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Backpropagation

{ Dunno if this is completely correct, I'm learning this as I'm writing it. There may be errors. ~drummyfish }

Backpropagation, or backprop, is an algorithm, based on the chain rule of derivation, used in training neural networks; it computes the partial derivative (or gradient) of the function of the network's error so that we can perform a gradient descent, i.e. update the weights towards lowering the network's error. It computes the analytical derivative (theoretically you could estimate a derivative numerically, but that's not so accurate and can be too computationally expensive). Backpropagation is one of the most common methods for training neural networks but it is NOT the only possible one -- there are many more such as evolutionary programming. It is called backpropagation because it works backwards and propagates the error from the output towards the input, due to how the chain rule works, and it's efficient by reusing already computed values.

Details

Consider the following neural network:

     w000     w100
  x0------y0------z0
    \    /  \    /  \
     \  /    \  /    \
      \/w010  \/w11O  \_E
      /\w001  /\w1O1  /
     /  \    /  \    /
    /    \  /    \  /
  x1------y1------z1
     w011     w111

It has an input layer (neurons x0, x1), a hidden layer (neurons y0, y1) and an output layer (neurons z0, z1). For simplicity there are no biases (biases can easily be added as input neurons that are always on). At the end there is a total error E computed from the networks's output against the desired output (training data).

Let's say the total error is computed as the squared error: E = squared_error(z0) + squared_error(z1) = 1/2 * (z0 - z0_desired)^2 + 1/2 * (z1 - z1_desired)^2.

We can see each non-input neuron as a function. E.g. the neuron z0 is a function z0(x) = z0(a(z0s(x))) where:

If you don't know what the fuck is going on see neural networks first.

What is our goal now? To find the partial derivative of the whole network's total error function (at the current point defined by the weights), or in other words the gradient at the current point. I.e. from the point of view of the total error (which is just a number output by this system), the network is a function of 8 variables (weights w000, w001, ...) and we want to find a derivative of this function in respect to each of these variables (that's what a partial derivative is) at the current point (i.e. with current values of the weights). This will, for each of these variables, tell us how much (at what rate and in which direction) the total error changes if we change that variable by certain amount. Why do we need to know this? So that we can do a gradient descent, i.e. this information is kind of a direction in which we want to move (change the weights and biases) towards lowering the total error (making the network compute results which are closer to the training data). So all in all the goal is to find derivatives (just numbers, slopes) with respect to w000, w001, w010, ... w111.

Could we do this without backpropagation? Yes -- we can use numerical algorithms to estimate derivatives, the simplest one would be to just try to change each weight, one by one, by some small number, let's say dw, and see how much such change changes the output error. I.e. we would sample the error function in all directions which could give us an idea of the slope in each direction. However this would be pretty slow, we would have to reevaluate the whole neural network as many times as there are weights. Backpropagation can do this much more efficiently.

Backpropagation is based on the chain rule, a rule of derivation that equates the derivative of a function composition (functions inside other functions) to a product of derivatives. This is important because by converting the derivatives to a product we will be able to reuse the individual factors and so compute very efficiently and quickly.

Let's write derivative of f(x) with respect to x as D{f(x),x}. The chain rule says that:

D{f(g(x)),x} = D{f(g(x)),g(x)} * D{g(x),x}

Notice that this can be applied to any number of composed functions, the product chain just becomes longer.

Let's get to the computation. Backpropagation work by going "backwards" from the output towards the input. So, let's start by computing the derivative against the weight w100. It will be a specific number; let's call it 'w100. Derivative of a sum is equal to the sum of derivatives:

'w100 = D{E,w100} = D{squared_error(z0),w100} + D{squared_error(z0),w100} = D{squared_error(z0),w100} + 0

(The second part of this sum became 0 because with respect to w100 it is a constant.)

Now we can continue and utilize the chain rule:

'w100 = D{E,w100} = D{squared_error(z0),w100} = D{squared_error(z0(a(z0s))),w100} = D(squared_error(z0),z0) * D{a(z0s),z0s} * d{z0s,w100}

We'll now skip the intermediate steps, they should be easy if you can do derivatives. The final results is:

'w100 = (z0_desired - z0) * (z0s * (1 - z0s)) * y0

Now we have computed the derivative against w100. In the same way can compute 'w101, 'w110 and 'w111 (weights leading to the output layer).

Now let's compute the derivative in respect to w000, i.e. the number 'w000. We will proceed similarly but the computation will be different because the weight w000 affects both output neurons ('z0' and 'z1'). Again, we'll use the chain rule.

w000 = D{E,w000} = D(E,y0) * D{a(y0s),y0s} * D{y0s,w000}

D(E,y0) = D{squared_error(z0),y0} + D{squared_error(z1),y0}

Let's compute the first part of the sum:

D{squared_error(z0),y0} = D{squared_error(z0),z0s} * D{squared_error(z0s),y0}

D{squared_error(z0),z0s} = D{squared_error(z0),z0} * D{a(z0s)),z0s}

Note that this last equation uses already computed values which we can reuse. Finally:

D{squared_error(z0s),y0} = D{squared_error(w100 * y0 + w110 * y1),y0} = w100

And we get:

D{squared_error(z0),y0} = D{squared_error(z0),z0} * D{a(z0s)),z0s} * w100

And so on until we get all the derivatives.

Once we have them, we multiply them all by some value (learning rate, a distance by which we move in the computed direction) and subtract them from the current weights by which we perform the gradient descent and lower the total error.

Note that here we've only used one training sample, i.e. the error E was computed from the network against a single desired output. If more example are used in a single update step, they are usually somehow averaged.


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