JAX internals: The jaxpr language

JAX internals: The jaxpr language #

Jaxprs are JAX’s internal intermediate representation (IR) of programs. They are explicitly typed, functional, first-order, and in algebraic normal form (ANF).

Conceptually, one can think of JAX transformations, such asjax.jit() orjax.grad(), as first trace-specializing the Python function to be transformed into a small and well-behaved intermediate form that is then interpreted with transformation-specific interpretation rules.

One of the reasons JAX can pack so much power into such a small software package is that it starts with a familiar and flexible programming interface (Python with NumPy) and it uses the actual Python interpreter to do most of the heavy lifting to distill the essence of the computation into a simple statically-typed expression language with limited higher-order features.

That language is the jaxpr language. The jaxpr term syntax looks as follows:

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>,...

Not all Python programs can be processed this way, but it turns out that many scientific computing and machine learning programs can.

Before you proceed, remember that not all JAX transformations literally materialize a jaxpr as described above. Some of them, such as differentiation or batching, will apply transformations incrementally during tracing. Nevertheless, if one wants to understand how JAX works internally, or to make use of the result of JAX tracing, it is useful to understand jaxprs.

`jax.core.ClosedJaxpr`#

A jaxpr instance represents a function with one or more typed parameters (input variables) and one or more typed results. The results depend only on the input variables; there are no free variables captured from enclosing scopes. The inputs and outputs have types, which in JAX are represented as abstract values.

There are two related representations in the code for jaxprs,jax.core.Jaxpr andjax.core.ClosedJaxpr. Ajax.core.ClosedJaxpr represents a partially-appliedjax.core.Jaxpr, and is what you obtain when you usejax.make_jaxpr() to inspect jaxprs. It has the following fields:

jaxpr: is ajax.core.Jaxpr representing the actual computation content of the function (described below).
consts is a list of constants.

The most interesting part of theClosedJaxpr is the actual execution content, represented as ajax.core.Jaxpr as printed using the following grammar:

jaxpr::={lambdaVar*;Var+.letEqn*in[Expr+]}

where:

The parameters of the jaxpr are shown as two lists of variables separated by;:
- The first set of variables are the ones that have been introduced to stand for constants that have been hoisted out. These are called theconstvars, and in ajax.core.ClosedJaxpr theconsts field holds corresponding values.
- The second list of variables, calledinvars, correspond to the inputs of the traced Python function.
Eqn* is a list of equations, defining intermediate variables referring to intermediate expressions. Each equation defines one or more variables as the result of applying a primitive on some atomic expressions. Each equation uses only input variables and intermediate variables defined by previous equations.
Expr+: is a list of output atomic expressions (literals or variables) for the jaxpr.

Equations are printed as follows:

Eqn::=letVar+=Primitive[Param*]Expr+

where:

Var+ are one or more intermediate variables to be defined as the output of a primitive invocation (some primitives can return multiple values).
Expr+ are one or more atomic expressions, each either a variable or a literal constant. A special variableunitvar or literalunit, printed as*, represents a value that is not needed in the rest of the computation and has been elided. That is, units are just placeholders.
Param* are zero or more named parameters to the primitive, printed in square brackets. Each parameter is shown asName=Value.

Most jaxpr primitives are first-order (they take just one or more Expr as arguments):

Primitive:=add|sub|sin|mul|...

The most common jaxpr primitives are documented in thejax.lax module.

For example, here is the jaxpr produced for the functionfunc1 below:

fromjaximportmake_jaxprimportjax.numpyasjnpdeffunc1(first,second):temp=first+jnp.sin(second)*3.returnjnp.sum(temp)print(make_jaxpr(func1)(jnp.zeros(8),jnp.ones(8)))

{lambda; a:f32[8] b:f32[8].letc:f32[8] = sin b    d:f32[8] = mul c 3.0:f32[]    e:f32[8] = add a d    f:f32[] = reduce_sum[axes=(0,) out_sharding=None] ein(f,) }

Here there are no constvars,a andb are the input variables and they correspond respectively tofirst andsecond function parameters. The scalar literal3.0 is kept inline. Thereduce_sum primitive has named parametersaxes andinput_shape, in addition to the operande.

Note that even though execution of a program that calls into JAX builds a jaxpr, Python-level control-flow and Python-level functions execute normally. This means that just because a Python program contains functions and control-flow, the resulting jaxpr does not have to contain control-flow or higher-order features.

For example, when tracing the functionfunc3 JAX will inline the call toinner and the conditionalifsecond.shape[0]>4, and will produce the same jaxpr as before:

deffunc2(inner,first,second):temp=first+inner(second)*3.returnjnp.sum(temp)definner(second):ifsecond.shape[0]>4:returnjnp.sin(second)else:assertFalsedeffunc3(first,second):returnfunc2(inner,first,second)print(make_jaxpr(func3)(jnp.zeros(8),jnp.ones(8)))

{lambda; a:f32[8] b:f32[8].letc:f32[8] = sin b    d:f32[8] = mul c 3.0:f32[]    e:f32[8] = add a d    f:f32[] = reduce_sum[axes=(0,) out_sharding=None] ein(f,) }

Handling pytrees#

In jaxpr there are no tuple types; instead primitives take multiple inputs and produce multiple outputs. When processing a function that has structured inputs or outputs, JAX will flatten those and in jaxpr they will appear as lists of inputs and outputs. For more details, refer to thePytrees tutorial.

For example, the following code produces an identical jaxpr to what you saw earlier (with two input vars, one for each element of the input tuple):

deffunc4(arg):# The `arg` is a pair.temp=arg[0]+jnp.sin(arg[1])*3.returnjnp.sum(temp)print(make_jaxpr(func4)((jnp.zeros(8),jnp.ones(8))))

{lambda; a:f32[8] b:f32[8].letc:f32[8] = sin b    d:f32[8] = mul c 3.0:f32[]    e:f32[8] = add a d    f:f32[] = reduce_sum[axes=(0,) out_sharding=None] ein(f,) }

Constant variables (vars)#

Some values in jaxprs are constants, in that their value does not depend on the jaxpr’s arguments. When these values are scalars they are represented directly in the jaxpr equations. Non-scalar array constants are instead hoisted out to the top-level jaxpr, where they correspond to constant variables (“constvars”). These constvars differ from the other jaxpr parameters (“invars”) only as a bookkeeping convention.

Higher-order JAX primitives#

Jaxpr includes several higher-order JAX primitives. They are more complicated because they include sub-jaxprs.

`cond` primitive (conditionals)#

JAX traces through normal Python conditionals. To capture a conditional expression for dynamic execution, one must use thejax.lax.switch() andjax.lax.cond() constructors, which have the signatures:

lax.switch(index:int,branches:Sequence[A->B],operand:A)->Blax.cond(pred:bool,true_body:A->B,false_body:A->B,operand:A)->B

Both of these will bind a primitive calledcond internally. Thecond primitive in jaxprs reflects the more general signature oflax.switch(): it takes an integer denoting the index of the branch to execute (clamped into valid indexing range).

For example:

fromjaximportlaxdefone_of_three(index,arg):returnlax.switch(index,[lambdax:x+1.,lambdax:x-2.,lambdax:x+3.],arg)print(make_jaxpr(one_of_three)(1,5.))

{lambda; a:i32[] b:f32[].letc:i32[] = convert_element_type[new_dtype=int32 weak_type=False] a    d:i32[] = clamp 0:i32[] c 2:i32[]    e:f32[] = cond[      branches=(        {lambda; f:f32[].let g:f32[] = add f 1.0:f32[]in(g,) }        {lambda; h:f32[].let i:f32[] = sub h 2.0:f32[]in(i,) }        {lambda; j:f32[].let k:f32[] = add j 3.0:f32[]in(k,) }      )    ] d bin(e,) }

Thecond primitive has a number of parameters:

branches are jaxprs that correspond to the branch functionals. In this example, those functionals each take one input variable, corresponding tox.
linear is a tuple of booleans that is used internally by the auto-differentiation machinery to encode which of the input parameters are used linearly in the conditional.

The above instance of the cond primitive takes two operands. The first one (d) is the branch index, thenb is the operand (arg) to be passed to whichever jaxpr inbranches is selected by the branch index.

Another example, usingjax.lax.cond():

fromjaximportlaxdeffunc7(arg):returnlax.cond(arg>=0.,lambdaxtrue:xtrue+3.,lambdaxfalse:xfalse-3.,arg)print(make_jaxpr(func7)(5.))

{lambda; a:f32[].letb:bool[] = ge a 0.0:f32[]    c:i32[] = convert_element_type[new_dtype=int32 weak_type=False] b    d:f32[] = cond[      branches=(        {lambda; e:f32[].let f:f32[] = sub e 3.0:f32[]in(f,) }        {lambda; g:f32[].let h:f32[] = add g 3.0:f32[]in(h,) }      )    ] c ain(d,) }

In this case, the boolean predicate is converted to an integer index (0 or 1), andbranches are jaxprs that correspond to the false and true branch functionals, in that order. Again, each function takes one input variable, corresponding toxfalse andxtrue respectively.

The following example shows a more complicated situation when the input to the branch functionals is a tuple, and thefalse branch functional contains a constantjnp.ones(1) that is hoisted as aconstvar.

deffunc8(arg1,arg2):# Where `arg2` is a pair.returnlax.cond(arg1>=0.,lambdaxtrue:xtrue[0],lambdaxfalse:jnp.array([1])+xfalse[1],arg2)print(make_jaxpr(func8)(5.,(jnp.zeros(1),2.)))

{lambdaa:i32[1]; b:f32[] c:f32[1] d:f32[].lete:bool[] = ge b 0.0:f32[]    f:i32[] = convert_element_type[new_dtype=int32 weak_type=False] e    g:f32[1] = cond[      branches=(        {lambda; h:i32[1] i:f32[1] j:f32[].letk:f32[1] = convert_element_type[new_dtype=float32 weak_type=True] h            l:f32[1] = add k jin(l,) }        {lambda; m:i32[1] n:f32[1] o:f32[].letin(n,) }      )    ] f a c din(g,) }

`while` primitive#

Just like for conditionals, Python loops are inlined during tracing. If you want to capture a loop for dynamic execution, you must use one of several special operations,jax.lax.while_loop() (a primitive) andjax.lax.fori_loop() (a helper that generates a while_loop primitive):

lax.while_loop(cond_fun:(C->bool),body_fun:(C->C),init:C)->Clax.fori_loop(start:int,end:int,body:(int->C->C),init:C)->C

In the above signature,C stands for the type of the loop “carry” value. For example, here is an examplefori_loop:

importnumpyasnpdeffunc10(arg,n):ones=jnp.ones(arg.shape)# A constant.returnlax.fori_loop(0,n,lambdai,carry:carry+ones*3.+arg,arg+ones)print(make_jaxpr(func10)(np.ones(16),5))

{lambda; a:f32[16] b:i32[].letc:f32[16] = broadcast_in_dim 1.0:f32[]    d:f32[16] = add a c    _:i32[] _:i32[] e:f32[16] = while[      body_jaxpr={lambda; f:f32[16] g:f32[16] h:i32[] i:i32[] j:f32[16].letk:i32[] = add h 1:i32[]          l:f32[16] = mul f 3.0:f32[]          m:f32[16] = add j l          n:f32[16] = add m gin(k, i, n) }      body_nconsts=2      cond_jaxpr={lambda; o:i32[] p:i32[] q:f32[16].letr:bool[] = lt o pin(r,) }      cond_nconsts=0    ] c a 0:i32[] b din(e,) }

Thewhile primitive takes 5 arguments:ca0bd, as follows:

0 constants forcond_jaxpr (sincecond_nconsts is 0)
2 constants forbody_jaxpr (c, anda)
3 parameters for the initial value of carry

`scan` primitive#

JAX supports a special form of loop over the elements of an array (with statically known shape). The fact that there are a fixed number of iterations makes this form of looping easily reverse-differentiable. Such loops are constructed with thejax.lax.scan() function:

lax.scan(body_fun:(C->A->(C,B)),init_carry:C,in_arr:Array[A])->(C,Array[B])

This is written in terms of aHaskell type signature:C is the type of thescan carry,A is the element type of the input array(s), andB is the element type of the output array(s).

For the example consider the functionfunc11 below:

deffunc11(arr,extra):ones=jnp.ones(arr.shape)#  A constantdefbody(carry,aelems):# carry: running dot-product of the two arrays# aelems: a pair with corresponding elements from the two arraysae1,ae2=aelemsreturn(carry+ae1*ae2+extra,carry)returnlax.scan(body,0.,(arr,ones))print(make_jaxpr(func11)(np.ones(16),5.))

{lambda; a:f32[16] b:f32[].letc:f32[16] = broadcast_in_dim 1.0:f32[]    d:f32[] e:f32[16] = scan[      _split_transpose=False      jaxpr={lambda; f:f32[] g:f32[] h:f32[] i:f32[].letj:f32[] = mul h i          k:f32[] = convert_element_type[new_dtype=float32 weak_type=False] g          l:f32[] = add k j          m:f32[] = convert_element_type[new_dtype=float32 weak_type=False] f          n:f32[] = add l min(n, g) }      length=16      linear=(False, False, False, False)      num_carry=1      num_consts=1      reverse=False      unroll=1    ] b 0.0:f32[] a cin(d, e) }

Thelinear parameter describes for each of the input variables whether they are guaranteed to be used linearly in the body. Once thescan goes through linearization, more arguments will be linear.

Thescan primitive takes 4 arguments:b0.0ac, of which:

One is the free variable for the body
One is the initial value of the carry
The next 2 are the arrays over which the scan operates

`(p)jit` primitive#

The call primitive arises from JIT compilation, and it encapsulates a sub-jaxpr along with parameters that specify the backend and the device on which the computation should run. For example:

fromjaximportjitdeffunc12(arg):@jitdefinner(x):returnx+arg*jnp.ones(1)# Include a constant in the inner function.returnarg+inner(arg-2.)print(make_jaxpr(func12)(1.))

{lambda; a:f32[].letb:f32[] = sub a 2.0:f32[]    c:f32[1] = jit[      name=inner      jaxpr={lambda; a:f32[] b:f32[].letd:f32[1] = broadcast_in_dim 1.0:f32[]          e:f32[] = convert_element_type[new_dtype=float32 weak_type=False] a          f:f32[1] = mul e d          g:f32[] = convert_element_type[new_dtype=float32 weak_type=False] b          c:f32[1] = add g fin(c,) }    ] a b    h:f32[] = convert_element_type[new_dtype=float32 weak_type=False] a    i:f32[1] = add h cin(i,) }

Contents

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JAX internals: The jaxpr language

Contents

JAX internals: The jaxpr language#

jax.core.ClosedJaxpr#

Handling pytrees#

Constant variables (vars)#

Higher-order JAX primitives#

cond primitive (conditionals)#

while primitive#

scan primitive#