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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Custom derivative rules for JAX-transformable Python functions
mattjj@ Mar 19 2020, last updated Oct 14 2020
There are two ways to define differentiation rules in JAX
Step1:  Custom VJPs with jax.custom_vjp
Step2:  Example problems
To get an idea of what problems jax.custom_jvp and jax.custom_vjp are meant to solve, let's go over a few examples. A more thorough introduction to the jax.custom_jvp and jax.custom_vjp APIs is in the next section.
Numerical stability
One application of jax.custom_jvp is to improve the numerical stability of differentiation.
Say we want to write a function called log1pexp, which computes $x \mapsto \log ( 1 + e^x )$. We can write that using jax.numpy
Step3:  Since it's written in terms of jax.numpy, it's JAX-transformable
Step4:  But there's a numerical stability problem lurking here
Step5:  That doesn't seem right! After all, the derivative of $x \mapsto \log (1 + e^x)$ is $x \mapsto \frac{e^x}{1 + e^x}$, and so for large values of $x$ we'd expect the value to be about 1.
We can get a bit more insight into what's going on by looking at the jaxpr for the gradient computation
Step6:  Stepping through how the jaxpr would be evaluated, we can see that the last line would involve multiplying values that floating point math will round to 0 and $\infty$, respectively, which is never a good idea. That is, we're effectively evaluating lambda x
Step7:  Here's a defjvps convenience wrapper to express the same thing
Step8:  Enforcing a differentiation convention
A related application is to enforce a differentiation convention, perhaps at a boundary.
Consider the function $f 
Step9:  As a mathematical function on $\mathbb{R}$ (the full real line), $f$ is not differentiable at zero (because the limit defining the derivative doesn't exist from the left). Correspondingly, autodiff produces a nan value
Step10:  But mathematically if we think of $f$ as a function on $\mathbb{R}_+$ then it is differentiable at 0 [Rudin's Principles of Mathematical Analysis Definition 5.1, or Tao's Analysis I 3rd ed. Definition 10.1.1 and Example 10.1.6]. Alternatively, we might say as a convention we want to consider the directional derivative from the right. So there is a sensible value for the Python function grad(f) to return at 0.0, namely 1.0. By default, JAX's machinery for differentiation assumes all functions are defined over $\mathbb{R}$ and thus doesn't produce 1.0 here.
We can use a custom JVP rule! In particular, we can define the JVP rule in terms of the derivative function $x \mapsto \frac{\sqrt{x} + 2}{2(\sqrt{x} + 1)^2}$ on $\mathbb{R}_+$,
Step11:  Here's the convenience wrapper version
Step12:  Gradient clipping
While in some cases we want to express a mathematical differentiation computation, in other cases we may even want to take a step away from mathematics to adjust the computation autodiff performs. One canonical example is reverse-mode gradient clipping.
For gradient clipping, we can use jnp.clip together with a jax.custom_vjp reverse-mode-only rule
Step13:  Python debugging
Another application that is motivated by development workflow rather than numerics is to set a pdb debugger trace in the backward pass of reverse-mode autodiff.
When trying to track down the source of a nan runtime error, or just examine carefully the cotangent (gradient) values being propagated, it can be useful to insert a debugger at a point in the backward pass that corresponds to a specific point in the primal computation. You can do that with jax.custom_vjp.
We'll defer an example until the next section.
Implicit function differentiation of iterative implementations
This example gets pretty deep in the mathematical weeds!
Another application for jax.custom_vjp is reverse-mode differentiation of functions that are JAX-transformable (by jit, vmap, ...) but not efficiently JAX-differentiable for some reason, perhaps because they involve lax.while_loop. (It's not possible to produce an XLA HLO program that efficiently computes the reverse-mode derivative of an XLA HLO While loop because that would require a program with unbounded memory use, which isn't possible to express in XLA HLO, at least without side-effecting interactions through infeed/outfeed.)
For example, consider this fixed_point routine which computes a fixed point by iteratively applying a function in a while_loop
Step14:  This is an iterative procedure for numerically solving the equation $x = f(a, x)$ for $x$, by iterating $x_{t+1} = f(a, x_t)$ until $x_{t+1}$ is sufficiently close to $x_t$. The result $x^$ depends on the parameters $a$, and so we can think of there being a function $a \mapsto x^(a)$ that is implicitly defined by equation $x = f(a, x)$.
We can use fixed_point to run iterative procedures to convergence, for example running Newton's method to calculate square roots while only executing adds, multiplies, and divides
Step15:  We can vmap or jit the function as well
Step16:  We can't apply reverse-mode automatic differentiation because of the while_loop, but it turns out we wouldn't want to anyway
Step17:  We can check our answers by differentiating jnp.sqrt, which uses a totally different implementation
Step18:  A limitation to this approach is that the argument f can't close over any values involved in differentiation. That is, you might notice that we kept the parameter a explicit in the argument list of fixed_point. For this use case, consider using the low-level primitive lax.custom_root, which allows for deriviatives in closed-over variables with custom root-finding functions.
Basic usage of jax.custom_jvp and jax.custom_vjp APIs
Use jax.custom_jvp to define forward-mode (and, indirectly, reverse-mode) rules
Here's a canonical basic example of using jax.custom_jvp, where the comments use
Haskell-like type signatures
Step19:  In words, we start with a primal function f that takes inputs of type a and produces outputs of type b. We associate with it a JVP rule function f_jvp that takes a pair of inputs representing the primal inputs of type a and the corresponding tangent inputs of type T a, and produces a pair of outputs representing the primal outputs of type b and tangent outputs of type T b. The tangent outputs should be a linear function of the tangent inputs.
You can also use f.defjvp as a decorator, as in
```python
@custom_jvp
def f(x)
Step20:  For automatic transposition to work, the JVP rule's output tangents must be linear as a function of the input tangents. Otherwise a transposition error is raised.
Multiple arguments work like this
Step21:  The defjvps convenience wrapper lets us define a JVP for each argument separately, and the results are computed separately then summed
Step22:  Here's a defjvps example with multiple arguments
Step23:  As a shorthand, with defjvps you can pass a None value to indicate that the JVP for a particular argument is zero
Step24:  Calling a jax.custom_jvp function with keyword arguments, or writing a jax.custom_jvp function definition with default arguments, are both allowed so long as they can be unambiguously mapped to positional arguments based on the function signature retrieved by the standard library inspect.signature mechanism.
When you're not performing differentiation, the function f is called just as if it weren't decorated by jax.custom_jvp
Step25:  The custom JVP rule is invoked during differentiation, whether forward or reverse
Step26:  Notice that f_jvp calls f to compute the primal outputs. In the context of higher-order differentiation, each application of a differentiation transform will use the custom JVP rule if and only if the rule calls the original f to compute the primal outputs. (This represents a kind of fundamental tradeoff, where we can't make use of intermediate values from the evaluation of f in our rule and also have the rule apply in all orders of higher-order differentiation.)
Step27:  You can use Python control flow with jax.custom_jvp
Step28:  Use jax.custom_vjp to define custom reverse-mode-only rules
While jax.custom_jvp suffices for controlling both forward- and, via JAX's automatic transposition, reverse-mode differentiation behavior, in some cases we may want to directly control a VJP rule, for example in the latter two example problems presented above. We can do that with jax.custom_vjp
Step29:  In words, we again start with a primal function f that takes inputs of type a and produces outputs of type b. We associate with it two functions, f_fwd and f_bwd, which describe how to perform the forward- and backward-passes of reverse-mode autodiff, respectively.
The function f_fwd describes the forward pass, not only the primal computation but also what values to save for use on the backward pass. Its input signature is just like that of the primal function f, in that it takes a primal input of type a. But as output it produces a pair, where the first element is the primal output b and the second element is any "residual" data of type c to be stored for use by the backward pass. (This second output is analogous to PyTorch's save_for_backward mechanism.)
The function f_bwd describes the backward pass. It takes two inputs, where the first is the residual data of type c produced by f_fwd and the second is the output cotangents of type CT b corresponding to the output of the primal function. It produces an output of type CT a representing the cotangents corresponding to the input of the primal function. In particular, the output of f_bwd must be a sequence (e.g. a tuple) of length equal to the number of arguments to the primal function.
So multiple arguments work like this
Step30:  Calling a jax.custom_vjp function with keyword arguments, or writing a jax.custom_vjp function definition with default arguments, are both allowed so long as they can be unambiguously mapped to positional arguments based on the function signature retrieved by the standard library inspect.signature mechanism.
As with jax.custom_jvp, the custom VJP rule comprised by f_fwd and f_bwd is not invoked if differentiation is not applied. If function is evaluated, or transformed with jit, vmap, or other non-differentiation transformations, then only f is called.
Step31:  Forward-mode autodiff cannot be used on the jax.custom_vjp function and will raise an error
Step32:  If you want to use both forward- and reverse-mode, use jax.custom_jvp instead.
We can use jax.custom_vjp together with pdb to insert a debugger trace in the backward pass
Step33:  ```python
jax.grad(foo)(3.)
<ipython-input-113-b19a2dc1abf7>(12)debug_bwd()
-> return g
(Pdb) p x
DeviceArray(9., dtype=float32)
(Pdb) p g
DeviceArray(-0.91113025, dtype=float32)
(Pdb) q
```
More features and details
Working with list / tuple / dict containers (and other pytrees)
You should expect standard Python containers like lists, tuples, namedtuples, and dicts to just work, along with nested versions of those. In general, any pytrees are permissible, so long as their structures are consistent according to the type constraints. 
Here's a contrived example with jax.custom_jvp
Step34:  And an analogous contrived example with jax.custom_vjp
Step35:  Handling  non-differentiable arguments
Some use cases, like the final example problem, call for non-differentiable arguments like function-valued arguments to be passed to functions with custom differentiation rules, and for those arguments to also be passed to the rules themselves. In the case of fixed_point, the function argument f was such a non-differentiable argument. A similar situation arises with jax.experimental.odeint.
jax.custom_jvp with nondiff_argnums
Use the optional nondiff_argnums parameter to jax.custom_jvp to indicate arguments like these. Here's an example with jax.custom_jvp
Step36:  Notice the gotcha here
Step37:  jax.custom_vjp with nondiff_argnums
A similar option exists for jax.custom_vjp, and, similarly, the convention is that the non-differentiable arguments are passed as the first arguments to the _bwd rule, no matter where they appear in the signature of the original function. The signature of the _fwd rule remains unchanged - it is the same as the signature of the primal function. Here's an example
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Python Code:
import jax.numpy as jnp
from jax import custom_jvp
@custom_jvp
def f(x, y):
  return jnp.sin(x) * y
@f.defjvp
def f_jvp(primals, tangents):
  x, y = primals
  x_dot, y_dot = tangents
  primal_out = f(x, y)
  tangent_out = jnp.cos(x) * x_dot * y + jnp.sin(x) * y_dot
  return primal_out, tangent_out
from jax import jvp, grad
print(f(2., 3.))
y, y_dot = jvp(f, (2., 3.), (1., 0.))
print(y)
print(y_dot)
print(grad(f)(2., 3.))
# Equivalent alternative using the defjvps convenience wrapper
@custom_jvp
def f(x, y):
  return jnp.sin(x) * y
f.defjvps(lambda x_dot, primal_out, x, y: jnp.cos(x) * x_dot * y,
          lambda y_dot, primal_out, x, y: jnp.sin(x) * y_dot)
print(f(2., 3.))
y, y_dot = jvp(f, (2., 3.), (1., 0.))
print(y)
print(y_dot)
print(grad(f)(2., 3.))
Explanation: Custom derivative rules for JAX-transformable Python functions
mattjj@ Mar 19 2020, last updated Oct 14 2020
There are two ways to define differentiation rules in JAX:
using jax.custom_jvp and jax.custom_vjp to define custom differentiation rules for Python functions that are already JAX-transformable; and
defining new core.Primitive instances along with all their transformation rules, for example to call into functions from other systems like solvers, simulators, or general numerical computing systems.
This notebook is about #1. To read instead about #2, see the notebook on adding primitives.
For an introduction to JAX's automatic differentiation API, see The Autodiff Cookbook. This notebook assumes some familiarity with jax.jvp and jax.grad, and the mathematical meaning of JVPs and VJPs.
TL;DR
Custom JVPs with jax.custom_jvp
End of explanation
from jax import custom_vjp
@custom_vjp
def f(x, y):
  return jnp.sin(x) * y
def f_fwd(x, y):
# Returns primal output and residuals to be used in backward pass by f_bwd.
  return f(x, y), (jnp.cos(x), jnp.sin(x), y)
def f_bwd(res, g):
  cos_x, sin_x, y = res # Gets residuals computed in f_fwd
  return (cos_x * g * y, sin_x * g)
f.defvjp(f_fwd, f_bwd)
print(grad(f)(2., 3.))
Explanation: Custom VJPs with jax.custom_vjp
End of explanation
import jax.numpy as jnp
def log1pexp(x):
  return jnp.log(1. + jnp.exp(x))
log1pexp(3.)
Explanation: Example problems
To get an idea of what problems jax.custom_jvp and jax.custom_vjp are meant to solve, let's go over a few examples. A more thorough introduction to the jax.custom_jvp and jax.custom_vjp APIs is in the next section.
Numerical stability
One application of jax.custom_jvp is to improve the numerical stability of differentiation.
Say we want to write a function called log1pexp, which computes $x \mapsto \log ( 1 + e^x )$. We can write that using jax.numpy:
End of explanation
from jax import jit, grad, vmap
print(jit(log1pexp)(3.))
print(jit(grad(log1pexp))(3.))
print(vmap(jit(grad(log1pexp)))(jnp.arange(3.)))
Explanation: Since it's written in terms of jax.numpy, it's JAX-transformable:
End of explanation
print(grad(log1pexp)(100.))
Explanation: But there's a numerical stability problem lurking here:
End of explanation
from jax import make_jaxpr
make_jaxpr(grad(log1pexp))(100.)
Explanation: That doesn't seem right! After all, the derivative of $x \mapsto \log (1 + e^x)$ is $x \mapsto \frac{e^x}{1 + e^x}$, and so for large values of $x$ we'd expect the value to be about 1.
We can get a bit more insight into what's going on by looking at the jaxpr for the gradient computation:
End of explanation
from jax import custom_jvp
@custom_jvp
def log1pexp(x):
  return jnp.log(1. + jnp.exp(x))
@log1pexp.defjvp
def log1pexp_jvp(primals, tangents):
  x, = primals
  x_dot, = tangents
  ans = log1pexp(x)
  ans_dot = (1 - 1/(1 + jnp.exp(x))) * x_dot
  return ans, ans_dot
print(grad(log1pexp)(100.))
print(jit(log1pexp)(3.))
print(jit(grad(log1pexp))(3.))
print(vmap(jit(grad(log1pexp)))(jnp.arange(3.)))
Explanation: Stepping through how the jaxpr would be evaluated, we can see that the last line would involve multiplying values that floating point math will round to 0 and $\infty$, respectively, which is never a good idea. That is, we're effectively evaluating lambda x: (1 / (1 + jnp.exp(x))) * jnp.exp(x) for large x, which effectively turns into 0. * jnp.inf.
Instead of generating such large and small values, hoping for a cancellation that floats can't always provide, we'd rather just express the derivative function as a more numerically stable program. In particular, we can write a program that more closely evaluates the equal mathematical expression $1 - \frac{1}{1 + e^x}$, with no cancellation in sight.
This problem is interesting because even though our definition of log1pexp could already be JAX-differentiated (and transformed with jit, vmap, ...), we're not happy with the result of applying standard autodiff rules to the primitives comprising log1pexp and composing the result. Instead, we'd like to specify how the whole function log1pexp should be differentiated, as a unit, and thus arrange those exponentials better.
This is one application of custom derivative rules for Python functions that are already JAX transformable: specifying how a composite function should be differentiated, while still using its original Python definition for other transformations (like jit, vmap, ...).
Here's a solution using jax.custom_jvp:
End of explanation
@custom_jvp
def log1pexp(x):
  return jnp.log(1. + jnp.exp(x))
log1pexp.defjvps(lambda t, ans, x: (1 - 1/(1 + jnp.exp(x))) * t)
print(grad(log1pexp)(100.))
print(jit(log1pexp)(3.))
print(jit(grad(log1pexp))(3.))
print(vmap(jit(grad(log1pexp)))(jnp.arange(3.)))
Explanation: Here's a defjvps convenience wrapper to express the same thing:
End of explanation
def f(x):
  return x / (1 + jnp.sqrt(x))
Explanation: Enforcing a differentiation convention
A related application is to enforce a differentiation convention, perhaps at a boundary.
Consider the function $f : \mathbb{R}+ \mapsto \mathbb{R}+$ with $f(x) = \frac{x}{1 + \sqrt{x}}$, where we take $\mathbb{R}_+ = [0, \infty)$. We might implement $f$ as a program like this:
End of explanation
print(grad(f)(0.))
Explanation: As a mathematical function on $\mathbb{R}$ (the full real line), $f$ is not differentiable at zero (because the limit defining the derivative doesn't exist from the left). Correspondingly, autodiff produces a nan value:
End of explanation
@custom_jvp
def f(x):
  return x / (1 + jnp.sqrt(x))
@f.defjvp
def f_jvp(primals, tangents):
  x, = primals
  x_dot, = tangents
  ans = f(x)
  ans_dot = ((jnp.sqrt(x) + 2) / (2 * (jnp.sqrt(x) + 1)**2)) * x_dot
  return ans, ans_dot
print(grad(f)(0.))
Explanation: But mathematically if we think of $f$ as a function on $\mathbb{R}_+$ then it is differentiable at 0 [Rudin's Principles of Mathematical Analysis Definition 5.1, or Tao's Analysis I 3rd ed. Definition 10.1.1 and Example 10.1.6]. Alternatively, we might say as a convention we want to consider the directional derivative from the right. So there is a sensible value for the Python function grad(f) to return at 0.0, namely 1.0. By default, JAX's machinery for differentiation assumes all functions are defined over $\mathbb{R}$ and thus doesn't produce 1.0 here.
We can use a custom JVP rule! In particular, we can define the JVP rule in terms of the derivative function $x \mapsto \frac{\sqrt{x} + 2}{2(\sqrt{x} + 1)^2}$ on $\mathbb{R}_+$,
End of explanation
@custom_jvp
def f(x):
  return x / (1 + jnp.sqrt(x))
f.defjvps(lambda t, ans, x: ((jnp.sqrt(x) + 2) / (2 * (jnp.sqrt(x) + 1)**2)) * t)
print(grad(f)(0.))
Explanation: Here's the convenience wrapper version:
End of explanation
from functools import partial
from jax import custom_vjp
@custom_vjp
def clip_gradient(lo, hi, x):
  return x  # identity function
def clip_gradient_fwd(lo, hi, x):
  return x, (lo, hi)  # save bounds as residuals
def clip_gradient_bwd(res, g):
  lo, hi = res
  return (None, None, jnp.clip(g, lo, hi))  # use None to indicate zero cotangents for lo and hi
clip_gradient.defvjp(clip_gradient_fwd, clip_gradient_bwd)
import matplotlib.pyplot as plt
from jax import vmap
t = jnp.linspace(0, 10, 1000)
plt.plot(jnp.sin(t))
plt.plot(vmap(grad(jnp.sin))(t))
def clip_sin(x):
  x = clip_gradient(-0.75, 0.75, x)
  return jnp.sin(x)
plt.plot(clip_sin(t))
plt.plot(vmap(grad(clip_sin))(t))
Explanation: Gradient clipping
While in some cases we want to express a mathematical differentiation computation, in other cases we may even want to take a step away from mathematics to adjust the computation autodiff performs. One canonical example is reverse-mode gradient clipping.
For gradient clipping, we can use jnp.clip together with a jax.custom_vjp reverse-mode-only rule:
End of explanation
from jax.lax import while_loop
def fixed_point(f, a, x_guess):
  def cond_fun(carry):
    x_prev, x = carry
    return jnp.abs(x_prev - x) > 1e-6
  def body_fun(carry):
    _, x = carry
    return x, f(a, x)
  _, x_star = while_loop(cond_fun, body_fun, (x_guess, f(a, x_guess)))
  return x_star
Explanation: Python debugging
Another application that is motivated by development workflow rather than numerics is to set a pdb debugger trace in the backward pass of reverse-mode autodiff.
When trying to track down the source of a nan runtime error, or just examine carefully the cotangent (gradient) values being propagated, it can be useful to insert a debugger at a point in the backward pass that corresponds to a specific point in the primal computation. You can do that with jax.custom_vjp.
We'll defer an example until the next section.
Implicit function differentiation of iterative implementations
This example gets pretty deep in the mathematical weeds!
Another application for jax.custom_vjp is reverse-mode differentiation of functions that are JAX-transformable (by jit, vmap, ...) but not efficiently JAX-differentiable for some reason, perhaps because they involve lax.while_loop. (It's not possible to produce an XLA HLO program that efficiently computes the reverse-mode derivative of an XLA HLO While loop because that would require a program with unbounded memory use, which isn't possible to express in XLA HLO, at least without side-effecting interactions through infeed/outfeed.)
For example, consider this fixed_point routine which computes a fixed point by iteratively applying a function in a while_loop:
End of explanation
def newton_sqrt(a):
  update = lambda a, x: 0.5 * (x + a / x)
  return fixed_point(update, a, a)
print(newton_sqrt(2.))
Explanation: This is an iterative procedure for numerically solving the equation $x = f(a, x)$ for $x$, by iterating $x_{t+1} = f(a, x_t)$ until $x_{t+1}$ is sufficiently close to $x_t$. The result $x^$ depends on the parameters $a$, and so we can think of there being a function $a \mapsto x^(a)$ that is implicitly defined by equation $x = f(a, x)$.
We can use fixed_point to run iterative procedures to convergence, for example running Newton's method to calculate square roots while only executing adds, multiplies, and divides:
End of explanation
print(jit(vmap(newton_sqrt))(jnp.array([1., 2., 3., 4.])))
Explanation: We can vmap or jit the function as well:
End of explanation
from jax import vjp
@partial(custom_vjp, nondiff_argnums=(0,))
def fixed_point(f, a, x_guess):
  def cond_fun(carry):
    x_prev, x = carry
    return jnp.abs(x_prev - x) > 1e-6
  def body_fun(carry):
    _, x = carry
    return x, f(a, x)
  _, x_star = while_loop(cond_fun, body_fun, (x_guess, f(a, x_guess)))
  return x_star
def fixed_point_fwd(f, a, x_init):
  x_star = fixed_point(f, a, x_init)
  return x_star, (a, x_star)
def fixed_point_rev(f, res, x_star_bar):
  a, x_star = res
  _, vjp_a = vjp(lambda a: f(a, x_star), a)
  a_bar, = vjp_a(fixed_point(partial(rev_iter, f),
                             (a, x_star, x_star_bar),
                             x_star_bar))
  return a_bar, jnp.zeros_like(x_star)
  
def rev_iter(f, packed, u):
  a, x_star, x_star_bar = packed
  _, vjp_x = vjp(lambda x: f(a, x), x_star)
  return x_star_bar + vjp_x(u)[0]
fixed_point.defvjp(fixed_point_fwd, fixed_point_rev)
print(newton_sqrt(2.))
print(grad(newton_sqrt)(2.))
print(grad(grad(newton_sqrt))(2.))
Explanation: We can't apply reverse-mode automatic differentiation because of the while_loop, but it turns out we wouldn't want to anyway: instead of differentiating through the implementation of fixed_point and all its iterations, we can exploit the mathematical structure to do something that is much more memory-efficient (and FLOP-efficient in this case, too!). We can instead use the implicit function theorem [Prop A.25 of Bertsekas's Nonlinear Programming, 2nd ed.], which guarantees (under some conditions) the existence of the mathematical objects we're about to use. In essence, we linearize at the solution and solve those linear equations iteratively to compute the derivatives we want.
Consider again the equation $x = f(a, x)$ and the function $x^$. We want to evaluate vector-Jacobian products like $v^\mathsf{T} \mapsto v^\mathsf{T} \partial x^(a_0)$.
At least in an open neighborhood around the point $a_0$ at which we want to differentiate, let's assume that the equation $x^(a) = f(a, x^(a))$ holds for all $a$. Since the two sides are equal as functions of $a$, their derivatives must be equal as well, so let's differentiate both sides:
$\qquad \partial x^(a) = \partial_0 f(a, x^(a)) + \partial_1 f(a, x^(a))  \partial x^(a)$.
Setting $A = \partial_1 f(a_0, x^(a_0))$ and $B = \partial_0 f(a_0, x^(a_0))$, we can write the quantity we're after more simply as
$\qquad \partial x^(a_0) = B + A \partial x^(a_0)$,
or, by rearranging,
$\qquad \partial x^*(a_0) = (I - A)^{-1} B$.
That means we can evaluate vector-Jacobian products like
$\qquad v^\mathsf{T} \partial x^*(a_0) = v^\mathsf{T} (I - A)^{-1} B = w^\mathsf{T} B$,
where $w^\mathsf{T} = v^\mathsf{T} (I - A)^{-1}$, or equivalently $w^\mathsf{T} = v^\mathsf{T} + w^\mathsf{T} A$, or equivalently $w^\mathsf{T}$ is the fixed point of the map $u^\mathsf{T} \mapsto v^\mathsf{T} + u^\mathsf{T} A$. That last characterization gives us a way to write the VJP for fixed_point in terms of a call to fixed_point! Moreover, after expanding $A$ and $B$ back out, we can see we need only to evaluate VJPs of $f$ at $(a_0, x^*(a_0))$.
Here's the upshot:
End of explanation
print(grad(jnp.sqrt)(2.))
print(grad(grad(jnp.sqrt))(2.))
Explanation: We can check our answers by differentiating jnp.sqrt, which uses a totally different implementation:
End of explanation
from jax import custom_jvp
import jax.numpy as jnp
# f :: a -> b
@custom_jvp
def f(x):
  return jnp.sin(x)
# f_jvp :: (a, T a) -> (b, T b)
def f_jvp(primals, tangents):
  x, = primals
  t, = tangents
  return f(x), jnp.cos(x) * t
f.defjvp(f_jvp)
from jax import jvp
print(f(3.))
y, y_dot = jvp(f, (3.,), (1.,))
print(y)
print(y_dot)
Explanation: A limitation to this approach is that the argument f can't close over any values involved in differentiation. That is, you might notice that we kept the parameter a explicit in the argument list of fixed_point. For this use case, consider using the low-level primitive lax.custom_root, which allows for deriviatives in closed-over variables with custom root-finding functions.
Basic usage of jax.custom_jvp and jax.custom_vjp APIs
Use jax.custom_jvp to define forward-mode (and, indirectly, reverse-mode) rules
Here's a canonical basic example of using jax.custom_jvp, where the comments use
Haskell-like type signatures:
End of explanation
from jax import grad
print(grad(f)(3.))
print(grad(grad(f))(3.))
Explanation: In words, we start with a primal function f that takes inputs of type a and produces outputs of type b. We associate with it a JVP rule function f_jvp that takes a pair of inputs representing the primal inputs of type a and the corresponding tangent inputs of type T a, and produces a pair of outputs representing the primal outputs of type b and tangent outputs of type T b. The tangent outputs should be a linear function of the tangent inputs.
You can also use f.defjvp as a decorator, as in
```python
@custom_jvp
def f(x):
  ...
@f.defjvp
def f_jvp(primals, tangents):
  ...
```
Even though we defined only a JVP rule and no VJP rule, we can use both forward- and reverse-mode differentiation on f. JAX will automatically transpose the linear computation on tangent values from our custom JVP rule, computing the VJP as efficiently as if we had written the rule by hand:
End of explanation
@custom_jvp
def f(x, y):
  return x ** 2 * y
@f.defjvp
def f_jvp(primals, tangents):
  x, y = primals
  x_dot, y_dot = tangents
  primal_out = f(x, y)
  tangent_out = 2 * x * y * x_dot + x ** 2 * y_dot
  return primal_out, tangent_out
print(grad(f)(2., 3.))
Explanation: For automatic transposition to work, the JVP rule's output tangents must be linear as a function of the input tangents. Otherwise a transposition error is raised.
Multiple arguments work like this:
End of explanation
@custom_jvp
def f(x):
  return jnp.sin(x)
f.defjvps(lambda t, ans, x: jnp.cos(x) * t)
print(grad(f)(3.))
Explanation: The defjvps convenience wrapper lets us define a JVP for each argument separately, and the results are computed separately then summed:
End of explanation
@custom_jvp
def f(x, y):
  return x ** 2 * y
f.defjvps(lambda x_dot, primal_out, x, y: 2 * x * y * x_dot,
          lambda y_dot, primal_out, x, y: x ** 2 * y_dot)
print(grad(f)(2., 3.))
print(grad(f, 0)(2., 3.))  # same as above
print(grad(f, 1)(2., 3.))
Explanation: Here's a defjvps example with multiple arguments:
End of explanation
@custom_jvp
def f(x, y):
  return x ** 2 * y
f.defjvps(lambda x_dot, primal_out, x, y: 2 * x * y * x_dot,
          None)
print(grad(f)(2., 3.))
print(grad(f, 0)(2., 3.))  # same as above
print(grad(f, 1)(2., 3.))
Explanation: As a shorthand, with defjvps you can pass a None value to indicate that the JVP for a particular argument is zero:
End of explanation
@custom_jvp
def f(x):
  print('called f!')  # a harmless side-effect
  return jnp.sin(x)
@f.defjvp
def f_jvp(primals, tangents):
  print('called f_jvp!')  # a harmless side-effect
  x, = primals
  t, = tangents
  return f(x), jnp.cos(x) * t
from jax import vmap, jit
print(f(3.))
print(vmap(f)(jnp.arange(3.)))
print(jit(f)(3.))
Explanation: Calling a jax.custom_jvp function with keyword arguments, or writing a jax.custom_jvp function definition with default arguments, are both allowed so long as they can be unambiguously mapped to positional arguments based on the function signature retrieved by the standard library inspect.signature mechanism.
When you're not performing differentiation, the function f is called just as if it weren't decorated by jax.custom_jvp:
End of explanation
y, y_dot = jvp(f, (3.,), (1.,))
print(y_dot)
print(grad(f)(3.))
Explanation: The custom JVP rule is invoked during differentiation, whether forward or reverse:
End of explanation
grad(grad(f))(3.)
Explanation: Notice that f_jvp calls f to compute the primal outputs. In the context of higher-order differentiation, each application of a differentiation transform will use the custom JVP rule if and only if the rule calls the original f to compute the primal outputs. (This represents a kind of fundamental tradeoff, where we can't make use of intermediate values from the evaluation of f in our rule and also have the rule apply in all orders of higher-order differentiation.)
End of explanation
@custom_jvp
def f(x):
  if x > 0:
    return jnp.sin(x)
  else:
    return jnp.cos(x)
@f.defjvp
def f_jvp(primals, tangents):
  x, = primals
  x_dot, = tangents
  ans = f(x)
  if x > 0:
    return ans, 2 * x_dot
  else:
    return ans, 3 * x_dot
print(grad(f)(1.))
print(grad(f)(-1.))
Explanation: You can use Python control flow with jax.custom_jvp:
End of explanation
from jax import custom_vjp
import jax.numpy as jnp
# f :: a -> b
@custom_vjp
def f(x):
  return jnp.sin(x)
# f_fwd :: a -> (b, c)
def f_fwd(x):
  return f(x), jnp.cos(x)
# f_bwd :: (c, CT b) -> CT a
def f_bwd(cos_x, y_bar):
  return (cos_x * y_bar,)
f.defvjp(f_fwd, f_bwd)
from jax import grad
print(f(3.))
print(grad(f)(3.))
Explanation: Use jax.custom_vjp to define custom reverse-mode-only rules
While jax.custom_jvp suffices for controlling both forward- and, via JAX's automatic transposition, reverse-mode differentiation behavior, in some cases we may want to directly control a VJP rule, for example in the latter two example problems presented above. We can do that with jax.custom_vjp:
End of explanation
from jax import custom_vjp
@custom_vjp
def f(x, y):
  return jnp.sin(x) * y
def f_fwd(x, y):
  return f(x, y), (jnp.cos(x), jnp.sin(x), y)
def f_bwd(res, g):
  cos_x, sin_x, y = res
  return (cos_x * g * y, -sin_x * g)
f.defvjp(f_fwd, f_bwd)
print(grad(f)(2., 3.))
Explanation: In words, we again start with a primal function f that takes inputs of type a and produces outputs of type b. We associate with it two functions, f_fwd and f_bwd, which describe how to perform the forward- and backward-passes of reverse-mode autodiff, respectively.
The function f_fwd describes the forward pass, not only the primal computation but also what values to save for use on the backward pass. Its input signature is just like that of the primal function f, in that it takes a primal input of type a. But as output it produces a pair, where the first element is the primal output b and the second element is any "residual" data of type c to be stored for use by the backward pass. (This second output is analogous to PyTorch's save_for_backward mechanism.)
The function f_bwd describes the backward pass. It takes two inputs, where the first is the residual data of type c produced by f_fwd and the second is the output cotangents of type CT b corresponding to the output of the primal function. It produces an output of type CT a representing the cotangents corresponding to the input of the primal function. In particular, the output of f_bwd must be a sequence (e.g. a tuple) of length equal to the number of arguments to the primal function.
So multiple arguments work like this:
End of explanation
@custom_vjp
def f(x):
  print("called f!")
  return jnp.sin(x)
def f_fwd(x):
  print("called f_fwd!")
  return f(x), jnp.cos(x)
def f_bwd(cos_x, y_bar):
  print("called f_bwd!")
  return (cos_x * y_bar,)
f.defvjp(f_fwd, f_bwd)
print(f(3.))
print(grad(f)(3.))
from jax import vjp
y, f_vjp = vjp(f, 3.)
print(y)
print(f_vjp(1.))
Explanation: Calling a jax.custom_vjp function with keyword arguments, or writing a jax.custom_vjp function definition with default arguments, are both allowed so long as they can be unambiguously mapped to positional arguments based on the function signature retrieved by the standard library inspect.signature mechanism.
As with jax.custom_jvp, the custom VJP rule comprised by f_fwd and f_bwd is not invoked if differentiation is not applied. If function is evaluated, or transformed with jit, vmap, or other non-differentiation transformations, then only f is called.
End of explanation
from jax import jvp
try:
  jvp(f, (3.,), (1.,))
except TypeError as e:
  print('ERROR! {}'.format(e))
Explanation: Forward-mode autodiff cannot be used on the jax.custom_vjp function and will raise an error:
End of explanation
import pdb
@custom_vjp
def debug(x):
  return x  # acts like identity
def debug_fwd(x):
  return x, x
def debug_bwd(x, g):
  import pdb; pdb.set_trace()
  return g
debug.defvjp(debug_fwd, debug_bwd)
def foo(x):
  y = x ** 2
  y = debug(y)  # insert pdb in corresponding backward pass step
  return jnp.sin(y)
Explanation: If you want to use both forward- and reverse-mode, use jax.custom_jvp instead.
We can use jax.custom_vjp together with pdb to insert a debugger trace in the backward pass:
End of explanation
from collections import namedtuple
Point = namedtuple("Point", ["x", "y"])
@custom_jvp
def f(pt):
  x, y = pt.x, pt.y
  return {'a': x ** 2,
          'b': (jnp.sin(x), jnp.cos(y))}
@f.defjvp
def f_jvp(primals, tangents):
  pt, = primals
  pt_dot, =  tangents
  ans = f(pt)
  ans_dot = {'a': 2 * pt.x * pt_dot.x,
             'b': (jnp.cos(pt.x) * pt_dot.x, -jnp.sin(pt.y) * pt_dot.y)}
  return ans, ans_dot
def fun(pt):
  dct = f(pt)
  return dct['a'] + dct['b'][0]
pt = Point(1., 2.)
print(f(pt))
print(grad(fun)(pt))
Explanation: ```python
jax.grad(foo)(3.)
<ipython-input-113-b19a2dc1abf7>(12)debug_bwd()
-> return g
(Pdb) p x
DeviceArray(9., dtype=float32)
(Pdb) p g
DeviceArray(-0.91113025, dtype=float32)
(Pdb) q
```
More features and details
Working with list / tuple / dict containers (and other pytrees)
You should expect standard Python containers like lists, tuples, namedtuples, and dicts to just work, along with nested versions of those. In general, any pytrees are permissible, so long as their structures are consistent according to the type constraints. 
Here's a contrived example with jax.custom_jvp:
End of explanation
@custom_vjp
def f(pt):
  x, y = pt.x, pt.y
  return {'a': x ** 2,
          'b': (jnp.sin(x), jnp.cos(y))}
def f_fwd(pt):
  return f(pt), pt
def f_bwd(pt, g):
  a_bar, (b0_bar, b1_bar) = g['a'], g['b']
  x_bar = 2 * pt.x * a_bar + jnp.cos(pt.x) * b0_bar
  y_bar = -jnp.sin(pt.y) * b1_bar
  return (Point(x_bar, y_bar),)
f.defvjp(f_fwd, f_bwd)
def fun(pt):
  dct = f(pt)
  return dct['a'] + dct['b'][0]
pt = Point(1., 2.)
print(f(pt))
print(grad(fun)(pt))
Explanation: And an analogous contrived example with jax.custom_vjp:
End of explanation
from functools import partial
@partial(custom_jvp, nondiff_argnums=(0,))
def app(f, x):
  return f(x)
@app.defjvp
def app_jvp(f, primals, tangents):
  x, = primals
  x_dot, = tangents
  return f(x), 2. * x_dot
print(app(lambda x: x ** 3, 3.))
print(grad(app, 1)(lambda x: x ** 3, 3.))
Explanation: Handling  non-differentiable arguments
Some use cases, like the final example problem, call for non-differentiable arguments like function-valued arguments to be passed to functions with custom differentiation rules, and for those arguments to also be passed to the rules themselves. In the case of fixed_point, the function argument f was such a non-differentiable argument. A similar situation arises with jax.experimental.odeint.
jax.custom_jvp with nondiff_argnums
Use the optional nondiff_argnums parameter to jax.custom_jvp to indicate arguments like these. Here's an example with jax.custom_jvp:
End of explanation
@partial(custom_jvp, nondiff_argnums=(0, 2))
def app2(f, x, g):
  return f(g((x)))
@app2.defjvp
def app2_jvp(f, g, primals, tangents):
  x, = primals
  x_dot, = tangents
  return f(g(x)), 3. * x_dot
print(app2(lambda x: x ** 3, 3., lambda y: 5 * y))
print(grad(app2, 1)(lambda x: x ** 3, 3., lambda y: 5 * y))
Explanation: Notice the gotcha here: no matter where in the argument list these parameters appear, they're placed at the start of the signature of the corresponding JVP rule. Here's another example:
End of explanation
@partial(custom_vjp, nondiff_argnums=(0,))
def app(f, x):
  return f(x)
def app_fwd(f, x):
  return f(x), x
def app_bwd(f, x, g):
  return (5 * g,)
app.defvjp(app_fwd, app_bwd)
print(app(lambda x: x ** 2, 4.))
print(grad(app, 1)(lambda x: x ** 2, 4.))
Explanation: jax.custom_vjp with nondiff_argnums
A similar option exists for jax.custom_vjp, and, similarly, the convention is that the non-differentiable arguments are passed as the first arguments to the _bwd rule, no matter where they appear in the signature of the original function. The signature of the _fwd rule remains unchanged - it is the same as the signature of the primal function. Here's an example:
End of explanation
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