This question already has answers here:
How does a back-propagation training algorithm work?
(4 answers)
Closed 6 years ago.
I've recently completed Professor Ng's Machine Learning course on Coursera, but I have some problem with understanding backpropagation algorithm. so I try to read Bishop codes for backpropagation using sigmoid function. I searched and found clean codes which try to explain what backpropagation does, but still have problem with understanding codes
can any one explain me what does really backpropagation do? and also explain codes for me?
here is the code that I found in the github and I mentioned it before
You have an error of the network. And first step of backpropagation is to compute a portion of guilt for each neuron in network. Your goal is to describe an error as dependence of weights(parameter which you can change). So backprop equation is partial derivation error/weights.
First step: error signal = (desired result - output of output neuron) x derivationactivation(x)
where x is input of output neuron. That is portion of guilt for output neuron.
Next step is compute a portion of guilt for hidden units. First part of this step is summation of error signals of next layer x weights which connect hidden unit with next layer unit. And rest is partial derivation of activation function. error signal = sum(nextlayererror x weight)x derivationactivation(x).
Final step is adaptation of weights.
wij = errorsignal_i x learning_rate x output_of_neuron_j
My implementation of BP in Matlab
NN
Related
I'm currently working through the SciML tutorials workshop exercises for the Julia language (https://tutorials.sciml.ai/html/exercises/01-workshop_exercises.html). Specifically, I'm stuck on exercise 6 part 3, which involves training a neural network to approximate the system of equations
function lotka_volterra(du,u,p,t)
x, y = u
α, β, δ, γ = p
du[1] = dx = α*x - β*x*y
du[2] = dy = -δ*y + γ*x*y
end
The goal is to replace the equation for du[2] with a neural network: du[2] = NN(u, p)
where NN is a neural net with parameters p and inputs u.
I have a set of sample data that the network should try to match. The loss function is the squared difference between the network model's output and that sample data.
I defined my network with
NN = Chain(Dense(2,30), Dense(30, 1)). I can get Flux.train! to run, but the problem is that sometimes the initial parameters for the neural network result in a loss on the order of 10^20 and so training never converges. My best try got the loss down from about 2000 initially to about 20 using the ADAM optimizer over about 1000 iterations, but I can't seem to do any better.
How can I make sure my network is consistently trainable, and is there a way to get better convergence?
How can I make sure my network is consistently trainable, and is there a way to get better convergence?
See the FAQ page on techniques for improving convergence. In a nutshell, the single shooting approach of most ML papers is very unstable and does not work on most practical problems, but there are a litany of techniques to help out. One of the best ones is multiple shooting, which optimizes only short bursts (in parallel) along the time series.
But training on a small interval and growing the interval works, also using more stable optimizers (BFGS) can work. You can also weigh the loss function so that earlier times mean more. Lastly, you can minibatch in a way similar to multiple shooting, i.e. start from a data point and only solve to the next (in fact, if you actually look at the original neural ODE paper NumPy code, they do not do the algorithm as explained but instead do this form of sampling to stabilize the spiral ODE training).
This question already has an answer here:
How to set the same initial seed random numbers in Matlab?
(1 answer)
Closed 6 years ago.
I run ANN on MATLAB, and the output of ANN is not consistent every time I run it? How to overcome this problem. I used same data and ANN structure.
clear;
clc;
load ('C:\USers\ARMA\Desktop\DATA.txt');
data=DATA;
N=length(data);
DT=data;
X=DT(1:N,1:2);
Y=DT(1:N,3);
H=3;
net=newff(minmax(X),[H,1],{'logsig','purelin'},'traingdx');
net=init(net);
net.trainparam.Ir=0.9;
net.trainparam.mc=0.1;
net.trainparam.epochs=10000;
net.trainparam.goal=0.001;
net.trainparam.show=1000;
[net,tr]=train(net,X,Y);
plotperform(tr)
The ANN toolbox uses a randomised initial values as initial weights and biases. So apparently the results are sensitive to them.
You need to fix them before training to achieve similar results.
So I'm starting with machine learning and Artificial Neural Networks and I found this article The Nature of Code that introduces to Artificial Neural Networks and the idea of a Perceptron.
Through the article they show you how to create a Perceptron that is able to discriminate between points positioned above or below a line based on the function:
f(x) = 2x + 1
I developed my own Perceptron in Swift and used XCode Playgrounds to illustrate my Perceptron performance.
The perceptron takes 3 inputs: x, y and bias (always 1). The weights of the 3 inputs are generated at random, and after some training they are adjusted.
This first graphic shows the value of the first weight over trainings. As you can see, the value stabilizes at the end, and this is a proof that the Perceptron learned to discriminate points:
The second graphic represents the function line and all the training points (selected at random). The green points are the ones that the Perceptron predicted well, whereas the red ones are the wrong predictions:
As you can see, almost all of the red dots are situated in the inverse function:
f(x) = -2x - 1
My question is why this line of error dots appear. I thought that at a certain point all the weights would be stabilized and that the Perceptron performance would be 100%, but it never does. Is this because of a code bug or ANN always have this tiny interval of error?
Any explanation will be welcomed, although keep in mind that I'm a newbie at ML and ANN.
Thank you very much.
I'm reading Neural Networks and Deep Learning (first two chapters), and I'm trying to follow along and build my own ANN to classify digits from the MNIST data set.
I've been scratching my head for several days now, since my implementation peaks out at ~57% accuracy at classifying digits from the test set (some 5734/10000) after 10 epochs (accuracy for the training set stagnates after the tenth epoch, and accuracy for the test set deteriorates presumably because of over-fitting).
I'm using nearly the same configuration as in the book: 2-layer feedforward ANN (784-30-10) with all layers fully connected; standard sigmoid activation functions; quadratic cost function; weights are initialized the same way (taken from a Gaussian distribution with mean 0 and standard deviation 1)
The only differences being that I'm using online training instead of batch/mini-batch training and a learning rate of 1.0 instead of 3.0 (I have tried mini-batch training + learning rate of 3.0 though)
And yet, my implementation doesn't pass the 60% percentile after a bunch of epochs where as in the book the ANN goes above %90 just after the first epoch with pretty much the exact same configuration.
At first I messed up implementing the backpropagation algorithm, but after reimplementing backpropagation differently three times, with the exactly the same results in each reimplementation, I'm stumped...
An example of the results the backpropagation algorithm is producing:
With a simpler feedforward network with the same configuration mentioned above (online training + learning rate of 1.0): 3 input neurons, 2 hidden neurons and 1 output neuron.
The initial weights are initialized as follows:
Layer #0 (3 neurons)
Layer #1 (2 neurons)
- Neuron #1: weights=[0.1, 0.15, 0.2] bias=0.25
- Neuron #2: weights=[0.3, 0.35, 0.4] bias=0.45
Layer #2 (1 neuron)
- Neuron #1: weights=[0.5, 0.55] bias=0.6
Given an input of [0.0, 0.5, 1.0], the output is 0.78900331.
Backpropagating for the same input and with the desired output of 1.0 gives the following partial derivatives (dw = derivative wrt weight, db = derivative wrt bias):
Layer #0 (3 neurons)
Layer #1 (2 neurons)
- Neuron #1: dw=[0, 0.0066968054, 0.013393611] db=0.013393611
- Neuron #2: dw=[0, 0.0061298212, 0.012259642] db=0.012259642
Layer #2 (1 neuron)
- Neuron #1: dw=[0.072069918, 0.084415339] db=0.11470326
Updating the network with those partial derivatives yields a corrected output value of 0.74862305.
If anyone would be kind enough to confirm the above results, it would help me tremendously as I've pretty much ruled out backpropagation being faulty as the reason for the problem.
Did anyone tackling the MNIST problem ever come across this problem?
Even suggestions for things I should check would help since I'm really lost here.
Doh..
Turns out nothing was wrong with my backpropagation implementation...
The problem was that I read the images into a signed char (in C++) array, and the pixel values overflowed, so that when I divided by 255.0 to normalize the input vectors into the range of 0.0-1.0, I actually got negative values... ;-;
So basically I spent some four days debugging and reimplementing the same thing when the problem was somewhere else entirely.
edit:
A more pointed question:
What is the derivative of softmax to be used in my gradient descent?
This is more or less a research project for a course, and my understanding of NN is very/fairly limited, so please be patient :)
I am currently in the process of building a neural network that attempts to examine an input dataset and output the probability/likelihood of each classification (there are 5 different classifications). Naturally, the sum of all output nodes should add up to 1.
Currently, I have two layers, and I set the hidden layer to contain 10 nodes.
I came up with two different types of implementations
Logistic sigmoid for hidden layer activation, softmax for output activation
Softmax for both hidden layer and output activation
I am using gradient descent to find local maximums in order to adjust the hidden nodes' weights and the output nodes' weights. I am certain in that I have this correct for sigmoid. I am less certain with softmax (or whether I can use gradient descent at all), after a bit of researching, I couldn't find the answer and decided to compute the derivative myself and obtained softmax'(x) = softmax(x) - softmax(x)^2 (this returns an column vector of size n). I have also looked into the MATLAB NN toolkit, the derivative of softmax provided by the toolkit returned a square matrix of size nxn, where the diagonal coincides with the softmax'(x) that I calculated by hand; and I am not sure how to interpret the output matrix.
I ran each implementation with a learning rate of 0.001 and 1000 iterations of back propagation. However, my NN returns 0.2 (an even distribution) for all five output nodes, for any subset of the input dataset.
My conclusions:
I am fairly certain that my gradient of descent is incorrectly done, but I have no idea how to fix this.
Perhaps I am not using enough hidden nodes
Perhaps I should increase the number of layers
Any help would be greatly appreciated!
The dataset I am working with can be found here (processed Cleveland):
http://archive.ics.uci.edu/ml/datasets/Heart+Disease
The gradient you use is actually the same as with squared error: output - target. This might seem surprising at first, but the trick is that a different error function is minimized:
(- \sum^N_{n=1}\sum^K_{k=1} t_{kn} log(y_{kn}))
where log is the natural logarithm, N depicts the number of training examples and K the number of classes (and thus units in the output layer). t_kn depicts the binary coding (0 or 1) of the k'th class in the n'th training example. y_kn the corresponding network output.
Showing that the gradient is correct might be a good exercise, I haven't done it myself, though.
To your problem: You can check whether your gradient is correct by numerical differentiation. Say you have a function f and an implementation of f and f'. Then the following should hold:
(f'(x) = \frac{f(x - \epsilon) - f(x + \epsilon)}{2\epsilon} + O(\epsilon^2))
please look at sites.google.com/site/gatmkorn for the open-source Desire simulation program.
For the Windows version, /mydesire/neural folder has several softmax classifiers, some with softmax-specific gradient-descent algorithm.
In the examples, this works nicely for a simplemcharacter-recognition task.
ASee also
Korn, G.A.: Advanced dynamic-system Simulation, Wiley 2007
GAK
look at the link:
http://www.youtube.com/watch?v=UOt3M5IuD5s
the softmax derivative is: dyi/dzi= yi * (1.0 - yi);