Performance of a Novel Chaotic Firefly Algorithm with Enhanced Exploration for Tackling Global Optimization Problems: Application for Dropout Regularization
Abstract
:1. Introduction
- A novel modified FA algorithm was implemented by specifically targeting the known flaws of the basic implementation of the FA approach;
- The devised algorithm was later utilized to help establish the proper dropout value and enhancing the CNN accuracy;
- Other well-known swarm intelligence metaheuristics for CNN dropout regularization challenge were further investigated.
2. Preliminaries and Related Works
3. Proposed Method
3.1. The Original Firefly Algorithm
3.2. Motivation for Improvements
3.3. Novel FA Metaheuristics
- Explicit exploration mechanism based on the solution’s exhaustiveness;
- gBest chaotic local search (CLS) strategy.
3.3.1. Explicit Exploration Mechanism
3.3.2. The gBest CLS Strategy
3.3.3. Chaotic FA with Enhanced Exploration Pseudo-Code
Algorithm 1 The CFAEE pseudo-code |
Initialize main metaheuristics control parameters N and T |
Initialize search space parameters D, and |
Initialize CFAEE control parameters , , , , K and |
Generate initial random population using Equation (15) in the search space |
whiledo |
for to N do |
for to i do |
if then |
Move solution z in the direction of individual i in D dimensions (Equation (12)) |
Attractiveness changes with distance r as exp[] (Equation (10)) |
Evaluate new solution, replace the worse individual with better one and update intensity of light (fitness) |
end if |
end for |
end for |
if then |
Replace all solutions for which with random ones using Equation (15) |
else |
Replace all solutions for which with guided replacement using Equation (16) |
for to K do |
Perform gBest CLS around the using Equations (17)–(19) and generate |
Retain better solution between and |
end for |
end if |
Update and according to Equations (14) and (20), respectively |
end while |
Return the best individual from the population |
Post-process results and perform visualization |
3.3.4. The CFAEE Complexity and Drawbacks
4. Bound-Constrained Benchmark Simulations
4.1. Experimental Setup
4.2. Benchmark Problem Set 1
4.3. Benchmark Problem Set 2
5. Dropout Estimation Simulations
5.1. Basic Experimental Setup
- MNIST—consists of images of handwritten digits “0–9”; it is divided into 60,000 training and 10,000 testing observations; image size pixels gray-scale (http://yann.lecun.com/exdb/mnist/, accessed on 10 October 2021);
- Fashion-MNIST—dataset of Zalando’s article images; it is comprised of different clothing images divided into 10 classes; it is split into 60,000 and 10,000 images used for training and testing, respectively; image size pixels (https://github.com/zalandoresearch/fashion-mnist, accessed on 10 October 2021);
- Semeion—includes a total of 1593 handwritten digits “0–9” images collected from 80 persons; digits are written accurately (normal way) and inaccurately (fast way); the original dataset is not split into training and testing; image size grayscale and each pixel is binarized (https://archive.ics.uci.edu/ml/datasets/Semeion+Handwritten+Digit, accessed on 10 October 2021);
- USPS—contains handwritten digits “0–9” images obtained from the envelopes of the United States Postal Service; dataset is split into 7291 training and 2007 testing images; image size gray-scale (http://statweb.stanford.edu/tibs/ElemStatLearn/datasets/zip.info.txt, accessed on 10 October 2021);
- CIFAR-10—consists of various images from 10 classes; subset of 80 million tiny images retrieved and collected by Alex Krizhevsky, Vinod Nair, and Geoffrey Hinton; divided into 50,000 images for training and 10,000 images for testing; image size color-scale (http://www.cs.toronto.edu/kriz/cifar.html, accessed on 10 October 2021).
5.2. Results, Comparative Analysis, and Discussion
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Parameter and Notation | Value |
---|---|
Number of solution N | 20 (benchmark1), 30 (benchmark2) |
Maximum number of () | 160,000 (benchmark1), 15,030 (benchmark2) |
Absorption coefficient | 1.0 |
Attractiveness at | 1.0 |
Randomization (step) | changes according to Equation (14) |
Initial value of step | 0.5 |
Minimum value of step | 0.1 |
Solutions’ exhaustiveness | |
CLS strategy step number K | 4 |
CLS strategy | changes according to Equation (20) |
Parameter |
ID | Name | Search Range | Formulation | Optimum |
---|---|---|---|---|
f1 | Sphere | min | 0 | |
f2 | Moved Axis Function | min | 0 | |
f3 | Griewank | min | 0 | |
f4 | Rastrigin | min | 0 | |
f5 | The Schwefel’s Problem 1.2 | min | 0 | |
f6 | Ackley | min , where | 0 | |
f7 | Powell Sum | min | 0 | |
f8 | Sum Squares | min | 0 | |
f9 | Schwefel 2.22 | min | 0 | |
f10 | Powell Singular | min | 0 | |
f11 | Alpine | min | 0 | |
f12 | Inverse Cosine-Wave Function | min | −D+1 | |
f13 | Pathological | min | 0 | |
f14 | Discus | min | 0 | |
f15 | Happy Cat | min , where | 0 | |
f16 | Drop-Wave Function | min | −1 | |
f17 | Schaffer 2 | min | 0 | |
f18 | Camel Function-Three Hump | min | 0 |
Function | Algorithm | Best Value | Worst Value | Mean Value | Function | Algorithm | Best Value | Worst Value | Mean Value |
---|---|---|---|---|---|---|---|---|---|
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0 0 0 0 0 0 | 0 0.531452 0 0.116521 0 0 | 0 0.151967 0 0.067858 0 0 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0 0 0 0 0 0 | 0 0.735625 0 0.79956 0 0 | 0 0.327158 0 0.431151 0 0 | ||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0 0 0 0 0 0 | 0 5.74765 0 4.005821 0 0 | 0 1.32645 0 1.003456 0 0 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0 0 0 0 0 | 0 14.95923 0 9.18410 0 0 | 0 2.736795 0 3.381069 0 0 | ||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0 0 0 0 0 0 | 0 0.451043 0 0.224532 0 0 | 0 0.145892 0 0.131779 0 0 | ||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 5.969720 1.12073 0 0 2.537912 2.142703 0 | 5.969720 16.96541 4.352192 0 22.243001 27.135292 0 | 5.969720 7.962931 2.160021 0 10.984211 11.528380 0 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | −3.007700 −7.416352 −9 −9 −9 −7.982860 −9 | −3.007700 −6.100051 −8.154811 −9 −6.738521 −5.318621 −9 | −3.007740 −6.821470 −8.837092 −9 −7.182860 −6.730021 −9 | ||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0 0 0 0 0 0 | 0 1.216521 0 0.668310 0 0 | 0 0.371654 0 0.315237 0 0 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0.502000 0 0 0 | 0.502000 0 0 | 0.502000 0 0 | ||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 1.156728 1.197652 | 0.363197 0.569403 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | 643.025312 0 0 0 0 0 | 697.974622 0 0.634750 0 0.177280 0 0 | 644.124100 0 0 0 0 | |||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | 1.753800 0 0 0 0.622315 0 0 | 1.753800 0.432198 0.453921 0 0.978813 0.635291 0 | 1.753800 0.168663 0 0.860170 0.160825 0 | ||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0 0 0 0 0 0 | 0 1.413521 0 0.727350 0 0 | 0 0.237733 0 0.355913 0 0 |
Function | Algorithm | Best Value | Worst Value | Mean Value | Function | Algorithm | Best Value | Worst Value | Mean Value |
---|---|---|---|---|---|---|---|---|---|
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0 13.62752 9.542603 0 0.143725 0 | 20.168334 17.455290 0.329325 0.335712 6.25 | 17.05007 14.08179 0.237264 0.194190 1.41 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0 15.668320 7.896512 0 0.759455 0.663490 0 | 18.532451 13.652705 4.84 1.652710 2.0693 | 17.168332 11.634482 1.444582 1.444582 6.09 | ||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0 952.735293 495.234239 0 9.0823 8.458129 0 | 1292.759201 932.959210 27.288553 35.736666 3.52 | 1151.53123 831.976505 15.382611 19.345189 4.52 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | 740.533299 1297.755023 0 3.545252 5.3022 0 | 4352.542059 3675.442951 33.82541 28.982541 1.13 | 2953.135592 2626.920051 23.710392 17.315642 4.44 | ||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0.453243 0.331970 0 0 | 0.573032 0.511440 1.84 | 0.516954 0.446482 2.33 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | 13.115620 6.718345 0 0.245052 0.133675 0 | 16.344592 13.539203 0.731462 0.475093 8.54 | 14.957239 10.529380 0.417792 0.312399 1.03 | ||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 53.728352 91.368000 26.842502 0 10.562432 48.503233 0 | 53.728352 145.032962 47.888361 0.293775 70.887502 118.455291 0.163325 | 53.728352 131.851977 37.948270 52.398675 89.757932 2.31 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | −2.745302 −14.281512 −19.773059 −29 −27.135292 −19.932444 −29 | −2.738143 −10.236442 −14.387294 −28.981153 −23.462555 −13.572562 −28.975432 | −2.741055 −12.601748 −17.381692 −28.995732 −25.463931 −16.488942 −28.997240 | ||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0 188.932905 175.893044 0 1.363823 1.167251 0 | 249.742592 248.643292 5.757921 7.374155 7.35 | 229.451399 218.334752 3.464743 3.888032 5.65 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | 4.821110 4.34 | 4.854329 0.135155 0.131320 3.13 | 4.831800 7.09 | ||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 4.340823 3.011800 0.433632 0.447690 | 4.530665 3.725154 0.855143 2.853752 1.05 | 4.493832 3.383214 2.87 0.688177 1.1317517 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | 16.331 12.645 0 0.10623 0 | 22.498 20.226 0.50956 0.42646 1.15 | 19.8985 16.6886 0.261277 0.244281 2.29 | ||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 6.16e-05 123.295565 21.954921 0 4.57E-11 3.52e-08 0 | 6.18e-05 1158.432456 4319.824940 3.17e-38 1.38e-03 9.83e-03 4.91 | 541.478399 1345.324915 3.12 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | 2.3302 2.262251 0.723335 0 1.175841 1.307425 0 | 2.3302 2.348725 1.466781 0.637052 1.371513 1.743721 0.592563 | 2.3302 2.305134 1.060788 9.92 1.294690 1.524977 | ||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0 152.832522 103.285692 0 1.534341 1.561 0 | 3.17e-04 275.964302 214.365219 4.781903 5.175238 2.22 | 235.952315 173.448925 3.255770 3.270697 9.29 |
Function | Algorithm | Best Value | Worst Value | Mean Value | Function | Algorithm | Best Value | Worst Value | Mean Value |
---|---|---|---|---|---|---|---|---|---|
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0 86.457552 84.743562 0 0.776975 6.0074 0 | 94.965352 101.550299 1.343821 9.086351 3.35 | 91.851742 95.331892 1.123621 7.397248 2.12 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | 7.270000 75.261423 61.271532 0 4.667233 19.453222 0 | 7.345340 79.183492 69.287492 0.282980 7.217744 29.786432 0.253300 | 7.272472 76.237822 65.957970 6.75 5.953690 24.393170 | ||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 20,155.732954 20,134.629495 0 183.584823 1439.319025 0 | 0.861329 23,097.569290 23,511.452949 2.732509 326.044592 2483.724942 0.525656 | 0.296353 21,435.685432 22,061.730052 0.263728 246.343291 1955.372492 0.255429 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0.113550 24,876.459003 24,626.324592 0 95.657422 688.787853 0 | 0.754291 29,942.359392 33,338.728942 0.769235 143.859235 1431.750099 0.621509 | 0.212700 27,295.176529 29,210.135929 116.538544 1058.681232 5.31 | ||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0.783852 0.746025 0 0.167315 0 | 0.354451 0.850443 0.837694 1.25 0.186983 | 0.195824 0.811365 0.799866 0.166489 2.13 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | 14.775500 68.413502 58.357421 0 1.295332 7.397541 0 | 15.145392 73.163592 65.772001 2.27 2.282315 11.648522 | 14.950132 70.657632 61.465342 1.686960 9.200357 3.02 | ||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 436.882200 638.513205 223.195002 0 113.543829 476.735252 0 | 551.395213 706.697495 263.465402 1.653533 213.352932 613.530234 1.293298 | 484.606492 668.543402 243.792502 0.589842 178.416452 558.464329 0.539520 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | -4.445052 −23.0423521 −44.356992 −99 −87.920501 −40.345210 −99 | −8.728848 −19.167452 −34.123586 −98.835492 −79.465202 −27.446501 −98.872555 | −6.178500 −21.911352 −37.588482 −98.947900 −83.891430 −36.452653 −98.962902 | ||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 2428.940492 4012.652903 3888.542030 0 33.682005 293.459724 0 | 2592.352049 4633.727049 4683.634029 0.418092 72.436405 518.965567 0.435304 | 2433.183441 4315.743555 4365.895902 53.696570 385.652334 4.67 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | 20.651103 0.162015 0.147652 0.142465 0.117900 3.65 | 20.770492 0.187683 0.191543 0.186110 0.169890 3.33 | 20.659945 0.169435 0.175026 0.163519 0.154551 9.66 | ||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 4.924155 4.140800 0.635644 3.269543 | 5.442632 4.543301 1.108742 4.236500 2.75 | 5.032945 4.412393 1.03 0.845798 3.732709 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | 93.691550 95.931103 0 0.880193 8.075650 0 | 105.629021 109.031902 1.358399 14.832029 2.83 | 100.355603 103.562900 6.53 1.099707 11.357613 | ||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0 21978.054329 0 | 3.45 | 6.65 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | 3.158222 3.186900 2.271819 0 1.756300 2.569432 0 | 3.158339 3.256833 2.492549 0.786523 1.896942 2.835301 0.725431 | 3.158275 3.225253 2.405293 0.451663 1.8131570 2.704312 0.417902 | ||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0.185455 0 36.945444 293.842003 0 | 0.499821 0.567650 73.345992 486.513050 0.475325 | 0.289011 53.455374 375.451515 3.19 |
Function | Algorithm | Best Value | Worst Value | Mean Value | Function | Algorithm | Best Value | Worst Value | Mean Value |
---|---|---|---|---|---|---|---|---|---|
FA VSSFA LFA GDAFA WFA CLFA CFAEE | −1 −1 −1 −1 −1 −1 −1 | −1 −1 −1 −1 −0.95357 −1 −1 | −1 −1 −1 −1 −0.997534 −1 −1 | FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0 0 0 0 0 0 0 | 0 0 0 0 0 0 | 0 0 0 0 0 0 | ||
FA VSSFA LFA GDAFA WFA CLFA CFAEE | 0 0 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 |
FA | VSSFA | LFA | WFA | CLFA | GDAFA | CFAEE | |
---|---|---|---|---|---|---|---|
Best 10 dim | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Mean 10 dim | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Worst 10 dim | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Total 10 dim | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Best 30 dim | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
Mean 30 dim | 0 | 0 | 0 | 0 | 0 | 2 | 13 |
Worst 30 dim | 0 | 0 | 0 | 0 | 0 | 3 | 12 |
Total 30 dim | 0 | 0 | 0 | 0 | 0 | 5 | 26 |
Best 100 dim | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
Mean 100 dim | 0 | 0 | 0 | 0 | 0 | 3 | 12 |
Worst 100 dim | 0 | 0 | 0 | 0 | 0 | 3 | 12 |
Total 100 dim | 0 | 0 | 0 | 0 | 0 | 6 | 25 |
GRAND TOTAL | 0 | 0 | 0 | 0 | 0 | 11 | 51 |
Function | CFAEE | GDAFA | FA | VSSFA | LFA | WFA | CLFA |
---|---|---|---|---|---|---|---|
f1 | 1.02 | 7.50 | |||||
f2 | |||||||
f3 | |||||||
f4 | |||||||
f5 | |||||||
f6 | 5.05 | 4.41 | 3.61 | ||||
f7 | |||||||
f8 | |||||||
f9 | 7.27 | 5.95 | |||||
f10 | |||||||
f11 | 1.68 | 9.20 | |||||
f12 | − | − | −6.18 | − | − | − | − |
f13 | |||||||
f14 | 1.10 | ||||||
f15 | 3.16 | 3.22 | 2.41 | 1.81 | 2.71 | ||
p-value | 3.125 | 4.39 | 2.13 | 3.05 | 3.05 | 3.05 | 3.05 |
ID | Name of the function | Class | Search Range | Optimum |
---|---|---|---|---|
F1 | Shifted and Rotated Bent Cigar Function | Unimodal | [−100, 100] | 100 |
F2 | Shifted and Rotated Sum of Different Power Function | Unimodal | [−100, 100] | 200 |
F3 | Shifted and Rotated Zakharov Function | Unimodal | [−100, 100] | 300 |
F4 | Shifted and Rotated Rosenbrock’s Function | Multimodal | [−100, 100] | 400 |
F5 | Shifted and Rotated Rastrigin’s Function | Multimodal | [−100, 100] | 500 |
F6 | Shifted and Rotated Expanded Scaffer’s Function | Multimodal | [−100, 100] | 600 |
F7 | Shifted and Rotated Lunacek Bi-Rastrigin Function | Multimodal | [−100, 100] | 700 |
F8 | Shifted and Rotated Non-Continuous Rastrigin’s Function | Multimodal | [−100, 100] | 800 |
F9 | Shifted and Rotated Lévy Function | Multimodal | [−100, 100] | 900 |
F10 | Shifted and Rotated Schwefel’s Function | Multimodal | [−100, 100] | 1000 |
F11 | Hybrid Function 1 (N = 3) | Hybrid | [−100, 100] | 1100 |
F12 | Hybrid Function 2 (N = 3) | Hybrid | [−100, 100] | 1200 |
F13 | Hybrid Function 3 (N = 3) | Hybrid | [−100, 100] | 1300 |
F14 | Hybrid Function 4 (N = 4) | Hybrid | [−100, 100] | 1400 |
F15 | Hybrid Function 5 (N = 4) | Hybrid | [−100, 100] | 1500 |
F16 | Hybrid Function 6 (N = 4) | Hybrid | [−100, 100] | 1600 |
F17 | Hybrid Function 6 (N = 5) | Hybrid | [−100, 100] | 1700 |
F18 | Hybrid Function 6 (N = 5) | Hybrid | [−100, 100] | 1800 |
F19 | Hybrid Function 6 (N = 5) | Hybrid | [−100, 100] | 1900 |
F20 | Hybrid Function 6 (N = 6) | Hybrid | [−100, 100] | 2000 |
F21 | Composition Function 1 (N = 3) | Composition | [−100, 100] | 2100 |
F22 | Composition Function 2 (N = 3) | Composition | [−100, 100] | 2200 |
F23 | Composition Function 3 (N = 4) | Composition | [−100, 100] | 2300 |
F24 | Composition Function 4 (N = 4) | Composition | [−100, 100] | 2400 |
F25 | Composition Function 5 (N = 5) | Composition | [−100, 100] | 2500 |
F26 | Composition Function 6 (N = 5) | Composition | [−100, 100] | 2600 |
F27 | Composition Function 7 (N = 6) | Composition | [−100, 100] | 2700 |
F28 | Composition Function 8 (N = 6) | Composition | [−100, 100] | 2800 |
F29 | Composition Function 9 (N = 3) | Composition | [−100, 100] | 2900 |
F30 | Composition Function 10 (N = 3) | Composition | [−100, 100] | 3000 |
Algorithm | F1 | F2 | F3 | F4 | F5 | |||||
---|---|---|---|---|---|---|---|---|---|---|
Mean | STD | Mean | STD | Mean | STD | Mean | STD | Mean | STD | |
IHHO | 26.921 | n/a | n/a | 3.02 | 52.152 | 4.03 | 2.607 | 3.251 | ||
HHO | n/a | n/a | 53.631 | 24.927 | ||||||
DE | n/a | n/a | 8.530 | 6.232 | ||||||
GOA | n/a | n/a | 61.300 | 19.48 | 16.803 | |||||
GWO | n/a | n/a | 10.705 | 8.543 | ||||||
MFO | n/a | n/a | 27.727 | 12.860 | ||||||
MVO | n/a | n/a | 46.451 | 1.392 | 9.888 | |||||
PSO | n/a | n/a | 65.409 | 10.318 | 7.305 | |||||
WOA | n/a | n/a | 69.033 | 17.46 | ||||||
SCA | n/a | n/a | 47.271 | 9.352 | ||||||
FA | n/a | n/a | 54.991 | 18.858 | 19.302 | |||||
CFAEE | 1.31 | 14.353 | n/a | n/a | 3.02 | 28.131 | 2.372 | 5.01 | 3.285 | |
Algorithm | F6 | F7 | F8 | F9 | F10 | |||||
Mean | STD | Mean | STD | Mean | STD | Mean | STD | Mean | STD | |
IHHO | 6.00 | 0.082 | 10.041 | 6.526 | 85.42 | |||||
HHO | 12.320 | 18.921 | 5.700 | |||||||
DE | 4.744 | 10.373 | 6.873 | |||||||
GOA | 10.295 | 11.375 | 14.512 | 93.212 | ||||||
GWO | 1.909 | 16.343 | 5.053 | 12.11 | ||||||
MFO | 2.411 | 22.655 | 13.786 | |||||||
MVO | 4.365 | 11.278 | 12.216 | 9.00 | 0.012 | |||||
PSO | 3.539 | 9.008 | 5.982 | 9.00 | 0.003 | 1.50 | ||||
WOA | 13.695 | 23.692 | 17.470 | |||||||
SCA | 4.105 | 13.299 | 7.577 | 85.98 | ||||||
FA | 11.393 | 11.55 | 13.914 | 81.44 | ||||||
CFAEE | 6.00 | 0.051 | 7.23 | 11.391 | 8.08 | 5.422 | 42.11 | 1.25 | ||
Algorithm | F11 | F12 | F13 | F14 | F15 | |||||
Mean | STD | Mean | STD | Mean | STD | Mean | STD | Mean | STD | |
IHHO | 13.523 | 1.42 | 1.651 | |||||||
HHO | 45.729 | |||||||||
DE | 36.317 | 1.35 | 78.355 | 11.826 | 1.51 | 18.454 | ||||
GOA | 58.009 | |||||||||
GWO | 183.524 | |||||||||
MFO | 107.133 | |||||||||
MVO | 27.331 | |||||||||
PSO | 1.10 | 3.727 | 88.291 | |||||||
WOA | 82.415 | |||||||||
SCA | 96.535 | |||||||||
FA | 39.705 | |||||||||
CFAEE | 1.10 | 1.503 | 3.18 | 2.29 | 1.35 | 20.499 | 21.350 | 1.51 | 10.217 | |
Algorithm | F16 | F17 | F18 | F19 | F20 | |||||
Mean | STD | Mean | STD | Mean | STD | Mean | STD | Mean | STD | |
IHHO | 59.44 | 7.519 | 1.90 | 6.993 | 19.561 | |||||
HHO | 65.751 | 86.017 | ||||||||
DE | 41.15 | 19.514 | 1.84 | 23.298 | 23.711 | |||||
GOA | 74.824 | |||||||||
GWO | 38.759 | 73.994 | ||||||||
MFO | 65.311 | 72.321 | ||||||||
MVO | 46.126 | 86.303 | ||||||||
PSO | 1.65 | 65.364 | 16.123 | 35.410 | ||||||
WOA | 73.459 | |||||||||
SCA | 95.425 | 46.855 | ||||||||
FA | 71.303 | |||||||||
CFAEE | 86.359 | 1.71 | 8.442 | 21.565 | 1.90 | 8.717 | 2.01 | 9.443 | ||
Algorithm | F21 | F22 | F23 | F24 | F25 | |||||
Mean | STD | Mean | STD | Mean | STD | Mean | STD | Mean | STD | |
IHHO | 2.20 | 4.615 | 17.820 | 14.213 | 85.338 | |||||
HHO | 53.711 | 25.234 | 35.522 | 93.623 | 49.573 | |||||
DE | 78.104 | 17.513 | 15.163 | 2.66 | 69.502 | 15.543 | ||||
GOA | 56.877 | 23.536 | 57.833 | 32.598 | ||||||
GWO | 32.884 | 57.573 | 13.862 | 25.132 | 28.256 | |||||
MFO | 29.255 | 93.557 | 11.327 | 76.435 | 37.776 | |||||
MVO | 11.839 | 10.445 | 18.246 | 84.256 | ||||||
PSO | 49.783 | 72.300 | 76.143 | 33.735 | ||||||
WOA | 60.021 | 29.838 | 85.902 | |||||||
SCA | 65.229 | 66.636 | 45.449 | 11.548 | 37.291 | |||||
FA | 34.701 | 17.452 | 47.019 | |||||||
CFAEE | 2.20 | 48.552 | 2.26 | 13.040 | 2.55 | 21.929 | 2.81 | 95.429 | ||
Algorithm | F26 | F27 | F28 | F29 | F30 | |||||
Mean | STD | Mean | STD | Mean | STD | Mean | STD | Mean | STD | |
IHHO | 33.657 | 48.694 | 3.20 | 28.982 | ||||||
HHO | 51.306 | 85.653 | ||||||||
DE | 95.929 | 3.07 | 2.558 | 27.035 | 35.216 | |||||
GOA | 25.326 | 75.411 | ||||||||
GWO | 13.541 | 49.822 | ||||||||
MFO | 5.722 | 93.459 | 55.593 | |||||||
MVO | 21.875 | 75.139 | ||||||||
PSO | 31.830 | 62.374 | ||||||||
WOA | 48.124 | |||||||||
SCA | 13.152 | 89.259 | 48.339 | |||||||
FA | 27.015 | 31.117 | ||||||||
CFAEE | 2.86 | 48.690 | 3.13 | 3.20 | 27.914 | 2.22 | 1.44 |
Function | IHHO | HHO | DE | GOA | GWO | MFO | MVO | PSO | WOA | SCA | FA | CFAEE |
---|---|---|---|---|---|---|---|---|---|---|---|---|
F1 | 2 | 7 | 11 | 5 | 9 | 8 | 3 | 4 | 10 | 12 | 6 | 1 |
F3 | 1.5 | 7 | 10 | 3.5 | 8 | 12 | 3.5 | 6 | 11 | 9 | 5 | 1.5 |
F4 | 1 | 10 | 9 | 6 | 5 | 8 | 3 | 4 | 11 | 12 | 7 | 2 |
F5 | 2 | 9 | 11 | 5 | 4 | 8 | 3 | 6 | 10 | 12 | 7 | 1 |
F6 | 1.5 | 11 | 9 | 6 | 3 | 4 | 5 | 7 | 10 | 8 | 12 | 1.5 |
F7 | 8 | 11 | 12 | 4 | 5.5 | 7 | 3 | 2 | 9 | 10 | 5.5 | 1 |
F8 | 2 | 6.5 | 12 | 8 | 3 | 6.5 | 5 | 4 | 10 | 11 | 9 | 1 |
F9 | 8 | 10 | 12 | 5.5 | 3 | 9 | 1.5 | 1.5 | 11 | 7 | 5.5 | 4 |
F10 | 3 | 9 | 10 | 7 | 4 | 8 | 5 | 1 | 11 | 12 | 6 | 2 |
F11 | 3 | 6.5 | 4.5 | 8 | 12 | 10 | 4.5 | 1.5 | 9 | 11 | 6.5 | 1.5 |
F12 | 4 | 10 | 3 | 8 | 5 | 7 | 6 | 2 | 11 | 12 | 9 | 1 |
F13 | 3 | 11 | 1.5 | 10 | 7 | 9 | 4 | 5 | 8 | 12 | 6 | 1.5 |
F14 | 1 | 5 | 3 | 9 | 10 | 12 | 8 | 4 | 11 | 7 | 6 | 2 |
F15 | 4 | 10 | 1.5 | 9 | 8 | 11 | 7 | 3 | 12 | 5 | 6 | 1.5 |
F16 | 5.5 | 11 | 3 | 1 | 7.5 | 10 | 9 | 2 | 12 | 5.5 | 7.5 | 4 |
F17 | 3 | 7 | 4.5 | 12 | 4.5 | 6 | 8.5 | 2 | 10.5 | 8.5 | 10.5 | 1 |
F18 | 3 | 7 | 1 | 5 | 11 | 10 | 8 | 4 | 9 | 12 | 6 | 2 |
F19 | 1.5 | 10 | 3 | 6 | 11 | 8 | 7 | 4 | 12 | 9 | 5 | 1.5 |
F20 | 2 | 12 | 3 | 10 | 5 | 8 | 6.5 | 4 | 11 | 9 | 6.5 | 1 |
F21 | 1.5 | 12 | 3 | 7.5 | 7.5 | 9.5 | 9.5 | 4 | 11 | 5.5 | 5.5 | 1.5 |
F22 | 2 | 5 | 3 | 10 | 4 | 8 | 6.5 | 6.5 | 12 | 11 | 9 | 1 |
F23 | 2 | 12 | 6.5 | 8 | 4.5 | 6.5 | 9 | 3 | 10 | 11 | 4.5 | 1 |
F24 | 3 | 12 | 1 | 6 | 7.5 | 9 | 7.5 | 4 | 10 | 11 | 5 | 2 |
F25 | 2 | 9 | 4 | 6.5 | 8 | 10 | 5 | 3 | 11.5 | 11.5 | 6.5 | 1 |
F26 | 2 | 12 | 3.5 | 5 | 10 | 7 | 8.5 | 3.5 | 11 | 8.5 | 6 | 1 |
F27 | 12 | 11 | 1 | 7 | 5 | 3 | 5 | 8 | 10 | 9 | 5 | 2 |
F28 | 4 | 10 | 3 | 5 | 11 | 2 | 8 | 6.5 | 12 | 9 | 6.5 | 1 |
F29 | 1.5 | 11 | 3.5 | 10 | 5 | 8 | 8 | 3.5 | 12 | 6 | 8 | 1.5 |
F30 | 2 | 11 | 3 | 6 | 7 | 8 | 4 | 9 | 10 | 12 | 5 | 1 |
Average Ranking | 3.138 | 9.483 | 5.362 | 6.862 | 6.724 | 8.017 | 5.914 | 4.069 | 10.621 | 9.603 | 6.655 | 1.552 |
Rank | 2 | 10 | 4 | 8 | 7 | 9 | 5 | 3 | 12 | 11 | 6 | 1 |
Function | IHHO | HHO | DE | GOA | GWO | MFO | MVO | PSO | WOA | SCA | FA | CFAEE |
---|---|---|---|---|---|---|---|---|---|---|---|---|
F1 | 2 | 7 | 347 | 5 | 9 | 8 | 3 | 4 | 346 | 348 | 6 | 1 |
F3 | 56.5 | 63 | 327 | 58.5 | 323 | 334 | 58.5 | 61 | 328 | 326 | 60 | 56.5 |
F4 | 144 | 226 | 211 | 177 | 164 | 192 | 152 | 157 | 255 | 278 | 183 | 146 |
F5 | 138 | 213 | 242 | 194 | 174 | 206 | 169 | 196 | 241 | 245 | 197 | 135 |
F6 | 153.5 | 235 | 218 | 173 | 158 | 161 | 165 | 180 | 232 | 212 | 270 | 153.5 |
F7 | 193 | 264 | 266 | 145 | 155.5 | 184 | 140 | 136 | 244 | 246 | 155.5 | 131 |
F8 | 151 | 198.5 | 251 | 204 | 167 | 198.5 | 191 | 172 | 229 | 231 | 207 | 143 |
F9 | 141 | 310 | 318 | 89.5 | 80 | 285 | 78.5 | 78.5 | 313 | 91 | 89.5 | 87 |
F10 | 81 | 293 | 301 | 257 | 88 | 290 | 95 | 74 | 309 | 317 | 216 | 75 |
F11 | 114 | 159.5 | 127.5 | 185 | 300 | 271 | 127.5 | 103.5 | 265 | 280 | 159.5 | 103.5 |
F12 | 13 | 19 | 12 | 17 | 14 | 16 | 15 | 11 | 344 | 345 | 18 | 10 |
F13 | 43 | 331 | 39.5 | 324 | 54 | 321 | 45 | 46 | 314 | 337 | 53 | 39.5 |
F14 | 66 | 70 | 68 | 311 | 316 | 333 | 73 | 69 | 320 | 72 | 71 | 67 |
F15 | 52 | 329 | 48.5 | 322 | 303 | 335 | 65 | 51 | 336 | 55 | 62 | 48.5 |
F16 | 291.5 | 308 | 276 | 64 | 298.5 | 305 | 302 | 233 | 312 | 291.5 | 298.5 | 282 |
F17 | 122 | 225 | 181.5 | 269 | 181.5 | 205 | 239.5 | 113 | 262.5 | 239.5 | 262.5 | 107 |
F18 | 38 | 76 | 35 | 47 | 332 | 325 | 82 | 41 | 319 | 338 | 50 | 36 |
F19 | 28.5 | 44 | 30 | 33 | 330 | 37 | 34 | 31 | 339 | 42 | 32 | 28.5 |
F20 | 98 | 294 | 112 | 260.5 | 150 | 237.5 | 223 | 121 | 286 | 247.5 | 223 | 96 |
F21 | 99.5 | 281 | 129 | 227.5 | 227.5 | 252.5 | 252.5 | 162.5 | 272 | 209 | 209 | 99.5 |
F22 | 110 | 142 | 118 | 258 | 130 | 217 | 170.5 | 170.5 | 297 | 283 | 234 | 101 |
F23 | 126 | 279 | 202.5 | 223 | 178.5 | 202.5 | 237.5 | 134 | 247.5 | 260.5 | 178.5 | 102 |
F24 | 119.5 | 288 | 105 | 200.5 | 220 | 236 | 220 | 132.5 | 259 | 267.5 | 175.5 | 111 |
F25 | 117 | 243 | 168 | 214.5 | 230 | 256 | 195 | 139 | 274.5 | 274.5 | 214.5 | 93 |
F26 | 84 | 315 | 85.5 | 94 | 306 | 106 | 249.5 | 85.5 | 307 | 249.5 | 97 | 77 |
F27 | 284 | 277 | 119.5 | 175.5 | 148 | 132.5 | 148 | 200.5 | 267.5 | 220 | 148 | 125 |
F28 | 137 | 287 | 124 | 166 | 289 | 92 | 254 | 189.5 | 296 | 273 | 189.5 | 83 |
F29 | 108.5 | 295 | 115.5 | 209 | 123 | 187 | 187 | 115.5 | 304 | 162.5 | 187 | 108.5 |
F30 | 21 | 342 | 22 | 25 | 26 | 27 | 23 | 340 | 341 | 343 | 24 | 20 |
Average Ranking | 108.017 | 221.172 | 158.621 | 169.948 | 188.810 | 205.259 | 144.655 | 122.328 | 291.724 | 244.276 | 147.259 | 91.931 |
Rank | 2 | 10 | 6 | 7 | 8 | 9 | 4 | 3 | 12 | 11 | 5 | 1 |
Friedman Value | Critical Value | p-Value | Iman–Davenport Value | F Critical Value |
---|---|---|---|---|
1.820 |
Comparison | p_VALUES | Ranking | alpha = 0.05 | alpha = 0.1 | H1 | H2 |
---|---|---|---|---|---|---|
CFAEE vs. HHO | 0 | 0 | 0.00455 | 0.00909 | TRUE | TRUE |
CFAEE vs. WOA | 0 | 1 | 0.00500 | 0.01000 | TRUE | TRUE |
CFAEE vs. SCA | 0 | 2 | 0.00556 | 0.01111 | TRUE | TRUE |
CFAEE vs. MFO | 3 | 0.00625 | 0.01250 | TRUE | TRUE | |
CFAEE vs. GOA | 4 | 0.00714 | 0.01429 | TRUE | TRUE | |
CFAEE vs. GWO | 5 | 0.00833 | 0.01667 | TRUE | TRUE | |
CFAEE vs. FA | 6 | 0.01000 | 0.02000 | TRUE | TRUE | |
CFAEE vs. MVO | 7 | 0.01250 | 0.02500 | TRUE | TRUE | |
CFAEE vs. DE | 8 | 0.01667 | 0.03333 | TRUE | TRUE | |
CFAEE vs. PSO | 9 | 0.02500 | 0.05000 | TRUE | TRUE | |
CFAEE vs. IHHO | 10 | 0.05000 | 0.10000 | FALSE | FALSE |
Dataset | Train Set | Validation Set | Testing Set |
---|---|---|---|
MNIST | 20.000 (64) | 40.000 (100) | 10.000 (100) |
Fashion-MNIST | 20.000 (64) | 40.000 (100) | 10.000 (100) |
Semeion | 200 (2) | 400 (400) | 993 (993) |
USPS | 2.406 (32) | 4.885 (977) | 2.007 (2.007) |
CIFAR-10 | 20.000 (100) | 30.000 (100) | 10.000 (100) |
Dataset | Epochs | ||||
---|---|---|---|---|---|
MNIST | 0.01 | 0.9 | 0.0005 | [0, 1] | 10.000 |
Fashion-MNIST | 0.01 | 0.9 | 0.0005 | [0, 1] | 10.000 |
Semeion | 0.001 | 0,9 | 0.0005 | [0, 1] | 10.000 |
USPS | 0.01 | 0.9 | 0.0005 | [0, 1] | 10.000 |
CIFAR-10 | 0.001 | 0.9 | 0.004 | [0, 1] | 4.000 |
Algorithm | Parameters |
---|---|
BA [67] | , , , |
CS [68] | , , |
PSO [69] | , , |
EHO [70] | , , , |
WOA [53] | linearly decreasing from 2 to 0, linearly decreasing from −1 to −2, b=1 |
SCA [51] | , linearly decreasing from 2 to 0 |
SSA [72] | non-linearly decreasing from 2 to 0, and rand from [0, 1] |
GOA [52] | c linearly decreasing from 1 to 0 |
BBO [71] | , , |
FA [1] | , , |
Method | MNIST | Fashion-MNIST | Semeion | USPS | CIFAR-10 | |||||
---|---|---|---|---|---|---|---|---|---|---|
acc. | acc. | acc. | acc. | acc. | ||||||
Caffe | 99.07 | 0 | 91.71 | 0 | 97.62 | 0 | 95.80 | 0 | 71.47 | 0 |
Dropout Caffe | 99.18 | 0.5 | 92.53 | 0.5 | 98.14 | 0.5 | 96.21 | 0.5 | 72.08 | 0.5 |
BA | 99.14 | 0.491 | 92.56 | 0.505 | 98.35 | 0.692 | 96.45 | 0.762 | 71.49 | 0.633 |
CS | 99.14 | 0.489 | 92.41 | 0.491 | 98.21 | 0.544 | 96.31 | 0.715 | 71.21 | 0.669 |
PSO | 99.16 | 0.493 | 92.38 | 0.481 | 97.79 | 0.371 | 96.33 | 0.725 | 71.51 | 0.621 |
EHO | 99.13 | 0.475 | 92.36 | 0.470 | 98.11 | 0.481 | 96.24 | 0.682 | 71.15 | 0.705 |
WOA | 99.15 | 0.489 | 92.43 | 0.493 | 98.23 | 0.561 | 96.32 | 0.722 | 71.23 | 0.685 |
SCA | 99.17 | 0.496 | 92.53 | 0.501 | 98.25 | 0.580 | 96.29 | 0.705 | 71.54 | 0.597 |
SSA | 99.19 | 0.499 | 92.63 | 0.527 | 98.31 | 0.642 | 96.41 | 0.753 | 71.58 | 0.529 |
GOA | 99.16 | 0.492 | 92.44 | 0.494 | 98.15 | 0.513 | 96.15 | 0.481 | 70.95 | 0.849 |
BBO | 99.13 | 0.474 | 92.35 | 0.468 | 98.16 | 0.515 | 96.17 | 0.483 | 71.08 | 0.768 |
FA | 99.18 | 0.495 | 92.58 | 0.511 | 98.29 | 0.619 | 96.42 | 0.758 | 71.55 | 0.583 |
CFAEE | 99.26 | 0.529 | 92.73 | 0.570 | 98.46 | 0.719 | 96.88 | 0.845 | 72.32 | 0.388 |
Function | CFAEE | Caffe | DropoutCaffe | BA | CS | PSO | EHO | WOA | SCA | SSA | GOA | BBO | FA |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
MNIST | 0.74 | 0.9 | 0.82 | 0.86 | 0.86 | 0.8 | 0.87 | 0.85 | 0.83 | 0.81 | 0.84 | 0.87 | 0.82 |
Fashion-MNIST | 7.27 | 8.29 | 7.47 | 7.44 | 7.59 | 7.62 | 7.64 | 7.57 | 7.47 | 7.37 | 7.56 | 7.65 | 7.42 |
Semeion | 1.54 | 2.38 | 1.86 | 1.65 | 1.79 | 2.21 | 1.89 | 1.77 | 1.75 | 1.69 | 1.85 | 1.84 | 1.71 |
USPS | 3.12 | 4.2 | 3.79 | 3.55 | 3.69 | 3.67 | 3.76 | 3.68 | 3.71 | 3.59 | 3.85 | 3.83 | 3.58 |
CIFAR-10 | 27.68 | 28.53 | 27.92 | 28.51 | 28.79 | 28.5 | 28.85 | 28.77 | 28.46 | 28.42 | 29.05 | 28.92 | 28.45 |
p-value | 3.125 | 3.125 | 3.125 | 3.125 | 3.125 | 3.125 | 3.125 | 3.125 | 3.125 | 3.125 | 3.125 | 3.125 | 3.125 |
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Bacanin, N.; Stoean, R.; Zivkovic, M.; Petrovic, A.; Rashid, T.A.; Bezdan, T. Performance of a Novel Chaotic Firefly Algorithm with Enhanced Exploration for Tackling Global Optimization Problems: Application for Dropout Regularization. Mathematics 2021, 9, 2705. https://doi.org/10.3390/math9212705
Bacanin N, Stoean R, Zivkovic M, Petrovic A, Rashid TA, Bezdan T. Performance of a Novel Chaotic Firefly Algorithm with Enhanced Exploration for Tackling Global Optimization Problems: Application for Dropout Regularization. Mathematics. 2021; 9(21):2705. https://doi.org/10.3390/math9212705
Chicago/Turabian StyleBacanin, Nebojsa, Ruxandra Stoean, Miodrag Zivkovic, Aleksandar Petrovic, Tarik A. Rashid, and Timea Bezdan. 2021. "Performance of a Novel Chaotic Firefly Algorithm with Enhanced Exploration for Tackling Global Optimization Problems: Application for Dropout Regularization" Mathematics 9, no. 21: 2705. https://doi.org/10.3390/math9212705
APA StyleBacanin, N., Stoean, R., Zivkovic, M., Petrovic, A., Rashid, T. A., & Bezdan, T. (2021). Performance of a Novel Chaotic Firefly Algorithm with Enhanced Exploration for Tackling Global Optimization Problems: Application for Dropout Regularization. Mathematics, 9(21), 2705. https://doi.org/10.3390/math9212705