Internals

Transforming Code

While a real anvil is made for reshaping metal, this package is a tool for reshaping code. We refer to such a rewriting of code as a transformation, of which there are three types:

R $\rightarrow$ AnvlGraph: Generic R functions are too complicated to handle, so the first step in {anvl} is always to convert them into a computational anvl::Graph object via tracing. Such a AnvlGraph is similar to JAXExpr objects in JAX. It operates only on AnvlArray objects and applies anvl::Primitive operations to them.
AnvlGraph $\rightarrow$ AnvlGraph: It is possible to transform AnvlGraphs into other AnvlGraphs. Their purpose is to change the functionality of the code. At the time of writing, there is essentially only one such transformation, namely reverse-mode automatic differentiation via gradient().
AnvlGraph $\rightarrow$ Executable: In order to perform the actual computation, the AnvlGraph needs to be converted into an executable. The main backend is XLA (via stablehlo and pjrt). There is also an experimental quickr backend.

Tracing R Functions into Graphs

All functionality in the {anvl} package is centered around the anvl::Graph class. While it is in principle possible to create AnvlGraphs by hand, these are usually created by tracing R functions. In general, when we want to convert some code into another form (in our case, R Code into a AnvlGraph), there are two approaches:

Static analysis, which would require operating on the abstract syntax tree (AST) of the code.
Dynamic analysis (aka “tracing”), which executes the code and records selected operations.

The former approach is followed by the {quickr} package, while we go with tracing. We start with a simple, yet illustrative example that either adds or multiplies two inputs x and y depending on the value of op.

library(anvl)
f <- function(x, y, op) {
  if (op == "add") {
    nv_add(x, y)
  } else if (op == "mul") {
    nv_mul(x, y)
  } else {
    stop("Unsupported operation")
  }
}

To do this, we use anvl::trace_fn(), which takes in an R function and a list of AbstractArray inputs that specify the types of the inputs.

aten <- nv_aval("f32", c())
aten

## AbstractArray(dtype=f32, shape=)

graph <- trace_fn(f, list(x = aten, y = aten, op = "mul"))
graph

## <AnvlGraph>
##   Inputs:
##     %x1: f32[]
##     %x2: f32[]
##   Body:
##     %1: f32[] = mul(%x1, %x2)
##   Outputs:
##     %1: f32[]

The output of trace_fn() is now a AnvlGraph object that represents the computation. The fields of the AnvlGraph are:

inputs, which are GraphNodes that represent the inputs to the function.
outputs, which are GraphNodes that represent the outputs of the function.
calls, which are PrimitiveCalls that take in GraphNodes (and parameters) and produce output GraphNodes.
in_tree, out_tree, which we will cover later (do we??)

What happens during trace_fn() is that a new GraphDescriptor is created and the inputs x and y are converted into anvl::GraphBox objects. Then, the function f is simply evaluated with the GraphBox objects as inputs. During this evaluation, we need to distinguish between two cases:

A “standard” R function is called: Here, nothing special happens and the function is simply evaluated.
An anvl function is called: Here, the operation that underlies the function is recorded in the GraphDescriptor.

The evaluation of the if statement is an example for the first category. Because we set op = "mul", only the second branch is executed. Then, we are calling nv_mul, which attaches a PrimitiveCall that represents the multiplication of the two arrays to the $calls of the GraphDescriptor. Note that the nv_mul is itself not primitive, but performs some type promotion and broadcasting if needed, before calling into the primitive prim_mul.

A PrimitiveCall object consists of the following fields:

primitive: The primitive function that was called.
inputs: The inputs to the primitive function.
params: The parameters (non-arrays) to the primitive function.
outputs: The outputs of the primitive function.

When the evaluation of f is complete, the $outputs field of the GraphDescriptor is set and the AnvlGraph is subsequently created from the GraphDescriptor. The only difference between the AnvlGraph and the GraphDescriptor is that the latter has some utility fields that are useful during graph creation, but for the purposes of this tutorial, you can think of them as being the same.

Transforming Graphs into other Graphs

Once the R function is staged out into a simpler format, it is ready to be transformed. The {anvl} package does not in any way dictate how such a AnvlGraph to AnvlGraph transformation can be implemented. For most interesting transformations, however, we need to store some information for each {anvl} primitive function. In the case of the gradient, we need to store the derivative rules. For this, the anvl::AnvlPrimitive metadata object attached to each primitive has a rules field that can be populated. The derivative rules are stored as functions under the "reverse" name. Each primitive is an exported prim_* function; [[ on it reads a rule:

prim_mul[["reverse"]]

## function (inputs, outputs, grads, .required) 
## {
##     lhs <- inputs[[1L]]
##     rhs <- inputs[[2L]]
##     grad <- grads[[1L]]
##     list(if (.required[[1L]]) prim_mul(grad, rhs), if (.required[[2L]]) prim_mul(grad, 
##         lhs))
## }
## <environment: namespace:anvl>

The anvl::transform_gradient function uses these rules to compute the gradient of a function. For this specific transformation, we are walking the graph backwards and apply the derivative rules, which will append the “reverse pass” to the graph. Besides the forward graph, the transformation takes in the wrt argument, which specifies with respect to which arguments to compute the gradient.

bwd_graph <- transform_gradient(graph, wrt = c("x", "y"))
bwd_graph

## <AnvlGraph>
##   Inputs:
##     %x1: f32[]
##     %x2: f32[]
##   Constants:
##     %c1: f32[]
##   Body:
##     %1: f32[] = mul(%x1, %x2)
##     %2: f32[] = mul(%c1, %x2)
##     %3: f32[] = mul(%c1, %x1)
##   Outputs:
##     %2: f32[]
##     %3: f32[]

Lowering a Graph

In order to execute a AnvlGraph, we need to convert it into a – wait for it – executable. Here, we show how to compile using the XLA backend. First, we will translate the AnvlGraph into the StableHLO representation via the {stablehlo} package. Then, we will compile this program using the XLA compiler that is accessible via the {pjrt} package.

Like for the gradient transformation, the rules of how to do this transformation are attached to each primitive.

prim_mul[["stablehlo"]]

## function (lhs, rhs) 
## {
##     list(stablehlo::hlo_multiply(lhs, rhs))
## }
## <environment: namespace:anvl>

The anvl::stablehlo function will create a stablehlo::Func object and will sequentially translate the PrimitiveCalls into StableHLO operations.

func <- stablehlo(graph)[[1L]]
func

## func.func @main (%0: tensor<f32>, %1: tensor<f32>) -> tensor<f32> {
## %2 = stablehlo.multiply %0, %1 : tensor<f32>
## return %2 : tensor<f32>
## }

Now, we can compile the function via pjrt_compile().

hlo_str <- stablehlo::repr(func)
program <- pjrt::pjrt_program(src = hlo_str, format = "mlir")
exec <- pjrt::pjrt_compile(program)

To run the function, we need to extract the underlying buffers from the arrays before passing them to the executable, which will output a PJRTBuffer that we can easily convert to an AnvlArray.

x <- nv_scalar(3, "f32")
y <- nv_scalar(4, "f32")
out <- pjrt::pjrt_execute(exec, x$data, y$data)
out

## PJRTBuffer 
##  12
## [ CPUf32{} ]

nv_array(out)

## AnvlArray
##  12
## [ CPUf32{} ]

The User Interface

In the previous section, we have shown how the transformations are implemented under the hood. The actual user interface is a little more convenient and follows the JAX interface.

`jit()`

The jit() function allows to convert a regular R function into a Just-In-Time compiled function that can be executed on AnvlArrays. We apply it to our simple example function, where we mark the non-array parameter op as “static”. This means that the value of this parameter needs to be known at compile time.

f_jit <-  jit(f, static = "op")
f_jit(x, y, "add")

## AnvlArray
##  7
## [ CPUf32{} ]

One might think that jit() first calls trace_fn(), then runs stablehlo(), followed by pjrt_compile(). This is, however, not what is happening, as this requires the input types to be known. Instead, f_jit is a “lazy” function that will only perform these steps once the inputs are provided. However, if those steps were applied every time the f_jit function is called, this would be very inefficient, because tracing and compiling takes some time. Therefore, the function f_jit also contains a cache (implemented as an xlamisc::LRUCache), which will check whether there is already a compiled executable for the given inputs. For this, the types of all AnvlArrays need to match exactly (data type and shape) and all static arguments need to be identical. For example, if we run the function with AnvlArrays of the same type, but different values, the function won’t be recompiled, which we can see by checking the size of the cache, which is already 1, because we have called it on x and y above.

cache_size <- function(f) environment(f)$cache$size
cache_size(f_jit)

## [1] 1

After calling it with arrays of the same types and identical static argument values, the size of the cache remains 1:

f_jit(nv_scalar(-99, "f32"), nv_scalar(2, "f32"), "add")

## AnvlArray
##  -97
## [ CPUf32{} ]

cache_size(f_jit)

## [1] 1

When we execute the function with arrays of different dtype or shape, the function will be recompiled:

f_jit(nv_scalar(1, "i32"), nv_scalar(2, "i32"), "add")

## AnvlArray
##  3
## [ CPUi32{} ]

cache_size(f_jit)

## [1] 2

Also, if we provide different values for static arguments, the function will be recompiled:

f_jit(nv_scalar(1, "f32"), nv_scalar(2, "f32"), "mul")

## AnvlArray
##  2
## [ CPUf32{} ]

cache_size(f_jit)

## [1] 3

`gradient()`

Just like jit(), gradient() also returns a function that will lazily create the graph and transform it, once the inputs are provided.

g <- gradient(f, wrt = c("x", "y"))

To actually compute the gradient, we wrap it in jit():

g_jit <- jit(g, static = "op")
g_jit(x, y, "add")

## $x
## AnvlArray
##  1
## [ CPUf32{} ] 
## 
## $y
## AnvlArray
##  1
## [ CPUf32{} ]

We can also use g inside another function:

h <- function(x, y) {
  z <- nv_add(x, y)
  g(z, x, "mul")
}
h_jit <- jit(h)
h_jit(x, y)

## $x
## AnvlArray
##  3
## [ CPUf32{} ] 
## 
## $y
## AnvlArray
##  7
## [ CPUf32{} ]

So, what is happening here? Once the inputs x and y are provided to h_jit, a new GraphDescriptor is created and the inputs x and y are converted into GraphBox objects. Then, the addition of x and y is recorded in the GraphDescriptor. The call into g() is a bit more involved. First, a new GraphDescriptor is created and the forward computation of g is recorded. Subsequently, the reverse pass will be added to the descriptor, after which it will be converted into a AnvlGraph. This AnvlGraph will then be inlined into the parent GraphDescriptor (representing the whole function h), which is then converted into the main AnvlGraph. We can look at this graph below, where trace_fn internally converts the AnvlArrays x and y into their abstract representation.

h_graph <- trace_fn(h, list(x = x, y = y))
h_graph

## <AnvlGraph>
##   Inputs:
##     %x1: f32[]
##     %x2: f32[]
##   Constants:
##     %c1: f32[]
##   Body:
##     %1: f32[] = add(%x1, %x2)
##     %2: f32[] = mul(%1, %x1)
##     %3: f32[] = mul(%c1, %x1)
##     %4: f32[] = mul(%c1, %1)
##   Outputs:
##     %3: f32[]
##     %4: f32[]

Afterwards, this graph is lowered to stableHLO and subsequently compiled.

More Internals

Constant Handling

Constants are handled specially in {anvl}. Consider the program below:

y <- nv_array(rnorm(1000000L))
graph <- trace_fn(function(x) {
  x + y + 1
}, list(x = nv_scalar(1L)))
graph

## <AnvlGraph>
##   Inputs:
##     %x1: i32[]
##   Constants:
##     %c1: f32[1000000]
##   Body:
##     %1: f32[] = convert [dtype = f32, ambiguous = FALSE] (%x1)
##     %2: f32[1000000] = broadcast_in_dim [shape = 1000000, broadcast_dimensions = <any>] (%1)
##     %3: f32[1000000] = add(%2, %c1)
##     %4: f32?[1000000] = broadcast_in_dim [shape = 1000000, broadcast_dimensions = <any>] (1:f32?)
##     %5: f32[1000000] = add(%3, %4)
##   Outputs:
##     %5: f32[1000000]

Here, y is a closed-over constant and it is included in the $constants field of the graph, just like the literal 1.

graph$constants

## [[1]]
## GraphValue(ConcreteArray(f32, (1000000)))

When compiling such a program to stableHLO, constants are treated differently depending on their shape (we follow JAX’s approach here). That is, constants with 1 element are inlined into the program, whereas other constants are added as inputs to the stableHLO program. This is because inlining large constants into the executable is inefficient. However, if we didn’t inline small scalars, the compiler would be unable to do constant folding.

out <- stablehlo(graph)
out[[1L]]

## func.func @main (%0: tensor<1000000xf32>, %1: tensor<i32>) -> tensor<1000000xf32> {
## %2 = "stablehlo.convert" (%1): (tensor<i32>) -> (tensor<f32>)
## %3 = "stablehlo.broadcast_in_dim" (%2) {
## broadcast_dimensions = array<i64>
## }: (tensor<f32>) -> (tensor<1000000xf32>)
## %4 = stablehlo.add %3, %0 : tensor<1000000xf32>
## %5 = "stablehlo.constant" () {
## value = dense<1.00000000e+00> : tensor<f32>
## }: () -> (tensor<f32>)
## %6 = "stablehlo.broadcast_in_dim" (%5) {
## broadcast_dimensions = array<i64>
## }: (tensor<f32>) -> (tensor<1000000xf32>)
## %7 = stablehlo.add %4, %6 : tensor<1000000xf32>
## return %7 : tensor<1000000xf32>
## }

out[[2L]]

## [[1]]
## GraphValue(ConcreteArray(f32, (1000000)))

Also, before compiling, we remove unused constants. Captured constants can become unused when we apply code transformations like below, where the gradient of the function w.r.t. x does not depend on the captured y:

f <- function(x) {
x + y
}
transform_gradient(trace_fn(f, list(x = nv_scalar(1))))

In principle, the compiler is able to do this itself, but because we pass constants as inputs to the program, we need to handle it ourselves.

Further note that:

R Literals are immediately embedded as literals into the program.
Currently, constants with the same value (that refer to different AnvlArrays) are not deduplicated, which we might change in the future.

Device Inference in `jit()`

Device handling in jit() is quite complicated. Some things that are important to be aware of:

We don’t know the inferred device just from looking at the input as we might have something like: jit(\(x) x + nv_scalar(1, device = "cuda")) where we might only learn about the device during tracing. This means the data is only converted at the end.
There are different backends. There might be a function like jit(\(dev) nv_scalar(1, device = dev), backend = "auto"). But with the current implementation of jit(), the tracing is handled by the backend’s jit method, so we need to determine the backend from the input arguments. Therefore, the device = device_arg("dev") needs to be specified:
```
f <- jit(\(dev) nv_scalar(1, device = dev), backend = "auto", device = device_arg("dev"))
f(nv_device("cpu", "xla"))
```
If we would make the main jit() function already trace, we could determine the backend during the tracing, but this is not really needed yet.
device = device_arg() is only accepted together with backend = NULL or backend = "auto". A concrete backend combined with device_arg() is rejected, because with a concrete backend the device can simply be passed via a static argument.

Nested Inputs and Outputs

TODO ## Dichotomy of anvl functions

Here, we will dig deeper into the dichotomy of {anvl} functions such as prim_add. In the Getting Started vignette, we have learned that these functions can either be called directly on AnvlArrays to transform data, or used within jit() blocks to build up programs. Here, we will explain what this actually does and why this is possible.

The core problem this dichotomy solves is that it is a mental burden to always keep two versions os an {anvl} function:

The jit()ted version that can be used to transform arrays.
The non-jit()ted one that can be used to build up programs.

With our implementation, the following is possible:

library(anvl)
prim_add(nv_scalar(1), nv_scalar(2))

## AnvlArray
##  3
## [ CPUf32{} ]

times_2 <- jit(function(x) {
  nv_mul(x, 2)
})

times_4 <- jit(function(x) {
  times_2(times_2(x))
})

times_2(nv_scalar(2))

## AnvlArray
##  4
## [ CPUf32{} ]

times_4(nv_scalar(2))

## AnvlArray
##  8
## [ CPUf32{} ]

Otherwise, we would need the following:

times_2_r <- function(x) {
  nv_mul(x, 2)
}
times_2_jit <- jit(times_2_r)
times_4 <- jit(function(x) {
  times_2_r(times_2_r(x))
})

This is rather cumbersome, as there are always two versions of a function and the first solution is preferable. Internally, we have implemented this by wrapping every primitive function in jit() and making a jit()ted function behave differently depending on whether we are in another jit() call or not.

If we are in a jit() call, and call into a function jit(f), internally f is evaluated, and the function is re-traced. Otherwise, the standard jit path is followed.

However, for the {anvl} API this now means that special care needs to be taken that everything works in jit-mode and in eager-mode. The most important points are:

Canonicalize inputs at the start using as_anvl_array(s)
Propagate device from inputs:
1. For functions with dynamic inputs: use nv_*_like for constant creation and pass input operands
2. For functions without dynamic inputs, add device arg and pass it to constant creators.