I figured I'd answer a self-contained post here for anyone that's interested. This will be using the notation described here.
Introduction
The idea behind backpropagation is to have a set of "training examples" that we use to train our network. Each of these has a known answer, so we can plug them into the neural network and find how much it was wrong.
For example, with handwriting recognition, you would have lots of handwritten characters alongside what they actually were. Then the neural network can be trained via backpropagation to "learn" how to recognize each symbol, so then when it's later presented with an unknown handwritten character it can identify what it is correctly.
Specifically, we input some training sample into the neural network, see how good it did, then "trickle backwards" to find how much we can change each node's weights and bias to get a better result, and then adjust them accordingly. As we continue to do this, the network "learns".
There are also other steps that may be included in the training process (for example, dropout), but I will focus mostly on backpropagation since that's what this question was about.
Partial derivatives
A partial derivative $\frac{\partial f}{\partial x}$ is a derivative of $f$ with respect to some variable $x$.
For example, if $f(x, y)=x^2 + y^2$, $\frac{\partial f}{\partial x}=2x$, because $y^2$ is simply a constant with respect to $x$. Likewise, $\frac{\partial f}{\partial y}= 2y$, because $x^2$ is simply a constant with respect to $y$.
A gradient of a function, designated $\nabla f$, is a function containing the partial derivative for every variable in f. Specifically:
$$\nabla f(v_1, v_2, ..., v_n) = \frac{\partial f}{\partial v_1 }\mathbf{e}_1 + \cdots + \frac{\partial f}{\partial v_n }\mathbf{e}_n$$,
where $e_i$ is a unit vector pointing in the direction of variable $v_1$.
Now, once we have computed the $\nabla f$ for some function $f$, if we are at position $(v_1, v_2, ..., v_n)$, we can "slide down" $f$ by going in direction $-\nabla f(v_1, v_2, ..., v_n)$.
With our example of $f(x, y)=x^2 + y^2$, the unit vectors are $e_1=(1, 0)$ and $e_2=(0, 1)$, because $v_1=x$ and $v_2=y$, and those vectors point in the direction of the $x$ and $y$ axes. Thus, $\nabla f(x, y) = 2x (1, 0) + 2y(0, 1)$.
Now, to "slide down" our function $f$, let's say we are at a point $(-2, 4)$. Then we would need to move in direction $-\nabla f(-2, -4)= -(2 \cdot -2 \cdot (1, 0) + 2 \cdot 4 \cdot (0, 1)) = -((-4, 0) + (0, 8))=(4, -8)$.
The magnitude of this vector will give us how steep the hill is (higher values means the hill is steeper). In this case, we have $\sqrt{4^2+(-8)^2}\approx 8.944$.
Hadamard Product
The Hadamard Product of two matrices $A, B \in R^{n\times m}$, is just like matrix addition, except instead of adding the matrices element-wise, we multiply them element-wise.
Formally, while matrix addition is $A + B = C$, where $C \in R^{n \times m}$ such that
$$C^i_j = A^i_j + B^i_j$$,
The Hadamard Product $A \odot B = C$, where $C \in R^{n \times m}$ such that
$$C^i_j = A^i_j \cdot B^i_j$$
Computing the gradients
(most of this section is from Neilsen's book).
We have a set of training samples, $(S, E)$, where $S_r$ is a single input training sample, and $E_r$ is the expected output value of that training sample. We also have our neural network, composed of weights $W$, and biases $B$. $r$ is used to prevent confusion from the $i$, $j$, and $k$ used in the definition of a feedforward network.
Next, we define a cost function, $C(W, B, S^r, E^r)$ that takes in our neural network and a single training example, and outputs how good it did.
Normally what is used is quadratic cost, which is defined by
$$C(W, B, S^r, E^r) = 0.5\sum\limits_j (a^L_j - E^r_j)^2$$
where $a^L$ is the output to our neural network, given input sample $S^r$
Then we want to find $\frac{\partial C}{\partial w^i_j}$ and $\frac{\partial C}{\partial b^i_j}$ for each node in our feedforward neural network.
We can call this the gradient of $C$ at each neuron because we consider $S^r$ and $E^r$ as constants, since we can't change them when we are trying to learn. And this makes sense - we want to move in a direction relative to $W$ and $B$ that minimizes cost, and moving in the negative direction of the gradient with respect to $W$ and $B$ will do this.
To do this, we define $\delta^i_j=\frac{\partial C}{\partial z^i_j}$ as the error of neuron $j$ in layer $i$.
We start with computing $a^L$ by plugging $S^r$ into our neural network.
Then we compute the error of our output layer, $\delta^L$, via
$$\delta^L_j = \frac{\partial C}{\partial a^L_j} \sigma^{ \prime}(z^L_j)$$.
Which can also be written as
$$\delta^L = \nabla_a C \odot \sigma^{ \prime}(z^L)$$.
Next, we find the error $\delta^i$ in terms of the error in the next layer $\delta^{i+1}$, via
$$\delta^i=((W^{i+1})^T \delta^{i+1}) \odot \sigma^{\prime}(z^i)$$
Now that we have the error of each node in our neural network, computing the gradient with respect to our weights and biases is easy:
$$\frac{\partial C}{\partial w^i_{jk}}=\delta^i_j a^{i-1}_k=\delta^i(a^{i-1})^T$$
$$\frac{\partial C}{\partial b^i_j} = \delta^i_j$$
Note that the equation for the error of the output layer is the only equation that's dependent on the cost function, so, regardless of the cost function, the last three equations are the same.
As an example, with quadratic cost, we get
$$\delta ^L = (a^L - E^r) \odot \sigma ^ {\prime}(z^L)$$
for the error of the output layer. and then this equation can be plugged into the second equation to get the error of the $L-1^{\text{th}}$ layer:
$$\delta^{L-1}=((W^{L})^T \delta^{L}) \odot \sigma^{\prime}(z^{L-1})$$
$$=((W^{L})^T ((a^L - E^r) \odot \sigma ^ {\prime}(z^L))) \odot \sigma^{\prime}(z^{L-1})$$
which we can repeat this process to find the error of any layer with respect to $C$, which then allows us to compute the gradient of any node's weights and bias with respect to $C$.
I could write up an explanation and proof of these equations if desired, though one can also find proofs of them here. I'd encourage anyone that is reading this to prove these themselves though, beginning with the definition $\delta^i_j=\frac{\partial C}{\partial z^i_j}$ and applying the chain rule liberally.
For some more examples, I made a list of some cost functions alongside their gradients here.
Gradient Descent
Now that we have these gradients, we need to use them learn. In the previous section, we found how to move to "slide down" the curve with respect to some point. In this case, because it's a gradient of some node with respect to weights and a bias of that node, our "coordinate" is the current weights and bias of that node. Since we've already found the gradients with respect to those coordinates, those values are already how much we need to change.
We don't want to slide down the slope at a very fast speed, otherwise we risk sliding past the minimum. To prevent this, we want some "step size" $\eta$.
Then, find the how much we should modify each weight and bias by, because we have already computed the gradient with respect to the current we have
$$\Delta w^i_{jk}= -\eta \frac{\partial C}{\partial w^i_{jk}}$$
$$\Delta b^i_j = -\eta \frac{\partial C}{\partial b^i_j}$$
Thus, our new weights and biases are
$$w^i_{jk} = w^i_{jk} + \Delta w^i_{jk}$$
$$b^i_j = b^i_j + \Delta b^i_j$$
Using this process on a neural network with only an input layer and an output layer is called the Delta Rule.
Stochastic Gradient Descent
Now that we know how to perform backpropagation for a single sample, we need some way of using this process to "learn" our entire training set.
One option is simply performing backpropagation for each sample in our training data, one at a time. This is pretty inefficient though.
A better approach is Stochastic Gradient Descent. Instead of performing backpropagation for each sample, we pick a small random sample (called a batch) of our training set, then perform backpropagation for each sample in that batch. The hope is that by doing this, we capture the "intent" of the data set, without having to compute the gradient of every sample.
For example, if we had 1000 samples, we could pick a batch of size 50, then run backpropagation for each sample in this batch. The hope is that we were given a large enough training set that it represents the distribution of the actual data we are trying to learn well enough that picking a small random sample is sufficient to capture this information.
However, doing backpropagation for each training example in our mini-batch isn't ideal, because we can end up "wiggling around" where training samples modify weights and biases in such a way that they cancel each other out and prevent them from getting to the minimum we are trying to get to.
To prevent this, we want to go to the "average minimum," because the hope is that, on average, the samples' gradients are pointing down the slope. So, after choosing our batch randomly, we create a mini-batch which is a small random sample of our batch. Then, given a mini-batch with $n$ training samples, and only update the weights and biases after averaging the gradients of each sample in the mini-batch.
Formally, we do
$$\Delta w^{i}_{jk} = \frac{1}{n}\sum\limits_r \Delta w^{ri}_{jk}$$
and
$$\Delta b^{i}_{j} = \frac{1}{n}\sum\limits_r \Delta b^{ri}_{j}$$
where $\Delta w^{ri}_{jk}$ is the computed change in weight for sample $r$, and $\Delta b^{ri}_{j}$ is the computed change in bias for sample $r$.
Then, like before, we can update the weights and biases via:
$$w^i_{jk} = w^i_{jk} + \Delta w^{i}_{jk}$$
$$b^i_j = b^i_j + \Delta b^{i}_{j}$$
This gives us some flexibility in how we want to perform gradient descent. If we have a function we are trying to learn with lots of local minima, this "wiggling around" behavior is actually desirable, because it means that we're much less likely to get "stuck" in one local minima, and more likely to "jump out" of one local minima and hopefully fall in another that is closer to the global minima. Thus we want small mini-batches.
On the other hand, if we know that there are very few local minima, and generally gradient descent goes towards the global minima, we want larger mini-batches, because this "wiggling around" behavior will prevent us from going down the slope as fast as we would like. See here.
One option is to pick the largest mini-batch possible, considering the entire batch as one mini-batch. This is called Batch Gradient Descent, since we are simply averaging the gradients of the batch. This is almost never used in practice, however, because it is very inefficient.
