# Learning Based Joint Coding-Modulation for Digital Semantic Communication
Systems ††thanks: This work is supported by the NSF of China under grant
62125108, 61901261, 12031011.

Yufei Bo  Department of Electronic Engineering
Shanghai Jiao Tong University
Shanghai, China
<EMAIL_ADDRESS>Yiheng Duan  Shuo Shao  Department of Electronic
Engineering
Shanghai Jiao Tong University
Shanghai, China
<EMAIL_ADDRESS>School of Cyber Science and Engineering
Shanghai Jiao Tong University
Shanghai, China
<EMAIL_ADDRESS>Meixia Tao  Department of Electronic Engineering
Shanghai Jiao Tong University
Shanghai, China
<EMAIL_ADDRESS>
###### Abstract

In learning-based semantic communications, neural networks have replaced
different building blocks in traditional communication systems. However, the
digital modulation still remains a challenge for neural networks. The
intrinsic mechanism of neural network based digital modulation is mapping
continuous output of the neural network encoder into discrete constellation
symbols, which is a non-differentiable function that cannot be trained with
existing gradient descend algorithms. To overcome this challenge, in this
paper we develop a joint coding-modulation scheme for digital semantic
communications with BPSK modulation. In our method, the neural network outputs
the likelihood of each constellation point, instead of having a concrete
mapping. A random code rather than a deterministic code is hence used, which
preserves more information for the symbols with a close likelihood on each
constellation point. The joint coding-modulation design can match the
modulation process with channel states, and hence improve the performance of
digital semantic communications. Experiment results show that our method
outperforms existing digital modulation methods in semantic communications
over a wide range of SNR, and outperforms neural network based analog
modulation method in low SNR regime.

###### Index Terms:

Semantic communications, variational autoencoder, digital modulation.

##  I Introduction

Recently, semantic communications are envisioned to be a key enabling
technique for future 6G communication networks[1]. As a paradigm shift beyond
Shannon, semantic communications only transmit necessary information relevant
to the specific task at the receiver, known as “semantic information”, and
therefore can significantly improve transmission efficiency. Semantic
communications are regarded as a truly intelligent communication framework.
They can find a wide range of applications such as smart transportation, video
conference, augmented reality (AR), interactive hologram, and intelligent
humanoid robots.

Thanks to the rapid development of machine learning, neural networks (NNs) are
powerful tools to learn and extract the hidden semantic information and have
been successfully applied in the design of various semantic communication
systems. Specifically, neural networks are most widely used to replace the
classic coding and decoding blocks. For example, the neural network is used as
source coding in the semantic communication system developed by Dai _et al_.
in [2]. Neural networks are also used as joint source channel coding (JSCC) in
[3, 4]. Meanwhile, neural networks can also achieve other functions such as
repeated requesting. For instance, Jiang _et al_.[5] integrate neural network
based Hybrid Automatic Repeat Request (HARQ) mechanism into semantic
communication systems.

Figure 1: The semantic communication system model.

Among different building blocks in modern wireless communications, digital
modulation remains a difficult component to be modeled by neural networks.
Analog modulation is used in most previous works on NN-based semantic
communication systems[4, 6, 7, 8], which does not match up with the real-world
digital systems. The difficulty of realizing NN-based digital modulation comes
from its intrinsic mechanism. The output of NN-based encoders is often real
numbers, and the modulator needs to establish a mapping from these real
numbers to discrete constellation symbols, which is equivalent to a non-
differentiable function. Hence this modulator cannot be achieved by neural
networks directly, since all the existing gradient descend algorithms will
fail with the non-differentiable mapping function.

To the best of our knowledge, there are two different methods to achieve
digital modulation in semantic communications. One straightforward idea is to
use a non-neural-network-based method for digital modulation. For example, an
8-bit quantizer is used in [3] to convert the continuous output of the JSCC
encoder to discrete symbol sequences, which can be transmitted through digital
communication systems. Another approach is to use an extra neural network to
generate the likelihood of constellations instead of a concrete one. For
example, Jiang _et al_. [5] use an MLP network with Sigmoid function to train
a non-uniform quantizer that has the minimum quantization error, and further
associate the quantized bits sequence with the constellation
$\left\\{+1,-1\right\\}$ for BPSK modulation. However, in both quantization-
based modulation methods, the modulator block works separately with the coding
block, which will decrease the transmission performance especially when the
channel noise situation is poor.

In this paper, we propose an NN-based joint coding-modulation scheme for
semantic communication systems with BPSK modulation. Specifically, the joint
coding-modulation (JCM) is modeled by an NN-based stochastic encoder in
conjunction with a random coding-based modulator. The NN learns the transfer
probability from source data, while the random coding generates the actual
modulated symbols for channel transmission based on the learned transfer
probability. Without intermediate steps and jointly trained in one unique
neural network, our method can better match up the coding and modulation
process with channel states.

In addition, the capacity-approaching property of random coding can preserve
more information for those symbols who has a marginal probability around
$0.5$. Moreover, the Gumbel-max reparameterization method[9] is used in random
code generation to alleviate the computation complexity of training the JCM
network. Accordingly, at the receiver side, two separate NNs are used to
reconstruct the source data and the semantic information, respectively, and
they are trained jointly with the neural network at the transmitter end.

Based on the proposed JCM scheme, we develop an end-to-end semantic
communication system using the Variational Autoencoder (VAE) architecture[10].
To this end, two theoretic contributions have been made on the loss function
design for the VAE network. First, we derive a variational inference lower
bound for the mutual information based neural network optimization objective
function. This lower bound is much easier to estimate and optimize than the
mutual information function when data are high dimensional variables, and can
be used in VAE networks. Second, we further extend the established lower bound
to the scenario of image semantic communications, and derive a loss function
that consists of cross entropy and mean square error distortion function. We
would like to point out that the meaning of theses two theoretic results are
beyond the engineering target in this paper. Both the mutual information
based[11] and data distortion based[4] loss functions are widely used in
semantic communications, but existing semantic rate distortion model [12] only
tells us that they are strongly related without telling us how exactly they
are related. Our results reveal a more concrete insight that this relationship
should be that one can lower bound the other in image semantic communications.

Experiment results on real-world datasets show that our method can inherit the
performance advantage of VAE in neural network based JSCC problems[7].
Specifically, our method outperforms existing methods for digital semantic
communications, namely the uniform quantizer [3] and the neural network based
quantizer [5]. Moreover, experiments also show that our method can have a
close performance with neural network based semantic communication systems
with analog modulation when the Signal-to-Noise-Ratio (SNR) is high, and a
better performance when SNR is low.

$\displaystyle I(\mathbf{S};\mathbf{\hat{Z}})+\lambda\cdot
I(\mathbf{\mathbf{X};\hat{Z}})\geq-\mathbb{E}_{p(\mathbf{\hat{z}}|\mathbf{z})}\mathbb{E}_{p_{en}(\mathbf{z}|\mathbf{x},\theta)}\mathbb{E}_{p(\mathbf{s},\mathbf{x})}\\{\mathrm{CE}\left[p(\mathbf{s}|\mathbf{\hat{z}})||p_{de,s}(\mathbf{\hat{s}}|\mathbf{\hat{z}},\phi)\right]+\lambda\cdot\mathrm{CE}\left[p(\mathbf{x}|\mathbf{\hat{z}})||p_{de,o}(\mathbf{\hat{x}}|\mathbf{\hat{z}},\psi)\right]\\}+K$
(2) $\displaystyle I(\mathbf{S};\mathbf{\hat{Z}})+\lambda\cdot
I(\mathbf{\mathbf{X};\hat{Z}})\geq-\mathbb{E}_{p(\mathbf{\hat{z}})}\\{\mathrm{CE}\left[p(\mathbf{s}|\mathbf{\hat{z}})||p_{de,s}(\mathbf{\hat{s}}|\mathbf{\hat{z}},\phi)\right]+\lambda\cdot\mathrm{CE}\left[p(\mathbf{x}|\mathbf{\hat{z}})||p_{de,o}(\mathbf{\hat{x}}|\mathbf{\hat{z}},\psi)\right]\\}+K$
(3)
$\displaystyle\mathcal{L}(\theta,\phi,\psi;\mathbf{X}^{N},\mathbf{S}^{N})=\frac{1}{N}\sum_{j=1}^{N}\left[\sum_{m=1}^{M}-p(\mathbf{s}_{j}=m|\mathbf{\hat{z}}_{j})\log
p_{de,s}(\mathbf{\hat{s}}_{j}=m|\mathbf{\hat{z}}_{j},\phi)+\lambda\cdot||\mathbf{x}_{j}-f_{\psi}(\mathbf{\hat{z}}_{j})||_{2}^{2}\right]$
(4)

##  II Problem Formulation

Fig. 1 shows the semantic communication system we focus on in this paper.
There is a source data $\mathbf{X}$, associated with its unknown semantic
information $\mathbf{S}$, to be communicated through a noisy channel for both
data recovery and semantic processing. There is also a knowledge base which
stores a sufficient number of empirical data tuples of
$(\mathbf{x},\mathbf{s})$, and is available to both the transmitter and the
receiver. Unlike the classic approach of separately designing coding and
modulation, the proposed joint coding-modulation is designed as a neural
network followed by a random coding. The NN is to map source data $\mathbf{X}$
to the transfer probability while the random coding generates discretely
modulated symbol $\mathbf{Z}$ based on this transfer probability. Accordingly,
the receiver consists of two neural networks to map from channel output
$\hat{\mathbf{Z}}$ to the reconstructed source data $\hat{\mathbf{X}}$ and
reconstructed semantic information $\hat{\mathbf{S}}$, respectively.

We consider a mandatory requirement for digital communication in our system,
where BPSK modulation is used for simplicity. High-order modulation shall be
considered in the future work. That is to say, the channel input, denoted as
$\mathbf{Z}=(Z_{1},Z_{2},...,Z_{n})$, is an $n$-length sequence taking values
from $\left\\{-1,+1\right\\}^{n}$. Hence the channel output can be written as
$\mathbf{\hat{Z}}=\mathbf{Z}+\boldsymbol{\varepsilon}$, where
$\boldsymbol{\varepsilon}\sim\mathcal{N}(\mathbf{0},\sigma^{2}\mathbf{I}_{n})$
is the channel noise.

Therefore, a Markov chain can be established as
$\mathbf{S}\to\mathbf{X}\to\mathbf{Z}\to\mathbf{\hat{Z}}\to(\mathbf{\hat{S}},\mathbf{\hat{X}})$.
As shown in Fig. 1, the transition between the observable source data
$\mathbf{X}$ and the channel input $\mathbf{Z}$ includes coding and
modulation, which are jointly realized by a neural network. This transition is
modeled as a stochastic joint encoder-modulator
$p_{en}(\mathbf{z}|\mathbf{x},\theta)$ in our paper, where $\theta$ is the
parameter of the neural network. As a remark, a functional encoder or
modulator can be regarded as a special case of the stochastic ones. At the
receiver end, we use two stochastic demodulator-decoders, namely
$p_{de,o}(\mathbf{\hat{x}}|\mathbf{\hat{z}},\psi)$ and
$p_{de,s}(\mathbf{\hat{s}}|\mathbf{\hat{z}},\phi)$ for the reconstruction of
the observable source data and the semantic information respectively, where
the parameters of these two networks are denoted as $\psi$ and $\phi$.

Our goal is to find a JCM network which can preserve as much information about
the semantic information and the source data as possible in the channel output
$\mathbf{\hat{Z}}$, and the corresponding receiver networks which can best
recover the semantic information and source data. Therefore, a mutual
information based theoretic objective function (shorted as “MI-OBJ”) can be
established as follows:

$I(\mathbf{S};\mathbf{\hat{Z}})+\lambda I(\mathbf{\mathbf{X};\hat{Z}}),$ (1)

where $\lambda$ is a trade-off hyperparameter. The MI-OBJ function is also
used in many previous works of semantic communications [7, 11]. By taking the
channel output $\mathbf{\hat{Z}}$ into consideration, the joint encoder-
modulator can have a better fitting with the channel states. However, this
function is sometimes not easy to directly optimize in practical scenarios,
and does not take the receiver into consideration. Hence a solution is given
in our Section III, which gives an alternative approximation of MI-OBJ with
the receiver design also merged in.

##  III Variational Learning for Joint Coding-Modulation

As mentioned, in some circumstances optimizing (1) is not “operational”. As
pointed out by Alemi _et al_. in [13], it is difficult to estimate the joint
distribution $p(\mathbf{x},\mathbf{\hat{z}})$ and
$p(\mathbf{s},\mathbf{\hat{z}})$ when $\mathbf{X}$, $\mathbf{Z}$ and
$\mathbf{\hat{Z}}$ are high dimensional variables, making MI-OBJ in (1)
intractable. Therefore, we hereby use the technique of variational
inference[14] to establish its lower bound and maximize this lower bound
instead, which is stated as our main theorem in Section III-A. Based on this
lower bound, we further derive a loss function for image transmission, as a
corollary in Section III-B. Moreover, we apply Gumble-max method for
generating random coding based modulation to fit with neural network training,
which is discussed in Section III-C.

###  III-A Variational Inference Lower Bound

######  Theorem 1 (VILB)

A variational inference lower bound of MI-OBJ in (1) is given by (2) , where
$K=H(\mathbf{S})+\lambda\cdot H(\mathbf{X})$ is a constant, $\lambda$ is a
trade-off hyperparameter and $\mathrm{CE}\left[\cdot\right]$ is the Cross
Entropy between two probability distributions.

Theorem 1 gives an operational lower bound of (1), where the neural networks,
namely the parameters of $\theta$, $\phi$ and $\psi$, can be trained over the
optimization process of this lower bound. According to [13], the cross entropy
on conditional distributions is much easier to be sampled and estimated than
the joint distribution and the mutual information function.

To prove Theorem 1, first we need to prove the following lemma, which is an
intermediate result that merges the receiver design into MI-OBJ.

###### Lemma 1

A simpler lower bound of MI-OBJ is given by (3), where
$K=H(\mathbf{S})+\lambda\cdot H(\mathbf{X})$.

###### Proof:

First, considering the term $I(\mathbf{S};\mathbf{\hat{Z}})$, we have:

$\displaystyle I(\mathbf{S};\mathbf{\hat{Z}})$
$\displaystyle=-H(\mathbf{S}|\mathbf{\hat{Z}})+H(\mathbf{S})$
$\displaystyle=\mathbb{E}_{p(\mathbf{s},\mathbf{\hat{z}})}\log\left[p(\mathbf{s}|\mathbf{\hat{z}})\frac{p_{de,s}(\mathbf{\hat{s}}|\mathbf{\hat{z}},\phi)}{p_{de,s}(\mathbf{\hat{s}}|\mathbf{\hat{z}},\phi)}\right]+H(\mathbf{S})$
$\displaystyle\overset{(a)}{=}\mathrm{KL}\left[p(\mathbf{s}|\mathbf{\hat{z}})||p_{de,s}(\mathbf{\hat{s}}|\mathbf{\hat{z}},\phi)\right]$
$\displaystyle\ \ \ \ +\mathbb{E}_{p(\mathbf{s},\mathbf{\hat{z}})}\log
p_{de,s}(\mathbf{\hat{s}}|\mathbf{\hat{z}},\phi)+H(\mathbf{S})$
$\displaystyle\overset{(b)}{\geq}\mathbb{E}_{p(\mathbf{s},\mathbf{\hat{z}})}\log
p_{de,s}(\mathbf{\hat{s}}|\mathbf{\hat{z}},\phi)+H(\mathbf{S})$ (5)
$\displaystyle\overset{(c)}{=}-\mathbb{E}_{p(\mathbf{\hat{z}})}\mathrm{CE}\left[p(\mathbf{s}|\mathbf{\hat{z}})||p_{de,s}(\mathbf{\hat{s}}|\mathbf{\hat{z}},\phi)\right]+H(\mathbf{S}),$
(6)

where $(a)$ follows the definition of KL divergence, and the semantic decoder
$p_{de,s}(\mathbf{\hat{s}}|\mathbf{\hat{z}},\phi)$ is expanded as a
variational approximation to the true posterior
$p(\mathbf{s}|\mathbf{\hat{z}})$; $(b)$ follows the non-negative property of
KL divergence; $(c)$ follows the definition of Cross Entropy.

Similarly, we have:

$I(\mathbf{X};\mathbf{\hat{Z}})\geq-\mathbb{E}_{p(\mathbf{\hat{z}})}\mathrm{CE}\left[p(\mathbf{x}|\mathbf{\hat{z}})||p_{de,o}(\mathbf{\hat{x}}|\mathbf{\hat{z}},\psi)\right]+H(\mathbf{X}).$
(7)

Putting (6) and (7) together finishes the proof of Lemma 1. ∎

To further prove Theorem 1, we consider the probability of $\mathbf{\hat{z}}$,
which is determined by the stochastic encoder:

$\displaystyle p(\mathbf{\hat{z}})=$
$\displaystyle\int_{\mathbf{s},\mathbf{x},\mathbf{z}}p(\mathbf{s},\mathbf{x},\mathbf{z},\mathbf{\hat{z}})d\mathbf{s}d\mathbf{x}d\mathbf{z}$
$\displaystyle=$
$\displaystyle\int_{\mathbf{s},\mathbf{x},\mathbf{z}}p(\mathbf{s},\mathbf{x})p_{en}(\mathbf{z}|\mathbf{x},\theta)p(\mathbf{\hat{z}|\mathbf{z}})d\mathbf{s}d\mathbf{x}d\mathbf{z}.$
(8)

For channel input symbols $\mathbf{z}=(z_{1},z_{2},...,z_{n})$, we model them
as mutually conditionally independent Bernoulli variables like that in [11]
with $z_{i}\sim B(p^{i}(\mathbf{x},\theta))$, where $p^{i}(\mathbf{x},\theta)$
is the probability of $z_{i}=1$. Due to the assumed independency among the
components in $\mathbf{z}$, their joint probability mass function (PMF) can be
express as:

$p_{en}(\mathbf{z}|\mathbf{x},\theta)=\prod_{i=1}^{n}[p^{i}(\mathbf{x},\theta)]^{\frac{1}{2}(1+z_{i})}[(1-p^{i}(\mathbf{x},\theta))]^{\frac{1}{2}(1-z_{i})}.$
(9)

By introducing (8) and the transition probability (9) into Lemma 1, we can
finish the proof of Theorem 1.

###  III-B Loss Function Design for Image Semantic Communications

Based on Theorem 1, a corresponding loss function for image semantic
communication can be further derived.

######  Corollary 1 (Loss function for image source data)

Consider image semantic communication scenarios with image classification as
semantics, the loss function can be written as (4) , where
$(\mathbf{X}^{N},\mathbf{S}^{N})=\left\\{(\mathbf{x}_{j},\mathbf{s}_{j})\right\\}_{j=1}^{N}$
denotes the batch of training data; $M$ is the total number of classes;
$f_{\psi}(\mathbf{\hat{z}}_{j})$ is the mean of the Gaussian-parameterized
stochastic decoder for the observable source data reconstruction, which comes
from a special assumption for image source data that we will later elaborate;
$||\cdot||$ denotes the $l_{2}$ norm;
$\mathbf{z}_{j}=y_{\theta}(\mathbf{x}_{j},\mathbf{g}_{j})$ is the
constellation symbol randomly generated by source data $\mathbf{x}_{j}$ and
auxiliary random variable $\mathbf{g}_{j}$, which will be introduced later in
Subsection III-C.

###### Proof:

We first introduce some widely recognized assumptions of image source data,
and then we further derive the loss function by applying these assumptions to
(2) and replacing the probability function with the empirical distribution.

As pointed out in [11], the source data $\mathbf{X}$ and its reconstruction
$\mathbf{\hat{X}}$ can be modeled as multivariate factorized Gaussian
distributions with isotropic covariance. Thus, we can assume that

$\displaystyle
p(\mathbf{x|\hat{z}})=\mathcal{N}(\boldsymbol{\mu},\sigma_{1}^{2}\mathbf{I}_{m\times
m}),$ (10) $\displaystyle
p_{de,o}(\mathbf{\hat{x}|\hat{z}},\psi)=\mathcal{N}(f_{\psi}(\mathbf{\hat{z}}),\sigma_{2}^{2}\mathbf{I}_{m\times
m}),$ (11)

where $\boldsymbol{\mu}$ is the true pixel value of the source data (usually
normalized between $0$ and $1$), $\sigma_{1}$ and $\sigma_{2}$ are precision
parameters treated as constants, and $f_{\psi}(\mathbf{\hat{z}})$ is the
output of the stochastic decoder for the reconstruction of the source data.

By taking (10) and (11) into Theorem 1, (III-B) can be further obtained. It
shows the variational lower bound of $I(\mathbf{X};\mathbf{\hat{Z}})$, where
$(a)$ follows the same step as in (5), and
$p_{de,o}(\mathbf{\hat{x}}|\mathbf{\hat{z}},\psi)$ is the neural network
parameterized variational approximation to the true posterior
$p(\mathbf{x}|\mathbf{\hat{z}})$; $(b)$ follows the expansion of the
expectation; $(c)$ follows the repeated use of the equation
$\mathbf{x}^{\mathrm{T}}\mathrm{A}\mathbf{x}=\mathrm{tr}(\mathrm{A}\mathbf{x}\mathbf{x}^{\mathrm{T}})$
with $a$ and $b$ being two constants.

The semantic information is set as the class of the images, and thus is
categorically distributed. By replacing probability function with empirical
distribution, we have

$\displaystyle\frac{1}{N}\sum_{j=1}^{N}\sum_{m=1}^{M}-p(\mathbf{s}_{j}=m|\mathbf{\hat{z}}_{j})\log
p_{de,s}(\mathbf{\hat{s}}_{j}=m|\mathbf{\hat{z}}_{j},\phi)$ (12)
$\displaystyle\rightarrow\mathbb{E}_{p(\mathbf{\hat{z}})}\mathrm{CE}\left[p(\mathbf{s}|\mathbf{\hat{z}})||p_{de,s}(\mathbf{\hat{s}}|\mathbf{\hat{z}},\phi)\right]$
$\displaystyle\frac{1}{N}\sum_{j=1}^{N}||\mathbf{x}_{j}-f_{\psi}(\mathbf{\hat{z}}_{j})||_{2}^{2}\rightarrow\mathbb{E}_{p(\mathbf{\hat{z}_{j}})}||\boldsymbol{\mu}_{1}-f_{\psi}(\mathbf{\hat{z}})||_{2}^{2}$
, (13)

where “$\rightarrow$” denotes the convergence due to the law of large numbers,
and $M$ denotes the total number of classes. Thus, we complete the proof of
Corollary 1. ∎

As a remark, in Corollay 1, the loss function is a weighted sum of two terms.
One term is the cross entropy between semantic information and its
reconstruction, and the other is the mean square error of the reconstructed
observable data. Thus, it is a loss function based on data distortion
function, which has been widely used in previous works such as [4, 7].
Essentially, Corollary 1 shows that the data distortion function based loss
function in (4) is a lower bound of the MI-OBJ in (1), when specified to image
semantic communication scenarios. This conclusion gives a more concrete bridge
between these two types of loss functions. Meanwhile, since deriving Corollary
1 uses the property of image data, it also partially explains why in text
semantic communications the loss functions in (4) is not often used.

$\displaystyle I(\mathbf{X};\mathbf{\hat{Z}})$
$\displaystyle=-H(\mathbf{X}|\mathbf{\hat{Z}})+H(\mathbf{X})$
$\displaystyle\overset{(a)}{\geq}\mathbb{E}_{p(\mathbf{\hat{z}})}\mathbb{E}_{p(\mathbf{x}|\mathbf{\hat{z}})}\log
p_{de,o}(\mathbf{\hat{x}}|\mathbf{\hat{z}},\psi)+H(\mathbf{X})$
$\displaystyle\overset{(b)}{=}\int_{\mathbf{\hat{z}}}p(\mathbf{\hat{z}})d\mathbf{\hat{z}}\int_{\mathbf{x}}\frac{1}{(2\pi)^{\frac{m}{2}}\sigma_{1}}e^{-\frac{1}{2}(\mathbf{x}-\mathbf{\boldsymbol{\mu}})^{T}\frac{1}{\sigma_{1}^{2}}\mathbf{I}(\mathbf{x}-\mathbf{\boldsymbol{\mu}})}\log\frac{1}{(2\pi)^{\frac{m}{2}}\sigma_{2}}e^{-\frac{1}{2}(\mathbf{\hat{x}}-f_{\psi}(\mathbf{\hat{z}}))^{T}\frac{1}{\sigma_{2}^{2}}\mathbf{I}(\mathbf{\hat{x}}-f_{\psi}(\mathbf{\hat{z}}))}d\mathbf{x}+H(\mathbf{X})$
$\displaystyle\overset{(c)}{=}\int_{\mathbf{\hat{z}}}p(\mathbf{\hat{z}})(b-a\left[\boldsymbol{\mu}_{1}-f_{\psi}(\mathbf{\hat{z}}))^{T}(\boldsymbol{\mu}_{1}-f_{\psi}(\mathbf{\hat{z}}))\right]d\mathbf{\hat{z}}+H(\mathbf{X})$
$\displaystyle\overset{(d)}{=}-a\cdot\mathbb{E}_{p(\mathbf{\hat{z}})}||\boldsymbol{\mu}_{1}-f_{\psi}(\mathbf{\hat{z}})||_{2}^{2}+b+H(\mathbf{X})$
(15)

###  III-C Gumble-max Method for Random Coding

In order to fit with the neural network training process, a Gumble-max method
is applied, along with its corresponding gradient estimation method. Hence in
this subsection, the random code generation and gradient estimation by using
Gumble-max method will be discussed.

$\bullet$ _Random code generation_

To get a codeword $\mathbf{z}$, we need to sample from the stochastic encoder
$p_{en}(\mathbf{z}|\mathbf{x},\theta)$, so that the empirical distribution of
the random code matches its probability distribution. It has been proven true
for using Gumbel-max method [15]. Specifically, in Gumbel-max we denote the
sampling process as $\mathbf{z}=y_{\theta}(\mathbf{x},\mathbf{g})$, where each
symbol $z_{i}$ is generated by the following rule:

$\displaystyle
z_{i}=\underset{\left\\{+1,-1\right\\}}{\mathrm{argmax}}\left\\{\log
p^{i}(\mathbf{x},\theta)+g^{i}_{1},\log\left[1-p^{i}(\mathbf{x},\theta)\right]+g^{i}_{2}\right\\},$
(14)

where $g^{i}_{1}$ and $g^{i}_{2}$ are independent Gumbel variables with
distribution

$g^{i}_{1},g^{i}_{2}\sim\mathrm{Gumbel}(0,1).$

In operation, we first draw a sample from $u\sim\mathrm{Uniform}(0,1)$, then
use the inverse transform and generate the Gumbel-distributed variable as
$g=-\log(-\log(u))$[9].

$\bullet$ _Gradient estimation_

Estimating the gradient of the lower bound (2) w.r.t the encoder parameters
$\theta$ is tricky, since for one thing, the sampling process is non-
differentiable; for another, the usual Monte Carlo gradient estimator exhibits
high variance[10].

To solve the first problem, the non-differentiable argmax operation in (14)
will be approximated by the Softmax funtion[9]. The reparameterization trick
widely applied in VAE[10] solves the second problem. Through the additional
Gumbel noise $\mathbf{g}$ and the transformation
$\mathbf{z}=y_{\theta}(\mathbf{x},\mathbf{g})$, the Monte Carlo gradient
estimator of $\theta$ w.r.t the expectation of an arbitrary function
$h(\mathbf{z})$ now becomes:

$\displaystyle\nabla_{\theta}\mathbb{E}_{p_{en}(\mathbf{z}|\mathbf{x},\theta)}\left[h(\mathbf{z})\right]$
$\displaystyle=\nabla_{\theta}\mathbb{E}_{p(\mathbf{g})}\left[h(y_{\theta}(\mathbf{x},\mathbf{g}))\right]$
$\displaystyle=\mathbb{E}_{p(\mathbf{g})}\nabla_{\theta}\left[h(y_{\theta}(\mathbf{x},\mathbf{g}))\right]$
$\displaystyle\simeq\frac{1}{N}\sum_{m=1}^{N}\nabla_{\theta}\left[h(y_{\theta}(\mathbf{x},\mathbf{g}_{m}))\right].$

The expectation is no longer with respect to the encoder parameters $\theta$
but the Gumbel noise, resulting a low-variance gradient estimator[10].

##  IV Experiment Results

In this section, we use simulations to test the performance of our proposed
method. The experiments are conducted on CIFAR10 dataset, where the class of
the images is set as the semantic information. Three conventional methods are
used as the benchmarks, including two digital modulation methods and one
analog modulation method.

###  IV-A Experiment Settings

####  IV-A1 Datasets

Our experiments are performed on the CIFAR10 dataset which includes 10 classes
of $32\times 32$ color images with a training set of 50000 examples and a test
set of 10000 examples.

####  IV-A2 Benchmarks

We compare the performance of our proposed joint coding-modulation scheme with
BPSK modulation with three benchmarks: the neural network based analog
modulation (abbreviated as “Analog”), the 8-bit quantizer for BPSK modulation
(abbreviated as “8-bit Uniform”) and learning-based quantization for BPSK
modulation (abbreviated as “1-bit NN”). The details of the baseline methods
are listed as follows.

  * •

_“Analog” method_ : This method uses a neural network with four Resnet
blocks[16] for encoding, a 4-layer Spinal[17] architecture for semantic
information decoding and a 3-layer trans-convolution architecture for source
data decoding. The output of the encoding network is a sequence of real
numbers, and then is sent into AWGN channel directly. As a remark, this method
does not obey the digital communication requirement of our system model.

  * •

_“8-bit Uniform” method_ : A same neural network architecture as Analog method
is used here. However, the output of the encoding network is sent to an 8-bit
uniform quantizer to convert into bit sequence. Then the “0” bits are
associated with “-1”, and the “1” bits are associated with “+1” in BPSK
modulation.

  * •

_“1-bit NN” method_ : A same neural network architecture as Analog method is
used for joint source-channel coding. Then an one-layer MLP with a Sigmoid
nonlinearity is applied as the 1-bit quantizer, which outputs the probability
of $b_{i}=1$ for each bit. A hard decision is then applied to the output (if
the probability is over 0.5, then this bit is decided as 1) so as to attain a
bit sequence. An association from bit sequence to BPSK constellations are also
made, which is similar to the 8-bit method.

####  IV-A3 Neural Network of Our Method

The neural network architecture of our method is the same as the analog
method. All details of loss function design, training methods and gradient
estimation follows Section III.

###  IV-B Performance Comparison

Figure 2: Accuracy against SNR (code length n=1536). Figure 3: PSNR against
SNR (code length n=1536).

Fig. 2 and Fig. 3 plot the performance of all four methods against SNR, with
channel input sequence length $n$=1536. As a general trend, our method
outperforms baseline methods when SNR is low. Meanwhile, when SNR is high, the
performance of all methods will converge, but our method still remains
competitive, only slightly less than the Analog method.

Fig. 2 shows the semantic information accuracy performance against SNR. It can
be seen that our method has stable performance with SNR changing, while the
performance of all other methods will be largely affected if SNR is low.
Therefore our method has an obvious performance advantage in low SNR regime.
For example, when SNR=-10 dB, the accuracy of our method can reach $81.32\%$
while the second place Analog method can only achieves $73.63\%$.

Fig. 3 shows the PSNR of observable source data reconstruction against SNR.
Similar with the accuracy performance, our method keeps a performance
advantage in most cases. Only when SNR=14 dB and SNR=18 dB, the PSNR
performance of our method is slightly worse than the Analog method. When
SNR=14 dB, the PSNR of our method is $22.13$ dB while $22.74$ dB for Analog
method; when SNR=18 dB, the PSNR of our method is $22.23$ dB while $23.34$ dB
for Analog method.

Fig. 4 and Fig. 5 plot the performance of all four methods against channel
input sequence length $n$ where the channel SNR is set to be -2 dB. Generally,
for both semantic information reconstruction and observable source data
reconstruction, our method always outperforms three baseline methods.

Fig. 4 shows the semantic information accuracy performance against sequence
length. It can be seen that when SNR=-2 dB, the accuracy performance of our
method is close to that of Analog method, and greatly exceeds the other two
methods. When $n$=512, our proposed method reaches a accuracy performance of
$74.67\%$ while the 1-bit NN method and 8-bit Uniform method respectively have
a performance of $68.49\%$ and $34.49\%$.

Fig. 5 shows the PSNR performance of reconstructing observable source data
against sequence length, for all four methods. As we can see, our method
outperforms all three benchmarks in all code length. When SNR=-2 dB and
$n$=512, the gap between our proposed method and the second place 1-bit NN
method is about $0.8$ dB.

Figure 4: Accuracy against code length n. Figure 5: PSNR against code length
n.

##  V Conclusion

In this paper, we propose a learning-based joint coding-modulation scheme for
semantic communications with digital modulation. Experiment results show that
our method can outperform all existing methods for digital semantic
communications. Meanwhile, new loss functions are proposed, which are more
operational for high dimensional data and reveal the relationship between
mutual information based loss function and data distortion function based loss
function.

The method we proposed can be further extended to higher-order digital
modulation systems, such as QPSK or 8QAM, which will be explored in our future
work.

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###  E.3 Gradients

The HMC sampler requires the derivatives of ${\psi}{}$ with respect to both
$\boldsymbol{x}$ and ${\sf{K}}$ parameters. We calculate

$\frac{\partial{\psi}{}}{\partial\boldsymbol{x}_{\ell\bar{m}i}}=-\epsilon_{\bar{m}}\sum_{j=i}^{n}{\sf{L}}_{\ell
ji}[{\sf{Y}}^{{\textrm{T}}{}}{\sf{R}}^{{\textrm{T}}{}}{\sf{N}}^{-1}(\boldsymbol{d}-{\sf{R}}{\sf{Y}}\boldsymbol{a})]_{\ell\bar{m}j}+\boldsymbol{x}_{\ell\bar{m}i}\,$
(75)

and

$\begin{split}\frac{\partial{\psi}{}}{\partial{\sf{K}}_{\ell
ij}}&=-{\sf{D}}_{\ell
ij}\sum_{\bar{m}}\epsilon_{\bar{m}}\left[{\sf{Y}}^{{\textrm{T}}{}}{\sf{R}}^{{\textrm{T}}{}}(\boldsymbol{d}-{\sf{R}}{\sf{Y}}\boldsymbol{a})\right]_{\ell\bar{m}i}\boldsymbol{x}_{\ell\bar{m}j}+\delta_{ij}^{K}(i-2-n-2q)\,.\end{split}$
(76)

Here ${\sf{D}}$ is as defined in Eq. 66.

###  E.4 Hessian

For the initial step sizes of the HMC sampler we require the diagonal of the
Hessian matrix (see Sect. 4). We calculate

$\frac{\partial^{2}{\psi}{}}{\partial\boldsymbol{x}_{\ell\bar{m}i}\boldsymbol{x}_{\ell\bar{m}i}}=\epsilon_{\bar{m}}^{2}\sum_{j=i}^{n}\sum_{k=i}^{n}{\sf{L}}_{\ell
ji}{\sf{L}}_{\ell
ki}\left[{\sf{Y}}^{{\textrm{T}}{}}{\sf{R}}^{{\textrm{T}}{}}{\sf{N}}^{-1}{\sf{R}}{\sf{Y}}\right]_{\ell\bar{m}j,\ell\bar{m}k}+1$
(77)

and

$\begin{split}\frac{\partial^{2}{\psi}{}}{\partial{\sf{K}}_{\ell
ij}\partial{\sf{K}}_{\ell ij}}=&{\sf{D}}_{\ell
ij}^{2}\sum_{\bar{m}}\sum_{\bar{m}^{\prime}}\epsilon_{\bar{m}}\epsilon_{\bar{m}^{\prime}}[{\sf{Y}}^{{\textrm{T}}{}}{\sf{R}}^{{\textrm{T}}{}}{\sf{N}}^{-1}{\sf{R}}{\sf{Y}}]_{\ell\bar{m}i,\ell\bar{m}^{\prime}i}\,\boldsymbol{x}_{\ell\bar{m}j}\boldsymbol{x}_{\ell\bar{m}^{\prime}j}\\\
&\,-\delta_{ij}^{K}{\sf{D}}_{\ell
ij}\sum_{\bar{m}}\epsilon_{\bar{m}}[{\sf{Y}}^{{\textrm{T}}{}}{\sf{R}}^{{\textrm{T}}{}}{\sf{N}}^{-1}(\boldsymbol{d}-{\sf{R}}{\sf{Y}}{\sf{L}}\boldsymbol{x})]_{\ell\bar{m}i}\boldsymbol{x}_{\ell\bar{m}j}\,.\end{split}$
(78)

We may estimate the diagonal terms of
${\sf{L}}^{{\textrm{T}}{}}{\sf{Y}}^{{\textrm{T}}{}}{\sf{R}}^{{\textrm{T}}{}}{\sf{N}}^{-1}{\sf{R}}{\sf{Y}}{\sf{L}}$
(where ${\sf{L}}$ is a block diagonal matrix in which each ${\sf{L}}_{\ell}$
is repeated $2\ell+1$ times) using Monte Carlo integration as described in
Appendix D, and this technique allows us to calculate Eq. (77). The double sum
in Eq. (78) makes this expression more challenging to calculate efficiently
and we therefore estimate it by numerical differentiation.


# Context-dependent communication under environmental constraints

Krzysztof Główka
Human Interactivity and Language Lab
Faculty of Psychology
University of Warsaw
Poland
<EMAIL_ADDRESS>&Julian Zubek
Human Interactivity and Language Lab
Faculty of Psychology
University of Warsaw
Poland
<EMAIL_ADDRESS>&Joanna Rączaszek-Leonardi
Human Interactivity and Language Lab
Faculty of Psychology
University of Warsaw
Poland
<EMAIL_ADDRESS>The corresponding author

###### Abstract

There is significant evidence that real-world communication cannot be reduced
to sending signals with context-independent meaning. In this work, based on a
variant of the classical Lewis, (1969) signaling model, we explore the
conditions for the emergence of context-dependent communication in a situated
scenario. In particular, we demonstrate that pressure to minimise the
vocabulary size is sufficient for such emergence. At the same time, we study
the environmental conditions and cognitive capabilities that enable contextual
disambiguation of symbol meanings. We show that environmental constraints on
the receiver’s referent choice can be unilaterally exploited by the sender,
without disambiguation capabilities on the receiver’s end. Consistent with
common assumptions, the sender’s awareness of the context appears to be
required for contextual communication. We suggest that context-dependent
communication is a situated multilayered phenomenon, crucially influenced by
environment properties such as distribution of contexts. The model developed
in this work is a demonstration of how signals may be ambiguous out of
context, but still allow for near-perfect communication accuracy.

_Keywords_ referential game, signalling game, signalling problem, language
emergence, pragmatics, context, context-dependent communication, ambiguity,
evolution strategies

##  1 Introduction

In language science and philosophy, there is a marked tension between the
perceived invariance of the meaning of natural language and its inevitable
interaction with the actual context of usage. In this vein, it is common to
distinguish between semantics, concerned with the standalone meaning, and
pragmatics, investigating situational context (for an overview, see Korta and
Perry, (2020)). In much of linguistic work, the focus is on semantics, while
pragmatics is peripheral (Steffensen and Harvey,, 2018). Although such a
relatively narrow framing of natural language may have been useful for a
number of research goals, there is ample evidence that, in real-world
communication, standalone meaning is an exception rather than a rule. Human
interlocutors seem to readily produce messages that are ambiguous (Wasow et
al.,, 2005), relying on the context to complete their meaning (Barwise and
Perry,, 1983). In fact, humans understand language in context so effortlessly
that potential ambiguities are often only noticed by the linguist (Ferreira,,
2008).

In this work, we focus on two questions. First, what pressure is sufficient
for the emergence of context-dependent communication (why may it be preferred
over unambiguous communication)? Second, what environmental conditions and
cognitive capabilities enable such communication? To address these issues, we
turn to agent-based models as tools to examine various factors shaping
language evolution in simplified and idealised simulations (Steels,, 2011;
Cangelosi and Parisi,, 2012). We adopt a variant of the Lewis, (1969)
signaling game – a cooperative scenario in which the sender’s task is to
communicate the referent that the receiver must identify in a limited set.
Within this model, we introduce the communication context as a set of possible
referents, selected at random in each communication episode.

Even in the simple scenario of the signaling game, context may play a crucial
role in at least two aspects: (a) it offers additional information that can be
exploited by the sender and the receiver, and (b) context directly constrains
the receiver’s action choice, which may impact the agents’ communicative
strategy regardless of the receiver’s information-processing capabilities. We
can illustrate this distinction by building upon an example given by Searle,
(1980): when someone hears the utterance “we need to cut it” while standing on
a lawn, they are likely to think about a mower. The same sentence uttered
during a birthday party can be interpreted as a request to fetch a knife to
cut a cake. However, when the sender knows that the right cutting tool, a
mower or a knife, is already present at the receiver’s hand, they can be sure
that the receiver will understand the utterance correctly even without knowing
the context, as the possible course of action is already narrowed down.

While previous research on context-dependent communication has usually assumed
agents’ nontrivial internal information-processing capabilities, we focus on
the enabling constraints external to the agents and carry out experiments with
the receiver’s minimal skills. We observe that the following conditions are
sufficient for the development of context-dependent communication in this
setting: (a) information on the receiver’s context is provided to the sender
and (b) pressure is placed on the agents to minimise their vocabulary. Under
these conditions, agents develop a language that is partly ambiguous and rely
on the context for disambiguation rather than invent a fully precise language.

##  2 Why does linguistic ambiguity arise?

Linguists’ focus on context-independent language is reflected and arguably
sustained by the commonly adopted metaphors of communication as a transmission
problem and the resulting treatment of language as a code (Reddy,, 1979). The
accompanying pervasive assumption of stable correspondence between language
utterances and their meaning has roots in the long history of focus on
language generativity (Katz and Fodor,, 1963). More pragmatically and
ecologically-oriented works argue for a different picture, where natural
language utterances are commonly ambiguous out of context – but they are
successfully understood by humans based on the shared situational context
(Sperber and Wilson,, 1986). As Barwise and Perry, (1983) indicated, while it
is important that we can express an infinite number of propositions with a
finite number of elements, due to the possibility of combination and
recursion, it is equally amazing that we can express an infinite number of
meanings with one expression due to its dependence on context. Arguably, the
conventional meaning of expressions can be extended to novel contexts due to
their ambiguity (Brochhagen,, 2015); and expression meaning is created in
interaction (Chater and Christiansen,, 2022), where language has the role of
steering or controlling interactive events rather than transmitting clear and
bounded meanings (Rączaszek-Leonardi,, 2012). These directions in the
development of the theory of language, from the transmission model to viewing
language as a creative, situated interaction control, make understanding of
the nature of context and its contribution to linguistic meaning and structure
an urgent task 111Paradoxically, from such a standpoint it seems more
challenging to account for the partial meaning context-independence than to
acknowledge the effects of situational context; for example, how it is
possible that we have a sense that utterances refer to something seemingly
unrelated to the current circumstances, possibly distant in time and space
(Rączaszek-Leonardi et al.,, 2018)..

Piantadosi et al., (2012) propose that linguistic ambiguity arises under
pressure for communicative efficiency. They consider the scenario of a sender
and receiver in a shared context. If the context is at least slightly
informative about the possible meanings of the sender’s message, the sender
does not need to convey full information for the receiver to disambiguate the
message, that is, infer its meaning from the context. Consequently, if symbol
production is costly and the receiver’s inference is relatively cheap, it is
suboptimal on the sender’s part to ensure that their utterance can only be
interpreted in exactly one way. A shorter, partially ambiguous utterance will
be preferred and will still be informative enough for the receiver.
Additionally, if there are differences in the production cost of certain
symbols, the sender may choose more economic, but more ambiguous ones.

While the bottleneck indicated by Piantadosi and colleagues concerned language
production, Kirby et al., (2015) argue for an informational bottleneck in
cultural transmission. They demonstrate that natural limitations in the
cognitive abilities of language learners impose pressure on simplicity on a
language system. Thus, despite the pressure for informativeness 222We follow
the use of terms in Winters et al., (2018); the original terms were
compressibility and expressivity (Kirby et al.,, 2015)., arising from the use
of language, languages will never become perfectly informative about the
intended meaning. Rather, they will gravitate towards a point of optimal
trade-off between the competing pressures, where a degree of context
dependence is to be expected. Under such a trade-off, language ambiguities can
make communication less effective but still functional in general (Santana,,
2014; O’Connor,, 2015).

Intuitively, if language meaning relies on the context, then the composition
of contexts, in which the interlocutors find themselves, should also play a
role for ambiguity. Imagine a simple interaction between a shop assistant and
a customer asking for a specific product visible on the display. Here,
successful communication requires discrimination of the intended referent from
other possible referents (e.g. Olson, (1970)). Winters et al., (2018)
demonstrated that in this kind of situated setting, more predictable contexts
allow for minimising information transmission through linguistic means. They
manipulated the sender’s access to context and contextual object distribution
in artificial language experiments, where pairs of participants were tasked to
communicate about the target referent. When the possible referents in the
context were random, the emergent signaling system was highly autonomous.
However, when objects within each context were always of the same colour and
the sender could see the receiver’s context, the emergent languages became
ambiguous, not encoding colour information, and thereby relying on the stable
object distribution in the context.

In summary, several factors seem to crucially contribute to the emergence of
ambiguous language. One of them is the situational context shared between
interlocutors, which is a source of additional information independent of what
is conveyed by language. The second is an additional pressure that makes the
language not only informative but also reasonably simple. The last is the
regularity of the environment, which determines the relevant distinctions to
be made by language.

##  3 Context in signalling games

The models relevant for this work follow the general schema of the influential
Lewis, (1969) signaling game, where the sender and receiver learn to
communicate about objects in their environment through rewards for
communication success. In the original formulation of the game, as well as in
its many extensions, there is no shared context between the agents. Under such
circumstances, the only protocols that guarantee communication success are
those that unequivocally map objects to signals, and they predictably emerge
in simulations of evolutionary dynamics with realistic perturbations (Skyrms,,
2010; Santana,, 2014).

In some more recent models of visually grounded language emergence, neural
agents faced an image discrimination task with shared context that directly
constrained the referent choice (e.g. Lazaridou et al., 2016a ; Lazaridou et
al., 2016b ; Lazaridou et al., (2018); Havrylov and Titov, (2017); Bouchacourt
and Baroni, (2018)). The size of the context ranged from just two objects
(e.g. Lazaridou et al., 2016b ) to $128$ (Havrylov and Titov,, 2017). In these
works, due to their engineering motivation, the only optimisation goal for
agent policies was communication success. The results of Guo et al., (2021)
suggest that the constrained choice of referents can itself lead to the
emergence of context-dependent languages. Nevertheless, context dependence was
not controlled for nor reported in these works.

Some works in evolutionary game theory investigated the origins of language
ambiguity in signaling games where additional costs were included in the
reward function. This included an additional cost imposed on the sender for
using a larger vocabulary (Santana,, 2014; Mühlenbernd,, 2021) and / or the
cost related to disambiguation imposed on the receiver (Mühlenbernd,, 2021).
The communicating agents were provided with shared contextual information. The
results show that partial ambiguity is evolutionarily the most likely strategy
when the sender’s signaling cost is higher or comparable to the receiver’s
disambiguation cost. In a similar vein, Brochhagen, (2020) showed that the
context can be exploited by the sender to use its preferred but more ambiguous
forms. In these game-theoretic studies, the receiver had the ability to infer
the intended meaning of the sender, in congruence with Piantadosi et al.,
(2012). However, the direct impact of the environment on context-dependent
communication was not accounted for because of the unlimited referent choice.

While the receiver’s inference is undoubtedly important for context-dependent
communication (Frank and Goodman,, 2012), direct environmental constraint on
the choice of possible referents substantially modifies the task (Winters et
al.,, 2018; Guo et al.,, 2021). In a situated communicative setting with a
constrained referent choice, the speaker’s signal needs to be only as precise
as to discriminate the referent from other present objects, and the receiver’s
task is arguably less challenging, as the agent’s inference is partly
“offloaded” to the environment. Therefore, to investigate the environment’s
role for context dependence, we use a model in which agents have minimal
skills and in which the pressure for vocabulary size and sender’s access to
context will be manipulated.

##  4 Contextual Lewis signalling game

Let us define a contextual variant of the probabilistic Lewis signalling
model. There are two agents: sender and receiver. The sender perceives one of
the alternative states of the world $s\in S$, and emits one of the available
signals $\sigma\in\Sigma$ with probability $f(\sigma|s)$. The receiver chooses
a response $r\in R$, based on the received signal, with probability
$g(r|\sigma)$. Upon receiving a correct response $r$ for a given state $s$
both agents are rewarded. Agent policies are represented through functions $f$
and $g$ and can be modelled, for example, using neural networks. Through
adaptive learning, agents can reach an equilibrium in which signals are used
for functional communication. Signals are conventional in the sense that
$s$–$\sigma$ and $\sigma$–$r$ mappings are arbitrary but must be consistent.
We can say that the signalling protocol established in a Lewis signalling game
is unambiguous if there is a unique mapping from signals $\sigma$ to states
$s$. Such a setting and its many varieties have been a kind of general testing
ground for investigating various aspects of language emergence.

Within recent neural models, a game that follows this schema without extending
it with contextual elements can be called reconstruction game, following Guo
et al., (2021). In these works, the receiver’s output belongs to the same
space as the sender’s input, namely, the response belongs to the set of
possible world states: $r\in S$ (e.g. Chaabouni et al., (2020); Kharitonov and
Baroni, (2020)). In the reconstruction game, the receiver has to reconstruct
the target solely based on the received signal – there are no contextual cues
of any kind. In contrast, in another popular scenario, which we call
discrimination game 333Guo et al., (2021) have called this scenario simply
referential game. However, this name has been used across a wide variety of
contexts and fields. To avoid confusion and capture the essence of those
scenarios, we choose the term discrimination. the receiver’s task is to
identify the target from among the provided set of objects which constitutes
the context. We will use the word context in this narrow technical sense. The
world state is multi-object, that is, if $O$ is the set of all possible
objects, the world state consists of the context $C\subset O$ and the target
$t\in C$ – what is communicated is only a part of the world state.

Within a discrimination game, we can consider a receiver that is not “aware”
of the context, but is still constrained by it, since only some objects can be
selected. For such a case, we need to define an auxiliary selection function:

$\iota_{C}(\mathbf{z})_{i}=\begin{cases}\dfrac{e^{z_{i}}}{\sum_{r_{j}\in
C}e^{z_{j}}}&\text{if }r_{i}\in C\\\ 0&\text{if }r_{i}\not\in C\end{cases}$

The function $\iota_{C}$ operates on a vector of probabilities $\mathbf{z}$
corresponding to different responses $r\in O$. It selects only valid responses
that belong to a given context and normalizes the probabilities.

On a spectrum of perceptual availability of the context, we can distinguish
the following variants of the sender–receiver pairs playing a discrimination
game, from the least to the most contextual:

  1. 1.

Neither sender nor receiver have access to the context. Sender’s signal is
generated with probability $f(\sigma|t)$, receiver’s response is generated
with probability $\iota_{C}(g(r|\sigma))$.

  2. 2.

Sender has access to the context, but the receiver does not. Sender’s signal
is generated with probability $f(\sigma|t,C)$, receiver’s response is
generated with probability $\iota_{C}(g(r|\sigma))$.

  3. 3.

Both sender and receiver have access to the context. Sender’s signal is
generated with probability $f(\sigma|t,C)$, receiver’s response is generated
with probability $g(r|\sigma,C)$.

Under variant 1) agents cannot use contextual information at all. The optimal
solution is to use unambiguous signalling, where every world state is
represented by a unique signal. Under variant 3) sender and receiver share
context, and may use it as a common ground. This, in principle, opens a
possibility for ambiguous, yet efficient, communication protocols, such that
the sender emits the signal conditional on the context and the receiver
disambiguates the signal in a process of inference within the particular
context (Piantadosi et al.,, 2012).

Variant 2) is an interesting middle ground between these two possibilities.
Since sender knows which objects are available for the receiver to choose
from, it may safely use an ambiguous signal when other probable referents of
the signal are not present in the current context. Signalling protocol that is
highly ambiguous in the entire object space may still be optimal for the vast
majority of contexts. Thus, accurate context-dependent communication may
emerge even without inferential capabilities on the receiver’s side. This is
the variant that we implement in our simulations.

##  5 Model architecture

Agents are modelled as multi-layer neural networks. We consider two agent
pairs, corresponding to variant 1 and 2 from section 4, including the same
type of receiver but differing with respect to the sender’s access to the
context beyond the target object. Importantly, the receiver is designed to be
unable to infer signal meaning in context.

Target-only (T) sender. A simple feed-forward network of fully-connected
layers, which receives only the target as the input and maps it to
probabilities of activation of each symbol (Fig. 1). Since this sender cannot
access the context, it shares no common information with the receiver, and
thus it is not expected to be able to communicate in a context-dependent
manner in any setting.

(a)  | (b)
---|---
Figure 1: The target-only (T) sender (a) and example vector encodings of
object properties (b).

Target-in-context (TC) sender . The input of this network is all objects in
the context, including the target. It receives input sequentially (Fig. 2). As
non-target objects are input to the network in subsequent steps, the target is
passed along each of them. Such an input structure enables direct comparisons
between the target and each of the objects. This network has an LSTM memory
cell, which constitutes the first layer of the network. After the last input
is provided, LSTM output is passed through fully-connected layers to produce
activation probabilities for each symbol. Since this sender architecture is
informed by the receiver’s context, it is expected to be able to communicate
in a context-dependent manner.

Figure 2: Target-in-context (TC) sender

Receiver The receiver is a simple network that generates the decision on the
choice of the object within a single forward pass (Fig. 3). It is inspired by
and very similar to the receiver architecture used by Lazaridou et al., 2016a
in that each object may only be compared with the received signal
independently from other objects. Each object is processed by the same
sequence of network layers along with the signal emitted by the sender, which
produces what may be interpreted as an alignment score between the input
object and the received signal. All such scores are combined with a softmax
operation into a probability distribution over objects, from which the final
selection is sampled. In simple terms, this receiver most often selects the
object that best matches the received signal.

Figure 3: The receiver

###  5.1 Optimization method

Each iteration in a simulation run consists of $n$ episodes. The fitness of
the agents is based on their accuracy and vocabulary size.

Accuracy of a sender-receiver pair characterised by a parameter vector
$\theta$ is defined as $Acc(\theta)=\frac{1}{n}\sum_{i=1}^{n}G_{i}$, where
$G_{i}$ is the reward for the $i$-th episode. $G_{i}=1$ if and only if the
receiver selects the target object in the $i$-th episode; otherwise $G_{i}=0$.

Vocabulary size is defined in a continuous way based on the usage of different
signals across all episodes in an iteration. If $\sigma^{j}$ is the $j$-th
symbol (or bit) of the signal vector, $1\leq j\leq m$, then:

$\textit{Voc}(\sigma)=\frac{1}{m}\sum_{j=1}^{m}H(\sigma^{j})\\\
=-\frac{1}{m}\sum_{j=1}^{m}(P(\sigma^{j}=0)\log_{2}P(\sigma^{j}=0)\\\
+P(\sigma^{j}=1)\log_{2}P(\sigma^{j}=1))$

Intuitively, it means that a language with 10 words that uses only 5 of them
in most episodes has a smaller vocabulary than one with 10 words that are
equally frequent words. The measure is made continuous for the purpose of
convenient gradient optimisation.

Fitness The fitness function calculated for the agent parameters $\theta$
takes the form:

$F(\theta)=\frac{1}{n}\sum_{i=1}^{n}G_{i}-p_{voc}\textit{Voc}(\sigma)$

where $p_{\textit{voc}}$ is the penalty coefficient (can be equal to zero).

The model parameters $\theta$ are iteratively optimised to maximise the
fitness of the agents. We chose one of the black-box optimisation methods –
the evolution strategies (Salimans et al.,, 2017). We define the model
parameters $\theta$ as a concatenation of the sender and receiver parameters,
the weights and biases of their corresponding networks. At first, the model
parameters are initialised randomly. In every iteration, a mini-batch of new
parameters $\theta$ is generated by sampling noise $\epsilon$ from the normal
distribution and adding it to the parameter vector. We also use mirror
sampling (Brockhoff et al.,, 2010), that is, for every sampled perturbation
$\epsilon$, an inverse perturbation $-\epsilon$ is generated. The perturbed
parameters are then evaluated to approximate the gradient of the fitness
function $F$:

$\nabla_{\theta}\mathbb{E}_{\epsilon\sim\mathcal{N}(0,I)}F(\theta+\sigma\epsilon)=\frac{1}{\sigma}\mathbb{E}_{\epsilon\sim\mathcal{N}(0,I)}\left[\epsilon
F(\theta+\sigma\epsilon)\right]$

The gradient approximated in this way is used for optimisation using gradient
ascent.

###  5.2 Measures of communication context-dependence

To measure how contextual the emerging language is, for each iteration, we
calculate several complementary metrics.

Target certainty. This measure is the reverse of the notion of ambiguity, that
is, it represents how much each symbol corresponds to only a single target.

At the same time, target certainty is the out-of-context baseline to be
compared with actual accuracy. It is the highest accuracy possible in a
counterfactual situation such that the sender emits the same signals but the
receiver’s choice is neither constrained nor informed by context. Therefore,
the best strategy is to select the most probable target given the given
signal. We expect that the more the language becomes context-dependent, the
more “imperfect” it will be in the eyes of an observer that considers only the
received signal.

$\frac{1}{n}\sum_{i=1}^{n}\max_{\textit{o}\in\textit{O}}P(T=\textit{o}|\sigma_{i})$

Signal certainty. This is an analogue of target certainty on the sender’s end
– a measure of how much each target perceived by the sender maps to a single
signal.

$\frac{1}{n}\sum_{i=1}^{n}\max_{\sigma\in\Sigma}P(\sigma|t_{i})$

Maximal contextless interpretation accuracy. This measure is devised to detect
nontrivial context processing by the receiver. It corresponds to accuracy in a
counterfactual situation where the receiver is neither constrained nor
informed by the context and still “does its best”. In this scenario, the
sender’s signal emission is unchanged, but the receiver selects the most
probable target in the current context given the received signal alone. If the
real accuracy exceeds this hypothetical quantity, it means that the receiver
conditions its response on the context in a more complex way than choosing the
most frequent signal referent considering each object $o$ present in the
context.

$\frac{1}{n}\sum_{i=1}^{n}1\textit{ iff }t_{i}=\arg\max_{\textit{o}\in
c_{i}}P(T=\textit{o}|\sigma_{i})\textit{ else }0$

For a full picture of the emergent communication, we additionally calculate
information-theoretic measures based on Shannon’s mutual information (Cover
and Thomas,, 2012), here denoted by $I$. They quantify context’s participation
in the signal generation on the part of the sender and in the interpretation
of its meaning on the part of the receiver – which here amounts to the
selection of the object from the context. For brevity, exact values of these
metrics are reported in Appendix A.

Sender context information gain.

This is the amount of information gained about the signal when the whole
context is considered beyond the target. When $C$, $T$ and $\sigma$ correspond
to random variables denoting context, target, and signal emitted, we define
the metric as follows: $I(\sigma;C)-I(\sigma;T)$

Receiver context information gain.

This is the amount of information gained about the receiver response when the
context is considered beyond the received signal. When $R$ is the receiver’s
response random variable, this metric is defined as follows:
$I(R;C,\sigma)-I(R;\sigma)$

a)  | b)
---|---
c)  | d)
Figure 4: Metric plots for Experiment 1: T and TC sender with no vocabulary
penalty. The dark area around curves marks the standard error.

##  6 Simulation results

We run our model in the world where all objects have $2$ categorical
properties, both with $3$ possible values. For illustration purposes, we can
consider them as shape and colour. The context size was set to $3$. We tested
our models in two main conditions: with unlimited vocabulary and with a
vocabulary size penalty. The experiments were carried out with both T and TC
senders (see section 5), paired with the same receiver type. We ran each
simulation $10$ times for $50k$ iterations. The exact model hyperparameters
used in both experiments and final metric values are specified in Appendix A.

###  6.1 Experiment 1: Unlimited vocabulary

The goal of the first experiment was to check whether agents could develop
contextual communication without additional pressure beyond the pressure for
communication accuracy. The results can tell us whether the agent
architectures used in this study (a) have any inherent tendency to develop
contextual languages and (b) can reach maximum accuracy in a favourable
setting. In this experiment, the vocabulary penalty coefficient
$p_{\textit{voc}}=0$.

Regardless of the sender type, whether T or TC, all agent pairs converge to
top accuracy (Fig. 4a). The developed languages are fully unambiguous, as
shown by the maximum value of target certainty – given the signal, the
receiver selects the correct object unequivocally regardless of the remaining
context. Interestingly, TC sender, unlike T sender, does not always send the
same signal given the target (Fig. 4c). However, this does not amount to
context-dependent communication, since at the receiver’s end the signal always
unequivocally indicates the target, which is shown by the maximum value of
target certainty (Fig. 4d). Let us note that, nevertheless, receiver context
information gain is non-zero, nearing $1.2$ for both receiver types: the
context is informative about the receiver’s response as it constrains it
directly.

###  6.2 Experiment 2: Penalised vocabulary size

In this experiment, the fitness of the agents is reduced proportionally to the
size of their vocabulary. According to previous research (Santana,, 2014;
Mühlenbernd,, 2021), this should introduce pressure for contextual
communication. We expected that TC sender’s access to context should allow it
to “get away” with ambiguity, which would lead to languages with a smaller
vocabulary. T sender, since it does not perceive the receiver’s context,
should not be able to communicate in a context-dependent manner despite the
pressure that favours context-dependence. We tested two different levels of
the vocabulary penalty coefficient $p_{\textit{voc}}$: $0.05$ and $0.1$.

a)  | b)
---|---
c)  | d)
Figure 5: Metric plots for Experiment 2: T and TC sender combined with 2
different levels of vocabulary penalty. The dark area around curves marks the
standard error.

All emergent signal systems were influenced by pressure to minimise their
vocabulary size, which now decreased from values slightly above $4$ in
Experiment 1 to values ranging from $1.4$ to $2.6$ (Fig. 5b). Significant
signal ambiguity, indicated by reduced target certainty (Fig. 5d), appeared in
all conditions except “T + 0.05pVoc”. Figure 6 qualitatively describes the
strategy used by the TC sender, while Figure 7 additionally shows some of the
language-meaning links from the receiver’s perspective.

For all agents with ambiguous signals, the ambiguity came at the price of
slightly reduced accuracy (Fig. 5a). Nevertheless, the accuracy was still
relatively high (above $0.88$) compared to the target certainty ($0.49$ to
$0.78$) (Fig. 5d). This means that the signal alone without contextual
information and constrained object choice would only allow for much lower
accuracy. This effect could be observed to some degree for both sender types.
The achieved accuracy did not exceed the maximum contextless interpretation
accuracy for either type of sender (see Appendix A for exact values),
confirming that the receiver architecture did not allow for any sort of
contextual inference.

There are differences between T and TC senders. At near-identical fitness
levels, TC sender managed to reduce its vocabulary in comparison to T sender.
While T sender could only produce symbols based on the target, TC sender’s
signal certainty values are below maximum (Fig. 5c) as the emitted signal can
be conditional on the context. The target certainty was much lower for TC than
for T sender for both penalty levels. In other words, TC sender managed to
“reuse” the same signals to a greater degree than T sender, which resulted in
greater ambiguity of the signals actually used.

---
Figure 6: Signal emission probabilities for TC sender in Experiment 2 for an
exemplary target and exemplary contexts. Given yellow triangle target, the
sender always sends one of two signals. Our inspection of the resulting
protocol shows that depending on the context, the sender’s signal follows one
of two distinct distributions, exemplified here. 1) For most contexts, signal
$a$ is sent with maximum probability, but 2) if green triangle also appears in
the context, one of the two signals, either $a$ or $b$ is sent with near-equal
probability.
---
Figure 7: Meaning probabilities for TC sender in Experiment 2 for exemplary
signal $a$ and exemplary contexts. Signal $a$, across all contexts, always
refers to one of three objects. The bottom two contexts show two different
cases of how potentially conflicting meanings are resolved. When both green
triangle and green circle are present, the signal is dysfunctionally ambiguous
as it may refer to either of them. When both green circle and yellow triangle
are present, there is one conventionalized referent: yellow triangle.

##  7 Discussion

We investigated a situated communicative scenario in which the environment
directly constrains the receiver’s response to the objects in the current
context and thus constrains its interpretation of the signal. In this setting,
we examined the consequences of pressure to minimise vocabulary. Our results
are consistent with previous modelling work in that the penalty on vocabulary
size induced out-of-context language ambiguity (Santana,, 2014; O’Connor,,
2015; Mühlenbernd,, 2021). Furthermore, they show the ambiguity that arises as
a trade-off between accuracy and minimisation of vocabulary size, in
accordance with O’Connor, (2015) predictions that a degree of inaccuracy is
optimal for a certain ratio between the cost of vocabulary size and the size
of the meaning space to be expressed. In Experiment 2, with a large enough
vocabulary size penalty, the accuracy dropped below the maximum value for both
sender types.

In our model, unlike in the above works, effective disambiguation of signal
meaning depends on the environmental constraints on the referent choice, and
not on the receiver’s inferential capabilities (Piantadosi et al.,, 2012). The
effectiveness of disambiguation can be seen in that the accuracy reached
levels higher than our out-of-context baseline – target certainty – for both
types of senders. While the target-in-context sender (TC) did not outperform
the target-only sender (T) in terms of raw accuracy, it developed a more
ambiguous language with a smaller vocabulary that was still reasonably
effective. This was the case even though the receiver was always context-
agnostic. The context-aware sender was able to capitalise on the situated
context constraining receiver’s actions. This kind of context-dependent
signaling constitutes a variant of audience design that does not rely on
simulating inferential processes at the receiver’s end, as in some previous
works (Batali,, 1998; Brochhagen,, 2020). The need for context awareness on
the sender’s side is consistent with empirical experiments by Winters et al.,
(2018), where the most sophisticated languages that encoded only the
contextually relevant semantic distinctions emerged only when the sender was
able to perceive the context.

Disambiguation mechanisms similar to those present in our models arguably play
a major role in shaping languages; natural languages evolve under pressures
that favour ambiguity, such as minimising speaker effort (Piantadosi et al.,,
2012) or language simplicity (Kirby et al.,, 2015), and these mechanisms may
allow ambiguity to enter the language without sacrificing communication
success. Although constrained referent choice in the absence of the sender’s
context awareness does not offer a systematic disambiguation mechanism, it may
still greatly reduce communication errors and thus pressure for meaning
precision. As the two meanings of the word “bat” – the baseball bat and the
flying mammal – are unlikely to be relevant in the same context, the word
itself may safely remain ambiguous. When the sender becomes aware of the
context, different or additional linguistic means can be used to specify the
meaning. This was illustrated by the context-aware sender’s behaviour in our
model and reflected in an experiment by Ferreira et al., (2005), where
participants were tasked to describe a scene. Perceiving both the flying
mammal and a baseball bat in one scene, the participants often used an
additional specifier for one of the objects, producing descriptions such as
“bat and baseball bat”. Clearly, complex human inferential capabilities that
involve both sender and receiver, as considered in pragmatics (e.g. Korta and
Perry, (2020)), will likely make the ambiguous language even more effective.

While context-dependent communication has usually been portrayed as relying on
processes internal to the communicating agents, we demonstrate that a
nontrivial part of the job of disambiguation is performed by the environment
itself (Brooks,, 1990; Wilson and Golonka,, 2013). Consequently, it seems that
the composition of contexts in which we find ourselves plays a key role. Guo
et al., (2021) show in a computational model that languages developed by
neural agents in signaling games with medium context size encode more
information than those emerging within a smaller or larger context. We can
speculate that another source of environmentally induced ambiguity may be
simply the frequency of referents. Huttegger, (2007) shows that for an uneven
distribution of referent types, the population is more likely to stay in a
suboptimal ambiguous signaling strategy. As some types need to be communicated
less frequently, the potential payoff of assigning them a unique symbol is
reduced. In our simulations, we used a very simple environment with uniform
distribution over the communicated targets and uncorrelated object occurrence
within the context. More complex environments with predictable regularities
open new possibilities. If two particular objects never co-occur (the baseball
bat and the flying mammal), the same signal can be potentially used for both
of them without causing communication errors. Winters et al., (2018) show that
the predictable composition of the context is, in fact, exploited by human
participants who, consequently, arrive at simpler languages than those
emerging for fully random contexts.

There may be more mechanisms responsible for handling ambiguity in language
than is often assumed. The phenomena discussed so far concerned one-
directional communication. In contrast, human interlocutors engage in
dialogues where they explicitly ask for disambiguation (Healey et al.,, 2018)
or even co-construct meaning together (Kempson et al.,, 2019). Looking from
this angle, we discover that not only sharing context, but also aggregating
information from individually available contexts is important for dialogically
co-constructed messages (Rączaszek-Leonardi et al.,, 2014). These multiple
mechanisms can remedy meaning uncertainty when precision is vital; the very
same uncertainty can allow for language flexibility, which is indispensable on
other occasions (see Section 2).

There are a couple of directions where the current work can be developed.
First, future research could examine the effects of various context
properties, such as the distribution of objects in the environment. In our
work, we used only a uniform distribution; moving away from it would affect
relative benefits of distinguishing certain object properties and also
introduce regularities that could be exploited by agents in their
disambiguation strategies. Second, the analysis could be extended to answer
questions concerning emerging language structure, an issue that we have left
aside in our work. As in natural languages a neighbouring word often
sufficiently disambiguates an otherwise ambiguous expression (Ferreira,,
2008), we could expect that complexifying signals would render their
individual components even more ambiguous. One possible amendment to the model
would be allowing agents to emit compound signals of variable length. Finally,
the current model could be altered to allow bi-directional exchange between
agents (in the vein of Kottur et al., (2017); Bancerek and Zubek, (2023)).
This would allow us to investigate dialogical mechanisms of disambiguation and
their interactions with situational context in constructing flexible language.

##  8 Conclusion

In this work, we developed a model in which the pressure that is conducive to
the emergence of context-dependent communication was combined with the
environmentally constrained interpretation of the signal by the receiver. In
doing this, we complemented previous modelling work, which focused on how
communicating agents can exploit shared information by means of inference. To
the best of our knowledge, this is also the first work where environmentally
constrained communication was modelled in conjunction with a vocabulary size
cost. Our results point to the fact that context-dependent communication is
likely a multilayered phenomenon, which should be analysed not only in terms
of individual inferential capabilities, but also in terms of the agents-
environment relation. In line with other authors of recent research on
linguistic context dependence, we urge that out-of-context signal ambiguity
should not be readily equated with proneness to error but rather understood as
a pervasive property of real-life situated communication, resulting in
communicative efficiency and flexibility.

## Acknowledgements

This work was funded by the National Science Centre, Poland (OPUS 15 grant,
2018/29/B/HS1/00884). The experiments were partly run on the supercomputers of
the Interdisciplinary Centre for Mathematical and Computational Modelling at
the University of Warsaw (computational grant G86-1039). The metrics in the
experimental results were tracked via Weights & Biases (Biewald,, 2020).

## Declarations of interest

The authors declare that they have no competing interests.

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##  Appendix A Appendix

###  A.1 Model hyperparameters

This section lists the details of network architecture and hyperparameters
used for both experiments in this work. All non-LTSM network layers are
interleaved with ReLU activations. TC sender consists of a single LSTM layer
and $2$ linear layers. T sender and the receiver network are both made of $3$
linear layers. The hyperparameter values are manually selected to allow smooth
optimisation. Hyperparameters below describe all networks in this work:

Network structure:

  * •

The size of all dense layers: $50$

  * •

The size of the hidden state and the cell state for LSTM cells: $50$

Optimisation with evolution strategies

  * •

mini-batch size: $50$

  * •

episodes per iteration: $400$

  * •

sample noise standard deviation $\sigma$: $0.1$

  * •

gradient optimiser: AdamW with weight decay $0.01$ and learning rate dependent
on the sender type:

    * –

$0.02$ for the receiver

    * –

$0.05$ for TC sender and $0.02$ for T sender

The results of Lazaridou et al., (2018) suggest that a too low signal length
may make the model more prone to getting stuck in the local minima; we set a
relatively large signal length $10$, to ensure that such a problem does not
occur.

###  A.2 Experiment results

Table 1: Final metric values for Experiment 1 run | metric | mean | std | min | max
---|---|---|---|---|---
T | fitness | 1.000 | 0.001 | 0.998 | 1.000
TC | fitness | 0.999 | 0.001 | 0.996 | 1.000
T | accuracy | 1.000 | 0.001 | 0.998 | 1.000
TC | accuracy | 0.999 | 0.001 | 0.996 | 1.000
T | signal vocabulary size | 4.250 | 0.961 | 3.181 | 5.605
TC | signal vocabulary size | 4.378 | 0.648 | 3.372 | 5.048
T | receiver context information gain | 1.220 | 0.005 | 1.211 | 1.226
TC | receiver context information gain | 1.218 | 0.005 | 1.212 | 1.228
T | max contextless interpretation accuracy | 1.000 | 0.000 | 1.000 | 1.000
TC | max contextless interpretation accuracy | 1.000 | 0.000 | 0.999 | 1.000
T | sender context information gain | 0.035 | 0.027 | 0.002 | 0.081
TC | sender context information gain | 0.780 | 0.274 | 0.250 | 1.128
T | signal entropy | 3.217 | 0.048 | 3.158 | 3.294
TC | signal entropy | 4.231 | 0.368 | 3.504 | 4.732
T | signal certainty | 0.982 | 0.014 | 0.962 | 0.999
TC | signal certainty | 0.682 | 0.104 | 0.563 | 0.896
T | target certainty | 1.000 | 0.000 | 0.999 | 1.000
TC | target certainty | 0.999 | 0.001 | 0.996 | 1.000
Table 2: Final metric values for Experiment 2 run | metric | mean | std | min | max
---|---|---|---|---|---
T + 0.05pVoc | fitness | 0.872 | 0.005 | 0.865 | 0.878
T + 0.1pVoc | fitness | 0.743 | 0.007 | 0.729 | 0.748
TC + 0.05pVoc | fitness | 0.869 | 0.007 | 0.859 | 0.880
TC + 0.1pVoc | fitness | 0.744 | 0.010 | 0.730 | 0.765
T + 0.05pVoc | accuracy | 0.999 | 0.001 | 0.998 | 1.000
T + 0.1pVoc | accuracy | 0.937 | 0.012 | 0.915 | 0.944
TC + 0.05pVoc | accuracy | 0.973 | 0.010 | 0.960 | 0.983
TC + 0.1pVoc | accuracy | 0.889 | 0.027 | 0.863 | 0.963
T + 0.05pVoc | signal vocabulary size | 2.552 | 0.095 | 2.406 | 2.685
T + 0.1pVoc | signal vocabulary size | 1.937 | 0.044 | 1.853 | 1.967
TC + 0.05pVoc | signal vocabulary size | 2.089 | 0.113 | 1.990 | 2.297
TC + 0.1pVoc | signal vocabulary size | 1.444 | 0.191 | 1.331 | 1.982
T + 0.05pVoc | receiver context information gain | 1.218 | 0.006 | 1.207 | 1.227
T + 0.1pVoc | receiver context information gain | 1.129 | 0.018 | 1.092 | 1.146
TC + 0.05pVoc | receiver context information gain | 1.196 | 0.010 | 1.178 | 1.209
TC + 0.1pVoc | receiver context information gain | 1.108 | 0.026 | 1.084 | 1.179
T + 0.05pVoc | max contextless interpretation accuracy | 1.000 | 0.000 | 1.000 | 1.000
T + 0.1pVoc | max contextless interpretation accuracy | 0.944 | 0.010 | 0.922 | 0.950
TC + 0.05pVoc | max contextless interpretation accuracy | 0.975 | 0.010 | 0.962 | 0.984
T + 0.05pVoc | sender context information gain | 0.002 | 0.001 | 0.000 | 0.004
T + 0.1pVoc | sender context information gain | 0.003 | 0.003 | 0.001 | 0.011
TC + 0.05pVoc | sender context information gain | 0.123 | 0.060 | 0.052 | 0.208
TC + 0.1pVoc | sender context information gain | 0.259 | 0.052 | 0.126 | 0.305
T + 0.05pVoc | signal entropy | 3.157 | 0.003 | 3.154 | 3.162
T + 0.1pVoc | signal entropy | 2.673 | 0.093 | 2.494 | 2.727
TC + 0.05pVoc | signal entropy | 2.913 | 0.083 | 2.787 | 3.036
TC + 0.1pVoc | signal entropy | 2.065 | 0.247 | 1.908 | 2.764
T + 0.05pVoc | signal certainty | 0.999 | 0.001 | 0.998 | 1.000
T + 0.1pVoc | signal certainty | 0.999 | 0.002 | 0.994 | 1.000
TC + 0.05pVoc | signal certainty | 0.946 | 0.032 | 0.899 | 0.983
TC + 0.1pVoc | signal certainty | 0.896 | 0.026 | 0.855 | 0.955
T + 0.05pVoc | target certainty | 1.000 | 0.001 | 0.999 | 1.000
T + 0.1pVoc | target certainty | 0.776 | 0.042 | 0.696 | 0.800
TC + 0.05pVoc | target certainty | 0.843 | 0.060 | 0.769 | 0.892
TC + 0.1pVoc | target certainty | 0.490 | 0.101 | 0.450 | 0.778