~

AAAI2021/tex/experiments.tex

@@ -2,14 +2,6 @@
 %\section{Agent Capacity vs. Compositionality}
 %\label{ssec:exp}
 
-\begin{figure}[t]
-  \centering \includegraphics[width=0.99\columnwidth]{fig/Figure6_Compostionality_of_symbolic_language.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/Figure7_The_ratio_of_high_compositional_language.pdf}
@@ -29,6 +21,7 @@
 
 \begin{table}[b]
   \centering
+  \small
   \caption{The Chi-square test between high-compositionality and agent capacity.}
   \label{tab:exp10}
   \begin{tabular}{cccc}

AAAI2021/tex/theory2.tex

@@ -11,6 +11,13 @@ MIS is the similarity between an identity matrix and the mutual information matr
 \end{figure}
 
 
+\begin{figure}[t]
+  \centering
+  \includegraphics[width=0.8\columnwidth]{fig/Figure5_An_emergent_language.pdf}
+  \caption{An emergent language that the unilateral metrics cannot measure its non-compositionality. Notice that given $s_1 = \mathrm{a}$, the listener can neither determine the shape nor the color without the knowledge about $s_0$.}
+  \label{fig:unilateral}
+\end{figure}
+
 Before giving the definition of MIS, we first model the agents in the referential games. As shown in Figure~\ref{fig:modeling}, the listener and speaker in the referential game are connected in tandem. The speaker agent can be regard as a channel, whose input is a concept $c = (c_0, c_1)$ and output is a symbol $s = (s_0, s_1)$. The listener agent can be regard as another channel, whose input is a symbol $s = (s_0, s_1)$ and output is a predict result $\hat{t} = (\hat{c}_0, \hat{c}_1)$. Since the output of the listener only depends on the symbol $s$, we can model the policy of the speaker agent and the listener agent by the probability distribution $P(s = (s_0, s_1) | t = (c_0, c_1))$ and $P(\hat{t} = (\hat{c}_0, \hat{c}_1) | s_0, s_1)$, respectively.
 
 Now we can analyse the information of the concepts preserved in the transmission process given the symbol transmitted, i.e. the conditional mutual information $I\left(t,\hat{t}|s\right)$. Whenever a stable language emerged, the speaker and the listener consistently use a specific symbol $s$ to refer to a specific object $t$. Therefore we can safely say $I\left(t,\hat{t}|s\right) = I\left(t,\hat{t}|s_{t,\hat{t}}\right)$ where $s_{t,\hat{t}}=\max_s\left\{P\left(\hat{t}|s\right)P\left(s|t\right)\right\}$. This conditional mutual information can be obtained by Equation~\ref{eq:cmi}.
@@ -33,15 +40,16 @@ R\left(c_0,s_0\right) & R\left(c_0,s_0\right)
 \end{equation}
 Each column of $M$ correspond to the semantic information carried by one symbol. In a perfectly compositional language, each symbol represents one specific concept exclusively. Therefore, the similarity between the columns of $M$ and a one-hot vector is align with the compositionality of the emergent language.
 
-
 \begin{figure}[t]
-  \centering
-  \includegraphics[width=0.8\columnwidth]{fig/Figure5_An_emergent_language.pdf}
-  \caption{An emergent language that the unilateral metrics cannot measure its non-compositionality. Notice that given $s_1 = \mathrm{a}$, the listener can neither determine the shape nor the color without the knowledge about $s_0$.}
-  \label{fig:unilateral}
+  \centering \includegraphics[width=0.99\columnwidth]{fig/Figure6_Compostionality_of_symbolic_language.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}
 
 
+
 Finally, we define \emph{raw mutual information similarity} ($\mathit{MIS}_0$)
 as the average cosine similarity of $M$ columns and one-hot vectors, as
 Equation~\ref{eq:mis2}. Furthermore, $\mathit{MIS}$ is the normalized mutual