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When walking about the countryside of Italy, the people will not hesitate to tell you that __JAX__ has [_"una anima di pura programmazione funzionale"_](https://www.sscardapane.it/iaml-backup/jax-intro/).
__JAX__ is a language for __expressing__ and __composing____transformations__ of numerical programs. __JAX__ is also able to __compile__ numerical programs for CPU or accelerators (GPU/TPU).
JAX transformation and compilation are designed to work only on Python functions that are functionally pure: all the input data is passed through the function parameters, all the results are output through the function results. A pure function will always return the same result if invoked with the same inputs.
Here are some examples of functions that are not functionally pure for which JAX behaves differently than the Python interpreter. Note that these behaviors are not guaranteed by the JAX system; the proper way to use JAX is to use it only on functionally pure Python functions.
It is not recommended to use iterators in any JAX function you want to `jit` or in any control-flow primitive. The reason is that an iterator is a python object which introduces state to retrieve the next element. Therefore, it is incompatible with JAX functional programming model. In the code below, there are some examples of incorrect attempts to use iterators with JAX. Most of them return an error, but some give unexpected results.
Instead, JAX offers a _functional_ array update using the [`.at` property on JAX arrays](https://jax.readthedocs.io/en/latest/_autosummary/jax.numpy.ndarray.at.html#jax.numpy.ndarray.at).
️⚠️ inside `jit`'d code and `lax.while_loop` or `lax.fori_loop` the __size__ of slices can't be functions of argument _values_ but only functions of argument _shapes_ -- the slice start indices have no such restriction. See the below __Control Flow__ Section for more information on this limitation.
JAX's array update functions, unlike their NumPy versions, operate out-of-place. That is, the updated array is returned as a new array and the original array is not modified by the update.
However, inside __jit__-compiled code, if the __input value__`x` of `x.at[idx].set(y)` is not reused, the compiler will optimize the array update to occur _in-place_.
For more details on indexed array updates, see the [documentation for the `.at` property](https://jax.readthedocs.io/en/latest/_autosummary/jax.numpy.ndarray.at.html#jax.numpy.ndarray.at).
However, raising an error from code running on an accelerator can be difficult or impossible. Therefore, JAX must choose some non-error behavior for out of bounds indexing (akin to how invalid floating point arithmetic results in `NaN`). When the indexing operation is an array index update (e.g. `index_add` or `scatter`-like primitives), updates at out-of-bounds indices will be skipped; when the operation is an array index retrieval (e.g. NumPy indexing or `gather`-like primitives) the index is clamped to the bounds of the array since __something__ must be returned. For example, the last value of the array will be returned from this indexing operation:
If you would like finer-grained control over the behavior for out-of-bound indices, you can use the optional parameters of [`ndarray.at`](https://jax.readthedocs.io/en/latest/_autosummary/jax.numpy.ndarray.at.html); for example:
Note that due to this behavior for index retrieval, functions like `jnp.nanargmin` and `jnp.nanargmax` return -1 for slices consisting of NaNs whereas Numpy would throw an error.
Note also that, as the two behaviors described above are not inverses of each other, reverse-mode automatic differentiation (which turns index updates into index retrievals and vice versa) [will not preserve the semantics of out of bounds indexing](https://github.com/google/jax/issues/5760). Thus it may be a good idea to think of out-of-bounds indexing in JAX as a case of [undefined behavior](https://en.wikipedia.org/wiki/Undefined_behavior).
This is a deliberate design choice, because passing lists or tuples to traced functions can lead to silent performance degradation that might otherwise be difficult to detect.
For example, consider the following permissive version of `jnp.sum` that allows list inputs:
The output is what we would expect, but this hides potential performance issues under the hood. In JAX's tracing and JIT compilation model, each element in a Python list or tuple is treated as a separate JAX variable, and individually processed and pushed to device. This can be seen in the jaxpr for the ``permissive_sum`` function above:
Each entry of the list is handled as a separate input, resulting in a tracing & compilation overhead that grows linearly with the size of the list. To prevent surprises like this, JAX avoids implicit conversions of lists and tuples to arrays.
If you would like to pass a tuple or list to a JAX function, you can do so by first explicitly converting it to an array:
You're used to _stateful_ pseudorandom number generators (PRNGs) from numpy and other libraries, which helpfully hide a lot of details under the hood to give you a ready fountain of pseudorandomness:
Underneath the hood, numpy uses the [Mersenne Twister](https://en.wikipedia.org/wiki/Mersenne_Twister) PRNG to power its pseudorandom functions. The PRNG has a period of $2^{19937}-1$ and at any point can be described by __624 32-bit unsigned ints__ and a __position__ indicating how much of this "entropy" has been used up.
This pseudorandom state vector is automagically updated behind the scenes every time a random number is needed, "consuming" 2 of the uint32s in the Mersenne twister state vector:
The problem with magic PRNG state is that it's hard to reason about how it's being used and updated across different threads, processes, and devices, and it's _very easy_ to screw up when the details of entropy production and consumption are hidden from the end user.
The Mersenne Twister PRNG is also known to have a [number](https://cs.stackexchange.com/a/53475) of problems, it has a large 2.5kB state size, which leads to problematic [initialization issues](https://dl.acm.org/citation.cfm?id=1276928). It [fails](http://www.pcg-random.org/pdf/toms-oneill-pcg-family-v1.02.pdf) modern BigCrush tests, and is generally slow.
JAX instead implements an _explicit_ PRNG where entropy production and consumption are handled by explicitly passing and iterating PRNG state. JAX uses a modern [Threefry counter-based PRNG](https://github.com/google/jax/blob/main/docs/jep/263-prng.md) that's __splittable__. That is, its design allows us to __fork__ the PRNG state into new PRNGs for use with parallel stochastic generation.
If you just want to apply `grad` to your python functions, you can use regular python control-flow constructs with no problems, as if you were using [Autograd](https://github.com/hips/autograd) (or Pytorch or TF Eager).
When we `jit`-compile a function, we usually want to compile a version of the function that works for many different argument values, so that we can cache and reuse the compiled code. That way we don't have to re-compile on each function evaluation.
For example, if we evaluate an `@jit` function on the array `jnp.array([1., 2., 3.], jnp.float32)`, we might want to compile code that we can reuse to evaluate the function on `jnp.array([4., 5., 6.], jnp.float32)` to save on compile time.
To get a view of your Python code that is valid for many different argument values, JAX traces it on _abstract values_ that represent sets of possible inputs. There are [multiple different levels of abstraction](https://github.com/google/jax/blob/main/jax/_src/abstract_arrays.py), and different transformations use different abstraction levels.
By default, `jit` traces your code on the `ShapedArray` abstraction level, where each abstract value represents the set of all array values with a fixed shape and dtype. For example, if we trace using the abstract value `ShapedArray((3,), jnp.float32)`, we get a view of the function that can be reused for any concrete value in the corresponding set of arrays. That means we can save on compile time.
But there's a tradeoff here: if we trace a Python function on a `ShapedArray((), jnp.float32)` that isn't committed to a specific concrete value, when we hit a line like `if x < 3`, the expression `x < 3` evaluates to an abstract `ShapedArray((), jnp.bool_)` that represents the set `{True, False}`. When Python attempts to coerce that to a concrete `True` or `False`, we get an error: we don't know which branch to take, and can't continue tracing! The tradeoff is that with higher levels of abstraction we gain a more general view of the Python code (and thus save on re-compilations), but we require more constraints on the Python code to complete the trace.
The good news is that you can control this tradeoff yourself. By having `jit` trace on more refined abstract values, you can relax the traceability constraints. For example, using the `static_argnums` argument to `jit`, we can specify to trace on concrete values of some arguments. Here's that example function again:
In effect, the loop gets statically unrolled. JAX can also trace at _higher_ levels of abstraction, like `Unshaped`, but that's not currently the default for any transformation
️⚠️ **functions with argument-__value__ dependent shapes**
These control-flow issues also come up in a more subtle way: numerical functions we want to __jit__ can't specialize the shapes of internal arrays on argument _values_ (specializing on argument __shapes__ is ok). As a trivial example, let's make a function whose output happens to depend on the input variable `length`.
Lastly, if your function has global side-effects, JAX's tracer can cause weird things to happen. A common gotcha is trying to print arrays inside __jit__'d functions:
There are more options for control flow in JAX. Say you want to avoid re-compilations but still want to use control flow that's traceable, and that avoids un-rolling large loops. Then you can use these 4 structured control flow primitives:
JAX code used within transforms like `jax.jit`, `jax.vmap`, `jax.grad`, etc. requires all output arrays and intermediate arrays to have static shape: that is, the shape cannot depend on values within other arrays.
For example, if you were implementing your own version of `jnp.nansum`, you might start with something like this:
The problem is that the size of `x_without_nans` is dependent on the values within `x`, which is another way of saying its size is *dynamic*.
Often in JAX it is possible to work-around the need for dynamically-sized arrays via other means.
For example, here it is possible to use the three-argument form of `jnp.where` to replace the NaN values with zeros, thus computing the same result while avoiding dynamic shapes:
This will cause computations to error-out immediately on production of a NaN. Switching this option on adds a nan check to every floating point type value produced by XLA. That means values are pulled back to the host and checked as ndarrays for every primitive operation not under an `@jit`. For code under an `@jit`, the output of every `@jit` function is checked and if a nan is present it will re-run the function in de-optimized op-by-op mode, effectively removing one level of `@jit` at a time.
There could be tricky situations that arise, like nans that only occur under a `@jit` but don't get produced in de-optimized mode. In that case you'll see a warning message print out but your code will continue to execute.
If the nans are being produced in the backward pass of a gradient evaluation, when an exception is raised several frames up in the stack trace you will be in the backward_pass function, which is essentially a simple jaxpr interpreter that walks the sequence of primitive operations in reverse. In the example below, we started an ipython repl with the command line `env JAX_DEBUG_NANS=True ipython`, then ran this:
The nan generated was caught. By running `%debug`, we can get a post-mortem debugger. This also works with functions under `@jit`, as the example below shows.
When this code sees a nan in the output of an `@jit` function, it calls into the de-optimized code, so we still get a clear stack trace. And we can run a post-mortem debugger with `%debug` to inspect all the values to figure out the error.
⚠️ You shouldn't have the NaN-checker on if you're not debugging, as it can introduce lots of device-host round-trips and performance regressions!
At the moment, JAX by default enforces single-precision numbers to mitigate the Numpy API's tendency to aggressively promote operands to `double`. This is the desired behavior for many machine-learning applications, but it may catch you by surprise!
While `jax.numpy` makes every attempt to replicate the behavior of numpy's API, there do exist corner cases where the behaviors differ.
Many such cases are discussed in detail in the sections above; here we list several other known places where the APIs diverge.
- For binary operations, JAX's type promotion rules differ somewhat from those used by NumPy. See [Type Promotion Semantics](https://jax.readthedocs.io/en/latest/type_promotion.html) for more details.
- When performing unsafe type casts (i.e. casts in which the target dtype cannot represent the input value), JAX's behavior may be backend dependent, and in general may diverge from NumPy's behavior. Numpy allows control over the result in these scenarios via the `casting` argument (see [`np.ndarray.astype`](https://numpy.org/devdocs/reference/generated/numpy.ndarray.astype.html)); JAX does not provide any such configuration, instead directly inheriting the behavior of [XLA:ConvertElementType](https://www.tensorflow.org/xla/operation_semantics#convertelementtype).
Here is an example of an unsafe cast with differing results between NumPy and JAX:
