply_add_prim = Traced<ShapedArray(float32[])>
|<- square_add_prim = Traced<ShapedArray(float32[])>
call multiply_add_lowering(LoweringRuleContext(module_context=ModuleContext(context=<jax._src.interpreters.mlir.JaxIrContext object at 0x7fb25457adc0>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7fb254582930>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7fb2545828d0>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7fb2557a28e0>, platforms=('cpu',), backend=<jaxlib.xla_extension.Client object at 0x7fb258145080>, axis_context=ShardingContext(num_devices=1, device_assignment=None, abstract_mesh=None), keepalives=[], channel_iterator=count(1), host_callbacks=[], shape_poly_state=<jax._src.interpreters.mlir.ShapePolyLoweringState object at 0x7fb2557a2920>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x56102ed269d0>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_1057/347789876.py":27:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1057/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_1057/2145028508.py":2:0) at "<module>"("/tmp/ipykernel_1057/2145028508.py":1:0)))))))))))}, location_cache={(<code object multiply_add_prim at 0x7fb2580fbaa0, file "/tmp/ipykernel_1057/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0)), (<code object func_wrapper at 0x7fb2580face0, file "/tmp/ipykernel_1057/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0)), (<code object multiply_add_value_and_jvp at 0x7fb2557addc0, file "/tmp/ipykernel_1057/347789876.py", line 3>, 36): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_1057/347789876.py":27:0)), (<code object square_add_prim at 0x7fb2580fb730, file "/tmp/ipykernel_1057/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_1057/1751132419.py":17:0)), (<code object <lambda> at 0x7fb2557ae760, file "/tmp/ipykernel_1057/2145028508.py", line 1>, 10): loc("<lambda>"("/tmp/ipykernel_1057/2145028508.py":2:0)), (<code object <module> at 0x7fb2557ad840, file "/tmp/ipykernel_1057/2145028508.py", line 1>, 16): loc("<module>"("/tmp/ipykernel_1057/2145028508.py":1:0))}, canonical_name_cache={'/tmp/ipykernel_1057/1751132419.py': '/tmp/ipykernel_1057/1751132419.py', '/tmp/ipykernel_1057/1393342955.py': '/tmp/ipykernel_1057/1393342955.py', '/tmp/ipykernel_1057/347789876.py': '/tmp/ipykernel_1057/347789876.py', '/tmp/ipykernel_1057/2145028508.py': '/tmp/ipykernel_1057/2145028508.py'}, is_user_file_cache={'/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/source_info_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/interpreters/partial_eval.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/core.py': False, '/tmp/ipykernel_1057/1751132419.py': True, '/tmp/ipykernel_1057/1393342955.py': True, '/tmp/ipykernel_1057/347789876.py': True, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/interpreters/ad.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/api_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/linear_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/api.py': False, '/tmp/ipykernel_1057/2145028508.py': True, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/profiler.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/pjit.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/traceback_util.py': False}), lowering_parameters=LoweringParameters(override_lowering_rules=None, global_constant_computation=False, for_export=False, export_ignore_forward_compatibility=False)), name_stack=NameStack(stack=(Scope(name='jit(<lambda>)'), Scope(name='jit(main)'), Transform(name='jvp'))), primitive=multiply_add, avals_in=[ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True)], avals_out=[ShapedArray(float32[])], tokens_in=<jax._src.interpreters.mlir.TokenSet object at 0x7fb2545902e0>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None, threefry_partitionable=True, cur_abstract_mesh=None, xla_metadata={}), platforms=None), Value(<block argument> of type 'tensor<f32>' at index: 0), Value(<block argument> of type 'tensor<f32>' at index: 0), Value(<block argument> of type 'tensor<f32>' at index: 1))
|<- multiply_add_lowering = [<jaxlib.mlir._mlir_libs._mlir.ir.OpResult object at 0x7fb255761bb0>]
call multiply_add_lowering(LoweringRuleContext(module_context=ModuleContext(context=<jax._src.interpreters.mlir.JaxIrContext object at 0x7fb25457adc0>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7fb254582930>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7fb2545828d0>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7fb2557a28e0>, platforms=('cpu',), backend=<jaxlib.xla_extension.Client object at 0x7fb258145080>, axis_context=ShardingContext(num_devices=1, device_assignment=None, abstract_mesh=None), keepalives=[], channel_iterator=count(1), host_callbacks=[], shape_poly_state=<jax._src.interpreters.mlir.ShapePolyLoweringState object at 0x7fb2557a2920>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x56102ed269d0>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_1057/347789876.py":27:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1057/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_1057/2145028508.py":2:0) at "<module>"("/tmp/ipykernel_1057/2145028508.py":1:0))))))))))), <jaxlib.xla_extension.Traceback object at 0x56102edd0550>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_1057/347789876.py":41:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1057/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_1057/2145028508.py":2:0) at "<module>"("/tmp/ipykernel_1057/2145028508.py":1:0)))))))))))}, location_cache={(<code object multiply_add_prim at 0x7fb2580fbaa0, file "/tmp/ipykernel_1057/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0)), (<code object func_wrapper at 0x7fb2580face0, file "/tmp/ipykernel_1057/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0)), (<code object multiply_add_value_and_jvp at 0x7fb2557addc0, file "/tmp/ipykernel_1057/347789876.py", line 3>, 36): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_1057/347789876.py":27:0)), (<code object square_add_prim at 0x7fb2580fb730, file "/tmp/ipykernel_1057/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_1057/1751132419.py":17:0)), (<code object <lambda> at 0x7fb2557ae760, file "/tmp/ipykernel_1057/2145028508.py", line 1>, 10): loc("<lambda>"("/tmp/ipykernel_1057/2145028508.py":2:0)), (<code object <module> at 0x7fb2557ad840, file "/tmp/ipykernel_1057/2145028508.py", line 1>, 16): loc("<module>"("/tmp/ipykernel_1057/2145028508.py":1:0)), (<code object multiply_add_value_and_jvp at 0x7fb2557addc0, file "/tmp/ipykernel_1057/347789876.py", line 3>, 86): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_1057/347789876.py":41:0))}, canonical_name_cache={'/tmp/ipykernel_1057/1751132419.py': '/tmp/ipykernel_1057/1751132419.py', '/tmp/ipykernel_1057/1393342955.py': '/tmp/ipykernel_1057/1393342955.py', '/tmp/ipykernel_1057/347789876.py': '/tmp/ipykernel_1057/347789876.py', '/tmp/ipykernel_1057/2145028508.py': '/tmp/ipykernel_1057/2145028508.py'}, is_user_file_cache={'/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/source_info_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/interpreters/partial_eval.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/core.py': False, '/tmp/ipykernel_1057/1751132419.py': True, '/tmp/ipykernel_1057/1393342955.py': True, '/tmp/ipykernel_1057/347789876.py': True, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/interpreters/ad.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/api_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/linear_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/api.py': False, '/tmp/ipykernel_1057/2145028508.py': True, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/profiler.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/pjit.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/traceback_util.py': False}), lowering_parameters=LoweringParameters(override_lowering_rules=None, global_constant_computation=False, for_export=False, export_ignore_forward_compatibility=False)), name_stack=NameStack(stack=(Scope(name='jit(<lambda>)'), Scope(name='jit(main)'), Transform(name='jvp'))), primitive=multiply_add, avals_in=[ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True)], avals_out=[ShapedArray(float32[])], tokens_in=<jax._src.interpreters.mlir.TokenSet object at 0x7fb254590430>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None, threefry_partitionable=True, cur_abstract_mesh=None, xla_metadata={}), platforms=None), Value(<block argument> of type 'tensor<f32>' at index: 0), Value(<block argument> of type 'tensor<f32>' at index: 2), Value(<block argument> of type 'tensor<f32>' at index: 3))
|<- multiply_add_lowering = [<jaxlib.mlir._mlir_libs._mlir.ir.OpResult object at 0x7fb25804f470>]
call multiply_add_lowering(LoweringRuleContext(module_context=ModuleContext(context=<jax._src.interpreters.mlir.JaxIrContext object at 0x7fb25457adc0>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7fb254582930>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7fb2545828d0>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7fb2557a28e0>, platforms=('cpu',), backend=<jaxlib.xla_extension.Client object at 0x7fb258145080>, axis_context=ShardingContext(num_devices=1, device_assignment=None, abstract_mesh=None), keepalives=[], channel_iterator=count(1), host_callbacks=[], shape_poly_state=<jax._src.interpreters.mlir.ShapePolyLoweringState object at 0x7fb2557a2920>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x56102ed269d0>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_1057/347789876.py":27:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1057/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_1057/2145028508.py":2:0) at "<module>"("/tmp/ipykernel_1057/2145028508.py":1:0))))))))))), <jaxlib.xla_extension.Traceback object at 0x56102edd0550>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_1057/347789876.py":41:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1057/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_1057/2145028508.py":2:0) at "<module>"("/tmp/ipykernel_1057/2145028508.py":1:0))))))))))), <jaxlib.xla_extension.Traceback object at 0x56102edb2c60>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_1057/347789876.py":41:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1057/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_1057/2145028508.py":2:0) at "<module>"("/tmp/ipykernel_1057/2145028508.py":1:0)))))))))))}, location_cache={(<code object multiply_add_prim at 0x7fb2580fbaa0, file "/tmp/ipykernel_1057/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0)), (<code object func_wrapper at 0x7fb2580face0, file "/tmp/ipykernel_1057/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0)), (<code object multiply_add_value_and_jvp at 0x7fb2557addc0, file "/tmp/ipykernel_1057/347789876.py", line 3>, 36): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_1057/347789876.py":27:0)), (<code object square_add_prim at 0x7fb2580fb730, file "/tmp/ipykernel_1057/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_1057/1751132419.py":17:0)), (<code object <lambda> at 0x7fb2557ae760, file "/tmp/ipykernel_1057/2145028508.py", line 1>, 10): loc("<lambda>"("/tmp/ipykernel_1057/2145028508.py":2:0)), (<code object <module> at 0x7fb2557ad840, file "/tmp/ipykernel_1057/2145028508.py", line 1>, 16): loc("<module>"("/tmp/ipykernel_1057/2145028508.py":1:0)), (<code object multiply_add_value_and_jvp at 0x7fb2557addc0, file "/tmp/ipykernel_1057/347789876.py", line 3>, 86): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_1057/347789876.py":41:0)), (<code object multiply_add_value_and_jvp at 0x7fb2557addc0, file "/tmp/ipykernel_1057/347789876.py", line 3>, 88): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_1057/347789876.py":41:0))}, canonical_name_cache={'/tmp/ipykernel_1057/1751132419.py': '/tmp/ipykernel_1057/1751132419.py', '/tmp/ipykernel_1057/1393342955.py': '/tmp/ipykernel_1057/1393342955.py', '/tmp/ipykernel_1057/347789876.py': '/tmp/ipykernel_1057/347789876.py', '/tmp/ipykernel_1057/2145028508.py': '/tmp/ipykernel_1057/2145028508.py'}, is_user_file_cache={'/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/source_info_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/interpreters/partial_eval.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/core.py': False, '/tmp/ipykernel_1057/1751132419.py': True, '/tmp/ipykernel_1057/1393342955.py': True, '/tmp/ipykernel_1057/347789876.py': True, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/interpreters/ad.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/api_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/linear_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/api.py': False, '/tmp/ipykernel_1057/2145028508.py': True, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/profiler.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/pjit.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/traceback_util.py': False}), lowering_parameters=LoweringParameters(override_lowering_rules=None, global_constant_computation=False, for_export=False, export_ignore_forward_compatibility=False)), name_stack=NameStack(stack=(Scope(name='jit(<lambda>)'), Scope(name='jit(main)'), Transform(name='jvp'))), primitive=multiply_add, avals_in=[ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True), ShapedArray(float32[])], avals_out=[ShapedArray(float32[])], tokens_in=<jax._src.interpreters.mlir.TokenSet object at 0x7fb254590490>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None, threefry_partitionable=True, cur_abstract_mesh=None, xla_metadata={}), platforms=None), Value(<block argument> of type 'tensor<f32>' at index: 2), Value(<block argument> of type 'tensor<f32>' at index: 0), Value(%3 = "stablehlo.add"(%2, %arg3) : (tensor<f32>, tensor<f32>) -> tensor<f32>))
|<- multiply_add_lowering = [<jaxlib.mlir._mlir_libs._mlir.ir.OpResult object at 0x7fb2545878b0>]

请注意，首先，您以抽象方式评估 multiply_add_value_and_jvp，这反过来又以抽象方式评估原始和切线评估（总共 3 次调用 ma 原语）。然后，您编译该原语的 3 次出现。

反向微分

如果您现在尝试使用反向微分，您会注意到 JAX 首先使用 multiply_add_value_and_jvp 来计算抽象值的前向微分，但随后会遇到 NotImplementedError。

在计算反向微分时，JAX 首先对前向微分代码 multiply_add_value_and_jvp 进行抽象评估，以获取计算输出切线的原语的跟踪。

• 请注意，JAX 使用微分点的具体值和切线的抽象值执行此抽象评估。

• 请注意，对于与 ma 的第三个参数相对应的切线，JAX 使用特殊的抽象切线值 Zero。这反映了您没有对 square_add_prim 的第二个参数求导，该参数流向 multiply_add_prim 的第三个参数。

• 另请注意，在切线的抽象评估期间，您将值 0.0 作为第三个参数的切线传递。这是因为在 multiply_add_value_and_jvp 的定义中使用了 make_zero 函数。

# This is reverse differentiation w.r.t. the first argument of `square_add_prim`
with expectNotImplementedError():
  api.grad(square_add_prim)(2., 10.)

call square_add_prim(Traced<ShapedArray(float32[], weak_type=True)>, 10.0)
  call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>, 10.0)
    call multiply_add_value_and_jvp((2.0, 2.0, 10.0), (Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>, Zero(ShapedArray(float32[], weak_type=True))))
      Primal evaluation:
      call multiply_add_prim(2.0, 2.0, 10.0)
        call multiply_add_impl(2.0, 2.0, 10.0)
        |<- multiply_add_impl = 14.0
      |<- multiply_add_prim = 14.0
      Tangent evaluation:
      call multiply_add_prim(2.0, Traced<ShapedArray(float32[], weak_type=True)>, 0.0)
        call multiply_add_abstract_eval(ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True))
        |<- multiply_add_abstract_eval = ShapedArray(float32[])
      |<- multiply_add_prim = Traced<ShapedArray(float32[])>
      call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>, 2.0, Traced<ShapedArray(float32[])>)
        call multiply_add_abstract_eval(ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True), ShapedArray(float32[]))
        |<- multiply_add_abstract_eval = ShapedArray(float32[])
      |<- multiply_add_prim = Traced<ShapedArray(float32[])>
    |<- multiply_add_value_and_jvp = (14.0, Traced<ShapedArray(float32[])>)
  |<- multiply_add_prim = Traced<ShapedArray(float32[])>
|<- square_add_prim = Traced<ShapedArray(float32[])>
Found expected exception:

Traceback (most recent call last):
  File "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/interpreters/ad.py", line 391, in get_primitive_transpose
    return primitive_transposes[p]
KeyError: multiply_add

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

Traceback (most recent call last):
  File "/home/docs/.asdf/installs/python/3.10.15/lib/python3.10/runpy.py", line 196, in _run_module_as_main
    return _run_code(code, main_globals, None,
  File "/home/docs/.asdf/installs/python/3.10.15/lib/python3.10/runpy.py", line 86, in _run_code
    exec(code, run_globals)
  File "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/ipykernel_launcher.py", line 18, in <module>
    app.launch_new_instance()
jax._src.source_info_util.JaxStackTraceBeforeTransformation: NotImplementedError: Transpose rule (for reverse-mode differentiation) for 'multiply_add' not implemented

The preceding stack trace is the source of the JAX operation that, once transformed by JAX, triggered the following exception.

--------------------

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

Traceback (most recent call last):
  File "/tmp/ipykernel_1057/2155094905.py", line 3, in <module>
    api.grad(square_add_prim)(2., 10.)
  File "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/traceback_util.py", line 180, in reraise_with_filtered_traceback
    return fun(*args, **kwargs)
  File "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/api.py", line 396, in grad_f
    _, g = value_and_grad_f(*args, **kwargs)
NotImplementedError: Transpose rule (for reverse-mode differentiation) for 'multiply_add' not implemented

上面的错误是因为 JAX 缺少使用前向微分代码来计算反向微分的部分。

转置

如前所述，在计算反向微分时，JAX 会获取使用前向微分计算切线的原语的跟踪。然后，JAX 会以抽象方式向后解释此跟踪，并为每个原语应用转置规则。

为了理解发生了什么，考虑一个更简单的函数示例 f(x, y) = x * y + y。假设需要在点 (2., 4.) 处进行微分。JAX 将从输入 xt 和 yt 的切线生成 ft 的以下 JVP 切线计算。

   a = xt * 4.
   b = 2. * yt
   c = a + b
   ft = c + yt

根据构造，切线计算始终是输入切线的线性函数。切线计算中可能出现的唯一非线性运算符是乘法，但其中一个操作数是常量。

JAX 将通过反向处理 JVP 计算来生成反向微分计算。对于切线计算中的每个操作，它会使用操作结果的余切，累积该操作使用的变量的余切。

  # Initialize cotangents of inputs and intermediate variables:
  xct = yct = act = bct = cct = 0.
  # Initialize cotangent of the output:
  fct = 1.
  # Process `ft = c + yt`:
  cct += fct
  yct += fct
  # Process `c = a + b`:
  act += cct
  bct += cct
  # Process `b = 2. * yt`:
  yct += 2. * bct
  # Process `a = xt * 4.`:
  xct += act * 4.

可以验证此计算产生 xct = 4. 和 yct = 3.，它们是函数 f 的偏导数。

对于 JVP 计算中可能出现的每个原语，JAX 知道如何转置它。从概念上讲，如果原语 p(x, y, z) 对于 x 的常量值，在参数 y 和 z 中是线性的，例如，p(x, y, z) = y*cy + z*cz，则原语的转置为

p_transpose(out_ct, x, _, _) = (None, out_ct*cy, out_ct*cz)

请注意，p_transpose 接受原语输出的余切和与原语的每个参数对应的值。对于线性参数，转置获得一个未定义的 _ 值，对于其他参数，它获得实际的常量。转置为原语的每个参数返回一个余切值，对于常量参数，返回值 None。

特别地

 add_transpose(out_ct, _, _) = (out_ct, out_ct)
 mult_transpose(out_ct, x, _) = (None, x * out_ct)
 mult_transpose(out_ct, _, y) = (out_ct * y, None)

@trace("multiply_add_transpose")
def multiply_add_transpose(ct, x, y, z):
  """Evaluates the transpose of a linear primitive.

  This method is only used when computing the backward gradient following 
  `value_and_jvp`, and is only needed for primitives that are used in the JVP 
  calculation for some other primitive. You need a transposition for `multiply_add_prim`, 
  because you have used `multiply_add_prim` in the computation of the `output_tangent` in 
  `multiply_add_value_and_jvp`.

  In this case, multiply_add is not a linear primitive. However, it is used linearly 
  w.r.t. tangents in `multiply_add_value_and_jvp`:
       `output_tangent(xt, yt, zt) = multiply_add_prim(xt, y, multiply_add_prim(x, yt, zt))`.

  Always one of the first two multiplicative arguments is a constant.

  Args:
      ct: The cotangent of the output of the primitive.
      x, y, z: The values of the arguments. The arguments that are used linearly
        get an ad.UndefinedPrimal value. The other arguments get a constant
        value.

  Returns:
      A tuple with the cotangent of the inputs, with the value None
      corresponding to the constant arguments.
  """
  if not ad.is_undefined_primal(x):
    # This use of multiply_add is with a constant "x".
    assert ad.is_undefined_primal(y)
    ct_y = ad.Zero(y.aval) if type(ct) is ad.Zero else multiply_add_prim(x, ct, lax.zeros_like_array(x))
    res = None, ct_y, ct
  else:
    # This use of multiply_add is with a constant "y".
    assert ad.is_undefined_primal(x)
    ct_x = ad.Zero(x.aval) if type(ct) is ad.Zero else multiply_add_prim(ct, y, lax.zeros_like_array(y))
    res = ct_x, None, ct
  return res

ad.primitive_transposes[multiply_add_p] = multiply_add_transpose

现在您可以完成 grad 的运行

assert api.grad(square_add_prim)(2., 10.) == 4.

call square_add_prim(Traced<ShapedArray(float32[], weak_type=True)>, 10.0)
  call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>, 10.0)
    call multiply_add_value_and_jvp((2.0, 2.0, 10.0), (Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>, Zero(ShapedArray(float32[], weak_type=True))))
      Primal evaluation:
      call multiply_add_prim(2.0, 2.0, 10.0)
        call multiply_add_impl(2.0, 2.0, 10.0)
        |<- multiply_add_impl = 14.0
      |<- multiply_add_prim = 14.0
      Tangent evaluation:
      call multiply_add_prim(2.0, Traced<ShapedArray(float32[], weak_type=True)>, 0.0)
        call multiply_add_abstract_eval(ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True))
        |<- multiply_add_abstract_eval = ShapedArray(float32[])
      |<- multiply_add_prim = Traced<ShapedArray(float32[])>
      call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>, 2.0, Traced<ShapedArray(float32[])>)
        call multiply_add_abstract_eval(ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True), ShapedArray(float32[]))
        |<- multiply_add_abstract_eval = ShapedArray(float32[])
      |<- multiply_add_prim = Traced<ShapedArray(float32[])>
    |<- multiply_add_value_and_jvp = (14.0, Traced<ShapedArray(float32[])>)
  |<- multiply_add_prim = Traced<ShapedArray(float32[])>
|<- square_add_prim = Traced<ShapedArray(float32[])>
call multiply_add_transpose(1.0, UndefinedPrimal(ShapedArray(float32[], weak_type=True)), 2.0, UndefinedPrimal(ShapedArray(float32[])))
  call multiply_add_prim(1.0, 2.0, 0.0)
    call multiply_add_impl(1.0, 2.0, 0.0)
    |<- multiply_add_impl = 2.0
  |<- multiply_add_prim = 2.0
|<- multiply_add_transpose = (2.0, None, 1.0)
call multiply_add_transpose(1.0, 2.0, UndefinedPrimal(ShapedArray(float32[], weak_type=True)), 0.0)
  call multiply_add_prim(2.0, 1.0, 0.0)
    call multiply_add_impl(2.0, 1.0, 0.0)
    |<- multiply_add_impl = 2.0
  |<- multiply_add_prim = 2.0
|<- multiply_add_transpose = (None, 2.0, 1.0)

请注意对 multiply_add_transpose 的两次调用。它们对应于在 multiply_add_value_and_jvp 中计算 output_tangent 时对 multiply_add_prim 的两次使用。第一次转置调用对应于最后一次使用 multiply_add_prim：multiply_add_prim(xt, y, ...)，其中 y 是常量 2.0。

反向微分的 JIT

请注意，multiply_add_value_and_jvp 的抽象评估仅使用抽象值。同时，在没有 JIT 的情况下，您使用了 ConcreteArray。

assert api.jit(api.grad(square_add_prim))(2., 10.) == 4.

call square_add_prim(Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>)
  call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>)
    call multiply_add_value_and_jvp((Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>), (Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>, Zero(ShapedArray(float32[], weak_type=True))))
      Primal evaluation:
      call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>)
        call multiply_add_abstract_eval(ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True))
        |<- multiply_add_abstract_eval = ShapedArray(float32[])
      |<- multiply_add_prim = Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>
      Tangent evaluation:
      call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>)
        call multiply_add_abstract_eval(ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True))
        |<- multiply_add_abstract_eval = ShapedArray(float32[])
      |<- multiply_add_prim = Traced<ShapedArray(float32[])>
      call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[])>)
        call multiply_add_abstract_eval(ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True), ShapedArray(float32[]))
        |<- multiply_add_abstract_eval = ShapedArray(float32[])
      |<- multiply_add_prim = Traced<ShapedArray(float32[])>
    |<- multiply_add_value_and_jvp = (Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[])>)
  |<- multiply_add_prim = Traced<ShapedArray(float32[])>
|<- square_add_prim = Traced<ShapedArray(float32[])>
call multiply_add_transpose(Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>, UndefinedPrimal(ShapedArray(float32[], weak_type=True)), Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, UndefinedPrimal(ShapedArray(float32[])))
  call multiply_add_prim(Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>)
    call multiply_add_abstract_eval(ShapedArray(float32[]), ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True))
    |<- multiply_add_abstract_eval = ShapedArray(float32[])
  |<- multiply_add_prim = Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>
|<- multiply_add_transpose = (Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>, None, Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>)
call multiply_add_transpose(Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, UndefinedPrimal(ShapedArray(float32[], weak_type=True)), Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>)
  call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace>)
    call multiply_add_abstract_eval(ShapedArray(float32[], weak_type=True), ShapedArray(float32[]), ShapedArray(float32[], weak_type=True))
    |<- multiply_add_abstract_eval = ShapedArray(float32[])
  |<- multiply_add_prim = Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>
|<- multiply_add_transpose = (None, Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[])>with<DynamicJaxprTrace>)
call multiply_add_lowering(LoweringRuleContext(module_context=ModuleContext(context=<jax._src.interpreters.mlir.JaxIrContext object at 0x7fb2545a7440>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7fb25459e520>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7fb25459f640>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7fb254591500>, platforms=('cpu',), backend=<jaxlib.xla_extension.Client object at 0x7fb258145080>, axis_context=ShardingContext(num_devices=1, device_assignment=None, abstract_mesh=None), keepalives=[], channel_iterator=count(1), host_callbacks=[], shape_poly_state=<jax._src.interpreters.mlir.ShapePolyLoweringState object at 0x7fb254592170>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x56102edb75c0>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_1057/347789876.py":41:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1057/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("<module>"("/tmp/ipykernel_1057/3085343041.py":1:0) at "run_code"("/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py":3577:0)))))))))))}, location_cache={(<code object multiply_add_prim at 0x7fb2580fbaa0, file "/tmp/ipykernel_1057/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0)), (<code object func_wrapper at 0x7fb2580face0, file "/tmp/ipykernel_1057/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0)), (<code object multiply_add_value_and_jvp at 0x7fb2557addc0, file "/tmp/ipykernel_1057/347789876.py", line 3>, 88): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_1057/347789876.py":41:0)), (<code object square_add_prim at 0x7fb2580fb730, file "/tmp/ipykernel_1057/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_1057/1751132419.py":17:0)), (<code object <module> at 0x7fb2557acbe0, file "/tmp/ipykernel_1057/3085343041.py", line 1>, 18): loc("<module>"("/tmp/ipykernel_1057/3085343041.py":1:0)), (<code object run_code at 0x7fb29538b050, file "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py", line 3541>, 76): loc("run_code"("/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py":3577:0))}, canonical_name_cache={'/tmp/ipykernel_1057/1751132419.py': '/tmp/ipykernel_1057/1751132419.py', '/tmp/ipykernel_1057/1393342955.py': '/tmp/ipykernel_1057/1393342955.py', '/tmp/ipykernel_1057/347789876.py': '/tmp/ipykernel_1057/347789876.py', '/tmp/ipykernel_1057/3085343041.py': '/tmp/ipykernel_1057/3085343041.py', '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py': '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py'}, is_user_file_cache={'/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/source_info_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/interpreters/partial_eval.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/core.py': False, '/tmp/ipykernel_1057/1751132419.py': True, '/tmp/ipykernel_1057/1393342955.py': True, '/tmp/ipykernel_1057/347789876.py': True, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/interpreters/ad.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/api_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/linear_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/profiler.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/api.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/traceback_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/pjit.py': False, '/tmp/ipykernel_1057/3085343041.py': True, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py': True}), lowering_parameters=LoweringParameters(override_lowering_rules=None, global_constant_computation=False, for_export=False, export_ignore_forward_compatibility=False)), name_stack=NameStack(stack=(Scope(name='jit(square_add_prim)'), Scope(name='jit(main)'), Transform(name='transpose'), Transform(name='jvp'))), primitive=multiply_add, avals_in=[ShapedArray(float32[]), ShapedArray(float32[], weak_type=True), ShapedArray(float32[], weak_type=True)], avals_out=[ShapedArray(float32[])], tokens_in=<jax._src.interpreters.mlir.TokenSet object at 0x7fb254590e20>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None, threefry_partitionable=True, cur_abstract_mesh=None, xla_metadata={}), platforms=None), Value(%0 = "stablehlo.constant"() <{value = dense<1.000000e+00> : tensor<f32>}> : () -> tensor<f32>), Value(<block argument> of type 'tensor<f32>' at index: 0), Value(%1 = "stablehlo.constant"() <{value = dense<0.000000e+00> : tensor<f32>}> : () -> tensor<f32>))
|<- multiply_add_lowering = [<jaxlib.mlir._mlir_libs._mlir.ir.OpResult object at 0x7fb254587d70>]
call multiply_add_lowering(LoweringRuleContext(module_context=ModuleContext(context=<jax._src.interpreters.mlir.JaxIrContext object at 0x7fb2545a7440>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7fb25459e520>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7fb25459f640>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7fb254591500>, platforms=('cpu',), backend=<jaxlib.xla_extension.Client object at 0x7fb258145080>, axis_context=ShardingContext(num_devices=1, device_assignment=None, abstract_mesh=None), keepalives=[], channel_iterator=count(1), host_callbacks=[], shape_poly_state=<jax._src.interpreters.mlir.ShapePolyLoweringState object at 0x7fb254592170>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x56102edb75c0>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_1057/347789876.py":41:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1057/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("<module>"("/tmp/ipykernel_1057/3085343041.py":1:0) at "run_code"("/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py":3577:0))))))))))), <jaxlib.xla_extension.Traceback object at 0x56102ee5eec0>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_1057/347789876.py":41:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1057/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("<module>"("/tmp/ipykernel_1057/3085343041.py":1:0) at "run_code"("/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py":3577:0)))))))))))}, location_cache={(<code object multiply_add_prim at 0x7fb2580fbaa0, file "/tmp/ipykernel_1057/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0)), (<code object func_wrapper at 0x7fb2580face0, file "/tmp/ipykernel_1057/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0)), (<code object multiply_add_value_and_jvp at 0x7fb2557addc0, file "/tmp/ipykernel_1057/347789876.py", line 3>, 88): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_1057/347789876.py":41:0)), (<code object square_add_prim at 0x7fb2580fb730, file "/tmp/ipykernel_1057/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_1057/1751132419.py":17:0)), (<code object <module> at 0x7fb2557acbe0, file "/tmp/ipykernel_1057/3085343041.py", line 1>, 18): loc("<module>"("/tmp/ipykernel_1057/3085343041.py":1:0)), (<code object run_code at 0x7fb29538b050, file "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py", line 3541>, 76): loc("run_code"("/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py":3577:0)), (<code object multiply_add_value_and_jvp at 0x7fb2557addc0, file "/tmp/ipykernel_1057/347789876.py", line 3>, 86): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_1057/347789876.py":41:0))}, canonical_name_cache={'/tmp/ipykernel_1057/1751132419.py': '/tmp/ipykernel_1057/1751132419.py', '/tmp/ipykernel_1057/1393342955.py': '/tmp/ipykernel_1057/1393342955.py', '/tmp/ipykernel_1057/347789876.py': '/tmp/ipykernel_1057/347789876.py', '/tmp/ipykernel_1057/3085343041.py': '/tmp/ipykernel_1057/3085343041.py', '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py': '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py'}, is_user_file_cache={'/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/source_info_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/interpreters/partial_eval.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/core.py': False, '/tmp/ipykernel_1057/1751132419.py': True, '/tmp/ipykernel_1057/1393342955.py': True, '/tmp/ipykernel_1057/347789876.py': True, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/interpreters/ad.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/api_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/linear_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/profiler.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/api.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/traceback_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/pjit.py': False, '/tmp/ipykernel_1057/3085343041.py': True, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py': True}), lowering_parameters=LoweringParameters(override_lowering_rules=None, global_constant_computation=False, for_export=False, export_ignore_forward_compatibility=False)), name_stack=NameStack(stack=(Scope(name='jit(square_add_prim)'), Scope(name='jit(main)'), Transform(name='transpose'), Transform(name='jvp'))), primitive=multiply_add, avals_in=[ShapedArray(float32[], weak_type=True), ShapedArray(float32[]), ShapedArray(float32[], weak_type=True)], avals_out=[ShapedArray(float32[])], tokens_in=<jax._src.interpreters.mlir.TokenSet object at 0x7fb2545927d0>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None, threefry_partitionable=True, cur_abstract_mesh=None, xla_metadata={}), platforms=None), Value(<block argument> of type 'tensor<f32>' at index: 0), Value(%0 = "stablehlo.constant"() <{value = dense<1.000000e+00> : tensor<f32>}> : () -> tensor<f32>), Value(%1 = "stablehlo.constant"() <{value = dense<0.000000e+00> : tensor<f32>}> : () -> tensor<f32>))
|<- multiply_add_lowering = [<jaxlib.mlir._mlir_libs._mlir.ir.OpResult object at 0x7fb25874dab0>]

批处理

批处理转换将逐点计算转换为向量上的计算。如果您现在尝试，您将收到一个 NotImplementedError。

# The arguments are two vectors instead of two scalars.
with expectNotImplementedError():
  api.vmap(square_add_prim, in_axes=0, out_axes=0)(np.array([2., 3.]),
                                               np.array([10., 20.]))

call square_add_prim(Traced<ShapedArray(float32[])>, Traced<ShapedArray(float32[])>)
  call multiply_add_prim(Traced<ShapedArray(float32[])>, Traced<ShapedArray(float32[])>, Traced<ShapedArray(float32[])>)

Found expected exception:

Traceback (most recent call last):
  File "/tmp/ipykernel_1057/1080163607.py", line 3, in <module>
    api.vmap(square_add_prim, in_axes=0, out_axes=0)(np.array([2., 3.]),
  File "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/traceback_util.py", line 180, in reraise_with_filtered_traceback
    return fun(*args, **kwargs)
  File "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/api.py", line 1001, in vmap_f
    out_flat = batching.batch(
NotImplementedError: Batching rule for 'multiply_add' not implemented

您需要指示 JAX 如何评估原语的批处理版本。在这种特殊情况下，multiply_add_prim 已经对输入向量的任何维度逐点运行，因此批处理版本可以使用相同的 multiply_add_prim 实现。

from jax.interpreters import batching

@trace("multiply_add_batch")
def multiply_add_batch(vector_arg_values, batch_axes):
  """Computes the batched version of the primitive.
  
  This must be a JAX-traceable function.
  
  Since the `multiply_add primitive` already operates point-wise on arbitrary
  dimension tensors, to batch it you can use the primitive itself. This works as
  long as both the inputs have the same dimensions and are batched along the
  same axes. The result is batched along the axis that the inputs are batched.

  Args:
    vector_arg_values: A tuple of two arguments, each being a tensor of matching
      shape.
    batch_axes: The axes that are being batched. See vmap documentation.

  Returns:
    A tuple of the result, and the result axis that was batched. 
  """
  assert batch_axes[0] == batch_axes[1]
  assert batch_axes[0] == batch_axes[2]
  _trace("Using multiply_add to compute the batch:")
  res = multiply_add_prim(*vector_arg_values)
  return res, batch_axes[0]


batching.primitive_batchers[multiply_add_p] = multiply_add_batch

assert np.allclose(api.vmap(square_add_prim, in_axes=0, out_axes=0)(
  np.array([2., 3.]),
  np.array([10., 20.])),
  [14., 29.])

call square_add_prim(Traced<ShapedArray(float32[])>, Traced<ShapedArray(float32[])>)
  call multiply_add_prim(Traced<ShapedArray(float32[])>, Traced<ShapedArray(float32[])>, Traced<ShapedArray(float32[])>)
    call multiply_add_batch(([2. 3.], [2. 3.], [10. 20.]), (0, 0, 0))
      Using multiply_add to compute the batch:
      call multiply_add_prim([2. 3.], [2. 3.], [10. 20.])
        call multiply_add_impl([2. 3.], [2. 3.], [10. 20.])
        |<- multiply_add_impl = [14. 29.]
      |<- multiply_add_prim = [14. 29.]
    |<- multiply_add_batch = ([14. 29.], 0)
  |<- multiply_add_prim = Traced<ShapedArray(float32[])>
|<- square_add_prim = Traced<ShapedArray(float32[])>

批处理的 JIT

下面是将 JIT 应用于批处理的示例

assert np.allclose(api.jit(api.vmap(square_add_prim, in_axes=0, out_axes=0))
                    (np.array([2., 3.]),
                     np.array([10., 20.])),
                    [14., 29.])

call square_add_prim(Traced<ShapedArray(float32[])>, Traced<ShapedArray(float32[])>)
  call multiply_add_prim(Traced<ShapedArray(float32[])>, Traced<ShapedArray(float32[])>, Traced<ShapedArray(float32[])>)
    call multiply_add_batch((Traced<ShapedArray(float32[2])>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[2])>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[2])>with<DynamicJaxprTrace>), (0, 0, 0))
      Using multiply_add to compute the batch:
      call multiply_add_prim(Traced<ShapedArray(float32[2])>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[2])>with<DynamicJaxprTrace>, Traced<ShapedArray(float32[2])>with<DynamicJaxprTrace>)
        call multiply_add_abstract_eval(ShapedArray(float32[2]), ShapedArray(float32[2]), ShapedArray(float32[2]))
        |<- multiply_add_abstract_eval = ShapedArray(float32[2])
      |<- multiply_add_prim = Traced<ShapedArray(float32[2])>with<DynamicJaxprTrace>
    |<- multiply_add_batch = (Traced<ShapedArray(float32[2])>with<DynamicJaxprTrace>, 0)
  |<- multiply_add_prim = Traced<ShapedArray(float32[])>
|<- square_add_prim = Traced<ShapedArray(float32[])>
call multiply_add_lowering(LoweringRuleContext(module_context=ModuleContext(context=<jax._src.interpreters.mlir.JaxIrContext object at 0x7fb2545a79c0>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7fb2545c4040>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7fb2545c40d0>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7fb254591230>, platforms=('cpu',), backend=<jaxlib.xla_extension.Client object at 0x7fb258145080>, axis_context=ShardingContext(num_devices=1, device_assignment=None, abstract_mesh=None), keepalives=[], channel_iterator=count(1), host_callbacks=[], shape_poly_state=<jax._src.interpreters.mlir.ShapePolyLoweringState object at 0x7fb254591db0>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x56102f1d6620>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("multiply_add_batch"("/tmp/ipykernel_1057/1827752256.py":25:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1057/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0) at callsite("<module>"("/tmp/ipykernel_1057/1392464762.py":1:0) at "run_code"("/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py":3577:0)))))))))))}, location_cache={(<code object multiply_add_prim at 0x7fb2580fbaa0, file "/tmp/ipykernel_1057/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_1057/1751132419.py":12:0)), (<code object func_wrapper at 0x7fb2580face0, file "/tmp/ipykernel_1057/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_1057/1393342955.py":48:0)), (<code object multiply_add_batch at 0x7fb2557af260, file "/tmp/ipykernel_1057/1827752256.py", line 3>, 52): loc("multiply_add_batch"("/tmp/ipykernel_1057/1827752256.py":25:0)), (<code object square_add_prim at 0x7fb2580fb730, file "/tmp/ipykernel_1057/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_1057/1751132419.py":17:0)), (<code object <module> at 0x7fb2557aee40, file "/tmp/ipykernel_1057/1392464762.py", line 1>, 48): loc("<module>"("/tmp/ipykernel_1057/1392464762.py":1:0)), (<code object run_code at 0x7fb29538b050, file "/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py", line 3541>, 76): loc("run_code"("/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py":3577:0))}, canonical_name_cache={'/tmp/ipykernel_1057/1751132419.py': '/tmp/ipykernel_1057/1751132419.py', '/tmp/ipykernel_1057/1393342955.py': '/tmp/ipykernel_1057/1393342955.py', '/tmp/ipykernel_1057/1827752256.py': '/tmp/ipykernel_1057/1827752256.py', '/tmp/ipykernel_1057/1392464762.py': '/tmp/ipykernel_1057/1392464762.py', '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py': '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py'}, is_user_file_cache={'/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/source_info_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/interpreters/partial_eval.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/core.py': False, '/tmp/ipykernel_1057/1751132419.py': True, '/tmp/ipykernel_1057/1393342955.py': True, '/tmp/ipykernel_1057/1827752256.py': True, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/interpreters/batching.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/linear_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/api.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/traceback_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/api_util.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/profiler.py': False, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/jax/_src/pjit.py': False, '/tmp/ipykernel_1057/1392464762.py': True, '/home/docs/checkouts/readthedocs.org/user_builds/jax/envs/latest/lib/python3.10/site-packages/IPython/core/interactiveshell.py': True}), lowering_parameters=LoweringParameters(override_lowering_rules=None, global_constant_computation=False, for_export=False, export_ignore_forward_compatibility=False)), name_stack=NameStack(stack=(Scope(name='jit(square_add_prim)'), Scope(name='jit(main)'), Transform(name='vmap'))), primitive=multiply_add, avals_in=[ShapedArray(float32[2]), ShapedArray(float32[2]), ShapedArray(float32[2])], avals_out=[ShapedArray(float32[2])], tokens_in=<jax._src.interpreters.mlir.TokenSet object at 0x7fb25575acb0>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None, threefry_partitionable=True, cur_abstract_mesh=None, xla_metadata={}), platforms=None), Value(<block argument> of type 'tensor<2xf32>' at index: 0), Value(<block argument> of type 'tensor<2xf32>' at index: 0), Value(<block argument> of type 'tensor<2xf32>' at index: 1))
|<- multiply_add_lowering = [<jaxlib.mlir._mlir_libs._mlir.ir.OpResult object at 0x7fb25874e030>]