Is there any way to generate pseudo-random numbers to less precision and thus speed the process up?
Another thing is that I know it saves time if random numbers are generated all at once (e.g. rand(100,1000)), instead of one by one. Could someone explain why this is true?
If you have a CUDA-capable GPU, you can do random number generation on it, as it's supposed to be much faster... Specifically Philox4x32-10:
parallel.gpu.rng(0, 'Philox4x32-10');
R = gpuArray.rand(sZ,'single'); % run this for more info: doc('gpuArray/rand')
MATLAB actually implements more than one random number generator. They differ significantly in terms of execution time and in terms of "randomness" (I think, but I didn't verify). However, I understand from your question that speed is more important for you.
% 'twister' is the default in MATLAB Versions 7.4 and later
tic();
for i=1:1000000
rand('twister');
end
toc();
%Elapsed time is 2.912960 seconds.
% 'state' is the default in MATLAB versions 5 through 7.3
tic();
for i=1:1000000
rand('state');
end
toc();
% Elapsed time is 2.162040 seconds.
% 'seed' is the default in MATLAB version 4
tic();
for i=1:1000000
rand('seed');
end
toc();
% Elapsed time is 0.758830 seconds.
Important note: I ran the script above with an rather old version of MATLAB (v.7.6, a.k.a. R2008a). In newer versions, the syntax rand(generator) is discouraged . Instead, you should use the function rng(seed, generator) (online documentation). As a side effect, rng(seed, generator) gives you even more random number generators to choose from. Check the documentation for details.
Regarding the second question: Whatever generator you pick, generating many random numbers at once will always be faster than generating many single random numbers. This is because MATLAB's internals are heavily optimized for parallel processing.
tic();
for i=1:100000
rand();
end
toc();
% Elapsed time is 0.024388 seconds.
tic();
rand(100, 1000);
toc();
% Elapsed time is 0.000680 seconds.
Since R2015a the rng function for configuring and seeding the global generator has a 'simdTwister' option that uses a faster "SIMD-oriented Fast Mersenne Twister" algorithm:
rng(1,'twister');
R = rand(1e4); % Warmup for timing
tic
R = rand(1e4);
toc
rng(1,'simdTwister');
R = rand(1e4); % Warmup for timing
tic
R = rand(1e4);
toc
This will probably be the fastest builtin generator for your system (excepting the possibility of GPU-based generators). On my computer it's a little more than twice as fast as the default Mersenne Twister algorithm for large arrays.
Related
function w=oja(X, varargin)
% get the dimensionality
[m n] = size(X);
% random initial weights
w = randn(m,1);
options = struct( ...
'rate', .00005, ...
'niter', 5000, ...
'delta', .0001);
options = getopt(options, varargin);
success = 0;
% run through all input samples
for iter = 1:options.niter
y = w'*X;
for ii = 1:n
% y is a scalar, not a vector
w = w + options.rate*(y(ii)*X(:,ii) - y(ii)^2*w);
end
end
if (any(~isfinite(w)))
warning('Lost convergence; lower learning rate?');
end
end
size(X)= 400 153600
This code implements oja's rule and runs slow. I am not able to vectorize it any more. To make it run faster I wanted to do computations on the GPU, therefore I changed
X=gpuArray(X)
But the code instead ran slower. The computation used seems to be compatible with GPU. Please let me know my mistake.
Profile Code Output:
Complete details:
https://drive.google.com/file/d/0B16PrXUjs69zRjFhSHhOSTI5RzQ/view?usp=sharing
This is not a full answer on how to solve it, but more an explanation why GPUs does not speed up, but actually enormously slow down your code.
GPUs are fantastic to speed up code that is parallel, meaning that they can do A LOT of things at the same time (i.e. my GPU can do 30070 things at the same time, while a modern CPU cant go over 16). However, GPU processors are very slow! Nowadays a decent CPU has around 2~3Ghz speed while a modern GPU has 700Mhz. This means that a CPU is much faster than a GPU, but as GPUs can do lots of things at the same time they can win overall.
Once I saw it explained as: What do you prefer, A million dollar sports car or a scooter? A million dolar car or a thousand scooters? And what if your job is to deliver pizza? Hopefully you answered a thousand scooters for this last one (unless you are a scooter fan and you answered the scooters in all of them, but that's not the point). (source and good introduction to GPU)
Back to your code: your code is incredibly sequential. Every inner iteration depends in the previous one and the same with the outer iteration. You can not run 2 of these in parallel, as you need the result from one iteration to run the next one. This means that you will not get a pizza order until you have delivered the last one, thus what you want is to deliver 1 by 1, as fast as you can (so sports car is better!).
And actually, each of these 1 line equations is incredibly fast! If I run 50 of them in my computer I get 13.034 seconds on that line which is 1.69 microseconds per iteration (7680000 calls).
Thus your problem is not that your code is slow, is that you call it A LOT of times. The GPU will not accelerate this line of code, because it is already very fast, and we know that CPUs are faster than GPUs for these kind of things.
Thus, unfortunately, GPUs suck for sequential code and your code is very sequential, therefore you can not use GPUs to speed up. An HPC will neither help, because every loop iteration depends in the previous one (no parfor :( ).
So, as far I can say, you will need to deal with it.
I've been experimenting with the GPU support of Matlab (v2014a). The notebook I'm using to test my code has a NVIDIA 840M build in.
Since I'm new to GPU computing with Matlab, I started out with a few simple examples and observed a strange scalability behavior. Instead of increasing the size of my problem, I simply put a forloop around my computation. I expected the time for the computation, to scale with the number of iterations, since the problem size itself does not increase. This was also true for smaller numbers of iterations, however at a certain point the time does not scale as expected, instead I observe a huge increase in computation time. Afterwards, the problem continues to scale again as expected.
The code example started from a random walk simulation, but I tried to produce an example that is easy and still shows the problem.
Here's what my code does. I initialize two matrices as sin(alpha)and cos(alpha). Then I loop over the number of iterations from 2**1to 2**15. I then repead the computation sin(alpha)^2 and cos(alpha)^2and add them up (this was just to check the result). I perform this calculation as often as the number of iterations suggests.
function gpu_scale
close all
NP = 10000;
NT = 1000;
ALPHA = rand(NP,NT,'single')*2*pi;
SINALPHA = sin(ALPHA);
COSALPHA = cos(ALPHA);
GSINALPHA = gpuArray(SINALPHA); % move array to gpu
GCOSALPHA = gpuArray(COSALPHA);
PMAX=15;
for P = 1:PMAX;
for i=1:2^P
GX = GSINALPHA.^2;
GY = GCOSALPHA.^2;
GZ = GX+GY;
end
end
The following plot, shows the computation time in a log-log plot for the case that I always double the number of iterations. The jump occurs when doubling from 1024 to 2048 iterations.
The initial bump for two iterations might be due to initialization and is not really relevant anyhow.
I see no reason for the jump between 2**10 and 2**11 computations, since the computation time should only depend on the number of iterations.
My question: Can somebody explain this behavior to me? What is happening on the software/hardware side, that explains this jump?
Thanks in advance!
EDIT: As suggested by Divakar, I changed the way I time my code. I wasn't sure I was using gputimeit correctly. however MathWorks suggests another possible way, namely
gd= gpuDevice();
tic
% the computation
wait(gd);
Time = toc;
Using this way to measure my performance, the time is significantly slower, however I don't observe the jump in the previous plot. I added the CPU performance for comparison and keept both timings for the GPU (wait / no wait), which can be seen in the following plot
It seems, that the observed jump "corrects" the timining in the direction of the case where I used wait. If I understand the problem correctly, then the good performance in the no wait case is due to the fact, that we do not wait for the GPU to finish completely. However, then I still don't see an explanation for the jump.
Any ideas?
I'm trying to vectorize or make this loop run faster (it's a minimal code):
n=1000;
L=100;
x=linspace(-L/2,L/2);
V1=rand(n);
for i=1:length(x)
for k=1:n
for j=1:n
V2(j,k)=V1(j,k)*log(2/L)*tan(pi/L*(x(i)+L/2)*j);
end
end
V3(i,:)=sum(V2);
end
would appreciate you help.
An alternative to vectorization, is to recognize the expensive operations in the code and somehow reduce them. For instance, the log(2/L) is called 100*1000*1000 times with input that does not depend on any of the three for loops. If we calculate this value outside of the for loops, then we can use it instead:
logResult = log(2/L);
and
V2(j,k)=V1(j,k)*log(2/L)*tan(pi/L*(x(i)+L/2)*j);
becomes
V2(j,k)=(V1(j,k)*logResult*tan(pi/L*(x(i)+L/2)*j));
Likewise, the code calls the tan function the same 100*1000*1000 times. Note how this calculation, tan(pi/L*(x(i)+L/2)*j) does not depend on k. And so if we calculate these values outside of the for loops, we can reduce this calculation by 1000 times:
tanValues = zeros(lenx,n);
for i=1:lenx
for j=1:n
tanValues(i,j) = tan(pi/L*(x(i)+L/2)*j);
end
end
and the calculation for V2(j,k) becomes
V2(j,k)=V1(j,k)*logResult*tanValues(i,j);
Also, memory can be pre-allocated to the V2 and V3 matrices to avoid the internal resizing that occurs on each iteration. Just do the following outside the for loops
V2 = zeros(n,n);
V3 = zeros(lenx,n);
Using tic and toc reduces the original execution from ~14 seconds to ~6 on my workstation. This is still three times slower than natan's solution which is ~2 seconds for me.
here's a vectorized solution using meshgrid, bsxfun and repmat:
% fast preallocation
jj(n,n)=0; B(n,n,L)=0; V3(L,n)=0;
lg=log(2/L);
% the vectorizaion part
jj=meshgrid(1:n);
B=bsxfun(#times,ones(n),permute(x,[3 1 2]));
V3=squeeze(sum(lg*repmat(V1,1,1,numel(x)).*tan(bsxfun(#times,jj',pi/L*(B+L/2))),1)).';
Running your code at my computer using tic\toc took ~25 seconds. The bsxfun code took ~4.5 seconds...
The algorithm repeats the same thing again-and-again. I expected to get the same time in each trial but I got very unexpected times for the four identical trials
in which I expected the curves to be identical but they act totally differently. The reason is probably in the tic/toc precision.
What kind of profiling/timing tools should I use in Matlab?
What am I doing wrong in the below code? How reliable is the tic/toc profiling?
Anyway to guarantee consistent results?
Algorithm
A=[];
for ii=1:25
tic;
timerval=tic;
AlgoCalculatesTheSameThing();
tElapsed=toc(timerval);
A=[A,tElapsed];
end
You should try timeit.
Have a look at this related question:
How to benchmark Matlab processes?
A snippet from Sam Roberts answer to the other question:
It handles many subtle issues related to benchmarking MATLAB code for you, such as:
ensuring that JIT compilation is used by wrapping the benchmarked code in a function
warming up the code
running the code several times and averaging
Have a look at this question for discussion regarding warm up:
Why does Matlab run faster after a script is "warmed up"?
Update:
Since timeit was first submitted at the fileexchange, the source code is available here and can be studied and analyzed (as opposed to most other MATLAB functions).
From the header of timeit.m:
% TIMEIT handles automatically the usual benchmarking procedures of "warming
% up" F, figuring out how many times to repeat F in a timing loop, etc.
% TIMEIT also compensates for the estimated time-measurement overhead
% associated with tic/toc and with calling function handles. TIMEIT returns
% the median of several repeated measurements.
You can go through the function step-by-step. The comments are very good and descriptive in my opinion. It is of course possible that Mathworks has changed parts of the code, but the overall functionality is there.
For instance, to account for the time it takes to run tic/toc:
function t = tictocTimeExperiment
% Call tic/toc 100 times and return the average time required.
It is later substracted from the total time.
The following is said regarding number of computations:
function t = roughEstimate(f, num_f_outputs)
% Return rough estimate of time required for one execution of
% f(). Basic warmups are done, but no fancy looping, medians,
% etc.
This rough estimate is used to determine how many times the computations should run.
If you want to change the number of computation times, you can modify the timeit function yourself, as it is available. I would recommend you to save it as my_timeit, or something else, so that you avoid overwriting the built-in version.
Qualitatively there are large differences between the same runs. I did the same four trials as in the question and tested them with the methods suggested so far and I created my own version of the timeit timeitH because the timeit has too large standard deviation between different trials. The timeitH returns far more robust results to other methods because it warm ups the code similarly to the timeit and then it has increased the amount of outer loops in the original timeit from 11 to 50.
The below has the four trials done with the three different methods. The closer the curves are to each other, the better.
TimeitH: results pretty good!
Some observations.
timeit: result smoothed but bumps
tic/toc: easy to adjust for larger cases to get the standard deviation smaller in computation times but no warming up
timeitH: download the code and change 60th line to num_outer_iterations = 50; to get smoother results
In summarum
I think the timeitH is the best candidate here, yet only tested in evaluating sparse polynomials. The timeit and tic/toc like 500 times do not result into robust results.
Timeit
500 trials and average with tic/toc
Algorithm for the 500 trials with tic/toc
for ii=1:25
numTrials = 500;
tic;
for ii=1:numTrials
AlgoCalculatesTheSameThing();
end
tTotal = toc;
tElapsed = tTotal/numTrials;
A=[A,tElapsed];
end
Is the time for AlgoCalculatesTheSameThing() relatively short (fractions of sec or a few sec) or long (multi-minutes or hours)? If the former I would suggest doing it more like this: move your timing functions outside your loop, then compute averages:
A=[];
numTrials = 25;
tic;
for ii=1:numTrials
AlgoCalculatesTheSameThing();
end
tTotal = toc;
tAvg = tTotal/numTrials;
If the event is short enough (fraction of sec) then you should also increase the value of numTrials to 100s or even 1000s.
You have to consider that with any timing function there will be error bars (like in any other measurement). If the event your timing is short enough, the uncertainties in your measurement can be relatively big, keeping in mind that the resolution of tic and toc also has some finite value.
More discussion on the accuracy of tic and toc can be found here.
You need to work out these uncertainties for your specific application, so do experiments: perform averages over a number of trials and then compute the standard deviation to get a sense of the "scatter" or uncertainly in your results.
I am wondering about the big performance difference of a fft and a simple addition on a GPU using Matlab. I would expect that a fft is slower on the GPU than a simple addition. But why is it the other way around? Any suggestions?
a=rand(2.^20,1);
a=gpuArray(a);
b=gpuArray(0);
c=gpuArray(1);
tic % should take a long time
for k=1:1000
fft(a);
end
toc % Elapsed time is 0.085893 seconds.
tic % should be fast, but isn't
for k=1:1000
b=b+c;
end
toc % Elapsed time is 1.430682 seconds.
It is also interesting to note that the computational time for the addition (second loop) decreases if I reduce the length of the vetor a.
EDIT
If I change the order of the two loops, i.e. if the addition is done first, the addition takes 0.2 seconds instead of 1.4 seconds. The FFT time is still the same.
I'm guessing that Matlab isn't actually running the fft because the output is not used anywhere. Also, in your simple addition loop, each iteration depends on the previous one, so it has to run serially.
I don't know why the order of the loops matters. Maybe it has something to do with cleaning up the GPU memory after the first loop. You could try calling pause(1) between the loops to let your computer get back to an idle state before the second loop. That may make your timing more consistent.
I don't have a 2012b MATLAB with GPU to hand to check this but I think that you are missing a wait() command. In 2012a, MATLAB introduced asynchronous GPU calculations. So, when you send something to the GPU it doesn't wait until its finished before moving on in code. Try this:
mygpu=gpuDevice(1);
a=rand(2.^20,1);
a=gpuArray(a);
b=gpuArray(0);
c=gpuArray(1);
tic % should take a long time
for k=1:1000
fft(a);
end
wait(mygpu); %Wait until the GPU has finished calculating before moving on
toc
tic % should be fast
for k=1:1000
b=b+c;
end
wait(mygpu); %Wait until the GPU has finished calculating before moving on
toc
The computation time of the addition should no longer depend on when its carried out. Would you mind checking and getting back to me please?