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@ -276,11 +276,11 @@ The choice of convolution for image classification tasks is not
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arbitrary. ... auge... bla bla
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\subsection{Limitations of the Gradient Descent Algorithm}
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% \subsection{Limitations of the Gradient Descent Algorithm}
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-Hyperparameter guesswork
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-Problems navigating valleys -> momentum
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-Different scale of gradients for vars in different layers -> ADAdelta
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% -Hyperparameter guesswork
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% -Problems navigating valleys -> momentum
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% -Different scale of gradients for vars in different layers -> ADAdelta
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\subsection{Stochastic Training Algorithms}
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@ -368,20 +368,21 @@ The results of the network being trained with gradient descent and
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stochastic gradient descent for 20 epochs are given in Figure~\ref{fig:sgd_vs_gd}
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and Table~\ref{table:sgd_vs_gd}
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\input{Plots/SGD_vs_GD.tex}
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Here it can be seen that the network trained with stochstic gradient
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descent is more accurate after the first epoch than the ones trained
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with gradient descent after 20 epochs.
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This is due to the former using a batch size of 32 and thus having
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made 1.875 updates to the weights
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after the first epoch in comparison to one update . While each of
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after the first epoch in comparison to one update. While each of
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these updates uses a approximate
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gradient calculated on the subset it performs far better than the
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network using true gradients when training for the same mount of time.
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\todo{vergleich training time}
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\input{Plots/SGD_vs_GD.tex}
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\clearpage
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\subsection{Modified Stochastic Gradient Descent}
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\subsection{\titlecap{modified stochastic gradient descent}}
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There is a inherent problem in the sensitivity of the gradient descent
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algorithm regarding the learning rate $\gamma$.
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The difficulty of choosing the learning rate can be seen
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@ -434,7 +435,7 @@ They all scale the gradient for the update depending of past gradients
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for each weight individually.
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The algorithms are build up on each other with the adaptive gradient
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algorithm (ADAGRAD, \textcite{ADAGRAD})
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algorithm (\textsc{AdaGrad}, \textcite{ADAGRAD})
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laying the base work. Here for each parameter update the learning rate
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is given my a constant
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$\gamma$ is divided by the sum of the squares of the past partial
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@ -456,11 +457,11 @@ algorithm is given in Algorithm~\ref{alg:ADAGRAD}.
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1, \dots,p$\;
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Apply Update: $x_{t+1} \leftarrow x_t + \Delta x_t$\;
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}
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\caption{ADAGRAD}
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\caption{\textls{ADAGRAD}}
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\label{alg:ADAGRAD}
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\end{algorithm}
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Building on ADAGRAD \textcite{ADADELTA} developed the ... (ADADELTA)
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Building on \textsc{AdaGrad} \textcite{ADADELTA} developed the ... (ADADELTA)
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in order to improve upon the two main drawbacks of ADAGRAD, being the
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continual decay of the learning rate and the need for a manually
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selected global learning rate $\gamma$.
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@ -476,22 +477,70 @@ if the parameter vector had some hypothetical units they would be matched
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by these of the parameter update $\Delta x_t$. This proper
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\todo{erklärung unit}
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\begin{algorithm}[H]
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\SetAlgoLined
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\KwInput{Decay Rate $\rho$, Constant $\varepsilon$}
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\KwInput{Initial parameter $x_1$}
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Initialize accumulation variables $E[g^2]_0 = 0, E[\Delta x^2]_0 =0$\;
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\For{$t \in \left\{1,\dots,T\right\};\, t+1$}{
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Compute Gradient: $g_t$\;
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Accumulate Gradient: $[E[g^2]_t \leftarrow \rho D[g^2]_{t-1} +
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(1-\rho)g_t^2$\;
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Compute Update: $\Delta x_t \leftarrow -\frac{\sqrt{E[\Delta
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x^2]_{t-1} + \varepsilon}}{\sqrt{E[g^2]_t + \varepsilon}} g_t$\;
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Accumulate Updates: $E[\Delta x^2]_t \leftarrow \rho E[\Delta
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x^2]_{t-1} + (1+p)\Delta x_t^2$\;
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Apply Update: $x_{t+1} \leftarrow x_t + \Delta x_t$\;
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}
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\caption{ADADELTA, \textcite{ADADELTA}}
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\label{alg:gd}
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\end{algorithm}
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While the stochastic gradient algorithm is less susceptible to local
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extrema than gradient descent the problem still persists especially
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with saddle points. \textcite{DBLP:journals/corr/Dauphinpgcgb14}
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A approach to the problem of ``getting stuck'' in saddle point or
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local minima/maxima is the addition of momentum to SDG. Instead of
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using the actual gradient for the parameter update a average over the
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past gradients is used. In order to avoid the need to SAVE the past
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values usually a exponentially decaying average is used resulting in
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Algorithm~\ref{alg_momentum}. This is comparable of following the path
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of a marble with mass rolling down the SLOPE of the error
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function. The decay rate for the average is comparable to the TRÄGHEIT
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of the marble.
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This results in the algorithm being able to escape ... due to the
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build up momentum from approaching it.
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\begin{itemize}
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\item ADAM
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\item momentum
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\item ADADETLA \textcite{ADADELTA}
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\item ADADETLA \textcite{ADADELTA}
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\end{itemize}
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\begin{algorithm}[H]
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\SetAlgoLined
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\KwInput{Decay Rate $\rho$, Constant $\varepsilon$}
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\KwInput{Learning Rate $\gamma$, Decay Rate $\rho$}
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\KwInput{Initial parameter $x_1$}
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Initialize accumulation variables $m_0 = 0$\;
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\For{$t \in \left\{1,\dots,T\right\};\, t+1$}{
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Compute Gradient: $g_t$\;
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Accumulate Gradient: $m_t \leftarrow \rho m_{t-1} + (1-\rho) g_t$\;
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Compute Update: $\Delta x_t \leftarrow -\gamma m_t$\;
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Apply Update: $x_{t+1} \leftarrow x_t + \Delta x_t$\;
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}
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\caption{SDG with momentum}
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\label{alg:gd}
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\end{algorithm}
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Problems / Improvements ADAM \textcite{rADAM}
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\begin{algorithm}[H]
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\SetAlgoLined
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\KwInput{Stepsize $\alpha$}
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\KwInput{Decay Parameters $\beta_1$, $\beta_2$}
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Initialize accumulation variables $E[g^2]_0 = 0, E[\Delta x^2]_0 =0$\;
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\For{$t \in \left\{1,\dots,T\right\};\, t+1$}{
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Compute Gradient: $g_t$\;
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@ -503,10 +552,12 @@ with saddle points. \textcite{DBLP:journals/corr/Dauphinpgcgb14}
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x^2]_{t-1} + (1+p)\Delta x_t^2$\;
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Apply Update: $x_{t+1} \leftarrow x_t + \Delta x_t$\;
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}
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\caption{ADADELTA, \textcite{ADADELTA}}
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\caption{ADAM, \cite{ADAM}}
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\label{alg:gd}
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\end{algorithm}
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\input{Plots/sdg_comparison.tex}
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% \subsubsubsection{Stochastic Gradient Descent}
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@ -533,7 +584,31 @@ by introducing noise into the training of the model. This is a
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successful strategy for ofter models as well, the a conglomerate of
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descision trees grown on bootstrapped trainig samples benefit greatly
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of randomizing the features available to use in each training
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iteration (Hastie, Bachelorarbeit??). The way noise is introduced into
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iteration (Hastie, Bachelorarbeit??).
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There are two approaches to introduce noise to the model during
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learning, either by manipulating the model it self or by manipulating
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the input data.
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\subsubsection{Dropout}
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If a neural network has enough hidden nodes to model a training set
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accuratly
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Similarly to decision trees and random forests training multiple
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models on the same task and averaging the predictions can improve the
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results and combat overfitting. However training a very large
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number of neural networks is computationally expensive in training
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as well as testing. In order to make this approach feasible
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\textcite{Dropout1} introduced random dropout.
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Here for each training iteration from a before specified (sub)set of nodes
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randomly chosen ones are deactivated (their output is fixed to 0).
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During training
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Instead of using different models and averaging them randomly
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deactivated nodes are used to simulate different networks which all
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share the same weights for present nodes.
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A simple but effective way to introduce noise to the model is by
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deactivating randomly chosen nodes in a layer
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The way noise is introduced into
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the model is by deactivating certain nodes (setting the output of the
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node to 0) in the fully connected layers of the convolutional neural
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networks. The nodes are chosen at random and change in every
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@ -543,7 +618,12 @@ iteration, this practice is called Dropout and was introduced by
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\todo{Vergleich verschiedene dropout größen auf MNSIT o.ä., subset als
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training set?}
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\subsubsection{Effectively for small training sets}
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\subsubsection{Effectivety for small training sets}
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For some applications (medical problems with small amount of patients)
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the available data can be highly limited. In the following the impact
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on highly reduced training sets has been ... for ... and the results
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are given in Figure ...
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%%% Local Variables:
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%%% mode: latex
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Tobias Arndt
4 years ago