Fast Element-Wise power to non-integer value
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I am working on a non-linear optimization problem using fmincon. My current setup delivers the best results so far in simulations.
Now I am working on optimizing my code and I have a bottle-neck in computation time in the expression:
(1-(1-x.^a).^b);
where 'x' is a vector of Nx1 observations, with N "medium-large" (> 30000 rows). Moreover, 'a' and 'b' are non-integer parameters > 0 (to be fitted)
f is the function that I am minimizing, thus, it's called by the optimizer several times (90000 in my current setting) resulting in a significant bottleneck in the overall procedure.
Are there any ways to improve the speed of this calculation? According to the code, profiler, this line is responsible for 25% of the total computation time in seconds. A replicable example is shown below:
%Example
N = 30000;
x = linspace(0,1,N)'; %vector of N x 1
a = 1.2; %positive non-integer power
b = 0.3; %positive non-integer power
f3 = @() (1-(1-x.^a).^b);
timeit(f3)
Although it seems fast, if the function is called by the optimization procedure 90000 times, then the total time is around 0.0029*90000 = 261 seconds
Question:
Is there any way to make this computation faster? So far, I've tried
- bsxfun, however it's included in the .^ command for newer releases
- Increase maxNumCompThreads to 8
- Relax optimization Tolerance levels
- Tried other optimizers (such as lsqnonlin) but deliver worst estimates
- write the expression as logarithm and then take exponential (it seems slower in preliminar trials)
Thank you for your comments
All best!
4 个评论
Bruno Luong
2023-2-23
编辑：Bruno Luong
2023-2-23
If you want to find 2 parameters to fit 30000 data, you could probably some sort of agregate your data and fit with a less overdertermined system, and quicker to evaluate.
You could use such strategy i a hirarchy of problems: to approach the solution, then using the solution to starting point of another lesser agregated modified problem, and make the convergence with less number of iterations than 90000.
John D'Errico
2023-2-23
编辑：John D'Errico
2023-2-23
Big problems take big time. Why would they have implemented a slower method than possible of raising a number to a power?
Note that for large arrays, MATLAB will already automatically use multiple cores. However, 30000 is not really that large. In a quick test, I tried a test with a 20000x20000 matrix, then raised those elements to a non-integer power. It took that large of a matrix before I started to see multiple cores being seriously used.
Alfonso Silva
2023-2-23
Walter Roberson
2023-2-23
[I was wondering whether it would be effective to taylor f3 to a reasonable number of terms. I experimented with taylor around b = 0.5 (midpoint of the range), tested at 0.1 0.2 0.3 0.4... and the results for order 10 or order 20 were not encouraging.
采纳的回答
I don't know your optimization problem, but this helps saving function evaluations if objective function and constraints share the same function:
5 个评论
John D'Errico
2023-2-23
Sadly, there are probably no constraints beyond bounds, if lsqnonlin was also an option.
Torsten
2023-2-23
Ah, I missed that. Thank you @John D'Errico
Alfonso Silva
2023-2-23
Alfonso Silva
2023-3-2
Walter Roberson
2023-3-2
Hmmm I am surprised that guide document does not refer to memoize()
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