[Rd] accelerating matrix multiply

[Cohn, Robert S]

> Do you have R code (including set.seed(.) if relevant) to show on how to generate
> the large square matrices you've mentioned in the beginning?  So we get to some
> reproducible benchmarks?


Hi Martin,

Here is the program I used. I only generate 2 random numbers and reuse them to make the benchmark run faster. Let me know if there is something I can do to help--alternate benchmarks, tests, experiments with compilers other than icc.

MKL LAPACK behavior is undefined for NaN's so I left the check in, just made it more efficient on a CPU with SIMD. Thanks for looking at this.

set.seed (1)
m <- 30000
n <- 30000
A <- matrix (runif(2),nrow=m,ncol=n)
B <- matrix (runif(2),nrow=m,ncol=n)
print(typeof(A[1,2]))
print(A[1,2])

# Matrix multiply
system.time (C <- B %*% A)
system.time (C <- B %*% A)
system.time (C <- B %*% A)

-----Original Message-----
From: Martin Maechler [mailto:maechler at stat.math.ethz.ch] 
Sent: Tuesday, January 10, 2017 8:59 AM
To: Cohn, Robert S <robert.s.cohn at intel.com>
Cc: r-devel at r-project.org
Subject: Re: [Rd] accelerating matrix multiply

>>>>> Cohn, Robert S <robert.s.cohn at intel.com>
>>>>>     on Sat, 7 Jan 2017 16:41:42 +0000 writes:

> I am using R to multiply some large (30k x 30k double) matrices on a 
> 64 core machine (xeon phi).  I added some timers to src/main/array.c 
> to see where the time is going. All of the time is being spent in the 
> matprod function, most of that time is spent in dgemm. 15 seconds is 
> in matprod in some code that is checking if there are NaNs.

> > system.time (C <- B %*% A)
> nancheck: wall time 15.240282s
>    dgemm: wall time 43.111064s
>  matprod: wall time 58.351572s
>     user   system  elapsed 
> 2710.154   20.999   58.398
> 
> The NaN checking code is not being vectorized because of the early 
> exit when NaN is detected:
> 
> 	/* Don't trust the BLAS to handle NA/NaNs correctly: PR#4582
> 	 * The test is only O(n) here.
> 	 */
> 	for (R_xlen_t i = 0; i < NRX*ncx; i++)
> 	    if (ISNAN(x[i])) {have_na = TRUE; break;}
> 	if (!have_na)
> 	    for (R_xlen_t i = 0; i < NRY*ncy; i++)
> 		if (ISNAN(y[i])) {have_na = TRUE; break;}
> 
> I tried deleting the 'break'. By inspecting the asm code, I verified 
> that the loop was not being vectorized before, but now is vectorized. 
> Total time goes down:
> 
> system.time (C <- B %*% A)
> nancheck: wall time  1.898667s
>    dgemm: wall time 43.913621s
>  matprod: wall time 45.812468s
>     user   system  elapsed 
> 2727.877   20.723   45.859
> 
> The break accelerates the case when there is a NaN, at the expense of 
> the much more common case when there isn't a NaN. If a NaN is 
> detected, it doesn't call dgemm and calls its own matrix multiply, 
> which makes the NaN check time insignificant so I doubt the early exit 
> provides any benefit.
> 
> I was a little surprised that the O(n) NaN check is costly compared to 
> the O(n**2) dgemm that follows. I think the reason is that nan check 
> is single thread and not vectorized, and my machine can do 2048 
> floating point ops/cycle when you consider the cores/dual issue/8 way 
> SIMD/muladd, and the constant factor will be significant for even 
> large matrices.
> 
> Would you consider deleting the breaks? I can submit a patch if that 
> will help. Thanks.
> 
> Robert

Thank you Robert for bringing the issue up ("again", possibly).
Within R core, some have seen somewhat similar timing on some platforms (gcc) .. but much less dramatical differences e.g. on macOS with clang.

As seen in the source code you cite above, the current implementation was triggered by a nasty BLAS bug .. actually also showing up only on some platforms, possibly depending on runtime libraries in addition to the compilers used.

Do you have R code (including set.seed(.) if relevant) to show on how to generate the large square matrices you've mentioned in the beginning?  So we get to some reproducible benchmarks?

With best regards,
Martin Maechler