.. _sec_calculus:
Calculus
========
Finding the area of a polygon had remained mysterious until at least
:math:`2,500` years ago, when ancient Greeks divided a polygon into
triangles and summed their areas. To find the area of curved shapes,
such as a circle, ancient Greeks inscribed polygons in such shapes. As
shown in :numref:`fig_circle_area`, an inscribed polygon with more
sides of equal length better approximates the circle. This process is
also known as the *method of exhaustion*.
.. _fig_circle_area:
.. figure:: ../img/polygon_circle.svg
Find the area of a circle with the method of exhaustion.
In fact, the method of exhaustion is where *integral calculus* (will be
described in :numref:`sec_integral_calculus`) originates from. More
than :math:`2,000` years later, the other branch of calculus,
*differential calculus*, was invented. Among the most critical
applications of differential calculus, optimization problems consider
how to do something *the best*. As discussed in
:numref:`subsec_norms_and_objectives`, such problems are ubiquitous in
deep learning.
In deep learning, we *train* models, updating them successively so that
they get better and better as they see more and more data. Usually,
getting better means minimizing a *loss function*, a score that answers
the question “how *bad* is our model?” This question is more subtle than
it appears. Ultimately, what we really care about is producing a model
that performs well on data that we have never seen before. But we can
only fit the model to data that we can actually see. Thus we can
decompose the task of fitting models into two key concerns: i)
*optimization*: the process of fitting our models to observed data; ii)
*generalization*: the mathematical principles and practitioners’ wisdom
that guide as to how to produce models whose validity extends beyond the
exact set of data points used to train them.
To help you understand optimization problems and methods in later
chapters, here we give a very brief primer on differential calculus that
is commonly used in deep learning.
Derivatives and Differentiation
-------------------------------
We begin by addressing the calculation of derivatives, a crucial step in
nearly all deep learning optimization algorithms. In deep learning, we
typically choose loss functions that are differentiable with respect to
our model’s parameters. Put simply, this means that for each parameter,
we can determine how rapidly the loss would increase or decrease, were
we to *increase* or *decrease* that parameter by an infinitesimally
small amount.
Suppose that we have a function
:math:`f: \mathbb{R} \rightarrow \mathbb{R}`, whose input and output are
both scalars. The *derivative* of :math:`f` is defined as
.. math:: f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h},
:label: eq_derivative
if this limit exists. If :math:`f'(a)` exists, :math:`f` is said to be
*differentiable* at :math:`a`. If :math:`f` is differentiable at every
number of an interval, then this function is differentiable on this
interval. We can interpret the derivative :math:`f'(x)` in
:eq:`eq_derivative` as the *instantaneous* rate of change of
:math:`f(x)` with respect to :math:`x`. The so-called instantaneous rate
of change is based on the variation :math:`h` in :math:`x`, which
approaches :math:`0`.
To illustrate derivatives, let’s experiment with an example. Define
:math:`u = f(x) = 3x^2-4x`.
.. code:: python
%matplotlib inline
import d2l
from IPython import display
from mxnet import np, npx
npx.set_np()
def f(x):
return 3 * x ** 2 - 4 * x
By setting :math:`x=1` and letting :math:`h` approach :math:`0`, the
numerical result of :math:`\frac{f(x+h) - f(x)}{h}` in
:eq:`eq_derivative` approaches :math:`2`. Though this experiment is
not a mathematical proof, we will see later that the derivative
:math:`u'` is :math:`2` when :math:`x=1`.
.. code:: python
def numerical_lim(f, x, h):
return (f(x + h) - f(x)) / h
h = 0.1
for i in range(5):
print('h=%.5f, numerical limit=%.5f' % (h, numerical_lim(f, 1, h)))
h *= 0.1
.. parsed-literal::
:class: output
h=0.10000, numerical limit=2.30000
h=0.01000, numerical limit=2.03000
h=0.00100, numerical limit=2.00300
h=0.00010, numerical limit=2.00030
h=0.00001, numerical limit=2.00003
Let’s familiarize ourselves with a few equivalent notations for
derivatives. Given :math:`y = f(x)`, where :math:`x` and :math:`y` are
the independent variable and the dependent variable of the function
:math:`f`, respectively. The following expressions are equivalent:
.. math:: f'(x) = y' = \frac{dy}{dx} = \frac{df}{dx} = \frac{d}{dx} f(x) = Df(x) = D_x f(x),
where symbols :math:`\frac{d}{dx}` and :math:`D` are *differentiation
operators* that indicate operation of *differentiation*. We can use the
following rules to differentiate common functions:
- :math:`DC = 0` (:math:`C` is a constant),
- :math:`Dx^n = nx^{n-1}` (the *power rule*, :math:`n` is any real
number),
- :math:`De^x = e^x`,
- :math:`D\ln(x) = 1/x.`
To differentiate a function that is formed from a few simpler functions
such as the above common functions, the following rules can be handy for
us. Suppose that functions :math:`f` and :math:`g` are both
differentiable and :math:`C` is a constant, we have the *constant
multiple rule*
.. math:: \frac{d}{dx} [Cf(x)] = C \frac{d}{dx} f(x),
the *sum rule*
.. math:: \frac{d}{dx} [f(x) + g(x)] = \frac{d}{dx} f(x) + \frac{d}{dx} g(x),
the *product rule*
.. math:: \frac{d}{dx} [f(x)g(x)] = f(x) \frac{d}{dx} [g(x)] + g(x) \frac{d}{dx} [f(x)],
and the *quotient rule*
.. math:: \frac{d}{dx} \left[\frac{f(x)}{g(x)}\right] = \frac{g(x) \frac{d}{dx} [f(x)] - f(x) \frac{d}{dx} [g(x)]}{[g(x)]^2}.
Now we can apply a few of the above rules to find
:math:`u' = f'(x) = 3 \frac{d}{dx} x^2-4\frac{d}{dx}x = 6x-4`. Thus, by
setting :math:`x = 1`, we have :math:`u' = 2`: this is supported by our
earlier experiment in this section where the numerical result approaches
:math:`2`. This derivative is also the slope of the tangent line to the
curve :math:`u = f(x)` when :math:`x = 1`.
To visualize such an interpretation of derivatives, we will use
``matplotlib``, a popular plotting library in Python. To configure
properties of the figures produced by ``matplotlib``, we need to define
a few functions. In the following, the ``use_svg_display`` function
specifies the ``matplotlib`` package to output the svg figures for
sharper images.
.. code:: python
# Saved in the d2l package for later use
def use_svg_display():
"""Use the svg format to display a plot in Jupyter."""
display.set_matplotlib_formats('svg')
We define the ``set_figsize`` function to specify the figure sizes. Note
that here we directly use ``d2l.plt`` since the import statement
``from matplotlib import pyplot as plt`` has been marked for being saved
in the ``d2l`` package in the preface.
.. code:: python
# Saved in the d2l package for later use
def set_figsize(figsize=(3.5, 2.5)):
"""Set the figure size for matplotlib."""
use_svg_display()
d2l.plt.rcParams['figure.figsize'] = figsize
The following ``set_axes`` function sets properties of axes of figures
produced by ``matplotlib``.
.. code:: python
# Saved in the d2l package for later use
def set_axes(axes, xlabel, ylabel, xlim, ylim, xscale, yscale, legend):
"""Set the axes for matplotlib."""
axes.set_xlabel(xlabel)
axes.set_ylabel(ylabel)
axes.set_xscale(xscale)
axes.set_yscale(yscale)
axes.set_xlim(xlim)
axes.set_ylim(ylim)
if legend:
axes.legend(legend)
axes.grid()
With these :math:`3` functions for figure configurations, we define the
``plot`` function to plot multiple curves succinctly since we will need
to visualize many curves throughout the book.
.. code:: python
# Saved in the d2l package for later use
def plot(X, Y=None, xlabel=None, ylabel=None, legend=[], xlim=None,
ylim=None, xscale='linear', yscale='linear',
fmts=['-', 'm--', 'g-.', 'r:'], figsize=(3.5, 2.5), axes=None):
"""Plot data points."""
d2l.set_figsize(figsize)
axes = axes if axes else d2l.plt.gca()
# Return True if X (ndarray or list) has 1 axis
def has_one_axis(X):
return (hasattr(X, "ndim") and X.ndim == 1 or isinstance(X, list)
and not hasattr(X[0], "__len__"))
if has_one_axis(X):
X = [X]
if Y is None:
X, Y = [[]] * len(X), X
elif has_one_axis(Y):
Y = [Y]
if len(X) != len(Y):
X = X * len(Y)
axes.cla()
for x, y, fmt in zip(X, Y, fmts):
if len(x):
axes.plot(x, y, fmt)
else:
axes.plot(y, fmt)
set_axes(axes, xlabel, ylabel, xlim, ylim, xscale, yscale, legend)
Now we can plot the function :math:`u = f(x)` and its tangent line
:math:`y = 2x - 3` at :math:`x=1`, where the coefficient :math:`2` is
the slope of the tangent line.
.. code:: python
x = np.arange(0, 3, 0.1)
plot(x, [f(x), 2 * x - 3], 'x', 'f(x)', legend=['f(x)', 'Tangent line (x=1)'])
.. figure:: output_calculus_694dfd_13_0.svg
Partial Derivatives
-------------------
So far we have dealt with the differentiation of functions of just one
variable. In deep learning, functions often depend on *many* variables.
Thus, we need to extend the ideas of differentiation to these
*multivariate* functions.
Let :math:`y = f(x_1, x_2, \ldots, x_n)` be a function with :math:`n`
variables. The *partial derivative* of :math:`y` with respect to its
:math:`i^\mathrm{th}` parameter :math:`x_i` is
.. math:: \frac{\partial y}{\partial x_i} = \lim_{h \rightarrow 0} \frac{f(x_1, \ldots, x_{i-1}, x_i+h, x_{i+1}, \ldots, x_n) - f(x_1, \ldots, x_i, \ldots, x_n)}{h}.
To calculate :math:`\frac{\partial y}{\partial x_i}`, we can simply
treat :math:`x_1, \ldots, x_{i-1}, x_{i+1}, \ldots, x_n` as constants
and calculate the derivative of :math:`y` with respect to :math:`x_i`.
For notation of partial derivatives, the following are equivalent:
.. math:: \frac{\partial y}{\partial x_i} = \frac{\partial f}{\partial x_i} = f_{x_i} = f_i = D_i f = D_{x_i} f.
Gradients
---------
We can concatenate partial derivatives of a multivariate function with
respect to all its variables to obtain the *gradient* vector of the
function. Suppose that the input of function
:math:`f: \mathbb{R}^n \rightarrow \mathbb{R}` is an
:math:`n`-dimensional vector
:math:`\mathbf{x} = [x_1, x_2, \ldots, x_n]^\top` and the output is a
scalar. The gradient of the function :math:`f(\mathbf{x})` with respect
to :math:`\mathbf{x}` is a vector of :math:`n` partial derivatives:
.. math:: \nabla_{\mathbf{x}} f(\mathbf{x}) = \bigg[\frac{\partial f(\mathbf{x})}{\partial x_1}, \frac{\partial f(\mathbf{x})}{\partial x_2}, \ldots, \frac{\partial f(\mathbf{x})}{\partial x_n}\bigg]^\top,
where :math:`\nabla_{\mathbf{x}} f(\mathbf{x})` is often replaced by
:math:`\nabla f(\mathbf{x})` when there is no ambiguity.
Let :math:`\mathbf{x}` be an :math:`n`-dimensional vector, the following
rules are often used when differentiating multivariate functions:
- For all :math:`\mathbf{A} \in \mathbb{R}^{m \times n}`,
:math:`\nabla_{\mathbf{x}} \mathbf{A} \mathbf{x} = \mathbf{A}^\top`,
- For all :math:`\mathbf{A} \in \mathbb{R}^{n \times m}`,
:math:`\nabla_{\mathbf{x}} \mathbf{x}^\top \mathbf{A} = \mathbf{A}`,
- For all :math:`\mathbf{A} \in \mathbb{R}^{n \times n}`,
:math:`\nabla_{\mathbf{x}} \mathbf{x}^\top \mathbf{A} \mathbf{x} = (\mathbf{A} + \mathbf{A}^\top)\mathbf{x}`,
- :math:`\nabla_{\mathbf{x}} \|\mathbf{x} \|^2 = \nabla_{\mathbf{x}} \mathbf{x}^\top \mathbf{x} = 2\mathbf{x}`.
Similarly, for any matrix :math:`\mathbf{X}`, we have
:math:`\nabla_{\mathbf{X}} \|\mathbf{X} \|_F^2 = 2\mathbf{X}`. As we
will see later, gradients are useful for designing optimization
algorithms in deep learning.
Chain Rule
----------
However, such gradients can be hard to find. This is because
multivariate functions in deep learning are often *composite*, so we may
not apply any of the aforementioned rules to differentiate these
functions. Fortunately, the *chain rule* enables us to differentiate
composite functions.
Let’s first consider functions of a single variable. Suppose that
functions :math:`y=f(u)` and :math:`u=g(x)` are both differentiable,
then the chain rule states that
.. math:: \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}.
Now let’s turn our attention to a more general scenario where functions
have an arbitrary number of variables. Suppose that the differentiable
function :math:`y` has variables :math:`u_1, u_2, \ldots, u_m`, where
each differentiable function :math:`u_i` has variables
:math:`x_1, x_2, \ldots, x_n`. Note that :math:`y` is a function of
:math:`x_1, x_2, \ldots, x_n`. Then the chain rule gives
.. math:: \frac{dy}{dx_i} = \frac{dy}{du_1} \frac{du_1}{dx_i} + \frac{dy}{du_2} \frac{du_2}{dx_i} + \cdots + \frac{dy}{du_m} \frac{du_m}{dx_i}
for any :math:`i = 1, 2, \ldots, n`.
Summary
-------
- Differential calculus and integral calculus are two branches of
calculus, where the former can be applied to the ubiquitous
optimization problems in deep learning.
- A derivative can be interpreted as the instantaneous rate of change
of a function with respect to its variable. It is also the slope of
the tangent line to the curve of the function.
- A gradient is a vector whose components are the partial derivatives
of a multivariate function with respect to all its variables.
- The chain rule enables us to differentiate composite functions.
Exercises
---------
1. Plot the function :math:`y = f(x) = x^3 - \frac{1}{x}` and its
tangent line when :math:`x = 1`.
2. Find the gradient of the function
:math:`f(\mathbf{x}) = 3x_1^2 + 5e^{x_2}`.
3. What is the gradient of the function
:math:`f(\mathbf{x}) = \|\mathbf{x}\|_2`?
4. Can you write out the chain rule for the case where
:math:`u = f(x, y, z)` and :math:`x = x(a, b)`, :math:`y = y(a, b)`,
and :math:`z = z(a, b)`?
`Discussions `__
-------------------------------------------------
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.. |image0| image:: ../img/qr_calculus.svg