FYS-STK4155/doc/pub/week41/ipynb/week41.ipynb

Week 41 Neural networks and constructing a neural network code

Morten Hjorth-Jensen, Department of Physics, University of Oslo

Date: Week 41

Plan for week 41, October 7-11

Material for the lecture on Monday October 7, 2024

1. Neural Networks, setting up the basic steps, from the simple perceptron model to the multi-layer perceptron model.

2. Building our own Feed-forward Neural Network

• Video of lecture notes at https://youtu.be/pMRUbf9E-gM

• Whiteboard notes at https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2024/NotesOctober7.pdf

Readings and Videos:

1. These lecture notes

2. Rashcka et al chapter 11

3. For neural networks we recommend Goodfellow et al chapter 6.

a. Neural Networks demystified at https://www.youtube.com/watch?v=bxe2T-V8XRs&list=PLiaHhY2iBX9hdHaRr6b7XevZtgZRa1PoU&ab_channel=WelchLabs

2. Building Neural Networks from scratch at https://www.youtube.com/watch?v=Wo5dMEP_BbI&list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&ab_channel=sentdex

3. Video on Neural Networks at https://www.youtube.com/watch?v=CqOfi41LfDw

4. Video on the back propagation algorithm at https://www.youtube.com/watch?v=Ilg3gGewQ5U

We also recommend Michael Nielsen's intuitive approach to the neural networks and the universal approximation theorem, see the slides at http://neuralnetworksanddeeplearning.com/chap4.html.

Material for the active learning sessions on Tuesday and Wednesday

• Exercise on writing your own stochastic gradient and gradient descent codes. This exercise continues next week with studies of automatic differentiation

• One lecture at the beginning of each session on the material from weeks 39 and 40 and how to write your own gradient descent code

• Discussion of project 2

• Your task before the sessions: revisit the material from weeks 39 and 40 and in particular the material from week 40 on stochastic gradient descent

Lecture Monday October 7

Introduction to Neural networks

Artificial neural networks are computational systems that can learn to perform tasks by considering examples, generally without being programmed with any task-specific rules. It is supposed to mimic a biological system, wherein neurons interact by sending signals in the form of mathematical functions between layers. All layers can contain an arbitrary number of neurons, and each connection is represented by a weight variable.

Artificial neurons

The field of artificial neural networks has a long history of development, and is closely connected with the advancement of computer science and computers in general. A model of artificial neurons was first developed by McCulloch and Pitts in 1943 to study signal processing in the brain and has later been refined by others. The general idea is to mimic neural networks in the human brain, which is composed of billions of neurons that communicate with each other by sending electrical signals. Each neuron accumulates its incoming signals, which must exceed an activation threshold to yield an output. If the threshold is not overcome, the neuron remains inactive, i.e. has zero output.

This behaviour has inspired a simple mathematical model for an artificial neuron.

\begin{equation}
 y = f\left(\sum_{i=1}^n w_ix_i\right) = f(u)
\label{artificialNeuron} \tag{1}
\end{equation}

Here, the output y of the neuron is the value of its activation function, which have as input a weighted sum of signals x_i, \dots ,x_n received by n other neurons.

Conceptually, it is helpful to divide neural networks into four categories:

1. general purpose neural networks for supervised learning,

2. neural networks designed specifically for image processing, the most prominent example of this class being Convolutional Neural Networks (CNNs),

3. neural networks for sequential data such as Recurrent Neural Networks (RNNs), and

4. neural networks for unsupervised learning such as Deep Boltzmann Machines.

In natural science, DNNs and CNNs have already found numerous applications. In statistical physics, they have been applied to detect phase transitions in 2D Ising and Potts models, lattice gauge theories, and different phases of polymers, or solving the Navier-Stokes equation in weather forecasting. Deep learning has also found interesting applications in quantum physics. Various quantum phase transitions can be detected and studied using DNNs and CNNs, topological phases, and even non-equilibrium many-body localization. Representing quantum states as DNNs quantum state tomography are among some of the impressive achievements to reveal the potential of DNNs to facilitate the study of quantum systems.

In quantum information theory, it has been shown that one can perform gate decompositions with the help of neural.

The applications are not limited to the natural sciences. There is a plethora of applications in essentially all disciplines, from the humanities to life science and medicine.

Neural network types

An artificial neural network (ANN), is a computational model that consists of layers of connected neurons, or nodes or units. We will refer to these interchangeably as units or nodes, and sometimes as neurons.

It is supposed to mimic a biological nervous system by letting each neuron interact with other neurons by sending signals in the form of mathematical functions between layers. A wide variety of different ANNs have been developed, but most of them consist of an input layer, an output layer and eventual layers in-between, called hidden layers. All layers can contain an arbitrary number of nodes, and each connection between two nodes is associated with a weight variable.

Neural networks (also called neural nets) are neural-inspired nonlinear models for supervised learning. As we will see, neural nets can be viewed as natural, more powerful extensions of supervised learning methods such as linear and logistic regression and soft-max methods we discussed earlier.

Feed-forward neural networks

The feed-forward neural network (FFNN) was the first and simplest type of ANNs that were devised. In this network, the information moves in only one direction: forward through the layers.

Nodes are represented by circles, while the arrows display the connections between the nodes, including the direction of information flow. Additionally, each arrow corresponds to a weight variable (figure to come). We observe that each node in a layer is connected to all nodes in the subsequent layer, making this a so-called fully-connected FFNN.

Convolutional Neural Network

A different variant of FFNNs are convolutional neural networks (CNNs), which have a connectivity pattern inspired by the animal visual cortex. Individual neurons in the visual cortex only respond to stimuli from small sub-regions of the visual field, called a receptive field. This makes the neurons well-suited to exploit the strong spatially local correlation present in natural images. The response of each neuron can be approximated mathematically as a convolution operation. (figure to come)

Convolutional neural networks emulate the behaviour of neurons in the visual cortex by enforcing a local connectivity pattern between nodes of adjacent layers: Each node in a convolutional layer is connected only to a subset of the nodes in the previous layer, in contrast to the fully-connected FFNN. Often, CNNs consist of several convolutional layers that learn local features of the input, with a fully-connected layer at the end, which gathers all the local data and produces the outputs. They have wide applications in image and video recognition.

Recurrent neural networks

So far we have only mentioned ANNs where information flows in one direction: forward. Recurrent neural networks on the other hand, have connections between nodes that form directed cycles. This creates a form of internal memory which are able to capture information on what has been calculated before; the output is dependent on the previous computations. Recurrent NNs make use of sequential information by performing the same task for every element in a sequence, where each element depends on previous elements. An example of such information is sentences, making recurrent NNs especially well-suited for handwriting and speech recognition.

Other types of networks

There are many other kinds of ANNs that have been developed. One type that is specifically designed for interpolation in multidimensional space is the radial basis function (RBF) network. RBFs are typically made up of three layers: an input layer, a hidden layer with non-linear radial symmetric activation functions and a linear output layer (''linear'' here means that each node in the output layer has a linear activation function). The layers are normally fully-connected and there are no cycles, thus RBFs can be viewed as a type of fully-connected FFNN. They are however usually treated as a separate type of NN due the unusual activation functions.

Multilayer perceptrons

One uses often so-called fully-connected feed-forward neural networks with three or more layers (an input layer, one or more hidden layers and an output layer) consisting of neurons that have non-linear activation functions.

Such networks are often called multilayer perceptrons (MLPs).

Why multilayer perceptrons?

According to the Universal approximation theorem, a feed-forward neural network with just a single hidden layer containing a finite number of neurons can approximate a continuous multidimensional function to arbitrary accuracy, assuming the activation function for the hidden layer is a non-constant, bounded and monotonically-increasing continuous function.

Note that the requirements on the activation function only applies to the hidden layer, the output nodes are always assumed to be linear, so as to not restrict the range of output values.

Illustration of a single perceptron model and a multi-perceptron model

Figure 1: In a) we show a single perceptron model while in b) we dispay a network with two hidden layers, an input layer and an output layer.

Examples of XOR, OR and AND gates

Let us first try to fit various gates using standard linear regression. The gates we are thinking of are the classical XOR, OR and AND gates, well-known elements in computer science. The tables here show how we can set up the inputs x_1 and x_2 in order to yield a specific target y_i.

In [1]:

"""
Simple code that tests XOR, OR and AND gates with linear regression
"""

import numpy as np
# Design matrix
X = np.array([ [1, 0, 0], [1, 0, 1], [1, 1, 0],[1, 1, 1]],dtype=np.float64)
print(f"The X.TX  matrix:{X.T @ X}")
Xinv = np.linalg.pinv(X.T @ X)
print(f"The invers of X.TX  matrix:{Xinv}")

# The XOR gate 
yXOR = np.array( [ 0, 1 ,1, 0])
ThetaXOR  = Xinv @ X.T @ yXOR
print(f"The values of theta for the XOR gate:{ThetaXOR}")
print(f"The linear regression prediction  for the XOR gate:{X @ ThetaXOR}")


# The OR gate 
yOR = np.array( [ 0, 1 ,1, 1])
ThetaOR  = Xinv @ X.T @ yOR
print(f"The values of theta for the OR gate:{ThetaOR}")
print(f"The linear regression prediction  for the OR gate:{X @ ThetaOR}")


# The OR gate 
yAND = np.array( [ 0, 0 ,0, 1])
ThetaAND  = Xinv @ X.T @ yAND
print(f"The values of theta for the AND gate:{ThetaAND}")
print(f"The linear regression prediction  for the AND gate:{X @ ThetaAND}")

What is happening here?

Does Logistic Regression do a better Job?

In [2]:

%matplotlib inline

"""
Simple code that tests XOR and OR gates with linear regression
and logistic regression
"""

import matplotlib.pyplot as plt
from sklearn.linear_model import LogisticRegression
import numpy as np

# Design matrix
X = np.array([ [1, 0, 0], [1, 0, 1], [1, 1, 0],[1, 1, 1]],dtype=np.float64)
print(f"The X.TX  matrix:{X.T @ X}")
Xinv = np.linalg.pinv(X.T @ X)
print(f"The invers of X.TX  matrix:{Xinv}")

# The XOR gate 
yXOR = np.array( [ 0, 1 ,1, 0])
ThetaXOR  = Xinv @ X.T @ yXOR
print(f"The values of theta for the XOR gate:{ThetaXOR}")
print(f"The linear regression prediction  for the XOR gate:{X @ ThetaXOR}")


# The OR gate 
yOR = np.array( [ 0, 1 ,1, 1])
ThetaOR  = Xinv @ X.T @ yOR
print(f"The values of theta for the OR gate:{ThetaOR}")
print(f"The linear regression prediction  for the OR gate:{X @ ThetaOR}")


# The OR gate 
yAND = np.array( [ 0, 0 ,0, 1])
ThetaAND  = Xinv @ X.T @ yAND
print(f"The values of theta for the AND gate:{ThetaAND}")
print(f"The linear regression prediction  for the AND gate:{X @ ThetaAND}")

# Now we change to logistic regression


# Logistic Regression
logreg = LogisticRegression()
logreg.fit(X, yOR)
print("Test set accuracy with Logistic Regression for OR gate: {:.2f}".format(logreg.score(X,yOR)))

logreg.fit(X, yXOR)
print("Test set accuracy with Logistic Regression for XOR gate: {:.2f}".format(logreg.score(X,yXOR)))


logreg.fit(X, yAND)
print("Test set accuracy with Logistic Regression for AND gate: {:.2f}".format(logreg.score(X,yAND)))

Not exactly impressive, but somewhat better.

Adding Neural Networks

In [3]:

# and now neural networks with Scikit-Learn and the XOR

from sklearn.neural_network import MLPClassifier
from sklearn.datasets import make_classification
X, yXOR = make_classification(n_samples=100, random_state=1)
FFNN = MLPClassifier(random_state=1, max_iter=300).fit(X, yXOR)
FFNN.predict_proba(X)
print(f"Test set accuracy with Feed Forward Neural Network  for XOR gate:{FFNN.score(X, yXOR)}")

Mathematics of deep learning

Two recent books online.

1. The Modern Mathematics of Deep Learning, by Julius Berner, Philipp Grohs, Gitta Kutyniok, Philipp Petersen, published as Mathematical Aspects of Deep Learning, pp. 1-111. Cambridge University Press, 2022

2. Mathematical Introduction to Deep Learning: Methods, Implementations, and Theory, Arnulf Jentzen, Benno Kuckuck, Philippe von Wurstemberger

Reminder on books with hands-on material and codes

• Sebastian Rashcka et al, Machine learning with Sickit-Learn and PyTorch

Reading recommendations

1. Rashkca et al., chapter 11, jupyter-notebook sent separately, from GitHub

2. Goodfellow et al, chapter 6 and 7 contain most of the neural network background.

Mathematics of deep learning and neural networks

Neural networks, in its so-called feed-forward form, where each iterations contains a feed-forward stage and a back-propgagation stage, consist of series of affine matrix-matrix and matrix-vector multiplications. The unknown parameters (the so-called biases and weights which deternine the architecture of a neural network), are uptaded iteratively using the so-called back-propagation algorithm. This algorithm corresponds to the so-called reverse mode of automatic differentation.

Basics of an NN

A neural network consists of a series of hidden layers, in addition to the input and output layers. Each layer l has a set of parameters \boldsymbol{\Theta}^{(l)}=(\boldsymbol{W}^{(l)},\boldsymbol{b}^{(l)}) which are related to the parameters in other layers through a series of affine transformations, for a standard NN these are matrix-matrix and matrix-vector multiplications. For all layers we will simply use a collective variable \boldsymbol{\Theta}.

It consist of two basic steps:

1. a feed forward stage which takes a given input and produces a final output which is compared with the target values through our cost/loss function.

2. a back-propagation state where the unknown parameters \boldsymbol{\Theta} are updated through the optimization of the their gradients. The expressions for the gradients are obtained via the chain rule, starting from the derivative of the cost/function.

These two steps make up one iteration. This iterative process is continued till we reach an eventual stopping criterion.

Overarching view of a neural network

The architecture of a neural network defines our model. This model aims at describing some function f(\boldsymbol{x} which represents some final result (outputs or tagrget values) given a specific inpput \boldsymbol{x}. Note that here \boldsymbol{y} and \boldsymbol{x} are not limited to be vectors.

The architecture consists of

1. An input and an output layer where the input layer is defined by the inputs \boldsymbol{x}. The output layer produces the model ouput \boldsymbol{\tilde{y}} which is compared with the target value \boldsymbol{y}

2. A given number of hidden layers and neurons/nodes/units for each layer (this may vary)

3. A given activation function \sigma(\boldsymbol{z}) with arguments \boldsymbol{z} to be defined below. The activation functions may differ from layer to layer.

4. The last layer, normally called output layer has normally an activation function tailored to the specific problem

5. Finally we define a so-called cost or loss function which is used to gauge the quality of our model.

The optimization problem

The cost function is a function of the unknown parameters \boldsymbol{\Theta} where the latter is a container for all possible parameters needed to define a neural network

If we are dealing with a regression task a typical cost/loss function is the mean squared error

C(\boldsymbol{\Theta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\right)\right\}.

This function represents one of many possible ways to define the so-called cost function. Note that here we have assumed a linear dependence in terms of the paramters \boldsymbol{\Theta}. This is in general not the case.

Parameters of neural networks

For neural networks the parameters \boldsymbol{\Theta} are given by the so-called weights and biases (to be defined below).

The weights are given by matrix elements w_{ij}^{(l)} where the superscript indicates the layer number. The biases are typically given by vector elements representing each single node of a given layer, that is b_j^{(l)}.

Other ingredients of a neural network

Having defined the architecture of a neural network, the optimization of the cost function with respect to the parameters \boldsymbol{\Theta}, involves the calculations of gradients and their optimization. The gradients represent the derivatives of a multidimensional object and are often approximated by various gradient methods, including

1. various quasi-Newton methods,

2. plain gradient descent (GD) with a constant learning rate \eta,

3. GD with momentum and other approximations to the learning rates such as

• Adapative gradient (ADAgrad)

• Root mean-square propagation (RMSprop)

• Adaptive gradient with momentum (ADAM) and many other

4. Stochastic gradient descent and various families of learning rate approximations

Other parameters

In addition to the above, there are often additional hyperparamaters which are included in the setup of a neural network. These will be discussed below.

Universal approximation theorem

The universal approximation theorem plays a central role in deep learning. Cybenko (1989) showed the following:

Let \sigma be any continuous sigmoidal function such that

\sigma(z) = \left\{\begin{array}{cc} 1 & z\rightarrow \infty\\ 0 & z \rightarrow -\infty \end{array}\right.

Given a continuous and deterministic function F(\boldsymbol{x}) on the unit cube in $d$-dimensions F\in [0,1]^d, x\in [0,1]^d and a parameter \epsilon >0, there is a one-layer (hidden) neural network f(\boldsymbol{x};\boldsymbol{\Theta}) with \boldsymbol{\Theta}=(\boldsymbol{W},\boldsymbol{b}) and $\boldsymbol{W}\in \mathbb{R}^{m\times n}$ and \boldsymbol{b}\in \mathbb{R}^{n}, for which

\vert F(\boldsymbol{x})-f(\boldsymbol{x};\boldsymbol{\Theta})\vert < \epsilon \hspace{0.1cm} \forall \boldsymbol{x}\in[0,1]^d.

Some parallels from real analysis

For those of you familiar with for example the Stone-Weierstrass theorem for polynomial approximations or the convergence criterion for Fourier series, there are similarities in the derivation of the proof for neural networks.

The approximation theorem in words

Any continuous function y=F(\boldsymbol{x}) supported on the unit cube in $d$-dimensions can be approximated by a one-layer sigmoidal network to arbitrary accuracy.

Hornik (1991) extended the theorem by letting any non-constant, bounded activation function to be included using that the expectation value

\mathbb{E}[\vert F(\boldsymbol{x})\vert^2] =\int_{\boldsymbol{x}\in D} \vert F(\boldsymbol{x})\vert^2p(\boldsymbol{x})d\boldsymbol{x} < \infty.

Then we have

\mathbb{E}[\vert F(\boldsymbol{x})-f(\boldsymbol{x};\boldsymbol{\Theta})\vert^2] =\int_{\boldsymbol{x}\in D} \vert F(\boldsymbol{x})-f(\boldsymbol{x};\boldsymbol{\Theta})\vert^2p(\boldsymbol{x})d\boldsymbol{x} < \epsilon.

More on the general approximation theorem

None of the proofs give any insight into the relation between the number of of hidden layers and nodes and the approximation error \epsilon, nor the magnitudes of \boldsymbol{W} and \boldsymbol{b}.

Neural networks (NNs) have what we may call a kind of universality no matter what function we want to compute.

It does not mean that an NN can be used to exactly compute any function. Rather, we get an approximation that is as good as we want.

Class of functions we can approximate

The class of functions that can be approximated are the continuous ones. If the function F(\boldsymbol{x}) is discontinuous, it won't in general be possible to approximate it. However, an NN may still give an approximation even if we fail in some points.

Setting up the equations for a neural network

The questions we want to ask are how do changes in the biases and the weights in our network change the cost function and how can we use the final output to modify the weights and biases?

To derive these equations let us start with a plain regression problem and define our cost function as

{\cal C}(\boldsymbol{\Theta})  =  \frac{1}{2}\sum_{i=1}^n\left(y_i - \tilde{y}_i\right)^2,

where the $y_i$s are our n targets (the values we want to reproduce), while the outputs of the network after having propagated all inputs \boldsymbol{x} are given by \boldsymbol{\tilde{y}}_i.

Layout of a neural network with three hidden layers

Figure 1:

Definitions

With our definition of the targets \boldsymbol{y}, the outputs of the network \boldsymbol{\tilde{y}} and the inputs \boldsymbol{x} we define now the activation z_j^l of node/neuron/unit j of the $l$-th layer as a function of the bias, the weights which add up from the previous layer l-1 and the forward passes/outputs \hat{a}^{l-1} from the previous layer as

z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l,

where b_k^l are the biases from layer l. Here M_{l-1} represents the total number of nodes/neurons/units of layer l-1. The figure in the whiteboard notes illustrates this equation. We can rewrite this in a more compact form as the matrix-vector products we discussed earlier,

\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l.

Inputs to the activation function

With the activation values \boldsymbol{z}^l we can in turn define the output of layer l as \boldsymbol{a}^l = f(\boldsymbol{z}^l) where f is our activation function. In the examples here we will use the sigmoid function discussed in our logistic regression lectures. We will also use the same activation function f for all layers and their nodes. It means we have

a_j^l = \sigma(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}.

Derivatives and the chain rule

From the definition of the activation z_j^l we have

\frac{\partial z_j^l}{\partial w_{ij}^l} = a_i^{l-1},

and

\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l.

With our definition of the activation function we have that (note that this function depends only on z_j^l)

\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=\sigma(z_j^l)(1-\sigma(z_j^l)).

Derivative of the cost function

With these definitions we can now compute the derivative of the cost function in terms of the weights.

Let us specialize to the output layer l=L. Our cost function is

{\cal C}(\boldsymbol{\Theta}^L)  =  \frac{1}{2}\sum_{i=1}^n\left(y_i - \tilde{y}_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - y_i\right)^2,

The derivative of this function with respect to the weights is

\frac{\partial{\cal C}(\boldsymbol{\Theta}^L)}{\partial w_{jk}^L}  =  \left(a_j^L - y_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}},

The last partial derivative can easily be computed and reads (by applying the chain rule)

\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1}.

Simpler examples first, and automatic differentiation

In order to understand the back propagation algorithm and its derivation (an implementation of the chain rule), let us first digress with some simple examples. These examples are also meant to motivate the link with back propagation and automatic differentiation.

Reminder on the chain rule and gradients

If we have a multivariate function f(x,y) where x=x(t) and y=y(t) are functions of a variable t, we have that the gradient of f with respect to t (without the explicit unit vector components)

\frac{df}{dt} = \begin{bmatrix}\frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \end{bmatrix} \begin{bmatrix}\frac{\partial x}{\partial t} \\ \frac{\partial y}{\partial t} \end{bmatrix}=\frac{\partial f}{\partial x} \frac{\partial x}{\partial t} +\frac{\partial f}{\partial y} \frac{\partial y}{\partial t}.

Multivariable functions

If we have a multivariate function f(x,y) where x=x(t,s) and y=y(t,s) are functions of the variables t and s, we have that the partial derivatives

\frac{\partial f}{\partial s}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial s},

and

\frac{\partial f}{\partial t}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial t}.

the gradient of f with respect to t and s (without the explicit unit vector components)

\frac{df}{d(s,t)} = \begin{bmatrix}\frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \end{bmatrix} \begin{bmatrix}\frac{\partial x}{\partial s}  &\frac{\partial x}{\partial t} \\ \frac{\partial y}{\partial s} & \frac{\partial y}{\partial t} \end{bmatrix}.

Automatic differentiation through examples

A great introduction to automatic differentiation is given by Baydin et al., see https://arxiv.org/abs/1502.05767.

Automatic differentiation is a represented by a repeated application of the chain rule on well-known functions and allows for the calculation of derivatives to numerical precision. It is not the same as the calculation of symbolic derivatives via for example SymPy, nor does it use approximative formulae based on Taylor-expansions of a function around a given value. The latter are error prone due to truncation errors and values of the step size \Delta.

Simple example

Our first example is rather simple,

f(x) =\exp{x^2},

with derivative

f'(x) =2x\exp{x^2}.

We can use SymPy to extract the pertinent lines of Python code through the following simple example

In [4]:

from __future__ import division
from sympy import *
x = symbols('x')
expr = exp(x*x)
simplify(expr)
derivative = diff(expr,x)
print(python(expr))
print(python(derivative))

Smarter way of evaluating the above function

If we study this function, we note that we can reduce the number of operations by introducing an intermediate variable

a = x^2,

leading to

f(x) = f(a(x)) = b= \exp{a}.

We now assume that all operations can be counted in terms of equal floating point operations. This means that in order to calculate f(x) we need first to square x and then compute the exponential. We have thus two floating point operations only.

Reducing the number of operations

With the introduction of a precalculated quantity a and thereby f(x) we have that the derivative can be written as

f'(x) = 2xb,

which reduces the number of operations from four in the orginal expression to two. This means that if we need to compute f(x) and its derivative (a common task in optimizations), we have reduced the number of operations from six to four in total.

Note that the usage of a symbolic software like SymPy does not include such simplifications and the calculations of the function and the derivatives yield in general more floating point operations.

Chain rule, forward and reverse modes

In the above example we have introduced the variables a and b, and our function is

f(x) = f(a(x)) = b= \exp{a},

with a=x^2. We can decompose the derivative of f with respect to x as

\frac{df}{dx}=\frac{df}{db}\frac{db}{da}\frac{da}{dx}.

We note that since b=f(x) that

Warning:

Output truncated. This notebook contains too many cells to display efficiently.