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I've already seen many articles about this topic and Backpropagation In Convolutional Neural Networks by Jefkine seems to be the best. Although, as author said,

For the purposes of simplicity we shall use the case where the input image is grayscale i.e single channel C = 1.

Also, he uses stride = 1 and assumes only 1 filter, for the same purpose.

These are the final equations for backpropagation (taken from the article):

Equations

The autor's notation explained:

Notation

I figured out how to do the forward pass with stride, depth and more filters, but couldn't do the same with the backpropagation. Do you know where to put the 3rd dimension, stride and filters number in those equations?

Also, how to backpropagate the bias (assuming there's 1 bias per filter)?

Let me know if something is unclear.

Thanks in advance.

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  • $\begingroup$ If you are conversant with calculus simply write down the equation for a layer and try to backpropagate, see which variables are affecting the output, try to differentiate w.r.t them, use relu for simplicity, this will require quite a long answer to be answered nicely, $\endgroup$
    – DuttaA
    Jul 22, 2018 at 16:59
  • $\begingroup$ @DuttaA The point is I am not, I think I'll mess something up, the indices etc. Could you do that? The final equations would be sufficient, I don't need to know all the calculus process. $\endgroup$
    – anx199
    Jul 22, 2018 at 17:12
  • $\begingroup$ Can't help you, I am not familiar with latex and I implemented it only once, it's quite cumbersome $\endgroup$
    – DuttaA
    Jul 22, 2018 at 17:22
  • $\begingroup$ becominghuman.ai/… $\endgroup$
    – DuttaA
    Jul 22, 2018 at 17:23
  • $\begingroup$ medium.com/@2017csm1006/… $\endgroup$
    – DuttaA
    Jul 22, 2018 at 17:26

1 Answer 1

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This is a frequently asked question (see SO, Quora, Reddit). I must admit that I haven't fully understood the generalized Brackprop of convolution on 3D volumes, especially when using the chain of calculus without using computational graphs. But I will give it a try.

I will use mostly the notation of the DL book (scalar $a$, matrix $\boldsymbol{M}$, $\boldsymbol{\mathsf{M}}$ etc.) and for some quantities the notation of Nielsen's book, e.g., $\delta^l_{...}$ for the neuron error at layer $l$ and I will consider the convolutional layer to be a linear layer, i.e., the output is just the result of the convolution without the non-linearity $z^l_{...}$. This is not relevant since it does not influence the dimensionality of the operations. The convolution operations are one-based index.

Furthermore, we will consider that the error at the output of the convolutional layer is known, so there is no need to consider any other layer. Further, to unclutter notation, we will consider one instance batches and we will omit the batch dimension. Finally, we will assume padding zero and stride one (sorry, but otherwise, it becomes way too complicated).

Now, we have:

  1. $l$ is the $l$th layer where $l=1$ is the first layer and $l=\ell$ is the last layer.

  2. The Input tensor $\boldsymbol{\mathsf{X}} \in \mathbb{R}^{n \times n \times n}$ with dimensions width $w$, height $h$ and three channels $c$.

  3. The Kernel tensor $\boldsymbol{\mathsf{K}} \in \mathbb{R}^{n \times n \times n \times n}$ with dimensions width $m$, height $n$, three channels $c$, and $k$ kernels that will produce $k$ feature maps.

  4. Every kernel $k$ in $\boldsymbol{\mathsf{K}}_{:,:,:,k}$ has weights $w_{...}$.

  5. The derivative of the loss function $J$ w.r.t. an input activation $z^l_{x,y,z}$, say, is defined as

    \begin{equation} \delta^l_{x,y,z} = \frac{\partial J}{\partial \; z^l_{x,y,z}} \end{equation}

First, we need to restate the equations in the corresponding form:

The 3D convolution acts such as one element of the output $\mathsf{M}{\,}^{l}_{i,j,k}$ is the scalar resulting from multiplying and adding the corresponding elements of the input with one kernel which weights are $w^l_{m,n,c,k}$. I.e.,

\begin{align} {\big( (\boldsymbol{\mathsf{I}} * \boldsymbol{\mathsf{K}})(i,j,k) \big)}^l &= \sum_{c} \sum_{m} \sum_{n} x^{l - 1}_{i + m - 1,j + n - 1,c} \cdot w^l_{m,n,c,k} \label{eq:cross_correlation_3d_1}\\ z^l_{i,j,k} &= {\big( (\boldsymbol{\mathsf{I}} * \boldsymbol{\mathsf{K}})(i,j,k) \big)}^l + b^l\\ \mathsf{M}{\,}^{l}_{i,j,k} &= z^l_{i,j,k}.\label{eq:cross_correlation_3d_3} \end{align}

Here $i,j,k$ are the output iterators.

The strategy we will follow is the one explained by Nielsen with some priority changes, i.e., first we will derive the backpropagation rule ($\delta^l_{\cdot,\cdot,\cdot}$ in term of $\delta^{l + 1}_{\cdot,\cdot,\cdot}$). Second, derive the error of the weights (derivative of the loss w.r.t. the kernel weights) in terms of $\delta^{l + 1}_{\cdot,\cdot,\cdot}$.

The 2D case derivation suggests that, in the end, we will have some kind of convolution of the layer error with the kernels.

Let us start with the error of one element at layer $\delta^l_{m,n,u}$ w.r.t. to the error at layer $\delta^{l + 1}_{i,j,k}$. In order to apply the chain rule, we have to identify which output elements $z^{l + 1}_{i,j,k}$ are affected by the input element $z^l_{m,n,u}$, so that we can add the partial derivatives. One important observation is that one input element affects all feature maps (albeit not all elements), so we must sum over all output channels $k$. Another (trivial?) observation is that the multi-channel $\boldsymbol{\mathsf{K}}$ does not move across the depth dimension, therefore the index does not contain offsets. After having considered the channels, we must further consider the effect of the input element $z^l_{m,n,u}$ in one specific output channel $z^l_{:,:,k}$. Using the 2D analogy, we can say that given a specific channels $k$, the input element affects some elements in that channel, namely the elements fixed at the position $m,n$, and that are the results of the kernel having been slit by by $-a, -b$ (input perspective, we introduce auxiliary indices $a$ and $b$). Finally, we know that all kernels have the same dimensionality (they are packed in one 4D-tensor), thus, we can use the same iterators $a,b$ for all kernels. These observations lead to

\begin{align} \delta^l_{m,n,u} &= \sum_{k} \sum_a \sum_b \frac{\partial J}{\partial \; z^{l + 1}_{m - a + 1,n - b + 1,k}} \cdot \frac{\partial z^{l + 1}_{m - a + 1,n - b + 1,k}}{\partial z^l_{m,n,u}}\label{eq:delta1}\tag{1}\\ &= \sum_{k} \sum_a \sum_b \delta^{l + 1}_{m - a + 1,n - b + 1,k} \cdot \frac{\partial z^{l + 1}_{m - a + 1,n - b + 1,k}}{\partial z^l_{m,n,u}}.\label{eq:delta2}\tag{2} \end{align}

That was one of the most difficult steps, where I could have committed errors. Feel free to correct me.

Here, we substituted, as usual, the error $\frac{\partial J}{\partial \; z^{l + 1}_{m - a + 1,n - b + 1,k}}$ by its definition $\delta^{l + 1}_{m - a + 1,n - b + 1,k}$.

You know where this is heading.

We now focus on the second term on the right and expand it. Here, again we introduce auxiliary indices $p, q, r$, and use $x^l_{...} = z^l_{...}$ (as mentioned, we deal with linear layers).

\begin{align} \frac{\partial z^{l + 1}_{m - a + 1,n - b + 1,k}}{\partial z^l_{m,n,u}} &= \frac{\partial}{\partial z^l_{m,n,u}} \Big( \sum_{p} \sum_{q} \sum_{r} z^{l}_{(m - a + 1) + p - 1,(n - b + 1) + q - 1,r} \cdot w^{l + 1}_{p,q,r,k}\Big)\label{eq1}\tag{3}\\ &= \frac{\partial}{\partial z^l_{m,n,u}} \Big( \sum_{p} \sum_{q} \sum_{r} z^{l}_{m - a + p,n - b + q,r} \cdot w^{l + 1}_{p,q,r,k}\Big)\label{eq2}\tag{4}\\ &= \frac{\partial}{\partial z^l_{m,n,u}} \Big( z^{l}_{m,n,u} \cdot w^{l + 1}_{a,b,u,k}\Big)\label{eq3}\tag{5}\\ &= w^{l + 1}_{a,b,u,k}.\label{eq4}\tag{6} \end{align}

In Eq. \ref{eq1} we expanded as mentioned above, then in Eq. \ref{eq2} we reduced the indices as $(m - a + 1) + p - 1 = m - a + p, (n - b + 1) + q - 1 = n - b + q$. In Eq. \ref{eq3} we observe that all partial derivatives w.r.t. $z^l_{m,n,u}$, except when $a = p, b = q, r = u$. Accordingly, the weights must have indices $a,b,u,k$. Then, plugging Eq. \ref{eq4} in Eq.\ref{eq:delta2}, we have:

\begin{align} \delta^l_{m,n,u} &= \sum_{k} \sum_a \sum_b \delta^{l + 1}_{m - a + 1,n - b + 1,k} \cdot w^{l + 1}_{a,b,u,k}.\label{eq:final1}\tag{7} \end{align}

The second most difficult part is how to interpret Eq.\ref{eq:final1}. As mentioned above, it is clear that this is some kind of convolution between the error at the end of the convolutional layer (the error it receives from the upper layer) with the kernel tensor. One thing we can do is transform (rotate?) the kernel to rearrange the indices, i.e.,

\begin{align} \delta^l_{m,n,u} &= \sum_{k} \sum_a \sum_b \delta^{l + 1}_{m - a + 1,n - b + 1,k} \cdot w^{l + 1}_{a,b,k,u}.\label{eq:final2}\tag{8} \end{align}

In Eq.\ref{eq:final2}, we can see that we have to collect all partial derivatives corresponding to elements in all channels $k$ having been generated by channel $u$. This is somehow not intuitive and difficult to picture (at least for me) because all input channels are combined/merged to create a feature map. So, how can we untangle a feature map to collect specific the input elements of one channel that were melted to form the feature map? It turns out that one can if you think in algebraic terms. The output is a tensor, and therefore, provided one has the right indices, the operation can be conducted. This standard 3D convolution may surely be expressed as a 3D cross-correlation. Ideas are welcome.

An interesting concrete example is given by Aggarwal 2018 on page 355:

In order to understand the transposition above, consider a situation in which we use 20 filters on the 3-channel RGB volume in order to create an output volume of depth 20. While backpropagating, we will need to take a gradient volume of depth 20 and transform to a gradient volume of depth 3. Therefore, we need to create 3 filters for backpropagation, each of which is for the red, green, and blue colors. We pull out the 20 spatial slices from the 20 filters that are applied to the red color invert them using the the approach of Figure 8.7, and then create a single 20-depth filter for backpropagating gradients with respect to the red slice. Similar approaches are used for the green and blue slices. The transposition and inversion in Equation 8.3 corresponds to these operations.

The derivation of the derivatives of the cost function w.r.t. the kernel weights and bias are left as exercise ;)

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