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I try to really internalize the way backpropagation works. I made up different networks with increasing complexity and wrote the formulas to it. However, I have some difficulties with the matrix notation. Hope anyone can help me.

My network has 2 input, 3 hidden and 2 output neurons. enter image description here

I wrote up the matricesenter image description here

The loss is MSE: $ L = \frac {1}{2} \sum (y_{hat} - y_{true})^2$

The derivative of the Loss with respect to the weight matrix $W^{(3)}$ should have the same dimensions like $W^{(3)}$ to update each entry with (stochastic) gradient descent.

$\frac {\partial L}{\partial W^{(3)}} = \frac {\partial L}{\partial a^{(3)}} \frac {\partial a^{(3)}}{\partial z^{(3)}} \frac {\partial z^{(3)}}{\partial W^{(3)}} = (a^{(3)} - y) \odot a^{(3)} \odot (1 - a^{(3)}) a^{(2)T}$

First question, is that correct to transpose $a^{(2)}$ since otherwise the dimension would not work out?

Now for the second weight matrix, where I cannot figure out what is wrong with the dimensions:

$\frac {\partial L}{\partial W^{(3)}} = \frac {\partial L}{\partial a^{(3)}} \frac {\partial a^{(3)}}{\partial z^{(3)}} \frac {\partial z^{(3)}}{\partial a^{(2)}} \frac {\partial a^{(2)}}{\partial z^{(2)}} \frac {\partial a^{(2)}}{\partial z^{(2)}} \frac {\partial z^{(2)}}{\partial W^{(2)}} = (a^{(3)} - y) \odot a^{(3)} \odot (1 - a^{(3)}) W^{(3)} (1,1,1)^T a^{(1)T}$

I get 2x1 2x3 3x1 1x2...

I wrote just $(1,1,1)$ assuming that the the $z = (z_1, z_2, z_3)$ are greater than 0.

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For your first question, yes, transposing $a^{(2)}$ will do the job since for each entry of the matrix $w_{ij}^{(3)}$, the derivative includes the multiplier $a_{i}^{(2)}$. So, $a_{1}^{(2)}$ will be in the first column, $a_{2}^{(2)}$ will be in the second column and so on. This is directly achieved by multiplying $\delta$ (the first three terms) by ${a_2^{(2)}}^T$Note that in your indexing, $w_{ij}$ denotes $j$-th row and $i$-th column.

Second one is a bit tricky. First of all, you're using denominator layout, so a vector of size $m$ divided by another vector of size $n$ has derivative of size $n\times m$. Typically, numerator layout is more common.

It's all about Layouts

Let's say we have a scalar loss $L$, and two vectors $a,z$ which have dimensions $m,n$ respectively. In $\frac{\partial L}{\partial z}=\frac{\partial L}{\partial a}\frac{\partial a}{\partial z}$, according to denominator layout, first one produces $m\times 1$ vector, and second one produces $n\times m$ matrix, so dimensions mismatch. If it was numerator layout, we'd have $1\times m$ times $m\times n$, and get $1\times n$ gradient, still consistent with the layout definition.

This is why you should append to the left as you move forward in denominator layout (because we're actually transposing a matrix multiplication in changing the layout: $(AB)^T=B^TA^T$): $$\underbrace{\frac{\partial L}{\partial z}}_{n\times 1}=\underbrace{\frac{\partial a}{\partial z}}_{n\times m}\underbrace{\frac{\partial L}{\partial a}}_{m\times 1}$$

So, $$\frac {\partial L}{\partial W_{ij}^{(2)}} = \underbrace{\frac {\partial z^{(2)}}{\partial W_{ij}^{(2)}}}_{1\times 3} \underbrace{\frac {\partial a^{(2)}}{\partial z^{(2)}}}_{3\times 3} \underbrace{\frac {\partial z^{(3)}}{\partial a^{(2)}}}_{3\times 2} \underbrace{\frac {\partial L}{\partial z^{(3)}}}_{2\times 1}$$

And everything matches. I've changed two more things:

  • Some of these calculations can be merged and optimised using element-wise multiplications, e.g. the term $\frac {\partial a^{(2)}}{\partial z^{(2)}}$ produces a 3x3 output, but it's a diagonal matrix. This is actually what you've done while calculating gradients in the last layer.
  • I've used $W_{ij}$ because it's easier to think. $\frac{\partial z}{\partial W}$ is a 3D tensor, since numerator is a vector and denominator is a matrix. After finding the expressions for each $W_{ij}$ and placing them into the gradient matrix one by one according to denominator layout, you can take out common multiplications and write the final formula.
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Please let me derive for quite a general case. I think the deriviation is somewhere out there, but not easy for me to find. Thus I upload here.

The notations I follow is based on a CS229 note.

Definitions. Let $J$ be a loss function of a neural network to minimize. Let $x\in \mathbb{R}^{d_0}$ be a single sample (input). Define its deep activations in a cascaded way as follows: $$ z^{(l)} = W^{(l)} a^{(l-1)} + b^{(l)} $$ and $a^{(l)} = g^{(l)}(z^{(l)})$ for $l=1, \dots, L$ where $L$ is the number of layers. Here, $W^{(l)} \in \mathbb{R}^{d_{l} \times d_{l-1}} $ and $b^{(l)} \in \mathbb{R}^{d_l} $ are parameters of the neural network model that we need to optimize. $a^{(l)} \in \mathbb{R}^{d_l} $ is the $l$-th layer activation output, and $g^{(l)} : \mathbb{R}^{d_l} \to \mathbb{R}^{d_l} $ is the corresponding activation function (such as ReLU). The zeroth layer activation is set as the input $a^{(0)} = x$.

Convention. Except for the final layer $L$, the activation funciton is the same for all hidden layers $l <L$; $g^{(l)} = g$. Also, we assume $g$ is pointwise computation such as ReLU or sigmoid.

Another convention is that the loss function $J$ is defined such that $$ \nabla_{z^{(L)}} J = z^{(L)} - y $$ where $y$ is the target that our neural network model needs to predict. Note that the gradient $\nabla_{z^{(L)}} J = z^{(L)}$ is a column vector $$ \nabla_{z^{(L)}} J = z^{(L)} = \begin{bmatrix} \frac{\partial J}{\partial z^{(L)}_1} \\ \vdots \\ \frac{\partial J}{\partial z^{(L)}_{d_L}} \end{bmatrix} \in \mathbb{R}^{d_L}. $$

Motivation. Backpropagation is simply a combination of chain rule applications. Observe first that $\frac{\partial J}{\partial W^{(L)}_{ij}}$ is decomposed into $$ \frac{\partial J}{\partial W^{(L)}_{ij}} = \sum_k \frac{\partial J}{\partial z^{(L)}_k} \frac{\partial z^{(L)}_k}{\partial W^{(L)}_{ij}} = \frac{\partial J}{\partial z^{(L)}_i} \frac{\partial z^{(L)}_i}{\partial W^{(L)}_{ij}}. $$ (Note that the summand $\frac{\partial z^{(L)}_k}{\partial W^{(L)}_{ij}}$ cancels out since $z_k^{(L)}$ does not depend on $W^{(L)}_{ij}$.) Note that the above is computable since we know $\frac{\partial J}{\partial z^{(L)}_i}$ and $\frac{\partial z^{(L)}_i}{\partial W^{(L)}_{ij}} = a^{(L-1)}_j $ is the forward activation output.

Derivation. We generalize this principle to early layers one by one. In particular, $$ \frac{\partial J}{\partial W^{(l)}_{ij}} = \frac{\partial J}{\partial z^{(l)}_i} \frac{\partial z^{(l)}_i}{\partial W^{(l)}_{ij}} = \frac{\partial J}{\partial z^{(l)}_i} a^{(l-1)}_j. \tag{1a} $$ as $ z^{(l)}_i = \sum_j W^{(l)}_{ij} a^{(l-1)}_j $. Similarly, $$ \frac{\partial J}{\partial b^{(l)}_{i}} = \frac{\partial J}{\partial z^{(l)}_i}. \tag{1b} $$

Now, it suffices to obtain $\frac{\partial J}{\partial z^{(l)}_i}$, which is obatined by applying chain rule in the backward direction (hence the name backpropagation); in particular, we have $$ \frac{\partial J}{\partial z^{(l)}_i} = \sum_k \frac{\partial J}{\partial z^{(l+1)}_k} \frac{\partial z^{(l+1)}_k}{\partial z^{(l)}_i} \tag{2}, $$ which is $$ \frac{\partial J}{\partial z^{(l)}_i} = \sum_k \frac{\partial J}{\partial z^{(l+1)}_k} \frac{\partial z^{(l+1)}_k}{\partial a^{(l)}_i} \frac{\partial a^{(l)}_i}{\partial z^{(l)}_i} = \sum_k \frac{\partial J}{\partial z^{(l+1)}_k} W^{(l+1)}_{ki} g'(z^{(l)}_i) $$ as $\frac{\partial z^{(l+1)}_k}{\partial a^{(l)}_i} = W^{(l+1)}_{ki}$ and $\frac{\partial a^{(l)}_i}{\partial z^{(l)}_i} = g'(z^{(l)}_i)$. (Here $g'$ is the derivative of $g$.)

In the matrix form, the above equations (1a), (1b), and (2) can be stated as follows: $$ \nabla_{W^{(l)}} J = \nabla_{z^{(l)}} J \cdot a^{(l-1)T}, \tag{1a-m} $$ $$ \nabla_{b^{(l)}} J = \nabla_{z^{(l)}} J, \tag{1a-m} $$ $$ \nabla_{z^{(l)}} J = (W^{(l+1)T} \cdot \nabla_{z^{(l+1)}} J) \odot g'(z^{(l)}) \tag{2-m} $$ Here, $\cdot$ is the matrix multiplication, $\odot$ is Hadamard product, and $\nabla_{W^{(l)}} J$ is the matrix that contains $\frac{\partial J}{\partial W^{(L)}_{ij}}$ as its $i$-th row $j$-th column element.

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The mathematics of backpropagation can be developed in a coordinate free framework. The proper setup is provided by differentiable neural networks over Hilbert spaces. This means that the inputs, outputs and weights are elements of open sets in Hilbert spaces, and the derivatives are bounded linear transformations. This can then be specialized to the case of Euclidean spaces (finite dimensional inner product spaces) with derivatives that are linear transformations (boundedness is automatic). A further specialization is to the numerical spaces Rn with derivatives that are Jacobian matrices, this case being of specific interest for AI, particularly in machine/deep learning.

Transposition of operators —or of matrices if you prefer— plays a crucial role. The transpose appears naturally when deriving the quadratic error function and expressing the gradient in terms of the output error.

With the coordinate free version of backpropagation together with appropriate definitions of units, layers and networks the notation simplifies enough to make possible an explicit and simultaneous treatment of all the units and layers.

The following articles can be helpful:

https://hal.science/hal-02867460v2/

https://hal.science/hal-02949853/

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