# 2.5. Automatic Differentiation¶

As we have explained in Section 2.4, differentiation is a crucial step in nearly all deep learning optimization algorithms. While the calculations for taking these derivatives are straightforward, requiring only some basic calculus, for complex models, working out the updates by hand can be a pain (and often error-prone).

The `autograd`

package expedites this work by automatically
calculating derivatives, i.e., *automatic differentiation*. And while
many other libraries require that we compile a symbolic graph to take
automatic derivatives, `autograd`

allows us to take derivatives while
writing ordinary imperative code. Every time we pass data through our
model, `autograd`

builds a graph on the fly, tracking which data
combined through which operations to produce the output. This graph
enables `autograd`

to subsequently backpropagate gradients on command.
Here, *backpropagate* simply means to trace through the *computational
graph*, filling in the partial derivatives with respect to each
parameter.

```
from mxnet import autograd, np, npx
npx.set_np()
```

## 2.5.1. A Simple Example¶

As a toy example, say that we are interested in differentiating the
function \(y = 2\mathbf{x}^{\top}\mathbf{x}\) with respect to the
column vector \(\mathbf{x}\). To start, let’s create the variable
`x`

and assign it an initial value.

```
x = np.arange(4)
x
```

```
array([0., 1., 2., 3.])
```

Note that before we even calculate the gradient of \(y\) with respect to \(\mathbf{x}\), we will need a place to store it. It is important that we do not allocate new memory every time we take a derivative with respect to a parameter because we will often update the same parameters thousands or millions of times and could quickly run out of memory.

Note also that a gradient of a scalar-valued function with respect to a
vector \(\mathbf{x}\) is itself vector-valued and has the same shape
as \(\mathbf{x}\). Thus it is intuitive that in code, we will access
a gradient taken with respect to `x`

as an attribute of the
`ndarray`

`x`

itself. We allocate memory for an `ndarray`

’s
gradient by invoking its `attach_grad`

method.

```
x.attach_grad()
```

After we calculate a gradient taken with respect to `x`

, we will be
able to access it via the `grad`

attribute. As a safe default,
`x.grad`

is initialized as an array containing all zeros. That is
sensible because our most common use case for taking gradient in deep
learning is to subsequently update parameters by adding (or subtracting)
the gradient to maximize (or minimize) the differentiated function. By
initializing the gradient to an array of zeros, we ensure that any
update accidentally executed before a gradient has actually been
calculated will not alter the parameters’ value.

```
x.grad
```

```
array([0., 0., 0., 0.])
```

Now let’s calculate \(y\). Because we wish to subsequently calculate gradients, we want MXNet to generate a computational graph on the fly. We could imagine that MXNet would be turning on a recording device to capture the exact path by which each variable is generated.

Note that building the computational graph requires a nontrivial amount
of computation. So MXNet will only build the graph when explicitly told
to do so. We can invoke this behavior by placing our code inside an
`autograd.record`

scope.

```
with autograd.record():
y = 2 * np.dot(x, x)
y
```

```
array(28.)
```

Since `x`

is an `ndarray`

of length 4, `np.dot`

will perform an
inner product of `x`

and `x`

, yielding the scalar output that we
assign to `y`

. Next, we can automatically calculate the gradient of
`y`

with respect to each component of `x`

by calling `y`

’s
`backward`

function.

```
y.backward()
```

If we recheck the value of `x.grad`

, we will find its contents
overwritten by the newly calculated gradient.

```
x.grad
```

```
array([ 0., 4., 8., 12.])
```

The gradient of the function \(y = 2\mathbf{x}^{\top}\mathbf{x}\)
with respect to \(\mathbf{x}\) should be \(4\mathbf{x}\). Let’s
quickly verify that our desired gradient was calculated correctly. If
the two `ndarray`

s are indeed the same, then the equality between
them holds at every position.

```
x.grad == 4 * x
```

```
array([ True, True, True, True])
```

If we subsequently compute the gradient of another variable whose value
was calculated as a function of `x`

, the contents of `x.grad`

will
be overwritten.

```
with autograd.record():
y = x.sum()
y.backward()
x.grad
```

```
array([1., 1., 1., 1.])
```

## 2.5.2. Backward for Non-Scalar Variables¶

Technically, when `y`

is not a scalar, the most natural interpretation
of the gradient of `y`

(a vector of length \(m\)) with respect to
`x`

(a vector of length \(n\)) is the *Jacobian* (an
\(m\times n\) matrix). For higher-order and higher-dimensional `y`

and `x`

, the Jacobian could be a gnarly high-order tensor.

However, while these more exotic objects do show up in advanced machine
learning (including in deep learning), more often when we are calling
backward on a vector, we are trying to calculate the derivatives of the
loss functions for each constituent of a *batch* of training examples.
Here, our intent is not to calculate the Jacobian but rather the sum of
the partial derivatives computed individually for each example in the
batch.

Thus when we invoke `backward`

on a vector-valued variable `y`

,
which is a function of `x`

, MXNet assumes that we want the sum of the
gradients. In short, MXNet will create a new scalar variable by summing
the elements in `y`

, and compute the gradient of that scalar variable
with respect to `x`

.

```
with autograd.record():
y = x * x # y is a vector
y.backward()
u = x.copy()
u.attach_grad()
with autograd.record():
v = (u * u).sum() # v is a scalar
v.backward()
x.grad == u.grad
```

```
array([ True, True, True, True])
```

## 2.5.3. Detaching Computation¶

Sometimes, we wish to move some calculations outside of the recorded
computational graph. For example, say that `y`

was calculated as a
function of `x`

, and that subsequently `z`

was calculated as a
function of both `y`

and `x`

. Now, imagine that we wanted to
calculate the gradient of `z`

with respect to `x`

, but wanted for
some reason to treat `y`

as a constant, and only take into account the
role that `x`

played after `y`

was calculated.

Here, we can call `u = y.detach()`

to return a new variable `u`

that
has the same value as `y`

but discards any information about how `y`

was computed in the computational graph. In other words, the gradient
will not flow backwards through `u`

to `x`

. This will provide the
same functionality as if we had calculated `u`

as a function of `x`

outside of the `autograd.record`

scope, yielding a `u`

that will be
treated as a constant in any `backward`

call. Thus, the following
`backward`

function computes the partial derivative of `z = u * x`

with respect to `x`

while treating `u`

as a constant, instead of the
partial derivative of `z = x * x * x`

with respect to `x`

.

```
with autograd.record():
y = x * x
u = y.detach()
z = u * x
z.backward()
x.grad == u
```

```
array([ True, True, True, True])
```

Since the computation of `y`

was recorded, we can subsequently call
`y.backward()`

to get the derivative of `y = x * x`

with respect to
`x`

, which is `2 * x`

.

```
y.backward()
x.grad == 2 * x
```

```
array([ True, True, True, True])
```

Note that attaching gradients to a variable `x`

implicitly calls
`x = x.detach()`

. If `x`

is computed based on other variables, this
part of computation will not be used in the `backward`

function.

```
y = np.ones(4) * 2
y.attach_grad()
with autograd.record():
u = x * y
u.attach_grad() # Implicitly run u = u.detach()
z = 5 * u - x
z.backward()
x.grad, u.grad, y.grad
```

```
(array([-1., -1., -1., -1.]), array([5., 5., 5., 5.]), array([0., 0., 0., 0.]))
```

## 2.5.4. Computing the Gradient of Python Control Flow¶

One benefit of using automatic differentiation is that even if building
the computational graph of a function required passing through a maze of
Python control flow (e.g., conditionals, loops, and arbitrary function
calls), we can still calculate the gradient of the resulting variable.
In the following snippet, note that the number of iterations of the
`while`

loop and the evaluation of the `if`

statement both depend on
the value of the input `a`

.

```
def f(a):
b = a * 2
while np.linalg.norm(b) < 1000:
b = b * 2
if b.sum() > 0:
c = b
else:
c = 100 * b
return c
```

Again to compute gradients, we just need to `record`

the calculation
and then call the `backward`

function.

```
a = np.random.normal()
a.attach_grad()
with autograd.record():
d = f(a)
d.backward()
```

We can now analyze the `f`

function defined above. Note that it is
piecewise linear in its input `a`

. In other words, for any `a`

there
exists some constant scalar `k`

such that `f(a) = k * a`

, where the
value of `k`

depends on the input `a`

. Consequently `d / a`

allows
us to verify that the gradient is correct.

```
a.grad == d / a
```

```
array(True)
```

## 2.5.5. Training Mode and Prediction Mode¶

As we have seen, after we call `autograd.record`

, MXNet logs the
operations in the following block. There is one more subtle detail to be
aware of. Additionally, `autograd.record`

will change the running mode
from *prediction mode* to *training mode*. We can verify this behavior
by calling the `is_training`

function.

```
print(autograd.is_training())
with autograd.record():
print(autograd.is_training())
```

```
False
True
```

When we get to complicated deep learning models, we will encounter some algorithms where the model behaves differently during training and when we subsequently use it to make predictions. We will cover these differences in detail in later chapters.

## 2.5.6. Summary¶

MXNet provides the

`autograd`

package to automate the calculation of derivatives. To use it, we first attach gradients to those variables with respect to which we desire partial derivatives. We then record the computation of our target value, execute its`backward`

function, and access the resulting gradient via our variable’s`grad`

attribute.We can detach gradients to control the part of the computation that will be used in the

`backward`

function.The running modes of MXNet include training mode and prediction mode. We can determine the running mode by calling the

`is_training`

function.

## 2.5.7. Exercises¶

Why is the second derivative much more expensive to compute than the first derivative?

After running

`y.backward()`

, immediately run it again and see what happens.In the control flow example where we calculate the derivative of

`d`

with respect to`a`

, what would happen if we changed the variable`a`

to a random vector or matrix. At this point, the result of the calculation`f(a)`

is no longer a scalar. What happens to the result? How do we analyze this?Redesign an example of finding the gradient of the control flow. Run and analyze the result.

Let \(f(x) = \sin(x)\). Plot \(f(x)\) and \(\frac{df(x)}{dx}\), where the latter is computed without exploiting that \(f'(x) = \cos(x)\).

In a second-price auction (such as in eBay or in computational advertising), the winning bidder pays the second-highest price. Compute the gradient of the final price with respect to the winning bidder’s bid using

`autograd`

. What does the result tell you about the mechanism? If you are curious to learn more about second-price auctions, check out the paper by Edelman et al. [Edelman et al., 2007].