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z50~@;}nHxEF zb!wWn+@Zcoku~x1%|jI28OXGYOY47i+qJIV#$p_S$4c5uq+Oos7}XKHN=S326WqHY z=BHQcrseG7t6fv*iouI&L=1b#>Em&?H}kE3AwRX3AK6YinEr^U=%_QOMl{=BueQ%h zyRb;|HTA0!-Tb_@NI41Yxy-z~SsQ%FZCd?1mEKNKqB>i0^W&FGc+HpEhxv^U@_pp; zi2)jC_^5b#7E5EnKsOD-Z1nn-$jh30FH7ZL5}KoG4_K^Jtm5R3g_nLFu|&EEhH%En z1xqh=7DU;;`cB)g=qt9=%f#YHEf})A5>sn-l3Ktrk+z{yLNnU*JD2WxI9Clt#kKNp yE;WF4cMU!klq?~=l*%{F$~^&lwX7OIoet{Zz*GzYc7?r7>k>HY&d8Ajm% diff --git a/semester6/iml/parts/04_networks.tex b/semester6/iml/parts/04_networks.tex index fcf0f3b..f504fdf 100644 --- a/semester6/iml/parts/04_networks.tex +++ b/semester6/iml/parts/04_networks.tex @@ -222,9 +222,57 @@ $$ In Minibatch GD, this becomes:\\ \subtext{Where $\mathcal{S} \subset \{1,\ldots,n\}$} $$ - \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) + \Theta^{t+1} = \Theta^t - \eta_t\cdot\nabla_{\Theta^t} \Biggl( \frac{1}{|\mathcal{S}|}\sum_{i\in \mathcal{S}} l\Bigl( \Theta^t; x_i,y_i \Bigr) \Biggr) $$ {\footnotesize - \remark An advantage: If $\Theta^t$ approaches a stationary point (which isn't the global minimu), GD will converge, but MB-GD may not converge. -} \ No newline at end of file + \remark An advantage: If $\Theta^t$ approaches a stat. point (which isn't the global minimum), GD will converge, but MB-GD may not converge. +} + +\subsubsection{Vanishing \& Exploding Gradients} +$$ + \nabla_{\Theta^t} \Biggl( \frac{1}{|\mathcal{S}|}\sum_{i\in \mathcal{S}} l\Bigl( \Theta^t; x_i,y_i \Bigr) \Biggr) = \frac{1}{|\mathcal{S}|}\sum_{i \in \mathcal{S}}\Bigl( \nabla_{\Theta^t} l(\Theta^t;x_i,y_i) \Bigr) +$$ +The terms $\nabla_{\Theta^z} l\Bigl(\Theta^t;x_i,y_i\Bigr)$ are composed of $\nabla_{\textbf{W}^{(l)}}l\Bigl(\textbf{W}^{(l)};x_i,y_i\Bigr)$. + +\textbf{Problem}: Optimization might fail if: +$$ + \Bigl\Vert \nabla_{\textbf{W}^{(l)}}l \Bigr\Vert \to \infty \qquad\text{or}\qquad \Bigl\Vert \nabla_{\textbf{W}^{(l)}}l \Bigr\Vert \to 0 +$$ + +\newpage + +\textbf{Solution}: $\Vert \nabla_{\textbf{W}^{(l)}}l \Vert$ depends linearly on $\text{diag}\Bigl( \psi'(z^{(l)}) \Bigr)$, so the choice of $\psi$ ($\psi'$) can be used to constrain $\Vert \nabla_{\textbf{W}^{(l)}}l \Vert$\\ +\subtext{Generally, the gradient follows the behaviour of $\psi'$} + +% Script contains examples for this on Sigmoid & ReLU. Generally, the properties we need are visible by inspection of the derivatives graph. + +Which features do we want $\psi$ ($\psi'$) to fullfil? +\begin{itemize} + \item $\psi'$ should be fast to calculate + \item $\psi'$ should be non-zero (and not get too close) +\end{itemize} + +\remark The $\Bigl\Vert \nabla_{\textbf{W}^{(l)}}l \Bigr\Vert$ may still vanish for any $\psi$. + +\subsubsection{Random Weight Initialization} +\begin{align*} + \nabla_{\textbf{W}^{(l)}}l &= \frac{\partial l}{\partial f}\cdot\frac{\partial f}{\partial h^{(l)}}\cdot\frac{\partial h^{(l)}}{\partial z^{(l)}}\cdot\frac{\partial z^{(l)}}{\partial \textbf{W}^{(l)}} \\ + &= \frac{\partial l}{\partial f}\cdot\frac{\partial f}{\partial h^{(l)}}\cdot\text{diag}\Bigl( \psi'(z^{(l)}) \Bigr)\cdot\begin{bmatrix} + (h^{(l-1)})^\top \\ + \vdots \\ + (h^{(l-1)})^\top + \end{bmatrix} +\end{align*} +So, the gradient $\Vert \nabla_{\textbf{W}^{(l)}}l \Vert$ also depends on $\Vert h^{(l-1)} \Vert$.\\ +\subtext{For $l \in \{1,\ldots,L\}$} + + +\textbf{Problem}: It might be that $\Vert h^{(l-1)} \Vert \to \infty$ or $\Vert h^{(l-1)} \Vert \to 0$. + +\textbf{Solution}: Set $h^{(l-1)}$ randomly, bound mean $\mu$ and var. $\sigma^2$.\\ +\subtext{There is no generally optimal bound for $\sigma^2$, it depends on the NN.} + +Some useful distributions for common $\psi$: + +% Table in script \ No newline at end of file