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a/doc/src/week41/week3.do.txt b/doc/src/week41/week3.do.txt new file mode 100644 index 000000000..9480a011f --- /dev/null +++ b/doc/src/week41/week3.do.txt @@ -0,0 +1,896 @@ +TITLE: January 29-February 2 : Advanced machine learning and data analysis for the physical sciences +AUTHOR: Morten Hjorth-Jensen {copyright, 1999-present|CC BY-NC} at Department of Physics and Center for Computing in Science Education, University of Oslo, Norway & Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University, East Lansing, Michigan, USA +DATE: January 30 + + + +!split +===== Overview of third week ===== + +!bblock + o Discussion of possible projects + o Review of neural networks and automatic differentiation + o Discussion of codes + o "Video of lecture":"https://youtu.be/OUTFo0oJadU" + o "Link to material for project suggestions":"https://github.com/CompPhysics/AdvancedMachineLearning/tree/main/doc/Projects/2024/ProjectProposals" + +!eblock + + +!split +===== Mathematics of deep learning ===== + +!bblock Two recent books online +o "The Modern Mathematics of Deep Learning, by Julius Berner, Philipp Grohs, Gitta Kutyniok, Philipp Petersen":"https://arxiv.org/abs/2105.04026", published as "Mathematical Aspects of Deep Learning, pp. 1-111. Cambridge University Press, 2022":"https://doi.org/10.1017/9781009025096.002" + +o "Mathematical Introduction to Deep Learning: Methods, Implementations, and Theory, Arnulf Jentzen, Benno Kuckuck, Philippe von Wurstemberger":"https://doi.org/10.48550/arXiv.2310.20360" +!eblock + + +!split +===== Reminder on books with hands-on material and codes ===== +!bblock +* "Sebastian Rashcka et al, Machine learning with Sickit-Learn and PyTorch":"https://sebastianraschka.com/blog/2022/ml-pytorch-book.html" +* "David Foster, Generative Deep Learning with TensorFlow":"https://www.oreilly.com/library/view/generative-deep-learning/9781098134174/ch01.html" +* "Bali and Gavras, Generative AI with Python and TensorFlow 2":"https://github.com/PacktPublishing/Hands-On-Generative-AI-with-Python-and-TensorFlow-2" +!eblock + +All three books have GitHub addresses from where one can download all codes. We will borrow most of the material from these three texts as well as +from Goodfellow, Bengio and Courville's text "Deep Learning":"https://www.deeplearningbook.org/" + + +!split +===== Reading recommendations ===== + +o Rashkca et al., chapter 11, jupyter-notebook sent separately, from "GitHub":"https://github.com/rasbt/machine-learning-book" +o Goodfellow et al, chapter 6 and 7 contain most of the neural network background. + +!split +===== 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. + +!split +===== 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 +$\bm{\Theta}^{(l)}=(\bm{W}^{(l)},\bm{b}^{(l)})$ 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 $\bm{\Theta}$. + +It consist of two basic steps: +o 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. +o a back-propagation state where the unknown parameters $\bm{\Theta}$ 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. + + +!split +===== Overarching view of a neural network ===== + +The architecture of a neural network defines our model. This model +aims at describing some function $f(\bm{x}$ which represents +some final result (outputs or tagrget values) given a specific inpput +$\bm{x}$. Note that here $\bm{y}$ and $\bm{x}$ are not limited to be +vectors. + +The architecture consists of +o An input and an output layer where the input layer is defined by the inputs $\bm{x}$. The output layer produces the model ouput $\bm{\tilde{y}}$ which is compared with the target value $\bm{y}$ +o A given number of hidden layers and neurons/nodes/units for each layer (this may vary) +o A given activation function $\sigma(\bm{z})$ with arguments $\bm{z}$ to be defined below. The activation functions may differ from layer to layer. +o The last layer, normally called _output_ layer has normally an activation function tailored to the specific problem +o Finally we define a so-called cost or loss function which is used to gauge the quality of our model. + + +!split +===== The optimization problem ===== + +The cost function is a function of the unknown parameters +$\bm{\Theta}$ 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 +!bt +\[ +C(\bm{\Theta})=\frac{1}{n}\left\{\left(\bm{y}-\bm{X}\bm{\theta}\right)^T\left(\bm{y}-\bm{X}\bm{\theta}\right)\right\}. +\] +!et +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 $\bm{\Theta}$. This is in general not the case. + + +!split +===== Parameters of neural networks ===== +For neural networks the parameters +$\bm{\Theta}$ 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 $b_j^{(l)}$. + +!split +===== 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 $\bm{\Theta}$, +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 +o various quasi-Newton methods, +o plain gradient descent (GD) with a constant learning rate $\eta$, +o 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 +o Stochastic gradient descent and various families of learning rate approximations + +!split +===== 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. + + +!split +===== Universal approximation theorem ===== + +The universal approximation theorem plays a central role in deep +learning. "Cybenko (1989)":"https://link.springer.com/article/10.1007/BF02551274" showed +the following: + +!bblock +Let $\sigma$ be any continuous sigmoidal function such that +!bt +\[ +\sigma(z) = \left\{\begin{array}{cc} 1 & z\rightarrow \infty\\ 0 & z \rightarrow -\infty \end{array}\right. +\] +!et +Given a continuous and deterministic function $F(\bm{x})$ 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 +$f(\bm{x};\bm{\Theta})$ with $\bm{\Theta}=(\bm{W},\bm{b})$ and $\bm{W}\in +\mathbb{R}^{m\times n}$ and $\bm{b}\in \mathbb{R}^{n}$, for which +!bt +\[ +\vert F(\bm{x})-f(\bm{x};\bm{\Theta})\vert < \epsilon \hspace{0.1cm} \forall \bm{x}\in[0,1]^d. +\] +!et + +!eblock + +!split +===== Some parallels from real analysis ===== + +For those of you familiar with for example the "Stone-Weierstrass +theorem":"https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem" +for polynomial approximations or the convergence criterion for Fourier +series, there are similarities in the derivation of the proof for +neural networks. + +!split +===== The approximation theorem in words ===== + +_Any continuous function $y=F(\bm{x})$ supported on the unit cube in +$d$-dimensions can be approximated by a one-layer sigmoidal network to +arbitrary accuracy._ + +"Hornik (1991)":"https://www.sciencedirect.com/science/article/abs/pii/089360809190009T" extended the theorem by letting any non-constant, bounded activation function to be included using that the expectation value +!bt +\[ +\mathbb{E}[\vert F(\bm{x})\vert^2] =\int_{\bm{x}\in D} \vert F(\bm{x})\vert^2p(\bm{x})d\bm{x} < \infty. +\] +!et +Then we have +!bt +\[ +\mathbb{E}[\vert F(\bm{x})-f(\bm{x};\bm{\Theta})\vert^2] =\int_{\bm{x}\in D} \vert F(\bm{x})-f(\bm{x};\bm{\Theta})\vert^2p(\bm{x})d\bm{x} < \epsilon. +\] +!et + + +!split +===== 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 $\bm{W}$ and $\bm{b}$. + +Neural networks (NNs) have what we may call a kind of universality no matter what function we want to compute. + +!bblock +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. +!eblock + +!split +===== Class of functions we can approximate ===== + +!bblock +The class of functions that can be approximated are the continuous ones. +If the function $F(\bm{x})$ 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. +!eblock + + + +!split +===== 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 + +!bt +\[ +{\cal C}(\bm{\Theta}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - \tilde{y}_i\right)^2, +\] +!et + +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 $\bm{x}$ are given by $\bm{\tilde{y}}_i$. + + +!split +===== Layout of a neural network with three hidden layers ===== + +FIGURE: [figures/nn1.pdf, width=900 frac=1.0] + +!split +===== Definitions ===== + +With our definition of the targets $\bm{y}$, the outputs of the +network $\bm{\tilde{y}}$ and the inputs $\bm{x}$ 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 + + +!bt +\[ +z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l, +\] +!et + +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, + +!bt +\[ +\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l. +\] +!et + +!split +===== Inputs to the activation function ===== + +With the activation values $\bm{z}^l$ we can in turn define the +output of layer $l$ as $\bm{a}^l = f(\bm{z}^l)$ 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 + +!bt +\[ +a_j^l = \sigma(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}. +\] +!et + + + + +!split +===== Derivatives and the chain rule ===== + +From the definition of the activation $z_j^l$ we have +!bt +\[ +\frac{\partial z_j^l}{\partial w_{ij}^l} = a_i^{l-1}, +\] +!et +and +!bt +\[ +\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l. +\] +!et + +With our definition of the activation function we have that (note that this function depends only on $z_j^l$) +!bt +\[ +\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=\sigma(z_j^l)(1-\sigma(z_j^l)). +\] +!et + + +!split +===== 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 +!bt +\[ +{\cal C}(\bm{\Theta}^L) = \frac{1}{2}\sum_{i=1}^n\left(y_i - \tilde{y}_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - y_i\right)^2, +\] +!et +The derivative of this function with respect to the weights is + +!bt +\[ +\frac{\partial{\cal C}(\bm{\Theta}^L)}{\partial w_{jk}^L} = \left(a_j^L - y_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, +\] +!et +The last partial derivative can easily be computed and reads (by applying the chain rule) +!bt +\[ +\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1}. +\] +!et + + + +!split +===== 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":"https://en.wikipedia.org/wiki/Automatic_differentiation". + +!split +===== 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) +!bt +\[ +\frac{df}{dt} = \begin{bmatrix}\frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \end{bmatrix} \begin{bmatrix}\frac{\partial x}{\partial t} \\ \frac{\partial y}{\partial t} \end{bmatrix}=\frac{\partial f}{\partial x} \frac{\partial x}{\partial t} +\frac{\partial f}{\partial y} \frac{\partial y}{\partial t}. +\] +!et + + +!split +===== 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 +!bt +\[ +\frac{\partial f}{\partial s}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial s}, +\] +!et +and +!bt +\[ +\frac{\partial f}{\partial t}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial t}. +\] +!et + +the gradient of $f$ with respect to $t$ and $s$ (without the explicit unit vector components) +!bt +\[ +\frac{df}{d(s,t)} = \begin{bmatrix}\frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \end{bmatrix} \begin{bmatrix}\frac{\partial x}{\partial s} &\frac{\partial x}{\partial t} \\ \frac{\partial y}{\partial s} & \frac{\partial y}{\partial t} \end{bmatrix}. +\] +!et + + + +!split +===== Automatic differentiation through examples ===== + +A great introduction to automatic differentiation is given by Baydin et al., see URL:"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 $\Delta$. + +!split +===== Simple example ===== + +Our first example is rather simple, +!bt +\[ +f(x) =\exp{x^2}, +\] +!et +with derivative +!bt +\[ +f'(x) =2x\exp{x^2}. +\] +!et +We can use SymPy to extract the pertinent lines of Python code through the following simple example +!bc pycod +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)) +!ec + +!split +===== 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 +!bt +\[ +a = x^2, +\] +!et +leading to +!bt +\[ +f(x) = f(a(x)) = b= \exp{a}. +\] +!et + +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. + +!split +===== 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 + +!bt +\[ +f'(x) = 2xb, +\] +!et + +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. + +!split +===== Chain rule, forward and reverse modes ===== + +In the above example we have introduced the variables $a$ and $b$, and our function is +!bt +\[ +f(x) = f(a(x)) = b= \exp{a}, +\] +!et +with $a=x^2$. We can decompose the derivative of $f$ with respect to $x$ as +!bt +\[ +\frac{df}{dx}=\frac{df}{db}\frac{db}{da}\frac{da}{dx}. +\] +!et + +We note that since $b=f(x)$ that +!bt +\[ +\frac{df}{db}=1, +\] +!et +leading to +!bt +\[ +\frac{df}{dx}=\frac{db}{da}\frac{da}{dx}=2x\exp{x^2}, +\] +!et +as before. + + +!split +===== Forward and reverse modes ===== + +We have that +!bt +\[ +\frac{df}{dx}=\frac{df}{db}\frac{db}{da}\frac{da}{dx}, +\] +!et +which we can rewrite either as +!bt +\[ +\frac{df}{dx}=\left[\frac{df}{db}\frac{db}{da}\right]\frac{da}{dx}, +\] +!et +or +!bt +\[ +\frac{df}{dx}=\frac{df}{db}\left[\frac{db}{da}\frac{da}{dx}\right]. +\] +!et + +The first expression is called reverse mode (or back propagation) +since we start by evaluating the derivatives at the end point and then +propagate backwards. This is the standard way of evaluating +derivatives (gradients) when optimizing the parameters of a neural +network. In the context of deep learning this is computationally +more efficient since the output of a neural network consists of either +one or some few other output variables. + +The second equation defines the so-called _forward mode_. + + +!split +===== More complicated function ===== + +We increase our ambitions and introduce a slightly more complicated function +!bt +\[ +f(x) =\sqrt{x^2+exp{x^2}}, +\] +!et +with derivative +!bt +\[ +f'(x) =\frac{x(1+\exp{x^2})}{\sqrt{x^2+exp{x^2}}}. +\] +!et +The corresponding SymPy code reads +!bc pycod +from __future__ import division +from sympy import * +x = symbols('x') +expr = sqrt(x*x+exp(x*x)) +simplify(expr) +derivative = diff(expr,x) +print(python(expr)) +print(python(derivative)) +!ec + +!split +===== Counting the number of floating point operations ===== + +A simple count of operations shows that we need five operations for +the function itself and ten for the derivative. Fifteen operations in total if we wish to proceed with the above codes. + +Can we reduce this to +say half the number of operations? + +!split +===== Defining intermediate operations ===== + +We can indeed reduce the number of operation to half of those listed in the brute force approach above. +We define the following quantities +!bt +\[ +a = x^2, +\] +!et +and +!bt +\[ +b = \exp{x^2} = \exp{a}, +\] +!et +and +!bt +\[ +c= a+b, +\] +!et +and +!bt +\[ +d=f(x)=\sqrt{c}. +\] +!et + +!split +===== New expression for the derivative ===== + +With these definitions we obtain the following partial derivatives +!bt +\[ +\frac{\partial a}{\partial x} = 2x, +\] +!et +and +!bt +\[ +\frac{\partial b}{\partial a} = \exp{a}, +\] +!et +and +!bt +\[ +\frac{\partial c}{\partial a} = 1, +\] +!et +and +!bt +\[ +\frac{\partial c}{\partial b} = 1, +\] +!et +and +!bt +\[ +\frac{\partial d}{\partial c} = \frac{1}{2\sqrt{c}}, +\] +!et +and finally +!bt +\[ +\frac{\partial f}{\partial d} = 1. +\] +!et + +!split +===== Final derivatives ===== +Our final derivatives are thus +!bt +\[ +\frac{\partial f}{\partial c} = \frac{\partial f}{\partial d} \frac{\partial d}{\partial c} = \frac{1}{2\sqrt{c}}, +\] +!et +!bt +\[ +\frac{\partial f}{\partial b} = \frac{\partial f}{\partial c} \frac{\partial c}{\partial b} = \frac{1}{2\sqrt{c}}, +\] +!et +!bt +\[ +\frac{\partial f}{\partial a} = \frac{\partial f}{\partial c} \frac{\partial c}{\partial a}+ +\frac{\partial f}{\partial b} \frac{\partial b}{\partial a} = \frac{1+\exp{a}}{2\sqrt{c}}, +\] +!et +and finally +!bt +\[ +\frac{\partial f}{\partial x} = \frac{\partial f}{\partial a} \frac{\partial a}{\partial x} = \frac{x(1+\exp{a})}{\sqrt{c}}, +\] +!et +which is just +!bt +\[ +\frac{\partial f}{\partial x} = \frac{x(1+b)}{d}, +\] +!et +and requires only three operations if we can reuse all intermediate variables. + + +!split +===== In general not this simple ===== + +In general, see the generalization below, unless we can obtain simple +analytical expressions which we can simplify further, the final +implementation of automatic differentiation involves repeated +calculations (and thereby operations) of derivatives of elementary +functions. + +!split +===== Automatic differentiation ===== + +We can make this example more formal. Automatic differentiation is a +formalization of the previous example (see graph). + +We define $\bm{x}\in x_1,\dots, x_l$ input variables to a given function $f(\bm{x})$ and $x_{l+1},\dots, x_L$ intermediate variables. + +In the above example we have only one input variable, $l=1$ and four intermediate variables, that is +!bt +\[ +\begin{bmatrix} x_1=x & x_2 = x^2=a & x_3 =\exp{a}= b & x_4=c=a+b & x_5 = \sqrt{c}=d \end{bmatrix}. +\] +!et + +Furthemore, for $i=l+1, \dots, L$ (here $i=2,3,4,5$ and $f=x_L=d$), we +define the elementary functions $g_i(x_{Pa(x_i)})$ where $x_{Pa(x_i)}$ are the parent nodes of the variable $x_i$. + +In our case, we have for example for $x_3=g_3(x_{Pa(x_i)})=\exp{a}$, that $g_3=\exp{()}$ and $x_{Pa(x_3)}=a$. + +!split +===== Chain rule ===== + +We can now compute the gradients by back-propagating the derivatives using the chain rule. +We have defined +!bt +\[ +\frac{\partial f}{\partial x_L} = 1, +\] +!et +which allows us to find the derivatives of the various variables $x_i$ as +!bt +\[ +\frac{\partial f}{\partial x_i} = \sum_{x_j:x_i\in Pa(x_j)}\frac{\partial f}{\partial x_j} \frac{\partial x_j}{\partial x_i}=\sum_{x_j:x_i\in Pa(x_j)}\frac{\partial f}{\partial x_j} \frac{\partial g_j}{\partial x_i}. +\] +!et + +Whenever we have a function which can be expressed as a computation +graph and the various functions can be expressed in terms of +elementary functions that are differentiable, then automatic +differentiation works. The functions may not need to be elementary +functions, they could also be computer programs, although not all +programs can be automatically differentiated. + +!split +===== First network example, simple percepetron with one input ===== + +As yet another example we define now a simple perceptron model with +all quantities given by scalars. We consider only one input variable +$x$ and one target value $y$. We define an activation function +$\sigma_1$ which takes as input + +!bt +\[ +z_1 = w_1x+b_1, +\] +!et +where $w_1$ is the weight and $b_1$ is the bias. These are the +parameters we want to optimize. The output is $a_1=\sigma(z_1)$ (see +graph from whiteboard notes). This output is then fed into the +_cost/loss_ function, which we here for the sake of simplicity just +define as the squared error + +!bt +\[ +C(x;w_1,b_1)=\frac{1}{2}(a_1-y)^2. +\] +!et + +!split +===== Layout of a simple neural network with no hidden layer ===== + +FIGURE: [figures/simplenn1.png, width=900 frac=1.0] + + + +!split +===== Optimizing the parameters ===== + +In setting up the feed forward and back propagation parts of the +algorithm, we need now the derivative of the various variables we want +to train. + +We need +!bt +\[ +\frac{\partial C}{\partial w_1} \hspace{0.1cm}\mathrm{and}\hspace{0.1cm}\frac{\partial C}{\partial b_1}. +\] +!et + +Using the chain rule we find +!bt +\[ +\frac{\partial C}{\partial w_1}=\frac{\partial C}{\partial a_1}\frac{\partial a_1}{\partial z_1}\frac{\partial z_1}{\partial w_1}=(a_1-y)\sigma_1'x, +\] +!et +and +!bt +\[ +\frac{\partial C}{\partial b_1}=\frac{\partial C}{\partial a_1}\frac{\partial a_1}{\partial z_1}\frac{\partial z_1}{\partial b_1}=(a_1-y)\sigma_1', +\] +!et +which we later will just define as +!bt +\[ +\frac{\partial C}{\partial a_1}\frac{\partial a_1}{\partial z_1}=\delta_1. +\] +!et + + +!split +===== Adding a hidden layer ===== + +We change our simple model to (see graph) +a network with just one hidden layer but with scalar variables only. + +Our output variable changes to $a_2$ and $a_1$ is now the output from the hidden node and $a_0=x$. +We have then +!bt +\[ +z_1 = w_1a_0+b_1 \hspace{0.1cm} \wedge a_1 = \sigma_1(z_1), +\] +!et +!bt +\[ +z_2 = w_2a_1+b_2 \hspace{0.1cm} \wedge a_2 = \sigma_2(z_2), +\] +!et +and the cost function +!bt +\[ +C(x;\bm{\Theta})=\frac{1}{2}(a_2-y)^2, +\] +!et +with $\bm{\Theta}=[w_1,w_2,b_1,b_2]$. + + +!split +===== Layout of a simple neural network with one hidden layer ===== + +FIGURE: [figures/simplenn2.png, width=900 frac=1.0] + + + +!split +===== The derivatives ===== + +The derivatives are now, using the chain rule again + +!bt +\[ +\frac{\partial C}{\partial w_2}=\frac{\partial C}{\partial a_2}\frac{\partial a_2}{\partial z_2}\frac{\partial z_2}{\partial w_2}=(a_2-y)\sigma_2'a_1=\delta_2a_1, +\] +!et +!bt +\[ +\frac{\partial C}{\partial b_2}=\frac{\partial C}{\partial a_2}\frac{\partial a_2}{\partial z_2}\frac{\partial z_2}{\partial b_2}=(a_2-y)\sigma_2'=\delta_2, +\] +!et +!bt +\[ +\frac{\partial C}{\partial w_1}=\frac{\partial C}{\partial a_2}\frac{\partial a_2}{\partial z_2}\frac{\partial z_2}{\partial a_1}\frac{\partial a_1}{\partial z_1}\frac{\partial z_1}{\partial w_1}=(a_2-y)\sigma_2'a_1\sigma_1'a_0, +\] +!et +!bt +\[ +\frac{\partial C}{\partial b_1}=\frac{\partial C}{\partial a_2}\frac{\partial a_2}{\partial z_2}\frac{\partial z_2}{\partial a_1}\frac{\partial a_1}{\partial z_1}\frac{\partial z_1}{\partial b_1}=(a_2-y)\sigma_2'\sigma_1'=\delta_1. +\] +!et + +Can you generalize this to more than one hidden layer? + + +!split +===== Important observations ===== + +!bblock +From the above equations we see that the derivatives of the activation +functions play a central role. If they vanish, the training may +stop. This is called the vanishing gradient problem, see discussions below. If they become +large, the parameters $w_i$ and $b_i$ may simply go to infinity. This +is referenced as the exploding gradient problem. +!eblock + + + + +!split +===== The training ===== + +The training of the parameters is done through various gradient descent approximations with + +!bt +\[ +w_{i}\leftarrow w_{i}- \eta \delta_i a_{i-1}, +\] +!et +and +!bt +\[ +b_i \leftarrow b_i-\eta \delta_i, +\] +!et +with $\eta$ is the learning rate. + +One iteration consists of one feed forward step and one back-propagation step. Each back-propagation step does one update of the parameters $\bm{\Theta}$. + +For the first hidden layer $a_{i-1}=a_0=x$ for this simple model. + +