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\title{Simulation of \cite{Grainger245} with Rescorla Wagner equations}
\shorttitle{Grainger et al. (2012) simulation with RW equations}
\author{Robert Geirhos (3827808), Klara Grethen (3899962), \\David-Elias Künstle (3822829), Felicia Saar (3818590),\\Julia Maier (3879869), Marlene Weller (3837283), Anne-Kathrin Mahlke (3897867)}
\affiliation{Linguistics for Cognitive Science Course, University of Tübingen}

\abstract{We try to simulate the results of a word learning experiments with baboons. To that end we use ndl, which is based on the Rescorla-Wagner learning model. The learning parameters by themselves re not able to make learning slow enough to be coparable to the monkeys, which is why we introduced a random parameter that makes the models take random guesses in 65\% of the trials. That way, we can successfully model the monkeys' performance.}

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%\cite{}
\section{Introduction}
Computational models have been used for a very long time in an attempt to represent actual events and phenomena. These models help scientists formulate their hypotheses more precisely, and to test these by applying the model to different situations. The models are also used to make predictions about future behavior and events.

One scientific field, where computational models are becoming increasingly important, is linguistics, especially the field of language learning. This is where our modelling attempt can be placed as well.

In \citeyear{Grainger245}, \citeauthor{Grainger245} tested Baboons on their ability to learn and recognize words as opposed to non-words, when presented to them in written form on a computer screen. Their goal was to show that it is possible to process orthographic information without knowledge of its semantic component and by that, that it ist possible to learn to 'read' (to a certain extend), without prior language knowledge.

To achieve that goal, they trained Baboons (who had, of course, no knowledge of human language) to discriminate words from non-words that were presented on a computer screen. The Baboons were able to independently start blocks of 100 trials in which they would be presented with 50 non-words, 25 already known words and always the same unknown word in the other 25 trials in random order. They reacted by pressing one of two buttons on the screen and were rewarded with food every time they answered correctly. A word was regarded as known once 80\% of the responses to it in one block of trials were correct. It was then added to the group of already known words and used as such in subsequent trials. The difference between the words and the non-words was, for the monkeys, that words appeared repeatedly, while one single non-word was only shown very few times.

The results show that correctness of the responses for both words and nonwords grew above chance very quickly, while word accuracy was slightly higher overall. It also became clear, that the monkeys did not only recognize words because of their appearing more often, but also were able to find patterns and by that recognize new words as words quite quickly.

We decided to try to model the results from \textcite{Grainger245} using Naive Discriminative Learning (NDL), which is a concept of modelling learning (and also an R-package) based on the Rescorla-Wagner model \parencite{rescorla1972theory} and the equilibrium equations by Danks (2003).

\subsection{Naive Discrimination Learning}
Since the first experiments in modern learning theory by Ivan Pavlov it's observable, that learning is not only making associations between co-occurring cues and outcomes but discriminating which cues predict the presence and the absence of a outcome \parencite{baayen2015abstraction}.

A naive discrimination learning (\emph{ndl}) model as a 2-layer network implementation of the established learning rules described by \cite{rescorla1972theory} fulfills this and was applied successfully in the language context (e.g. \cite{baayen2016comprehension}).

\section{Simulations}

\subsection{Stimuli}
For stimuli we used the words given in the supplemetary material of the original paper. The list contained 307 four-letter words and 7832 non-words, each also made up of four letters. In every trial, the word or non-word was presented split into overlapping trigrams (for example for the word atom: \#at, ato, tom, om\#), one trigram after the other, as proposed by Baayen et al. (2016). 

\subsection{Experimental Code}

The simulation code is split in three parts, the creation of the trials, the learning of the monkey and the analysis of the learning results, implemented in the \emph{R Programming Language} \parencite{Rcore}.

\subsubsection{Trial creation}
The algorithm follows in general the structure defined in the reference paper and supplemental materials and described above.
The word-nonword corpus is the one used by the monkey DAN in \cite{Grainger245}.

The lack of information leaded to our own design decision in some edge cases.
Trials will always be created in blocks of 100.
To ensure this constraint the new word block part can be replaced by learned words if there is no new word left in the corpus and vice versa if there's no word learned the learned word part will be filled by the new word.
The new words, learned words and nonwords get picked randomly out of their pool with repetition allowed.

\subsubsection{Monkey learning}
After a block the presented new word can be marked as learned by the definition in \cite{Grainger245}. The rescorla wagner learner therefore has to learn a block, return the guesses and then continue learning with the next block.
This is not easily possible with \emph{ndl} \parencite{Rndl} where for we implemented a rescorla wagner learner ourself.

Since preliminary experiments showed that the monkeys performed with very high accuracies (>90\%), we decided to introduce a random parameter $ r $ in the experiment, defined as the fraction of times the monkey would make a random guess instead of an experience-based prediction.

\subsubsection{Data analysis}
To compare the accuracy with different learning rates we used not only standard tools like linear regression models \emph{(lm)} and \emph{anova} \parencite{Rcore} but also more advanced non linear general additive models \emph{(GAM)} provided by the package \emph{mgcv} \parencite{Rmgcv} compared and visualized with \emph{itsadug} \parencite{Ritsadug}

\subsection{Choice of Parameters}
\subsubsection{Number of Trials}
The six monkeys in the original experiment participated in a different number of trials (min: 43.041, max: 61.142, mean: 52.812). For the sake of simplicity, we presented exactly 50.000 trials in each of our experiments.
\subsubsection{Random Parameter} The random parameter $ r $ was set to 0.65, which proved to be reasonable value. That means, in 65\% of the cases the monkey would guess for either word or nonword with equal probabilities. Therefore, the maximum possible performance $ p_{max} $ is:
$$ p_{max} = 1 - \frac{r}{2} = 0.675$$
In other words, the maximum possible performance is no longer 1.0 (for a very intelligent monkey) but rather restricted by $ r $. If a monkey's performance is slightly better than $ p_{max} $, this is assured to be due to chance.
\subsubsection{Alpha and Beta} Both $ \alpha $ and $ \beta $ were our independent variables which we manipulated over the course of the experiments. We gathered data for every possible combination of $ \alpha $ and $ \beta $ values within an equally spaced range from 0.0 to 0.3. A total of 15 values for each $ \alpha $ and $ \beta $ were combined to $ 15*15 = 225 $ possible combinations. Since $ \alpha $ and $ \beta $ were internally multiplied to a single value, we expected the outcome to be symmetrical due to the commutativity of the multiplication operation and therefore calculated each combination of $ \alpha $ and $ \beta $ only once, which we used as a trick to improve the overall runtime. Therefore, $\sum_{i=1}^{15}i = 120$ combinations  remained to be explored.
\subsubsection{Lambda}
The independent variable $\lambda$ represents the maximum activation in the Rescorla-Wagner model and therefore limits the learning.
It makes it possible to modulate saliency of a stimulus. A more salient stimulus could not only have higher learning rates but also a higher maximum activation. In the original experiment the stimulus were same colored words and nonwords with four letters on an equally colored background. We assume the single words and nonwords are equally salient and keep therefore $\lambda$ constant (1). 

\subsection{Running Parallelized Experiments}

Running an experiment with a single combination of $ \alpha $ and $ \beta $ on a normal desktop computer took about 75 minutes. Therefore, the parameter space one could explore within a reasonable amount of time was quite restricted. We decided to write a parallelized version of the code to reduce the overall runtime. Using the R packages foreach \parencite{Rforeach}, parallel \parencite{Rparallel} and doParallel \parencite{RdoParallel}, restructured the experiment. Since conflicts can easily occur when more than one core is trying to access a shared data structure at the same time, we implemented a parallelized version that is able to run without even containing critical sections. Instead, each thread has its own data structure, a .txt file, and in the end the results are harvested and combined. This version of the experiment ran on a cluster with 15 cores, each performing a total amount of eight experiments. Altogether, 120 combinations of $ \alpha $ and $ \beta $ were explored overnight, which would have taken about 150 hours in a non-parallelized version.

\section{Results}
The number of words learned by the actual monkeys ranged between 87 and 308. With the chosen range for $\alpha$ and $\beta$, we obtained between 275 and 307 learned words, however, it is important to note that we only presented 307 words, so the model reached maximum learning potential. The general accuracy for the real monkeys lay between 71.14\% and 79.81\%, while our accuracies moved between 0.60 and 0.68. Accuracies for word and non-word decisions are similar in both cases. 

\begin{figure*}[ht]
  \centering
  \includegraphics[width=0.9\textwidth]{../plots/plot_accuracy}
  \caption{
    Top row shows model output  accuracies in dependence of modulated alpha and beta.
    Second row visualizes corresponding nonlinear regressions (GAM).
    Accuracy seem to approximate a maximal accuracy with growing alpha, beta parameter.
    In the GAM you see the small influence of one of the parameter.
    Therefore this resoults could be approximated with just one nonlinear parameter.
  }
  \label{fig:accuracy}
\end{figure*}

\begin{figure*}[ht]
  \centering
  \includegraphics[width=0.9\textwidth]{../plots/plot_numwords}
  \caption{
    The left plot shows the raw num words learned of the model with modulated parameter (alpha, beta).
    The model performs always quite well, just several parameter value result in lower perfomance. The corresponding nonlinear regression plot (middle) doesn't mirror a first hipothesis of growing words learned with growing parameter values.
    This is not necessarily a consequense of a wrong hypothesis but of a wrong regression model because of the weak data with very high frequency of around 305 learned words but almost no other number of learned words (right plot).
}
  \label{fig:numwords}
\end{figure*}


\section{Discussion}

The results show that our model is actually too good for the actual monkeys. Only the random parameter we introduced made it possible to obtain similar results as in the original experiment. When trying to account for the unequality only by lowering the learning rates, we encountered a restriction in form of the need to use floating-point numbers, which might have led to unforeseeable behaviour. Therefore, we chose to use the random parameter instead.

Unfortunately, some information on the exact conduct of the original experiment was missing in the paper, so we had to guess some of the details. For expample, it was not made clear what a block of trials would have looked like in the first few blocks, when there were no already known words to be used in the corresponding 25\% of the block.

We were also slightly unhappy with the definition of a word being learned, which was when the word had 80\% accuracy of recognition. We would expect this definition to become proplematic when a word was 'almost' learned, but not quite reaching the 80\%. In the next block with that word, the learning would be a lot quicker than for an actually new word. It might be a good idea to monitor and save the knowledge level concerning one specific word an measuring the actual number of reptitions a word needed to become known.

Concerning our code, there are a few measurements that could be taken to improve it, too. As mentioned above, we parallelised the process because it would have otherwise taken far too long to calculate. It would be very interesting to look into ways to make the program run even faster, therefore enabling more trials to be run and therefore resulting in more data and exacter results.

Shortening running times would also make it possible to re-run the program with more words to see if there are changes in the later learning process which we now could not explore due to lack of words. The mode of presentation could be reassessed, as well as whether the number of letters changes the behaviour of the model.

Lastly, of course, different models could be used in the experiment, to see if other models fit the results of the actual monkeys better.

\newpage

\printbibliography{}

\appendix

\onecolumn

\section{Complete Results}
Here are the complete results of our experiments. The abbreviations used are:
\begin{APAitemize}
\item \#Trials: Number of trials
\item \#LearnedW: Number of learned words
\item \#PresW: Number of presented words
\item GenAcc: General accuracy
\item WAcc: Word accuracy
\item NWAcc: Nonword accuracy
\end{APAitemize}

\input{result_tables.tex}

\lstinputlisting[language=R]{../baboonSimulation.R}

\end{document}