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Streaming Combinators and Extracting Flat Parallelism

Posted on June 25, 2017 by Troels Henriksen

This Tuesday I presented Futhark at PLDI, a large academic conference on programming language design and implementation. The associated paper is available (paper PDF), but makes for somewhat dry reading. This blog post covers the main points of the presentation, demonstrated by examples, in a way that may be more accessible than the paper. This is a bit of a long post, but logically divided into two parts. The first half introduces the basics of the Futhark language and shows its application to a small problem. The second half contains a more complicated example of how nested parallelism is turned into GPU-friendly flat parallelism. There is also a short conclusion at the end with measurements of how Futhark-generated code compares in performance to hand-written GPU programs. Scroll to the end if you enjoy looking at numbers and graphs.

Problem Statement

It is old news that parallel computers have become mainstream. It is also widely acknowledged that we still lack good models and languages for programming them. The problem becomes even more severe when we leave the relatively benign world of multi-core CPUs and begin to consider such devices as GPUs, which have evolved into massively parallel machines for general-purpose computation. Saturating a GPU typically requires tens of thousands of lightweight threads with minimal communication or shared state. Traditional sequential and imperative languages are a poor fit.

Functional languages are often mentioned as a solution to the parallelism problem, as they do not rely as much on evaluation order or side effects. For example, a map function can process each element in parallel, and a reduce can shrink an array to one element through repeated parallel application of a function to pairs of elements in the input array.

Unfortunately, life is not that easy. While the functional style indeed makes parallelism explicit, safe and correct, it is not automatically a good fit for a GPU. One major problem is that GPUs support only a restricted form of parallelism (efficiently, at least). The parallelism must be flat, meaning that one parallel thread may not spawn more threads. Further, code running on the GPU may not allocate memory - all allocation must be done prior to starting GPU execution, by controller code running on the CPU. There are further restrictions on how memory must be accessed to obtain high efficiency, and a myriad of other hindrances such as too little stack to implement recursion, and the absence of function pointers. By itself, functional programming does not address these problems, and existing functional languages like OCaml or Haskell are too large and expressive to ever run well on a device as restricted as a GPU.

But all is not lost: the right solution, we believe, is to carefully design a small functional language that has just the features and restrictions to enable the construction of an aggressively optimising compiler capable of generating GPU code. This compiler will worry about all the low-level details, leaving the programmer solely concerned with how best to express the parallelism in their algorithm. We have spent the last four years constructing such a language and compiler, and we call it Futhark.

The Language

Futhark is a small data-parallel purely functional array language that resembles a cross between Haskell and OCaml. Parallelism is expressed through built-in second-order array combinators (SOACs) that resemble higher-order functions known from functional programming. For example, we can define a function that takes a one-dimensional array of integers of input and increments each by two:

let add_two [n] (a: [n]i32): [n]i32 = map (+2) a

Note the shape declarations: we indicate that there is some size n, and that both the result and input to the function are one-dimensional arrays of size n. We do not have to pass the size n explicitly when we call the function; rather it is automatically inferred when we pass the array argument for a.

As another example, this function sums an integer array:

let sum [n] (a: [n]i32): i32 = reduce (+) 0 a

The reduce construct takes a function that must be associative, and possess an identity element (0 is the identity for addition).

We can write a function that uses sum to sum the rows of a two-dimensional array:

let sum_rows (as: [n][m]i32): [n]i32 = map sum as

The sum_rows function makes use of nested parallelism, as a reduce (via sum) is used inside of a map. Mapping nested parallelism to efficient flat parallelism is one of the most important tasks performed by the Futhark compiler, and the last part of this post concerns itself with how that is done.

A few more constructs must be introduced before we can look at real Futhark code. First, we can use iota and replicate to create new arrays:

iota 5           ⇒        [0,1,2,3,4]
replicate 3 1337 ⇒ [1337, 1337, 1337]

Futhark has an important restriction regarding arrays: only regular arrays are permitted. An array is regular if all its elements have the same shape. Thus, [[1,2],[3,4]] is regular, but [[1,2], [3]] is not.

Second, Futhark provides special syntax for expressing sequential loops:

loop (x = 1) for i < n do
  x * (i + 1)

The meaning of the above expression is to first assign x the value 1, then execute the loop body n times, each time binding x to the result, and eventually return the final value of x as the result of the entire loop. There is nothing magical about for-loops. They are equal in power to the tail-recursive functions we are used to from other functional languages, but they make life a little easier for the compiler, as we shall see.

Case Study: K-Means Clustering

To introduce some of Futhark’s more novel constructs, as well as the transformations performed by the compiler, let us take a look at k-means clustering. The problem is to assign a collection of n points in some d-dimensional space to k clusters, such that the distance from a point to the centre of its cluster is minimised.

According to Wikipedia, the overall algorithm is as follows (images by I, Weston.pace, CC BY-SA 3.0):

  1. k initial cluster means (here k=3) are randomly generated within the data domain:

    The first step of K-means.

  2. k clusters are created by associating every observation with the nearest cluster mean:

The second step of K-means.

  1. For each of the k clusters, compute the average of all points in the cluster, which then becomes the new mean for the cluster:

The third step of K-means.

  1. Steps (2) and (3) are repeated until convergence is reached:

The fourth step of K-means.

For this post, we shall focus on the main computation of the third step. Each point has been associated with a cluster (given by an integer), and we already know the number of points in each cluster. The problem is thus: given n points and assignments of each point to one of k clusters, compute the mean of each cluster. We can do this sequentially by keeping a running tally of cluster means in the form of a k×d array, traversing the n input points, and updating the tally at the appropriate index. We begin by defining a function for adding two points:

let add_points(x: [d]f32) (y: [d]f32): [d]f32 =
  map (+) x y

This is just vector addition. There is one quirk: in Futhark, the map construct can operate on any number of array inputs, somewhat resembling the zipWith of Haskell. We can now sequentially compute cluster means like this:

let cluster_means_seq (cluster_sizes: [k]i32)
                      (points: [n][d]f32)
                      (membership: [n]i32): [k][d]f32 =
  loop (acc = replicate k (replicate d 0.0)) for i < n do
    let p = points[i]
    let c = membership[i]
    let c_size = f32 cluster_sizes[c]
    let p' = map (/c_size) p
    in acc with [c] <- add_points acc[c] p'

The most interesting part here is the update of the acc array, which is done with an in-place update. Semantically, the construct a with [i] <- b produces an array with the same elements as a, except with the value b at index i. In most purely functional languages, this would require a copy of the array a, but Futhark uses a mechanism based on uniqueness types to permit the update to be in-place. Essentially, the type checker verifies that the “source” array a is never used on any execution path following the in-place update. This permits reuse of the memory where a is stored, without violating referential transparency. Thus, Futhark permits some (simple) sequential code to be expressed efficiently.

The cluster_means_seq function performs O(n×d) work, but has little parallelism. Let’s try doing a better job. The idea is to map each point to a partial accumulator - a k×d array that is all zero except at the position corresponding to the cluster of the point. We then perform a reduction of all the partial accumulators, using matrix addition as the operator:

let matrix_add (xss: [k][d]f32) (yss: [k][d]f32): [k][d]f32 =
  map (\xs ys -> map (+) xs ys) xss yss

let cluster_means_par(cluster_sizes: [k]i32)
                     (points: [n][d]f32)
                     (membership: [n]i32): [k][d]f32 =
  let increments : [n][k][d]i32 =
    map (\p c ->
           let a = replicate k (replicate d 0.0)
           let c_size = f32 cluster_sizes[c]
           let p' = map (/c_size) p
           in a with [c] <- p')
        points membership
  in reduce matrix_add (replicate k (replicate d 0.0)) increments

(In Futhark, an anonymous function is written as (\x -> ...), just as in Haskell.)

This version is fully parallel, which is great, but not work efficient, as it requires O(k×n×d), compared to the O(n×d) of the sequential version. The core issue is that cluster_means_par is too parallel. Real machines are not infinitely parallel, but have some maximum parallel capacity beyond which adding more threads simply means more overhead, without obtaining better hardware utilisation. Ideally, each thread sequentially processes some chunk of the input, followed by a parallel combination of the per-thread partial results. Thus, we pay only for the parallelism that we can profitably use:

A visualisation of how each thread processes a sequential
chunk of the input, followed by each per-thread result being
combined into a single result.

However, the ideal number of threads depends on the concrete hardware (and other runtime factors), and therefore should not be baked into the program by the programmer. The solution is to provide a language construct that encapsulates both efficient sequential execution, as well as how to combine per-thread results. In Futhark, this constructed is called stream_red for stream reduction, and is used like this:

let cluster_means_stream(cluster_sizes: [k]i32)
                        (points: [n][d]f32)
                        (membership: [n]i32): [k][d]f32 =
  let on_chunk [chunk_size]
               (points':     [chunk_size][d]f32)
               (membership': [chunk_size]i32) =
        cluster_means_seq cluster_sizes points' membership'
  in stream_red
       matrix_add on_chunk
       points membership

The combination function (which must be associative, as with reduction) is matrix addition. The local function on_chunk is called to sequentially process each chunk within a thread, and itself merely calls the cluster_means_seq function we defined above. The chunk_size may vary freely between threads, and is not known until runtime.

One nice property about the stream_red construct is that, if deemed necessary, the compiler can “recover” the fully parallel implementation by using n threads with a chunk size of 1, or the fully sequential implementation by setting the chunk size to n. In essence, stream_red provides a “dialable” amount of parallelism.

During compilation, the compiler will break up the stream_red into a per-thread part and an ordinary reduction:

let per_thread_results : [num_threads][k][d]f32 =
-- combine the per-thread results
let cluster_means =
  reduce (map (map (+)))
         (replicate k (replicate d 0.0))

(The (map (map (+))) part is presently not valid Futhark syntax, but is used here for simplicity - it’s just the matrix addition from above.)

I have left out the expression that computes per_thread_results, as it depends on the internal compiler representation. The num_threads variable is some value computed at run-time based on the hardware (and can be subject to tuning).

The reduction with (map (map (+))) is not great, as the intermediate k×d matrices are too large to fit in the GPUs fast on-chip memory (a kind of manually managed cache). Thus, the Futhark compiler will perform a transformation called Interchange Reduce With Inner Map (IRWIM), which moves the reduction inwards at the cost of a transposition:

let per_thread_results' : [k][d][num_threads]f32 =
  rearrange (1,2,0) per_thread_results
let cluster_means =
  map (map (reduce (+) 0.0)) per_thread_results'

The rearrange construct permutes the dimensions of an array, here transposing the outermost dimension of per_thread_results innermost - see how the type changes from [num_threads][k][d]f32 to [k][d][num_threads]f32.

The only problem now is that the two map-parallel dimensions are of size k and d, which is likely not enough to fully saturate the GPU. Fortunately, the compiler is smart enough to recognise that a reduce inside of maps corresponds to a pattern called a segmented reduction, which has an efficient implementation on GPUs (details in an upcoming paper!).

The implementation based on reduction streams is significantly faster than the fully parallel one. On an NVIDIA Tesla K40 GPU with k=5, n=10,000,000, d=3, the function cluster_means_par executes in 131.1ms, while cluster_means_stream executes in 17.6ms - a speedup of 7.6×.

Improving Available Parallelism via Loop Distribution and Interchange

Futhark as a language supports (regular) nested data parallelism, but GPUs prefer flat parallelism. A GPU kernel is lingo for a GPU program, which we can think of as a perfect map nest containing either sequential code, or specific known patterns of parallelism, like reduce. Only such known patterns can be executed by the GPU. The Futhark compiler therefore has the job of turning complicated nestings of parallel constructs into perfectly nested maps, each corresponding to a single GPU kernel. As an example, consider this fragment of code:

map (\xs -> let y = reduce (+) 0 xs
            in map (+y) xs)

We have an outermost map, the body of which contains further parallelism in the form of a reduce and another map. If we wish, we can simply parallelise the outermost map and compile the inner parallel operators to sequential code, thus producing one GPU kernel. This will limit the amount of parallelism we extract from the program, but if the outer map operates on enough elements, then that may well be the right choice. Alternatively, we can distribute each of the two inner SOACs to their own map nest:

let ys = map (\xs -> reduce (+) 0 xs) xss
in map (\xs y -> map (+y) xs) xss ys

Note how the intermediate result y has now been lift to an array of intermediate results ys, which is passed into the second map nest. This form corresponds to two GPU kernels, each providing more parallelism than the single one from before. Since a GPU typically requires kernels to contain tens of thousands of threads in order to fully utilise the hardware, this transformation is sometimes necessary.

For functional languages, this problem was in principle solved in the early 90s in NESL by Guy Blelloch. NESL defines a flattening algorithm (sometimes called vectorisation) that describes how to turn arbitrary nested data-parallelism into flat data-parallel operations. The flattening algorithm is universal, in that it always works (provided the language fulfils a few criteria, such as purity). Unfortunately, full flattening has a few problems:

  1. Always maximises available parallelism, even when not worthwhile (e.g innermost loops in a matrix multiplication).
  2. Wasteful of memory (fully flattened matrix multiplication requires O(n³) space).
  3. Destroys access pattern information, rendering locality-of-reference optimisations such as loop tiling hard or impossible.

Thus, the Futhark compiler takes a step back and uses a kernel extraction algorithm based on limited flattening. It’s not as universal as full flattening, but for those cases where you don’t need to fully maximise parallelism, it can generate substantially faster code (and the Futhark compiler could always fall back to full flattening if necessary).

The algorithm is based on the rich set of rewrite rules permitted by functional languages. For example, there is a well-known rule describing how to compose two maps into one:

map f ◦ map g ⇒ map (f ◦ g)

This rule is used in the Futhark compiler to perform loop fusion, but it can also be reversed to obtain fission:

map (f ◦ g) ⇒ map f ◦ map g

This, along with other higher-order rules (details in the paper), are applied by the compiler to extract perfect map nests. When and how to apply the rules is currently determined by heuristics in the compiler. As an example, let us consider the following contrived program:

let bss: [m][m]i32 =
  map (\(ps: [m]i32) (ps: [m]i32) ->
        loop (ws=ps) for i < n do
          map (\w -> w * 2) ws)

Let us assume that the array pss (the outermost input array) has type [m][m]f32, for some m. We could choose to simply parallelise the outermost map as a single kernel with m threads. Depending on the data set, this may be the best choice, but in this case the compiler will try to improve the amount of exposed parallelism. Specifically, the compiler will interchange the outer parallel map and the inner sequential loop:

let bss: [m][m]i32 =
  loop (wss=pss) for i < n do
    map (\ws ->
          map (\w -> w * 2) ws)

This interchange has made a perfect map nest (of size m×m) visible, which can be turned into a fully parallel GPU kernel. This kernel will be executed n times in total because of the now-outermost sequential loop. The question becomes: is executing an m×m kernel n times better than executing a size m kernel once, if each of those m threads run n iterations of a sequential loop? The answer depends on the exact values of n and m. If m is sufficiently large, then the GPU can be fully utilised with just m threads, but otherwise, full utilisation requires us to also exploit the innermost parallelism, even if it comes at the overhead of launching more kernels.

In the future, we intend to have the Futhark compiler generate several different versions of the program, based on different parallelisation decisions, and choose the best one at run-time, based on characteristics of the actual input data. For now, hand-picked heuristics are used.

Gotta Go Fast!

Futhark is based on the idea that a restricted language permits a more powerful compiler. However, we must be careful not to restrict the language so much that it becomes useless for its intended purpose. While Futhark is not designed for full application programming, it should be able to efficiently represent a broad set of parallel algorithms. To demonstrate Futhark’s power and flexibility, we have ported various published benchmarks and examples from hand-written OpenCL and CUDA GPU code to Futhark. The graphs below demonstrate the speedup of Futhark implementations of nine benchmarks ported from Rodinia, running on both an NVIDIA GTX 780 Ti GPU and an AMD W8100 GPU:

Speedup graphs.

Speedup graphs.

In all cases, Futhark performs acceptably. The point of a language such as Futhark is not to exceed the performance of highly optimised code painstakingly tuned by experts - that’s not really realistic -but instead to provide easily accessible performance that is ideally within a factor of two of the performance of hand-written code. The benchmark implementations, as well as several more, can be seen in our futhark-benchmarks repository.

Somewhat surprisingly, as we can see, code generated by the Futhark compiler is often faster than hand-written code. One reason is that GPU performance is a sensitive thing, in particular when it comes to memory access patterns. The transformations needed to obtain optimal performance are error-prone (and very tedious) to do by hand, but feasible to automate in a compiler. Another reason is that parallelisation opportunities are sometimes missed by the human programmer. This is the case for the NN benchmark, where a reduction was left sequential in the reference implementation, but parallelised in Futhark - the result is that the Futhark program is 16.3× faster on the GTX 780 Ti than the hand-written program.

In Conclusion

This post skips many details (read the paper!), but hopefully managed to communicate two main points:

  1. A restricted/specialised language permits a clever compiler that saves the programmer from worrying about low-level details
  2. An idea of what such an approximation to a sufficiently smart compiler actually does.

And here is a bonus third point:

  1. Futhark is a simple but fast language that you can try out yourself. We have previously written about how to inter-operate Futhark with Python, which can be used to create fancy interactive Futhark programs.

If you are curious about seeing more Futhark code, we have more examples to peruse.