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======= Convolutional Neural Networks =======
Convolutional neural networks (CNNs) were developed during the last
decade of the previous century, with a focus on character recognition
tasks. Nowadays, CNNs are a central element in the spectacular success
@@ -18,12 +19,53 @@ loss function (for example Softmax) on the last (fully-connected) layer
and all the tips/tricks we developed for learning regular Neural
Networks still apply (back propagation, gradient descent etc etc).
What is the difference? _CNN architectures make the explicit assumption that
_CNN architectures make the explicit assumption that
the inputs are images, which allows us to encode certain properties
into the architecture. These then make the forward function more
efficient to implement and vastly reduce the amount of parameters in
the network._
Here we provide only a superficial overview, for the more interested, we recommend highly the course
"IN5400 – Machine Learning for Image Analysis":"https://www.uio.no/studier/emner/matnat/ifi/IN5400/index-eng.html"
and the slides of "CS231":"http://cs231n.github.io/convolutional-networks/".
Another good read is the article here URL:"https://arxiv.org/pdf/1603.07285.pdf".
===== Neural Networks vs CNNs =====
Neural networks are defined as _affine transformations_, that is
a vector is received as input and is multiplied with a matrix of so-called weights (our unknown paramters) to produce an
output (to which a bias vector is usually added before passing the result
through a nonlinear activation function). This is applicable to any type of input, be it an
image, a sound clip or an unordered collection of features: whatever their
dimensionality, their representation can always be flattened into a vector
before the transformation.
However, when we consider images, sound clips and many other similar kinds of data, these data have an intrinsic
structure. More formally, they share these important properties:
* They are stored as multi-dimensional arrays (think of the pixels of a figure) .
* They feature one or more axes for which ordering matters (e.g., width and height axes for an image, time axis for a sound clip).
* One axis, called the channel axis, is used to access different views of the data (e.g., the red, green and blue channels of a color image, or the left and right channels of a stereo audio track).
These properties are not exploited when an affine transformation is applied; in
fact, all the axes are treated in the same way and the topological information
is not taken into account. Still, taking advantage of the implicit structure of
the data may prove very handy in solving some tasks, like computer vision and
speech recognition, and in these cases it would be best to preserve it. This is
where discrete convolutions come into play.
A discrete convolution is a linear transformation that preserves this notion of
ordering. It is sparse (only a few input units contribute to a given output
unit) and reuses parameters (the same weights are applied to multiple locations
in the input).
As an example, consider
an image of size $32\times 32\times 3$ (32 wide, 32 high, 3 color channels), so a
@@ -42,7 +84,6 @@ would quickly lead to possible overfitting.
FIGURE: [figslides/nn.jpeg, width=500 frac=0.6] A regular 3-layer Neural Network.
Convolutional Neural Networks take advantage of the fact that the
input consists of images and they constrain the architecture in a more
sensible way.
@@ -67,11 +108,14 @@ end of the CNN architecture we will reduce the full image into a
single vector of class scores, arranged along the depth
dimension.
FIGURE: [figslides/cnn.jpeg, width=500 frac=0.6] A CNN arranges its neurons in three dimensions (width, heigh#t, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D out#put volume of neuron activations. In this example, the red input layer holds the image, so its width and heigh#t would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels).
FIGURE: [figslides/cnn.jpeg, width=500 frac=0.6] A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels).
===== Layers used to build CNNs =====
A simple CNN is a sequence of layers, and every layer of a CNN
transforms one volume of activations to another through a
differentiable function. We use three main types of layers to build
@@ -89,6 +133,7 @@ A simple CNN for image classification could have the architecture:
CNNs transform the original image layer by layer from the original
pixel values to the final class scores.
@@ -103,8 +148,6 @@ are consistent with the labels in the training set for each image.
=== CNNs in brief ===
In summary:
* A CNN architecture is in the simplest case a list of Layers that transform the image volume into an output volume (e.g. holding the class scores)
@@ -114,6 +157,438 @@ In summary:
* Each Layer may or may not have additional hyperparameters (e.g. CONV/FC/POOL do, RELU doesn’t)
A dense neural network is representd by an affine operation (like matrix-matrix multiplication) where all parameters are included.
The key idea in CNNs for say imaging is that in images neighbor pixels tend to be related! So we connect
only neighboring neurons in the input instead of connecting all with the first hidden layer.
We say we perform a filtering (convolution is the mathematical operation).
===== Mathematics of CNNs =====
The mathematics of CNNs is based on the mathematical operation of
_convolution_. In mathematics (in particular in functional analysis),
convolution is represented by matheematical operation (integration,
summation etc) on two function in order to produce a third function
that expresses how the shape of one gets modified by the other.
Convolution has a plethora of applications in a variety of disciplines, spanning from statistics to signal processing, computer vision, solutions of differential equations,linear algebra, engineering, and yes, machine learning.
Mathematically, convolution is defined as follows (one-dimensional example):
Let us define a continuous function $y(t)$ given by
!bt
\[
y(t) = \int x(a) w(t-a) da,
\]
!et
where $x(a)$ represents a so-called input and $w(t-a)$ is normally called the weight function or kernel.
The above integral is written in a more compact form as
!bt
\[
y(t) = \left(x * w\right)(t).
\]
!et
The discretized version reads
!bt
\[
y(t) = \sum_{a=-\infty}^{a=\infty}x(a)w(t-a).
\]
!et
Computing the inverse of the above convolution operations is known as deconvolution.
How can we use this? And what does it mean? Let us study some familiar examples first.
=== Convolution Examples: Polynomial multiplication ===
We have already met such an example in project 1 when we tried to set
up the design matrix for a two-dimensional function. This was an
example of polynomial multiplication. Let us recast such a problem in terms of the convolution operation.
Let us look a the following polynomials to second and third order, respectively:
!bt
\[
p(t) = \alpha_0+\alpha_1 t+\alpha_2 t^2,
\]
!et
and
!bt
\[
s(t) = \beta_0+\beta_1 t+\beta_2 t^2+\beta_3 t^3.
\]
!et
The polynomial multiplication gives us a new polynomial of degree $5$
!bt
\[
z(t) = \delta_0+\delta_1 t+\delta_2 t^2+\delta_3 t^3+\delta_4 t^4+\delta_5 t^5.
\]
!et
Computing polynomial products can be implemented efficiently if we rewrite the more brute force multiplications using convolution.
We note first that the new coefficients are given as
!bt
\begin{split}
\delta_0=&\alpha_0\beta_0\\
\delta_1=&\alpha_1\beta_0+\alpha_1\beta_0\\
\delta_2=&\alpha_0\beta_2+\alpha_1\beta_1+\alpha_2\beta_0\\
\delta_3=&\alpha_1\beta_2+\alpha_2\beta_1+\alpha_0\beta_3\\
\delta_4=&\alpha_2\beta_2+\alpha_1\beta_3\\
\delta_5=&\alpha_2\beta_3.\\
\end{split}
!et
We note that $\alpha_i=0$ except for $i\in \left\{0,1,2\right\}$ and $\beta_i=0$ except for $i\in\left\{0,1,2,3\right\}$.
We can then rewrite the coefficients $\delta_j$ using a discrete convolution as
!bt
\[
\delta_j = \sum_{i=-\infty}^{i=\infty}\alpha_i\beta_{j-i}=(\alpha * \beta)_j,
\]
!et
or as a double sum with restriction $l=i+j$
!bt
\[
\delta_l = \sum_{ij}\alpha_i\beta_{j}.
\]
!et
Do you see a potential drawback with these equations?
Since we only have a finite number of $\alpha$ and $\beta$ values
which are non-zero, we can rewrite the above convolution expressions
as a matrix-vector multiplication
!bt
\[
\bm{\delta}=\begin{bmatrix}\alpha_0 & 0 & 0 & 0 \\
\alpha_1 & \alpha_0 & 0 & 0 \\
\alpha_2 & \alpha_1 & \alpha_0 & 0 \\
0 & \alpha_2 & \alpha_1 & \alpha_0 \\
0 & 0 & \alpha_2 & \alpha_1 \\
0 & 0 & 0 & \alpha_2
\end{bmatrix}\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3\end{bmatrix}.
\]
!et
The process is commutative and we can easily see that we can rewrite the multiplication in terms of a matrix holding $\beta$ and a vector holding $\alpha$.
In this case we have
!bt
\[
\bm{\delta}=\begin{bmatrix}\beta_0 & 0 & 0 \\
\beta_1 & \beta_0 & 0 \\
\beta_2 & \beta_1 & \beta_0 \\
\beta_3 & \beta_2 & \beta_1 \\
0 & \beta_3 & \beta_2 \\
0 & 0 & \beta_3
\end{bmatrix}\begin{bmatrix} \alpha_0 \\ \alpha_1 \\ \alpha_2\end{bmatrix}.
\]
!et
Note that the use of these matrices is for mathematical purposes only and not implementation purposes.
When implementing the above equation we do not encode (and allocate memory) the matrices explicitely.
We rather code the convolutions in the minimal memory footprint that they require.
Does the number of floating point operations change here when we use the commutative property?
=== Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms) ===
For problems with so-called harmonic oscillations, given by for example the following differential equation
!bt
\[
m\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x(t)=F(t),
\]
!et
where $F(t)$ is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations.
If one has several driving forces, $F(t)=\sum_n F_n(t)$, one can find
the particular solution to each $F_n$, $x_{pn}(t)$, and the particular
solution for the entire driving force is then given by a series like
!bt
\begin{equation}
x_p(t)=\sum_nx_{pn}(t).
\end{equation}
!et
This is known as the principle of superposition. It only applies when
the homogenous equation is linear. If there were an anharmonic term
such as $x^3$ in the homogenous equation, then when one summed various
solutions, $x=(\sum_n x_n)^2$, one would get cross
terms. Superposition is especially useful when $F(t)$ can be written
as a sum of sinusoidal terms, because the solutions for each
sinusoidal (sine or cosine) term is analytic.
Driving forces are often periodic, even when they are not
sinusoidal. Periodicity implies that for some time $\tau$
!bt
\begin{eqnarray}
F(t+\tau)=F(t).
\end{eqnarray}
!et
One example of a non-sinusoidal periodic force is a square wave. Many
components in electric circuits are non-linear, e.g. diodes, which
makes many wave forms non-sinusoidal even when the circuits are being
driven by purely sinusoidal sources.
The code here shows a typical example of such a square wave generated using the functionality included in the _scipy_ Python package. We have used a period of $\tau=0.2$.
!bc pycod
import numpy as np
import math
from scipy import signal
import matplotlib.pyplot as plt
# number of points
n = 500
# start and final times
t0 = 0.0
tn = 1.0
# Period
t = np.linspace(t0, tn, n, endpoint=False)
SqrSignal = np.zeros(n)
SqrSignal = 1.0+signal.square(2*np.pi*5*t)
plt.plot(t, SqrSignal)
plt.ylim(-0.5, 2.5)
plt.show()
!ec
For the sinusoidal example the
period is $\tau=2\pi/\omega$. However, higher harmonics can also
satisfy the periodicity requirement. In general, any force that
satisfies the periodicity requirement can be expressed as a sum over
harmonics,
!bt
\begin{equation}
F(t)=\frac{f_0}{2}+\sum_{n>0} f_n\cos(2n\pi t/\tau)+g_n\sin(2n\pi t/\tau).
\end{equation}
!et
We can write down the answer for
$x_{pn}(t)$, by substituting $f_n/m$ or $g_n/m$ for $F_0/m$. By
writing each factor $2n\pi t/\tau$ as $n\omega t$, with $\omega\equiv
2\pi/\tau$,
!bt
\begin{equation}
label{eq:fourierdef1}
F(t)=\frac{f_0}{2}+\sum_{n>0}f_n\cos(n\omega t)+g_n\sin(n\omega t).
\end{equation}
!et
The solutions for $x(t)$ then come from replacing $\omega$ with
$n\omega$ for each term in the particular solution,
!bt
\begin{eqnarray}
x_p(t)&=&\frac{f_0}{2k}+\sum_{n>0} \alpha_n\cos(n\omega t-\delta_n)+\beta_n\sin(n\omega t-\delta_n),\\
\nonumber
\alpha_n&=&\frac{f_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\
\nonumber
\beta_n&=&\frac{g_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\
\nonumber
\delta_n&=&\tan^{-1}\left(\frac{2\beta n\omega}{\omega_0^2-n^2\omega^2}\right).
\end{eqnarray}
!et
Because the forces have been applied for a long time, any non-zero
damping eliminates the homogenous parts of the solution, so one need
only consider the particular solution for each $n$.
The problem is considered solved if one can find expressions for the
coefficients $f_n$ and $g_n$, even though the solutions are expressed
as an infinite sum. The coefficients can be extracted from the
function $F(t)$ by
!bt
\begin{eqnarray}
label{eq:fourierdef2}
f_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\cos(2n\pi t/\tau),\\
\nonumber
g_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\sin(2n\pi t/\tau).
\end{eqnarray}
!et
To check the consistency of these expressions and to verify
Eq. (ref{eq:fourierdef2}), one can insert the expansion of $F(t)$ in
Eq. (ref{eq:fourierdef1}) into the expression for the coefficients in
Eq. (ref{eq:fourierdef2}) and see whether
!bt
\begin{eqnarray}
f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~\left\{
\frac{f_0}{2}+\sum_{m>0}f_m\cos(m\omega t)+g_m\sin(m\omega t)
\right\}\cos(n\omega t).
\end{eqnarray}
!et
Immediately, one can throw away all the terms with $g_m$ because they
convolute an even and an odd function. The term with $f_0/2$
disappears because $\cos(n\omega t)$ is equally positive and negative
over the interval and will integrate to zero. For all the terms
$f_m\cos(m\omega t)$ appearing in the sum, one can use angle addition
formulas to see that $\cos(m\omega t)\cos(n\omega
t)=(1/2)(\cos[(m+n)\omega t]+\cos[(m-n)\omega t]$. This will integrate
to zero unless $m=n$. In that case the $m=n$ term gives
!bt
\begin{equation}
\int_{-\tau/2}^{\tau/2}dt~\cos^2(m\omega t)=\frac{\tau}{2},
\end{equation}
!et
and
!bt
\begin{eqnarray}
f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~f_n/2\\
\nonumber
&=&f_n~\checkmark.
\end{eqnarray}
!et
The same method can be used to check for the consistency of $g_n$.
The code here uses the Fourier series applied to a
square wave signal. The code here
visualizes the various approximations given by Fourier series compared
with a square wave with period $T=0.2$ (dimensionless time), width $0.1$ and max value of the force $F=2$. We
see that when we increase the number of components in the Fourier
series, the Fourier series approximation gets closer and closer to the
square wave signal.
!bc pycod
import numpy as np
import math
from scipy import signal
import matplotlib.pyplot as plt
# number of points
n = 500
# start and final times
t0 = 0.0
tn = 1.0
# Period
T =0.2
# Max value of square signal
Fmax= 2.0
# Width of signal
Width = 0.1
t = np.linspace(t0, tn, n, endpoint=False)
SqrSignal = np.zeros(n)
FourierSeriesSignal = np.zeros(n)
SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T)
a0 = Fmax*Width/T
FourierSeriesSignal = a0
Factor = 2.0*Fmax/np.pi
for i in range(1,500):
FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T)
plt.plot(t, SqrSignal)
plt.plot(t, FourierSeriesSignal)
plt.ylim(-0.5, 2.5)
plt.show()
!ec
===== Two-dimensional Objects =====
We often use convolutions over more than one dimension at a time. If
we have a two-dimensional image $I$ as input, we can have a _filter_
defined by a two-dimensional _kernel_ $K$. This leads to an output $S$
!bt
\[
S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(m,n)K(i-m,j-n).
\]
!et
Convolution is a commutatitave process, which means we can rewrite this equation as
!bt
\[
S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i-m,j-n)K(m,n).
\]
!et
Normally the latter is more straightforward to implement in a machine elarning library since there is less variation in the range of values of $m$ and $n$.
Many deep learning libraries implement cross-correlation instead of convolution
!bt
\[
S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i+m,j-+)K(m,n).
\]
!et
===== More on Dimensionalities =====
In fields like signal processing (and imaging as well), one designs
so-called filters. These filters are defined by the convolutions and
are often hand-crafted. One may specify filters for smoothing, edge
detection, frequency reshaping, and similar operations. However with
neural networks the idea is to automatically learn the filters and use
many of them in conjunction with non-linear operations (activation
functions).
As an example consider a neural network operating on sound sequence
data. Assume that we an input vector $\bm{x}$ of length $d=10^6$. We
construct then a neural network with onle hidden layer only with
$10^4$ nodes. This means that we will have a weight matrix with
$10^4\times 10^6=10^{10}$ weights to be determined, together with $10^4$ biases.
Assume furthermore that we have an output layer which is meant to train whether the sound sequence represents a human voice (true) or something else (false).
It means that we have only one output node. But since this output node connects to $10^4$ nodes in the hidden layer, there are in total $10^4$ weights to be determined for the output layer, plus one bias. In total we have
!bt
\[
\mathrm{NumberParameters}=10^{10}+10^4+10^4+1 \approx 10^{10},
\]
!et
that is ten billion parameters to determine.
===== Further Dimensionality Remarks =====
In today’s architecture one can train such neural networks, however
this is a huge number of parameters for the task at hand. In general,
it is a very wasteful and inefficient use of dense matrices as
parameters. Just as importantly, such trained network parameters are
very specific for the type of input data on which they were trained
and the network is not likely to generalize easily to variations in
the input.
The main principles that justify convolutions is locality of
information and repetion of patterns within the signal. Sound samples
of the input in adjacent spots are much more likely to affect each
other than those that are very far away. Similarly, sounds are
repeated in multiple times in the signal. While slightly simplistic,
reasoning about such a sound example demonstrates this. The same
principles then apply to images and other similar data.
===== CNNs in more detail, building convolutional neural networks in Tensorflow and Keras =====
@@ -138,6 +613,7 @@ dataset of images, we require a 4D matrix or _tensor_. This tensor has the dimen
!et
=== The MNIST dataset again ===
The MNIST dataset consists of grayscale images with a pixel size of
$28\times 28$, meaning we require $28 \times 28 = 724$ weights to each
@@ -183,6 +659,7 @@ small regions, and this serves as input to the next convolutional
layer.
=== Systematic reduction ===
By systematically reducing the size of the input volume, through
convolution and pooling, the network should create representations of
@@ -194,7 +671,6 @@ then serves as input to the output layer, e.g. a softmax output for
classification.
=== Prerequisites: Collect and pre-process data ===
!bc pycod
# import necessary packages
@@ -240,7 +716,7 @@ plt.show()
!ec
=== Importing Keras and Tensorflow ===
!bc pycod
from tensorflow.keras import datasets, layers, models
from tensorflow.keras.layers import Input
@@ -297,6 +773,7 @@ lmbd_vals = np.logspace(-5, 1, 7)
!ec
!bc pycod
CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
@@ -316,8 +793,8 @@ for i, eta in enumerate(eta_vals):
print()
!ec
!split
=== Final visualization ===
!bc pycod
# visual representation of grid search
@@ -378,7 +855,6 @@ train_images, test_images = train_images / 255.0, test_images / 255.0
To verify that the dataset looks correct, let's plot the first 25 images from the training set and display the class name below each image.
!bc pycod
@@ -399,7 +875,7 @@ plt.show()
!ec
The 6 lines of code below define the convolutional base using a common pattern: a stack of Conv2D and MaxPooling2D layers.
The six lines of code below define the convolutional base using a common pattern: a stack of Conv2D and MaxPooling2D layers.
As input, a CNN takes tensors of shape (image_height, image_width, color_channels), ignoring the batch size. If you are new to these dimensions, color_channels refers to (R,G,B). In this example, you will configure our CNN to process inputs of shape (32, 32, 3), which is the format of CIFAR images. You can do this by passing the argument input_shape to our first layer.
@@ -420,6 +896,8 @@ You can see that the output of every Conv2D and MaxPooling2D layer is a 3D tenso
To complete our model, you will feed the last output tensor from the
convolutional base (of shape (4, 4, 64)) into one or more Dense layers
to perform classification. Dense layers take vectors as input (which
@@ -438,7 +916,6 @@ model.summary()
!ec
As you can see, our (4, 4, 64) outputs were flattened into vectors of shape (1024) before going through two Dense layers.
Compile and train the model.
!bc pycod
model.compile(optimizer='adam',
@@ -450,7 +927,7 @@ history = model.fit(train_images, train_labels, epochs=10,
!ec
Finally, we evaluate the model.
!bc pycod
plt.plot(history.history['accuracy'], label='accuracy')
@@ -468,99 +945,7 @@ print(test_acc)
===== Recurrent neural networks: Overarching view =====
Till now our focus has been, including convolutional neural networks
as well, on feedforward neural networks. The output or the activations
flow only in one direction, from the input layer to the output layer.
A recurrent neural network (RNN) looks very much like a feedforward
neural network, except that it also has connections pointing
backward.
RNNs are used to analyze time series data such as stock prices, and
tell you when to buy or sell. In autonomous driving systems, they can
anticipate car trajectories and help avoid accidents. More generally,
they can work on sequences of arbitrary lengths, rather than on
fixed-sized inputs like all the nets we have discussed so far. For
example, they can take sentences, documents, or audio samples as
input, making them extremely useful for natural language processing
systems such as automatic translation and speech-to-text.
=== A simple example ===
!bc pycod
# Start importing packages
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import tensorflow as tf
from tensorflow.keras import datasets, layers, models
from tensorflow.keras.layers import Input
from tensorflow.keras.models import Model, Sequential
from tensorflow.keras.layers import Dense, SimpleRNN, LSTM, GRU
from tensorflow.keras import optimizers
from tensorflow.keras import regularizers
from tensorflow.keras.utils import to_categorical
# convert into dataset matrix
def convertToMatrix(data, step):
X, Y =[], []
for i in range(len(data)-step):
d=i+step
X.append(data[i:d,])
Y.append(data[d,])
return np.array(X), np.array(Y)
step = 4
N = 1000
Tp = 800
t=np.arange(0,N)
x=np.sin(0.02*t)+2*np.random.rand(N)
df = pd.DataFrame(x)
df.head()
plt.plot(df)
plt.show()
values=df.values
train,test = values[0:Tp,:], values[Tp:N,:]
# add step elements into train and test
test = np.append(test,np.repeat(test[-1,],step))
train = np.append(train,np.repeat(train[-1,],step))
trainX,trainY =convertToMatrix(train,step)
testX,testY =convertToMatrix(test,step)
trainX = np.reshape(trainX, (trainX.shape[0], 1, trainX.shape[1]))
testX = np.reshape(testX, (testX.shape[0], 1, testX.shape[1]))
model = Sequential()
model.add(SimpleRNN(units=32, input_shape=(1,step), activation="relu"))
model.add(Dense(8, activation="relu"))
model.add(Dense(1))
model.compile(loss='mean_squared_error', optimizer='rmsprop')
model.summary()
model.fit(trainX,trainY, epochs=100, batch_size=16, verbose=2)
trainPredict = model.predict(trainX)
testPredict= model.predict(testX)
predicted=np.concatenate((trainPredict,testPredict),axis=0)
trainScore = model.evaluate(trainX, trainY, verbose=0)
print(trainScore)
index = df.index.values
plt.plot(index,df)
plt.plot(index,predicted)
plt.axvline(df.index[Tp], c="r")
plt.show()
!ec
doc/BookChapters/chapter13.do.txt
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@@ -922,7 +922,31 @@ plt.show()
===== Stochastic Gradient Descent =====
===== Stochastic Gradient Descent (SGD) =====
In stochastic gradient descent, the extreme case is the case where we
have only one batch, that is we include the whole data set.
This process is called Stochastic Gradient
Descent (SGD) (or also sometimes on-line gradient descent). This is
relatively less common to see because in practice due to vectorized
code optimizations it can be computationally much more efficient to
evaluate the gradient for 100 examples, than the gradient for one
example 100 times. Even though SGD technically refers to using a
single example at a time to evaluate the gradient, you will hear
people use the term SGD even when referring to mini-batch gradient
descent (i.e. mentions of MGD for “Minibatch Gradient Descent”, or BGD
for “Batch gradient descent” are rare to see), where it is usually
assumed that mini-batches are used. The size of the mini-batch is a
hyperparameter but it is not very common to cross-validate or bootstrap it. It is
usually based on memory constraints (if any), or set to some value,
e.g. 32, 64 or 128. We use powers of 2 in practice because many
vectorized operation implementations work faster when their inputs are
sized in powers of 2.
In our notes with SGD we mean stochastic gradient descent with mini-batches.
Stochastic gradient descent (SGD) and variants thereof address some of
the shortcomings of the Gradient descent method discussed above.
@@ -954,7 +978,6 @@ minibatches. We denote these minibatches by $B_k$ where
$k=1,\cdots,n/M$.
As an example, suppose we have $10$ data points $(\mathbf{x}_1,\cdots, \mathbf{x}_{10})$
and we choose to have $M=5$ minibathces,
then each minibatch contains two data points. In particular we have
@@ -991,11 +1014,12 @@ minibathces (n/M) is commonly referred to as an epoch. Thus it is
typical to choose a number of epochs and for each epoch iterate over
the number of minibatches, as exemplified in the code below.
!bc pycod
import numpy as np
n = 100 #100 datapoints
M = 5 #size of each minibatch
M = 5 #size of each mini-batche
m = int(n/M) #number of minibatches
n_epochs = 10 #number of epochs
@@ -1017,7 +1041,6 @@ cheaper since we sum over the datapoints in the $k-th$ minibatch and not
all $n$ datapoints.
A natural question is when do we stop the search for a new minimum?
One possibility is to compute the full gradient after a given number
of epochs and check if the norm of the gradient is smaller than some
@@ -1030,7 +1053,6 @@ compare the values of the cost function and keep the $\beta$ that
gave the lowest value.
Another approach is to let the step length $\gamma_j$ depend on the
number of epochs in such a way that it becomes very small after a
reasonable time such that we do not move at all.
@@ -1071,37 +1093,41 @@ print("gamma_j after %d epochs: %g" % (n_epochs,gamma_j))
!ec
We note that we have defined several hyperparameters. These are now the number of epochs, the number of mini-batches and the parameters $t_0$ and $t_1$.
=== Program for stochastic gradient ===
!bc pycod
# Importing various packages
# Importing various packages
from math import exp, sqrt
from random import random, seed
import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import SGDRegressor
m = 100
x = 2*np.random.rand(m,1)
y = 4+3*x+np.random.randn(m,1)
n = 100
x = 2*np.random.rand(n,1)
y = 4+3*x+np.random.randn(n,1)
X = np.c_[np.ones((m,1)), x]
X = np.c_[np.ones((n,1)), x]
XT_X = X.T @ X
theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)
print("Own inversion")
print(theta_linreg)
sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)
sgdreg.fit(x,y.ravel())
print("sgdreg from scikit")
print(sgdreg.intercept_, sgdreg.coef_)
# Hessian matrix
H = (2.0/n)* XT_X
EigValues, EigVectors = np.linalg.eig(H)
print(f"Eigenvalues of Hessian Matrix:{EigValues}")
theta = np.random.randn(2,1)
eta = 0.1
eta = 1.0/np.max(EigValues)
Niterations = 1000
for iter in range(Niterations):
gradients = 2.0/m*X.T @ ((X @ theta)-y)
gradients = 2.0/n*X.T @ ((X @ theta)-y)
theta -= eta*gradients
print("theta from own gd")
print(theta)
@@ -1111,8 +1137,9 @@ Xnew = np.c_[np.ones((2,1)), xnew]
ypredict = Xnew.dot(theta)
ypredict2 = Xnew.dot(theta_linreg)
n_epochs = 50
M = 5 #size of each minibatch
m = int(n/M) #number of minibatches
t0, t1 = 5, 50
def learning_schedule(t):
return t0/(t+t1)
@@ -1120,16 +1147,20 @@ def learning_schedule(t):
theta = np.random.randn(2,1)
for epoch in range(n_epochs):
# Can you figure out a better way of setting up the contributions to each batch?
for i in range(m):
random_index = np.random.randint(m)
xi = X[random_index:random_index+1]
yi = y[random_index:random_index+1]
gradients = 2 * xi.T @ ((xi @ theta)-yi)
random_index = M*np.random.randint(m)
xi = X[random_index:random_index+M]
yi = y[random_index:random_index+M]
gradients = (2.0/M)* xi.T @ ((xi @ theta)-yi)
eta = learning_schedule(epoch*m+i)
theta = theta - eta*gradients
print("theta from own sdg")
print(theta)
plt.plot(xnew, ypredict, "r-")
plt.plot(xnew, ypredict2, "b-")
plt.plot(x, y ,'ro')
@@ -1142,6 +1173,13 @@ plt.show()
!ec
In the above code, we have use replacement in setting up the
mini-batches. The discussion
"here":"https://sebastianraschka.com/faq/docs/sgd-methods.html" may be
useful. More material will be added later.
===== Momentum based GD =====
The stochastic gradient descent (SGD) is almost always used with a
@@ -1175,7 +1213,6 @@ earlier. An equivalent way of writing the updates is
where we have defined $\Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1}$.
Let us try to get more intuition from these equations. It is helpful
to consider a simple physical analogy with a particle of mass $m$
moving in a viscous medium with drag coefficient $\mu$ and potential
@@ -1205,7 +1242,6 @@ Rearranging this equation, we can rewrite this as
!et
Notice that this equation is identical to previous one if we identify
the position of the particle, $\mathbf{w}$, with the parameters
$\boldsymbol{\theta}$. This allows us to identify the momentum
@@ -1254,6 +1290,7 @@ One of the major advantages of NAG is that it allows for the use of a larger lea
In stochastic gradient descent, with and without momentum, we still
have to specify a schedule for tuning the learning rates $\eta_t$
as a function of time. As discussed in the context of Newton's
@@ -1272,10 +1309,9 @@ Hessians.
Recently, a number of methods have been introduced that accomplish
this by tracking not only the gradient, but also the second moment of
the gradient. These methods include AdaGrad, AdaDelta, RMS-Prop, and
the gradient. These methods include AdaGrad, AdaDelta, Root Mean Squared Propagation (RMS-Prop), and
ADAM.
=== RMS prop ===
In RMS prop, in addition to keeping a running average of the first
@@ -1301,6 +1337,8 @@ directions where the norm of the gradient is consistently large. This
greatly speeds up the convergence by allowing us to use a larger
learning rate for flat directions.
=== ADAM optimizer ===
A related algorithm is the ADAM optimizer. In ADAM, we keep a running
@@ -1359,6 +1397,7 @@ update rule for this parameter is given by
* _Adaptive optimization methods don't always have good generalization._ Recent studies have shown that adaptive methods such as ADAM, RMSPorp, and AdaGrad tend to have poor generalization compared to SGD or SGD with momentum, particularly in the high-dimensional limit (i.e. the number of parameters exceeds the number of data points). Although it is not clear at this stage why these methods perform so well in training deep neural networks, simpler procedures like properly-tuned SGD may work as well or better in these applications.
===== Automatic differentiation =====
"Automatic differentiation (AD)":"https://en.wikipedia.org/wiki/Automatic_differentiation",
@@ -1540,7 +1579,6 @@ might be easier to work with, as the output is closer to what one
could expect form a gradient-evaluting function.
!bc pycod
import autograd.numpy as np
from autograd import grad
@@ -1581,7 +1619,6 @@ print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x)))
!ec
!bc pycod
import autograd.numpy as np
from autograd import grad
@@ -1654,21 +1691,21 @@ print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical))
Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.
Autograd supports many features. However, there are some functions that are not supported (yet) by Autograd.
Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.
Assigning a value to the variable being differentiated with respect to is an example thereof.
Assigning a value to the variable being differentiated with respect to
!bc pycod
#import autograd.numpy as np
#from autograd import grad
#def f8(x): # Assume x is an array
# x[2] = 3
# return x*2
import autograd.numpy as np
from autograd import grad
def f8(x): # Assume x is an array
x[2] = 3
return x*2
#f8_grad = grad(f8)
f8_grad = grad(f8)
#x = 8.4
x = 8.4
#print("The derivative of f8 is:",f8_grad(x))
print("The derivative of f8 is:",f8_grad(x))
!ec
Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.
@@ -1718,10 +1755,186 @@ a /=b
!ec
More examples will be added, in particular how to compare autograd with own codes for the gradients.
===== Using Autograd with OLS =====
We conclude the part on optmization by showing how we can make codes
for linear regression and logistic regression using _autograd_. The
first example shows results with ordinary leats squares.
!bc pycod
# Using Autograd to calculate gradients for OLS
from random import random, seed
import numpy as np
import autograd.numpy as np
import matplotlib.pyplot as plt
from autograd import grad
def CostOLS(beta):
return (1.0/n)*np.sum((y-X @ beta)**2)
n = 100
x = 2*np.random.rand(n,1)
y = 4+3*x+np.random.randn(n,1)
X = np.c_[np.ones((n,1)), x]
XT_X = X.T @ X
theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
print("Own inversion")
print(theta_linreg)
# Hessian matrix
H = (2.0/n)* XT_X
EigValues, EigVectors = np.linalg.eig(H)
print(f"Eigenvalues of Hessian Matrix:{EigValues}")
theta = np.random.randn(2,1)
eta = 1.0/np.max(EigValues)
Niterations = 1000
# define the gradient
training_gradient = grad(CostOLS)
for iter in range(Niterations):
gradients = training_gradient(theta)
theta -= eta*gradients
print("theta from own gd")
print(theta)
xnew = np.array([[0],[2]])
Xnew = np.c_[np.ones((2,1)), xnew]
ypredict = Xnew.dot(theta)
ypredict2 = Xnew.dot(theta_linreg)
plt.plot(xnew, ypredict, "r-")
plt.plot(xnew, ypredict2, "b-")
plt.plot(x, y ,'ro')
plt.axis([0,2.0,0, 15.0])
plt.xlabel(r'$x$')
plt.ylabel(r'$y$')
plt.title(r'Random numbers ')
plt.show()
!ec
=== Including Stochastic Gradient Descent with Autograd ===
In this code we include the stochastic gradient descent approach discussed above. Note here that we specify which argument we are taking the derivative with respect to when using _autograd_.
!bc pycod
# Using Autograd to calculate gradients using SGD
# OLS example
from random import random, seed
import numpy as np
import autograd.numpy as np
import matplotlib.pyplot as plt
from autograd import grad
# Note change from previous example
def CostOLS(y,X,theta):
return np.sum((y-X @ theta)**2)
n = 100
x = 2*np.random.rand(n,1)
y = 4+3*x+np.random.randn(n,1)
X = np.c_[np.ones((n,1)), x]
XT_X = X.T @ X
theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
print("Own inversion")
print(theta_linreg)
# Hessian matrix
H = (2.0/n)* XT_X
EigValues, EigVectors = np.linalg.eig(H)
print(f"Eigenvalues of Hessian Matrix:{EigValues}")
theta = np.random.randn(2,1)
eta = 1.0/np.max(EigValues)
Niterations = 1000
# Note that we request the derivative wrt third argument (theta, 2 here)
training_gradient = grad(CostOLS,2)
for iter in range(Niterations):
gradients = (1.0/n)*training_gradient(y, X, theta)
theta -= eta*gradients
print("theta from own gd")
print(theta)
xnew = np.array([[0],[2]])
Xnew = np.c_[np.ones((2,1)), xnew]
ypredict = Xnew.dot(theta)
ypredict2 = Xnew.dot(theta_linreg)
plt.plot(xnew, ypredict, "r-")
plt.plot(xnew, ypredict2, "b-")
plt.plot(x, y ,'ro')
plt.axis([0,2.0,0, 15.0])
plt.xlabel(r'$x$')
plt.ylabel(r'$y$')
plt.title(r'Random numbers ')
plt.show()
n_epochs = 50
M = 5 #size of each minibatch
m = int(n/M) #number of minibatches
t0, t1 = 5, 50
def learning_schedule(t):
return t0/(t+t1)
theta = np.random.randn(2,1)
for epoch in range(n_epochs):
# Can you figure out a better way of setting up the contributions to each batch?
for i in range(m):
random_index = M*np.random.randint(m)
xi = X[random_index:random_index+M]
yi = y[random_index:random_index+M]
gradients = (1.0/M)*training_gradient(yi, xi, theta)
eta = learning_schedule(epoch*m+i)
theta = theta - eta*gradients
print("theta from own sdg")
print(theta)
!ec
=== And Logistic Regression ===
!bc pycod
import autograd.numpy as np
from autograd import grad
def sigmoid(x):
return 0.5 * (np.tanh(x / 2.) + 1)
def logistic_predictions(weights, inputs):
# Outputs probability of a label being true according to logistic model.
return sigmoid(np.dot(inputs, weights))
def training_loss(weights):
# Training loss is the negative log-likelihood of the training labels.
preds = logistic_predictions(weights, inputs)
label_probabilities = preds * targets + (1 - preds) * (1 - targets)
return -np.sum(np.log(label_probabilities))
# Build a toy dataset.
inputs = np.array([[0.52, 1.12, 0.77],
[0.88, -1.08, 0.15],
[0.52, 0.06, -1.30],
[0.74, -2.49, 1.39]])
targets = np.array([True, True, False, True])
# Define a function that returns gradients of training loss using Autograd.
training_gradient_fun = grad(training_loss)
# Optimize weights using gradient descent.
weights = np.array([0.0, 0.0, 0.0])
print("Initial loss:", training_loss(weights))
for i in range(100):
weights -= training_gradient_fun(weights) * 0.01
print("Trained loss:", training_loss(weights))
!ec
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@@ -29,9 +29,12 @@ parts:
numbered: true
chapters:
- file: chapter8.ipynb
- file: clustering.ipynb
- caption: Deep Learning Methods
numbered: true
chapters:
- file: chapter9.ipynb
- file: chapter10.ipynb
- file: chapter11.ipynb
- file: chapter12.ipynb
- file: chapter13.ipynb
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