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b/exercises/exercise1.tex @@ -1,11 +1,13 @@ \documentclass{article} \usepackage{graphicx} % Required for inserting images \usepackage{tikz} +\usetikzlibrary{positioning} \usepackage{amsmath} \usepackage{amssymb} \title{Very Deep Learning \\ Exercise 1} -\author{Niklas Eberts - 409829 } +\author{Niklas Eberts - 409829 \\ + Frederick Phillips - 404986} \date{\today} \begin{document} @@ -205,6 +207,56 @@ $d = w + c$ \subsection{Perform backpropagation on the same two computational graphs you chose in (iii).} +\subsubsection{Expression A} +\begin{center} + \begin{tikzpicture}[main/.style = {draw, circle}] + \node[main] (x) {$x$}; + \node[main] (y) [below of=x] {$y$}; + \node[main] (z) [below of=y] {$z$}; + \node[main] (w) [below of=z] {$w$}; + \node[main] (a) [below right of=y, above right of=z] {$a$}; + \node[main] (b) [above right of=a] {$b$}; + \node[main] (c) [below right of=b] {$c$}; + + \draw[<-] (y) -- node[midway, above right, sloped, pos=0.3] {-1} (a); + \draw[<-] (z) -- node[midway, above, sloped, pos=0.3] {5} (a); + \draw[<-] (a) -- node[midway, above, sloped, pos=0.4] {1} (b); + \draw[<-] (x) -- node[midway, above right, sloped, pos=0.3] {1} (b); + \draw[<-] (b) -- node[midway, above, sloped, pos=0.4] {1} (c); + \draw[<-] (w) -- node[midway, below, sloped, pos=0.5] {-2} (c); + \draw[->] (x) --++(0:-1cm) node[midway, above]{1}; + \draw[->] (y) --++(0:-1cm) node[midway, above]{-1}; + \draw[->] (z) --++(0:-1cm) node[midway, above]{5}; + \draw[->] (w) --++(0:-1cm) node[midway, above]{-2}; + \end{tikzpicture} +\end{center} + +\subsubsection{Expression C} +\begin{center} + \begin{tikzpicture}[main/.style = {draw, circle}, node distance=2cm] + \node[main] (x) {$x$}; + \node[main] (y) [below of=x] {$y$}; + \node[main] (z) [below of=y] {$z$}; + \node[main] (w) [below of=z] {$w$}; + \node[main] (a) [below right of=x, above right of=y] {$a$}; + \node[main] (b) [below right of=a] {$b$}; + \node[main] (c) [above right of=b] {$c$}; + \node[main] (d) [right of=w, below right of=c] {$d$}; + + \draw[<-] (x) -- node[midway, above right, sloped, pos=0.3] {-2.718} (a); + \draw[<-] (y) -- node[midway, above, sloped, pos=0.3] {-2.718} (a); + \draw[<-] (a) -- node[midway, above, sloped, pos=0.5] {-2.718} (b); + \draw[<-] (z) -- node[midway, above right, sloped, pos=0.2] {21.744} (b); + \draw[<-] (b) -- node[midway, above, sloped, pos=0.5] {2.718} (c); + \draw[<-] (w) -- node[midway, above, sloped, pos=0.5] {1} (d); + \draw[<-] (c) -- node[midway, above, sloped, pos=0.5] {1} (d); + \draw[->] (x) --++(0:-2cm) node[midway, above]{-2.718}; + \draw[->] (y) --++(0:-2cm) node[midway, above]{-2.718}; + \draw[->] (z) --++(0:-2cm) node[midway, above]{21.744}; + \draw[->] (w) --++(0:-2cm) node[midway, above]{1}; + \end{tikzpicture} +\end{center} + \section{Vanishing and Exploding Gradients} Consider a network with input $x \in \mathbb{R}$, 4 hidden layers, each having only one node, and one output $y \in \mathbb{R}$: \begin{center} @@ -230,12 +282,48 @@ The sigmoid function is given by \end{equation*} \subsection{What is the vanishing gradients problem?} +When gradients become extremely small during the backpropogation process, the partial derivations guiding the parameter updates tend to approach zero. As a result, the weights of the neural network remain largely unchanged, leading to a lack of meaningful learning or weight updates. This phenomenon is often referred to as the \textbf{vanishing gradient problem} and can hinder the training of deep neural networks. \subsection{Calculate the derivative $\sigma'$ of the sigmoid function. Determine the best possible lower and upper bounds for the derivative $\sigma'$.} +\begin{equation*} + \begin{split} + \sigma (x) &= \frac{1}{1+e^{-x}} \\ + \sigma' (x) &= \frac{d}{dx} \frac{1}{1+e^{-x}} \\ + &= \frac{d}{dx} (1+e^{-x})^{-1} \\ + &= -(1+e^{-x})^{-2} \cdot \frac{d}{dx} (1+e^{-x}) \\ + &= -(1+e^{-x})^{-2} \cdot (\frac{d}{dx}(1) + \frac{d}{dx}(e^{-x})) \\ + &= -(1+e^{-x})^{-2} \cdot (e^{-x} \cdot \frac{d}{dx}(-x)) \\ + &= -(1+e^{-x})^{-2} \cdot (e^{-x} \cdot (-1)) \\ + &= (1+e^{-x})^{-2} \cdot (e^{-x}) \\ + &= \frac{e^{-x}}{(1+e^{-x})^2} \\ + &= \frac{1}{(1+e^{-x})} \cdot \frac{e^{-x}}{(1+e^{-x})} \\ + &= \frac{1}{(1+e^{-x})} \cdot \frac{e^{-x}+1-1}{(1+e^{-x})} \\ + &= \frac{1}{(1+e^{-x})} \cdot (\frac{(1+e^{-x})}{(1+e^{-x})}-\frac{1}{(1+e^{-x})}) \\ + &= \frac{1}{(1+e^{-x})} \cdot (1-\frac{1}{(1+e^{-x})}) \\ + &= \sigma(x) \cdot (1 - \sigma (x)) + \end{split} +\end{equation*} \subsection{By using the chain rule, calculate the gradient $\frac{\delta y}{\delta x}$ and express your results using the weights $w_i$ and the derivative $\sigma'$} +\begin{equation*} + \begin{split} + \frac{dy}{dx} &= \frac{dy}{da_4} \cdot \frac{da_4}{da_3} \cdot \frac{da_3}{da_2} \cdot \frac{da_2}{da_1} \cdot \frac{da_1}{dx} \\ + &= \frac{da_5}{da_4} \cdot \frac{da_4}{da_3} \cdot \frac{da_3}{da_2} \cdot \frac{da_2}{da_1} \cdot \frac{da_1}{da_0} \\\\ + a_1 &= \sigma(w_1 \cdot a_0) \\ + a_2 &= \sigma(w_2 \cdot a_1) \\ + a_3 &= \sigma(w_3 \cdot a_2) \\ + a_4 &= \sigma(w_4 \cdot a_3) \\ + a_5 &= \sigma(w_5 \cdot a_4) \\\\ + \frac{da_5}{da_4} &= \sigma (a_4) \cdot (1-\sigma(a_4)) \cdot w_5 \\ + \frac{da_4}{da_3} &= \sigma (a_3) \cdot (1-\sigma(a_3)) \cdot w_4 \\ + \frac{da_3}{da_2} &= \sigma (a_2) \cdot (1-\sigma(a_2)) \cdot w_3 \\ + \frac{da_2}{da_1} &= \sigma (a_1) \cdot (1-\sigma(a_1)) \cdot w_2 \\ + \frac{da_1}{da_0} &= \sigma (a_0) \cdot (1-\sigma(a_0)) \cdot w_1 \\ + \end{split} +\end{equation*} \subsection{Do the upper or lower bounds of $\sigma'$ contribute to the vanishing gradients problem? If so, how?} +When the value is $0.25$, frequent multiplication of sigmoid activation results in gradients that tend to become extraordinary small, approaching $0$. This phenomenon, known as gradient saturation, particularly effects sigmoid activation when used across multiple layers in deep neural networks. For instance, if you have a deep neural network with multiple sigmoid activation layers, the repeated multiplications of small gradients can lead to the vanishing gradient problem hindering the effective training of the model. \subsection{What is meant by exploding gradients? Why do we not want the gradients to explode? When can sigmoid activations have an exploding gradient?} -- GitLab