On the Fast MM for GPUs

This month,  I read a paper about a Strassen-like MM for a GPU published in a distributed computing conference.  The paper is academic but it can be read quickly and it provides what it promises: the implementation of both Strassen and Winograd algorithms on a GPU.  I am always pleased to read that the community starts investigating Fast MM and in particular Winograd’s, this means that the HW is reaching the performance limit and an algorithmic view kicks in. Practitioners and more theoretical researchers start using old algorithms in this new arena and claim performance and error: being a new arena, GPUs are nice little computational engines, allows a freedom that rarely is provided.  It is like a field covered with fresh snow and we are the first stepping on it.

The literature liberty to compare the GPUs to a white snow field is a big one. The GPUs are the current computational engines and god-like capable scientist  such as prof. Dongarra and prof. Demmel are providing powerful Matrix Multiplication kernels and quite smart researchers with them for quite some time now. Some kernels are designed to be fast and others to be more accurate and less fast.  I noticed that  because the computational engines is new the algorithm is simplified: thread computation are simple row-by-column computations. It makes sense, these engines allows lots of thread and to make them independent the easy way out is the tiling of matrix C into independent computations. But this is not the core of my train of thoughts.

In the article that is the inspiration for this post, the authors show a 30% performance improvement and 200x error for both Strassen and Winograd algorithms.  The authors know want they want to show and provide a set of experiments to confirm this: they use power of two matrices to show performance and error analysis.

What is wrong with that? Really, nothing. Power of two are useful, the code is simple, the problem size double every time and the number-of-operation complexity increases by 8. Every thing is clean and it is easy. It easy to explain and the message is simple to understand. If you need performance and you are willing to loose 2 digits, the least you can achieve is just 30% speed up on matrices of size 16Kx 16K. Sorry, I forgot to tell you the problem size. Is it  big, uh?

Such a result and such a paper will scare any engineer working on the implementation of MM kernels: you need a huge/large problem size; when you do have a large one, instead of 100 seconds, the computation will take 70 seconds on these machines, and  you will have the result correct to the second position.  This is disarming, right?  I will bet that the final conclusion any reader will have is negative. The reader will not care to use any fast algorithms, simply put it, it will  not be worth it.

I think this is the course of being the first stepping on the fresh snow: if no new snow will cover the path traced, most engineers evaluating their options, especially who will not have time to read other papers or evaluate the performance on their own, they will have to skip Fast MM.

What frustrates me is the too narrow punch line. Despite the classic scientific approach, theory, algorithms, implementation, and experimental results,  I know that the paper presents one side of a coin, and there is the other one just right there, waiting patiently to be told. Furthermore, I know it has been told several times previously.

 

 

 

 

Matrix Multiply, All-Pairs Shortest Path, and Large Dense Graphs

Take a node in a graph and consider it as a user in twitter. A link between two users is the minimum retweet time. The link represents the close connection between users. You may be interested to compute the distance among all users. Why ? To determine clusters, to find the closest nit of friends, to find relations between celebrity, or because WTF we can.

In the era of large graphs it may be interesting to compute distances among nodes. If a weighed edge represents the distance between two nodes, we  could compute the all-pairs shortest path (APSP) so that to know, given any node, what is its distance from any other node in the graph. APSP and MM are computationally equivalent (“the design and analysis of computer algorithms”  Aho, Hopcroft, and Ullman 1974. This is a great book to have anyway, I stole my advisor’s copy) and we have shown that incidentally they are the same. I will clarify in the post.

Recursive algorithms are extremely appealing for the design of matrix computation because of their natural cache locality exploitation and for their intuitive formulation. In this section, we present a recursive D&C algorithm derived from Kleene’s algorithm Figure 1.(a). Notice that Kleene’s algorithm was originally designed to solve the transitive closure (TC) of an adjacency matrix. That is, finding whether or not there is a path connecting two nodes in directed graph. However, Kleene’s algorithm is also a standard algorithm/solution for the APSP. In fact, in a closed semi-ring TC and APSP are the same problem and Kleene’s algorithm is a solution (when the scalar operators ∗ and + are specified as in the following paragraph) and it determines every edge of the (shortest) path directly [2].

A brief description of Kleene’s algorithm follows. We divide the basic problem into two sub-problems, and we solve each subproblem directly using the Floyd-Warshall algorithm, see Figure 1.(a). Then, we perform several MMs to combine the results of the subproblems using a temporary matrix; where the scalar addition of two numbers is actually the minimum of the two numbers –i.e., a + b = min(a, b)– and the scalar multiplication of two numbers is the (regular arithmetic) addition –i.e., a ∗ b = a + b.

Figure 1.(a)                                             Figure 1.(b)

 

 

 

 

Figure 1.(c)

In this case, the MM is defined in a closed semi-ring and using the properties within, we reorganize the algorithm in Figure 1.(a) and obtain the R-Kleene algorithm in Figure 1.(b). Thus, R-Kleene is a solution for APSP in a closed semi-ring. Please, note that in literature it is suggested to use a power method using MM (log(n) times) but as you can see it is not necessary.

We can say that the ASPS can be formulated as a self-matrix multiplication. This finding is not that interesting per se:  MM is highly parallel while APSP was known as really sequential making it not very compelling for large graphs.

So if you like to compute the shortest path across all nodes in a large graph, relatively dense and usign a highly parallel algorithm: R-Kleene + MM is the algorithm for you hands down.

What is the connection with Fast MM ?

Fast MM are in general NOT applicable so that to speed up the MM. This is because Fast MMs deploy cancellation operations (substractions or addition by opposite). In the APSP problem the + operation is a min() operation that is not invertable. Let me clarify this point: take a+b = c where the operation + is the arithmetic addition, then having c, and having either operand we can get the other: b = c – a or  a = c – b. Unfortunately,  min() does have this basic property (infact this is a semiring dang it). In the literature, there are example show to extend a few problem from the semiring to a ring and making possible to use Fast MM.

If your links are discrete numbers (integers), there are simple way to extend the problem to a ring. I will love to see how fast algorithms will do here, in this case no numerical error to bother either.

For more information, please take a look at our Algorithmica paper.

 

The Unbearable Lighteness of Being Wrong


Every one seems to know that Fast MM based on Winograd-Like algorithm cannot be as accurate as the regular MM. A few believe that Winograd-Like algorithm are unstable numerically. A few believe that are stable for any practical purpose.  All have a very strong opinion. No one has a clear understanding.

In this post, I will talk about the contributions by Prof. Richard. P. Brent, who really started the investigation for Fast MM, Prof. Higham, one of the gods in numerical algorithms, and Prof. Bini who really brought forth the generalization analysis and estimate for Strassen like algorithms. To them, I must pay my dues and give credits. If I have any contribution, it is in the empirical measure of the accuracy of fast MM codes for large problem sizes where fast MM are actually applicable and useful in modern architectures.

Let me start with the main statement. The forward error bound for the conventional C=AB  MM with matrices of size nxn can be written as follows:

(1)   \begin{equation*} |{\bf  C - \dot{C} }| \leq nu{\bf |A||B|} + O(u^2) \end{equation*}

Where the term on the left of the inequality represents the component-wise difference  or the forward error between the result matrix and its practical computation. The right side is the bound and it says that the error is a function of the range of matrix A and B multiplied by the size of the matrices and the precision of the architecture u. When the matrices have large values, the error we commit in computing the MM is large and, as the problem gets bigger, the error increases linearly as the problem size.   To achieve the bound in Equation 1 the algorithm has to perform at least n^3  multiplications. Fast MM cannot have the same component-wise (this have been shown by Miller 1975) because they compute fewer multiplications. I am reading and digesting Miller’s original paper.

In practice, each component of the matrix C is the sum of n terms. Each component is independent and it can be computed independently. Optimizations such as tiling breaks the computations and interleaves them, but very often the original order of the additions remains unchanged (just interleaved with other sums).

If we take the matrices A and B with elements in the range [0,1], we can see that each component of the product |A||B| has bound n and we can easily bound the right side by u(n^2).

The main problem for the addition is when two large and similar operands cancel each other. For example,  1.0001*10^5 – 0.9999*10^5 = 20, it may be that due to the finite representation of the mantissa the actual computation return 0 (instead of 20). An error of 20 w.r.t. the operands is small, but w.r.t. to the result is very large. You can see as the value of the matrices comes to play when something goes wrong.  Small matrices will produce small errors (note that we cannot say the same thing about relative errors).

In practice, no component-wise bound is found for Fast MM and instead is presented a norm-wise bound. A component-wise provides an independent bound to each element of the result matrix. A norm-wise bound provides a single bound for all. A norm bound is weaker in this sense.

(2)   \begin{equation*} \|{\bf C -\dot{C}} \|\leq n^2u\|{\bf A}\|\|{\bf B}\| +O(u^2)  \end{equation*}

The norm of a matrix A is defined as

\|A\| \equiv \max_{i,j}|a_{ij}|.

The first thing you may notice between the two bounds in Equation 1 and 2 is the square factor n^2 factor. This is because ||AB|| <= n ||A||||B||. Again take the matrices with elements in [0,1] and  ||A|| = ||B || = 1.

By reading again the references, I finally understand the contributions by Brent and Bini, and thus I appreciate much better their research. Let see if I can convey here their message and shed any insight about the accuracy of Fast MM and its error:

(3)   \begin{equation*} \dot{\bf C} = {\bf C}+{\bf E} \end{equation*}

We want to estimate E or its norm ||E||.

Brent work out a bound like a real mathematicians (from the eye of an engineer). He starts with an assumption. Assume that

(4)   \begin{equation*} \|{\bf E}\| \leq uf(n)\|{\bf A}\|\|{\bf B}\| \end{equation*}

and the goal is to find f(n) for the specific algorithm: Strassen, Winograd or any of their variations. Brent uses Strassen and here we follow.

{\bf C}_0 = P_0 - P_2-P_4+P_6 \\ {\bf C}_1 = P_3 - P_0 \\ {\bf C}_2 = P_1 + P_2 \\ {\bf C}_3 = -P_1 - P_3-P_4+P_5

Where

P_0 = ({\bf A}_0 - {\bf A}_1){\bf B}_3 \\P_1 = ({\bf A}_2 - {\bf A}_3){\bf B}_0 \\P_2 = {\bf A}_3({\bf B}_0 + {\bf B}_2) \\P_3 = {\bf A}_0 ({\bf B}_0 + {\bf B}_3) \\P_4 = ({\bf A}_0 - {\bf A}_3)({\bf B}_3 - {\bf B}_0) \\P_5 = ({\bf A}_0 - {\bf A}_2)({\bf B}_0 + {\bf B}_1) \\P_6 = ({\bf A}_1 - {\bf A}_3)({\bf B}_2 + {\bf B}_3)

Of course, the P_i products are computed using Strassen recursively. If we consider for now the product P_0 =(A_0-A_1)B_3 and we assume that  for matrices of size (n/2)x(n/2) the error bound is true we have:

\dot{\bf P}_0 = {\bf P}_0+E_0 \\ \|E_0\| \leq u(\frac{n}{2}+f(\frac{n}{2}))(\|A_0\|+\|A_1\|)\|B_3\| \\ \|E_0\| \leq u(\frac{n}{2}+f(\frac{n}{2}))2ab

Note, I do not understand why we have the additive term n/2  in (n/2+f(n/2)), I believe we are using the norm of the products. In exactly the same manner and using the same bound we can state the bound for P_1, P_2, and P_3. For P_4, P_5, and P_6 we have the following bound:

\|E_4\| \leq u(n+f(\frac{n}{2}))2a2b

Adding up the terms for the submatrices of C, for example, C_0

\|\dot{\bf C}_0 - {\bf C}_0\| \leq \\ \|E_0\| +\|E_2\| +\|E_4\|+\|E_6\| + \\ u(3 \|P_0\|+ 3\|P_2\| +2\P_4\|  +\|P_6\|)

We commit an error computing the products and by adding them in the same order (the error of P_0 and P_2 is affecting 3 additions, P_4 is affecting 2). Please, work out the detail of the formula above, it is a useful exercise.

(5)   \begin{equation*} \|\dot{\bf C}_0 - {\bf C}_0\| \leq u(44\frac{n}{2}+12f(\frac{n}{2})ab  \end{equation*}

We have the recursive formula: f(1) =1 and f(n) = (44(n/2) +12f(n/2)), and the final solution is

(6)   \begin{equation*} \|{\bf E} \| \leq u 65n^{\log_2 12}\|{\bf A}\|\|{\bf B}\|  \end{equation*}

If we do not perform a recursion till a problem size of 1, but earlier, let’s say n_0 and we apply k times the recursion:

(7)   \begin{equation*} \|{\bf E }\| \leq u 4^kn^2\|{\bf A}\|\|{\bf B}\| \end{equation*}

—3M (WINOGRAD/STRASSEN) PIPE: our implementation of the 3M algorithm
as presented in Table II, where MM is implemented as STRASSEN PIPE, WINOGRAD
PIPE and thus there is software pipelining between MMs and MAs, and complex
matrices are stored as two real matrices

Brent suggested that each time we apply a recursion of Strassen, we are loosing 2 bits precision (up to three levels are common for my tests and it means about 6 bits and in practice is in between 3 and 4).

Working out with Higham’s bounds and using a few simple manipulation I could find a lower coefficient: 3^k (instead of 4^k).

As I write this post I really appreciate much better Brent’s work (1970), way ahead of his time, and even more the elegant and far reaching work by Bini and Lotti (1980), where they take the Brent Analysis and generalize it for families of Strassen’s like algorithms thus Winograd’s variants.

 

Interpretation.

Let us start with an observation. We take a matrix product like P_0 and use it in the computation of C_0 and C_1, any error on P_0 will affect both C_0 and C_1 equally; For Strassen algorithm, each P_i is used twice in different computations. We will not show it here, but a Winograd-variant will have a P_2 that is used in all C_i. Per se, it is not a big deal. The error of a single product will spread evenly to the results, increasing the unlucky possibility to add error from another product. The spread in itself is not dangerous.

Here it comes the core of this long post.

I have one number I would like to highlight:

  • The factor 12 in Equation 5.

The factor 12 is the result of four terms: 2 + 2 + 4 +4. As a rule of thumb, 2 means that one operand has the addition of two sub matrices such as A_0+A_1 (say). The factor 4 is when both operands have additions. If we perform a MA prior the product we are increasing the error of the computation because of any unlucky cancellation; that is,  the more  MAs before the products, the Larger the error.

I see two main problems:

  • The algorithms have about 4 MA post the product computations (2 more than the regular computation). We save operations but we create a longer computation chain that may, and does, increase the error accumulation. Because we are seeking cancellation we increase the injection of unwanted error.
  • The pre MA may have a significant effect of the error, because even if it is small, it will be amplified by the MM that follows. Unfortunately, this is something I cannot grasp very well. The MA is among the inputs, no fancy computations, just short set of additions. I cannot see how there could be any catastrophic cancellation.

In Bini and Lotti work, They took these two ideas and provided a generalization.

Enough food for thoughts.

 

 

 

Just to make a little sense about the legend:

—WOPT: Winograd’s algorithm with fewer MAs.

—WIDEAL: Winograd’s algorithm optimized for a pipeline execution (but not pipelined).

—GOTOS: MM implementation as available in GotoBLAS.

—BLAS MM or MM only: MM implementation row-by-column (this is used in the error analysis only).

—GOTOS 3M: 3M algorithm as available in GotoBLAS where matrices are stored as
complex matrices.

—3M (GOTOS/WINOGRAD/STRASSEN/ATLAS): our implementation of the 3M algorithm as presented in Table II, where MM is implemented as STRASSEN, WINOGRAD, GOTOS, or ATLAS and thus complex matrices are stored as two distinct real matrices.

We believe, this is the first attempt to show how in practice all these algorithms work and how they affect the final result and error. We do not justify the blind use of fast algorithms when accuracy is paramount; however, we want to make sure that fast algorithms are not rejected just because of an unfounded fear of their instability.

Pratical, Theoretical, or Plain Frustrating

In the previous post, I talked about a new Winograd’s algorithm and its novelty. In this post I will talk about its practical benefits.

I understand that a few performance advantages are really subjective, in the sense that they may not really add up to the level of practical advantages.  However, it is a fact that having a balanced computation is a necessary step to exploit the best speed up for the Winograd algorithm. In other words, if an algorithm is not balanced, it performs more computations and thus will be eventually slower. An educated question is “so what?“, it is really adding any performance? This question is an old one, since the appearance of Strassen’s algorithm researchers wondered and tested when these algorithms should be applied: you may find the same question reformulated for the taste of the audience: is it a theoretical result (good only on paper), a practical result  (it find application in real problems).

Winograd’s algorithm has two clear curses:

  1. Numerical, a few have the misconception that the algorithm is numerically instable, I will have a post in the future.
  2. Practicality, a few believe that the performance advantages are available only for either galactic problem sizes or for all sizes.

I use the  term galactic from Lipton Blog: http://rjlipton.wordpress.com/2010/10/23/galactic-algorithms/#comments

Today I will talk about the latter and probably the easier to discuss: is Strassen–Winograd’ss algorithm any practical? If you have time and look up the papers by who really implemented and tested the implementation of Strassen–Winograd’s algorithms, the results are often contrasting. There are two reasons: in time the hardware is becoming more and more complex and the efficient implementation of MM is trying hard to follow up the architecture evolution. Remember the question boils down to find the smallest problem size for which Fast MM are faster than the regular MM (and for regular I mean the high performance MM).

Let us try to go back in time and see how the change of HW may affect the performance of the classic MM and the Fast MM. In the past, machines had little memory and thus the problems they could compute were small. They had a relatively flat memory and slow. They had relatively a few registers if any, and often the operations of multiplication and addition were done in software without any dedicated HW (floating point units). In this scenario, multiplication were very expensive, while additions dirt cheap.  Fast MM addressed two main concerns: First, if you really wanted you could eliminate multiplications and thus simplifying  the HW computations;  Second, the memory was flat and expensive, if we could  reduce the number of operation, we could reduce the memory access and the time spent reading data. The real draw back for Fast MM was the need for more space (in registers and memory). So if the problem was a fit in memory, Fast MM were appealing and useful. Because of the cost of multiplications, Fast MM found application for extremely small problem sizes. In the literature, you may find that Fast MM were always applicable.

With integration and the event of the co-processor designed to perform complex operations in HW, the cost difference between an addition and multiplication became smaller, much smaller, but there was a small difference any way. However, the memory did not change much. In such a scenario, the reduction of operations was beneficial even for small problems but the difference shrank.  In this new architecture, communication among processor and co-processor start counting towards the total execution time.

When the co-processor became  the functional-unit within the processor (Floating Point Unit), multiplication cost was pretty much the cost of an addition.  In the previous architectures, Fast MM could trade off multiplications with additions directly. The benefits was immediate. Now the trading of the operations is not so appealing.

With the FPU, the bottle neck became the register file, the communication means between memory and the FPU. Fast MM use more space and thus more registers.  But if the registers are not enough and spills to memory happen, we pay dearly for those additions and now those addition cost as much as multiplications.  Fast MM will kick in only when the number of operations offset for the costs, now we need a problem size in the tens.

If you play with an O() notation and count the number of operations such as multiplication and additions and they cost equally, say 1, you will find that the problem size should be at least 12 to break even. In practice, you must have a larger problem because the possible spill costs.

The memory is still basically flat, thus counting operation works just fine and the O() notation is elegant and simple. The possible spills are expensive but if we are careful.

Notice that In this scenario, we start trading off Matrix Multiplications with Matrix Additions. We must start taking advantage that we trade a single Matrix Multipliciation O(N^3) with 15-18 Matrix Additions O(N^2). Of course, as side effect, we perform fewer multiplications as in the previous cases, but we need a problem size for which this is advantageous. Just for fun, for N=10, the cost of a MM is 1000 and the cost of a MA is 100.  N=13 seems a good start for searching the right break even point.

With the integration of caches and memory hierarchies in a processor, the situation has  got tricky. If an application reuses data and these data are stored in caches, in practice we reduce the waiting time to access: we pay dearly for the first access, but then we can amortize the total cost because the following accesses are cheap: 3 cycles.

The MM if implemented carefully can exploit optimal reuse and thus can take advantage of caches extremely well. With caches and deep pipeline, MM can reach peak performance.

Fast MM are penalized by memory hierarchy because MA have no data reuse and their performance is basically bound to the throughput of the memory level where the data are stored. In my experience, caches are rarely big enough so that Fast MM and MM are competitive. In my experience, only when in memory FastMM can start kicking some FLOPS.

Again, Fast MMs trade off one MM for 15-18 MAs. The break even point for modern architectures varies. For one architecture, 4 core 2 processors 3.2 GHz Xeon Based system, Fast MM and MM break even at N=7,000.This system has a very poor memory bandwitdth one bus and a single common memory. Thus, a break even point of 7,000 says a  lot about the architecture more than about the algorithm. For other architectures with the same or better peak performance, the break even point is in the range 2,000 — 3,500.

An interesting question is related the future architectures. How FastMM will face off against the future Multicore and multi-processor systems? What about GPUs?

About GPUs I have no clue because I am not an expert. But if they do not have an memory hierarchy or caches are big enough, I can see application and great potential for FastMM even for moderate problem sizes. About multicore, we will need larger problem sizes. What I like is that with the appearance of larger machines, there is  the tendency of solving larger problems. In such a case, what is practical and what is not will be defined by the application where MM are used.

 

 

 

 

 

 

Is this adaptive-Winograd MM really new?

Once upon a time I showed the adaptive Strassen-Winograd algorithm presented in the previous post, Prof. Nicolau (the co-author) asked why this did not come up in the literature before. Why is it new?

Let us breath and let the question sink in ….  I always find these questions really fun(ny) because they are unanswerable. I will not know that ever, the previous authors know that, probably. Unfortunately, researchers do not publish or better they won’t publish it (like a car will not start without a battery) and, often, they want to forget why something did not work out. I will feel  like a moron to publish a paper saying that “Well, I liked this problem, but I am an idiot and I could not solve it, better luck to you all“. It is not good for your career and  for your self esteem, it is just not right.

Ironically, here I give an answer even though I may out of my element. If you try to reproduce the previous algorithm without paying attention at the matrix sizes of the temporary results, very likely you will have an incorrect algorithm. I know this because I tried to take the original algorithm and apply it. I failed miserably. Very likely you would fail at first and I guess this is what happened for previous practitioners. Previous researchers had to try, they may had the same problems but went for the path with least resistance. For a solution they understood.

When in CMU, I could chat with one professor who was involved in the implementation of Winograd (not the balanced one). I could finally ask the unanswerable question. What I discovered is kind of illuminating how we think: his main goal was to apply matrix tensors for the description of fast algorithms (Winograd) and he really did not care about a balanced or not balanced algorithm. In CMU I was working on the implementation of the Fast Fourier Transforms (FFT) and matrix tensors are powerful tools for the description of FFT implementations —there is a tensor in every SPIRAL paper :).

Why is this illuminating?

I cared for a balanced Winograd algorithm because I wanted to use a set of  tools that  would fit a balanced computation: optimality of the computation, simplicity, and exploiting data locality in caches. So I went through a path to show that I could use such tools. The previous researcher probably never consider my problem because they had other tools they wanted to use. They may have never tried to have a balanced computation, thus never failed.

Conclusions:

When somebody ask you an answerable question that you cannot know the answer; for example, your co-author, your wife, your boss or your daughter,  you must give an answer, hopefully in the future you will stumble upon the right one.

 

 

 

 

 

Adaptive-Balanced Winograd

Once I thought I knew Strassen-Winograd’s algorithm. Well, I did not.  Probably, you do not too. In this post, I am going to discuss a variation of the classic algorithm

Take the Strassen’s algorithm presented in wikipedia and change the notation just slightly:

We partition A, B and C into equally sized block matrices

C = \begin{bmatrix}C_0 & C_1\\ C_2& C_3 \end{bmatrix}, A = \begin{bmatrix}A_0 & A_1\\ A_2& A_3 \end{bmatrix}, \text{ and } B = \begin{bmatrix}B_0 & B_1\\ B_2& B_3 \end{bmatrix}

with  C_i, A_i, B_i \in \mathbb{R} ^{2^n\times 2^n}

Now comes the important part. We define new matrices

P_1 = (A_0+A_3)*(B_0+B_3), \\ P_2 = (A_2+A_3)*B_0,  \\ P_3 = A_0*(B_1-B3), \\ P_4 = A_3*(B_2-B3), \\ P_5 = (A_0+A_1)*B3, \\ P_6=(A_2-A0)*(B_0+B_1), \text{ and  }\\ P_7 = (A_1- A_3)*(B_2+B3)

which are then used to express the submatrices of C in terms of these products Ps We eliminated one matrix multiplication and reduced the number of multiplications to 7 as

C = \begin{bmatrix} C_0=P_1+P_4-P_5+P_7 & C_1 = P_3+P_5 \\ C_2=P2+P4 & C_3= P_1-P_2+P_3+P_6 \end{bmatrix}

The first question that came to my mind  was: Why power of two matrices?

Intuitively, I knew  that if the matrices are power of two  the division process could be done recursively and stop when only when the operands are single-element matrices; that is, scalars. In fact, it is easy to write a recurrence equations:

(1)   \begin{equation*}   T(2^n) = \begin{cases} 7T(2^{n-1})+18O(2^{n-1}) & n>0 \\ 1 & n==1 \end{cases} \end{equation*}

Equation 2 states that a matrix multiplication is reduces into 7 MMs and 18 matrix additions (MAs) recursively. The definition of the matrix sizes simplifies the algorithm and, more importantly, it simplifies the recurrence equation:

(2)   \begin{equation*}   T(2^n) =  O((2^n)^{\log_{2}7}) \end{equation*}

If you try to look up implementations of the algorithm above, you will find solutions for any matrix sizes:

First: The most common implementation is based on a preliminary division of the matrices into quadrants as above, but where the left-top is the largest even quadrant: for example, A \in \mathbb{R}^{m \times n} where m=2k+1 and n=2j+1 are odd, then we can do a first division by setting A_0 \in   \mathbb{R}^{2k \times 2j}.  In practice, we can apply once Strassen’s recursive approach for the product  C_0 = A_0*B_0. The remain of the computation is carried out as a sequence of Matrix-Vector and VEctor-Vector operations

    \[  C_0 += A_1*B_2, \\ C_1= A_0*B_1 +A_1*B3,  \\ C_2 = A_2*B_0 + A_3*B_2, \\ C_3 = A_2*B_1+A_3*B_3. \]

Notice that right most quadrant of the result matrix C is a scalar and requires O(n) operations, the rest takes 3O(nm) operations (like 3 MAs) and  O(n+m).

Second: We can pad with zeros the matrices to the closest power-of-two size. This is academics.

Third: We can pad the matrices once so that A and B are matrices with even sizes, then we can apply Strassen’s once. Recursively, we can pad the matrices as we see fit during the recursion.

The first solution is the most common in the literature and real libraries (e.g., IBM ESSL). The second one is simply academics. The third one is metioned and if I remember correctly has the cute nickname of peeling, but considered inferior to the first one.

Personally, I find the first solution uninspiring and we have found an elegant and more efficient solution, which is closer to the third one. A reviewer suggested the nick name of  dynamic padding and also suggested that this was found already but not published. Probable.

Before I introduce the algorithm let me provide two teasers that provide the basic idea of our solution:

  • Notice: the best way to exploit the speed up by Strassen/Winograd algorithm is by braking the matrices into to balanced and square submatrices. This assures fewer operations and we can avoid the Matrix-Vector operations. A balanced division provides the most efficient algorithm so that to get closer to the performance of the original algorithm.
  • Notice: Any high performance implementation of the Strassen–Winograd algorithm requires 2 or 3 temporary matrices to store matrix additions for As, Bs, and Cs. We do not need to pad matrices as long as we redefine what is a Matrix addition for uneven matrices.

Now that I gave away the two main idea, Let’s go one step at a time and present a better algorithm.

We partition A, B and C into equally sized block matrices

C = \begin{bmatrix}C_0 & C_1\\ C_2& C_3 \end{bmatrix}\in \mathbb{R}^{m\times p}, \\ A = \begin{bmatrix}A_0 & A_1\\ A_2& A_3 \end{bmatrix} \in \mathbb{R}^{m\times n}, \text{ and } \\ B = \begin{bmatrix}B_0 & B_1\\ B_2& B_3 \end{bmatrix} \in \mathbb{R}^{n\times p}

with the large quadrant

C_0 \in \mathbb{R}^{\lceil\frac{m}{2}\rceil\times \lceil\frac{p}{2}\rceil}, A_0  \in \mathbb{R}^{\lceil\frac{m}{2}\rceil\times \lceil\frac{p}{2}\rceil},  B_0  \in \mathbb{R}^{\lceil\frac{n}{2}\rceil\times \lceil\frac{p}{2}\rceil}

and small quadrant C_3 \in \mathbb{R}^{\lfloor\frac{m}{2}\rfloor\times \lfloor\frac{p}{2}\rfloor},A_3  \in \mathbb{R}^{\lfloor\frac{m}{2}\rfloor\times \lfloor\frac{p}{2}\rfloor}, B_3  \in \mathbb{R}^{\lfloor\frac{n}{2}\rfloor\times \lfloor\frac{p}{2}\rfloor}.

The only original thing we are going to do is to define the MA for operations such as S = A_0 +A_3. Everything is like we pad A with an bottom row of  zeros and a left column with zero. In practice, the definition  of matrix addition is simple, it follows real unoptimized code …

 

// C = A + B
void add_t(DEF(c), DEF(a), DEF(b)) {

  int i,j,x,y;

   /* minimum sizes */
   x = min(a.m,b.m); y = min(a.n,b.n);

   //# pragma omp parallel for
   for (i=0; i<x; i++) {
     /* core of the computation */
     for (j=0;j<y;j++)  E_(c.data,i,j,c.M,c.N) = a.beta*E_(a.data,i,j,a.M,a.N) + b.beta*E_(b.data,i,j,b.M,b.N);

     if (y<a.n)  E_(c.data,i,j,c.M,c.N) =  a.beta*E_(a.data,i,j,a.M,a.N) ; /* A is larger than B */
     else if (y<b.n) E_(c.data,i,j,c.M,c.N) = b.beta*E_(b.data,i,j,b.M,b.N); /* B is larger than A */
	}
   /* last row */
   if (x<a.m)  {/* A is taller than B */
     for (j=0;j<a.n;j++)  E_(c.data,i,j,c.M,c.N)  = a.beta*E_(a.data,i,j,a.M,a.N);
   }
   if (x<b.m)  {/* B is taller than A */
     for (j=0;j<b.n;j++)   E_(c.data,i,j,c.M,c.N) = b.beta*E_(b.data,i,j,b.M,b.N);
   }
   //   c.beta = 1;
}
  \mathbf{A} = \begin{bmatrix} \mathbf{A}_{1,1} & \mathbf{A}_{1,2} \\ \mathbf{A}_{2,1} & \mathbf{A}_{2,2} \end{bmatrix} \mbox { , } \mathbf{B} = \begin{bmatrix} \mathbf{B}_{1,1} & \mathbf{B}_{1,2} \\ \mathbf{B}_{2,1} & \mathbf{B}_{2,2} \end{bmatrix} \mbox { , } \mathbf{C} = \begin{bmatrix} \mathbf{C}_{1,1} & \mathbf{C}_{1,2} \\ \mathbf{C}_{2,1} & \mathbf{C}_{2,2} \end{bmatrix}

with

\mathbf{A}_{i,j}, \mathbf{B}_{i,j}, \mathbf{C}_{i,j} \in R^{2^{n-1} \times 2^{n-1}}

then

\mathbf{C}_{1,1} = \mathbf{A}_{1,1} \mathbf{B}_{1,1} + \mathbf{A}_{1,2} \mathbf{B}_{2,1}
\mathbf{C}_{1,2} = \mathbf{A}_{1,1} \mathbf{B}_{1,2} + \mathbf{A}_{1,2} \mathbf{B}_{2,2}
\mathbf{C}_{2,1} = \mathbf{A}_{2,1} \mathbf{B}_{1,1} + \mathbf{A}_{2,2} \mathbf{B}_{2,1}
\mathbf{C}_{2,2} = \mathbf{A}_{2,1} \mathbf{B}_{1,2} + \mathbf{A}_{2,2} \mathbf{B}_{2,2}

With this construction we have not reduced the number of multiplications. We still need 8 multiplications to calculate the Ci,j matrices, the same number of multiplications we need when using standard matrix multiplication.

Now comes the important part. We define new matrices

\mathbf{M}_{1} := (\mathbf{A}_{1,1} + \mathbf{A}_{2,2}) (\mathbf{B}_{1,1} + \mathbf{B}_{2,2})
\mathbf{M}_{2} := (\mathbf{A}_{2,1} + \mathbf{A}_{2,2}) \mathbf{B}_{1,1}
\mathbf{M}_{3} := \mathbf{A}_{1,1} (\mathbf{B}_{1,2} - \mathbf{B}_{2,2})
\mathbf{M}_{4} := \mathbf{A}_{2,2} (\mathbf{B}_{2,1} - \mathbf{B}_{1,1})
\mathbf{M}_{5} := (\mathbf{A}_{1,1} + \mathbf{A}_{1,2}) \mathbf{B}_{2,2}
\mathbf{M}_{6} := (\mathbf{A}_{2,1} - \mathbf{A}_{1,1}) (\mathbf{B}_{1,1} + \mathbf{B}_{1,2})
\mathbf{M}_{7} := (\mathbf{A}_{1,2} - \mathbf{A}_{2,2}) (\mathbf{B}_{2,1} + \mathbf{B}_{2,2})

which are then used to express the Ci,j in terms of Mk. Because of our definition of the Mk we can eliminate one matrix multiplication and reduce the number of multiplications to 7 (one multiplication for each Mk) and express the Ci,j as

\mathbf{C}_{1,1} = \mathbf{M}_{1} + \mathbf{M}_{4} - \mathbf{M}_{5} + \mathbf{M}_{7}
\mathbf{C}_{1,2} = \mathbf{M}_{3} + \mathbf{M}_{5}
\mathbf{C}_{2,1} = \mathbf{M}_{2} + \mathbf{M}_{4}
\mathbf{C}_{2,2} = \mathbf{M}_{1} - \mathbf{M}_{2} + \mathbf{M}_{3} + \mathbf{M}_{6}

Ok, We defined the way we decompose the matrices. We defined how we add matrices. Now here is our algorithm:

Adaptive Winograd

S = A_2 + A_3 \\ T = B_1 + B_0 \\C_3 = U, C_1 = U \\ \\ U  = A_0 {*_w}B_0 \\C_0  = U \\ \\ C_0 {+=} A_1 {*_w} B_2    \\ \\S = S + A_0\\T = B_3 - T \\U {+=} S {*_w} T \\C_1 {+=}  U \\ \\S = S + A_1   \\C_1 {+=} S {*_w} B_3  \\ \\T = B_2 - T   \\C_2 = A_3 {*_w} T \\ \\ S = A_0 + A_2 \\T = B_3 + B_1 \\ U {+=} S  {*_w} T\\C_3 {+=} U\\C_2 {+=} U

 

and the code verbatim:

// Winograd's matrix multiply                                                                                                                                                      
// Notation and order taken from                                                                                                                                                   
// http://www.cs.duke.edu/~alvy/papers/sc98/index.htm                                                                                                                               

int wmul(DEF(c), DEF(a), DEF(b)) {
  c.beta =1;
  if (a.m<= LEAF || a.n<= LEAF || b.n<=LEAF) {
      // Go GotoBLAS or ATLAS
      CMC(USE(c), = , USE(a),  mm_leaf_computation , USE(b));
  }
  else {

    Matrix s = {0, S0(a.m,a.n),S0(a.m,a.n)   ,a.trans,a.beta};
    Matrix t = {0, S0(b.m,b.n),S0(b.m,b.n)   ,b.trans,b.beta};
    Matrix p  = {0, S0(c.m,c.n),S0(c.m,c.n)  ,'n',1};                                                                                                                 
    Matrix u2  = {0, S0(c.m,c.n),S0(c.m,c.n) ,'n',1};                                                                                                                 
    Matrix tc0 = Q0(c),tc1 = Q1(c), tc2 =Q2(c),tc3=Q3(c);
    Matrix ta0 = Q0(a),ta1 = Q1(a), ta2 =Q2(a),ta3=Q3(a);
    Matrix tb0 = Q0(b),tb1 = Q1(b), tb2 =Q2(b),tb3=Q3(b);

    s.data  = (Mat *) CALLOC(s);
    t.data  = (Mat *) CALLOC(t);
    p.data  =  (Mat *) CALLOC(p);
    u2.data =  (Mat *) CALLOC(u2);
    /* P1 */
    /* P = A0*B0   */ CMC (RQ0(u2,c), =, ta0, wmul, tb0);
    /* C0  = P     */ copy(tc0    , u2);
    /* U2  = P           copy(u2, p); */

    /* P2 */
    /* P = A1 * B2 */  CMC(p,   =, ta1, wmul,   tb2);
    /* C0  += P    */  CMC(tc0, =, tc0, s_add,  p);

    /* P3 */
    /* S = A2 + A3 */  CMC(RQ2(s,a), =, ta2,      s_add,   ta3);
    /* T = B1 - B0 */  CMC(t,        =, tb1,      s_sub,   tb0);
    /* P = S * T   */  CMC(RQ2(p,c), =, RQ2(s,a), wmul ,   t);
    /* C3  =  P    */  copy(tc3, RQ3(p,c));
    /* C1  =  P    */  copy(tc1, RQ3(p,c));

    /* P4 */
    /* S  = S - A0 */  CMC(s,   =, RQ2(s,a),   s_sub,  ta0);
    /* T  = B3 - T */  CMC(t,   =, tb3,        s_sub,  t);
    /* P = S*T   */    CMC(p,   =, s,          wmul, t);
    /* U3 =U2 +=P */   CMC(u2,  =, u2,         s_add,  p);
    /* C1 +=U2, */     CMC(tc1, =, RQ3(tc1,c), s_add,  RQ1(u2,c));

    /* P6 */
    /* S = A1 - S */   CMC(s,        =, ta1,       s_sub,   s);
    /* P = S * B3 */   CMC(RQ1(p,c), =, RQ1(s,a),  wmul,    tb3);
    /* C1  += P   */   CMC(tc1,      =, tc1,       s_add,   RQ1(p,c));

    /* P7 */
    /* T = B2 - T */   CMC(RQ2(t,b), =, tb2,    s_sub,  RQ2(t,b));
    /* P = A3*T  */    CMC(RQ2(p,c), =, ta3,    wmul,   RQ2(t,b));
    /* C2 = P    */    copy(tc2, RQ2(p,c));

    /* P5 */
    /* S = A0 - A2 */  CMC(s,        =, ta0,       s_sub,  ta2);
    /* T = B3 - B1 */  CMC(RQ1(t,b), =, tb3,       s_sub,  tb1);
    /* P = S*T     */  CMC(RQ1(p,c), =, s,         wmul,   RQ1(t,b));
    /* U3  += P    */  CMC(u2,       =, u2,        s_add,  RQ1(p,c));
    /* C3  += U3   */  CMC(tc3,      =, tc3,       s_add,  RQ3(u2,c));
    /* C2  += U3   */  CMC(tc2,      =, RQ2(u2,c), s_add,  tc2);
    FREE(s.data);
    FREE(t.data);
    FREE(p.data);
    FREE(u2.data);

  }
  dept--;
  return recursive;

}

Conclusions

After such a boring, full of notations, post how can we summarize ? What is the punch line ?

Despise the long history of Strassen-Winograd algorithm, we present here the true generalization for every matrix sizes and shape. Yep, the algorithm can be applied to rectangular matrices as well and as it is. (well this is a white lie, for rectangular matrices we need to change one line of the program).

Reference

If you have a Matrix Multiply Kernel that achieves 90-95% peak performance

As I interview people (e.g., engineers) or I get interviewed for a new project or a new job, I can barely stop wondering why we have this unfounded belief that we are the best in what we are doing. See what I mean by reading the entire post.

For the implementation of Matrix Multiplication, and thus the implementation of the BLAS library, I have seen several project and libraries. Two of them are very impressive: ATLAS by Whaley and GotoBLAS by Goto.  As a context, BLAS is the building block for vector and matrix operations. Most of scientific computing applications are  based upon this library and the operations within it.  LAPACK is built on BLAS. For a funny connection, if you use python and you used scipy (scipy is built on top of LAPACK and BLAS).  ATLAS or GotoBLAS are high performance implementation of the BLAS and often written in FORTRAN (the inner kernel in assembly).

In turn, the general matrix multiplication (GEMM) is THE matrix operation within BLAS and the scientific computing work horse (GEMM is the work horse for QR-factorization which is the work horse for the solution of linear systems).

What is my point then? There was a time, when I thought I could do a better job than the other guys, probably because I thought I was better. The problem is that I do NOT have the chops to do half of the job that  Goto or Whaley did and are doing. My implementation of the general MM has been close to  but always behind these two top designers and developers.

As the time passes by and I see other engineers/students attempt to climb the high peak of GEMM performance, I see the allure. The application is simple to explain and present. Any developer can write a working code using a single liner in C. This is like playing football, the beautiful game, it is simple to explain and play, everybody can play. Writing high performance GEMM is like playing football. I can play a few tricks but I cannot even stand close to a professional player like  Zinadine Zidan (I mention him because we have the same age).

How come I can claim I can improve the performance of GEMM and also have the fastest MM?  Because I understood that there are different algorithms and there is space for different algorithms.  Personally, I am better in thinking and writing recursive algorithms. In another post, I will present the code itself and I believe it is beautiful. The simplicity of the code is embarrassing and you may wonder if there was any contribution to start with.

Winograd–Strassen algorithms have useful applications to build on top of these high performance GEMM. Think for one second,  when the GEMM reaches 90-95 % of the machine peak performance, 150 GFLOPS and more, even the original author has little space to improve GEMM. Actually, gaining anything further will not be worth it. A new machine will come along and the process to fit the code to the new system  will restart.

I believe that any significant improvement of the GEMM by software alone is by the application of new algorithms: Winograd–Strassen recursive algorithms to start.  These new algorithms do not substitute ATLAS nor GotoBLAS GEMM implementations. They build on top of them. They need the fastest implementation of the MM in order to achieve the fastest implementation.

I will draw two conclusions for this post: one optimistic and one cynical.

[optimistic] I can have dinner with Whaley or Goto tonight and I could  feel comfortable (I do not know how conformable they could be though).  I discovered I did not need to compete and actually I can help them to extend their algorithms to performance they could not achieve otherwise. Together we will succeed, divided we will fail.

[cynical] I discovered late in time the old adagio: “when you cannot beat them, either join them or …”. In this case, my ego was the real obstacle because it was Big but not BIG enough.

The morale of this post? When you have a MM kernel that achieves 95% peak performance, do not be afraid to use it :O. You better invest your time to learn about your problem than to show to the world that you can write better code.  In fact,  very likely you will not. And, if you do write better code, you will be able to improve performance by  little, as much as  5%. In contrast, I could improve performance by 27% with no extra effort.

 

Parallel (Winograd) Fast MM

The following is taken from the abstract of my latest paper to be published in Transaction in Mathematical Software:

We have found  a simple and efficient methodology for the development, tuning, and installation of matrix algorithms such as the hybrid Strassen’s and Winograd’s fast matrix multiply or their combination with the
3M algorithm for complex matrices (i.e., hybrid: a recursive algorithm as Strassen’s until a highly tuned
BLAS matrix multiplication allows performance advantages).

We have found  that modern symmetric multiprocessor (SMP) architectures present old and new challenges that can be addressed by the combination of an algorithm design with careful and natural parallelism exploitation at the function level (optimizations) such as function-call parallelism, function percolation, and function software pipelining. We have three contributions: first, we investigated the performance overview for double and double complex (published) and single and single complex (private)  precision matrices for state-of-the-art SMP systems; second, we introduce new algorithm implementations: a variant of the 3M algorithm and two new different schedules of Winograd’s matrix multiplication (achieving up to 20% speed up w.r.t. regular matrix multiplication). About the latter Winograd’s algorithms: one is designed to minimize the number of matrix additions and the other to minimize the computation latency of matrix additions; third, we apply software pipelining and threads allocation to all the algorithms and we show how that yields up to 10% further performance improvements.

In this work and for one architecture (Nehalem) we achieve  the ideal performance speedup using Winograd/Strassen like algorithms. This means that for one architecture we have the fastest implementation of the Winograd algorithm.