Certain Derivations using the Chain Rule for the Backpropagation Algorithm

I'm currently trying to teach myself how to code a neural network and I want as good of an understanding of the math as is possible for someone who got a D- in calculus over 15 years ago and has not taken a math class since. Suffice it to say I am quite a math beginner but have been reading/watching videos for weeks now. I have an ok understanding of much of what goes into the algorithms but - largely because I struggle with the Chain Rule - I am absolutely stumped by the backpropagation algorithm, specifically how the to derive the error term.

All of the equations I'm using can be found here:

This is what I know:

This is the error equation representing sum of squared error. $E = \frac{1}{2} $${\sum\limits_{k}} (t_k - a_k)^2 In a neural network basically you have an output that is associated with some weights. We want to change the weights to reduce the error. \bigtriangleup W \propto- \frac{\partial E}{\partial W} Ok fine so far. {\partial E} is not directly related to the weights so I get we have to use the chain rule here. 'E' is a function of a_k which is itself a function of a net input (net_k) which is ITSELF a function of the weights. So According to the link ultimately the equation for backpropagation is as follows: {\frac{\partial E}{\partial a_k}}\times {\frac{\partial a_k}{\partial net_k}}\times {\frac{\partial net_k}{\partial w_{jk}}} Ok so I get {\frac{\partial E}{\partial a_k}} - you use the power rule. You multiply 2 x .5 for it to cancel and the summation also cancels so the derivative here is just -(t_k - a_k). I get this so far. So a_k is the activation function which for a sigmoid is {\frac{1}{1+e^{-z}}} and this is where I lose it. So the sigmoid derivative is a_k(1-a_k) but I have no idea why. The link presents the sigmoid equation as {(1+e^{-net_k})^{-1}} which leads to {\frac{\partial a_k}{\partial net_k}} = {\frac{e^{-net_k}}{(1+e^{-net_k})^{2}}} but I don't understand this derivation. Could someone walk me through it, possibly by explaining what rules you're using to get here? Finally, when performing {\frac{\partial net_k}{\partial w_{jk}}} the article states: Note that only one term of the net summation will have a non-zero derivative: again the one associated with the particular weight we are considering. This means that certain weights are going to have derivatives of 0 so we only need to consider...I think...the weights on...'j'? Clearly I'm not sure here. Then they state: {\frac{\partial net_k}{\partial w_{jk}}} = {\frac{\partial(w_{kj}a_j)}{\partial{w_{kj}}} = a_j} Conceptually I understand the idea behind partial derivatives is to take whatever is not in the equation and treat it as a constant so in this example we are treating a_j as a constant. Why then is the derivative a_j? Shouldn't it be derived as 0? Additionally, where did the w's go? What kind of rules are being used in the derivation of these variables? I'm not sure if one uses the power rule/quotient rule/etc. (obviously not but just as examples) to get this result. And then to get the actual weight change you just multiply all these results right? I am clearly an absolute math idiot. If someone could come down to my level and help me out here I would appreciate it beyond words. • Together with the chain rule you need the so-called total derivative, that if g(t) = f(x(t),y(t)) then g'(t) = x'(t) \frac{\partial f}{\partial x}[x(t),y(t)]+y'(t) \frac{\partial f}{\partial y}[x(t),y(t)]. This allows to compute all the partial derivatives \frac{\partial E}{\partial w_{jk}} and to run a gradient descent. Try first with a toy example : two input neurons, two hidden neurons, two output neurons, no bias (thus 8 weights) and take some time to draw everything on a paper and find the the partial derivatives. – reuns Aug 30 '17 at 22:47 • In fact a neural network is a big function of many variables : the input, the weights, the output of the hidden neurons.. All those variables are indeed functions of each other. So all you need is to write each of those functions precisely and compute the partial derivatives using the chain rule and the total derivative. – reuns Aug 30 '17 at 23:00 • Focus on doing one thing at a time. That is why the chain rule is important here : separation of the layers of abstraction. As networks grow, it becomes impossible no matter how good you are at calculus to keep the whole expressions you work with in your head. It becomes necessary to split it up. – mathreadler Aug 30 '17 at 23:08 1 Answer The derivative of that sigmoid is a good example for where we can practice the chain rule.$$\cases{f(x) = \frac 1 {1+e^{-x}}\\ f(g) =\frac 1 {g}\\g(h) = 1+e^{h(x)}\\ h(x)=-x}\frac {\partial{f}}{\partial x} = \frac{\partial f}{\partial g}\times\frac{\partial g}{\partial h}\times \frac{\partial h}{\partial x}$$The first one is -1/g^2 the second is e^{h(x)}, third -1 so substituting and simplifying$$-\frac {-e^{-x}} {(1+e^{-x})^2} =\frac {e^{-x}} {(1+e^{-x})^2}$$This is equal if we set$x = net_k$. • Thank you. I'd like to just repeat what you said to make sure I'm understanding.${\partial {e^{-x}}} = e^{-x}$so in essence, for${\frac{\partial g}{\partial x}}$the constant 1 will have a derivative of 0 added to${e^{-x}}$. Is the negative in front of the e because we moved a -1 from the exponent in front of the 'x'? If so it would leave 'e' raised to the power of -2x via the power rule so I'm not sure this is correct. Then for${\frac{\partial f}{\partial g}}$is the idea that you use the quotient rule? Where the denominator is${g(x)^2}$? Can I ask how the -1 got to the numerator? – chainhomelow Aug 31 '17 at 16:41 •$1+e^{-x}$has derivative$-e^{-x}$, because of the$-$before$x$the internal detivative is$-1$. So we can make one more step of the chain rule if we want.$1+e^{h(x)}, h(x) = -x$– mathreadler Aug 31 '17 at 16:44 • @chainhomelow I made an edit now to make it more clear with a new innermost function$h(x)\$. – mathreadler Aug 31 '17 at 16:59
• Thank you so much, mathreadler. That was very clear and I appreciate your efforts very much. I was able to derive the rest of the function from your example, actually. I'll type the whole thing out soon. Thank you again. – chainhomelow Sep 1 '17 at 22:04
• @chainhomelow: Feel free to click "accept answer" if you are happy with it. – mathreadler Sep 2 '17 at 8:05