Sigmoid Derivative

This proof is to answer @Christian Sanchez question regarding how my code calculates the Sigmoid activation derivative easily:

https://www.mathworks.com/matlabcentral/fileexchange/54076-mlp-neural-network-with-backpropagation?s_tid=mlc_fex_email_activity#feedback-31e1e281-f332-4e40-ba59-5e9c3c8cdc5d

Neural Networks Backpropagation Algorithm in Arabic

Neural Networks Backpropagation Algorithm in Arabic - Lectures from April 2017

Ranked top 5% percent in Kaggle Distracted Driver Competition

Kaggle State Farm Distracted Driver Detection competition has just ended, and I ranked within top 5% (64th out of 1450 participating teams, winner's got $65,000).My approach is mainly based on Deep Learning (trained 20 very deep models) but still applies Computer Vision strategies to reduce neural network distraction.A brief description about the system is in the image below:

References:[1] Rajen Bhatt, Abhinav Dhall, 'Skin Segmentation Dataset', UCI Machine Learning Repository.
[2] X. Zhu, D. Ramanan. "Face detection, pose estimation and landmark localization in the wild" Computer Vision and Pattern Recognition (CVPR) Providence, Rhode Island, June 2012.
[3] Simonyan, Karen, and Andrew Zisserman. "Very deep convolutional networks for large-scale image recognition." arXiv preprint arXiv:1409.1556 (2014).

Notes:

• The competition was very challenging, we did not do some costly annotations nor used test data in any form of learning (even semi-supervised) nor annotation.
  We treat images independently, while some participants learn from test data and take advantage of the fact that test images are originally sampled from recorded videos. So, they do some sort of test videos reconstruction and hence image classification makes use of temporal context.
  Also, some participants annotate their training data and some crowdsources the annotation. Which is either too much work or needs money.
  Our system is more general than such systems, even that they are doing better than ours in leaderboard; they are indirectly over-fitting the competition test data.
• The average loss metric used in competition leaderboard ranking doesn't directly reflect the system accuracy. I believe all top 100 systems classification accuracies are higher than 99%, but the loss metric reflects how were you confident in your classification, which is harder. Only a single misclassification with high confidence, will give a very bad average loss.
• Face detection for such problem is hard, the well-known Haar Cascades surely fail. The example in the image above is easy, but normally driver face is seen from side.
• I've tried lots of strategies for upper body Human Pose Estimation (Calvin) and scene/driver segmentation (this and this) but didn't achieve good results.
• I used data augmentation. Because the training data size is not large.
• The camera is not calibrated, and changes orientation. The system should be intelligent enough to handle this.
• I know many interesting details and results (like the following image visualizing the most important area that affected network decision) are missing here, but I will be happy to answer any of your questions about any details.

Collision Avoidance and Navigation with Evolutionary Neural Network

Download Source Code (MATLAB)

1- Project Description

This is a fully configurable MATLAB project that implements and provides simulation for vehicle self-learning of collision avoidance and navigation with a rangefinder sensor using an evolutionary artificial neural network. The neural network guides the vehicle around the environment and a genetic algorithm is used to pick and breed generations of more intelligent vehicles.

The vehicle uses a rangefinder sensor that calculates N intersections depths with the environment and then feeds these N values as inputs to the neural network. The inputs are then passed through a neural network and finally to an output layer of 2 neurons: a left and right steering force. These forces are used to turn the vehicle by deciding the vehicle steering angle.

Each vehicle represents a different chromosome in a generation (or a unique set weight for the neural net) which are evaluated and potentially carried through to the next generation by a fitness score. The fitness score has different definition in each of my three experiments for collision avoidance and navigation self-learning.

2- Software Configurations

• The vehicles dimensions and its wheels (base and dimensions)
• Rangefinder range and number of beams
• The environment
• Neural network architecture
• Number of vehicles and their replacement strategy
• The generic algorithm parameters: mutation probability, crossover probability, crossover site probability distribution, population size, selection strategy, replacement strategy …

3- Simple 2D vehicle steering physics

Given the vehicle speed and simulation time tick Δt the travelled distance L per a single time step is calculated. Given wheel base, vehicle position P, heading θ, and distance travelled per time step L, the new vehicle position Pnew and heading θnew are calculated as shown in figure 1. Video 1 shows a simulation result.

Figure 1. Simple 2D vehicle steering physics

Video 1. Simple 2D vehicle steering physics in action

4- Self-learning Navigation Experiment

Fitness function is chosen to be the distance that the vehicle traveled along the track before it collides with track boundaries. I was surprised by how fast vehicles learn navigation without any human interaction! In less than 50 generations with each generation having a population of 200 chromosomes, and with a neural network of only 3 hidden layers, perfect navigation is learnt! Mutation probability is 0.1, crossover probability is 1, cross over site follows the normal distribution: ~N(95%,5%), selection is based on tournaments of size 10 candidates, and all children replace their parents replacement strategy is adopted.

There is an interesting observation here. For the track map shown in figure 2, the vehicle took 12 generations to learn how to successfully turn in the first critical location A marked by red circle in the figure. Once the vehicle learns that, it achieves a huge fitness increase by implicitly learning how drive through all the following tricky turns in the track. This fact is demonstrated in figure 3 plot. This interesting because it is similar to the way humans learn things. The same effect happens for the vehicle to learn how to turn by 180° in the critical learning location B.

Figure 2. A track critical learning locations

As in figure 3, after 12 generations, the vehicle tries to learn how to turn by 180° in the critical learning location B, so it modifies its behavior but in a way that makes it fail to pass through the critical location A. This is why the fitness decreases again after it has increased, and that repeats until the vehicle learns to avoid such bad behavior by itself. However, the vehicle still fails to turn by 180°, and this is why the fitness function saturates. Actually, the road is too narrow for the vehicle to learn how to achieve that tricky 180° turn in a small number of learning generations. In another experiment, I modified the track to have a wider width (30 meters width instead of 12 meters, new map is also shown in figure 4). In only 16 iterations the vehicle learned to do that tricky 180° turn and navigate through the map almost forever without colliding! (Specifically, the car travelled the whole track more than 100 times until I stopped it manually.)

Figure 3. Fitness function per generation for figure 3 track set-up

Figure 4. Fitness function per generation for figure 3 track set-up with a wider track

Video 2. Navigation Self-learning

For the track map of figure 5, the time traveled by the vehicle before crash for each generation is shown for different rangefinder sensor number of beams. Moderate number of beams (5 beams performed best) is proven to be the better. Figure 6 shows the same information for different rangefinder sensor ranges. The higher the sensor range is proven to be the better.

Figure 5. Fitness per generation for different number of rangefinder sensor number of beams

Figure 6. Fitness per generation for different number of rangefinder sensor ranges

It’s important to mention that to prevent vehicles from rotating around themselves, a trick that is described later (section 6) in this report is used.

5- Can vehicle learn route to a specific destination?

With a simple modification to the fitness function, such that the fitness function becomes the subtraction of the vehicle drive time before collision and the Euclidean distance between the vehicle position and the destination location just before collision, the vehicle easily learns its route to the destination. In my recorded video for this experiment, it took the vehicle only 7 generations to learn its route to a far destination!

Video 3. Self-learning route to a specific destination

Figure 7. Vehicle learns to decide which turn to take to reach the destination correctly

6- Self-learning Collision Avoidance Experiment

Fitness for each vehicle is simply to survive. A vehicle dies and starts from a random location if it collides with track boundaries or with another vehicle. It’s important to penalize the vehicle responsible for the accident when a collision happens as shown in figure 8. I came to that simple role: when a collision happens, ask the question: “Would crash still happen if a vehicle x is the only vehicle that moved at collision time step?”. If the answer is yes, vehicle x is a reason for that accident, and should be penalized.

Figure 8. Collision penalization. Two examples with two vehicles before and after the accident time step

It is interesting to discover that vehicles started to learn bad habits to survive. Each vehicle learned to rotate around itself such that it avoids colliding with track boundaries and other vehicles! Figure 9 show such behavior. To cope with that, the fitness function is modified such that if a vehicle “gets smart” and starts to rotate around itself, it is penalized with a fitness of zero. That was a banality that is good enough for vehicles not to adopt such a bad habit. The standard deviation of vehicle position can easily detect such behavior.

Figure 9. Vehicles learn bad habit too!

Eventually, vehicles learned to avoid collision. Videos 4 and 5 show the experiment results for early and late generations respectively. The fascinating thing is that no human has told the vehicles how to drive and avoid collision! The video for late generation is recorded while cars are in generations 33, 20, 12, 12, 31, 21, 18, and 14 respectively. A different replacement strategy is adopted to achieve such good performance; the new population is composed of the best 90% children chromosomes in addition to 10% of the best chromosomes from all the vehicles.

Video 4. Collision avoidance experiment for early generations

Video 5. Collision avoidance experiment for late generations

MLP Neural Network with Backpropagation [MATLAB Code]

This is an implementation for Multilayer Perceptron (MLP) Feed Forward Fully Connected Neural Network with a Sigmoid activation function. The training is done using the Backpropagation algorithm with options for Resilient Gradient Descent, Momentum Backpropagation, and Learning Rate Decrease. The training stops when the Mean Square Error (MSE) reaches zero or a predefined maximum number of epochs is reached.

Four example data for training and testing are included with the project. They are generated by SharkTime Sharky Neural Network.

1- Download Code
http://www.mathworks.com/matlabcentral/fileexchange/54076-mlp-neural-network-with-backpropagation

2- Network architecture & Training Parameters:
The code configuration parameters are as follows:

1- Numbers of hidden layers and neurons per hidden layer. It’s represented by the variable nbrOfNeuronsInEachHiddenLayer. To have a neural network with 3 hidden layers with number of neurons 4, 10, and 5 respectively; that variable is set to [4 10 5].
2- Number of output layer nits. Usually the number of output units is equal to the number of classes, but it still can be less (≤ log2(nbrOfClasses)). It’s represented by the variable nbrOfOutUnits. The number of input layer units is obtained from the training samples dimension.
3- The selection if the sigmoid activation function is unipolar or polar. It’s represented by the variable unipolarBipolarSelector.
4- The learning rate η.
5- The maximum number of epochs at which the training stops unless MSE reaches zero. It’s represented by the variable nbrOfEpochs_max.
6- Option to enable or disable Momentum Backpropagation. It’s represented by the variable enable_learningRate_momentum.
7- The Momentum Backpropagation rate α. It’s represented by the variable momentum_alpha.
8- Option to enable or disable Resilient Gradient Descent. It’s represented by the variable enable_resilient_gradient_descent.
9- The Resilient Gradient Descent parameters: η+, η-, Δmin, Δmax, represented by the variables learningRate_plus, learningRate_negative, deltas_min, and deltas_max.

10- Option to enable or disable Learning Rate Decrease. It’s represented by the variable enable_decrease_learningRate.
11- The Learning Rate Decrease parameters: and . It’s represented by the variables learningRate_decreaseValue and min_learningRate.

The code also contains a parameter for drawing the decision boundary separating the classes and the MSE curve. The number of epochs after which a figure is drawn and saved on the machine is specified. The figures are saved in a folder named Results besides the m files. This parameter is represented by the variable draw_each_nbrOfEpochs. The variable dataFileName takes the Sharky input points file name as string.

3- Results:
In all of the following four test cases, MCCR=1. The stop condition is that MSE reaches zero in all the cases. A unipolar sigmoid function is chosen.

A) Linear Points Case:

Figure 1. Network: 2-4-2 , Unipolar Sigmoid Activation, No Options, η=0.15.

B) Circle Points Case:

Figure2. Network: 2-10-2 , Unipolar Sigmoid Activation, Resilient Gradient Descent, η+=1.2, η-=0.5, Δmin=10^-6, Δmax=50.

C) Wave Points Case:

Figure 3. Network: 2-10-10-2 , Unipolar Sigmoid Activation, Resilient Gradient Descent, η+=1.2, η-=0.5, Δmin=10^-6, Δmax=50.

D) Spiral Points Case:

Figure 4. Network: 2-10-10-2 , Unipolar Sigmoid Activation, Resilient Gradient Descent, η+=1.2, η-=0.5, Δmin=10^-6, Δmax=50.

The following image shows a magnification for the decision boundary with better resolution.

Figure 5. Magnified decision boundary with better resolution for Spiral points case. 

The following video shows the solving of the Two Spirals problem.

Video 1. Solving the Two Spirals Problem.

Solving Two Spirals Problem with Multilayer Perceptron [Video]

Video for Solving Two Spirals Problem with Multilayer Perceptron Neural Networks

I will post the code implementation for the network and the backpropagation training trick used to solve it soon. More details about the problem are in this paper.