Posts Tagged ‘Direct Compute’

Vector –Matrix Inner Product with Computer Shader and C++ AMP

Large vector-matrix inner products by the GPU are 250 times faster than straight forward CPU implementations on my PC. Using C++ AMP or a Compute Shader the GPU realized a performance of over 30 gFLOPS. That is a huge increase, but my GPU has a “computational power” (whatever that may be) of 1 teraFLOP, and 30 gFLOPS is still a long way from 1000 gFLOPS.

This article presents a general architectural view of the GPU and some details of a particular exemplar: the Ati Radeon HD5750. Then code examples follow that show various approaches to large vector-matrix products. Of course the algorithms at the end of the article are the fastest. It is also the simplest.

Unified View of the GPU Architecture

Programming the GPU is based on an architectural view of the GPU. The purpose of this architectural view is to provide a unified perspective on GPUs from various vendors, hence with different hardware setup. It is this unified architecture that’s being programmed against using DirectX11. A good source of information on Direct Compute and Compute Shaders is the Microsoft Direct Compute BLog. The architecture described below is based on information from Chas Boyd’s talk at PDC09, as published on Channel9. Of course, this blog post only presents some fragments of the information found there.

A GPU is considered to be build from a number of SIMD cores. SIMD means: Single Instruction Multiple Data. By the way, the pictures below are hyperlinks to their source.

The idea is that a single instruction is executed on a lot of data, in parallel. The SIMD processing unit is particularly fit for “data parallel” algorithms. A GPU may consist of 32 SIMD cores (yes, the image shows 40 cores) that access memory with 32 floats at a time (128 bit bus width). Typically the processor runs at 1Ghz, and has a (theoretical) computational power of about 1 TeraFLOP.

A SIMD core uses several kinds of memory:

  • 16 Kbyte of (32-bit) registers. Used for local variables
  • 8 Kbyte SIMD shared memory, L1 cache.
  • L2 cache

The GPU as a whole has typically 1Gb of general RAM. Memory access bandwidth is typically of order 100GBit/s.

Programming Model

A GPU is programmed using a Compute Shader or C++ AMP. Developers can write compute shaders in HLSL (Looks like C) to be executed on the GPU. AMD is a C++ library. The GPU can run up to 1024 threads per SIMD. A thread is a line of execution through code. The SIMD shared memory is shared among the threads of a SIMD. It is programmable in the sense that you can declare variables (arrays) as “groupshared” and they will be stored in the Local Data Share. Note however, that over-allocation will spill the variables to general RAM, thus reducing performance. Local variables in shader code will be stored in registers.

Tactics

The GPU architecture suggests programming tactics that will optimize performance.

  1. Do your program logic on the CPU, send the data to the GPU for operations that apply to (about) all of the data and contain a minimal number of alternative processing paths.
  2. Load as much data as possible into the GPU general RAM, so as to prevent the GPU waiting for data from CPU memory.
  3. Declare registers to store isolated local variables
  4. Cache data that you reuse in “groupshared” Memory. Don’t cache data you don’t reuse. Keep in mind that you can share cached data among the threads of a single group only.
  5. Use as much threads as possible. This requires you use only small amounts of cache memory per thread.
  6. Utilize the GPU as efficiently as possible by offering much more threads to it than it can process in a small amount of time.
  7. Plan the use of threads and memory ahead, then experiment to optimize.

Loading data from CPU memory into GPU memory passes the PCIe bridge which has a bandwidth, typically of order 1GBit/s; that is, it is a bottleneck.

So, you really like to load as much data onto GPU memory before executing your code.

The trick in planning your parallelism is to chop up (schedule, that is J ) the work in SIMD size chunks. You can declare groups of threads; the size of the groups and the number of groups. A group is typically executed by a single SIMD. To optimize performance, use Group Shared Memory, and set up the memory consumption of your thread group so it will fit into the available Group Shared Memory. That is: restrict the number of threads per group, and make sure you have a sufficient number of groups. Thread groups are three dimensional. My hypothesis at this time is that it is best to fit the dimensionality of the thread groups to match the structure of the end result. More about this below. Synchronization of the threads within a thread group flushes the GroupShared Memory of the SIMD.

A register typically has a lifetime that is bound to a thread. Individual threads are member of several groups – depending on how you program stuff. So, intermediate results aggregated by thread groups can be stored in registers.

Does My ATI Radeon HD5750 GPU Look Like This Architecture… A Bit?

The picture below (from here) is of the HD5770, which has 10 SIMD cores, one more than the HD5750.

What do we see here?

  • SIMD engines. We see 10 cores for the HD5770, but there are 9 in the HD5750. Each core consists of 16 red blocks (streaming cores) and 4 yellow blocks (texture units).
  • Registers (light red lines between the red blocks).
  • L1 Textures caches, 18Kbyte per SIMD.
  • Local Data Share, 32 Kbyte per SIMD.
  • L2 caches, 8 Kbyte each.

Not visible is the 1Gb general RAM.

The processing unit runs at 700Mhz, memory runs at 1,150Mhz. Over clocking is possible however. The computational power is 1,008 TeraFLOP. Memory bandwidth is 73.6 GBit/s.

So, my GPU is quite a lot less powerful than the reference model. At first, a bit disappointing but on the other hand: much software I write for this GPU cannot run on the PCs of most people I know – their PCs are too old.

Various Approaches to Vector-Matrix Multiplication

Below we will see a number of approaches to vector-matrix multiplication discussed. The will include measurements of time and capacity. So, how do we execute the code and what do we measure?

Times measured include a number of iterations that each multiply the vector by the matrix. Usually this is 100 iterations, but fast alternatives get 1000 iterations. The faster the alternative, the more we are interested in variance and overhead.

Measurements:

  • Do not include data upload and download times.
  • Concern an equal data load, 12,288 input elements if the alternative can handle it.
  • Correctness check; computation is also performed by CPU code, reference code.
  • Run a release build from Visual Studio, without debugging.
  • Allow AMP programs get a warming up run.

Vector-Matrix Product by CPU: Reference Measurement

In order to determine the performance gain, we measure the time it takes the CPU to perform the product. The algorithm, hence the code is straightforward:

In this particular case rows = cols = 12,288. The average over 100 runs is 2,452 ms, or 2.45 seconds. This amounts to a time performance of 0.12 gFLOPS (giga FLOPS: FLoating point Operations Per Second). We restrict floating point operations to addition and multiplication (yes, that includes subtraction and division). We calculate gFLOPS as:

2 / ms x Rows / 1000 x Cols / 1000, where ms is the average time in milliseconds.

The result of the test is correct.

Parallel Patterns Library

Although this blog post is about GPU performance, I took a quick look at PPL performance. We then see a performance gain of a factor 2, but the result is incorrect, that is, the above code leads to indeterminacy in a parallel_for loop. I left it at that, for now.

Matrix-Matrix Product

We can of course, view a vector as a matrix with a single column. The C++ AMP documentation has a running code example of a matrix multiplication. There is also an accompanying compute shader analog.

AMP

To the standard AMP example I’ve added some optimizing changes, and measured the performance. The AMP code look like this:

Here: amp is an alias for the Concurrency namespace. The tile size TS has been set to 32, which is the maximum; the product of the dimensional extents of a compute domain should not exceed 1024. The extent of the compute domain has been changed to depend on B, the matrix, instead of the output vector. The loop that sums element products has been unrolled in order to further improve performance.

As mentioned above, we start with a warming up. As is clear from the code we do not measure data transport to and from the GPU. Time measurements are over 100 iterations. The average run time obtained is 9,266.6 ms, hence 0.01 gFLOPS. The result after the test run was correct.

The data load is limited to 7*1024 = 7,168; that is 8*1024 is unstable.

Compute Shader

The above code was adapted to also run as a compute shader. The code looks like this:

The variables Group_SIZE_X and Group_SIZE_Y are passed into the shader at compile time, and are set to 32 each.

Time measurements are over 100 iterations. The average run time obtained is 11,468.3 ms, hence 0.01 gFLOPS. The result after the test run was correct. The data load is limited to 7*1024 = 7,168; that is 8*1024 is unstable.

Analysis

The performance of the compute shader is slightly worse that the AMP variant. Analysis with the Visual Studio 11 Concurrency Visualizer shows that work by the GPU in case of the compute shader program is executes in small spurts, separated by small periods of idleness, whereas in the AMP program the work is executed by the GPU in one contiguous period of time.

Nevertheless, performance is bad, worse than the CPU alternative. Why? Take a look at the picture below:

For any value of t_idx.global[0] – which is based on the extent of the matrix- that is unequal to zero, vector A does not have a value. So, in fact, if N is the number of elements in the vector, we do O( N3)retrievals but only O(N2) computations. So, we need an algorithm that is based on the extent of a vector, say the output vector.

Vector-Matrix Product

Somehow, it proved easier to develop the vector-matrix product as a compute shader. This is in spite of the fact that unlike AMP, it is not possible (yet?) to trace a running compute shader in Visual Studio. The idea of the algorithm is that we tile the vector in one dimension, and the matrix in two, thus obtaining the effect that the vector tile can be reused in multiplications with the matrix tile.

Compute Shader

A new compute shader was developed. This compute shader caches vector and matrix data in Group Shared memory. The HLSL code looks like this:

This program can handle much larger amounts of data. Indeed, this program runs problem free for a vector of 12,288 elements and a total data size of 576 Mbyte. Using an input vector of 12,288 elements, with total data size of 576 Mbyte. The time performance is 10.3 ms per run, averaged over 1,000 runs, which amounts to 29.3 gFLOPS. The result of the final run was reported to be correct.

AMP

In analogy to the compute shader above I wrote (and borrowed 🙂 ) a C++ AMP program. The main method looks like this:

The matrix is a vector with size * size elements. He tile size was chosen to be 128, because that setting yields optimal performance. The program was run on an input vector of 12,288 elements again, with total data size of 576 Mbyte. The time performance is 10.1 ms per run, averaged over 1000 runs, which amounts to 30.0 gFLOPS. The result of the final run was reported to be correct.

Analysis

We see here that the performance has much improved. When compared to the reference case, we can now do it (in milliseconds) 2,452 : 10.1 = 243 : 1, hence 243 times faster.

Simpler

Then, I read an MSDN Magazine article on AMP tiling by Daniel Moth, and it reminded me that caching is useless if you do not reuse the data. Well, the above algorithm does not reuse the cached matrix data. So I adapted the Compute Shader program to retrieve matrix data from central GPU memory directly. The HLSL code looks like this:

Note the tileSize of 512(!). This program was run for a vector of 12,288 elements and a total data size of 576 Mbyte. The time performance is again 10.3 ms for a multiplication which amounts to 29,3 gFLOPS (averaged over 1000 runs). The result of the final run was reported to be correct. So, indeed, caching the matrix data does not add any performance improvement.

AMP

For completeness, the AMP version:

Time performance is optimal for a tile size of 128, in case the number of vector elements is 12,288. We obtain an average run time of 9.7 ms (averaged over 1,000 runs), and a corresponding 31.1 gFLOPS. The result of the final run was correct. This program is 2452 / 9.7 = 252.8 times as fast as the reference implementation.

Conclusions

Developing an algorithm for vector-matrix inner product has demonstrated comparable performance for Compute Shaders and AMP, but much better tooling support for AMP: we can step through AMP code while debugging, and the Concurrency Visualizer has an AMP line. This better tool support helped very well in analyzing performance of a first shot at the algorithm. The final algorithm proved over 250 times faster than a straight forward CPU program for the same functionality.

Detailed knowledge of the GPU architecture, or the hardware model, proved of limited value. When trying to run the program with either the maximum nr of threads per group, or the maximum amount of data per Group Shared Memory, I ran into parameter value limits, instabilities, performance loss, and incorrect results. I guess, you will have to leave the detailed optimization to the GPU driver and to the AMP compiler.

One question keeps bothering me though: Where is my TeraFLOP?

I mean, Direct Compute was introduced with the slogan “A teraFLOP for every one of us”, AMP is built on top of Direct Compute, and my GPU has a computational power of 1.08 TeraFLOP. Am I not ‘one of us’?

C++ AMP Performance and Compute Shader Performance

Edit (April 23rd 2012):

The AMP team has updated the N-Body Simulation code to turn it into a clean port that relates to the Compute Shader original in a comprehensible way. Now it has comparable performance to the original (optimized) version (both versions do >330 gFLOPS at >30 fps for 23,040 particles on my pc).

I’m impressed. For one, by the attitude of the AMP people that energetically reacted to issues which other people / teams might well have dismissed as unimportant. Then there is the point that you get maximum performance from a set of very powerfull processors with code that is very short compared to the direct compute code you had to write otherwise, and this code, by AMP design, is very elegant as well.

Of course, there is a risk in short and elegant code: subtle differences in code can make substantial differences in performance, hence developing AMP code is rather knowledge intensive. But I kind of like that.

Edit (April 16th 2012):

The results below were brought to the C++ AMP forum for discussion. Daniel Moth advised to update the driver of the graphics card. This update made a tremendous difference for two of the three programs mentioned below for which now C++ AMP performance is equal to or better than Compute Shader performance.

The discussion on the N-Body Simulation program, which is heavily optimized in the Compute Shader version is still open, mainly because the required information is not available yet. I expect that also in this case C++ AMP will prove to be equipotent to Compute Shader programs.

Now, what have we learned from this exercise? For one, a lot about Compute Shader optimization and the mechanisms of GPU computing performance. This is an interesting and instructive subject. I also have learned that C++ AMP performance is comparable to Compute Shader performance. However, I do not (yet) understand if and how this will always and necessarily be the case, and that still itches a bit.

Results as they are standing now:

 

Program

AMP

CS

 

Guide

 

 

Average time (ms, 10 it.)

 

2,650

2,995

gFLOPS

 

36.9

32.7

Max. Data Load (Kb)

 

714,432

691,200

 

Vector Addition

 

 

Average time (ms, 10 it.)

 

6,017

8,155

gFLOPS

 

0.03

0.02

Max. Data Load (Kb)

 

1,781,248

2,039,056

 

N-Body Simulation

 

 

Number of Particles

 

16,128

16,128

Frame rate

 

44.4

63.4

gFLOPS

 

229

329

Up to date, I find that Compute Shader based programs outperform C++ APM programs both in time and space. Results of example programs I explored, which have been created by the respective product teams tend to show substantially better performance by the Compute Shader programs. These programs are the N-Body Simulation Sample; Basic Summation; and the matrix multiplication programs from the “C++ AMP for the DirectCompute Programmer” guide. Hyperlinks are provided in the sections below.

So, the question is: can there be an AMP program that performs substantially better in time and space on, let’s say, large matrix multiplication (or large matrix-vector multiplication) than a Compute Shader program? C++ AMP has been built upon Direct Compute, so the answer is: not likely.

Should we, alternatively, draw the conclusion that a direct compute program categorically has better performance?

N-Body Simulation

The first pair of programs compared, consisted of:

Performance is expressed in gFLOPS. The code for the gFLOPS was copied from the C++ AMP version to the Compute Shader version. I also changed the Compute Shader version to make it write gFLOPS and the number of particles to the screen.

First, I tweaked the particle count parameter to get the best gFLOP count from either program; they both peak at 16,128 particles on my PC. Then the following results (gFLOPS) were obtained for release builds, running without debugging (this was also the configuration in the comparisons below).

C++ AMP Compute Shader More (%) Less(%)
Number of particles 16,128 16,128
Frames per second 43.46 57.38 32.03 24.26
gFLOPS 226.07 298.51 32.04 24.27

A note on the More and Less columns: The Compute Shader version delivers 32.03% more frames per second, and the C++ AMP version 24.26% less. So crudely: the Compute Shader version is about 30% faster.

Vector Addition

The second pair of programs compared consisted of:

The C++ AMP code was adapted as follows:

  • It was made to work with the same structs as the BasicCompute11 sample. This struct consists of an int and a float.
  • The arrays were made global variables.
  • A loop was added to fill the input arrays.
  • The verification code from the BasicCompute11 sample was added.

For timing, timing code was added to both programs. This timing code is from this post in the Parallel Programming in Native Code blog.

For timing measurements the code was adapted as follows: In the Compute Shader program timing covers code from the Dispatch call to the Map call. In the AMP program timing covers the lambda expression, and an added array_view::Synchronize() call on the “sum” array_view.

In experiments I first pushed the size until, in the case of the Compute Shader version, the output of the result verifying code became “failure”,

and in the case of the C++ AMP program, it either didn’t compile or produced a runtime error.

Then I measured time and gFLOPS. The experiments yielded the following result.

C++ AMP Compute Shader More (%) Less(%)
Number array elements 76*10^6 87*10^6 14.47 12.64
Total data size (Kb) 1,781,250 2,039,062.5
Time (ms) 6,868 8,182
gFLOPS 0.022 0.021

gFLOPS were measured as: 2*n / (10^6 * ms), where n is the number of elements in an array.

It seems to me that the time results are too similar to call them different. The Compute Shader version has a slight space advantage.

Note that since the total data size in both cases is larger than the RAM the graphics card has on board, there is some automatic sectioning going on.

Matrix Multiplication

Both programs in this comparison come from the C++ AMP for the DirectCompute Programmer guide. This guide can be obtained from a post on the official MSDN Parallel Programming in Native Code blog. The C++ AMP program is a transformation of the Compute Shader program.

The code for the starting point of the transformation is not entirely complete, so I added standard code from the BasicCompute11 Sample that loads and compiles the compute shader.

The following results were obtained.

C++ AMP Compute Shader More (%) Less(%)
Number array elements 4,608 7,616 65.28 39.50
Total data size (Kb) 248,832 679,728 173.17 63.39
Av Processing time (ms, 10 runs) 11,742 12,804
gFLOPS 8.3 34.5 315.66 75.94

Notes:

  • Both programs measure the time spent in the “mm” function, using the timing code referred to above. This includes uploading and offloading the data onto and from the GPU.
  • For both programs we have that any higher multiple of 64 in the number of array elements crashes the display driver.

  • gFLOPS are measured as: n^3 / (10^6 * ms) where:
  • n is the size of a matrix dimension (the matrices are square).
  • Ms is the averaged (over 10 iterations) measured processing time in milliseconds.

Conclusions

Three program pairs have been compared, informally and semi-systematically, for their performance in time and space.

In the case of the N-Body simulation, the data load was selected that is optimal for time performance. That resulted in an about 30% better time performance of the Compute Shader Program.

In the case of vector addition – about the simplest program imaginable in this context – the time performance was measured for maximum data load. This resulted in practically equal time performance for both programs. The Compute Shader version can load some more data.

Finally, the programs from the AMP guide for Compute Shader programmers were implemented, and the time performance was again measured for maximum data load. This resulted in a time performance of the Compute Shader that is three times as good as the time performance of the AMP program.

So, conclusion, it seems that if you want to get the max from your GPU, a Compute Shader is still the way to go.