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zP*OvQQUm-PzlHTabl|L%UIv*vrE%EW(BZDYv;5l=@6yq|hv!*?%uYVOs|$k_9EIJZ0fRZ|VD_KC009O5*YyTaA5U zhZI`1aNl-Ys|?Xn4-lQ2H0!@X)R+cj@+D-WRp3jW;QX`T zX|Uk%PeBEtTo5y33+f6V4pt@nl|n+Q@AuS$J(BUjXWBoDL8SDxK2(zdvLivZBY|>p v@Lq99P>L{Oia^?k$nO69@xcp%B#}BjaC$!g4}qDTnVk!cj7&^H9PWPsa;5P| diff --git a/semester6/iml/parts/04_networks.tex b/semester6/iml/parts/04_networks.tex index 3d3d59d..fcf0f3b 100644 --- a/semester6/iml/parts/04_networks.tex +++ b/semester6/iml/parts/04_networks.tex @@ -11,6 +11,12 @@ $$ Where $\phi(x,\Theta) = \Bigl( \phi_1(x;\theta_1),\ldots,\phi_m(x;\theta_m) \Bigr)$.\\ \subtext{$\theta_i$ is the $i$th row of $\Theta$, i.e. $\theta_i := (\Theta)_{i,:}$} +More compact, in terms of $\Theta$, which combines $w, \phi$: +$$ + \Theta^* := \underset{\Theta}{\text{arg min}}\Bigl( L(\Theta; \mathcal{D}) \Bigr) = \underset{\Theta}{\text{arg min}} \Biggl( \frac{1}{n} \sum_{i=1}^{n} l\Bigl( \Theta;x_i,y_i \Bigr) \Biggr) +$$ +\subtext{$\Theta$ may also encapsulate $w,\phi$ for multiple layers, depending on definition} + \subsection{Definitions} \definition \textbf{Activation Function}\\ @@ -94,10 +100,19 @@ These are called \textit{fully connected}, since every node in a layer is connec \subtext{\textit{Introduction to Machine Learning (2026), p. 183}} \end{center} -\definition \textbf{Forward Propagation} +\notation Weights: $\textbf{W}^{(i)} := \Bigl[ w_{k,l}^{(i)} \Bigr]$, Biases: $b^{(i)}_k$\\ +and $\Theta = \Bigl(\textbf{W}^{(1)},\ldots,\textbf{W}^{(L)}, b^{(1)},\ldots,b^{(L)}\Bigr)$ (All parameters)\\ +\subtext{$w_{k,l}^{(i)}$: "Weight at layer $i$ to node $k$ from node $l$"} + +\newpage +\subsection{Forward Propagation} + + +How can we make predictions, i.e. how can $\hat{f}$ be evaluated? + +\definition \textbf{Forward Propagation}\\ +\subtext{This is just the computation for $1$-layer ANN generalized for $L$ layers} -How can $\hat{f}$ be evaluated?\\ -\subtext{This is just the computation for $1$-layer ANN generalized} \begin{algorithm} \caption{Forward Propagation} @@ -108,4 +123,108 @@ How can $\hat{f}$ be evaluated?\\ } $f \gets \textbf{W}^{(L)}h^{(L-1)}+b^{(L)}$ \\ \Return f -\end{algorithm} \ No newline at end of file +\end{algorithm} + +\subsection{Backwards Propagation} + +How can we get all gradients needed for model training? + +\definition \textbf{Backwards Propagation} + +\textbf{Intuition}: An efficient way to get the gradients is to reuse results from forward prop. and previous steps. This works best when starting at the back, at $\nabla_{\textbf{W}^{(L)}}l$. + +\textbf{Goal}: $\nabla_{\textbf{W}^{(1)}}l,\ldots,\nabla_{\textbf{W}^{(L)}} l, \nabla_{b^{(1)}}l,\ldots,\nabla_{b^{(L)}}l$ + +\textbf{Step 1}: Calculate $\nabla_{\textbf{W}^{(L)}}l$, i.e. start from the back. +\begin{align*} + \nabla_{\textbf{W}^{(L)}}l &= \frac{\partial l}{\partial \textbf{W}^{L}} \\ + &= \frac{\partial l}{\partial f}\cdot\frac{\partial f}{\partial \textbf{W}^{(L)}} & \text{(Chain Rule)} \\ + &= \frac{\partial l}{\partial f}\cdot\begin{bmatrix} + \bigl( h^{(L-1)} \bigr)^\top \\ + \vdots \\ + \bigl( h^{(L-1)} \bigr)^\top + \end{bmatrix} & (f = \textbf{W}^{(L)}h^{(L-1)} + b^{(L)}) \\ + &= \nabla_f l \cdot\begin{bmatrix} + \bigl( h^{(L-1)} \bigr)^\top \\ + \vdots \\ + \bigl( h^{(L-1)} \bigr)^\top + \end{bmatrix} & \Biggl(\frac{\partial l}{\partial f} = \nabla_f l\Biggr) +\end{align*} +Notice how $h^{(L-1)}$ was computed during forward prop. + +\newpage + +\textbf{Step 2}: Calculate $\nabla_{\textbf{W}^{(L-1)}}l$. +\begin{align*} + \nabla_{\textbf{W}^{(L-1)}}l &= \underbrace{\frac{\partial l}{\partial f}}_{\text{(1)}}\cdot\underbrace{\frac{\partial f}{\partial h^{(L-1)}}}_{\text{(2)}}\cdot\underbrace{\frac{\partial h^{(L-1)}}{\partial z^{(L-1)}}}_\text{(3)}\cdot\underbrace{\frac{z^{(L-1)}}{\partial \textbf{W}^{(L-1)}}}_\text{(4)} & \text{(Chain Rule)} \\ +\end{align*} +\begin{enumerate} + \item Already done in Step 1. + \item Already done in forward propagation, equal to $\textbf{W}^{(L)}$: + $$ + f \overset{\text{def}}{=} \textbf{W}^{(L)}h^{(L-1)}+b^{(L)} \implies \frac{\partial f}{\partial h^{(L-1)}} = \textbf{W}^{(L)} + $$ + \item \textbf{Not done.} Needs to be calculated: + \begin{align*} + \frac{\partial h^{(L-1)}}{\partial z^{(L-1)}} &= \frac{\partial \psi\bigl( z^{(L-1)} \bigr)}{\partial z^{(L-1)}} \\ + &= \text{diag}\Bigl( \psi'\bigl( z^{(L-1)} \bigr) \Bigr) \\ + &= \begin{bmatrix} + \psi'\Bigl(z_1^{(L-1)}\Bigr) & 0 & \cdots & 0 \\ + 0 & \psi'\Bigl(z_2^{(L-1)}\Bigr) & \cdots & 0 \\ + \vdots & \vdots & \ddots & \vdots \\ + 0 & 0 & \cdots & \psi\Bigl(z_n^{(L-1)}\Bigr) \\ + \end{bmatrix} + \end{align*} + \item Already done in forward propagation, analogous to step 1. + $$ + \frac{\partial z^{(L-1)}}{\partial \textbf{W}^{(L-1)}} = + \begin{bmatrix} + \bigl( h^{(L-2)} \bigr)^\top \\ + \vdots \\ + \bigl( h^{(L-2)} \bigr)^\top + \end{bmatrix} + $$ +\end{enumerate} + +\textbf{Step $i \leq L$}: Calculate $\nabla_{\textbf{W}^{(L-i)}}l$ Analogoues to step 2.\\ +\subtext{The biases $\nabla_{b^{(l)}}l$ are analogous.} + +\newpage + +\subsection{Optimization} + +\textbf{Problem}: How can we train the model, i.e find $\Theta^*$? +$$ + \Theta^* := \underset{\Theta}{\text{arg min}}\Bigl( L(\Theta; \mathcal{D}) \Bigr) = \underset{\Theta}{\text{arg min}} \Biggl( \frac{1}{n} \sum_{i=1}^{n} l\Bigl( \Theta;x_i,y_i \Bigr) \Biggr) +$$ + +{\footnotesize + \remark $L(\Theta;\mathcal{D})$ is generally not convex.\\ + {\color{gray} + i.e. local minima, saddle points may exist + } + + \remark $\dim(\Theta)$ is the total param. count of NN, may be very large +} + +\textbf{Solution}: Gradient Descent (with optimizations) +\begin{itemize} + \item Stochastic Gradient Descent\\ + \subtext{(Why? $\dim(\Theta)$ is very large, $\nabla_\Theta l(\Theta;x_i,y_i)$ are expensive)} + \item Minibatch Gradient Descent\\ + \subtext{(Why? $\mathcal{D}$ may be very large, so there are \textit{many} gradients)} +\end{itemize} + +The standard GD update for $\Theta$ is: +$$ + \Theta^{t+1} = \Theta^t - \eta_t\cdot\nabla_\Theta L\Bigl( \Theta;\mathcal{D} \Bigr) +$$ +In Minibatch GD, this becomes:\\ +\subtext{Where $\mathcal{S} \subset \{1,\ldots,n\}$} +$$ + \Theta^{t+1} = \Theta^t - \eta_t\cdot\nabla_\Theta L\Biggl( \frac{1}{|\mathcal{S}|}\sum_{i\in \mathcal{S}} l\Bigl( \Theta^t; x_i,y_i \Bigr) \Biggr) +$$ + +{\footnotesize + \remark An advantage: If $\Theta^t$ approaches a stationary point (which isn't the global minimu), GD will converge, but MB-GD may not converge. +} \ No newline at end of file