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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:
 Just In Time Compilation with JAX
Authors
Step1:  The Understanding Jaxprs section of the documentation provides more information on the meaning of the above output.
Importantly, note how the jaxpr does not capture the side-effect of the function
Step2:  See how the printed x is a Traced object? That's the JAX internals at work.
The fact that the Python code runs at least once is strictly an implementation detail, and so shouldn't be relied upon. However, it's useful to understand as you can use it when debugging to print out intermediate values of a computation.
A key thing to understand is that jaxpr captures the function as executed on the parameters given to it. For example, if we have a conditional, jaxpr will only know about the branch we take
Step3:  JIT compiling a function
As explained before, JAX enables operations to execute on CPU/GPU/TPU using the same code.
Let's look at an example of computing a Scaled Exponential Linear Unit
(SELU), an
operation commonly used in deep learning
Step4:  The code above is sending one operation at a time to the accelerator. This limits the ability of the XLA compiler to optimize our functions.
Naturally, what we want to do is give the XLA compiler as much code as possible, so it can fully optimize it. For this purpose, JAX provides the jax.jit transformation, which will JIT compile a JAX-compatible function. The example below shows how to use JIT to speed up the previous function.
Step5:  Here's what just happened
Step6:  The problem is that we tried to condition on the value of an input to the function being jitted. The reason we can't do this is related to the fact mentioned above that jaxpr depends on the actual values used to trace it. 
The more specific information about the values we use in the trace, the more we can use standard Python control flow to express ourselves. However, being too specific means we can't reuse the same traced function for other values. JAX solves this by tracing at different levels of abstraction for different purposes.
For jax.jit, the default level is ShapedArray -- that is, each tracer has a concrete shape (which we're allowed to condition on), but no concrete value. This allows the compiled function to work on all possible inputs with the same shape -- the standard use case in machine learning. However, because the tracers have no concrete value, if we attempt to condition on one, we get the error above.
In jax.grad, the constraints are more relaxed, so you can do more. If you compose several transformations, however, you must satisfy the constraints of the most strict one. So, if you jit(grad(f)), f mustn't condition on value. For more detail on the interaction between Python control flow and JAX, see 🔪 JAX - The Sharp Bits 🔪
Step7:  If we really need to JIT a function that has a condition on the value of an input, we can tell JAX to help itself to a less abstract tracer for a particular input by specifying static_argnums or static_argnames. The cost of this is that the resulting jaxpr is less flexible, so JAX will have to re-compile the function for every new value of the specified static input. It is only a good strategy if the function is guaranteed to get limited different values.
Step8:  To specify such arguments when using jit as a decorator, a common pattern is to use python's functools.partial
Step9:  When to use JIT
In many of the examples above, jitting is not worth it
Step10:  This is because jax.jit introduces some overhead itself. Therefore, it usually only saves time if the compiled function is complex and you will run it numerous times. Fortunately, this is common in machine learning, where we tend to compile a large, complicated model, then run it for millions of iterations.
Generally, you want to jit the largest possible chunk of your computation; ideally, the entire update step. This gives the compiler maximum freedom to optimise.
Caching
It's important to understand the caching behaviour of jax.jit.
Suppose I define f = jax.jit(g). When I first invoke f, it will get compiled, and the resulting XLA code will get cached. Subsequent calls of f will reuse the cached code. This is how jax.jit makes up for the up-front cost of compilation.
If I specify static_argnums, then the cached code will be used only for the same values of arguments labelled as static. If any of them change, recompilation occurs. If there are many values, then your program might spend more time compiling than it would have executing ops one-by-one.
Avoid calling jax.jit inside loops. For most cases, JAX will be able to use the compiled, cached function in subsequent calls to jax.jit. However, because the cache relies on the hash of the function, it becomes problematic when equivalent functions are redefined. This will cause unnecessary compilation each time in the loop
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Python Code:
import jax
import jax.numpy as jnp
global_list = []
def log2(x):
  global_list.append(x)
  ln_x = jnp.log(x)
  ln_2 = jnp.log(2.0)
  return ln_x / ln_2
print(jax.make_jaxpr(log2)(3.0))
Explanation: Just In Time Compilation with JAX
Authors: Rosalia Schneider & Vladimir Mikulik
In this section, we will further explore how JAX works, and how we can make it performant.
We will discuss the jax.jit() transform, which will perform Just In Time (JIT) compilation
of a JAX Python function so it can be executed efficiently in XLA.
How JAX transforms work
In the previous section, we discussed that JAX allows us to transform Python functions. This is done by first converting the Python function into a simple intermediate language called jaxpr. The transformations then work on the jaxpr representation. 
We can show a representation of the jaxpr of a function by using jax.make_jaxpr:
End of explanation
def log2_with_print(x):
  print("printed x:", x)
  ln_x = jnp.log(x)
  ln_2 = jnp.log(2.0)
  return ln_x / ln_2
print(jax.make_jaxpr(log2_with_print)(3.))
Explanation: The Understanding Jaxprs section of the documentation provides more information on the meaning of the above output.
Importantly, note how the jaxpr does not capture the side-effect of the function: there is nothing in it corresponding to global_list.append(x). This is a feature, not a bug: JAX is designed to understand side-effect-free (a.k.a. functionally pure) code. If pure function and side-effect are unfamiliar terms, this is explained in a little more detail in 🔪 JAX - The Sharp Bits 🔪: Pure Functions.
Of course, impure functions can still be written and even run, but JAX gives no guarantees about their behaviour once converted to jaxpr. However, as a rule of thumb, you can expect (but shouldn't rely on) the side-effects of a JAX-transformed function to run once (during the first call), and never again. This is because of the way that JAX generates jaxpr, using a process called 'tracing'.
When tracing, JAX wraps each argument by a tracer object. These tracers then record all JAX operations performed on them during the function call (which happens in regular Python). Then, JAX uses the tracer records to reconstruct the entire function. The output of that reconstruction is the jaxpr. Since the tracers do not record the Python side-effects, they do not appear in the jaxpr. However, the side-effects still happen during the trace itself.
Note: the Python print() function is not pure: the text output is a side-effect of the function. Therefore, any print() calls will only happen during tracing, and will not appear in the jaxpr:
End of explanation
def log2_if_rank_2(x):
  if x.ndim == 2:
    ln_x = jnp.log(x)
    ln_2 = jnp.log(2.0)
    return ln_x / ln_2
  else:
    return x
print(jax.make_jaxpr(log2_if_rank_2)(jax.numpy.array([1, 2, 3])))
Explanation: See how the printed x is a Traced object? That's the JAX internals at work.
The fact that the Python code runs at least once is strictly an implementation detail, and so shouldn't be relied upon. However, it's useful to understand as you can use it when debugging to print out intermediate values of a computation.
A key thing to understand is that jaxpr captures the function as executed on the parameters given to it. For example, if we have a conditional, jaxpr will only know about the branch we take:
End of explanation
import jax
import jax.numpy as jnp
def selu(x, alpha=1.67, lambda_=1.05):
  return lambda_ * jnp.where(x > 0, x, alpha * jnp.exp(x) - alpha)
x = jnp.arange(1000000)
%timeit selu(x).block_until_ready()
Explanation: JIT compiling a function
As explained before, JAX enables operations to execute on CPU/GPU/TPU using the same code.
Let's look at an example of computing a Scaled Exponential Linear Unit
(SELU), an
operation commonly used in deep learning:
End of explanation
selu_jit = jax.jit(selu)
# Warm up
selu_jit(x).block_until_ready()
%timeit selu_jit(x).block_until_ready()
Explanation: The code above is sending one operation at a time to the accelerator. This limits the ability of the XLA compiler to optimize our functions.
Naturally, what we want to do is give the XLA compiler as much code as possible, so it can fully optimize it. For this purpose, JAX provides the jax.jit transformation, which will JIT compile a JAX-compatible function. The example below shows how to use JIT to speed up the previous function.
End of explanation
# Condition on value of x.
def f(x):
  if x > 0:
    return x
  else:
    return 2 * x
f_jit = jax.jit(f)
f_jit(10)  # Should raise an error. 
# While loop conditioned on x and n.
def g(x, n):
  i = 0
  while i < n:
    i += 1
  return x + i
g_jit = jax.jit(g)
g_jit(10, 20)  # Should raise an error. 
Explanation: Here's what just happened:
1) We defined selu_jit as the compiled version of selu.
2) We ran selu_jit once on x. This is where JAX does its tracing -- it needs to have some inputs to wrap in tracers, after all. The jaxpr is then compiled using XLA into very efficient code optimized for your GPU or TPU. Subsequent calls to selu_jit will now use that code, skipping our old Python implementation entirely. 
(If we didn't include the warm-up call separately, everything would still work, but then the compilation time would be included in the benchmark. It would still be faster, because we run many loops in the benchmark, but it wouldn't be a fair comparison.)
3) We timed the execution speed of the compiled version. (Note the use of block_until_ready(), which is required due to JAX's Asynchronous execution model).
Why can't we just JIT everything?
After going through the example above, you might be wondering whether we should simply apply jax.jit to every function. To understand why this is not the case, and when we should/shouldn't apply jit, let's first check some cases where JIT doesn't work.
End of explanation
# While loop conditioned on x and n with a jitted body.
@jax.jit
def loop_body(prev_i):
  return prev_i + 1
def g_inner_jitted(x, n):
  i = 0
  while i < n:
    i = loop_body(i)
  return x + i
g_inner_jitted(10, 20)
Explanation: The problem is that we tried to condition on the value of an input to the function being jitted. The reason we can't do this is related to the fact mentioned above that jaxpr depends on the actual values used to trace it. 
The more specific information about the values we use in the trace, the more we can use standard Python control flow to express ourselves. However, being too specific means we can't reuse the same traced function for other values. JAX solves this by tracing at different levels of abstraction for different purposes.
For jax.jit, the default level is ShapedArray -- that is, each tracer has a concrete shape (which we're allowed to condition on), but no concrete value. This allows the compiled function to work on all possible inputs with the same shape -- the standard use case in machine learning. However, because the tracers have no concrete value, if we attempt to condition on one, we get the error above.
In jax.grad, the constraints are more relaxed, so you can do more. If you compose several transformations, however, you must satisfy the constraints of the most strict one. So, if you jit(grad(f)), f mustn't condition on value. For more detail on the interaction between Python control flow and JAX, see 🔪 JAX - The Sharp Bits 🔪: Control Flow.
One way to deal with this problem is to rewrite the code to avoid conditionals on value. Another is to use special control flow operators like jax.lax.cond. However, sometimes that is impossible. In that case, you can consider jitting only part of the function. For example, if the most computationally expensive part of the function is inside the loop, we can JIT just that inner part (though make sure to check the next section on caching to avoid shooting yourself in the foot):
End of explanation
f_jit_correct = jax.jit(f, static_argnums=0)
print(f_jit_correct(10))
g_jit_correct = jax.jit(g, static_argnames=['n'])
print(g_jit_correct(10, 20))
Explanation: If we really need to JIT a function that has a condition on the value of an input, we can tell JAX to help itself to a less abstract tracer for a particular input by specifying static_argnums or static_argnames. The cost of this is that the resulting jaxpr is less flexible, so JAX will have to re-compile the function for every new value of the specified static input. It is only a good strategy if the function is guaranteed to get limited different values.
End of explanation
from functools import partial
@partial(jax.jit, static_argnames=['n'])
def g_jit_decorated(x, n):
  i = 0
  while i < n:
    i += 1
  return x + i
print(g_jit_decorated(10, 20))
Explanation: To specify such arguments when using jit as a decorator, a common pattern is to use python's functools.partial:
End of explanation
print("g jitted:")
%timeit g_jit_correct(10, 20).block_until_ready()
print("g:")
%timeit g(10, 20)
Explanation: When to use JIT
In many of the examples above, jitting is not worth it:
End of explanation
from functools import partial
def unjitted_loop_body(prev_i):
  return prev_i + 1
def g_inner_jitted_partial(x, n):
  i = 0
  while i < n:
    # Don't do this! each time the partial returns
    # a function with different hash
    i = jax.jit(partial(unjitted_loop_body))(i)
  return x + i
def g_inner_jitted_lambda(x, n):
  i = 0
  while i < n:
    # Don't do this!, lambda will also return
    # a function with a different hash
    i = jax.jit(lambda x: unjitted_loop_body(x))(i)
  return x + i
def g_inner_jitted_normal(x, n):
  i = 0
  while i < n:
    # this is OK, since JAX can find the
    # cached, compiled function
    i = jax.jit(unjitted_loop_body)(i)
  return x + i
print("jit called in a loop with partials:")
%timeit g_inner_jitted_partial(10, 20).block_until_ready()
print("jit called in a loop with lambdas:")
%timeit g_inner_jitted_lambda(10, 20).block_until_ready()
print("jit called in a loop with caching:")
%timeit g_inner_jitted_normal(10, 20).block_until_ready()
Explanation: This is because jax.jit introduces some overhead itself. Therefore, it usually only saves time if the compiled function is complex and you will run it numerous times. Fortunately, this is common in machine learning, where we tend to compile a large, complicated model, then run it for millions of iterations.
Generally, you want to jit the largest possible chunk of your computation; ideally, the entire update step. This gives the compiler maximum freedom to optimise.
Caching
It's important to understand the caching behaviour of jax.jit.
Suppose I define f = jax.jit(g). When I first invoke f, it will get compiled, and the resulting XLA code will get cached. Subsequent calls of f will reuse the cached code. This is how jax.jit makes up for the up-front cost of compilation.
If I specify static_argnums, then the cached code will be used only for the same values of arguments labelled as static. If any of them change, recompilation occurs. If there are many values, then your program might spend more time compiling than it would have executing ops one-by-one.
Avoid calling jax.jit inside loops. For most cases, JAX will be able to use the compiled, cached function in subsequent calls to jax.jit. However, because the cache relies on the hash of the function, it becomes problematic when equivalent functions are redefined. This will cause unnecessary compilation each time in the loop:
End of explanation
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