JAX 内部原理：基本操作

JAX 基本操作入门

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 操作符的封装）或 jax.core.Primitive("multiply_add") 来实现，如下所示。

JAX 能够获取此类原语操作序列，并通过其 Python 函数的可组合转换（例如 jax.jit()、 jax.grad() 和 jax.vmap()）对其进行转换。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要么是

• 数据属性的检查： 数据信息，例如形状或类型；或者

• JAX 原语： 这些是在本教程中介绍的 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。JAX 可追踪函数 可以被 JAX 使用抽象参数调用。例如，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 提供了与大多数 XLA 操作相对应的预定义原语，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 JAX 可追踪。

此外，JAX 原语集是可扩展的，因此您可以定义一个新的原语来封装该函数的行为，而不是使用预定义的 JAX 原语重新实现该函数。

考虑以下示例：您想为 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实现：

• 使用现有的 JAX 原语；或者

• 定义新的 JAX 基本操作

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，或者使用其他本身使用 JAX 原语编写的函数，例如 jax.lax() 模块中定义的那些函数

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

要了解 JAX 在内部如何使用原语，请添加一些辅助程序来跟踪函数调用

#@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

您可以不直接使用 jax.lax() 原语，而是使用其他已经用这些原语编写的函数，例如 jax.numpy 中的函数

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(...)（在本 colab 中进一步描述）调用 square_add_numpy 和 multiply_add_numpy。请务必记住，JAX 可追踪函数不仅能够对具体参数进行操作，还能对 JAX 可能用于抽象函数执行的特殊抽象参数进行操作。

只要该函数是用 JAX 原语编写的，JAX 的可追踪性属性就得到满足。

定义新的 JAX 原语

添加对乘加功能支持的正确方法是使用现有的 JAX 原语，如上所示。但是，为了演示 JAX 原语的工作原理，假设您想为 JAX 添加一个新的原语来执行乘加功能。

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_1037/1751132419.py:3: DeprecationWarning: jax.core.Primitive is deprecated. Use jax.extend.core.Primitive instead, and see https://jax.net.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，则会收到错误，因为您尚未告知 JAX 关于新原语的任何语义。

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_1037/2844449444.py", line 2, in <module>
    square_add_prim(2., 10.)
  File "/tmp/ipykernel_1037/1393342955.py", line 48, in func_wrapper
    res = func(*args)
  File "/tmp/ipykernel_1037/1751132419.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 NotImplementedError

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_1037/1813425700.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 338, 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 函数以及其他转换，JAX 首先仅使用参数的形状和类型对其进行抽象评估。这种抽象评估有多种用途

• 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以是 ShapedArray(float32[3])，或 ConcreteArray([1., 2., 3.])。在后一种情况下，JAX 使用包装为抽象值的实际具体值。

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，但您将收到另一个关于缺少实际 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_1037/1813425700.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 338, 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 编译通过将每个原语编译成 XLA 操作图来工作。

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 已经为大多数操作预定义了原语。但是，XLA 包含一个 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.net.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，下面 JAX 首先抽象地评估该函数，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 0x7f34947acfe0>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7f34947cfb30>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7f34947cfbb0>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7f34947cfb70>, backend_or_name=<jaxlib.xla_extension.Client object at 0x7f34982137c0>, platforms=('cpu',), 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 0x7f34947dc700>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x55d07ba1bdc0>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1037/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_1037/1570919344.py":1:0) at callsite("<module>"("/tmp/ipykernel_1037/1570919344.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 0x7f34981ef7e0, file "/tmp/ipykernel_1037/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0)), (<code object func_wrapper at 0x7f34981ee340, file "/tmp/ipykernel_1037/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0)), (<code object square_add_prim at 0x7f34981ed790, file "/tmp/ipykernel_1037/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_1037/1751132419.py":17:0)), (<code object <lambda> at 0x7f3494823730, file "/tmp/ipykernel_1037/1570919344.py", line 1>, 6): loc("<lambda>"("/tmp/ipykernel_1037/1570919344.py":1:0)), (<code object <module> at 0x7f3494823890, file "/tmp/ipykernel_1037/1570919344.py", line 1>, 16): loc("<module>"("/tmp/ipykernel_1037/1570919344.py":1:0)), (<code object run_code at 0x7f34d5376fa0, 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 0x7f34d5376e40, 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 0x7f34d5376ad0, 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 0x7f34d52356e0, 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_1037/1751132419.py': '/tmp/ipykernel_1037/1751132419.py', '/tmp/ipykernel_1037/1393342955.py': '/tmp/ipykernel_1037/1393342955.py', '/tmp/ipykernel_1037/1570919344.py': '/tmp/ipykernel_1037/1570919344.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_1037/1751132419.py': True, '/tmp/ipykernel_1037/1393342955.py': True, '/tmp/ipykernel_1037/1570919344.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 0x7f34947dcbe0>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None, threefry_partitionable=False, 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 0x7f34d015a270>]

下面是 jit 的另一个用法，您只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 0x7f34947adc60>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7f34947e3330>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7f34947e33b0>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7f34947e3370>, backend_or_name=<jaxlib.xla_extension.Client object at 0x7f34982137c0>, platforms=('cpu',), 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 0x7f34947dd9f0>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x55d07ba1bdc0>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1037/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_1037/4165789807.py":1:0) at callsite("<module>"("/tmp/ipykernel_1037/4165789807.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 0x7f34981ef7e0, file "/tmp/ipykernel_1037/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0)), (<code object func_wrapper at 0x7f34981ee340, file "/tmp/ipykernel_1037/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0)), (<code object square_add_prim at 0x7f34981ed790, file "/tmp/ipykernel_1037/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_1037/1751132419.py":17:0)), (<code object <lambda> at 0x7f34d45679f0, file "/tmp/ipykernel_1037/4165789807.py", line 1>, 6): loc("<lambda>"("/tmp/ipykernel_1037/4165789807.py":1:0)), (<code object <module> at 0x7f34d51fc240, file "/tmp/ipykernel_1037/4165789807.py", line 1>, 20): loc("<module>"("/tmp/ipykernel_1037/4165789807.py":1:0)), (<code object run_code at 0x7f34d5376fa0, 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 0x7f34d5376e40, 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 0x7f34d5376ad0, 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 0x7f34d52356e0, 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_1037/1751132419.py': '/tmp/ipykernel_1037/1751132419.py', '/tmp/ipykernel_1037/1393342955.py': '/tmp/ipykernel_1037/1393342955.py', '/tmp/ipykernel_1037/4165789807.py': '/tmp/ipykernel_1037/4165789807.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_1037/1751132419.py': True, '/tmp/ipykernel_1037/1393342955.py': True, '/tmp/ipykernel_1037/4165789807.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 0x7f34947dded0>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None, threefry_partitionable=False, 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 0x7f34982edc70>]

前向微分

JAX 以雅可比向量积 (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，则会收到错误，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_1037/459539105.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 1693, 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 1722, 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 0x7f3494617f10>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7f3498127b70>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7f3494625ff0>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7f3494625fb0>, backend_or_name=<jaxlib.xla_extension.Client object at 0x7f34982137c0>, platforms=('cpu',), 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 0x7f34947ded10>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x55d07a242b30>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_1037/347789876.py":27:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1037/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_1037/2145028508.py":2:0) at "<module>"("/tmp/ipykernel_1037/2145028508.py":1:0)))))))))))}, location_cache={(<code object multiply_add_prim at 0x7f34981ef7e0, file "/tmp/ipykernel_1037/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0)), (<code object func_wrapper at 0x7f34981ee340, file "/tmp/ipykernel_1037/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0)), (<code object multiply_add_value_and_jvp at 0x7f34d01664a0, file "/tmp/ipykernel_1037/347789876.py", line 3>, 36): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_1037/347789876.py":27:0)), (<code object square_add_prim at 0x7f34981ed790, file "/tmp/ipykernel_1037/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_1037/1751132419.py":17:0)), (<code object <lambda> at 0x7f3494821630, file "/tmp/ipykernel_1037/2145028508.py", line 1>, 10): loc("<lambda>"("/tmp/ipykernel_1037/2145028508.py":2:0)), (<code object <module> at 0x7f34948216e0, file "/tmp/ipykernel_1037/2145028508.py", line 1>, 16): loc("<module>"("/tmp/ipykernel_1037/2145028508.py":1:0))}, canonical_name_cache={'/tmp/ipykernel_1037/1751132419.py': '/tmp/ipykernel_1037/1751132419.py', '/tmp/ipykernel_1037/1393342955.py': '/tmp/ipykernel_1037/1393342955.py', '/tmp/ipykernel_1037/347789876.py': '/tmp/ipykernel_1037/347789876.py', '/tmp/ipykernel_1037/2145028508.py': '/tmp/ipykernel_1037/2145028508.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_1037/1751132419.py': True, '/tmp/ipykernel_1037/1393342955.py': True, '/tmp/ipykernel_1037/347789876.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_1037/2145028508.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 0x7f34947dddb0>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None, threefry_partitionable=False, 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 0x7f3494833230>]
call multiply_add_lowering(LoweringRuleContext(module_context=ModuleContext(context=<jax._src.interpreters.mlir.JaxIrContext object at 0x7f3494617f10>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7f3498127b70>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7f3494625ff0>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7f3494625fb0>, backend_or_name=<jaxlib.xla_extension.Client object at 0x7f34982137c0>, platforms=('cpu',), 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 0x7f34947ded10>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x55d07a242b30>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_1037/347789876.py":27:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1037/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_1037/2145028508.py":2:0) at "<module>"("/tmp/ipykernel_1037/2145028508.py":1:0))))))))))), <jaxlib.xla_extension.Traceback object at 0x55d07ba9fb40>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_1037/347789876.py":41:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1037/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_1037/2145028508.py":2:0) at "<module>"("/tmp/ipykernel_1037/2145028508.py":1:0)))))))))))}, location_cache={(<code object multiply_add_prim at 0x7f34981ef7e0, file "/tmp/ipykernel_1037/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0)), (<code object func_wrapper at 0x7f34981ee340, file "/tmp/ipykernel_1037/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0)), (<code object multiply_add_value_and_jvp at 0x7f34d01664a0, file "/tmp/ipykernel_1037/347789876.py", line 3>, 36): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_1037/347789876.py":27:0)), (<code object square_add_prim at 0x7f34981ed790, file "/tmp/ipykernel_1037/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_1037/1751132419.py":17:0)), (<code object <lambda> at 0x7f3494821630, file "/tmp/ipykernel_1037/2145028508.py", line 1>, 10): loc("<lambda>"("/tmp/ipykernel_1037/2145028508.py":2:0)), (<code object <module> at 0x7f34948216e0, file "/tmp/ipykernel_1037/2145028508.py", line 1>, 16): loc("<module>"("/tmp/ipykernel_1037/2145028508.py":1:0)), (<code object multiply_add_value_and_jvp at 0x7f34d01664a0, file "/tmp/ipykernel_1037/347789876.py", line 3>, 86): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_1037/347789876.py":41:0))}, canonical_name_cache={'/tmp/ipykernel_1037/1751132419.py': '/tmp/ipykernel_1037/1751132419.py', '/tmp/ipykernel_1037/1393342955.py': '/tmp/ipykernel_1037/1393342955.py', '/tmp/ipykernel_1037/347789876.py': '/tmp/ipykernel_1037/347789876.py', '/tmp/ipykernel_1037/2145028508.py': '/tmp/ipykernel_1037/2145028508.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_1037/1751132419.py': True, '/tmp/ipykernel_1037/1393342955.py': True, '/tmp/ipykernel_1037/347789876.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_1037/2145028508.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 0x7f34947dde70>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None, threefry_partitionable=False, 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 0x7f3494627c70>]
call multiply_add_lowering(LoweringRuleContext(module_context=ModuleContext(context=<jax._src.interpreters.mlir.JaxIrContext object at 0x7f3494617f10>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7f3498127b70>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7f3494625ff0>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7f3494625fb0>, backend_or_name=<jaxlib.xla_extension.Client object at 0x7f34982137c0>, platforms=('cpu',), 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 0x7f34947ded10>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x55d07a242b30>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_1037/347789876.py":27:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1037/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_1037/2145028508.py":2:0) at "<module>"("/tmp/ipykernel_1037/2145028508.py":1:0))))))))))), <jaxlib.xla_extension.Traceback object at 0x55d07ba9fb40>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_1037/347789876.py":41:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1037/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_1037/2145028508.py":2:0) at "<module>"("/tmp/ipykernel_1037/2145028508.py":1:0))))))))))), <jaxlib.xla_extension.Traceback object at 0x55d07bbe2830>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_1037/347789876.py":41:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1037/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_1037/2145028508.py":2:0) at "<module>"("/tmp/ipykernel_1037/2145028508.py":1:0)))))))))))}, location_cache={(<code object multiply_add_prim at 0x7f34981ef7e0, file "/tmp/ipykernel_1037/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0)), (<code object func_wrapper at 0x7f34981ee340, file "/tmp/ipykernel_1037/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0)), (<code object multiply_add_value_and_jvp at 0x7f34d01664a0, file "/tmp/ipykernel_1037/347789876.py", line 3>, 36): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_1037/347789876.py":27:0)), (<code object square_add_prim at 0x7f34981ed790, file "/tmp/ipykernel_1037/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_1037/1751132419.py":17:0)), (<code object <lambda> at 0x7f3494821630, file "/tmp/ipykernel_1037/2145028508.py", line 1>, 10): loc("<lambda>"("/tmp/ipykernel_1037/2145028508.py":2:0)), (<code object <module> at 0x7f34948216e0, file "/tmp/ipykernel_1037/2145028508.py", line 1>, 16): loc("<module>"("/tmp/ipykernel_1037/2145028508.py":1:0)), (<code object multiply_add_value_and_jvp at 0x7f34d01664a0, file "/tmp/ipykernel_1037/347789876.py", line 3>, 86): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_1037/347789876.py":41:0)), (<code object multiply_add_value_and_jvp at 0x7f34d01664a0, file "/tmp/ipykernel_1037/347789876.py", line 3>, 88): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_1037/347789876.py":41:0))}, canonical_name_cache={'/tmp/ipykernel_1037/1751132419.py': '/tmp/ipykernel_1037/1751132419.py', '/tmp/ipykernel_1037/1393342955.py': '/tmp/ipykernel_1037/1393342955.py', '/tmp/ipykernel_1037/347789876.py': '/tmp/ipykernel_1037/347789876.py', '/tmp/ipykernel_1037/2145028508.py': '/tmp/ipykernel_1037/2145028508.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_1037/1751132419.py': True, '/tmp/ipykernel_1037/1393342955.py': True, '/tmp/ipykernel_1037/347789876.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_1037/2145028508.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 0x7f34947dded0>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None, threefry_partitionable=False, 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 0x7f3494814870>]

请注意，首先，您抽象地评估 multiply_add_value_and_jvp，这反过来抽象地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 个实例。

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 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，以获得计算输出切线的原语轨迹。

• 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，JAX 将特殊的抽象切线值 Zero 用于与 ma 的第三个参数对应的切线。这反映了您不对 square_add_prim 的第二个参数求微分，该参数流向 multiply_add_prim 的第三个参数。

• 还请注意，在切线的抽象求值期间，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 366, 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_1037/2155094905.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 393, 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 缺少一个部分，使其无法使用前向微分代码来计算反向微分。

转置

如前所述，在计算反向微分时，JAX 会获得一个使用前向微分计算切线的原语轨迹。然后，JAX 会抽象地反向解释此轨迹，并对每个原语应用一个转置规则。

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。假设您需要在点 (2., 4.) 处求微分。JAX 将根据输入 xt 和 yt 的切线产生以下 ft 的 JVP 切线计算

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

通过构造，切线计算始终在输入切线中是线性的。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来累积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。

JAX 知道对于 JVP 计算中可能出现的每个原语如何对其进行转置。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，则原语的转置是

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对应于原语每个参数的值。对于线性参数，转置会得到一个未定义的 _ 值，而对于其他参数，它会得到实际的常量。转置为原语的每个参数返回一个余切值，其中为常量参数返回 None 值。

特别是

 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。对转置的第一次调用对应于对 multiply_add_prim 的最后一次使用：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 的抽象求值仅使用抽象值。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 0x7f349465c680>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7f34946672b0>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7f3494666770>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7f34947b0930>, backend_or_name=<jaxlib.xla_extension.Client object at 0x7f34982137c0>, platforms=('cpu',), 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 0x7f34947df340>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x55d07bcf0330>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_1037/347789876.py":41:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1037/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("<module>"("/tmp/ipykernel_1037/3085343041.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 0x7f34981ef7e0, file "/tmp/ipykernel_1037/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0)), (<code object func_wrapper at 0x7f34981ee340, file "/tmp/ipykernel_1037/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0)), (<code object multiply_add_value_and_jvp at 0x7f34d01664a0, file "/tmp/ipykernel_1037/347789876.py", line 3>, 88): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_1037/347789876.py":41:0)), (<code object square_add_prim at 0x7f34981ed790, file "/tmp/ipykernel_1037/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_1037/1751132419.py":17:0)), (<code object <module> at 0x7f3494821630, file "/tmp/ipykernel_1037/3085343041.py", line 1>, 18): loc("<module>"("/tmp/ipykernel_1037/3085343041.py":1:0)), (<code object run_code at 0x7f34d5376fa0, 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_1037/1751132419.py': '/tmp/ipykernel_1037/1751132419.py', '/tmp/ipykernel_1037/1393342955.py': '/tmp/ipykernel_1037/1393342955.py', '/tmp/ipykernel_1037/347789876.py': '/tmp/ipykernel_1037/347789876.py', '/tmp/ipykernel_1037/3085343041.py': '/tmp/ipykernel_1037/3085343041.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_1037/1751132419.py': True, '/tmp/ipykernel_1037/1393342955.py': True, '/tmp/ipykernel_1037/347789876.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_1037/3085343041.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 0x7f3494660e20>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None, threefry_partitionable=False, 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 0x7f3494625ab0>]
call multiply_add_lowering(LoweringRuleContext(module_context=ModuleContext(context=<jax._src.interpreters.mlir.JaxIrContext object at 0x7f349465c680>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7f34946672b0>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7f3494666770>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7f34947b0930>, backend_or_name=<jaxlib.xla_extension.Client object at 0x7f34982137c0>, platforms=('cpu',), 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 0x7f34947df340>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x55d07bcf0330>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_1037/347789876.py":41:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1037/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("<module>"("/tmp/ipykernel_1037/3085343041.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 0x55d07bc4a580>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_1037/347789876.py":41:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1037/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("<module>"("/tmp/ipykernel_1037/3085343041.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 0x7f34981ef7e0, file "/tmp/ipykernel_1037/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0)), (<code object func_wrapper at 0x7f34981ee340, file "/tmp/ipykernel_1037/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0)), (<code object multiply_add_value_and_jvp at 0x7f34d01664a0, file "/tmp/ipykernel_1037/347789876.py", line 3>, 88): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_1037/347789876.py":41:0)), (<code object square_add_prim at 0x7f34981ed790, file "/tmp/ipykernel_1037/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_1037/1751132419.py":17:0)), (<code object <module> at 0x7f3494821630, file "/tmp/ipykernel_1037/3085343041.py", line 1>, 18): loc("<module>"("/tmp/ipykernel_1037/3085343041.py":1:0)), (<code object run_code at 0x7f34d5376fa0, 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 0x7f34d01664a0, file "/tmp/ipykernel_1037/347789876.py", line 3>, 86): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_1037/347789876.py":41:0))}, canonical_name_cache={'/tmp/ipykernel_1037/1751132419.py': '/tmp/ipykernel_1037/1751132419.py', '/tmp/ipykernel_1037/1393342955.py': '/tmp/ipykernel_1037/1393342955.py', '/tmp/ipykernel_1037/347789876.py': '/tmp/ipykernel_1037/347789876.py', '/tmp/ipykernel_1037/3085343041.py': '/tmp/ipykernel_1037/3085343041.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_1037/1751132419.py': True, '/tmp/ipykernel_1037/1393342955.py': True, '/tmp/ipykernel_1037/347789876.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_1037/3085343041.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 0x7f3494661030>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None, threefry_partitionable=False, 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 0x7f3498a09bf0>]

批处理

批处理转换采用逐点计算并将其转换为向量上的计算。如果您现在尝试，您将得到一个 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_1037/1080163607.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 994, 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 已经针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

以下是将 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 0x7f349465dfd0>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7f3494666af0>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7f3494667ef0>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7f34946644f0>, backend_or_name=<jaxlib.xla_extension.Client object at 0x7f34982137c0>, platforms=('cpu',), 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 0x7f34946615a0>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x55d07bc48f50>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("multiply_add_batch"("/tmp/ipykernel_1037/1827752256.py":25:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_1037/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0) at callsite("<module>"("/tmp/ipykernel_1037/1392464762.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 0x7f34981ef7e0, file "/tmp/ipykernel_1037/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_1037/1751132419.py":12:0)), (<code object func_wrapper at 0x7f34981ee340, file "/tmp/ipykernel_1037/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_1037/1393342955.py":48:0)), (<code object multiply_add_batch at 0x7f3494823520, file "/tmp/ipykernel_1037/1827752256.py", line 3>, 52): loc("multiply_add_batch"("/tmp/ipykernel_1037/1827752256.py":25:0)), (<code object square_add_prim at 0x7f34981ed790, file "/tmp/ipykernel_1037/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_1037/1751132419.py":17:0)), (<code object <module> at 0x7f34947b87c0, file "/tmp/ipykernel_1037/1392464762.py", line 1>, 48): loc("<module>"("/tmp/ipykernel_1037/1392464762.py":1:0)), (<code object run_code at 0x7f34d5376fa0, 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_1037/1751132419.py': '/tmp/ipykernel_1037/1751132419.py', '/tmp/ipykernel_1037/1393342955.py': '/tmp/ipykernel_1037/1393342955.py', '/tmp/ipykernel_1037/1827752256.py': '/tmp/ipykernel_1037/1827752256.py', '/tmp/ipykernel_1037/1392464762.py': '/tmp/ipykernel_1037/1392464762.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_1037/1751132419.py': True, '/tmp/ipykernel_1037/1393342955.py': True, '/tmp/ipykernel_1037/1827752256.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_1037/1392464762.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 0x7f3494660cd0>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None, threefry_partitionable=False, 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 0x7f34b837a170>]