Graduationwoot

Dragon Notes

i

\( \newcommand{bvec}[1]{\overrightarrow{\boldsymbol{#1}}} \newcommand{bnvec}[1]{\overrightarrow{\boldsymbol{\mathrm{#1}}}} \newcommand{uvec}[1]{\widehat{\boldsymbol{#1}}} \newcommand{vec}[1]{\overrightarrow{#1}} \newcommand{\parallelsum}{\mathbin{\|}} \) \( \newcommand{s}[1]{\small{#1}} \newcommand{t}[1]{\text{#1}} \newcommand{tb}[1]{\textbf{#1}} \newcommand{ns}[1]{\normalsize{#1}} \newcommand{ss}[1]{\scriptsize{#1}} \newcommand{vpl}[]{\vphantom{\large{\int^{\int}}}} \newcommand{vplup}[]{\vphantom{A^{A^{A^A}}}} \newcommand{vplLup}[]{\vphantom{A^{A^{A^{A{^A{^A}}}}}}} \newcommand{vpLup}[]{\vphantom{A^{A^{A^{A^{A^{A^{A^A}}}}}}}} \newcommand{up}[]{\vplup} \newcommand{Up}[]{\vplLup} \newcommand{Uup}[]{\vpLup} \newcommand{vpL}[]{\vphantom{\Large{\int^{\int}}}} \newcommand{lrg}[1]{\class{lrg}{#1}} \newcommand{sml}[1]{\class{sml}{#1}} \newcommand{qq}[2]{{#1}_{\t{#2}}} \newcommand{ts}[2]{\t{#1}_{\t{#2}}} \) \( \newcommand{ds}[]{\displaystyle} \newcommand{dsup}[]{\displaystyle\vplup} \newcommand{u}[1]{\underline{#1}} \newcommand{tu}[1]{\underline{\text{#1}}} \newcommand{tbu}[1]{\underline{\bf{\text{#1}}}} \newcommand{bxred}[1]{\class{bxred}{#1}} \newcommand{Bxred}[1]{\class{bxred2}{#1}} \newcommand{lrpar}[1]{\left({#1}\right)} \newcommand{lrbra}[1]{\left[{#1}\right]} \newcommand{lrabs}[1]{\left|{#1}\right|} \newcommand{bnlr}[2]{\bn{#1}\left(\bn{#2}\right)} \newcommand{nblr}[2]{\bn{#1}(\bn{#2})} \newcommand{real}[1]{\Ree\{{#1}\}} \newcommand{Real}[1]{\Ree\left\{{#1}\right\}} \newcommand{abss}[1]{\|{#1}\|} \newcommand{umin}[1]{\underset{{#1}}{\t{min}}} \newcommand{umax}[1]{\underset{{#1}}{\t{max}}} \newcommand{und}[2]{\underset{{#1}}{{#2}}} \) \( \newcommand{bn}[1]{\boldsymbol{\mathrm{#1}}} \newcommand{bns}[2]{\bn{#1}_{\t{#2}}} \newcommand{b}[1]{\boldsymbol{#1}} \newcommand{bb}[1]{[\bn{#1}]} \) \( \newcommand{abs}[1]{\left|{#1}\right|} \newcommand{ra}[]{\rightarrow} \newcommand{Ra}[]{\Rightarrow} \newcommand{Lra}[]{\Leftrightarrow} \newcommand{rai}[]{\rightarrow\infty} \newcommand{ub}[2]{\underbrace{{#1}}_{#2}} \newcommand{ob}[2]{\overbrace{{#1}}^{#2}} \newcommand{lfrac}[2]{\large{\frac{#1}{#2}}\normalsize{}} \newcommand{sfrac}[2]{\small{\frac{#1}{#2}}\normalsize{}} \newcommand{Cos}[1]{\cos{\left({#1}\right)}} \newcommand{Sin}[1]{\sin{\left({#1}\right)}} \newcommand{Frac}[2]{\left({\frac{#1}{#2}}\right)} \newcommand{LFrac}[2]{\large{{\left({\frac{#1}{#2}}\right)}}\normalsize{}} \newcommand{Sinf}[2]{\sin{\left(\frac{#1}{#2}\right)}} \newcommand{Cosf}[2]{\cos{\left(\frac{#1}{#2}\right)}} \newcommand{atan}[1]{\tan^{-1}({#1})} \newcommand{Atan}[1]{\tan^{-1}\left({#1}\right)} \newcommand{intlim}[2]{\int\limits_{#1}^{#2}} \newcommand{lmt}[2]{\lim_{{#1}\rightarrow{#2}}} \newcommand{ilim}[1]{\lim_{{#1}\rightarrow\infty}} \newcommand{zlim}[1]{\lim_{{#1}\rightarrow 0}} \newcommand{Pr}[]{\t{Pr}} \newcommand{prop}[]{\propto} \newcommand{ln}[1]{\t{ln}({#1})} \newcommand{Ln}[1]{\t{ln}\left({#1}\right)} \newcommand{min}[2]{\t{min}({#1},{#2})} \newcommand{Min}[2]{\t{min}\left({#1},{#2}\right)} \newcommand{max}[2]{\t{max}({#1},{#2})} \newcommand{Max}[2]{\t{max}\left({#1},{#2}\right)} \newcommand{pfrac}[2]{\frac{\partial{#1}}{\partial{#2}}} \newcommand{pd}[]{\partial} \newcommand{zisum}[1]{\sum_{{#1}=0}^{\infty}} \newcommand{iisum}[1]{\sum_{{#1}=-\infty}^{\infty}} \newcommand{var}[1]{\t{var}({#1})} \newcommand{exp}[1]{\t{exp}\left({#1}\right)} \newcommand{mtx}[2]{\left[\begin{matrix}{#1}\\{#2}\end{matrix}\right]} \newcommand{nmtx}[2]{\begin{matrix}{#1}\\{#2}\end{matrix}} \newcommand{nmttx}[3]{\begin{matrix}\begin{align} {#1}& \\ {#2}& \\ {#3}& \\ \end{align}\end{matrix}} \newcommand{amttx}[3]{\begin{matrix} {#1} \\ {#2} \\ {#3} \\ \end{matrix}} \newcommand{nmtttx}[4]{\begin{matrix}{#1}\\{#2}\\{#3}\\{#4}\end{matrix}} \newcommand{mtxx}[4]{\left[\begin{matrix}\begin{align}&{#1}&\hspace{-20px}{#2}\\&{#3}&\hspace{-20px}{#4}\end{align}\end{matrix}\right]} \newcommand{mtxxx}[9]{\begin{matrix}\begin{align} &{#1}&\hspace{-20px}{#2}&&\hspace{-20px}{#3}\\ &{#4}&\hspace{-20px}{#5}&&\hspace{-20px}{#6}\\ &{#7}&\hspace{-20px}{#8}&&\hspace{-20px}{#9} \end{align}\end{matrix}} \newcommand{amtxxx}[9]{ \amttx{#1}{#4}{#7}\hspace{10px} \amttx{#2}{#5}{#8}\hspace{10px} \amttx{#3}{#6}{#9}} \) \( \newcommand{ph}[1]{\phantom{#1}} \newcommand{vph}[1]{\vphantom{#1}} \newcommand{mtxxxx}[8]{\begin{matrix}\begin{align} & {#1}&\hspace{-17px}{#2} &&\hspace{-20px}{#3} &&\hspace{-20px}{#4} \\ & {#5}&\hspace{-17px}{#6} &&\hspace{-20px}{#7} &&\hspace{-20px}{#8} \\ \mtxxxxCont} \newcommand{\mtxxxxCont}[8]{ & {#1}&\hspace{-17px}{#2} &&\hspace{-20px}{#3} &&\hspace{-20px}{#4}\\ & {#5}&\hspace{-17px}{#6} &&\hspace{-20px}{#7} &&\hspace{-20px}{#8} \end{align}\end{matrix}} \newcommand{mtXxxx}[4]{\begin{matrix}{#1}\\{#2}\\{#3}\\{#4}\end{matrix}} \newcommand{cov}[1]{\t{cov}({#1})} \newcommand{Cov}[1]{\t{cov}\left({#1}\right)} \newcommand{var}[1]{\t{var}({#1})} \newcommand{Var}[1]{\t{var}\left({#1}\right)} \newcommand{pnint}[]{\int_{-\infty}^{\infty}} \newcommand{floor}[1]{\left\lfloor {#1} \right\rfloor} \) \( \newcommand{adeg}[1]{\angle{({#1}^{\t{o}})}} \newcommand{Ree}[]{\mathcal{Re}} \newcommand{Im}[]{\mathcal{Im}} \newcommand{deg}[1]{{#1}^{\t{o}}} \newcommand{adegg}[1]{\angle{{#1}^{\t{o}}}} \newcommand{ang}[1]{\angle{\left({#1}\right)}} \newcommand{bkt}[1]{\langle{#1}\rangle} \) \( \newcommand{\hs}[1]{\hspace{#1}} \)

  UNDER CONSTRUCTION

Deep Learning:
Recurrent Neural Networks





RNN Base Model (I)

ANN architecture using temporal memory to process sequential data:

  • - Excels at sequential, time-series data
  • - Applied in natural language processing, signal anomaly detection, speech synthesis, music composition ...
RNN diagram

RNN Base Model (II)

RNN model is represented via NN time-steps \(\t{<}\t{t}\t{>}\), infeeding current inputs \(\bn{x}^{ < t > }\) and past activations \(\bn{a}^{ < t - 1 > }\) to predict:

\(\ds \begin{align} \bn{a}^{ < t > } & = g (\bn{W}_{aa}\bn{a}^{ < t-1 > } + \bn{W}_{ax}\bn{x}^{ < t > } + \bn{b}_a) \\ \widehat{\bn{y}}^{ < t > } & = g (\bn{W}_{ya}\bn{a}^{ < t > } + \bn{b}_y) \end{align}\)
\(\Leftrightarrow\)

\(\ds \begin{align} \bn{a}^{ < t > } & = g (\bn{W}_a [\bn{a}^{< t - 1 >}, \bn{x}^{ < t > }] + \bn{b}_a) \\ \widehat{\bn{y}}^{ < t > } & = g (\bn{W}_y \bn{a}^{ < t > } + \bn{b}_y) \end{align} \hspace{48px}\)
\(\bn{a}^{[l](i)< j >} = \) timestep \(j\) in training example \(i\) at layer \(l\) activation

RNN Architectures

RNNs can be built to handle different forms of input-output relations:


LSTM

Long Short-Term Memory enables RNN to remember key information over many timesteps:

\(\ds \begin{align} \tilde{\bn{c}}^{ < t > } & = \t{tanh}(\bn{W}_c[\bn{a}^{ < t > },\bn{x}^{ < t > }] + \bn{b}_c) \\ \bn{c}^{ < t > } & = \bn{\Gamma}_u \cdot \tilde{\bn{c}}^{ < t > } + \bn{\Gamma}_f \cdot \bn{c}^{ < t - 1> } \\ \bn{a}^{ < t > } & = \bn{\Gamma}_o \cdot \t{tanh}(\bn{c}^{ < t > }) \\ \widehat{\bn{y}}^{ < t > } & = \t{softmax}(\bn{W}_y\bn{a}^{ < t > } + \bn{b}_y) \\ \bn{\Gamma}_u & = \sigma(\bn{W}_u[\bn{a}^{ < t - 1> },\bn{x}^{ < t > }] + \bn{b}_u)\\ \bn{\Gamma}_f & = \sigma(\bn{W}_f[\bn{a}^{ < t - 1> },\bn{x}^{ < t > }] + \bn{b}_f)\\ \bn{\Gamma}_o & = \sigma(\bn{W}_o[\bn{a}^{ < t - 1> },\bn{x}^{ < t > }] + \bn{b}_o)\\ \end{align}\)
  • - Effectively aids with vanishing/exploding gradient
  • - More robust against gap length memory decay than GRU
$$d \Gamma_o^{\langle t \rangle} = da_{next}*\tanh(c_{next}) * \Gamma_o^{\langle t \rangle}*(1-\Gamma_o^{\langle t \rangle})\tag{7}$$ $$d\tilde c^{\langle t \rangle} = dc_{next}*\Gamma_u^{\langle t \rangle}+ \Gamma_o^{\langle t \rangle} (1-\tanh(c_{next})^2) * i_t * da_{next} * \tilde c^{\langle t \rangle} * (1-\tanh(\tilde c)^2) \tag{8}$$ $$d\Gamma_u^{\langle t \rangle} = dc_{next}*\tilde c^{\langle t \rangle} + \Gamma_o^{\langle t \rangle} (1-\tanh(c_{next})^2) * \tilde c^{\langle t \rangle} * da_{next}*\Gamma_u^{\langle t \rangle}*(1-\Gamma_u^{\langle t \rangle})\tag{9}$$ $$d\Gamma_f^{\langle t \rangle} = dc_{next}*\tilde c_{prev} + \Gamma_o^{\langle t \rangle} (1-\tanh(c_{next})^2) * c_{prev} * da_{next}*\Gamma_f^{\langle t \rangle}*(1-\Gamma_f^{\langle t \rangle})\tag{10}$$ ### 3.2.3 parameter derivatives $$ dW_f = d\Gamma_f^{\langle t \rangle} * \begin{pmatrix} a_{prev} \\ x_t\end{pmatrix}^T \tag{11} $$ $$ dW_u = d\Gamma_u^{\langle t \rangle} * \begin{pmatrix} a_{prev} \\ x_t\end{pmatrix}^T \tag{12} $$ $$ dW_c = d\tilde c^{\langle t \rangle} * \begin{pmatrix} a_{prev} \\ x_t\end{pmatrix}^T \tag{13} $$ $$ dW_o = d\Gamma_o^{\langle t \rangle} * \begin{pmatrix} a_{prev} \\ x_t\end{pmatrix}^T \tag{14}$$




Dragon Notes,   Est. 2018     About

By OverLordGoldDragon