Autodidax: 从零开始构建 JAX 核心

是否曾经想了解 JAX 的工作原理，但发现其实现看起来难以理解？那么，您很幸运！通过阅读本教程，您将学习 JAX 核心系统中的每一个重要思想。您甚至会了解我们的奇怪术语！

这是一个正在进行中的草稿。 仍然缺少一些重要的组成部分，将在第 5 部分和第 6 部分（以及更多？）中介绍。这里还有一些简化，我们尚未应用于主系统，但我们将这样做。

第 1 部分：将转换作为解释器：标准求值、jvp 和 vmap

我们想要转换看起来像这样的函数

def f(x):
  y = sin(x) * 2.
  z = - y + x
  return z

将诸如 sin 之类的函数以及作为中缀运算符（mul、add 和 neg）基础的算术运算视为原语操作，意味着处理的原子单元而不是组合。

“转换”意味着“不同的解释”。代替标准解释，即我们将原语操作应用于数值输入以产生数值输出，我们想要重写原语应用，并让不同的值流经我们的程序。例如，我们可能想要将每个原语的应用替换为 其 JVP 规则的应用，并让原始-切线对流经我们的程序。此外，我们希望能够组合多个转换，从而形成解释器堆栈。

JAX 核心机制

我们可以实现解释器堆栈，甚至可以在执行要转换的 Python 函数时动态释放它们。首先，让我们定义这些原语，以便我们可以拦截它们的应用程序

from typing import NamedTuple

class Primitive(NamedTuple):
  name: str

add_p = Primitive('add')
mul_p = Primitive('mul')
neg_p = Primitive("neg")
sin_p = Primitive("sin")
cos_p = Primitive("cos")
reduce_sum_p = Primitive("reduce_sum")
greater_p = Primitive("greater")
less_p = Primitive("less")
transpose_p = Primitive("transpose")
broadcast_p = Primitive("broadcast")

def add(x, y): return bind1(add_p, x, y)
def mul(x, y): return bind1(mul_p, x, y)
def neg(x): return bind1(neg_p, x)
def sin(x): return bind1(sin_p, x)
def cos(x): return bind1(cos_p, x)
def greater(x, y): return bind1(greater_p, x, y)
def less(x, y): return bind1(less_p, x, y)
def transpose(x, perm): return bind1(transpose_p, x, perm=perm)
def broadcast(x, shape, axes): return bind1(broadcast_p, x, shape=shape, axes=axes)
def reduce_sum(x, axis=None):
  if axis is None:
    axis = tuple(range(np.ndim(x)))
  if type(axis) is int:
    axis = (axis,)
  return bind1(reduce_sum_p, x, axis=axis)

def bind1(prim, *args, **params):
  out, = bind(prim, *args, **params)
  return out

我们稍后将设置数组数据类型和中缀运算符方法。

Primitive 只是一个具有名称的对象，我们为其附加了解释规则（每个转换一个）。bind 函数是我们的拦截点：它将根据参数如何在 tracer 中封装以及哪些解释器处于活动状态，来确定要应用哪个转换规则。

用户代码调用的函数（如 add 和 sin）只是对 bind 调用的封装器。这些封装器使我们能够控制参数如何传递给 bind，特别是我们遵循一个方便的内部约定：当我们调用 bind 时，我们将表示数组数据的值作为位置参数传递，并通过关键字传递元数据，例如 reduce_sum_p 的 axis 参数。这种调用约定简化了一些核心逻辑（因为例如，下面要定义的 Tracer 类的实例只能在 bind 的位置参数中出现）。封装器还可以提供文档字符串！

我们将活动解释器表示为堆栈。堆栈只是一个简单的 list，每个元素都是一个容器，其中包含整数级别（对应于元素在堆栈中的高度）、解释器类型（我们将其称为 trace_type）以及解释器需要的任何全局数据的可选字段。我们将每个元素称为 MainTrace，尽管“Interpreter”可能更具描述性。

from collections.abc import Sequence
from contextlib import contextmanager
from typing import Any

class MainTrace(NamedTuple):
  level: int
  trace_type: type['Trace']
  global_data: Any | None

trace_stack: list[MainTrace] = []
dynamic_trace: MainTrace | None = None  # to be employed in Part 3

@contextmanager
def new_main(trace_type: type['Trace'], global_data=None):
  level = len(trace_stack)
  main = MainTrace(level, trace_type, global_data)
  trace_stack.append(main)

  try:
    yield main
  finally:
    trace_stack.pop()

当我们即将应用转换时，我们将使用 new_main 将另一个解释器压入堆栈。然后，当我们在函数中应用原语时，我们可以认为 bind 首先由堆栈顶部的 trace（即级别最高的 trace）解释。如果第一个解释器本身在其原语的解释规则中绑定了其他原语，例如 sin_p 的 JVP 规则可能绑定 cos_p 和 mul_p，那么这些 bind 调用将由下一级解释器处理。

解释器堆栈的底部是什么？在底部，我们知道所有转换解释器都已完成，我们只想进行标准求值。因此，在底部，我们将放置一个求值解释器。

让我们勾勒出解释器的接口，该接口基于 Trace 和 Tracer 基类。Tracer 表示一个封装的值，可能携带一些解释器使用的额外上下文数据。Trace 处理将值封装到 Tracer 中，并处理原语应用。

class Trace:
  main: MainTrace

  def __init__(self, main: MainTrace) -> None:
    self.main = main

  def pure(self, val): assert False  # must override
  def lift(self, val): assert False  # must override

  def process_primitive(self, primitive, tracers, params):
    assert False  # must override

前两个方法是关于将值封装在 Tracer 中，Tracer 是在我们要转换的 Python 程序中流动的对象。最后一个方法是我们将用于解释原语应用程序的回调。

Trace 本身不包含任何数据，除了对其对应的 MainTrace 实例的引用。实际上，在应用转换期间，可能会创建和丢弃 Trace 的多个实例，而每次应用转换仅创建一个 MainTrace 实例。

至于 Tracer 本身，每个都携带一个抽象值（并将中缀运算符转发给它），其余的取决于转换。（Tracer 和 AbstractValue 之间的关系是，每个转换都有一个 Tracer，并且每个基本类型至少有一个 AbstractValue，例如数组。）

import numpy as np

class Tracer:
  _trace: Trace

  __array_priority__ = 1000

  @property
  def aval(self):
    assert False  # must override

  def full_lower(self):
    return self  # default implementation

  def __neg__(self): return self.aval._neg(self)
  def __add__(self, other): return self.aval._add(self, other)
  def __radd__(self, other): return self.aval._radd(self, other)
  def __mul__(self, other): return self.aval._mul(self, other)
  def __rmul__(self, other): return self.aval._rmul(self, other)
  def __gt__(self, other): return self.aval._gt(self, other)
  def __lt__(self, other): return self.aval._lt(self, other)
  def __bool__(self): return self.aval._bool(self)
  def __nonzero__(self): return self.aval._nonzero(self)

  def __getattr__(self, name):
    try:
      return getattr(self.aval, name)
    except AttributeError:
      raise AttributeError(f"{self.__class__.__name__} has no attribute {name}")

def swap(f): return lambda x, y: f(y, x)

class ShapedArray:
  array_abstraction_level = 1
  shape: tuple[int, ...]
  dtype: np.dtype

  def __init__(self, shape, dtype):
    self.shape = shape
    self.dtype = dtype

  @property
  def ndim(self):
    return len(self.shape)

  _neg = staticmethod(neg)
  _add = staticmethod(add)
  _radd = staticmethod(swap(add))
  _mul = staticmethod(mul)
  _rmul = staticmethod(swap(mul))
  _gt = staticmethod(greater)
  _lt = staticmethod(less)

  @staticmethod
  def _bool(tracer):
    raise Exception("ShapedArray can't be unambiguously converted to bool")

  @staticmethod
  def _nonzero(tracer):
    raise Exception("ShapedArray can't be unambiguously converted to bool")

  def str_short(self):
    return f'{self.dtype.name}[{",".join(str(d) for d in self.shape)}]'

  def __hash__(self):
    return hash((self.shape, self.dtype))

  def __eq__(self, other):
    return (type(self) is type(other) and
            self.shape == other.shape and self.dtype == other.dtype)

  def __repr__(self):
    return f"ShapedArray(shape={self.shape}, dtype={self.dtype})"

class ConcreteArray(ShapedArray):
  array_abstraction_level = 2
  val: np.ndarray

  def __init__(self, val):
    self.val = val
    self.shape = val.shape
    self.dtype = val.dtype

  @staticmethod
  def _bool(tracer):
    return bool(tracer.aval.val)

  @staticmethod
  def _nonzero(tracer):
    return bool(tracer.aval.val)

def get_aval(x):
  if isinstance(x, Tracer):
    return x.aval
  elif type(x) in jax_types:
    return ConcreteArray(np.asarray(x))
  else:
    raise TypeError(x)

jax_types = {bool, int, float,
             np.bool_, np.int32, np.int64, np.float32, np.float64, np.ndarray}

请注意，我们实际上有两个用于数组的 AbstractValue，表示不同的抽象级别。ShapedArray 表示具有给定形状和 dtype 的所有可能数组的集合。ConcreteArray 表示由单个数组值组成的单例集合。

现在我们已经设置了解释器堆栈、解释器的 Trace/Tracer API 和抽象值，我们可以回头实现 bind

def bind(prim, *args, **params):
  top_trace = find_top_trace(args)
  tracers = [full_raise(top_trace, arg) for arg in args]
  outs = top_trace.process_primitive(prim, tracers, params)
  return [full_lower(out) for out in outs]

主要操作是我们调用 find_top_trace 以确定哪个解释器应处理此原语应用。然后我们调用该顶层 trace 的 process_primitive，以便 trace 可以应用其解释规则。对 full_raise 的调用仅确保输入在顶层 trace 的 Tracer 实例中封装，而对 full_lower 的调用是一个可选的优化，以便我们尽可能多地从 Tracer 中解封装值。

import operator as op

def find_top_trace(xs) -> Trace:
  top_main = max((x._trace.main for x in xs if isinstance(x, Tracer)),
                 default=trace_stack[0], key=op.attrgetter('level'))
  if dynamic_trace and dynamic_trace.level > top_main.level:
    top_main = dynamic_trace
  return top_main.trace_type(top_main)

用文字来说，忽略第 3 部分之前的 dynamic_trace 步骤，find_top_trace 返回与输入上的 Tracer 关联的最高级别解释器，否则返回堆栈底部的解释器（至少目前始终是求值 trace）。这与上面的描述有所偏差，在上面的描述中，我们始终从运行堆栈顶部的解释器开始，然后逐步向下，应用堆栈中的每个解释器。相反，我们仅在原语绑定的输入参数在与该解释器对应的 Tracer 中封装时才应用解释器。这种优化使我们能够跳过不相关的转换，但默认了一个假设，即转换主要遵循数据依赖性（除了特殊的堆栈底部解释器，它解释一切）。

另一种选择是让堆栈中的每个解释器解释每个操作。这值得探索！JAX 的设计主要围绕数据依赖性，因为这对于自动微分非常自然，并且 JAX 的根基在于自动微分。但这可能过度拟合。

def full_lower(val: Any):
  if isinstance(val, Tracer):
    return val.full_lower()
  else:
    return val

def full_raise(trace: Trace, val: Any) -> Tracer:
  if not isinstance(val, Tracer):
    assert type(val) in jax_types
    return trace.pure(val)
  level = trace.main.level
  if val._trace.main is trace.main:
    return val
  elif val._trace.main.level < level:
    return trace.lift(val)
  elif val._trace.main.level > level:
    raise Exception(f"Can't lift level {val._trace.main.level} to {level}.")
  else:  # val._trace.level == level
    raise Exception(f"Different traces at same level: {val._trace}, {trace}.")

full_raise 中的逻辑用于将值封装到特定 Trace 的 Tracer 中，基于上下文调用 Trace 上的不同方法：Trace.pure 在非 Tracer 常量上调用，Trace.lift 在已经是来自较低级别解释器的 Tracer 的值上调用。这两个方法可以共享相同的实现，但是通过在核心逻辑中区分它们，我们可以向 Trace 子类提供更多信息。

这就是 JAX 核心的全部内容！现在我们可以开始添加解释器了。

求值解释器

我们将从最简单的解释器开始：求值解释器，它将位于解释器堆栈的底部。

class EvalTrace(Trace):
  pure = lift = lambda self, x: x  # no boxing in Tracers needed

  def process_primitive(self, primitive, tracers, params):
    return impl_rules[primitive](*tracers, **params)

trace_stack.append(MainTrace(0, EvalTrace, None))  # special bottom of the stack

# NB: in JAX, instead of a dict we attach impl rules to the Primitive instance
impl_rules = {}

impl_rules[add_p] = lambda x, y: [np.add(x, y)]
impl_rules[mul_p] = lambda x, y: [np.multiply(x, y)]
impl_rules[neg_p] = lambda x: [np.negative(x)]
impl_rules[sin_p] = lambda x: [np.sin(x)]
impl_rules[cos_p] = lambda x: [np.cos(x)]
impl_rules[reduce_sum_p] = lambda x, *, axis: [np.sum(x, axis)]
impl_rules[greater_p] = lambda x, y: [np.greater(x, y)]
impl_rules[less_p] = lambda x, y: [np.less(x, y)]
impl_rules[transpose_p] = lambda x, *, perm: [np.transpose(x, perm)]

def broadcast_impl(x, *, shape, axes):
  for axis in sorted(axes):
    x = np.expand_dims(x, axis)
  return [np.broadcast_to(x, shape)]
impl_rules[broadcast_p] = broadcast_impl

使用此解释器，我们可以求值用户函数

def f(x):
  y = sin(x) * 2.
  z = - y + x
  return z

print(f(3.0))

2.7177599838802657

哇！就像在一个大圈子里绕圈。但是这种间接的目的在于，现在我们可以添加一些真正的转换。

使用 jvp 的前向模式自动微分

首先，是一些辅助函数

import builtins

def zeros_like(val):
  aval = get_aval(val)
  return np.zeros(aval.shape, aval.dtype)

def unzip2(pairs):
  lst1, lst2 = [], []
  for x1, x2 in pairs:
    lst1.append(x1)
    lst2.append(x2)
  return lst1, lst2

def map(f, *xs):
  return list(builtins.map(f, *xs))

def zip(*args):
  fst, *rest = args = map(list, args)
  n = len(fst)
  for arg in rest:
    assert len(arg) == n
  return list(builtins.zip(*args))

前向模式自动微分的 Tracer 携带原始-切线对。Trace 应用 JVP 规则。

class JVPTracer(Tracer):
  def __init__(self, trace, primal, tangent):
    self._trace = trace
    self.primal = primal
    self.tangent = tangent

  @property
  def aval(self):
    return get_aval(self.primal)

class JVPTrace(Trace):
  pure = lift = lambda self, val: JVPTracer(self, val, zeros_like(val))

  def process_primitive(self, primitive, tracers, params):
    primals_in, tangents_in = unzip2((t.primal, t.tangent) for t in tracers)
    jvp_rule = jvp_rules[primitive]
    primal_outs, tangent_outs = jvp_rule(primals_in, tangents_in, **params)
    return [JVPTracer(self, x, t) for x, t in zip(primal_outs, tangent_outs)]

jvp_rules = {}

请注意，pure 和 lift 都将值打包到 JVPTracer 中，并带有最少的上下文，即零切线值。

让我们为原语添加一些 JVP 规则

def add_jvp(primals, tangents):
  (x, y), (x_dot, y_dot) = primals, tangents
  return [x + y], [x_dot + y_dot]
jvp_rules[add_p] = add_jvp

def mul_jvp(primals, tangents):
  (x, y), (x_dot, y_dot) = primals, tangents
  return [x * y], [x_dot * y + x * y_dot]
jvp_rules[mul_p] = mul_jvp

def sin_jvp(primals, tangents):
  (x,), (x_dot,) = primals, tangents
  return [sin(x)], [cos(x) * x_dot]
jvp_rules[sin_p] = sin_jvp

def cos_jvp(primals, tangents):
  (x,), (x_dot,) = primals, tangents
  return [cos(x)], [-sin(x) * x_dot]
jvp_rules[cos_p] = cos_jvp

def neg_jvp(primals, tangents):
  (x,), (x_dot,) = primals, tangents
  return [neg(x)], [neg(x_dot)]
jvp_rules[neg_p] = neg_jvp

def reduce_sum_jvp(primals, tangents, *, axis):
  (x,), (x_dot,) = primals, tangents
  return [reduce_sum(x, axis)], [reduce_sum(x_dot, axis)]
jvp_rules[reduce_sum_p] = reduce_sum_jvp

def greater_jvp(primals, tangents):
  (x, y), _ = primals, tangents
  out_primal = greater(x, y)
  return [out_primal], [zeros_like(out_primal)]
jvp_rules[greater_p] = greater_jvp

def less_jvp(primals, tangents):
  (x, y), _ = primals, tangents
  out_primal = less(x, y)
  return [out_primal], [zeros_like(out_primal)]
jvp_rules[less_p] = less_jvp

最后，我们添加一个转换 API 来启动 trace

def jvp_v1(f, primals, tangents):
  with new_main(JVPTrace) as main:
    trace = JVPTrace(main)
    tracers_in = [JVPTracer(trace, x, t) for x, t in zip(primals, tangents)]
    out = f(*tracers_in)
    tracer_out = full_raise(trace, out)
    primal_out, tangent_out = tracer_out.primal, tracer_out.tangent
  return primal_out, tangent_out

有了这个，我们就可以微分了！

x = 3.0
y, sin_deriv_at_3 = jvp_v1(sin, (x,), (1.0,))
print(sin_deriv_at_3)
print(cos(3.0))

-0.9899924966004454
-0.9899924966004454

def f(x):
  y = sin(x) * 2.
  z = - y + x
  return z

x, xdot = 3., 1.
y, ydot = jvp_v1(f, (x,), (xdot,))
print(y)
print(ydot)

2.7177599838802657
2.979984993200891

def deriv(f):
  return lambda x: jvp_v1(f, (x,), (1.,))[1]

print(deriv(sin)(3.))
print(deriv(deriv(sin))(3.))
print(deriv(deriv(deriv(sin)))(3.))
print(deriv(deriv(deriv(deriv(sin))))(3.))

-0.9899924966004454
-0.1411200080598672
0.9899924966004454
0.1411200080598672

def f(x):
  if x > 0.:  # Python control flow
    return 2. * x
  else:
    return x

print(deriv(f)(3.))
print(deriv(f)(-3.))

2.0
1.0

Pytrees 和扁平化用户函数的输入和输出

jvp_v1 的一个限制是它假设用户函数接受数组作为位置参数，并生成单个数组作为输出。如果它生成一个列表作为输出怎么办？或者接受嵌套容器作为输入？在堆栈的每一层处理输入和输出中所有可能的容器将是一件痛苦的事情。相反，我们可以封装用户函数，以便封装版本接受数组作为输入，并返回扁平数组列表作为输出。封装器只需要解扁平化其输入，调用用户函数，然后扁平化输出。

这是我们希望编写 jvp 的方式，假设用户始终为我们提供以数组作为输入并生成扁平数组列表作为输出的函数

def jvp_flat(f, primals, tangents):
  with new_main(JVPTrace) as main:
    trace = JVPTrace(main)
    tracers_in = [JVPTracer(trace, x, t) for x, t in zip(primals, tangents)]
    outs = f(*tracers_in)
    tracers_out = [full_raise(trace, out) for out in outs]
    primals_out, tangents_out = unzip2((t.primal, t.tangent) for t in tracers_out)
  return primals_out, tangents_out

为了支持在输入和输出中具有任意容器的用户函数，这是我们编写面向用户的 jvp 封装器的方式

def jvp(f, primals, tangents):
  primals_flat, in_tree = tree_flatten(primals)
  tangents_flat, in_tree2 = tree_flatten(tangents)
  if in_tree != in_tree2: raise TypeError
  f, out_tree = flatten_fun(f, in_tree)
  primals_out_flat, tangents_out_flat = jvp_flat(f, primals_flat, tangents_flat)
  primals_out = tree_unflatten(out_tree(), primals_out_flat)
  tangents_out = tree_unflatten(out_tree(), tangents_out_flat)
  return primals_out, tangents_out

请注意，我们必须将用户函数输出的树结构输送回 flatten_fun 的调用者。flatten_fun 实际运行用户函数后才能获得该信息，因此 flatten_fun 仅返回对可变单元格的引用，表示为 thunk。这些副作用是安全的，因为我们始终只运行用户函数一次。（这种安全机制是 linear_util.py 中 “linear” 名称的原因，在 线性类型的意义上。）

剩下的就是编写 tree_flatten、tree_unflatten 和 flatten_fun。

显示代码单元格来源 隐藏代码单元格来源

def flatten_fun(f, in_tree):
  store = Store()

  def flat_fun(*args_flat):
    pytree_args = tree_unflatten(in_tree, args_flat)
    out = f(*pytree_args)
    out_flat, out_tree = tree_flatten(out)
    store.set_value(out_tree)
    return out_flat

  return flat_fun, store

class Empty: pass
empty = Empty()

class Store:
  val = empty

  def set_value(self, val):
    assert self.val is empty
    self.val = val

  def __call__(self):
    return self.val

显示代码单元格来源 隐藏代码单元格来源

from collections.abc import Hashable, Iterable, Iterator
import itertools as it
from collections.abc import Callable

class NodeType(NamedTuple):
  name: str
  to_iterable: Callable
  from_iterable: Callable

def register_pytree_node(ty: type, to_iter: Callable, from_iter: Callable
                         ) -> None:
  node_types[ty] = NodeType(str(ty), to_iter, from_iter)

node_types: dict[type, NodeType] = {}
register_pytree_node(tuple, lambda t: (None, t), lambda _, xs: tuple(xs))
register_pytree_node(list,  lambda l: (None, l), lambda _, xs:  list(xs))
register_pytree_node(dict,
                     lambda d: map(tuple, unzip2(sorted(d.items()))),
                     lambda keys, vals: dict(zip(keys, vals)))

class PyTreeDef(NamedTuple):
  node_type: NodeType
  node_metadata: Hashable
  child_treedefs: tuple['PyTreeDef', ...]

class Leaf: pass
leaf = Leaf()

def tree_flatten(x: Any) -> tuple[list[Any], PyTreeDef]:
  children_iter, treedef = _tree_flatten(x)
  return list(children_iter), treedef

def _tree_flatten(x: Any) -> tuple[Iterable, PyTreeDef]:
  node_type = node_types.get(type(x))
  if node_type:
    node_metadata, children = node_type.to_iterable(x)
    children_flat, child_trees = unzip2(map(_tree_flatten, children))
    flattened = it.chain.from_iterable(children_flat)
    return flattened, PyTreeDef(node_type, node_metadata, tuple(child_trees))
  else:
    return [x], leaf

def tree_unflatten(treedef: PyTreeDef, xs: list[Any]) -> Any:
  return _tree_unflatten(treedef, iter(xs))

def _tree_unflatten(treedef: PyTreeDef, xs: Iterator) -> Any:
  if treedef is leaf:
    return next(xs)
  else:
    children = (_tree_unflatten(t, xs) for t in treedef.child_treedefs)
    return treedef.node_type.from_iterable(treedef.node_metadata, children)

通过这种 pytree 处理的 jvp 实现，我们现在可以处理任意输入和输出容器。这在未来的转换中也会派上用场！

def f(x):
  y = sin(x) * 2.
  z = - y + x
  return {'hi': z, 'there': [x, y]}

x, xdot = 3., 1.
y, ydot = jvp(f, (x,), (xdot,))
print(y)
print(ydot)

{'hi': np.float64(2.7177599838802657), 'there': [3.0, np.float64(0.2822400161197344)]}
{'hi': np.float64(2.979984993200891), 'there': [1.0, np.float64(-1.9799849932008908)]}

使用 vmap 的向量化批处理

首先，是几个辅助函数，一个用于从未映射的抽象值生成映射的抽象值（通过删除轴），另一个用于移动批处理维度

def mapped_aval(batch_dim, aval):
  shape = list(aval.shape)
  del shape[batch_dim]
  return ShapedArray(tuple(shape), aval.dtype)

def move_batch_axis(axis_size, src, dst, x):
  if src is not_mapped:
    target_shape = list(np.shape(x))
    target_shape.insert(dst, axis_size)
    return broadcast(x, target_shape, [dst])
  elif src == dst:
    return x
  else:
    return moveaxis(x, src, dst)

def moveaxis(x, src: int, dst: int):
  perm = [i for i in range(np.ndim(x)) if i != src]
  perm.insert(dst, src)
  return transpose(x, perm)

向量化批处理的 Tracer 携带一个批处理值和一个可选整数，指示哪个轴（如果有）是批处理轴。

from typing import Union

class NotMapped: pass
not_mapped = NotMapped()

BatchAxis = Union[NotMapped, int]

class BatchTracer(Tracer):
  def __init__(self, trace, val, batch_dim: BatchAxis):
    self._trace = trace
    self.val = val
    self.batch_dim = batch_dim

  @property
  def aval(self):
    if self.batch_dim is not_mapped:
      return get_aval(self.val)
    else:
      return mapped_aval(self.batch_dim, get_aval(self.val))

  def full_lower(self):
    if self.batch_dim is not_mapped:
      return full_lower(self.val)
    else:
      return self

class BatchTrace(Trace):
  pure = lift = lambda self, val: BatchTracer(self, val, not_mapped)

  def process_primitive(self, primitive, tracers, params):
    vals_in, bdims_in = unzip2((t.val, t.batch_dim) for t in tracers)
    vmap_rule = vmap_rules[primitive]
    val_outs, bdim_outs = vmap_rule(self.axis_size, vals_in, bdims_in, **params)
    return [BatchTracer(self, x, bd) for x, bd in zip(val_outs, bdim_outs)]

  @property
  def axis_size(self):
    return self.main.global_data

vmap_rules = {}

在这里，我们实现了可选的 Tracer.full_lower 方法，该方法允许我们在不需要批处理 tracer 时将其剥离，因为它不表示批处理值。

对于 BatchTrace，类似于 JVPTrace，方法 pure 和 lift 只是将值封装在 BatchTracer 中，并带有最少的上下文，在这种情况下，上下文是采用哨兵值 not_mapped 的 batch_dim。请注意，我们使用 MainTrace 的解释器全局数据字段来存储批处理轴大小。

接下来，我们可以为每个原语定义批处理解释器规则

from functools import partial

def binop_batching_rule(op, axis_size, vals_in, dims_in):
  (x, y), (x_bdim, y_bdim) = vals_in, dims_in
  if x_bdim != y_bdim:
    if x_bdim is not_mapped:
      x = move_batch_axis(axis_size, x_bdim, y_bdim, x)
      x_bdim = y_bdim
    else:
      y = move_batch_axis(axis_size, y_bdim, x_bdim, y)
  return [op(x, y)], [x_bdim]
vmap_rules[add_p] = partial(binop_batching_rule, add)
vmap_rules[mul_p] = partial(binop_batching_rule, mul)

def vectorized_unop_batching_rule(op, axis_size, vals_in, dims_in):
  (x,), (x_bdim,) = vals_in, dims_in
  return [op(x)], [x_bdim]
vmap_rules[sin_p] = partial(vectorized_unop_batching_rule, sin)
vmap_rules[cos_p] = partial(vectorized_unop_batching_rule, cos)
vmap_rules[neg_p] = partial(vectorized_unop_batching_rule, neg)

def reduce_sum_batching_rule(axis_size, vals_in, dims_in, *, axis):
  (x,), (x_bdim,) = vals_in, dims_in
  new_axis = tuple(ax + (x_bdim <= ax) for ax in axis)
  out_bdim = x_bdim - sum(ax < x_bdim for ax in axis)
  return [reduce_sum(x, new_axis)], [out_bdim]
vmap_rules[reduce_sum_p] = reduce_sum_batching_rule

最后，我们添加一个转换 API 来启动 trace

def vmap_flat(f, in_axes, *args):
  axis_size, = {x.shape[ax] for x, ax in zip(args, in_axes)
                if ax is not not_mapped}
  with new_main(BatchTrace, axis_size) as main:
    trace = BatchTrace(main)
    tracers_in = [BatchTracer(trace, x, ax) if ax is not None else x
                  for x, ax in zip(args, in_axes)]
    outs = f(*tracers_in)
    tracers_out = [full_raise(trace, out) for out in outs]
    vals_out, bdims_out = unzip2((t.val, t.batch_dim) for t in tracers_out)
  outs_transposed = [move_batch_axis(axis_size, bdim, 0, val_out)
                     for val_out, bdim in zip(vals_out, bdims_out)]
  return outs_transposed

def vmap(f, in_axes):
  def batched_f(*args):
    args_flat, in_tree = tree_flatten(args)
    in_axes_flat, in_tree2 = tree_flatten(in_axes)
    if in_tree != in_tree2: raise TypeError
    f_flat, out_tree = flatten_fun(f, in_tree)
    outs_flat = vmap_flat(f_flat, in_axes_flat, *args_flat)
    return tree_unflatten(out_tree(), outs_flat)
  return batched_f

def add_one_to_a_scalar(scalar):
  assert np.ndim(scalar) == 0
  return 1 + scalar

vector_in = np.arange(3.)
vector_out = vmap(add_one_to_a_scalar, (0,))(vector_in)

print(vector_in)
print(vector_out)

[0. 1. 2.]
[1. 2. 3.]

def jacfwd(f, x):
  pushfwd = lambda v: jvp(f, (x,), (v,))[1]
  vecs_in = np.eye(np.size(x)).reshape(np.shape(x) * 2)
  return vmap(pushfwd, (0,))(vecs_in)

def f(x):
  return sin(x)

jacfwd(f, np.arange(3.))

array([[ 1.        ,  0.        , -0.        ],
       [ 0.        ,  0.54030231, -0.        ],
       [ 0.        ,  0.        , -0.41614684]])

这就是 jvp 和 vmap 的全部内容！

第 2 部分：Jaxprs

接下来的转换是用于即时编译的 jit 和用于反向模式自动微分的 vjp。（grad 只是 vjp 的一个小封装器。）而 jvp 和 vmap 只需要每个 Tracer 携带一点额外的上下文，对于 jit 和 vjp，我们需要更丰富的上下文：我们需要表示程序。也就是说，我们需要 jaxpr！

Jaxpr 是 JAX 程序的内部中间表示。它们是显式类型、函数式、一阶且采用 ANF 形式。我们需要 jit 的程序表示，因为 jit 的目的是将计算移出 Python。对于任何我们想要移出的计算，我们需要能够将其表示为数据，并在我们追踪 Python 函数时构建它。同样，vjp 需要一种表示反向模式自动微分的反向传播计算的方法。我们将相同的 jaxpr 程序表示用于这两种需求。

（构建程序表示是最 自由 的 trace-transformation 类型，因此，除了处理原生 Python 控制流的问题外，任何转换都可以通过首先追踪到 jaxpr 然后解释 jaxpr 来实现。）

Jaxpr 数据结构

jaxpr 术语语法大致为

jaxpr ::=
  { lambda <binder> , ... .
    let <eqn>
        ...
    in ( <atom> , ... ) }

binder ::= <var>:<array_type>
var ::= a | b | c | ...
atom ::= <var> | <literal>
literal ::= <int32> | <int64> | <float32> | <float64>

eqn ::= <binder> , ... = <primitive> [ <params> ] <atom> , ...

类型语法为

jaxpr_type ::= [ <array_type> , ... ] -> [ <array_type> , ... ]
array_type ::= <dtype>[<shape>]
dtype ::= f32 | f64 | i32 | i64
shape ::= <int> , ...

我们如何将这些表示为 Python 数据结构？我们重用 ShapedArrays 来表示类型，并且可以使用一些 Python 结构来表示术语语法

class Var:
  aval: ShapedArray
  def __init__(self, aval): self.aval = aval

class Lit:
  val: Any
  aval: ShapedArray

  def __init__(self, val):
    self.aval = aval = raise_to_shaped(get_aval(val))
    self.val = np.array(val, aval.dtype)

Atom = Union[Var, Lit]

class JaxprEqn(NamedTuple):
  primitive: Primitive
  inputs: list[Atom]
  params: dict[str, Any]
  out_binders: list[Var]

class Jaxpr(NamedTuple):
  in_binders: list[Var]
  eqns: list[JaxprEqn]
  outs: list[Atom]

  def __hash__(self): return id(self)
  __eq__ = op.is_

def raise_to_shaped(aval):
  return ShapedArray(aval.shape, aval.dtype)

类型检查 jaxpr 涉及检查是否没有未绑定变量，变量是否仅绑定一次，以及对于每个方程，原语应用的类型是否与输出绑定器的类型匹配。

class JaxprType(NamedTuple):
  in_types:  list[ShapedArray]
  out_types: list[ShapedArray]

  def __repr__(self):
    in_types = ', '.join(aval.str_short() for aval in self.in_types)
    out_types = ', '.join(aval.str_short() for aval in self.out_types)
    return f'({in_types}) -> ({out_types})'

def typecheck_jaxpr(jaxpr: Jaxpr) -> JaxprType:
  env: set[Var] = set()

  for v in jaxpr.in_binders:
    if v in env: raise TypeError
    env.add(v)

  for eqn in jaxpr.eqns:
    in_types = [typecheck_atom(env, x) for x in eqn.inputs]
    out_types = abstract_eval_rules[eqn.primitive](*in_types, **eqn.params)
    for out_binder, out_type in zip(eqn.out_binders, out_types):
      if not out_type == out_binder.aval: raise TypeError
    for out_binder in eqn.out_binders:
      if out_binder in env: raise TypeError
      env.add(out_binder)

  in_types = [v.aval for v in jaxpr.in_binders]
  out_types = [typecheck_atom(env, x) for x in jaxpr.outs]
  return JaxprType(in_types, out_types)

def typecheck_atom(env: set[Var], x: Atom) -> ShapedArray:
  if isinstance(x, Var):
    if x not in env: raise TypeError("unbound variable")
    return x.aval
  elif isinstance(x, Lit):
    return raise_to_shaped(get_aval(x.val))
  else:
    assert False

我们可以使用简单的解释器将 jaxpr 表示的函数应用于参数。

def eval_jaxpr(jaxpr: Jaxpr, args: list[Any]) -> list[Any]:
  env: dict[Var, Any] = {}

  def read(x: Atom) -> Any:
    return env[x] if type(x) is Var else x.val

  def write(v: Var, val: Any) -> None:
    assert v not in env  # single-assignment
    env[v] = val

  map(write, jaxpr.in_binders, args)
  for eqn in jaxpr.eqns:
    in_vals = map(read, eqn.inputs)
    outs = bind(eqn.primitive, *in_vals, **eqn.params)
    map(write, eqn.out_binders, outs)
  return map(read, jaxpr.outs)

def jaxpr_as_fun(jaxpr: Jaxpr):
  return lambda *args: eval_jaxpr(jaxpr, args)

通过在解释器中使用 bind，此解释器本身是可追踪的。

使用追踪构建 jaxpr

现在我们有了作为数据结构的 jaxpr，我们需要方法从追踪 Python 代码中生成这些。通常，我们追踪到 jaxpr 的方式有两种变体；jit 使用一种，vjp 使用另一种。我们将从 jit 使用的一种开始，它也由控制流原语（如 lax.cond、lax.while_loop 和 lax.scan）使用。

def split_list(lst: list[Any], n: int) -> tuple[list[Any], list[Any]]:
  assert 0 <= n <= len(lst)
  return lst[:n], lst[n:]

def partition_list(bs: list[bool], l: list[Any]) -> tuple[list[Any], list[Any]]:
  assert len(bs) == len(l)
  lists = lst1, lst2 = [], []
  for b, x in zip(bs, l):
    lists[b].append(x)
  return lst1, lst2

# NB: the analogous class in JAX is called 'DynamicJaxprTracer'
class JaxprTracer(Tracer):
  __slots__ = ['aval']
  aval: ShapedArray

  def __init__(self, trace, aval):
    self._trace = trace
    self.aval = aval

# NB: the analogous class in JAX is called 'DynamicJaxprTrace'
class JaxprTrace(Trace):
  def new_arg(self, aval: ShapedArray) -> JaxprTracer:
    aval = raise_to_shaped(aval)
    tracer = self.builder.new_tracer(self, aval)
    self.builder.tracer_to_var[id(tracer)] = Var(aval)
    return tracer

  def get_or_make_const_tracer(self, val: Any) -> JaxprTracer:
    tracer = self.builder.const_tracers.get(id(val))
    if tracer is None:
      tracer = self.builder.new_tracer(self, raise_to_shaped(get_aval(val)))
      self.builder.add_const(tracer, val)
    return tracer
  pure = lift = get_or_make_const_tracer

  def process_primitive(self, primitive, tracers, params):
    avals_in = [t.aval for t in tracers]
    avals_out = abstract_eval_rules[primitive](*avals_in, **params)
    out_tracers = [self.builder.new_tracer(self, a) for a in avals_out]
    inputs = [self.builder.getvar(t) for t in tracers]
    outvars = [self.builder.add_var(t) for t in out_tracers]
    self.builder.add_eqn(JaxprEqn(primitive, inputs, params, outvars))
    return out_tracers

  @property
  def builder(self):
    return self.main.global_data

# NB: in JAX, we instead attach abstract eval rules to Primitive instances
abstract_eval_rules = {}

请注意，我们保留一个构建器对象作为解释器全局数据，该对象在构建 jaxpr 时跟踪变量、常量和方程。

class JaxprBuilder:
  eqns: list[JaxprEqn]
  tracer_to_var: dict[int, Var]
  const_tracers: dict[int, JaxprTracer]
  constvals: dict[Var, Any]
  tracers: list[JaxprTracer]

  def __init__(self):
    self.eqns = []
    self.tracer_to_var = {}
    self.const_tracers = {}
    self.constvals = {}
    self.tracers = []

  def new_tracer(self, trace: JaxprTrace, aval: ShapedArray) -> JaxprTracer:
    tracer = JaxprTracer(trace, aval)
    self.tracers.append(tracer)
    return tracer

  def add_eqn(self, eqn: JaxprEqn) -> None:
    self.eqns.append(eqn)

  def add_var(self, tracer: JaxprTracer) -> Var:
    assert id(tracer) not in self.tracer_to_var
    var = self.tracer_to_var[id(tracer)] = Var(tracer.aval)
    return var

  def getvar(self, tracer: JaxprTracer) -> Var:
    var = self.tracer_to_var.get(id(tracer))
    assert var is not None
    return var

  def add_const(self, tracer: JaxprTracer, val: Any) -> Var:
    var = self.add_var(tracer)
    self.const_tracers[id(val)] = tracer
    self.constvals[var] = val
    return var

  def build(self, in_tracers: list[JaxprTracer], out_tracers: list[JaxprTracer]
            ) -> tuple[Jaxpr, list[Any]]:
    constvars, constvals = unzip2(self.constvals.items())
    t2v = lambda t: self.tracer_to_var[id(t)]
    in_binders = constvars + [t2v(t) for t in in_tracers]
    out_vars = [t2v(t) for t in out_tracers]
    jaxpr = Jaxpr(in_binders, self.eqns, out_vars)
    typecheck_jaxpr(jaxpr)
    jaxpr, constvals = _inline_literals(jaxpr, constvals)
    return jaxpr, constvals

def _inline_literals(jaxpr: Jaxpr, consts: list[Any]) -> tuple[Jaxpr, list[Any]]:
  const_binders, other_binders = split_list(jaxpr.in_binders, len(consts))
  scalars = [type(x) in jax_types and not get_aval(x).shape for x in consts]
  new_const_binders, lit_binders = partition_list(scalars, const_binders)
  new_consts, lit_vals = partition_list(scalars, consts)
  literals = dict(zip(lit_binders, map(Lit, lit_vals)))
  new_eqns = [JaxprEqn(eqn.primitive, [literals.get(x, x) for x in eqn.inputs],
                       eqn.params, eqn.out_binders) for eqn in jaxpr.eqns]
  new_outs = [literals.get(x, x) for x in jaxpr.outs]
  new_jaxpr = Jaxpr(new_const_binders + other_binders, new_eqns, new_outs)
  typecheck_jaxpr(new_jaxpr)
  return new_jaxpr, new_consts

JaxprTrace.process_primitive 所需的规则本质上是原语应用的类型规则：给定原语、其参数和输入类型，规则必须生成输出类型，然后将其与输出 JaxprTracer 一起封装。我们可以将抽象求值规则用于相同的目的，即使它们可能更通用（因为抽象求值规则必须接受 ConcreteArray 输入，并且由于它们只需要返回可能输出集合的上限，因此它们也可以生成 ConcreteArray 输出）。我们将重用这些抽象求值规则用于其他 jaxpr 生成的追踪机制，其中潜在的额外通用性很有用。

def binop_abstract_eval(x: ShapedArray, y: ShapedArray) -> list[ShapedArray]:
  if not isinstance(x, ShapedArray) or not isinstance(y, ShapedArray):
    raise TypeError
  if raise_to_shaped(x) != raise_to_shaped(y): raise TypeError
  return [ShapedArray(x.shape, x.dtype)]

abstract_eval_rules[add_p] = binop_abstract_eval
abstract_eval_rules[mul_p] = binop_abstract_eval

def compare_abstract_eval(x: ShapedArray, y: ShapedArray) -> list[ShapedArray]:
  if not isinstance(x, ShapedArray) or not isinstance(y, ShapedArray):
    raise TypeError
  if x.shape != y.shape: raise TypeError
  return [ShapedArray(x.shape, np.dtype('bool'))]
abstract_eval_rules[greater_p] = compare_abstract_eval
abstract_eval_rules[less_p] = compare_abstract_eval

def vectorized_unop_abstract_eval(x: ShapedArray) -> list[ShapedArray]:
  return [ShapedArray(x.shape, x.dtype)]

abstract_eval_rules[sin_p] = vectorized_unop_abstract_eval
abstract_eval_rules[cos_p] = vectorized_unop_abstract_eval
abstract_eval_rules[neg_p] = vectorized_unop_abstract_eval

def reduce_sum_abstract_eval(x: ShapedArray, *, axis: tuple[int, ...]
                             ) -> list[ShapedArray]:
  axis_ = set(axis)
  new_shape = [d for i, d in enumerate(x.shape) if i not in axis_]
  return [ShapedArray(tuple(new_shape), x.dtype)]
abstract_eval_rules[reduce_sum_p] = reduce_sum_abstract_eval

def broadcast_abstract_eval(x: ShapedArray, *, shape: Sequence[int],
                            axes: Sequence[int]) -> list[ShapedArray]:
  return [ShapedArray(tuple(shape), x.dtype)]
abstract_eval_rules[broadcast_p] = broadcast_abstract_eval

为了检查我们 jaxpr 的实现，我们可以添加 make_jaxpr 转换和漂亮打印机

from functools import lru_cache

@lru_cache  # ShapedArrays are hashable
def make_jaxpr_v1(f, *avals_in):
  avals_in, in_tree = tree_flatten(avals_in)
  f, out_tree = flatten_fun(f, in_tree)

  builder = JaxprBuilder()
  with new_main(JaxprTrace, builder) as main:
    trace = JaxprTrace(main)
    tracers_in = [trace.new_arg(aval) for aval in avals_in]
    outs = f(*tracers_in)
    tracers_out = [full_raise(trace, out) for out in outs]
    jaxpr, consts = builder.build(tracers_in, tracers_out)
  return jaxpr, consts, out_tree()

显示代码单元格来源 隐藏代码单元格来源

from collections import defaultdict
import string

class PPrint:
  lines: list[tuple[int, str]]

  def __init__(self, lines):
    self.lines = lines

  def indent(self, indent: int) -> 'PPrint':
    return PPrint([(indent + orig_indent, s) for orig_indent, s in self.lines])

  def __add__(self, rhs: 'PPrint') -> 'PPrint':
    return PPrint(self.lines + rhs.lines)

  def __rshift__(self, rhs: 'PPrint') -> 'PPrint':
    if not rhs.lines: return self
    if not self.lines: return rhs
    indent, s = self.lines[-1]
    indented_block = rhs.indent(indent + len(s))
    common_line = s + ' ' * rhs.lines[0][0] + rhs.lines[0][1]
    return PPrint(self.lines[:-1]
                  + [(indent, common_line)]
                  + indented_block.lines[1:])

  def __str__(self) -> str:
    return '\n'.join(' ' * indent + s for indent, s in self.lines)

def pp(s: Any) -> PPrint:
  return PPrint([(0, line) for line in str(s).splitlines()])

def vcat(ps: list[PPrint]) -> PPrint:
  return sum(ps, pp(''))

def pp_jaxpr(jaxpr: Jaxpr) -> PPrint:
  namegen = (''.join(s) for r in it.count(1)
             for s in it.permutations(string.ascii_lowercase, r))
  names = defaultdict(lambda: next(namegen))
  in_binders = ', '.join(var_str(names, x) for x in jaxpr.in_binders)
  eqns = vcat([pp_eqn(names, e) for e in jaxpr.eqns])
  outs = ', '.join(names[v] if isinstance(v, Var) else str(v.val)
                   for v in jaxpr.outs)
  return (pp(f'{{ lambda {in_binders} .') +
          ((pp('let ') >> eqns) + pp(f'in ( {outs} ) }}')).indent(2))

def var_str(names: defaultdict[Var, str], v: Var) -> str:
  return f'{names[v]}:{v.aval.str_short()}'

def pp_eqn(names: defaultdict[Var, str], eqn: JaxprEqn) -> PPrint:
  rule = pp_rules.get(eqn.primitive)
  if rule:
    return rule(names, eqn)
  else:
    lhs = pp(' '.join(var_str(names, v) for v in eqn.out_binders))
    rhs = (pp(eqn.primitive.name) >> pp_params(eqn.params) >>
           pp(' '.join(names[x] if isinstance(x, Var) else str(x.val)
                       for x in eqn.inputs)))
    return lhs >> pp(' = ') >> rhs

def pp_params(params: dict[str, Any]) -> PPrint:
  items = sorted(params.items())
  if items:
    return pp(' [ ') >> vcat([pp(f'{k}={v}') for k, v in items]) >> pp(' ] ')
  else:
    return pp(' ')

Jaxpr.__repr__ = lambda self: str(pp_jaxpr(self))
pp_rules: dict[Primitive, Callable[..., PPrint]] = {}

jaxpr, consts, _ = make_jaxpr_v1(lambda x: 2. * x, raise_to_shaped(get_aval(3.)))
print(jaxpr)
print(typecheck_jaxpr(jaxpr))

{ lambda a:float64[] .
  let b:float64[] = mul 2.0 a
  in ( b ) }
(float64[]) -> (float64[])

但是这里有一个限制：由于 find_top_trace 通过数据依赖性运行，因此 make_jaxpr_v1 无法移出 Python 可调用对象执行的所有原语操作。例如

jaxpr, consts, _ = make_jaxpr_v1(lambda: mul(2., 2.))
print(jaxpr)

{ lambda  .
  let 
  in ( 4.0 ) }

这正是 omnistaging 修复的问题。我们希望确保始终应用由 make_jaxpr 启动的 JaxprTrace，无论绑定到 bind 的任何输入是否在相应的 JaxprTracer 实例中封装。我们可以通过使用第 1 部分中定义的 dynamic_trace 全局变量来实现这一点

@contextmanager
def new_dynamic(main: MainTrace):
  global dynamic_trace
  prev_dynamic_trace, dynamic_trace = dynamic_trace, main
  try:
    yield
  finally:
    dynamic_trace = prev_dynamic_trace

@lru_cache
def make_jaxpr(f: Callable, *avals_in: ShapedArray,
               ) -> tuple[Jaxpr, list[Any], PyTreeDef]:
  avals_in, in_tree = tree_flatten(avals_in)
  f, out_tree = flatten_fun(f, in_tree)

  builder = JaxprBuilder()
  with new_main(JaxprTrace, builder) as main:
    with new_dynamic(main):
      trace = JaxprTrace(main)
      tracers_in = [trace.new_arg(aval) for aval in avals_in]
      outs = f(*tracers_in)
      tracers_out = [full_raise(trace, out) for out in outs]
      jaxpr, consts = builder.build(tracers_in, tracers_out)
  return jaxpr, consts, out_tree()

jaxpr, consts, _ = make_jaxpr(lambda: mul(2., 2.))
print(jaxpr)

{ lambda  .
  let a:float64[] = mul 2.0 2.0
  in ( a ) }

使用 dynamic_trace 这种方式在概念上与暂存当前解释器堆栈并在底部使用 JaxprTrace 启动一个新堆栈相同。也就是说，不应用堆栈中低于 dynamic_trace 的解释器（因为 JaxprTrace.process_primitive 不调用 bind），但如果被追踪为 jaxpr 的 Python 可调用对象本身使用转换，则可以将这些转换推送到解释器堆栈中 JaxprTrace 之上。但是，临时暂存解释器堆栈会破坏系统状态。dynamic_trace 标签实现了相同的目标，同时保持系统状态更简单。

jaxpr 的介绍就到这里！有了 jaxpr，我们就可以实现 JAX 剩余的主要功能了。

第 3 部分：简化的 jit

虽然 jit 具有类似于转换的 API，因为它接受 Python 可调用对象作为参数，但在底层它实际上是一个高阶原语，而不是转换。当一个原语由函数参数化时，它是高阶的。

即时（“最终风格”）和分阶段（“初始风格”）处理

对于如何处理高阶原语，有两种选择。每种方法都需要不同的追踪方法，并会产生不同的权衡。

1. 即时处理，其中 bind 接受 Python 可调用对象作为参数。 我们延迟形成 jaxpr，直到尽可能晚的时候，即直到我们在解释器堆栈底部运行最终解释器时。这样，我们就可以在解释器堆栈底部交换一个 JaxprTrace，从而分阶段输出而不是执行所有原语操作。使用这种方法，堆栈中的转换会像往常一样在我们执行 Python 可调用对象时应用。这种方法实现起来可能非常棘手，但它尽可能通用，因为它允许高阶原语不提高其参数的抽象级别，因此允许数据相关的 Python 控制流。我们将这种方法称为使用“最终风格高阶原语”，采用我们目前使用的追踪时放电的“最终风格转换”。

2. 分阶段处理，其中 bind 接受 jaxpr 作为参数。 在我们调用 bind 之前，在原语包装器中，我们可以只使用 make_jaxpr 预先形成一个 jaxpr，并完全完成 Python 可调用对象的操作。在这种情况下，make_jaxpr 将其 JaxprTrace 放在解释器堆栈的顶部，并且堆栈中较低的转换（可能通过闭包 Tracer 进入）不会在我们追踪 Python 可调用对象时应用于它。（在 Python 可调用对象中应用的转换会像往常一样应用，被添加到 JaxprTrace 之上的堆栈中。）相反，堆栈中较低的转换稍后会应用于调用原语，并且调用原语的规则必须随后转换 jaxpr 本身。因为我们预先追踪到 jaxpr，所以这种方法不支持数据相关的 Python 控制流，但它更容易实现。我们将这种高阶原语称为“初始风格高阶原语”，并说它的 jaxpr 处理转换规则是“初始风格转换规则”。

后一种方法适用于 jit，因为我们不需要在用户提供的 Python 可调用对象中支持数据相关的 Python 控制流，因为 jit 的全部目的是将计算分阶段输出 Python，以便由 XLA 执行。（相比之下，custom_jvp 是一个高阶原语，我们希望在其中支持数据相关的 Python 控制流。）

从历史上看，我们在阅读了 typed tagless final interpreters 论文后，开始使用“初始风格”和“最终风格”术语，并开玩笑地将 JAX 称为“无类型 tagful 最终解释器”的实现。我们不声称要延续（或理解）这些术语背后的任何深刻含义；我们粗略地使用“初始风格”来表示“构建 AST 然后转换它”，我们使用“最终风格”来表示“在追踪时转换”。但这只是不精确但又固执的行话。

使用初始风格方法，这是面向用户的 jit 包装器

def jit(f):
  def f_jitted(*args):
    avals_in = [raise_to_shaped(get_aval(x)) for x in args]
    jaxpr, consts, out_tree = make_jaxpr(f, *avals_in)
    outs = bind(xla_call_p, *consts, *args, jaxpr=jaxpr, num_consts=len(consts))
    return tree_unflatten(out_tree, outs)
  return f_jitted

xla_call_p = Primitive('xla_call')

对于任何新的原语，我们需要为其提供转换规则，从其评估规则开始。当我们评估 xla_call 原语的应用时，我们希望将计算分阶段输出到 XLA。这涉及到将 jaxpr 转换为 XLA HLO 程序，将参数值传输到 XLA 设备，执行 XLA 程序，并将结果传输回来。我们将缓存 XLA HLO 编译，以便对于每个 jit 函数，它只需要为每个参数形状和 dtype 签名执行一次。

首先，是一些实用工具。

class IDHashable:
  val: Any

  def __init__(self, val):
    self.val = val

  def __hash__(self) -> int:
    return id(self.val)

  def __eq__(self, other):
    return type(other) is IDHashable and id(self.val) == id(other.val)

接下来，我们将定义 xla_call 的评估规则

import io
from jax.extend.mlir import ir
from jax.extend.mlir.dialects import func
from jax.extend.mlir.dialects import stablehlo as hlo
from jax._src import xla_bridge as xb

class MlirContext(NamedTuple):
  module: ir.Module
  symbol_table: ir.SymbolTable

def xla_call_impl(*args, jaxpr: Jaxpr, num_consts: int):
  consts, args = args[:num_consts], args[num_consts:]
  hashable_consts = tuple(map(IDHashable, consts))
  execute = xla_callable(IDHashable(jaxpr), hashable_consts)
  return execute(*args)
impl_rules[xla_call_p] = xla_call_impl

@lru_cache
def xla_callable(hashable_jaxpr: IDHashable,
                 hashable_consts: tuple[IDHashable, ...]):
  jaxpr: Jaxpr = hashable_jaxpr.val
  typecheck_jaxpr(jaxpr)
  consts = [x.val for x in hashable_consts]
  in_avals = [v.aval for v in jaxpr.in_binders[len(consts):]]

  with ir.Context() as ctx, ir.Location.unknown(ctx):
    hlo.register_dialect(ctx)
    m = ir.Module.create()
    c = MlirContext(m, ir.SymbolTable(m.operation))

    with ir.InsertionPoint(c.module.body):
      @func.func(*(aval_to_ir_type(aval) for aval in in_avals))
      def main(*params):
        return jaxpr_subcomp(c, jaxpr, _hlo_consts(consts) + params)

  output = io.StringIO()
  c.module.operation.print(file=output)
  compiled = xb.get_backend(None).compile(output.getvalue())
  return partial(execute_compiled, compiled, [v.aval for v in jaxpr.outs])

def _mlir_dtype(dtype: np.dtype) -> ir.Type:
  if np.issubdtype(dtype, np.signedinteger):
    return ir.IntegerType.get_signless(np.iinfo(dtype).bits)
  elif dtype == np.float32:
    return ir.F32Type.get()
  elif dtype == np.float64:
    return ir.F64Type.get()
  else:
    raise NotImplementedError("MLIR conversion not implemented for ", dtype)

def aval_to_ir_type(aval: ShapedArray) -> ir.Type:
  return ir.RankedTensorType.get(aval.shape, _mlir_dtype(aval.dtype))

def _hlo_const(x: Any) -> ir.Value:
  a = np.asarray(x)
  if a.dtype == np.bool_:
    return hlo.constant(ir.DenseElementsAttr.get(
      np.packbits(a, bitorder='little'), type=ir.IntegerType.get_signless(1),
      shape=a.shape))
  else:
    return hlo.constant(ir.DenseElementsAttr.get(a))

def _hlo_consts(consts: list[Any]) -> list[ir.Value]:
  unique_consts = {id(cnst): cnst for cnst in consts}
  ir_consts = {id_: _hlo_const(cnst) for id_, cnst in unique_consts.items()}
  return tuple(ir_consts[id(cnst)] for cnst in consts)

主要操作在 xla_callable 中，它使用 jaxpr_subcomp 将 jaxpr 编译为 XLA HLO 程序，然后返回一个可调用对象，该对象执行编译后的程序

def jaxpr_subcomp(c: MlirContext, jaxpr: Jaxpr, args: list[ir.Value]) -> list[ir.Value]:
  env: dict[Var, ir.Value] = {}

  def read(x: Atom) -> ir.Value:
    return env[x] if type(x) is Var else _hlo_const(np.asarray(x.val))

  def write(v: Var, val: ir.Value) -> None:
    env[v] = val

  map(write, jaxpr.in_binders, args)
  for eqn in jaxpr.eqns:
    in_avals = [x.aval for x in eqn.inputs]
    in_vals = map(read, eqn.inputs)
    out_avals = [x.aval for x in eqn.out_binders]
    rule = hlo_translations[eqn.primitive]
    assert all(isinstance(v, ir.Value) for v in in_vals), in_vals
    out_vals = rule(c, in_avals, out_avals, in_vals, **eqn.params)
    assert all(isinstance(v, ir.Value) for v in out_vals), out_vals
    map(write, eqn.out_binders, out_vals), out_vals
  return map(read, jaxpr.outs)

def execute_compiled(compiled, out_avals, *args):
  input_bufs = [input_handlers[type(x)](x) for x in args]
  out_bufs = compiled.execute(input_bufs)
  return [handle_result(aval, buf) for aval, buf in zip(out_avals, out_bufs)]

default_input_handler = xb.get_backend(None).buffer_from_pyval
input_handlers = {ty: default_input_handler for ty in
                  [bool, int, float, np.ndarray, np.float64, np.float32]}

def handle_result(aval: ShapedArray, buf):
  del aval  # Unused for now
  return np.asarray(buf)

hlo_translations = {}

请注意，jaxpr_subcomp 具有简单解释器的结构。这是一个常见的模式：我们处理 jaxpr 的方式通常是使用解释器。与任何解释器一样，我们需要为每个原语提供解释规则。

def direct_translation(op, c, in_avals, out_avals, in_vals):
  del c, in_avals, out_avals
  return [op(*in_vals)]

hlo_translations[add_p] = partial(direct_translation, hlo.add)
hlo_translations[mul_p] = partial(direct_translation, hlo.multiply)
hlo_translations[neg_p] = partial(direct_translation, hlo.negate)
hlo_translations[sin_p] = partial(direct_translation, hlo.sine)
hlo_translations[cos_p] = partial(direct_translation, hlo.cosine)

def compare_translation(op, c, in_avals, out_avals, in_vals):
  del c, out_avals
  return [hlo.compare(*in_vals, hlo.ComparisonDirectionAttr.get(op))]

hlo_translations[greater_p] = partial(compare_translation, "GT")
hlo_translations[less_p] = partial(compare_translation, "LT")

def reduce_sum_translation(c, in_avals, out_avals, in_vals, *, axis):
  del c
  (x_aval,), (out_aval,), (x,) = in_avals, out_avals, in_vals
  op = hlo.ReduceOp(
    [aval_to_ir_type(out_aval)], [x], [_hlo_const(np.array(0, x_aval.dtype))],
    axis)
  scalar_type = aval_to_ir_type(ShapedArray((), x_aval.dtype))
  reducer_region = op.body.blocks.append(scalar_type, scalar_type)
  with ir.InsertionPoint(reducer_region):
    hlo.return_([hlo.add(*reducer_region.arguments)])
  return op.results

hlo_translations[reduce_sum_p] = reduce_sum_translation

def broadcast_translation(c, in_avals, out_avals, in_vals, *, shape, axes):
  del c
  (x,), (out_aval,) = in_vals, out_avals
  dims_complement = [i for i in range(len(shape)) if i not in axes]
  return [hlo.broadcast_in_dim(aval_to_ir_type(out_aval), x, dims_complement)]
hlo_translations[broadcast_p] = broadcast_translation

有了这些，我们现在可以使用 jit 来分阶段输出、编译和执行带有 XLA 的程序了！

@jit
def f(x, y):
  print('tracing!')
  return sin(x) * cos(y)

z = f(3., 4.)  # 'tracing!' prints the first time
print(z)

tracing!
-0.09224219304455371

z = f(4., 5.)  # 'tracing!' doesn't print, compilation cache hit!
print(z)

-0.21467624978306993

@jit
def f(x):
  return reduce_sum(x, axis=0)

print(f(np.array([1., 2., 3.])))

6.0

def f(x):
  y = sin(x) * 2.
  z = - y + x
  return z

def deriv(f):
  return lambda x: jvp(f, (x,), (1.,))[1]

print(    deriv(deriv(f))(3.))
print(jit(deriv(deriv(f)))(3.))

0.2822400161197344
0.2822400161197344

与其实现 jit 以首先追踪到 jaxpr，然后再将 jaxpr 降低到 XLA HLO，看起来我们似乎可以跳过 jaxpr 步骤，并在追踪时直接降低到 HLO。也就是说，也许我们可以改为使用 Trace 和 Tracer 实现 jit，它们在每个原语绑定时增量地附加到 XLA HLO 图。现在这样做是正确的，但当我们引入编译后的 SPMD 计算时将不可能，因为在那里我们必须在编译程序之前知道所需的副本数量。

我们尚未为 xla_call_p 定义除评估规则之外的任何转换规则。也就是说，我们还不能执行 vmap-of-jit 或 jvp-of-jit 甚至 jit-of-jit。相反，jit 必须处于“顶层”。让我们解决这个问题！

def xla_call_jvp_rule(primals, tangents, *, jaxpr, num_consts):
  del num_consts  # Unused
  new_jaxpr, new_consts = jvp_jaxpr(jaxpr)
  outs = bind(xla_call_p, *new_consts, *primals, *tangents, jaxpr=new_jaxpr,
              num_consts=len(new_consts))
  n = len(outs) // 2
  primals_out, tangents_out = outs[:n], outs[n:]
  return primals_out, tangents_out
jvp_rules[xla_call_p] = xla_call_jvp_rule

@lru_cache
def jvp_jaxpr(jaxpr: Jaxpr) -> tuple[Jaxpr, list[Any]]:
  def jvp_traceable(*primals_and_tangents):
    n = len(primals_and_tangents) // 2
    primals, tangents = primals_and_tangents[:n], primals_and_tangents[n:]
    return jvp(jaxpr_as_fun(jaxpr), primals, tangents)

  in_avals = [v.aval for v in jaxpr.in_binders]
  new_jaxpr, new_consts, _ = make_jaxpr(jvp_traceable, *in_avals, *in_avals)
  return new_jaxpr, new_consts

def xla_call_vmap_rule(axis_size, vals_in, dims_in, *, jaxpr, num_consts):
  del num_consts  # Unused
  new_jaxpr, new_consts = vmap_jaxpr(jaxpr, axis_size, tuple(dims_in))
  outs = bind(xla_call_p, *new_consts, *vals_in, jaxpr=new_jaxpr,
              num_consts=len(new_consts))
  return outs, [0] * len(outs)
vmap_rules[xla_call_p] = xla_call_vmap_rule

@lru_cache
def vmap_jaxpr(jaxpr: Jaxpr, axis_size: int, bdims_in: tuple[BatchAxis, ...]
               ) -> tuple[Jaxpr, list[Any]]:
  vmap_traceable = vmap(jaxpr_as_fun(jaxpr), tuple(bdims_in))
  in_avals = [unmapped_aval(axis_size, d, v.aval)
              for v, d in zip(jaxpr.in_binders, bdims_in)]
  new_jaxpr, new_consts, _ = make_jaxpr(vmap_traceable, *in_avals)
  return new_jaxpr, new_consts

def unmapped_aval(axis_size: int, batch_dim: BatchAxis, aval: ShapedArray
                  ) -> ShapedArray:
  if batch_dim is not_mapped:
    return aval
  else:
    shape = list(aval.shape)
    shape.insert(batch_dim, axis_size)
    return ShapedArray(tuple(shape), aval.dtype)

def xla_call_abstract_eval_rule(*in_types, jaxpr, num_consts):
  del num_consts  # Unused
  jaxpr_type = typecheck_jaxpr(jaxpr)
  if not all(t1 == t2 for t1, t2 in zip(jaxpr_type.in_types, in_types)):
    raise TypeError
  return jaxpr_type.out_types
abstract_eval_rules[xla_call_p] = xla_call_abstract_eval_rule

def xla_call_translation(c, in_avals, out_avals, in_vals, *, jaxpr, num_consts):
  del num_consts, out_avals
  # Calling jaxpr_subcomp directly would inline. We generate a Call HLO instead.
  with ir.InsertionPoint(c.module.body):
    @func.func(*(aval_to_ir_type(aval) for aval in in_avals))
    def inner_xla_call(*params):
      return jaxpr_subcomp(c, jaxpr, params)
    name = c.symbol_table.insert(inner_xla_call.func_op)
  return func.CallOp(inner_xla_call.func_op, in_vals).results
hlo_translations[xla_call_p] = xla_call_translation

@jit
def f(x):
  print('tracing!')
  y = sin(x) * 2.
  z = - y + x
  return z

x, xdot = 3., 1.
y, ydot = jvp(f, (x,), (xdot,))
print(y)
print(ydot)

tracing!
2.7177599838802657
2.979984993200891

y, ydot = jvp(f, (x,), (xdot,))  # 'tracing!' not printed

ys = vmap(f, (0,))(np.arange(3.))
print(ys)

[ 0.         -0.68294197  0.18140515]

缺少的一个部分是数组的设备内存持久性。也就是说，我们已经定义了 handle_result 将结果作为 NumPy 数组传输回 CPU 内存，但通常最好避免传输结果，只是为了在下一次操作中将它们传输回来。我们可以通过引入 Array 类来实现这一点，它可以包装 XLA 缓冲区，并在其他方面进行鸭子类型化 numpy.ndarray。

def handle_result(aval: ShapedArray, buf):  # noqa: F811
  return Array(aval, buf)

class Array:
  buf: Any
  aval: ShapedArray

  def __init__(self, aval, buf):
    self.aval = aval
    self.buf = buf

  dtype = property(lambda self: self.aval.dtype)
  shape = property(lambda self: self.aval.shape)
  ndim  = property(lambda self: self.aval.ndim)

  def __array__(self): return np.asarray(self.buf)
  def __repr__(self):  return repr(np.asarray(self.buf))
  def __str__(self):   return str(np.asarray(self.buf))

  _neg = staticmethod(neg)
  _add = staticmethod(add)
  _radd = staticmethod(add)
  _mul = staticmethod(mul)
  _rmul = staticmethod(mul)
  _gt = staticmethod(greater)
  _lt = staticmethod(less)
input_handlers[Array] = lambda x: x.buf

jax_types.add(Array)

@jit
def f(x):
  y = sin(x) * 2.
  z = - y + x
  return z

x, xdot = 3., 1.
y, ydot = jvp(f, (x,), (xdot,))
print(y)
print(ydot)

2.7177599838802657
2.979984993200891

显示代码单元格来源 隐藏代码单元格来源

def pprint_xla_call(names: defaultdict[Var, str], eqn: JaxprEqn) -> PPrint:
  lhs = pp(' '.join(var_str(names, v) for v in eqn.out_binders))
  params_without_jaxpr = {k:v for k, v in eqn.params.items() if k != 'jaxpr'}
  rhs = (pp(eqn.primitive.name) >> pp_params(params_without_jaxpr) >>
         pp(' '.join(names[x] if isinstance(x, Var) else str(x.val)
                     for x in eqn.inputs)))
  return vcat([lhs >> pp(' = ') >> rhs,
               pp_jaxpr(eqn.params['jaxpr']).indent(2)])
pp_rules[xla_call_p] = pprint_xla_call

第 4 部分：linearize 和 vjp（以及 grad！）

linearize 和 vjp 自动微分函数构建于 jvp 之上，但也涉及 jaxpr。这是因为两者都涉及分阶段输出或延迟计算。

linearize

在 linearize 的情况下，我们希望分阶段输出 jvp 计算的线性部分。也就是说，用 类似 Haskell 的类型签名 来说，如果我们有 jvp : (a -> b) -> (a, T a) -> (b, T b)，那么我们写 linearize : (a -> b) -> a -> (b, T a -o T b)，使用 T a 来表示“a 的切线类型”，并使用“棒棒糖” -o 而不是箭头 -> 来表示线性函数。我们也用 jvp 定义 linearize 的语义

y, f_lin = linearize(f, x)
y_dot = f_lin(x_dot)

对于 (y, y_dot) 给出与以下代码相同的结果

y, y_dot = jvp(f, (x,), (x_dot,))

其中 f_lin 的应用不会重做任何线性化工作。我们将延迟的线性部分 f_lin : T a -o T b 表示为一个 jaxpr。

顺便说一句，既然我们有了线性箭头 -o，我们可以为 jvp 提供稍微更具信息量的类型

jvp : (a -> b) -> (UnrestrictedUse a, T a) -o (UnrestrictedUse b, T b)

在这里，我们写 UnrestrictedUse 只是为了表明我们有一个特殊的对，其中第一个元素可以以不受限制（非线性）的方式使用。结合线性箭头，这种表示法只是为了表达函数 jvp f 以非线性的方式使用其第一个输入，但以线性的方式使用其第二个输入，从而产生相应的非线性输出（可以以非线性的方式使用）并与线性输出配对。这种更精细的类型签名编码了 jvp f 中的数据依赖关系，这对于部分评估很有用。

要从 JVP 构建 f_lin jaxpr，我们需要执行部分评估：我们在追踪时评估所有原始值，但将切线计算分阶段输出到 jaxpr 中。这是我们构建 jaxpr 的第二种方式。但是，make_jaxpr 及其底层的 JaxprTrace/JaxprTracer 解释器旨在分阶段输出每个原语绑定，而第二种方法仅分阶段输出那些数据依赖于切线输入的原语绑定。

首先，是一些实用工具

def split_half(lst: list[Any]) -> tuple[list[Any], list[Any]]:
  assert not len(lst) % 2
  return split_list(lst, len(lst) // 2)

def merge_lists(which: list[bool], l1: list[Any], l2: list[Any]) -> list[Any]:
  l1, l2 = iter(l1), iter(l2)
  out = [next(l2) if b else next(l1) for b in which]
  assert next(l1, None) is next(l2, None) is None
  return out

接下来，我们将通过将 jvp 与通用的部分评估转换结合起来编写 linearize，接下来会添加该转换。

def linearize_flat(f, *primals_in):
  pvals_in = ([PartialVal.known(x) for x in primals_in] +
              [PartialVal.unknown(vspace(get_aval(x))) for x in primals_in])
  def f_jvp(*primals_tangents_in):
    primals_out, tangents_out = jvp(f, *split_half(primals_tangents_in))
    return [*primals_out, *tangents_out]
  jaxpr, pvals_out, consts = partial_eval_flat(f_jvp, pvals_in)
  primal_pvals, _ = split_half(pvals_out)
  assert all(pval.is_known for pval in primal_pvals)
  primals_out = [pval.const for pval in primal_pvals]
  f_lin = lambda *tangents: eval_jaxpr(jaxpr, [*consts, *tangents])
  return primals_out, f_lin

def linearize(f, *primals_in):
  primals_in_flat, in_tree = tree_flatten(primals_in)
  f, out_tree = flatten_fun(f, in_tree)
  primals_out_flat, f_lin_flat = linearize_flat(f, *primals_in_flat)
  primals_out = tree_unflatten(out_tree(), primals_out_flat)

  def f_lin(*tangents_in):
    tangents_in_flat, in_tree2 = tree_flatten(tangents_in)
    if in_tree != in_tree2: raise TypeError
    tangents_out_flat = f_lin_flat(*tangents_in_flat)
    return tree_unflatten(out_tree(), tangents_out_flat)

  return primals_out, f_lin

def vspace(aval: ShapedArray) -> ShapedArray:
  return raise_to_shaped(aval)  # TODO handle integers?

现在我们转向通用的部分评估转换。目标是接受一个 Python 可调用对象和输入列表（一些已知，一些未知），并生成：（1）可以从已知输入计算的所有输出，以及（2）一个 jaxpr，表示 Python 可调用对象计算的一部分，该部分只能在剩余输入已知后才能执行。

这种转换很难用类型签名来概括。如果我们假设输入函数的类型签名是 (a1, a2) -> (b1, b2)，其中 a1 和 a2 分别表示已知和未知输入，并且其中 b1 仅具有对 a1 的数据依赖性，而 b2 具有对 a2 的一些数据依赖性，那么我们可以写成

partial_eval : ((a1, a2) -> (b1, b2)) -> a1 -> exists r. (b1, r, (r, a2) -> b2)

用文字来说，给定类型为 a1 的输入值，partial_eval 生成类型为 b1 的输出，以及类型存在量化的“残差”值 r，表示在第二阶段完成计算所需的中间值。它还生成一个类型为 (r, a2) -> b2 的函数，该函数接受残差值以及剩余输入，并生成剩余输出。

我们喜欢将部分评估视为将一个计算“解压缩”为两个。例如，考虑这个 jaxpr

{ lambda a:float64[] .
  let b:float64[] = sin a
      c:float64[] = neg b
  in ( c ) }

JVP 的 jaxpr 看起来像

{ lambda a:float64[] b:float64[] .
  let c:float64[] = sin a
      d:float64[] = cos a
      e:float64[] = mul d b
      f:float64[] = neg c
      g:float64[] = neg e
  in ( f, g ) }

如果我们想象对这个 jaxpr 应用部分评估，其中第一个输入已知，第二个输入未知，我们最终会将 JVP jaxpr“解压缩”为原始 jaxpr 和切线 jaxpr

{ lambda a:float64[] .
  let c:float64[] = sin a
      d:float64[] = cos a
      f:float64[] = neg c
  in ( f, d ) }

{ lambda d:float64[] b:float64[] .
  let e:float64[] = mul d b
      g:float64[] = neg e
  in ( g ) }

第二个 jaxpr 表示我们希望从 linearize 获得的线性计算。

但是，与此 jaxpr 示例不同，我们希望在评估输入 Python 可调用对象时发生对已知值的计算。也就是说，与其为整个函数 (a1, a2) -> (b1, b2) 形成 jaxpr，首先分阶段输出 Python 中的所有操作，然后再弄清楚哪些可以立即评估，哪些必须延迟，我们只想为那些必须延迟的操作形成 jaxpr，因为它们依赖于未知输入。在自动微分的上下文中，此功能最终使我们能够处理诸如 grad(lambda x: x**2 if x > 0 else 0.) 之类的函数。Python 控制流之所以有效，是因为部分评估将原始计算保留在 Python 中。因此，我们的 Trace 和 Tracer 子类必须动态地分类哪些可以评估，哪些必须分阶段输出到 jaxpr 中。

首先，我们从 PartialVal 类开始，它表示一个可以是已知或未知的值

class PartialVal(NamedTuple):
  aval: ShapedArray
  const: Any | None

  @classmethod
  def known(cls, val: Any):
    return PartialVal(get_aval(val), val)

  @classmethod
  def unknown(cls, aval: ShapedArray):
    return PartialVal(aval, None)

  is_known   = property(lambda self: self.const is not None)
  is_unknown = property(lambda self: self.const is     None)

部分评估将接受表示输入的 PartialVal 列表，并返回 PartialVal 输出列表以及表示延迟计算的 jaxpr

def partial_eval_flat(f: Callable, pvals_in: list[PartialVal]
                      ) -> tuple[Jaxpr, list[PartialVal], list[Any]]:
  with new_main(PartialEvalTrace) as main:
    trace = PartialEvalTrace(main)
    tracers_in = [trace.new_arg(pval) for pval in pvals_in]
    outs = f(*tracers_in)
    tracers_out = [full_raise(trace, out) for out in outs]
    pvals_out = [t.pval for t in tracers_out]
    unk_tracers_in  = [t for t in tracers_in  if t.pval.is_unknown]
    unk_tracers_out = [t for t in tracers_out if t.pval.is_unknown]
    jaxpr, consts = tracers_to_jaxpr(unk_tracers_in, unk_tracers_out)
  return jaxpr, pvals_out, consts

接下来我们需要实现 PartialEvalTrace 及其 PartialEvalTracer。此解释器将在跟踪数据依赖性的同时动态构建 jaxpr。为此，它在 PartialEvalTracer 节点（表示分阶段输出的值）和 JaxprRecipe 节点（表示如何从其他值计算某些值的公式）之间构建二分有向无环图 (DAG)。一种配方是 JaxprEqnRecipe，对应于 JaxprEqn 的原语应用，但我们也有常量和 lambda 绑定器的配方类型。

from weakref import ref, ReferenceType

class LambdaBindingRecipe(NamedTuple):
  pass

class ConstRecipe(NamedTuple):
  val: Any

class JaxprEqnRecipe(NamedTuple):
  prim: Primitive
  tracers_in: list['PartialEvalTracer']
  params: dict[str, Any]
  avals_out: list[ShapedArray]
  tracer_refs_out: list['ReferenceType[PartialEvalTracer]']

JaxprRecipe = Union[LambdaBindingRecipe, ConstRecipe, JaxprEqnRecipe]

class PartialEvalTracer(Tracer):
  pval: PartialVal
  recipe: JaxprRecipe | None

  def __init__(self, trace, pval, recipe):
    self._trace = trace
    self.pval = pval
    self.recipe = recipe

  aval = property(lambda self: self.pval.aval)

  def full_lower(self):
    if self.pval.is_known:
      return full_lower(self.pval.const)
    return self

PartialEvalTrace 包含用于构建 JaxprRecipe 和 PartialEvalTracer 图的逻辑。每个参数对应于一个 LambdaBindingRecipe 叶节点，每个常量都是一个 ConstRecipe 叶节点，其中包含对常量的引用。所有其他 tracer 和配方都来自 process_primitive，它使用 JaxprEqnRecipe 形成 tracer。

对于大多数原语，process_primitive 逻辑很简单：如果所有输入都是已知的，那么我们可以将原语绑定到已知值（在 Python 中评估它）并避免形成对应于输出的 tracer。相反，如果任何输入是未知的，那么我们将其分阶段输出到一个 JaxprEqnRecipe 中，表示原语应用。要构建表示未知输出的 tracer，我们需要 avals，我们从抽象评估规则中获得它们。（请注意，tracer 引用 JaxprEqnRecipe，而 JaxprEqnRecipe 引用 tracer；我们通过使用 weakref 避免循环垃圾回收。）

该 process_primitive 逻辑适用于大多数原语，但 xla_call_p 需要递归处理。因此，我们在 partial_eval_rules 字典中特殊处理其规则。

class PartialEvalTrace(Trace):
  def new_arg(self, pval: PartialVal) -> Any:
    return PartialEvalTracer(self, pval, LambdaBindingRecipe())

  def lift(self, val: Any) -> PartialEvalTracer:
    return PartialEvalTracer(self, PartialVal.known(val), None)
  pure = lift

  def instantiate_const(self, tracer: PartialEvalTracer) -> PartialEvalTracer:
    if tracer.pval.is_unknown:
      return tracer
    else:
      pval = PartialVal.unknown(raise_to_shaped(tracer.aval))
      return PartialEvalTracer(self, pval, ConstRecipe(tracer.pval.const))

  def process_primitive(self, primitive, tracers, params):
    if all(t.pval.is_known for t in tracers):
      return bind(primitive, *map(full_lower, tracers), **params)
    rule = partial_eval_rules.get(primitive)
    if rule: return rule(self, tracers, **params)
    tracers_in = [self.instantiate_const(t) for t in tracers]
    avals_in = [t.aval for t in tracers_in]
    avals_out = abstract_eval_rules[primitive](*avals_in, **params)
    tracers_out = [PartialEvalTracer(self, PartialVal.unknown(aval), None)
                   for aval in avals_out]
    eqn = JaxprEqnRecipe(primitive, tracers_in, params, avals_out,
                         map(ref, tracers_out))
    for t in tracers_out: t.recipe = eqn
    return tracers_out

partial_eval_rules = {}

现在我们可以使用 PartialEvalTrace 构建 jaxpr 的图表示，我们需要一种机制将图表示转换为标准 jaxpr。jaxpr 对应于图的拓扑排序。

def tracers_to_jaxpr(tracers_in: list[PartialEvalTracer],
                     tracers_out: list[PartialEvalTracer]):
  tracer_to_var: dict[int, Var] = {id(t): Var(raise_to_shaped(t.aval))
                                   for t in tracers_in}
  constvar_to_val: dict[int, Any] = {}
  constid_to_var: dict[int, Var] = {}
  processed_eqns: set[int] = set()
  eqns: list[JaxprEqn] = []
  for t in toposort(tracers_out, tracer_parents):
    if isinstance(t.recipe, LambdaBindingRecipe):
      assert id(t) in set(map(id, tracers_in))
    elif isinstance(t.recipe, ConstRecipe):
      val = t.recipe.val
      var = constid_to_var.get(id(val))
      if var is None:
        aval = raise_to_shaped(get_aval(val))
        var = constid_to_var[id(val)] = Var(aval)
        constvar_to_val[var] = val
      tracer_to_var[id(t)] = var
    elif isinstance(t.recipe, JaxprEqnRecipe):
      if id(t.recipe) not in processed_eqns:
        eqns.append(recipe_to_eqn(tracer_to_var, t.recipe))
        processed_eqns.add(id(t.recipe))
    else:
      raise TypeError(t.recipe)

  constvars, constvals = unzip2(constvar_to_val.items())
  in_binders = constvars + [tracer_to_var[id(t)] for t in tracers_in]
  out_vars = [tracer_to_var[id(t)] for t in tracers_out]
  jaxpr = Jaxpr(in_binders, eqns, out_vars)
  typecheck_jaxpr(jaxpr)
  return jaxpr, constvals

def recipe_to_eqn(tracer_to_var: dict[int, Var], recipe: JaxprEqnRecipe
                  ) -> JaxprEqn:
  inputs = [tracer_to_var[id(t)] for t in recipe.tracers_in]
  out_binders = [Var(aval) for aval in recipe.avals_out]
  for t_ref, var in zip(recipe.tracer_refs_out, out_binders):
    if t_ref() is not None: tracer_to_var[id(t_ref())] = var
  return JaxprEqn(recipe.prim, inputs, recipe.params, out_binders)

def tracer_parents(t: PartialEvalTracer) -> list[PartialEvalTracer]:
  return t.recipe.tracers_in if isinstance(t.recipe, JaxprEqnRecipe) else []

显示代码单元格来源 隐藏代码单元格来源

def toposort(out_nodes: list[Any], parents: Callable[[Any], list[Any]]):
  if not out_nodes: return []
  out_nodes = remove_duplicates(out_nodes)

  child_counts = {}
  stack = list(out_nodes)
  while stack:
    node = stack.pop()
    if id(node) in child_counts:
      child_counts[id(node)] += 1
    else:
      child_counts[id(node)] = 1
      stack.extend(parents(node))
  for node in out_nodes:
    child_counts[id(node)] -= 1

  sorted_nodes = []
  childless_nodes = [node for node in out_nodes if not child_counts[id(node)]]
  while childless_nodes:
    node = childless_nodes.pop()
    sorted_nodes.append(node)
    for parent in parents(node):
      if child_counts[id(parent)] == 1:
        childless_nodes.append(parent)
      else:
        child_counts[id(parent)] -= 1

  sorted_nodes = sorted_nodes[::-1]
  check_toposort(sorted_nodes, parents)
  return sorted_nodes

def remove_duplicates(lst):
  seen = set()
  return [x for x in lst if id(x) not in seen and not seen.add(id(x))]

def check_toposort(nodes: list[Any], parents: Callable[[Any], list[Any]]):
  seen = set()
  for node in nodes:
    assert all(id(parent) in seen for parent in parents(node))
    seen.add(id(node))

现在我们可以线性化了！

y, sin_lin = linearize(sin, 3.)
print(y, sin(3.))
print(sin_lin(1.), cos(3.))

0.1411200080598672 0.1411200080598672
-0.9899924966004454 -0.9899924966004454

为了处理 linearize-of-jit，我们仍然需要为 xla_call_p 编写部分评估规则。除了 tracer 簿记之外，主要任务是对 jaxpr 执行部分评估，将其“解压缩”为两个 jaxpr。

实际上需要编写两个规则：一个用于追踪时部分评估，我们将其称为 xla_call_partial_eval，另一个用于 jaxpr 的部分评估，我们将其称为 xla_call_peval_eqn。

def xla_call_partial_eval(trace, tracers, *, jaxpr, num_consts):
  del num_consts  # Unused
  in_unknowns = [not t.pval.is_known for t in tracers]
  jaxpr1, jaxpr2, out_unknowns, num_res = partial_eval_jaxpr(jaxpr, in_unknowns)
  known_tracers, unknown_tracers = partition_list(in_unknowns, tracers)
  known_vals = [t.pval.const for t in known_tracers]
  outs1_res = bind(xla_call_p, *known_vals, jaxpr=jaxpr1, num_consts=0)
  outs1, res = split_list(outs1_res, len(jaxpr1.outs) - num_res)
  res_tracers = [trace.instantiate_const(full_raise(trace, x)) for x in res]
  outs2 = [PartialEvalTracer(trace, PartialVal.unknown(v.aval), None)
           for v in jaxpr2.outs]
  eqn = JaxprEqnRecipe(xla_call_p, res_tracers + unknown_tracers,
                       dict(jaxpr=jaxpr2, num_consts=0),
                       [v.aval for v in jaxpr2.outs], map(ref, outs2))
  for t in outs2: t.recipe = eqn
  return merge_lists(out_unknowns, outs1, outs2)
partial_eval_rules[xla_call_p] = xla_call_partial_eval

def partial_eval_jaxpr(jaxpr: Jaxpr, in_unknowns: list[bool],
                       instantiate: list[bool] | None = None,
                       ) -> tuple[Jaxpr, Jaxpr, list[bool], int]:
  env: dict[Var, bool] = {}
  residuals: set[Var] = set()

  def read(x: Atom) -> bool:
    return type(x) is Var and env[x]

  def write(unk: bool, v: Var) -> None:
    env[v] = unk

  def new_res(x: Atom) -> Atom:
    if type(x) is Var: residuals.add(x)
    return x

  eqns1, eqns2 = [], []
  map(write, in_unknowns, jaxpr.in_binders)
  for eqn in jaxpr.eqns:
    unks_in = map(read, eqn.inputs)
    rule = partial_eval_jaxpr_rules.get(eqn.primitive)
    if rule:
      eqn1, eqn2, unks_out, res = rule(unks_in, eqn)
      eqns1.append(eqn1); eqns2.append(eqn2); residuals.update(res)
      map(write, unks_out, eqn.out_binders)
    elif any(unks_in):
      inputs = [v if unk else new_res(v) for unk, v in zip(unks_in, eqn.inputs)]
      eqns2.append(JaxprEqn(eqn.primitive, inputs, eqn.params, eqn.out_binders))
      map(partial(write, True), eqn.out_binders)
    else:
      eqns1.append(eqn)
      map(partial(write, False), eqn.out_binders)
  out_unknowns = map(read, jaxpr.outs)
  if instantiate is not None:
    for v, uk, inst in zip(jaxpr.outs, out_unknowns, instantiate):
      if inst and not uk: new_res(v)
    out_unknowns = map(op.or_, out_unknowns, instantiate)

  residuals, num_res = list(residuals), len(residuals)
  assert all(type(v) is Var for v in residuals), residuals

  ins1, ins2 = partition_list(in_unknowns, jaxpr.in_binders)
  outs1, outs2 = partition_list(out_unknowns, jaxpr.outs)

  jaxpr1 = Jaxpr(ins1, eqns1, outs1 + residuals)
  jaxpr2 = Jaxpr(residuals + ins2, eqns2, outs2)
  typecheck_partial_eval_jaxpr(jaxpr, in_unknowns, out_unknowns, jaxpr1, jaxpr2)

  return jaxpr1, jaxpr2, out_unknowns, num_res

def typecheck_partial_eval_jaxpr(jaxpr, unks_in, unks_out, jaxpr1, jaxpr2):
  jaxprty = typecheck_jaxpr(jaxpr)    # (a1,  a2) -> (b1, b2 )
  jaxpr1ty = typecheck_jaxpr(jaxpr1)  #  a1       -> (b1, res)
  jaxpr2ty = typecheck_jaxpr(jaxpr2)  # (res, a2) -> b2

  a1, a2 = partition_list(unks_in, jaxprty.in_types)
  b1, b2 = partition_list(unks_out, jaxprty.out_types)
  b1_, res = split_list(jaxpr1ty.out_types, len(b1))
  res_, a2_ = split_list(jaxpr2ty.in_types, len(res))
  b2_ = jaxpr2ty.out_types

  if jaxpr1ty.in_types != a1: raise TypeError
  if jaxpr2ty.out_types != b2: raise TypeError
  if b1 != b1_: raise TypeError
  if res != res_: raise TypeError
  if a2 != a2_: raise TypeError
  if b2 != b2_: raise TypeError

partial_eval_jaxpr_rules = {}

def xla_call_peval_eqn(unks_in: list[bool], eqn: JaxprEqn,
                       ) -> tuple[JaxprEqn, JaxprEqn, list[bool], list[Var]]:
  jaxpr = eqn.params['jaxpr']
  jaxpr1, jaxpr2, unks_out, num_res = partial_eval_jaxpr(jaxpr, unks_in)
  ins1, ins2 = partition_list(unks_in, eqn.inputs)
  out_binders1, out_binders2 = partition_list(unks_out, eqn.out_binders)
  residuals = [Var(v.aval) for v in jaxpr2.in_binders[:num_res]]
  eqn1 = JaxprEqn(xla_call_p, ins1, dict(jaxpr=jaxpr1, num_consts=0),
                  out_binders1 + residuals)
  eqn2 = JaxprEqn(xla_call_p, residuals + ins2,
                  dict(jaxpr=jaxpr2, num_consts=0), out_binders2)
  return eqn1, eqn2, unks_out, residuals
partial_eval_jaxpr_rules[xla_call_p] = xla_call_peval_eqn

有了这些，我们可以随意组合 linearize 和 jit。

@jit
def f(x):
  y = sin(x) * 2.
  z = - y + x
  return z

y, f_lin = linearize(f, 3.)
y_dot = f_lin(1.)
print(y, y_dot)

2.7177599838802657 2.979984993200891

@jit
def f(x):
  y = sin(x) * 2.
  z = g(x, y)
  return z

@jit
def g(x, y):
  return cos(x) + y

y, f_lin = linearize(f, 3.)
y_dot = f_lin(1.)
print(y, y_dot)

-0.7077524804807109 -2.121105001260758

vjp 和 grad

vjp 转换的工作方式与 linearize 非常相似。它的类型签名是类似的

linearize : (a -> b) -> a -> (b, T a -o T b)
vjp       : (a -> b) -> a -> (b, T b -o T a)

唯一的区别是我们在返回线性计算部分之前对其进行转置，使其从类型 T a -o T b 变为类型 T b -o T a。也就是说，我们基本上将 vjp 实现为

def vjp(f, x):
  y, f_lin = linearize(f, x)
  f_vjp = lambda y_bar: transpose(f_lin)(y_bar)
  return y, f_vjp

由于我们以 jaxpr 的形式获得了线性计算，而不仅仅是 Python 可调用对象，因此我们可以将转置转换实现为 jaxpr 解释器。

def vjp_flat(f, *primals_in):
  pvals_in = ([PartialVal.known(x) for x in primals_in] +
              [PartialVal.unknown(vspace(get_aval(x))) for x in primals_in])
  primal_pvals_in, tangent_pvals_in = split_half(pvals_in)
  def f_jvp(*primals_tangents_in):
    primals_out, tangents_out = jvp(f, *split_half(primals_tangents_in))
    return [*primals_out, *tangents_out]
  jaxpr, pvals_out, consts = partial_eval_flat(f_jvp, pvals_in)  # linearize
  primal_pvals, _ = split_half(pvals_out)
  assert all(pval.is_known for pval in primal_pvals)
  primals_out = [pval.const for pval in primal_pvals]
  transpose_inputs = consts + [UndefPrimal(p.aval) for p in tangent_pvals_in]
  f_vjp = lambda *cts: eval_jaxpr_transposed(jaxpr, transpose_inputs, cts)
  return primals_out, f_vjp

def vjp(f, *primals_in):
  primals_in_flat, in_tree = tree_flatten(primals_in)
  f, out_tree = flatten_fun(f, in_tree)
  primals_out_flat, f_vjp_flat = vjp_flat(f, *primals_in_flat)
  primals_out = tree_unflatten(out_tree(), primals_out_flat)

  def f_vjp(*cotangents_out):
    cotangents_out_flat, _ = tree_flatten(cotangents_out)
    cotangents_in_flat = f_vjp_flat(*cotangents_out_flat)
    return tree_unflatten(in_tree, cotangents_in_flat)

  return primals_out, f_vjp

class UndefPrimal(NamedTuple):
  aval: ShapedArray

register_pytree_node(UndefPrimal,
                     lambda u: (u.aval, ()),
                     lambda aval, _: UndefPrimal(aval))

我们使用 UndefPrimal 实例来指示我们想要转置的参数。出现这些实例的原因是，通常，为了明确闭包值，我们希望将类型为 a -> b -o c 的函数转置为类型为 a -> c -o b 的函数。更一般地，函数相对于其为线性的输入可以分散在参数列表中。因此，我们使用 UndefPrimal 指示线性位置。我们将 UndefPrimal 注册为 pytree 节点，因为 pytree 机制提供了一种方便的方法来从参数列表中修剪掉这些占位符。

接下来，我们可以编写 eval_jaxpr_transposed，以及所有至少在一个参数中可以是线性的原语的转置规则。

# NB: the analogous function in JAX is called 'backward_pass'
def eval_jaxpr_transposed(jaxpr: Jaxpr, args: list[Any], cotangents: list[Any]
                          ) -> list[Any]:
  primal_env: dict[Var, Any] = {}
  ct_env: dict[Var, Any] = {}

  def read_primal(x: Atom) -> Any:
    return primal_env.get(x, UndefPrimal(x.aval)) if type(x) is Var else x.val

  def write_primal(v: Var, val: Any) -> None:
    if type(val) is not UndefPrimal:
      primal_env[v] = val

  def read_cotangent(v: Var) -> Any:
    return ct_env.pop(v, np.zeros(v.aval.shape, v.aval.dtype))

  def write_cotangent(x: Atom, val: Any):
    if type(x) is Var and val is not None:
      ct_env[x] = add(ct_env[x], val) if x in ct_env else val

  map(write_primal, jaxpr.in_binders, args)
  map(write_cotangent, jaxpr.outs, cotangents)
  for eqn in jaxpr.eqns[::-1]:
    primals_in = map(read_primal, eqn.inputs)
    cts_in = map(read_cotangent, eqn.out_binders)
    rule = transpose_rules[eqn.primitive]
    cts_out = rule(cts_in, *primals_in, **eqn.params)
    map(write_cotangent, eqn.inputs, cts_out)

  return [read_cotangent(v) for v, x in zip(jaxpr.in_binders, args)
          if type(x) is UndefPrimal]

transpose_rules = {}

def mul_transpose_rule(cts, x, y):
  z_bar, = cts
  assert (type(x) is UndefPrimal) ^ (type(y) is UndefPrimal)
  return [mul(z_bar, y), None] if type(x) is UndefPrimal else [None, mul(x, z_bar)]
transpose_rules[mul_p] = mul_transpose_rule

def neg_transpose_rule(cts, x):
  ybar, = cts
  assert type(x) is UndefPrimal
  return [neg(ybar)]
transpose_rules[neg_p] = neg_transpose_rule

def add_transpose_rule(cts, x, y):
  z_bar, = cts
  return [z_bar, z_bar]
transpose_rules[add_p] = add_transpose_rule

def reduce_sum_transpose_rule(cts, x, *, axis):
  y_bar, = cts
  return [broadcast(y_bar, x.aval.shape, axis)]
transpose_rules[reduce_sum_p] = reduce_sum_transpose_rule

def xla_call_transpose_rule(cts, *invals, jaxpr, num_consts):
  del num_consts  # Unused
  undef_primals = [type(x) is UndefPrimal for x in invals]
  transposed_jaxpr, new_consts = transpose_jaxpr(jaxpr, tuple(undef_primals))
  residuals, _ = partition_list(undef_primals, invals)
  outs = bind(xla_call_p, *new_consts, *residuals, *cts,
              jaxpr=transposed_jaxpr, num_consts=len(new_consts))
  outs = iter(outs)
  return [next(outs) if undef else None for undef in undef_primals]
transpose_rules[xla_call_p] = xla_call_transpose_rule

@lru_cache
def transpose_jaxpr(jaxpr: Jaxpr, undef_primals: tuple[bool, ...]
                    ) -> tuple[Jaxpr, list[Any]]:
  avals_in, avals_out = typecheck_jaxpr(jaxpr)
  traceable = partial(eval_jaxpr_transposed, jaxpr)
  args = [UndefPrimal(a) if u else a for a, u in zip(avals_in, undef_primals)]
  trans_jaxpr, consts, _ = make_jaxpr(traceable, tuple(args), tuple(avals_out))
  typecheck_jaxpr(trans_jaxpr)
  return trans_jaxpr, consts

现在我们可以线性化和转置了，我们终于可以编写 grad 了。

def grad(f):
  def gradfun(x, *xs):
    y, f_vjp = vjp(f, x, *xs)
    if np.shape(y) != (): raise TypeError
    x_bar, *_ = f_vjp(np.ones(np.shape(y), np.result_type(y)))
    return x_bar
  return gradfun

y, f_vjp = vjp(sin, 3.)
print(f_vjp(1.), cos(3.))

(np.float64(-0.9899924966004454),) -0.9899924966004454

def f(x):
  y = sin(x) * 2.
  z = - y + x
  return z

print(grad(f)(3.))

2.979984993200891

@jit
def f(x):
  y = x * 2.
  z = g(y)
  return z

@jit
def g(x):
  return cos(x) * 2.

print(grad(f)(3.))

1.1176619927957034

这是一个组合性压力测试

# from core_test.py fun_with_nested_calls_2
def foo(x):
  @jit
  def bar(y):
    def baz(w):
      q = jit(lambda x: y)(x)
      q = q + jit(lambda: y)()
      q = q + jit(lambda y: w + y)(y)
      q = jit(lambda w: jit(sin)(x) * y)(1.0) + q
      return q
    p, t = jvp(baz, (x + 1.0,), (y,))
    return t + (x * p)
  return bar(x)

def assert_allclose(*vals):
  for v1, v2 in zip(vals[:-1], vals[1:]):
    np.testing.assert_allclose(v1, v2)

ans1 = f(3.)
ans2 = jit(f)(3.)
ans3, _ = jvp(f, (3.,), (5.,))
ans4, _ = jvp(jit(f), (3.,), (5.,))
assert_allclose(ans1, ans2, ans3, ans4)

deriv1 = grad(f)(3.)
deriv2 = grad(jit(f))(3.)
deriv3 = jit(grad(jit(f)))(3.)
_, deriv4 = jvp(f, (3.,), (1.,))
_, deriv5 = jvp(jit(f), (3.,), (1.,))
assert_allclose(deriv1, deriv2, deriv3, deriv4, deriv5)

hess1 = grad(grad(f))(3.)
hess2 = grad(grad(jit(f)))(3.)
hess3 = grad(jit(grad(f)))(3.)
hess4 = jit(grad(grad(f)))(3.)
_, hess5 = jvp(grad(f), (3.,), (1.,))
_, hess6 = jvp(jit(grad(f)), (3.,), (1.,))
_, hess7 = jvp(jit(grad(f)), (3.,), (1.,))
assert_allclose(hess1, hess2, hess3, hess4, hess5, hess6, hess7)

第 5 部分：控制流原语 cond

接下来，我们将为分阶段输出的控制流添加高阶原语。这些原语类似于第 3 部分中的 jit（另一个高阶原语），但不同之处在于它们由多个可调用对象而不是仅由一个可调用对象参数化。

添加 cond

我们引入一个 cond 原语来表示在 jaxpr 内部有条件地应用一个函数或另一个函数。我们将 cond 的类型写为 Bool -> (a -> b) -> (a -> b) -> a -> b。换句话说，cond 接受一个布尔值（表示谓词）和两个类型相同的函数。根据谓词的值，它将一个或另一个函数应用于其最终参数。

在 Python 中，我们将其表示为一个函数，该函数本身接受两个函数作为参数。与 jit 一样，第一步是对其可调用参数调用 make_jaxpr，以将它们转换为 jaxpr。

def cond(pred, true_fn, false_fn, *operands):
  avals_in = [raise_to_shaped(get_aval(x)) for x in operands]
  true_jaxpr, true_consts, out_tree = make_jaxpr(true_fn, *avals_in)
  false_jaxpr, false_consts, out_tree_ = make_jaxpr(false_fn, *avals_in)
  if out_tree != out_tree_: raise TypeError
  true_jaxpr, false_jaxpr = _join_jaxpr_consts(
      true_jaxpr, false_jaxpr, len(true_consts), len(false_consts))
  if typecheck_jaxpr(true_jaxpr) != typecheck_jaxpr(false_jaxpr):
    raise TypeError
  outs = bind_cond(pred, *true_consts, *false_consts, *operands,
                   true_jaxpr=true_jaxpr, false_jaxpr=false_jaxpr)
  return tree_unflatten(out_tree, outs)
cond_p = Primitive('cond')

def _join_jaxpr_consts(jaxpr1: Jaxpr, jaxpr2: Jaxpr, n1: int, n2: int
                       ) -> tuple[Jaxpr, Jaxpr]:
  jaxpr1_type, jaxpr2_type = typecheck_jaxpr(jaxpr1), typecheck_jaxpr(jaxpr2)
  assert jaxpr1_type.in_types[n1:] == jaxpr2_type.in_types[n2:]
  consts1, rest1 = split_list(jaxpr1.in_binders, n1)
  consts2, rest2 = split_list(jaxpr2.in_binders, n2)
  new_jaxpr1 = Jaxpr(consts1 + consts2 + rest1, jaxpr1.eqns, jaxpr1.outs)
  new_jaxpr2 = Jaxpr(consts1 + consts2 + rest2, jaxpr2.eqns, jaxpr2.outs)
  return new_jaxpr1, new_jaxpr2

def bind_cond(pred, *args, true_jaxpr, false_jaxpr):
  assert len(args) == len(true_jaxpr.in_binders) == len(false_jaxpr.in_binders)
  return bind(cond_p, pred, *args, true_jaxpr=true_jaxpr, false_jaxpr=false_jaxpr)

我们要求 true_jaxpr 和 false_jaxpr 具有相同的类型，但由于它们可能闭包不同的常量（并且由于 jaxpr 只能表示闭合项，即不能有自由变量，而是经过闭包转换），我们需要使用辅助函数 _join_jaxpr_consts 使两个 jaxpr 的输入绑定器列表保持一致。（为了更经济，我们可以尝试识别具有相同形状的常量对，但我们只是连接常量列表。）

接下来，我们可以转向为 cond 添加解释器规则。其评估规则很简单。

def cond_impl(pred, *operands, true_jaxpr, false_jaxpr):
  if pred:
    return eval_jaxpr(true_jaxpr, operands)
  else:
    return eval_jaxpr(false_jaxpr, operands)
impl_rules[cond_p] = cond_impl

out = cond(True, lambda: 3, lambda: 4)
print(out)

3

对于其 JVP 和 vmap 规则，我们只需要调用我们为 jit 创建的相同的 jvp_jaxpr 和 vmap_jaxpr 实用程序，然后再次传递 _join_jaxpr_consts。

def cond_jvp_rule(primals, tangents, *, true_jaxpr, false_jaxpr):
  pred, *primals = primals
  _   , *tangents = tangents
  true_jaxpr , true_consts  = jvp_jaxpr(true_jaxpr)
  false_jaxpr, false_consts = jvp_jaxpr(false_jaxpr)
  true_jaxpr, false_jaxpr = _join_jaxpr_consts(
      true_jaxpr, false_jaxpr, len(true_consts), len(false_consts))
  assert typecheck_jaxpr(true_jaxpr) == typecheck_jaxpr(false_jaxpr)
  outs = bind_cond(pred, *true_consts, *false_consts, *primals, *tangents,
                   true_jaxpr=true_jaxpr, false_jaxpr=false_jaxpr)
  primals_out, tangents_out = split_half(outs)
  return primals_out, tangents_out
jvp_rules[cond_p] = cond_jvp_rule

out, out_tan = jvp(lambda x: cond(True, lambda: x * x, lambda: 0.), (1.,), (1.,))
print(out_tan)

2.0

def cond_vmap_rule(axis_size, vals_in, dims_in, *, true_jaxpr, false_jaxpr):
  pred    , *vals_in = vals_in
  pred_dim, *dims_in = dims_in
  if pred_dim is not not_mapped: raise NotImplementedError  # TODO
  true_jaxpr, true_consts = vmap_jaxpr(true_jaxpr, axis_size, tuple(dims_in))
  false_jaxpr, false_consts = vmap_jaxpr(false_jaxpr, axis_size, tuple(dims_in))
  true_jaxpr, false_jaxpr = _join_jaxpr_consts(
      true_jaxpr, false_jaxpr, len(true_consts), len(false_consts))
  assert typecheck_jaxpr(true_jaxpr) == typecheck_jaxpr(false_jaxpr)
  outs = bind_cond(pred, *true_consts, *false_consts, *vals_in,
                   true_jaxpr=true_jaxpr, false_jaxpr=false_jaxpr)
  return outs, [0] * len(outs)
vmap_rules[cond_p] = cond_vmap_rule

xs = np.array([1., 2., 3])
out = vmap(lambda x: cond(True, lambda: x + 1., lambda: 0.), (0,))(xs)
print(out)

[2. 3. 4.]

请注意，我们目前不支持谓词值本身是批量处理的情况。在主线 JAX 中，我们通过将条件转换为 select 原语 来处理这种情况。只要 true_fun 和 false_fun 不涉及任何副作用原语，这种转换在语义上是正确的。

这里未表示的另一件事（但在主线 JAX 中存在）是，对两个类型相同的 jaxpr 应用转换可能会导致类型不同的 jaxpr。例如，将主线 JAX 版本的 vmap_jaxpr 应用于恒等函数 jaxpr

{ lambda a:float32[] .
  let
  in ( a ) }

如果批量大小为 10，则会导致具有批量输出的 jaxpr，类型为 [float32[10]] -> [float32[10]]，而将其应用于零函数 jaxpr

{ lambda a:float32[] .
  let
  in ( 0. ) }

会导致具有非批量输出的 jaxpr，类型为 [float32[10]] -> [float32[]]。这是一种优化，旨在避免不必要地批量处理值。但这意味着在 cond 中，我们需要一个额外的步骤来连接两个转换后的 jaxpr，以使其具有一致的输出类型。我们不需要在此处执行此步骤，因为我们选择 vmap_jaxpr 始终批量处理前导轴上的所有输出。

接下来，我们可以转向抽象评估和 XLA 降低规则。

def cond_abstract_eval(pred_type, *in_types, true_jaxpr, false_jaxpr):
  if pred_type != ShapedArray((), np.dtype('bool')): raise TypeError
  jaxpr_type = typecheck_jaxpr(true_jaxpr)
  if jaxpr_type != typecheck_jaxpr(false_jaxpr):
    raise TypeError
  if not all(t1 == t2 for t1, t2 in zip(jaxpr_type.in_types, in_types)):
    raise TypeError
  return jaxpr_type.out_types
abstract_eval_rules[cond_p] = cond_abstract_eval

def cond_translation(c, in_avals, out_avals, in_vals, *, true_jaxpr, false_jaxpr):
  del in_avals  # Unused
  pred, *in_vals = in_vals

  op = hlo.IfOp([aval_to_ir_type(aval) for aval in out_avals], pred)
  with ir.InsertionPoint(op.true_branch.blocks.append()):
    hlo.return_(jaxpr_subcomp(c, true_jaxpr, in_vals))
  with ir.InsertionPoint(op.false_branch.blocks.append()):
    hlo.return_(jaxpr_subcomp(c, false_jaxpr, in_vals))
  return op.results

hlo_translations[cond_p] = cond_translation

out = jit(lambda: cond(False, lambda: 1, lambda: 2))()
print(out)

2

最后，为了支持反向模式自动微分，我们需要部分评估和转置规则。对于部分评估，我们需要引入另一个 jaxpr 操作实用程序 _join_jaxpr_res，以处理以下事实：对 true_fun 和 false_fun 应用部分评估通常会导致不同的残差。我们使用 _join_jaxpr_res 使转换后的 jaxpr 的输出类型保持一致（而 _join_jaxpr_consts 处理输入类型）。

def cond_partial_eval(trace, tracers, *, true_jaxpr, false_jaxpr):
  pred_tracer, *tracers = tracers
  assert pred_tracer.pval.is_known
  pred = pred_tracer.pval.const
  in_uks = [not t.pval.is_known for t in tracers]

  *jaxprs, out_uks, num_res = _cond_partial_eval(true_jaxpr, false_jaxpr, in_uks)
  t_jaxpr1, f_jaxpr1, t_jaxpr2, f_jaxpr2 = jaxprs

  known_tracers, unknown_tracers = partition_list(in_uks, tracers)
  known_vals = [t.pval.const for t in known_tracers]
  outs1_res = bind_cond(pred, *known_vals,
                        true_jaxpr=t_jaxpr1, false_jaxpr=f_jaxpr1)
  outs1, res = split_list(outs1_res, len(outs1_res) - num_res)
  pred_tracer_ = trace.instantiate_const(full_raise(trace, pred_tracer))
  res_tracers = [trace.instantiate_const(full_raise(trace, x)) for x in res]
  outs2 = [PartialEvalTracer(trace, PartialVal.unknown(v.aval), None)
           for v in t_jaxpr2.outs]
  eqn = JaxprEqnRecipe(cond_p, [pred_tracer_, *res_tracers, *unknown_tracers],
                       dict(true_jaxpr=t_jaxpr2, false_jaxpr=f_jaxpr2),
                       [v.aval for v in t_jaxpr2.outs], map(ref, outs2))
  for t in outs2: t.recipe = eqn
  return merge_lists(out_uks, outs1, outs2)
partial_eval_rules[cond_p] = cond_partial_eval

def _cond_partial_eval(true_jaxpr: Jaxpr, false_jaxpr: Jaxpr, in_uks: list[bool]
                       ) -> tuple[Jaxpr, Jaxpr, Jaxpr, Jaxpr, list[bool], int]:
  _, _, t_out_uks, _ = partial_eval_jaxpr(true_jaxpr , in_uks)
  _, _, f_out_uks, _ = partial_eval_jaxpr(false_jaxpr, in_uks)
  out_uks = map(op.or_, t_out_uks, f_out_uks)

  t_jaxpr1, t_jaxpr2, _, t_nres = partial_eval_jaxpr(true_jaxpr , in_uks, out_uks)
  f_jaxpr1, f_jaxpr2, _, f_nres = partial_eval_jaxpr(false_jaxpr, in_uks, out_uks)

  t_jaxpr1, f_jaxpr1 = _join_jaxpr_res(t_jaxpr1, f_jaxpr1, t_nres, f_nres)
  t_jaxpr2, f_jaxpr2 = _join_jaxpr_consts(t_jaxpr2, f_jaxpr2, t_nres, f_nres)
  assert typecheck_jaxpr(t_jaxpr1) == typecheck_jaxpr(f_jaxpr1)
  assert typecheck_jaxpr(t_jaxpr2) == typecheck_jaxpr(f_jaxpr2)
  num_res = t_nres + f_nres

  return t_jaxpr1, f_jaxpr1, t_jaxpr2, f_jaxpr2, out_uks, num_res

def _join_jaxpr_res(jaxpr1: Jaxpr, jaxpr2: Jaxpr, n1: int, n2: int
                    ) -> tuple[Jaxpr, Jaxpr]:
  jaxpr1_type, jaxpr2_type = typecheck_jaxpr(jaxpr1), typecheck_jaxpr(jaxpr2)
  out_types1, _ = split_list(jaxpr1_type.out_types, len(jaxpr1.outs) - n1)
  out_types2, _ = split_list(jaxpr2_type.out_types, len(jaxpr2.outs) - n2)
  assert out_types1 == out_types2
  outs1, res1 = split_list(jaxpr1.outs, len(jaxpr1.outs) - n1)
  outs2, res2 = split_list(jaxpr2.outs, len(jaxpr2.outs) - n2)
  zeros_like1 = [Lit(np.zeros(v.aval.shape, v.aval.dtype)) for v in res1]
  zeros_like2 = [Lit(np.zeros(v.aval.shape, v.aval.dtype)) for v in res2]
  new_jaxpr1 = Jaxpr(jaxpr1.in_binders, jaxpr1.eqns, outs1 + res1 + zeros_like2)
  new_jaxpr2 = Jaxpr(jaxpr2.in_binders, jaxpr2.eqns, outs2 + zeros_like1 + res2)
  return new_jaxpr1, new_jaxpr2

_, f_lin = linearize(lambda x: cond(True, lambda: x, lambda: 0.), 1.)
out = f_lin(3.14)
print(out)

3.14

def cond_peval_eqn(unks_in: list[bool], eqn: JaxprEqn,
                   ) -> tuple[JaxprEqn, JaxprEqn, list[bool], list[Atom]]:
  pred_unk, *unks_in = unks_in
  assert not pred_unk
  true_jaxpr, false_jaxpr = eqn.params['true_jaxpr'], eqn.params['false_jaxpr']
  *jaxprs, unks_out, num_res = _cond_partial_eval(true_jaxpr, false_jaxpr, unks_in)
  t_jaxpr1, f_jaxpr1, t_jaxpr2, f_jaxpr2 = jaxprs
  ins1, ins2 = partition_list(unks_in, eqn.inputs[1:])
  outs1, outs2 = partition_list(unks_out, eqn.out_binders)
  residuals, _ = split_list(t_jaxpr2.in_binders, num_res)
  eqn1 = JaxprEqn(cond_p, [eqn.inputs[0], *ins1],
                  dict(true_jaxpr=t_jaxpr1, false_jaxpr=f_jaxpr1),
                  outs1 + residuals)
  eqn2 = JaxprEqn(cond_p, [eqn.inputs[0], *residuals, *ins2],
                  dict(true_jaxpr=t_jaxpr2, false_jaxpr=f_jaxpr2),
                  outs2)
  res = [eqn.inputs[0], *residuals] if type(eqn.inputs[0]) is Var else residuals
  return eqn1, eqn2, unks_out, res
partial_eval_jaxpr_rules[cond_p] = cond_peval_eqn

_, f_lin = linearize(jit(lambda x: cond(True, lambda: x, lambda: 0.)), 1.)
out = f_lin(3.14)
print(out)

3.14

转置是 transpose_jaxpr 的相当直接的应用。

def cond_transpose_rule(cts, pred, *invals, true_jaxpr, false_jaxpr):
  undef_primals = tuple(type(x) is UndefPrimal for x in invals)
  true_jaxpr, true_consts = transpose_jaxpr(true_jaxpr, undef_primals)
  false_jaxpr, false_consts = transpose_jaxpr(false_jaxpr, undef_primals)
  true_jaxpr, false_jaxpr = _join_jaxpr_consts(
      true_jaxpr, false_jaxpr, len(true_consts), len(false_consts))
  res = [x for x in invals if type(x) is not UndefPrimal]
  outs = bind_cond(pred, *true_consts, *false_consts, *res, *cts,
                   true_jaxpr=true_jaxpr, false_jaxpr=false_jaxpr)
  outs = iter(outs)
  return [None] + [next(outs) if type(x) is UndefPrimal else None for x in invals]
transpose_rules[cond_p] = cond_transpose_rule

out = grad(lambda x: cond(True, lambda: x * x, lambda: 0.))(1.)
print(out)

2.0

显示代码单元格来源 隐藏代码单元格来源

def pprint_cond(names: defaultdict[Var, str], eqn: JaxprEqn) -> PPrint:
  true_jaxpr, false_jaxpr = eqn.params['true_jaxpr'], eqn.params['false_jaxpr']
  new_params = {k:v for k, v in eqn.params.items() if not k.endswith('jaxpr')}
  lhs = pp(' '.join(var_str(names, v) for v in eqn.out_binders))
  rhs = (pp(eqn.primitive.name) >> pp_params(new_params) >>
         pp(' '.join(names[x] if isinstance(x, Var) else str(x.val)
                     for x in eqn.inputs)))
  return vcat([lhs >> pp(' = ') >> rhs,
               pp_jaxpr(true_jaxpr).indent(2),
               pp_jaxpr(false_jaxpr).indent(2)])
pp_rules[cond_p] = pprint_cond