#### Introduction

PyTorch is a popular Deep Learning library which provides automatic differentiation for all operations on Tensors. It’s in-built output.backward() function computes the gradients for all composite variables that contribute to the output variable. Mysteriously, calling .backward() only works on scalar variables. When called on vector variables, an additional ‘gradient’ argument is required.

In the official PyTorch 0.4.0 tutorials,

If you want to compute the derivatives, you can call .backward() on a Tensor. If Tensor is a scalar (i.e. it holds a one element data), you don’t need to specify any arguments to backward(), however if it has more elements, you need to specify a gradient argument that is a tensor of matching shape.

Scalar variables: specifying a gradient argument

In most cases when training the neural network, .backward() is called by some loss or cost variable from which we wish to perform backpropagation of gradients down to the weight parameters in the neural network.

In a basic single layer sigmoid network, $\hat{y}$ is the predicted output, $\sigma$ is the sigmoid function, $W$ and $b$ are weight parameters in the neural network to be learned via updating weights through something like gradient descent.

\begin{align} \hat{y} &= \sigma(W.x + b) \\ losses &= (y - \hat{y})^2 \\ avgLoss &= \frac{1}{n}\sum_{i}^{n}losses \end{align}

In the following example, a single training batch that we are feeding into the network is given by $X \in \mathbb{R}^{n\times m}$, where $n$ is the number of instances, and $m$ is the number of dimensions of one training instance. The target variable can take on either $0$ or $1$, $y \in \{0,1\}$, and the output of the neural network is given by $Y \in \mathbb{R}^{n}$

After backpropagation from avg_loss, the values of W.grad are $(1.00000e-02 * [[-2.6718, -3.2673, -3.8628]])$ and the values of b.grad are $[[0.1481, -0.1038]]$.

We can then use these gradients to update the new value of $W$ and $b$ manually, or with in-built PyTorch modules like torch.nn.optimize.

Note that the gradient arguments here are implicit, it is simply torch.Tensor([1]). avg_loss.backward() is equivalent to avg_loss.backward(torch.Tensor([1]).

Vector variables: specifying a gradient argument

In the above examples, avg_loss is a scalar variable. If we attempted to backpropogate from losses.backward() this would throw the following: RuntimeError: grad can be implicitly created only for scalar outputs.

losses is a vector which contains the squared-error loss for each of the $\hat{y}$’s. That is, $losses = [loss^1, loss^2]$. If we wanted to call losses.backward() to the same effect as avg_loss.backward(), we would need to provide the gradient of losses with respect to avg_loss, $\frac{\delta(avgLoss)}{\delta(losses)}$ as an argument in backward. By chain rule,

$$\frac{\delta(avgLoss)}{\delta(W)} = \frac{\delta(avgLoss)}{\delta(losses)} \times \frac{\delta(losses)}{\delta(W)}$$

Note that W.grad is equivalent to $\frac{\delta(avgLoss)}{\delta(W)}$.

This translates to losses.backward(torch.Tensor([0.5, 0.5])), where the input argument, $\frac{\delta(avgLoss)}{\delta(losses)}$ can be obtained by:

\begin{align} avgLoss &= \frac{1}{2} [loss^1, loss^2] \\ \frac{\delta(avgLoss)}{\delta(losses)} &= [\frac{1}{2}, \frac{1}{2}] \end{align}

Running the code block with the gradient input gives the same W.grad and b.grad as the commented out code. The values of W.grad are $(1.00000e-02 * [[-2.6718, -3.2673, -3.8628]])$ and the values of b.grad are $[[0.1481, -0.1038]]$.

Reconstructing the Jacobian

Sometimes we may wish to obtain the Jacobian matrix of partial derivatives to capture the rate of change of each component of the output with respect to the input vector. The following example shows the Jacobian matrix of weights derivatives with respect to losses.

$$\frac{\delta(losses)}{\delta(W)} = \begin{bmatrix} \frac{\delta loss^1}{\delta w_1} & \cdots & \frac{\delta loss^n}{\delta w_1} \\ \vdots & \ddots & \vdots \\ \frac{\delta loss^1}{\delta w_m} & \cdots & \frac{\delta loss^n}{\delta w_m} \end{bmatrix}$$

Because .backward() requires gradient arguments as inputs and performs a matrix multiplication internally to give the output (see eq 4), the recommmended way to obtain the Jacobian is by feeding in a gradient input which accounts for that specific row of the Jacobian. This is done by providing a mask for the specific dimension in the gradient vector, i.e. the gradient vector has a non-zero value on that dimension, and 0 everywhere else.

The jacobian $\frac{\delta(losses)}{\delta W}$ is now $(1.00000e-02 * [[-4.1526, -6.2289, -8.3053], [1.4808, 2.9616, 4.4424]])$.

Summing the two rows element-wise of the Jacobian would simply result in W.grad. The masked gradient that we input to losses.backward(..) allows us to generate individual rows of the Jacobian matrix instead of summing all the rows in the matrix multiplication operation of eq 4.

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