Commit 614687f3 by haoyifan

haoyifan

parent 0de80caa
\begin{figure}[t]
\centering \includegraphics[width=0.99\columnwidth]{fig/Appendix_Figure1_MIS.pdf}
\caption{Compositionality of symbolic language under different parameters
($[\mu-\sigma,\mu+\sigma]$, where $\mu$ is the mean value and $\sigma$ is
the standard deviation).}
\label{fig:exp1}
\end{figure}
\begin{figure}[t]
\centering \includegraphics[width=0.99\columnwidth]{fig/Appendix_Figure2_Ratio.pdf}
\caption{The ratio of high compositional language. (a) $MIS>0.99$. (b)
$MIS>0.9$. }
\label{fig:exp2}
\end{figure}
\begin{table}[b]
\centering
\small
\caption{The Chi-square test between high-compositionality and agent capacity.}
\label{tab:exp10}
\begin{tabular}{cccc}
\toprule
\multicolumn{4}{c}{$H_0$: $\mathit{MIS} > 0.90$ is independent with $h_{\mathit{size}}$}\\
\midrule
Configuration & $\chi^2$ & $df$ & $p$-value \\
\midrule
$|M_0|=3,|M_1|=5,|V|=10$ & 87.20 & 10 & $1.72\times 10^{-13}$ \\
$|M_0|=4,|M_1|=4,|V|=10$ & 71.47 & 10 & $1.70\times 10^{-10}$ \\
\bottomrule
\multicolumn{4}{c}{\vspace{1em}}\\
\toprule
\multicolumn{4}{c}{$H_0$: $\mathit{MIS} > 0.99$ is independent with $h_{\mathit{size}}$}\\
\midrule
Configuration & $\chi^2$ & $df$ & $p$-value \\
\midrule
$|M_0|=3,|M_1|=5,|V|=10$ & 34.15 & 10 & $6.39\times 10^{-4}$ \\
$|M_0|=4,|M_1|=4,|V|=10$ & 38.26 & 10 & $1.39\times 10^{-4}$ \\
\bottomrule
\end{tabular}
\end{table}
\section{Appendix}
\label{sec:exp}
We exploit the relationship between agent capacity and the compositionality of
symbolic language that emerged in our natural referential game.
For various configuration of
vocabulary size, we fix $|M_0|=|M_1|=3$ and train the speaker-listener agents to emerge symbolic
language when varying the agent capacities, i.e., hidden layer size
($h_{size}$), from 6 to 100.
Figure~\ref{fig:exp1} reports the experimental results. It can be observed that
the mean value of MIS decreases as the value of $h_{size}$ increases. Taking the
configuration of vocabulary size $|V|=10$ as an example, the mean value of MIS
is around 0.8 when $h_{size}\le 20$; MIS significantly decreases to 0.75 when
$h_{size}$ increases from 20 to 40; MIS further reduces to 0.7 when $h_{size}$
increases from 40 to 100.
For different vocabulary sizes, the MIS shares the
similar behavior.
It is because symbols in low-compositional languages carry semantic information
about more concepts. As a result, higher capacity is required to characterize the
complex semantic information for low-compositional language to emerge.
In summary, lower agent capacity improves the possibility of
emerging high compositional symbolic language.
\subsection{Ratio of high compositional language.}
We further breakdown our results to investigate the importance of agent capacity
to the compositionality of symbolic language. Figure~\ref{fig:exp2} reports the
ratio of high compositional symbolic language in all emerged languages,
Figure~\ref{fig:exp2} (a) and (b) for $\mathit{MIS}>0.99$ and $\mathit{MIS}>0.9$, respectively. It
can be observed that the ratio of high compositional symbolic languages
decreases drastically with the increase of $h_{size}$.
Taking vocabulary size $|V|=4$ as an example, symbolic languages with
compositionality $\mathit{MIS}>0.99$ take $>$10\% mainly over all the emerged symbolic
languages, when $h_{size}<20$; the ratio reduces to 0\%$\sim$5\% when $h_{size}$
increases to 40; the ratio reduces around 3\% when $h_{size}$ goes beyond 40.
$\mathit{MIS}>0.9$ reports similar results.
Notably, when $h_{size}$ is large enough (e.g., $>40$), high compositional
symbolic language is hard to emerge in a natural referential game, for
easy-to-emerge low compositional symbolic language is sufficient in scenarios of
referential game.
On the other side, agents are enforced to use compositionality to express
more meanings, for the constraint from low capacity.
Additionally, we also perform $\chi^2$ test to check the statistical
significance between the high compositionality and agent
capacity. Table~\ref{tab:exp10} reports the $\chi^2$ test results for
$\mathit{MIS}>0.99$ and $\mathit{MIS}>0.9$, respectively. It can be observed that
for different vocabulary sizes, the p-value is always less than 0.05, which means
the high compositionality has a statistical significance related to agent
capacity.
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