JAX 内部：基本运算

JAX 基本运算简介

JAX 基本运算是一个 JAX 程序的基本计算单元。本文档解释了 JAX 基本运算必须支持的接口，以允许 JAX 执行其所有转换（这不是使用指南）。

例如，乘加运算可以用低级 jax.lax.* 基本运算（类似于 XLA 运算符包装器）或 jax.core.Primitive("multiply_add") 实现，如下所示。

JAX 能够获取此类基本运算的序列，并通过其对 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 可 JAX 追踪 的方式实现这些转换。这意味着当执行 Python 函数时，它对数据应用的唯一操作是：

• **数据属性检查：** 数据信息，如形状或类型；或者

• **JAX 基本运算：** 本教程中介绍的这些 JAX 特殊操作。

JAX 原语知道如何在具体数据值和抽象 JAX 值上进行运算。可 JAX 追踪的函数可以被 JAX 调用，并传入抽象参数。例如，一个 JAX 抽象值——ShapedArray(float32[2,2])——捕获了值的类型和形状，但没有捕获具体的数据值。

JAX 变换后的函数本身必须是可 JAX 追踪的函数，以确保这些变换是可组合的，例如像 jax.jit(jax.jacfwd(jax.grad(f)))。

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

此外，JAX 提供了基于 JAX 原语的 NumPy 函数实现。这意味着，使用 JAX 实现的 NumPy 的 Python 程序是可 JAX 追踪的，因此是可变换的。其他库可以通过基于 JAX 原语实现来使其可 JAX 追踪。

此外，JAX 原语集是可扩展的，因此，您可以定义一个封装函数行为的新原语，而不是根据预定义的 JAX 原语重新实现函数。

考虑以下示例：您希望为 JAX 添加对具有三个参数的乘加函数的支持，在数学上定义为 multiply_add(x, y, z) = x * y + z。此函数作用于 3 个形状相同的浮点值张量，并逐点执行运算。您可以通过以下方式实现：

• 使用现有的 JAX 原语；或

• 定义新的 JAX 基本运算

使用现有的 JAX 原语

定义新函数最简单的方法是根据 JAX 原语或根据其他本身使用 JAX 原语编写的函数（例如，在 jax.lax() 模块中定义的函数）来编写它们。

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，请添加一些用于追踪函数调用的辅助函数。

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

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

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

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

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

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

    return func_wrapper

  return trace_func

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

您可以使用其他已经根据这些原语编写的函数（例如 jax.numpy 中的函数），而不是直接使用 jax.lax() 原语。

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<ConcreteArray(2.0, dtype=float32, weak_type=True)>, 10.0)
  call multiply_add_numpy(Traced<ConcreteArray(2.0, dtype=float32, weak_type=True)>, Traced<ConcreteArray(2.0, dtype=float32, weak_type=True)>, 10.0)
  |<- multiply_add_numpy = Traced<ConcreteArray(14.0, dtype=float32, weak_type=True)>
|<- square_add_numpy = Traced<ConcreteArray(14.0, dtype=float32, weak_type=True)>
grad(square_add_numpy) =  4.0

请注意，在计算 jax.grad() 的过程中，JAX 会使用特殊参数 ConcreteArray(...) 调用 square_add_numpy 和 multiply_add_numpy（在下面的此 colab 中进一步描述）。务必记住，可 JAX 追踪的函数必须能够不仅在具体参数上运行，还能够在 JAX 可能用于抽象函数执行的特殊抽象参数上运行。

只要函数是用 JAX 原语编写的，就会满足 JAX 可追踪性属性。

定义新的 JAX 原语

添加对乘加的支持的正确方法是根据现有的 JAX 原语，如上所示。但是，为了演示 JAX 原语的工作原理，假设您希望为 JAX 添加一个新的原语来实现乘加功能。

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)

如果尝试调用新定义的函数，您将收到错误，因为您尚未告诉 JAX 关于新原语的语义。

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_838/2844449444.py", line 2, in <module>
    square_add_prim(2., 10.)
  File "/tmp/ipykernel_838/1393342955.py", line 48, in func_wrapper
    res = func(*args)
  File "/tmp/ipykernel_838/1751132419.py", line 17, in square_add_prim
    return multiply_add_prim(a, a, b)
NotImplementedError: Evaluation rule for 'multiply_add' not implemented

原始求值规则

@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(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>)
  call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>)

Found expected exception:

Traceback (most recent call last):
  File "/tmp/ipykernel_838/1813425700.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 331, in cache_miss
    outs, out_flat, out_tree, args_flat, jaxpr, attrs_tracked = _python_pjit_helper(
NotImplementedError: Abstract evaluation for 'multiply_add' not implemented

抽象求值规则

为了 JIT 化函数以及其他变换，JAX 首先仅使用参数的形状和类型对其进行抽象求值。这种抽象求值有多种用途

• 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运算的形状和类型。

例如，一个具有 3 个元素的向量的抽象可能是 ShapedArray(float32[3]) 或 ConcreteArray([1., 2., 3.])。在后一种情况下，JAX 使用作为抽象值包装的实际具体值。

from jax import core

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

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

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

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

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

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

如果您重新尝试应用 jit，您可以检查抽象求值如何进行，但您会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(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>)
  call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>)
    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(level=1/0)>
|<- square_add_prim = Traced<ShapedArray(float32[])>with<DynamicJaxprTrace(level=1/0)>

Found expected exception:

Traceback (most recent call last):
  File "/home/docs/.asdf/installs/python/3.10.14/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/3.10.14/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_838/1813425700.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 331, in cache_miss
    outs, out_flat, out_tree, args_flat, jaxpr, attrs_tracked = _python_pjit_helper(
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，可用于封装使用 C++ 定义的任意功能。

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

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

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

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

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

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

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

您现在将成功应用 jax.jit。请注意，JAX 首先对函数进行抽象求值，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(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>)
  call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>)
    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(level=1/0)>
|<- square_add_prim = Traced<ShapedArray(float32[])>with<DynamicJaxprTrace(level=1/0)>
call multiply_add_lowering(LoweringRuleContext(module_context=ModuleContext(context=<jaxlib.mlir._mlir_libs._site_initialize.<locals>.Context object at 0x7f16b84ef830>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7f16818199b0>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7f1681819070>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7f1681819170>, backend_or_name=<jaxlib.xla_extension.Client object at 0x7f1685486c80>, platforms=('cpu',), axis_context=ShardingContext(num_devices=1, device_assignment=None, mesh_shape=None), keepalives=[], channel_iterator=count(1), host_callbacks=[], shape_poly_state=<jax._src.interpreters.mlir.ShapePolyLoweringState object at 0x7f1682a491b0>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x55743e6a1560>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_838/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_838/1570919344.py":1:0) at callsite("<module>"("/tmp/ipykernel_838/1570919344.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 0x7f1682a33c00, file "/tmp/ipykernel_838/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0)), (<code object func_wrapper at 0x7f1685382810, file "/tmp/ipykernel_838/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0)), (<code object square_add_prim at 0x7f1682a33100, file "/tmp/ipykernel_838/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_838/1751132419.py":17:0)), (<code object <lambda> at 0x7f1682a6b3c0, file "/tmp/ipykernel_838/1570919344.py", line 1>, 6): loc("<lambda>"("/tmp/ipykernel_838/1570919344.py":1:0)), (<code object <module> at 0x7f1682a69580, file "/tmp/ipykernel_838/1570919344.py", line 1>, 16): loc("<module>"("/tmp/ipykernel_838/1570919344.py":1:0)), (<code object run_code at 0x7f16bdf5a080, 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 0x7f16bdf59f20, 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 0x7f16bdf59bb0, 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 0x7f16bde247c0, 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_838/1751132419.py': '/tmp/ipykernel_838/1751132419.py', '/tmp/ipykernel_838/1393342955.py': '/tmp/ipykernel_838/1393342955.py', '/tmp/ipykernel_838/1570919344.py': '/tmp/ipykernel_838/1570919344.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_838/1751132419.py': True, '/tmp/ipykernel_838/1393342955.py': True, '/tmp/ipykernel_838/1570919344.py': True, '/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 0x7f1682a99810>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None,threefry_partitionable=False),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 0x7f1682a53ef0>]

以下是 jit 的另一个用法，其中您仅相对于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(level=1/0)>, 10.0)
  call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, 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(level=1/0)>
|<- square_add_prim = Traced<ShapedArray(float32[])>with<DynamicJaxprTrace(level=1/0)>
call multiply_add_lowering(LoweringRuleContext(module_context=ModuleContext(context=<jaxlib.mlir._mlir_libs._site_initialize.<locals>.Context object at 0x7f16853ed030>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7f1681828470>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7f1681828430>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7f1681828070>, backend_or_name=<jaxlib.xla_extension.Client object at 0x7f1685486c80>, platforms=('cpu',), axis_context=ShardingContext(num_devices=1, device_assignment=None, mesh_shape=None), keepalives=[], channel_iterator=count(1), host_callbacks=[], shape_poly_state=<jax._src.interpreters.mlir.ShapePolyLoweringState object at 0x7f1682a9bd30>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x55743e6a1560>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_838/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_838/4165789807.py":1:0) at callsite("<module>"("/tmp/ipykernel_838/4165789807.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 0x7f1682a33c00, file "/tmp/ipykernel_838/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0)), (<code object func_wrapper at 0x7f1685382810, file "/tmp/ipykernel_838/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0)), (<code object square_add_prim at 0x7f1682a33100, file "/tmp/ipykernel_838/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_838/1751132419.py":17:0)), (<code object <lambda> at 0x7f16bddbf260, file "/tmp/ipykernel_838/4165789807.py", line 1>, 6): loc("<lambda>"("/tmp/ipykernel_838/4165789807.py":1:0)), (<code object <module> at 0x7f16bddbe600, file "/tmp/ipykernel_838/4165789807.py", line 1>, 20): loc("<module>"("/tmp/ipykernel_838/4165789807.py":1:0)), (<code object run_code at 0x7f16bdf5a080, 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 0x7f16bdf59f20, 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 0x7f16bdf59bb0, 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 0x7f16bde247c0, 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_838/1751132419.py': '/tmp/ipykernel_838/1751132419.py', '/tmp/ipykernel_838/1393342955.py': '/tmp/ipykernel_838/1393342955.py', '/tmp/ipykernel_838/4165789807.py': '/tmp/ipykernel_838/4165789807.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_838/1751132419.py': True, '/tmp/ipykernel_838/1393342955.py': True, '/tmp/ipykernel_838/4165789807.py': True, '/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 0x7f1681830220>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None,threefry_partitionable=False),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 0x7f1682a61c70>]

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，您将收到错误，因为您尚未告诉 JAX 如何微分 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<ConcreteArray(2.0, dtype=float32, weak_type=True)>, Traced<ConcreteArray(10.0, dtype=float32, weak_type=True)>)
  call multiply_add_prim(Traced<ConcreteArray(2.0, dtype=float32, weak_type=True)>, Traced<ConcreteArray(2.0, dtype=float32, weak_type=True)>, Traced<ConcreteArray(10.0, dtype=float32, weak_type=True)>)

Found expected exception:

Traceback (most recent call last):
  File "/tmp/ipykernel_838/459539105.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 1681, 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 1710, 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<ConcreteArray(2.0, dtype=float32, weak_type=True)>, Traced<ConcreteArray(10.0, dtype=float32, weak_type=True)>)
  call multiply_add_prim(Traced<ConcreteArray(2.0, dtype=float32, weak_type=True)>, Traced<ConcreteArray(2.0, dtype=float32, weak_type=True)>, Traced<ConcreteArray(10.0, dtype=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<ConcreteArray(14.0, dtype=float32)>
|<- square_add_prim = Traced<ConcreteArray(14.0, dtype=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(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>), (Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>))
      Primal evaluation:
      call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>)
        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(level=1/0)>
      Tangent evaluation:
      call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>)
        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(level=1/0)>
      call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[])>with<DynamicJaxprTrace(level=1/0)>)
        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(level=1/0)>
    |<- multiply_add_value_and_jvp = (Traced<ShapedArray(float32[])>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[])>with<DynamicJaxprTrace(level=1/0)>)
  |<- multiply_add_prim = Traced<ShapedArray(float32[])>
|<- square_add_prim = Traced<ShapedArray(float32[])>
call multiply_add_lowering(LoweringRuleContext(module_context=ModuleContext(context=<jaxlib.mlir._mlir_libs._site_initialize.<locals>.Context object at 0x7f168186e6b0>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7f16818739f0>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7f1681873970>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7f16853bf870>, backend_or_name=<jaxlib.xla_extension.Client object at 0x7f1685486c80>, platforms=('cpu',), axis_context=ShardingContext(num_devices=1, device_assignment=None, mesh_shape=None), keepalives=[], channel_iterator=count(1), host_callbacks=[], shape_poly_state=<jax._src.interpreters.mlir.ShapePolyLoweringState object at 0x7f1681830940>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x55743e7b6240>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_838/347789876.py":27:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_838/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_838/2145028508.py":2:0) at "<module>"("/tmp/ipykernel_838/2145028508.py":1:0)))))))))))}, location_cache={(<code object multiply_add_prim at 0x7f1682a33c00, file "/tmp/ipykernel_838/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0)), (<code object func_wrapper at 0x7f1685382810, file "/tmp/ipykernel_838/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0)), (<code object multiply_add_value_and_jvp at 0x7f1682a6a6b0, file "/tmp/ipykernel_838/347789876.py", line 3>, 36): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_838/347789876.py":27:0)), (<code object square_add_prim at 0x7f1682a33100, file "/tmp/ipykernel_838/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_838/1751132419.py":17:0)), (<code object <lambda> at 0x7f1682a6b310, file "/tmp/ipykernel_838/2145028508.py", line 1>, 10): loc("<lambda>"("/tmp/ipykernel_838/2145028508.py":2:0)), (<code object <module> at 0x7f1682a69210, file "/tmp/ipykernel_838/2145028508.py", line 1>, 16): loc("<module>"("/tmp/ipykernel_838/2145028508.py":1:0))}, canonical_name_cache={'/tmp/ipykernel_838/1751132419.py': '/tmp/ipykernel_838/1751132419.py', '/tmp/ipykernel_838/1393342955.py': '/tmp/ipykernel_838/1393342955.py', '/tmp/ipykernel_838/347789876.py': '/tmp/ipykernel_838/347789876.py', '/tmp/ipykernel_838/2145028508.py': '/tmp/ipykernel_838/2145028508.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_838/1751132419.py': True, '/tmp/ipykernel_838/1393342955.py': True, '/tmp/ipykernel_838/347789876.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/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_838/2145028508.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 0x7f16b850e3e0>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None,threefry_partitionable=False),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 0x7f1682a2e3f0>]
call multiply_add_lowering(LoweringRuleContext(module_context=ModuleContext(context=<jaxlib.mlir._mlir_libs._site_initialize.<locals>.Context object at 0x7f168186e6b0>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7f16818739f0>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7f1681873970>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7f16853bf870>, backend_or_name=<jaxlib.xla_extension.Client object at 0x7f1685486c80>, platforms=('cpu',), axis_context=ShardingContext(num_devices=1, device_assignment=None, mesh_shape=None), keepalives=[], channel_iterator=count(1), host_callbacks=[], shape_poly_state=<jax._src.interpreters.mlir.ShapePolyLoweringState object at 0x7f1681830940>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x55743e7b6240>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_838/347789876.py":27:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_838/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_838/2145028508.py":2:0) at "<module>"("/tmp/ipykernel_838/2145028508.py":1:0))))))))))), <jaxlib.xla_extension.Traceback object at 0x55743e731290>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_838/347789876.py":41:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_838/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_838/2145028508.py":2:0) at "<module>"("/tmp/ipykernel_838/2145028508.py":1:0)))))))))))}, location_cache={(<code object multiply_add_prim at 0x7f1682a33c00, file "/tmp/ipykernel_838/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0)), (<code object func_wrapper at 0x7f1685382810, file "/tmp/ipykernel_838/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0)), (<code object multiply_add_value_and_jvp at 0x7f1682a6a6b0, file "/tmp/ipykernel_838/347789876.py", line 3>, 36): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_838/347789876.py":27:0)), (<code object square_add_prim at 0x7f1682a33100, file "/tmp/ipykernel_838/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_838/1751132419.py":17:0)), (<code object <lambda> at 0x7f1682a6b310, file "/tmp/ipykernel_838/2145028508.py", line 1>, 10): loc("<lambda>"("/tmp/ipykernel_838/2145028508.py":2:0)), (<code object <module> at 0x7f1682a69210, file "/tmp/ipykernel_838/2145028508.py", line 1>, 16): loc("<module>"("/tmp/ipykernel_838/2145028508.py":1:0)), (<code object multiply_add_value_and_jvp at 0x7f1682a6a6b0, file "/tmp/ipykernel_838/347789876.py", line 3>, 86): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_838/347789876.py":41:0))}, canonical_name_cache={'/tmp/ipykernel_838/1751132419.py': '/tmp/ipykernel_838/1751132419.py', '/tmp/ipykernel_838/1393342955.py': '/tmp/ipykernel_838/1393342955.py', '/tmp/ipykernel_838/347789876.py': '/tmp/ipykernel_838/347789876.py', '/tmp/ipykernel_838/2145028508.py': '/tmp/ipykernel_838/2145028508.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_838/1751132419.py': True, '/tmp/ipykernel_838/1393342955.py': True, '/tmp/ipykernel_838/347789876.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/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_838/2145028508.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 0x7f1681830e20>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None,threefry_partitionable=False),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 0x7f16b8571030>]
call multiply_add_lowering(LoweringRuleContext(module_context=ModuleContext(context=<jaxlib.mlir._mlir_libs._site_initialize.<locals>.Context object at 0x7f168186e6b0>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7f16818739f0>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7f1681873970>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7f16853bf870>, backend_or_name=<jaxlib.xla_extension.Client object at 0x7f1685486c80>, platforms=('cpu',), axis_context=ShardingContext(num_devices=1, device_assignment=None, mesh_shape=None), keepalives=[], channel_iterator=count(1), host_callbacks=[], shape_poly_state=<jax._src.interpreters.mlir.ShapePolyLoweringState object at 0x7f1681830940>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x55743e7b6240>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_838/347789876.py":27:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_838/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_838/2145028508.py":2:0) at "<module>"("/tmp/ipykernel_838/2145028508.py":1:0))))))))))), <jaxlib.xla_extension.Traceback object at 0x55743e731290>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_838/347789876.py":41:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_838/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_838/2145028508.py":2:0) at "<module>"("/tmp/ipykernel_838/2145028508.py":1:0))))))))))), <jaxlib.xla_extension.Traceback object at 0x55743e97c660>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_838/347789876.py":41:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_838/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("<lambda>"("/tmp/ipykernel_838/2145028508.py":2:0) at "<module>"("/tmp/ipykernel_838/2145028508.py":1:0)))))))))))}, location_cache={(<code object multiply_add_prim at 0x7f1682a33c00, file "/tmp/ipykernel_838/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0)), (<code object func_wrapper at 0x7f1685382810, file "/tmp/ipykernel_838/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0)), (<code object multiply_add_value_and_jvp at 0x7f1682a6a6b0, file "/tmp/ipykernel_838/347789876.py", line 3>, 36): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_838/347789876.py":27:0)), (<code object square_add_prim at 0x7f1682a33100, file "/tmp/ipykernel_838/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_838/1751132419.py":17:0)), (<code object <lambda> at 0x7f1682a6b310, file "/tmp/ipykernel_838/2145028508.py", line 1>, 10): loc("<lambda>"("/tmp/ipykernel_838/2145028508.py":2:0)), (<code object <module> at 0x7f1682a69210, file "/tmp/ipykernel_838/2145028508.py", line 1>, 16): loc("<module>"("/tmp/ipykernel_838/2145028508.py":1:0)), (<code object multiply_add_value_and_jvp at 0x7f1682a6a6b0, file "/tmp/ipykernel_838/347789876.py", line 3>, 86): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_838/347789876.py":41:0)), (<code object multiply_add_value_and_jvp at 0x7f1682a6a6b0, file "/tmp/ipykernel_838/347789876.py", line 3>, 88): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_838/347789876.py":41:0))}, canonical_name_cache={'/tmp/ipykernel_838/1751132419.py': '/tmp/ipykernel_838/1751132419.py', '/tmp/ipykernel_838/1393342955.py': '/tmp/ipykernel_838/1393342955.py', '/tmp/ipykernel_838/347789876.py': '/tmp/ipykernel_838/347789876.py', '/tmp/ipykernel_838/2145028508.py': '/tmp/ipykernel_838/2145028508.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_838/1751132419.py': True, '/tmp/ipykernel_838/1393342955.py': True, '/tmp/ipykernel_838/347789876.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/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_838/2145028508.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 0x7f1682a9b0d0>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None,threefry_partitionable=False),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 0x7f16853b9d30>]

请注意，首先，您对 multiply_add_value_and_jvp 进行抽象求值，这反过来又会对原始值和切线求值进行抽象求值（ma 原语的总共 3 次调用）。然后，您编译原语的 3 次出现。

反向微分

如果您现在尝试使用反向微分，您会注意到 JAX 最初使用 multiply_add_value_and_jvp 计算抽象值的正向微分，但随后遇到 NotImplementedError。

在计算反向微分时，JAX 首先对前向微分代码 multiply_add_value_and_jvp 执行抽象求值，以获得计算输出切线的原语跟踪。

• 观察到 JAX 使用微分点的具体值和切线的抽象值执行此抽象求值。

• 请注意，JAX 使用特殊抽象切线值 Zero 作为 ma 的第三个参数对应的切线。这反映了您没有相对于 square_add_prim 的第二个参数进行微分，这会传递到 multiply_add_prim 的第三个参数。

• 还要注意，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<ConcreteArray(2.0, dtype=float32, weak_type=True)>, 10.0)
  call multiply_add_prim(Traced<ConcreteArray(2.0, dtype=float32, weak_type=True)>, Traced<ConcreteArray(2.0, dtype=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(ConcreteArray(2.0, dtype=float32, weak_type=True), ShapedArray(float32[], weak_type=True), ConcreteArray(0.0, dtype=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), ConcreteArray(2.0, dtype=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<ConcreteArray(14.0, dtype=float32)>
|<- square_add_prim = Traced<ConcreteArray(14.0, dtype=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 283, 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/3.10.14/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/3.10.14/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_838/2155094905.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 392, in grad_f
    _, g = value_and_grad_f(*args, **kwargs)
NotImplementedError: Transpose rule (for reverse-mode differentiation) for 'multiply_add' not implemented

上述错误是由于缺少 JAX 能够使用正向微分代码计算反向微分的部分。

转置

如前所述，在计算反向微分时，JAX 获取使用正向微分计算切线的原语跟踪。然后，**JAX 反向解释此跟踪**，并对每个原语应用**转置规则**。

要了解正在发生的事情，请考虑函数 f(x, y) = x * y + y 的一个更简单的示例。假设您需要在点 (2., 4.) 处进行微分。JAX 将从输入 xt 和 yt 的切线生成以下 ft 的 JVP 切线计算。

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

根据构造，切线计算在输入切线中始终是线性的。在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数。

JAX 将通过向后处理 JVP 计算来生成反向微分计算。对于切线计算中的每个操作，它使用操作结果的余切线累积操作使用的变量的余切线。

  # 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.

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

JAX 知道在 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) 在 x 的常数值下对于参数 y 和 z 是线性的，例如 p(x, y, z) = y*cy + z*cz，则原语的转置为

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

请注意，p_transpose 获取原始输出的余切值以及原始函数每个参数对应的值。对于线性参数，转置得到一个未定义的 _ 值，而对于其他参数，它获取实际的常量值。转置为原始函数的每个参数返回一个余切值，对于常量参数则返回 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<ConcreteArray(2.0, dtype=float32, weak_type=True)>, 10.0)
  call multiply_add_prim(Traced<ConcreteArray(2.0, dtype=float32, weak_type=True)>, Traced<ConcreteArray(2.0, dtype=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(ConcreteArray(2.0, dtype=float32, weak_type=True), ShapedArray(float32[], weak_type=True), ConcreteArray(0.0, dtype=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), ConcreteArray(2.0, dtype=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<ConcreteArray(14.0, dtype=float32)>
|<- square_add_prim = Traced<ConcreteArray(14.0, dtype=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 multiply_add_prim 的两次使用。第一次转置调用对应于 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 的抽象评估仅使用抽象值。同时，在没有 JIT 的情况下，您使用了 ConcreteArray。

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

call square_add_prim(Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>)
  call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>)
    call multiply_add_value_and_jvp((Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>), (Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>, Zero(ShapedArray(float32[], weak_type=True))))
      Primal evaluation:
      call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>)
        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(level=1/0)>
      Tangent evaluation:
      call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>)
        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)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, 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 = (Traced<ShapedArray(float32[])>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[])>)
  |<- multiply_add_prim = Traced<ShapedArray(float32[])>
|<- square_add_prim = Traced<ShapedArray(float32[])>
call multiply_add_transpose(Traced<ShapedArray(float32[])>with<DynamicJaxprTrace(level=1/0)>, UndefinedPrimal(ShapedArray(float32[], weak_type=True)), Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, UndefinedPrimal(ShapedArray(float32[])))
  call multiply_add_prim(Traced<ShapedArray(float32[])>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>)
    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(level=1/0)>
|<- multiply_add_transpose = (Traced<ShapedArray(float32[])>with<DynamicJaxprTrace(level=1/0)>, None, Traced<ShapedArray(float32[])>with<DynamicJaxprTrace(level=1/0)>)
call multiply_add_transpose(Traced<ShapedArray(float32[])>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, UndefinedPrimal(ShapedArray(float32[], weak_type=True)), Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>)
  call multiply_add_prim(Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[])>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[], weak_type=True)>with<DynamicJaxprTrace(level=1/0)>)
    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(level=1/0)>
|<- multiply_add_transpose = (None, Traced<ShapedArray(float32[])>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[])>with<DynamicJaxprTrace(level=1/0)>)
call multiply_add_lowering(LoweringRuleContext(module_context=ModuleContext(context=<jaxlib.mlir._mlir_libs._site_initialize.<locals>.Context object at 0x7f168188cdb0>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7f1681886c30>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7f1681886670>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7f1681886870>, backend_or_name=<jaxlib.xla_extension.Client object at 0x7f1685486c80>, platforms=('cpu',), axis_context=ShardingContext(num_devices=1, device_assignment=None, mesh_shape=None), keepalives=[], channel_iterator=count(1), host_callbacks=[], shape_poly_state=<jax._src.interpreters.mlir.ShapePolyLoweringState object at 0x7f16818328f0>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x55743e964900>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_838/347789876.py":41:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_838/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("<module>"("/tmp/ipykernel_838/3085343041.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 0x7f1682a33c00, file "/tmp/ipykernel_838/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0)), (<code object func_wrapper at 0x7f1685382810, file "/tmp/ipykernel_838/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0)), (<code object multiply_add_value_and_jvp at 0x7f1682a6a6b0, file "/tmp/ipykernel_838/347789876.py", line 3>, 88): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_838/347789876.py":41:0)), (<code object square_add_prim at 0x7f1682a33100, file "/tmp/ipykernel_838/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_838/1751132419.py":17:0)), (<code object <module> at 0x7f1682a68be0, file "/tmp/ipykernel_838/3085343041.py", line 1>, 18): loc("<module>"("/tmp/ipykernel_838/3085343041.py":1:0)), (<code object run_code at 0x7f16bdf5a080, 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_838/1751132419.py': '/tmp/ipykernel_838/1751132419.py', '/tmp/ipykernel_838/1393342955.py': '/tmp/ipykernel_838/1393342955.py', '/tmp/ipykernel_838/347789876.py': '/tmp/ipykernel_838/347789876.py', '/tmp/ipykernel_838/3085343041.py': '/tmp/ipykernel_838/3085343041.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_838/1751132419.py': True, '/tmp/ipykernel_838/1393342955.py': True, '/tmp/ipykernel_838/347789876.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/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_838/3085343041.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 0x7f1681831570>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None,threefry_partitionable=False),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 0x7f1681881bb0>]
call multiply_add_lowering(LoweringRuleContext(module_context=ModuleContext(context=<jaxlib.mlir._mlir_libs._site_initialize.<locals>.Context object at 0x7f168188cdb0>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7f1681886c30>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7f1681886670>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7f1681886870>, backend_or_name=<jaxlib.xla_extension.Client object at 0x7f1685486c80>, platforms=('cpu',), axis_context=ShardingContext(num_devices=1, device_assignment=None, mesh_shape=None), keepalives=[], channel_iterator=count(1), host_callbacks=[], shape_poly_state=<jax._src.interpreters.mlir.ShapePolyLoweringState object at 0x7f16818328f0>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x55743e964900>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_838/347789876.py":41:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_838/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("<module>"("/tmp/ipykernel_838/3085343041.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 0x55743eb38650>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("multiply_add_value_and_jvp"("/tmp/ipykernel_838/347789876.py":41:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_838/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("<module>"("/tmp/ipykernel_838/3085343041.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 0x7f1682a33c00, file "/tmp/ipykernel_838/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0)), (<code object func_wrapper at 0x7f1685382810, file "/tmp/ipykernel_838/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0)), (<code object multiply_add_value_and_jvp at 0x7f1682a6a6b0, file "/tmp/ipykernel_838/347789876.py", line 3>, 88): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_838/347789876.py":41:0)), (<code object square_add_prim at 0x7f1682a33100, file "/tmp/ipykernel_838/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_838/1751132419.py":17:0)), (<code object <module> at 0x7f1682a68be0, file "/tmp/ipykernel_838/3085343041.py", line 1>, 18): loc("<module>"("/tmp/ipykernel_838/3085343041.py":1:0)), (<code object run_code at 0x7f16bdf5a080, 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 0x7f1682a6a6b0, file "/tmp/ipykernel_838/347789876.py", line 3>, 86): loc("multiply_add_value_and_jvp"("/tmp/ipykernel_838/347789876.py":41:0))}, canonical_name_cache={'/tmp/ipykernel_838/1751132419.py': '/tmp/ipykernel_838/1751132419.py', '/tmp/ipykernel_838/1393342955.py': '/tmp/ipykernel_838/1393342955.py', '/tmp/ipykernel_838/347789876.py': '/tmp/ipykernel_838/347789876.py', '/tmp/ipykernel_838/3085343041.py': '/tmp/ipykernel_838/3085343041.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_838/1751132419.py': True, '/tmp/ipykernel_838/1393342955.py': True, '/tmp/ipykernel_838/347789876.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/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_838/3085343041.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 0x7f1681833910>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None,threefry_partitionable=False),xla_metadata={}, platforms=None), Value(<block argument> of type 'tensor<f32>' at index: 0), Value(%4 = "stablehlo.constant"() <{value = dense<1.000000e+00> : tensor<f32>}> : () -> tensor<f32>), Value(%5 = "stablehlo.constant"() <{value = dense<0.000000e+00> : tensor<f32>}> : () -> tensor<f32>))
|<- multiply_add_lowering = [<jaxlib.mlir._mlir_libs._mlir.ir.OpResult object at 0x7f16818a9cf0>]

批处理

批处理转换将逐点计算转换为向量上的计算。如果您现在尝试，将会得到一个 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_838/1080163607.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 986, in vmap_f
    out_flat = batching.batch(
NotImplementedError: Batching rule for 'multiply_add' not implemented

您需要指示 JAX 如何评估原始函数的批处理版本。在本例中，multiply_add_prim 已经对任何维度输入向量进行逐点运算，因此批处理版本可以使用相同的 multiply_add_prim 实现。

from jax.interpreters import batching

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

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

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


batching.primitive_batchers[multiply_add_p] = multiply_add_batch

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

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

批处理的 JIT

下面是将 JIT 应用于批处理的示例。

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

call square_add_prim(Traced<ShapedArray(float32[])>, Traced<ShapedArray(float32[])>)
  call multiply_add_prim(Traced<ShapedArray(float32[])>, Traced<ShapedArray(float32[])>, Traced<ShapedArray(float32[])>)
    call multiply_add_batch((Traced<ShapedArray(float32[2])>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[2])>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[2])>with<DynamicJaxprTrace(level=1/0)>), (0, 0, 0))
      Using multiply_add to compute the batch:
      call multiply_add_prim(Traced<ShapedArray(float32[2])>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[2])>with<DynamicJaxprTrace(level=1/0)>, Traced<ShapedArray(float32[2])>with<DynamicJaxprTrace(level=1/0)>)
        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(level=1/0)>
    |<- multiply_add_batch = (Traced<ShapedArray(float32[2])>with<DynamicJaxprTrace(level=1/0)>, 0)
  |<- multiply_add_prim = Traced<ShapedArray(float32[])>
|<- square_add_prim = Traced<ShapedArray(float32[])>
call multiply_add_lowering(LoweringRuleContext(module_context=ModuleContext(context=<jaxlib.mlir._mlir_libs._site_initialize.<locals>.Context object at 0x7f168188e430>, module=<jaxlib.mlir._mlir_libs._mlir.ir.Module object at 0x7f1681884170>, ip=<jaxlib.mlir._mlir_libs._mlir.ir.InsertionPoint object at 0x7f1681828d70>, symbol_table=<jaxlib.mlir._mlir_libs._mlir.ir.SymbolTable object at 0x7f16818283f0>, backend_or_name=<jaxlib.xla_extension.Client object at 0x7f1685486c80>, platforms=('cpu',), axis_context=ShardingContext(num_devices=1, device_assignment=None, mesh_shape=None), keepalives=[], channel_iterator=count(1), host_callbacks=[], shape_poly_state=<jax._src.interpreters.mlir.ShapePolyLoweringState object at 0x7f16b850ee00>, all_default_mem_kind=True, cached_primitive_lowerings={}, traceback_caches=TracebackCaches(traceback_cache={<jaxlib.xla_extension.Traceback object at 0x55743ec58910>: loc(callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("multiply_add_batch"("/tmp/ipykernel_838/1827752256.py":25:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("square_add_prim"("/tmp/ipykernel_838/1751132419.py":17:0) at callsite("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0) at callsite("<module>"("/tmp/ipykernel_838/1392464762.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 0x7f1682a33c00, file "/tmp/ipykernel_838/1751132419.py", line 5>, 10): loc("multiply_add_prim"("/tmp/ipykernel_838/1751132419.py":12:0)), (<code object func_wrapper at 0x7f1685382810, file "/tmp/ipykernel_838/1393342955.py", line 45>, 24): loc("func_wrapper"("/tmp/ipykernel_838/1393342955.py":48:0)), (<code object multiply_add_batch at 0x7f16818364a0, file "/tmp/ipykernel_838/1827752256.py", line 3>, 52): loc("multiply_add_batch"("/tmp/ipykernel_838/1827752256.py":25:0)), (<code object square_add_prim at 0x7f1682a33100, file "/tmp/ipykernel_838/1751132419.py", line 14>, 8): loc("square_add_prim"("/tmp/ipykernel_838/1751132419.py":17:0)), (<code object <module> at 0x7f1682a6a970, file "/tmp/ipykernel_838/1392464762.py", line 1>, 48): loc("<module>"("/tmp/ipykernel_838/1392464762.py":1:0)), (<code object run_code at 0x7f16bdf5a080, 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_838/1751132419.py': '/tmp/ipykernel_838/1751132419.py', '/tmp/ipykernel_838/1393342955.py': '/tmp/ipykernel_838/1393342955.py', '/tmp/ipykernel_838/1827752256.py': '/tmp/ipykernel_838/1827752256.py', '/tmp/ipykernel_838/1392464762.py': '/tmp/ipykernel_838/1392464762.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_838/1751132419.py': True, '/tmp/ipykernel_838/1393342955.py': True, '/tmp/ipykernel_838/1827752256.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/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_838/1392464762.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 0x7f16818bc310>, tokens_out=None, axis_size_env=None, dim_var_values=[], jaxpr_eqn_ctx=JaxprEqnContext(compute_type=None,threefry_partitionable=False),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 0x7f1681881470>]