# Goal-oriented compression for $L_{p}$-norm-type goal functions: Application
to power consumption scheduling

Yifei Sun  Hang Zou  Chao Zhang  Samson Lasaulce  Michel Kieffer

###### Abstract

Conventional data compression schemes aim at implementing a trade-off between
the rate required to represent the compressed data and the resulting
distortion between the original and reconstructed data. However, in more and
more applications, what is desired is not reconstruction accuracy but the
quality of the realization of a certain task by the receiver. In this paper,
the receiver task is modeled by an optimization problem whose parameters have
to be compressed by the transmitter. Motivated by applications such as the
smart grid, this paper focuses on a goal function which is of $L_{p}$-norm-
type. The aim is to design the precoding, quantization, and decoding stages
such that the maximum of the goal function obtained with the compressed
version of the parameters is as close as possible to the maximum obtained
without compression. The numerical analysis, based on real smart grid signals,
clearly shows the benefits of the proposed approach compared to the
conventional distortion-based compression paradigm.

###### keywords:

Data compression, Goal oriented communications, Quantization, Learning, Neural
networks, Precoding.

††journal: Journal of The Franklin Institute

[label1]organization=Universite Paris-Saclay - CNRS - CentraleSupelec - L2S,
city=Gif-sur-Yvette, postcode=F-91192, country=France

[label2]organization=Technology Innovation Institute,city=Abu Dhabi,
country=UAE

[label3]organization=School of Computer Science and Engineering, Central South
University,city=Changsha, postcode=410083, country=China
[label4]organization=Khalifa University,city=Abu Dhabi, postcode=127788,
country=UAE

[label5]organization=CRAN, CNRS-Universite de Lorraine,city=Nancy,
postcode=F-54000, country=France

General framework for designing compression methods for the $L_{p}$ norm
minimization problem.

Novel linear and nonlinear transformation schemes by taking into account the
performance degradation in terms of the $L_{p}$ norm induced by model
reduction.

Tailor the quantization rule to be goal-oriented by considering the impact of
the precoding and the final use of the compressed data.

Evaluation of the proposed coding schemes with a real dataset and show the
significant performance improvement compared to existing conventional
transformation and quantization techniques.

##  1 Introduction

With the development of new paradigms such as the industrial internet, the
internet of things (IoT), the smart grid, or networked controlled systems for
instance, networks become more and more distributed. Information exchanges
between the different devices are necessary to implement cooperation or
coordination and achieve a given goal. A huge amount of data is often
generated and transmitted, such as in the smart grid [1, 2]. Because of
practical limitations in terms of communication and computational resources,
it is important if not necessary to compress the exchanged data.

There exists a quite solid literature on the problem of data compression in
the smart grid. Compared to lossless compression (see e.g., [3]), lossy
compression achieves a higher compression ratio at the expense of degrading
the accuracy of data. Reference [4] transforms power quality event data to 2D,
showing the correlation between far away sample, which help to exploit the
redundancy of the data. [5] compresses the estimated fundamental component and
transient component by different compression techniques. A two-step
compression is proposed in [6]. The dimension of the data is firstly reduced
by using principal component analysis. The preserved components are further
compressed by using compressed sampling. Due to the limit in terms of rate,
quantization schemes [7] are often used to minimize distortion [8].

Most of the aforementioned lossy compression techniques try to minimize the
mean square error (MSE) while compressing the data. Alternative distortion
measures have been considered for a long time, see [7] and the references
therein. Indeed, it has been known that the MSE may be not suited for
assessing the performance of various image processing operations (_e.g._ ,
image segmentation or pattern recognition). Over the past years, the signal
processing and digital communications communities have realized that the
distortion-based data compression paradigm should be revisited [9, 10, 11, 12,
13, 14, 15] by adapting the compression scheme to the final use, task, or goal
pursued by the receiver. The attempt to incorporate aspects such as semantics
and effectiveness into communication theory is in fact not new. For instance,
reference [16] adopts a probabilistic logic approach to study the semantic
aspect whereas [17] develops a complexity theory approach of the goal-oriented
communication problem. Concerning the signal processing and communication
point of view, which is the one of interest for the present paper, the
literature of goal-oriented communications is still in full development. For
instance, a goal-oriented signal processing technique, based on a graph-based
semantic language and a goal filtering method, is proposed and applied to
specific goals in [13]. The goal-oriented/semantic communication concept has
also been developed to design the next cellular communication generation (see
e.g., [14, 18, 19, 20]).

Concerning the goal-oriented approach for the data compression problem, the
literature is equally relatively small but contains some relevant works that
the present work can be related to. For data compression also, the signal
processing community, has also been aware well before the recent publications
on the topic that large compression gains might be reaped when considering the
final use of the signal or image (see, _e.g._ , [21, 22, 23, 24] for the case
of speech compression). The goal-oriented quantization problem is posed and
formalized for the first time in [9] and is tackled in a deeper manner in
[25], [26], [10], [27], and in [28]. In [29], the problem of goal-oriented
quantization is studied for the control problem of rendez-vous in a multi-
agent setting. As for the goal-oriented precoding or data preprocessing
problem, it has been introduced independently in [10] and [30].

Compared to the most related works, the present paper provides significant
progresses into the design of goal-oriented data source encoders. The paper
presents a novel task-oriented compression scheme when the task can be modeled
by an optimization problem, which is a very relevant model for communications
and energy systems. Our approach precisely assumes the task can be represented
by a function (to be maximized) whose variables are the decisions to be taken
and whose parameters have to be compressed. The focus of this paper is on the
$L_{p}-$norm minimization problem, which is an important problem for example
for smart grid applications, and in particular, for the fundamental problem of
power consumption scheduling (_e.g._ , to minimize the total consumed energy
price, Joule losses, or the peak power). More specifically, for the goal-
oriented precoding or transform stage, the best linear solution is developed
and compared to a non-linear solution which is based on a convolution auto-
encoder. In contrast with [10], the proposed precoding scheme exploits the
structure of the decision function (an approximation of it to be precise).
Also, a goal-oriented quantizer is used not to quantize the input signal (as
in [29, 28]) but the goal-oriented precoder output. Additionally, the used
quantizer is not assumed to be a set of uniform scalar quantizers as in [10]
and also it is tailored to the goal, which is an $L_{p}$-norm-type utility
function. We also investigate the problem of knowing how the two compression
stages interact each other, to understand to what extent it is possible to
accumulate the gains of the goal-oriented paradigm when applied to the two
stages. Last but not least, the schemes are applied to real smart grid
measurements, which leads to a detailed numerical performance analysis and
discussing the design of goal-oriented precoding and quantization. The paper
does not only provide implementable coding schemes but also provides
quantitative elements behind the intuition that accounting for the impact of
compression noise on the task is beneficial. Our approach allows one to
provide e.g., an analytical characterization of an approximation of the
optimal precoder, a goal-oriented quantization algorithm which works for an
arbitrary utility function and not only for the $L_{p}$-norm. A purely data-
driven approach (based on neural networks) would not provide these structural
elements which can be exploited both for interpretations and making
implementation easier.

The rest of the paper is organized as follows. We formulate the problem in
Section 2. In Section 3 a linear approximation (LT) is considered for the
optimal decision function. A linear and nonlinear transformation (NLT) are
proposed in Section 4 and Section 5 respectively. We describe the proposed
goal-oriented quantizer in Section 6. An iterative approach optimizing the
linear transformation stage and the goal-oriented quantization stage is
proposed in Section 7. Section 8 provides the numerical performance analysis.
The paper is concluded in Section 9.

##  2 Problem formulation

Consider the following utility function

$u\left(x;\ell\right)=-||x+\ell||_{p}$ (1)

where
$x=\left[x_{1},x_{2},\dots,x_{N}\right]^{\mathrm{T}}\in\mathbb{R}_{+}^{N}$
represent the vector of decision variables,
$\ell=\left[\ell_{1},\ell_{2},\dots,\ell_{N}\right]^{\mathrm{T}}\in\mathbb{R}_{+}^{N}$
is a vector of parameters, and $\|\cdot\|_{p}$ is the $L_{p}$-norm, _i.e._ ,
for any $v\in\mathbb{R}^{n}$,
$\|v\|_{p}=\left(|v_{1}|^{p}+\dots+|v_{n}|^{p}\right){}^{1/p}$ with
$p\geqslant 1$. Assuming that the sum of the decision variables is lower-
bounded as

$\sum_{j=1}^{N}x_{j}\geqslant E$ (2)

where $E>0$ is a constant and that (1) has to be maximized, one obtains the
following optimization problem

$\displaystyle\underset{x}{\mathrm{maximize}}$ $\displaystyle\
u\left(x;\ell\right)$ (3) s.t. $\displaystyle\sum_{j=1}^{N}x_{j}-E\geqslant 0$
$\displaystyle x_{j}\geqslant 0,\ j=1,\dots,N.$

The solution of (3) is denoted as $x^{\star}\left(\ell\right)$. The
optimization problem (3) can model typical resource allocation problems in
several applications. For instance, in the power consumption scheduling
problem of energy systems, $\ell$ represents the non-controllable part of the
energy consumption of a given household over $N$ time slots (_e.g._ , consumed
by the lighting and cooking appliances, TV, computers), $x$ represents the
controllable part (_e.g._ , the desired state-of-charge of an electric vehicle
(EV)) to be allocated over the $N$ time slots by some decision-making entity
(scheduler), the constraint (2) corresponds to the minimum amount of energy
that has to be provided. The utility function thus corresponds to minus the
$L_{p}$-norm of the total consumption vector. When $p$ becomes large,
maximizing the considering utility amounts to minimizing the peak power. When
$p=2$ and $\ell$ and $x$ are interpreted as currents instead of powers, then
it corresponds to the Joule losses minimization problem.

We assume that only an approximated version $\widehat{\ell}\in\mathbb{R}^{N}$
of the non-controllable load vector $\ell$ is available to the scheduler which
takes the decision $x$. This assumption can be not only motivated by the
existence of limitations in terms of communication or computational resources
but also for a need in terms of privacy. A lossy compression technique is thus
implemented to remove some redundancy from the source signal $\ell$.
Considering $\widehat{\ell}$ in place of $\ell$ in (3), the resulting solution
becomes $x^{\star}\left(\widehat{\ell}\right)$.

Figure 1: Coding scheme targeting optimality loss minimization

Instead of minimizing the reconstruction error as in conventional compression
schemes, this paper aims to design a compression scheme for $\ell$ to mitigate
the performance degradation resulting from the distortion (compression noise)
introduced on $\widehat{\ell}$, _i.e._ , to minimize the expected optimality
loss

$\Gamma=\mathbb{E}_{\ell}\left[\left|u\left(x^{\star}\left(\ell\right);\ell\right)-u\left(x^{\star}\left(\widehat{\ell}\right);\ell\right)\right|^{2}\right],$
(4)

where the expectation is over $\ell$. To define properly the arguments of
$\Gamma$, it is first needed to define the different functions that need to be
optimized. These functions appear in Figure 1. In this setting, the
compression noise stems both from precoding/decoding and quantization. The
precoding mapping

$\begin{array}[]{cccc}g:&\mathbb{R}_{+}^{N}&\rightarrow&\mathbb{R}^{K}\\\
&\ell&\mapsto&\theta\end{array}$ (5)

transforms the parameter vector $\ell$ to a vector $\theta$ of dimension
$K\leqslant N$, and the dimensionality reduction induces some information
loss. A quantizer

$\begin{array}[]{cccc}q:&\mathbb{R}^{K}&\rightarrow&\mathbb{R}^{K}\\\
&\theta&\mapsto&\widehat{\theta}\end{array}$ (6)

follows the precoder, which constitutes a second source of the compression
noise. The quantized vector $\widehat{\theta}$ is transmitted and assumed
received without error by the decision-making entity. A decoding function

$\begin{array}[]{cccc}h:&\mathbb{R}^{K}&\rightarrow&\mathbb{R}^{N}\\\
&\widehat{\theta}&\mapsto&\widehat{\ell}\end{array}$ (7)

is then used to obtain an estimate $\widehat{\ell}$ of $\ell$.

The goal is to design the precoding, quantization, and decoding functions that
minimize the expected optimality loss (4)

$\left(g^{\star},q^{\star},h^{\star}\right)\in\arg\min_{\left(g,q,h\right)}\Gamma(g,q,h)=\mathbb{E}_{\ell}\left[\left|u\left(x^{\star}\left(\ell\right);\ell\right)-u(x^{\star}(\widehat{\ell});\ell)\right|^{2}\right]$
(8)

where $\widehat{\ell}=h(q(g(\ell)))$.

Solving (8) directly is generally a hard task computationally speaking. This
is the main motivation for searching for suboptimal solutions whose
determination involve affordable complexity. In this paper, the functions $g$,
$q$, and $h$ are therefore optimized separately. More precisely, the precoding
stage without quantization noise is designed. In contrast with conventional
linear transformation such as the Karhunen-Loève Transform (KLT), the
precoding stage aims at minimizing the optimality loss in terms of the utility
function $u$. The KLT is known to be the best linear transformed in terms of
MSE but this optimality no longer holds for the optimality loss. Second, the
precoding/decoding scheme is fixed and a goal-oriented quantization scheme is
proposed to mitigate the performance degradation brought by quantization
noise. At last, the precoding/decoding stage and the quantization stage are
optimized in an iterative manner.

##  3 Linear approximation of the optimal decision

The impact of quantization on the optimality loss, as defined by (4), can be
seen to depend on the utility function $u$ and the optimal decision function
$x^{\star}$. In [28], it has been proved formally (in the high-resolution
regime) how the regularity and smoothness properties of these functions impact
the optimality loss. To be able to exploit the optimal decision function in
the design of the precoding stage, we resort to a linear approximation of the
former. In this section, we thus first characterize the solution of the
optimization problem (3). Then, a linear approximation of this solution is
provided to make the problem tractable.

###  3.1 Optimal decision function

For a given value of $\ell$, the solution of (3) is provided by Proposition 1
and involves a water-filling approach, which is widely used [31, 32, 33].

###### Proposition 1

Consider a value of the parameter vector $\ell$, and assume, without loss of
generality, that

$\ell_{1}\leqslant\dots\leqslant\ell_{j}\leqslant\dots\leqslant\ell_{N}.$

Then the components of $x^{\star}\left(\ell\right)$ are obtained by

$x_{j}^{\star}=\left(\mu-\ell_{j}\right)^{+},$ (9)

where

$\mu=\frac{1}{n^{\star}}\left(E+\sum_{j=1}^{n^{\star}}\ell_{j}\right)$

indicates the water level, $\left(\cdot\right)^{+}=\max\left(\cdot,0\right)$,
and $n^{\star}$ is the number of non-zero components of
$x^{\star}\left(\ell\right)$ evaluated as

$n^{\star}=\max_{n\leqslant
N}\left\\{n:\left(n-1\right)\ell_{n}-\sum_{j=1}^{n-1}\ell_{j}\leqslant
E\right\\}.$ (10)

###### Proof 1

See A.1.

###  3.2 Linear approximation

The solution provided by Proposition 1 has a non-linear dependency in $\ell$,
and this dependency is not explicit in general. To circumvent this difficulty
we resort to a first order approximation of the optimal decision function. For
this purpose, we evaluate the sensitivity of $x^{\star}\left(\ell\right)$ with
respect to $\ell$, by considering the first-order Taylor expansion of
$x^{\star}$ around $\ell$

$\displaystyle x^{\star}\left(\ell+d\ell\right)$
$\displaystyle=x^{\star}\left(\ell\right)+\boldsymbol{H}(\ell)d\ell+o\left(d\ell\right)$
(11)

where

$\boldsymbol{H}(\ell)=\frac{\partial
x^{\star}}{\partial\ell^{\mathrm{T}}}\left(\ell\right)$

is the Jacobian matrix of $x^{\star}\left(\ell\right)$ obtained using
Proposition 2.

###### Proposition 2

Consider a value of the parameter vector $\ell$ such that $n^{\star}$ given by
(10) remains constant over some neighborhood $\mathcal{N}\left(\ell\right)$ of
$\ell$. Without loss of generality, assume that

$\ell_{1}\leqslant\dots\leqslant\ell_{n^{\star}}\leqslant\ell_{n^{\star}+1}\leqslant\dots\leqslant\ell_{N}.$

Then

$x^{\star}\left(\ell\right)=\boldsymbol{H}\left(\ell\right)\ell+b\left(\ell\right)$
(12)

with

$\displaystyle\boldsymbol{H}\left(\ell\right)=$
$\displaystyle\left(\begin{array}[]{ccccccc}-1+\frac{1}{n^{\star}}&\frac{1}{n^{\star}}&\text{$\cdots$}&\frac{1}{n^{\star}}&0&\cdots&0\\\
\frac{1}{n^{\star}}&-1+\frac{1}{n^{\star}}&\ddots&\vdots&\vdots&&\vdots\\\
\vdots&\ddots&\ddots&\vdots\frac{1}{n^{\star}}&\vdots&&\vdots\\\
\frac{1}{n^{\star}}&\cdots&\frac{1}{n^{\star}}&-1+\frac{1}{n^{\star}}&0&\cdots&0\\\
0&\cdots&\cdots&0&0&\cdots&0\\\ \vdots&&&\vdots&\vdots&\ddots&\vdots\\\
0&\cdots&\cdots&0&0&\cdots&0\end{array}\right)$

and

$b\left(\ell\right)=\left(\frac{E}{n^{\star}},\dots,\frac{E}{n^{\star}},0,\dots,0\right)^{\mathrm{T}}.$

###### Proof 2

See A.2.

##  4 Linear goal-oriented precoding

In this section, we investigate the problem of the determination of the best
linear precoding or transform to be applied to $\ell$, which is to approximate
$\ell$ by a weighted sum of a given number of vectors of a certain basis; the
number of vectors of this sum precisely corresponds to the number
$K\in\\{1,\dots,N\\}$ defined by (5). It is known that, for a given number of
basis vector, the KLT provides the best basis in the sense of the MSE
$\mathbb{E}_{\ell}(\|\ell-\widehat{\ell}\|^{2})$ (see, _e.g._ , [34]).
Nevertheless, this optimality result is no longer true in the presence of an
arbitrary goal and in particular for the optimality loss definition used in
this paper. The motivation of this section is therefore to propose a linear
precoding or transformation scheme which is matched to the $L_{p}$-norm
function.

In this section, to simplify the analysis, the effect of quantization noise is
assumed to be negligible. Under these assumptions, one has that
$\theta=g\left(\ell\right)=\boldsymbol{B}\ell$, and
$\widehat{\ell}=h\left(\theta\right)=\boldsymbol{B}^{\mathrm{T}}\theta$, where
$\boldsymbol{B}\in\mathbb{R}^{K\times N}$ is the precoding matrix and
$\boldsymbol{B}^{\mathrm{T}}$ the decoding matrix. Then the expected
optimality loss (4) becomes

$\Gamma\left(\boldsymbol{B}\right)=\mathbb{E}_{\ell}\left[\left|u\left(x^{\star}\left(\ell\right);\ell\right)-u(x^{\star}(\boldsymbol{B}^{\mathrm{T}}\boldsymbol{B}\ell);\ell)\right|^{2}\right].$
(13)

Two notational remarks are in order at this point. First, notice that in the
above, a small abuse of notation is employed for clarity. Since, the
optimality loss function is only considered with respect to the precoding
stage, the arguments $q$ and $h$ are removed from $\Gamma$, and since only
linear precoding is considered, the precoding function $g$ is replaced with
the matrix $\boldsymbol{B}$. Second, motivated by practical considerations, in
which exact statistics are not available but one has only access to a set of
measurements, the notation $\widehat{\Gamma}_{T}$ will be use to refer to the
empirical version of the optimality loss function $\Gamma$. Assuming that a
dataset
$\mathcal{L}=\\{\ell^{\left(1\right)},\ell^{\left(2\right)},\dots,\ell^{\left(T\right)}\\}$
of $T$ realizations or samples of $\ell$ is available, the empirical
optimality loss expresses as

$\displaystyle\widehat{\Gamma}_{T}\left(\boldsymbol{B}\right)=$
$\displaystyle\frac{1}{T}\sum_{i=1}^{T}\left|u(x^{\star}(\ell^{(i)});\ell^{(i)})-u(x^{\star}(\boldsymbol{B}^{\mathrm{T}}\boldsymbol{B}\ell^{(i)});\ell^{(i)})\right|^{2}.$
(14)

Using the linear form (12) introduced in Proposition 2, one obtains

$\displaystyle\widehat{\Gamma}_{T}\left(\boldsymbol{B}\right)$
$\displaystyle=\frac{1}{T}\sum_{i=1}^{T}\left|u(\boldsymbol{H}\left(\ell^{(i)}\right)\ell^{(i)}+b\left(\ell^{(i)}\right);\ell^{(i)})\right.$
$\displaystyle-\left.u\left(\boldsymbol{H}\left(\boldsymbol{B}^{\mathrm{T}}\boldsymbol{B}\ell^{(i)}\right)\boldsymbol{B}^{\mathrm{T}}\boldsymbol{B}\ell^{(i)}+b\left(\boldsymbol{B}^{\mathrm{T}}\boldsymbol{B}\ell^{(i)}\right);\ell^{(i)}\right)\right|^{2}.$
(15)

To minimize $\widehat{\Gamma}_{T}$ we resort to a gradient descent algorithm.
To simplify the computation procedure of the derivatives of
$\widehat{\Gamma}_{T}$, it is assumed that a small variation of
$\boldsymbol{B}$ does not change the entries of
$\boldsymbol{H}\left(\boldsymbol{B}^{\mathrm{T}}\boldsymbol{B}\ell^{(i)}\right)$
and $b\left(\boldsymbol{B}^{\mathrm{T}}\boldsymbol{B}\ell^{(i)}\right)$ for
all $i=1,\dots,T$. This assumption allows one to state the following
proposition.

###### Proposition 3

Consider $p\in\mathbb{N}^{+}$. The matrix containing the derivatives of
$\widehat{\Gamma}_{T}\left(\boldsymbol{B}\right)$ with respect to the
components of $\boldsymbol{B}$ is

$\nabla_{\boldsymbol{B}}\widehat{\Gamma}_{T}\left(\boldsymbol{B}\right)=\frac{1}{T}\sum_{i=1}^{T}C_{i}\left(\boldsymbol{B}\ell^{\left(i\right)}\beta_{i}^{\mathrm{T}}\boldsymbol{H}_{i}+\boldsymbol{B}\boldsymbol{H}_{i}^{\mathrm{T}}\beta_{i}\ell^{\left(i\right)T}\right)$
(16)

where

$C_{i}=2\left(u\left(x^{\star}\left(\ell^{\left(i\right)}\right);\ell^{\left(i\right)}\right)-u\left(x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right);\ell^{\left(i\right)}\right)\right)||x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right)+\ell^{\left(i\right)}||_{p}^{1-p}$
(17)
$\beta_{i}=\text{$\underbrace{\left(x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right)+\ell^{\left(i\right)}\right)\odot\cdots\odot\left(x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right)+\ell^{\left(i\right)}\right)}_{p-1}$}$
(18)

where $\odot$ indicates Hadamard product.

Moreover, when $p\rightarrow+\infty$, (16) boils down to

$\nabla_{\boldsymbol{B}}\widehat{\Gamma}_{T}\left(\boldsymbol{B}\right)=\frac{1}{T}\sum_{i=1}^{T}D_{i}\left(\boldsymbol{B}\ell^{\left(i\right)}s_{k\left(i\right)}^{\mathrm{T}}\boldsymbol{H}_{i}+\boldsymbol{B}\boldsymbol{H}_{i}^{\mathrm{T}}s_{k\left(i\right)}\ell^{\left(i\right)T}\right)$
(19)

where

$D_{i}=2\left(u\left(x^{\star}\left(\ell^{\left(i\right)}\right);\ell^{\left(i\right)}\right)-u\left(x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right);\ell^{\left(i\right)}\right)\right)$
(20) $s_{k\left(i\right)}=\left(\begin{array}[]{c}0_{(k\left(i\right)-1)\times
1}\\\ 1\\\ 0_{(N-k\left(i\right))\times 1}\end{array}\right)$ (21)

and

$k\left(i\right)=\arg\max_{k}x_{k}^{\star}\left(\widehat{\ell}^{\left(i\right)}\right)+\ell_{k}^{\left(i\right)}.$

###### Proof 3

See A.3

Using (16) or (19), a local minimization of
$\widehat{\Gamma}_{T}\left(\boldsymbol{B}\right)$ can be performed by using a
gradient descent algorithm (see Algorithm 1) to obtain
$\boldsymbol{B}^{\star}$. The search is initialized with the $K$ first vectors
of the KLT built from the empirical covariance matrix obtained by using the
vectors of the dataset $\mathcal{L}$.

Algorithm 1 Gradient descent search $\boldsymbol{B}^{*}$

0: Initialize $\boldsymbol{B}$ with the $K$ first vectors of Karhunen-Loève
transform

0: Initialize $i=0$,

while $i<it_{\max}$ and optimality loss reduced more than $0.01\%$ do

Compute the gradient:
$\boldsymbol{G}\leftarrow\nabla_{\boldsymbol{B}}\widehat{\Gamma}_{T}\left(\boldsymbol{B}\right)$

Perform line search to get $\lambda$ such that
$\widehat{\Gamma}_{T}\left(\boldsymbol{B}-\lambda\boldsymbol{G}\right)<\widehat{\Gamma}_{T}\left(\boldsymbol{B}\right)$

Update: $\boldsymbol{B}=\boldsymbol{B}-\lambda\boldsymbol{G}$

$i\leftarrow i+1$

end while

##  5 Nonlinear goal-oriented precoding

The use of linear transforms for data compression is largely motivated by
complexity issues. Nevertheless, linear transforms are generally not optimal.
They are optimal in terms of MSE when the input signal corresponds to
realizations of a (vector) Gaussian random variable [34]. A natural question
is thus to assess the benefits of a nonlinear transform in the presence of
goal functions such as the $L_{p}$-norm. This is why we consider here a larger
class of precoders for which the parameters can be learned. The aim of the
precoder is to obtain a latent representation $\theta\in\mathbb{R}^{K}$ of
$\ell$ as follows $\theta=g\left(\ell,\Phi\right)$, where $\Phi$ is a vector
of parameters. Then, the decoder takes a possibly quantized version of the
latent representation as input to obtain an estimate of $\widehat{\ell}$ as
$\widehat{\ell}=h\left(\theta,\Psi\right)$, where $\Psi$ is a vector of
parameters.

Auto-encoders appear as a natural tool to implement a nonlinear transform when
the purpose is to perform model reduction. Auto-encoders have been previously
developed for image compression [35, 36, 37] and are considered here to obtain
a suited low-dimension representation of $\ell$.

A convolutional auto-encoder consists of an encoding stage and a decoding
stage as illustrated in Figure 2. The encoding stage is simply a concatenation
of convolution layers followed by a fully connected layer in the end while the
decoding stage implements the inverse structure of the encoding one.

Figure 2: Structure of the considered convolutional auto-encoder to evaluate
the nonlinear goal-oriented precoding and decoding functions

The convolutional auto-encoder aims at searching a pair of
$\left(\Phi^{\star},\Psi^{\star}\right)$ minimizing the expected optimality
loss

$\left(\Phi^{\star},\Psi^{\star}\right)\in\arg\min_{\left(\Phi,\Psi\right)}\mathbb{E}_{\ell}\left[\left|u\left(x^{\star}\left(\ell\right);\ell\right)-u(x^{\star}(h\left(g\left(\ell,\Phi\right),\Psi\right));\ell)\right|^{2}\right].$
(22)

As in the previous section, we also make a small notational abuse, since
$\Psi$ and $\Phi$ are taken as the arguments of $\widehat{\Gamma}_{T}$.

Assume a set $\mathcal{L}$ of realizations of $\ell$ is available,
$\left(\Phi^{\star},\Psi^{\star}\right)$ can be trained through the loss
function defined as

$\widehat{\Gamma}_{T}\left(\Phi,\Psi\right)=\frac{1}{T}\sum_{i=1}^{T}\left|u(x^{\star}\left(\ell^{(i)}\right);\ell^{(i)})-u\left(x^{\star}\left(h\left(g\left(\ell^{(i)},\Phi\right),\Psi\right)\right);\ell^{(i)}\right)\right|^{2}.$
(23)

More precisely, as for the encoding stage in our simulation, one convolutional
layer involving two kernels with size $5$ is considered. Zero padding is
implemented to impose the output of each kernel having the same dimension as
the input. The outputs of two kernels are reshaped to a vector of dimension
$2N$. A fully connected layer between the $2N$ neurons and $K$ neurons is
considered to get the latent representation $\theta$. The decoding part is
inverse processing to reconstruct the data $\widehat{\ell}$ with dimension
$N$.

##  6 Goal-oriented quantization

In this section, we assume that the precoding and decoding functions
$g\left(\cdot\right)$ and $h\left(\cdot\right)$ are given. We focus on the
optimization of the quantizer design in order to minimize the optimality loss
for a given choice of $g\left(\cdot\right)$ and $h\left(\cdot\right)$.

For the choice of quantization schemes, vector quantization is considered in
this paper. Targeting signal reconstruction quality, using uniform scalar
quantizers followed by entropy coding constitutes an overwhelmingly popular
solution; it is adopted in neural compression schemes [35]. Nevertheless, when
the principal goal pursued is not to reconstruct the original data, it has
been shown that element-wise uniform scalar quantization (namely, using a bank
of scalar quantizers) could induce a significant performance degradation for
the task to be executed ([25, 28]). This can be explained by the fact that the
different elements of the vector of variables to be compressed can have a
markedly different influence on the final task; therefore, treating these
elements equally is generally suboptimal. As a consequence, we use vector
quantization in our scheme for task-oriented communication. Moreover, vector
quantization allows one to ensure a fixed-rate quantized output, in contrast
with auto-encoders accounting for rate constraints such as [35]. At last, our
choice is very well suited for the power consumption scheduling problem
exploiting smart meter measurements, which have a relatively small dimension.

A vector quantizer partitions the space $\mathbb{R}^{K}$ of the encoded
parameter $\theta=g\left(\ell\right)$ into several disjoint quantization
regions $\mathcal{C}_{1},\dots,\mathcal{C}_{M}$, _i.e._ ,

$\bigcup_{i=1}^{M}\mathcal{C}_{i}=\mathbb{R}^{K}$

and $\mathcal{C}_{i}\bigcap\mathcal{C}_{j}=\varnothing$ for all $i\neq j$. The
quantization function is such that

$q\left(\theta\right)=r_{i}\iff\theta\in\mathcal{C}_{i},$ (24)

where $\mathcal{R}=\left\\{r_{1},\dots,r_{M}\right\\}$ is the set of
representatives associated with the quantization regions
$\mathcal{C}_{1},\dots,\mathcal{C}_{M}$ respectively. Contrary to conventional
quantizers, which usually minimize the mean-square reconstruction error, in
what follows, we search for a pair
$\left(\mathcal{R}^{\star},\mathcal{C}^{\star}\right)$ that minimizes the
expected optimality loss

$\displaystyle\mathbb{E}_{\ell}\left[\left|u\left(x^{\star}\left(\ell\right);\ell\right)-u\left(x^{\star}\left(\widehat{\ell}\right);\ell\right)\right|^{2}\right]=$
$\displaystyle\sum_{m=1}^{M}\int_{\ell\in\mathcal{L}_{m}}\left|u\left(x^{\star}\left(\ell\right);\ell\right)-u\left(x^{\star}\left(h\left(r_{m}\right)\right);\ell\right)\right|^{2}\phi\left(\ell\right)\mathrm{d}\ell,$
(25)

where $\phi\left(\ell\right)$ is the probability density function of $\ell$
and

$\mathcal{L}_{m}\triangleq\\{\ell\in\mathbb{R}^{N}|g\left(\ell\right)\in\mathcal{C}_{m}\\},1\leqslant
m\leqslant M.$

Finding jointly $\left(\mathcal{R}^{\star},\mathcal{C}^{\star}\right)$ is not
trivial. Moreover, the evaluation of (25) considering $\phi\left(\ell\right)$
is complex, even if $\phi\left(\ell\right)$ is perfectly known. Consequently,
a practical algorithm is proposed which is similar to the decisional quantizer
proposed in [26]. The main steps of this algorithm are detailed in what
follows. As in Section 4, a set
$\mathcal{L}=\\{\ell^{\left(1\right)},\ell^{\left(2\right)},\dots,\ell^{\left(T\right)}\\}$
of realizations of the parameter $\ell$ is used to approximate (25).

Assume that at iteration $j$, a set of representatives
$\mathcal{R}^{\left(j\right)}=\left\\{r_{1}^{\left(j\right)},\dots,r_{M}^{\left(j\right)}\right\\}$
is available.

  1. 1.

From $\mathcal{R}^{\left(j\right)}$, partition the set $\mathcal{L}$ as

$\mathcal{L}_{m}^{\left(j\right)}=\left\\{\ell\in\mathcal{L}\mid\mathcal{E}\left(r_{m}^{\left(j\right)};\ell\right)=\min_{i}\mathcal{E}\left(r_{i}^{\left(j\right)};\ell\right)\right\\},1\leqslant
m\leqslant M,$ (26)

where

$\mathcal{E}\left(r_{m};\ell\right)=\left|u\left(x^{\star}\left(\ell\right);\ell\right)-u\left(x^{\star}\left(h\left(r_{m}\right)\right);\ell\right)\right|^{2}.$
(27)

  2. 2.

Update the set of representatives
$\mathcal{R}^{\left(j+1\right)}=\left\\{r_{1}^{\left(j+1\right)},\dots,r_{M}^{\left(j+1\right)}\right\\}$
as

$r_{m}^{\left(j+1\right)}\in\arg\min_{r}\sum_{\ell\in\mathcal{L}_{m}^{\left(j\right)}}\mathcal{E}\left(r;\ell\right),1\leqslant
m\leqslant M.$ (28)

For any $\ell\in\mathbb{R}_{+}^{K}$, the goal-oriented quantizer output is
then obtained as

$q^{\left(j+1\right)}\left(g\left(\ell\right)\right)=r_{m}^{\left(j+1\right)}$

with

$m=\arg\min_{i=1,\dots,M}\mathcal{E}\left(r_{i}^{\left(j+1\right)};\ell\right).$

  3. 3.

Evaluate the resulting estimate of the expected optimality loss as

$\Gamma^{\left(j+1\right)}=\sum_{m=1}^{M}\sum_{\ell\in\mathcal{L}_{m}^{\left(k\right)}}\left|u\left(x^{\star}\left(\ell\right);\ell\right)-u\left(x^{\star}\left(h\left(q^{\left(j+1\right)}\left(g\left(\ell\right)\right)\right)\right);\ell\right)\right|^{2}.$
(29)

Like Linde-Buzo-Gray (LBG) algorithm [38], Algorithm 2 performs this iterative
evaluation to determine a locally optimal goal-oriented quantizer. The set of
representatives $\mathcal{R}^{(0)}$ is initialized considering, $e.g.$,
randomly chosen elements of the set
$g\left(\mathcal{L}\right)=\left\\{g\left(\ell^{\left(1\right)}\right),g\left(\ell^{\left(2\right)}\right),\dots,g\left(\ell^{\left(T\right)}\right)\right\\}$.

Algorithm 2 Goal-oriented quantizer design algorithm

0: Utility function $u\left(x;\ell\right)$

0: $\mathcal{R}^{\star}$

0: $\mathcal{R}^{(0)}=\\{r_{1}^{(0)},\dots,r_{M}^{(0)}\\}$, $j=1$,
$\Gamma^{\left(0\right)}=\infty$

while $j\leqslant j_{\max}$ and optimality loss uced more than $0.01\%$ do

Evaluate $\mathcal{L}_{m}^{\left(j\right)},1\leqslant m\leqslant M,$ from
$\mathcal{R}^{(j-1)}$ using (26)

Evaluate $r_{m}^{\left(j+1\right)},$$m=1,\dots,M$, using (28)

Evaluate $\Gamma^{\left(j+1\right)}$ using (29)

$j=j+1$

end while

As LBG algorithm, Algorithm 2 may only converge to a local minimum of the
expected optimality loss. The convergence depends on the initialization of
$\mathcal{R}^{(0)}$.

##  7 Iterative optimization of linear transform and goal-oriented
quantization

Using Algorithm 1 and Algorithm 2, one obtains a pair of separately designed
precoder/decoder pair and quantizer. In Section 4, Algorithm 1 neglects the
impact of the quantization noise in the design of the linear precoder/decoder.
This section presents an iterative design, accounting for the quantization
noise in Algorithm 1, to further uce the optimality loss.

The effect of quantization in Section 6 is to add some noise $\eta$ to the
output of the precoder

$\displaystyle\widehat{\theta}$ $\displaystyle=q\left(\theta\right)$
$\displaystyle=\theta+\eta.$

We assume that $\eta$ is Gaussian with mean $\mu_{\eta}\in\mathbb{R}^{K}$ and
covariance matrix $\Sigma_{\eta}$. Considering the linear precoder/decoder
pair, the reconstructed signal becomes

$\displaystyle\widehat{\ell}=$
$\displaystyle\boldsymbol{B}^{\mathrm{T}}\widehat{\theta}$ (30)
$\displaystyle=$
$\displaystyle\boldsymbol{B}^{\mathrm{T}}\left(\theta+\eta\right)$
$\displaystyle=$
$\displaystyle\widetilde{\ell}+\boldsymbol{B}^{\mathrm{T}}\eta$

where
$\widetilde{\ell}=\boldsymbol{B}^{\mathrm{T}}\boldsymbol{B}\ell=\boldsymbol{B}^{\mathrm{T}}\theta$
is reconstructed from $\theta$ in absence of quantization.

Consider the $j$-th iteration of an iterative optimization algorithm of the
precoder/decoder pair and of the quantizer. Assume that a precoding matrix
$\boldsymbol{B}^{\left(j\right)}$, and a set of representatives
$\mathcal{R}^{\left(j\right)}$ have been obtained from Algorithms 1 and 2. The
characteristics of the quantization noise can be evaluated using the set
$\mathcal{L}$ of realizations of $\ell$ as

$\displaystyle\widehat{\mu}^{\left(j\right)}$
$\displaystyle=\frac{1}{T}\sum_{i=1}^{T}\eta^{\left(i\right)}$
$\displaystyle\widehat{\Sigma}^{\left(j\right)}$
$\displaystyle=\frac{1}{T}\sum_{i=1}^{T}\left(\eta^{\left(i\right)}-\widehat{\mu}_{\eta}\right)\left(\eta^{\left(i\right)}-\widehat{\mu}_{\eta}\right)^{\mathrm{T}}$

where

$\begin{aligned}
\eta^{\left(i\right)}=&q\left(\boldsymbol{B}^{\left(j\right)}\ell^{\left(i\right)}\right)-\boldsymbol{B}^{\left(j\right)}\ell^{\left(i\right)}\end{aligned}.$
(31)

At iteration $j+1$, the impact of the quantization noise can be introduced by
introducing for each $\ell^{\left(i\right)}\in\mathcal{L}$,
$\overline{\kappa}$ noisy reconstructions of $\ell^{\left(i\right)}$

$\widehat{\ell}^{\left(i,\kappa\right)}=\left(\boldsymbol{B}^{\left(j\right)}\right)^{\mathrm{T}}\boldsymbol{B}^{\left(j\right)}\ell^{\left(i\right)}+\left(\boldsymbol{B}^{\left(j\right)}\right)^{\mathrm{T}}\eta^{\left(i,\kappa\right)}$
(32)

where $\eta^{\left(i,\kappa\right)}$,
$1\leqslant\kappa\leqslant\overline{\kappa}$ are realizations of independent
and identically distributed Gaussian random vectors with mean
$\widehat{\mu}^{\left(j\right)}$ and covariance matrix
$\widehat{\Sigma}^{\left(j\right)}$. The noisy reconstructions
$\widehat{\ell}^{\left(i,\kappa\right)}$ are then used in the evaluation of
the gradient (16) or (19) to obtain a precoder/decoder optimization accounting
for the quantization noise. As a result, a matrix
$\boldsymbol{B}^{\left(j+1\right)}$ is obtained. Algorithm 2 is then applied
to get $\mathcal{R}^{\left(j+1\right)}$. Then
$\widehat{\mu}^{\left(j+1\right)}$ and $\widehat{\Sigma}^{\left(j+1\right)}$
can be evaluated.

The iterative optimization process is summarized in Algorithm 3. The
computational complexity of linear precoding part is $O\left(TN^{2}K\right)$,
the complexity of the goal-oriented quantization part is
$O\left(N_{rep}(TNK+C_{r})\right)$, where $N_{rep}$ is the overall number of
representatives and $C_{r}$ represents the complexity of computing the
representatives of the $L_{p}$ norm optimization problem.

Algorithm 3 Iterative optimization of the linear transformation and
quantization

0: Initial matrix $\boldsymbol{B}^{\left(0\right)}$ (KL basis), utility
function $u\left(x;\ell\right)$, initial $\mathcal{R}^{\left(0\right)}$,
$\mu^{\left(0\right)}$, and $\Sigma^{\left(0\right)}$;

0: $\boldsymbol{B}^{\star}$ and $\mathcal{R}^{\star}$;

0: $j=1$;

while $j\leqslant$$j_{\text{max}}$ and optimality loss reduced more than
$0.01\%$ do

Evaluate $\boldsymbol{B}^{\left(j\right)}$ using Algorithm 1 and (32)

Evaluate $\mathcal{R}^{\left(j\right)}$ using Algorithm 2

Evaluate $\mu^{\left(j\right)}$ and $\Sigma^{\left(j\right)}$

Evaluate $\Gamma^{\left(j\right)}$ using (29)

$j=j+1$

end while

##  8 Numerical performance analysis

In this section, we conduct a comprehensive numerical analysis to provide
insights into the benefits of goal-oriented compression. Our focus is on the
power consumption scheduling problem in which the decision maker aims to find
a controllable consumption vector $x$ minimizing the $L_{p}$ norm given a
(perfect) day-ahead forecast of the non-controllable vector $\ell$ (the case
of considering the forecast noise can be treated as an extension of this
work). Due to the communication resource limitation, the decision maker has
the sole knowledge of an approximated version of $\ell$, namely,
$\widehat{\ell}$. Our goal is to mitigate the performance degradation induced
by the deviation between $\widehat{\ell}$ and ${\ell}$. To assess the
influence of the compression loss on the goal, we define the relative squared
optimality loss (RSOL) of the compression scheme with respect to the ideal
case as follows:

$\rho_{C}(\%)=100\frac{\sum_{i=1}^{T}\left(U^{\left(i\right)}_{\mathrm{perfect}}-U^{\left(i\right)}_{C}\right)^{2}}{\sum_{i=1}^{T}\left(U^{\left(i\right)}_{\mathrm{perfect}}\right)^{2}},$
(33)

where $U^{\left(i\right)}$ is obtained from the realization
$\ell^{\left(i\right)}$, $C$ represents the compression scheme, and the
performance of the ideal case is obtained by assuming that the controllable
consumption $\ell$ is perfectly known by the receiver (that is, the decision-
maker).

To evaluate our approach in practical scenarios, consumption profiles from the
Ausgrid [39] database are used for all the considered schemes.  The data from
Ausgrid consist of daily energy consumptions sampled every half hour for one
year (from $01-07-2012$ to $30-06-2013$) for 300 users, thus the dimension of
the data is $48\times(365\times 300)$, that is $N=48$. Data have been randomly
shuffled. Then $80~{}\%$ of the data have been used to train the methods and
the remaining $20~{}\%$ for evaluation. The energy need in terms of
controllable consumption is set to $E=50$ kWh. As the system consists of
several stages including precoding, quantization and decoding, we present the
simulation results from different aspects for a comprehensive illustration.

###  8.1 Precoding

We first evaluate the benefit of using goal-oriented precoding schemes in the
scenario without quantization noise.  A KLT-based precoding scheme and an
auto-encoder (AE) serve as references. The AE scheme is an adaptation to the
compression for 1D vectors of the Tensorflow learned data compression network
[40], itself based on the [35]. The architecture consists of three parts: an
encoder that transforms the input vector into a lower-dimensional latent
variable; a decoder that reconstructs the original source vector from this
latent representation; a prior and entropy model between the encoder and
decoder to model the marginal distribution of the latent variable and
efficiently encode it to minimize the average code length. In this first part,
the AE is trained to minimize the reconstruction mean-square error (MSE) as
commonly used. When $K=1$, Figure 3 clearly demonstrates the improvement
provided by the proposed goal-oriented transformation schemes over the
conventional KLT and the AE. The nonlinear transformation scheme obtained from
our neural network architecture outperforms the solution given by Algo. 1,
since the activation function brings nonlinearity to the scheme. Moreover, one
can observe that the RSOL increases with $p$. This can be explained by the
fact that the denominator of RSOL defined by (33) usually decreases when $p$
increases. For the remaining simulations, we fix $p=\infty$, corresponding to
the peak power minimization problem.

Figure 3: Relative squared optimality loss ($\%$) v.s. the exponent of
$L_{p}$-norm ($p$) when $K=1$. Both goal-oriented linear transform (LT) and
goal-oriented non-linear transform (NLT) have a lower RSOL than the Karhuenen-
Loeve transform (KLT) and the auto-encoder (AE transform). Among all schemes,
the proposed NLT solution provides the best result. The influence of $p$ on
the RSOL value shows also the influence of the goal function on the benefits
of the goal-oriented approach.

To evaluate the influence of dimensionality reduction on the goal-oriented
transformation schemes, we plot the RSOL w.r.t. the value of $K$. For the KLT,
the orthogonal basis is obtained as eigenvectors of the covariance matrix, and
the basis vectors are sorted from the most to the least important in terms of
minimizing reconstruction error. One can observe that the proposed NLT scheme
outperforms other techniques. Moreover, while the AE structure leverages the
nonlinear capabilities of neural networks, its lack of task-specific focus
results in greater performance degradation when using a smaller $K$. This loss
in performance can be substantially mitigated by employing a latent space with
larger dimensions.

Figure 4: Relative squared optimality loss ($\%$) v.s. the dimension of the
precoding output signal space / latent space ($K$).

Additionally, to interpret the obtained gain in a more intuitive manner, we
choose one representative realization of $\ell$ from Ausgrid, and display the
series of $\ell$, and $x^{\star}(\widehat{\ell})+\ell$ with different
transformation methods in Figure 5. By approximating $\ell$ with a single
basis vector, one can observe that the shape of $\widehat{\ell}$ in goal-
oriented transformations is more similar to $\ell$ compared with the KLT. As a
result, there are fewer fluctuations in $x^{\star}(\widehat{\ell})+\ell$,
making it easier to minimize the $L_{p}$ norm using the proposed LT and NLT
schemes.

Figure 5: Daily energy consumption $\ell$ of one user and its approximations
with the corresponding utility entries obtained by three different methods
(when $K=1$). NLT has the best performance of minimization of peak value.

###  8.2 Quantization

After assessing the impact of transformation, we further study the goal-
oriented quantization with given transformation schemes. We use the above-
mentioned two transformation schemes, followed by a quantizer, and compare the
performance with different quantization techniques. When the precoding process
reduces the dimension of $\ell$ to $K=1$, we compare three quantization
techniques, namely, uniform quantization, LBG and GOQ. Figure 6 represents the
RSOL against the number of quantization bits. When implementing LT and uniform
quantization, we use the hardware-limited task-based quantization (HLTB)
algorithm proposed by [10]. Figure 6 shows the GOQ exploits the quantization
resources more efficiently such that the optimality loss can be reduced
significantly, especially with non-linear transformation.

Another important issue to discuss here is the choice of the reduced dimension
$K$ for  a given total bit budget. Intuitively, when the number of bits tends
to infinity, it is not necessary to reduce the dimension in the precoding
stage since each element of the transformed vector can be quantized with
negligible errors. Conversely, when the total bit budget is very small, a
significant reduction in dimensionality is expected, as these bits need to
transmit the most important features. Based on these observations, we explore
the tradeoff between information loss induced by dimensionality reduction and
information loss induced by quantization with finite bit budget. To determine
the optimal dimension $K$ that minimizes the optimality loss, we implement the
proposed linear transformation scheme and the GOQ method. Figure 7 illustrates
that the RSOL generally first decreases and then increases as $K$ becomes
larger, indicating the existence of an optimal dimension for the quantization
input. This can be explained by the fact that we are operating at a fixed
budget on the total number of quantization bits. Therefore, when $K$
increases, the number of bits per dimension decreases, hence the observed
phenomenon. This analysis highlights the importance of selecting an
appropriate $K$ for dimensionality reduction in the context of quantization,
balancing the reduction in information loss due to dimensionality reduction
with the information loss introduced by quantization. In addition, we address
the rate-relative square optimality loss tradeoff in Fig. 8 for different
values of $K$. For the AE, the compromise is tuned by properly weighting the
MSE and the rate in the loss function. The network structure has to be
retrained for each $K$ and each value of the weights. In contract with the AE
structure, our method can achieve a given level of RSOL with much less
transmission bits.

Figure 6: Evolution of the relative optimality loss ($\%$) with quantization
bit constraint ($N=48$). RSOL is reduced with larger budget of quantization
bits. Among these methods, NLT followed by a GOQ has the best performance.
Figure 7: Relative optimality loss v.s. the dimension of encoded space ($K$)
with a fixed quantization bit constraint for linear transformation and goal-
oriented quantization Figure 8: Relative squared optimality loss v.s. the
total rate as a function for different latent space dimensions ($K$) for the
linear transform followed by the goal-oriented vector quantizer and for the
auto-encoder structure adapted from [40].

###  8.3 Iterative algorithm

At last, we assess the performance of the proposed iterative algorithm by
optimizing the transformation scheme and the quantization rule in an
alternative way. By taking into account the interplay between quantization
noise and the transformation scheme, Figure 9 shows that the iterative
algorithm outperforms the aforementioned method using LT and GOQ once.
Although the iterative algorithm provides only a marginal improvement beyond
the initial optimization, it demonstrates the potential for further
enhancement through careful refinement of the transformation scheme and the
quantization rule. Regarding the computation cost, while iterative
optimization has its merits in situations where resources are abundant and
performance maximization is required, a single iteration suffices in resource-
limited scenarios, providing a pragmatic balance between performance and
computational efficiency.

Figure 9: Comparing the result of the iterative optimization algorithm to the
result without optimizing ($K=1,\ p=\infty$)

##  9 Conclusion

In this paper, the task of the receiver is modeled by an optimization problem.
The goal of the receiver is to minimize an $L_{p}$-norm performance metric.
For taking its decision, the receiver is assumed to have only access to a
compressed version of the parameters of the function (minus an $L_{p}$-norm)
to be maximized. The problem of designing a goal-oriented precoder which is
followed by a goal-oriented quantizer is addressed. By adopting a non-joint
design approach for these two stages and making appropriate approximations,
the problem becomes tractable. We provide both an interpretable linear
transform which exploits the knowledge of the utility function $u$ and optimal
decision, and another non-linear transform based on neural networks. Compared
to the KLT, the proposed linear transformation yields significant performance
gains in terms of relative optimality loss. It is also seen to what extent the
CNN-based nonlinear transformation performs better than the proposed linear
transformation. By moving from the KLT to the CNN-based goal-oriented
precoding, the relative optimality loss has been to drop from $20\%$ to values
as small as $1-2\%$, which shows all merits of matching the coding scheme to
the goal. It is also seen under which conditions, the benefit from using a
goal-oriented quantization can accumulate with those from the goal-oriented
precoding stage. All these positive results show the interest of adapting,
possibly on the fly, the coding scheme to the task of the receiver. The
proposed approach might be extended to other settings of practical interest.
For instance, when the goal function is not known but only realizations of it
are available. This would naturally lead to a reinforcement learning
perspective for the design of goal-oriented encoders. Also, in this paper, the
communication channel between the encoder and decoder is assumed to be
perfect, which leaves space for improvements in the presence of communication
noise as considered in [41, 42].

##  Appendix A Proofs

###  A.1 Proof of Proposition 1

We want to prove that the solution $x^{\star}$ which maximizes (1) under the
constraints (2) and (3) is a water-filling solution. For this, we first notice
that (1) is a concave function and then apply KKT conditions, which are
necessary and sufficient optimality conditions since (2) and (3) are affine
constraints.

Consider a given vector $\ell$ and assume without loss of generality that

$\ell_{1}\leqslant\ell_{2}\leqslant\dots\leqslant\ell_{j}\leqslant\ell_{j+1}\leqslant\dots\leqslant\ell_{N}.$
(34)

The considered optimization problem (3) is a convex problem since the Lp-norm
is convex and the constraint functions are affine. Introducing the Lagrangian

$\mathcal{L}=-\sum_{k=1}^{N}\left(x_{k}+\ell_{k}\right)^{p}-\delta\left(E-\sum_{k=1}^{N}x_{k}\right)-\sum_{k=1}^{N}\lambda_{k}x_{k}$
(35)

and applying KKT conditions, one obtains, as $p\geqslant 1$

$\frac{\partial\mathcal{L}}{\partial
x_{k}}=-p\left(x_{k}+\ell_{k}\right)^{p-1}+\delta-\lambda_{k},$ (36)

and

$\begin{cases}-p\left(x_{k}^{\star}+\ell_{k}\right)^{p-1}+\delta^{\star}-\lambda_{k}^{\star}&=0\\\
E-\sum_{i=1}^{N}x_{i}^{\star}&=0\\\ -x_{k}^{\star}&\leqslant 0\\\
\lambda_{k}^{\star}x_{k}^{\star}&=0.\end{cases}$ (37)

Getting rid of the slack variables $\left\\{\lambda_{k}^{\star}\right\\}$, one
obtains

$p\left(x_{k}^{\star}+\ell_{k}\right)^{p-1}=\delta^{\star}$ (38)
$x_{k}^{\star}=\left(\left(\frac{\delta^{\star}}{p}\right)^{\frac{1}{p-1}}-\ell_{k}\right)^{+}$
(39)

where $\left(a\right)^{+}=\max(a,0)$. This is therefore a water-filling
solution with water level
$\mu=\left(\frac{\delta^{\star}}{p}\right)^{\frac{1}{p-1}}$, thus

$x_{j}^{\star}=\left(\mu-\ell_{j}\right)^{+}.$ (40)

Let $n^{\star}$ be the number of non-zero entries of $x^{\star}$. According to
(34), one has $x_{j}^{\star}>0$ for $j=1,\dots,n^{\star}$ and
$x_{j}^{\star}=0$ for $j>n^{\star}$. Consequently, as

$\sum_{j=1}^{n^{*}}x_{j}^{\star}=E,$ (41)

one obtains

$\sum_{j=1}^{n^{*}}\left(\mu-\ell_{j}\right)=E,$ (42)

leading to

$\mu=\frac{1}{n^{\star}}\left(E+\sum_{j=1}^{n^{\star}}\ell_{j}\right).$ (43)

The value of $n^{\star}$ is then obtained as the largest value of $n$ such
that $\mu-\ell_{n}>0$

$\displaystyle n^{\star}$ $\displaystyle=\arg\max_{n}n$
$\displaystyle\text{s.t.
}\frac{1}{n}\left(E+\sum_{j=1}^{n}\ell_{j}\right)-\ell_{n}>0.$

###  A.2 Proof of Proposition 2

As seen in A.1,
$x^{\star}\left(\ell\right)=\left[x_{1}^{\star},x_{2}^{\star},\dots,x_{N}^{\star}\right]^{\mathrm{T}}$
is the function of $\ell$. As $n^{\star}\in\\{1,\dots,N\\}$, $2^{N}-1$
different subsets of $\mathbb{R}_{+}^{N}$ may be defined as

$\mathcal{M}_{\mathcal{I}}=\\{\ell\in\mathbb{R}_{+}^{N}\mid
x_{k}^{\star}\left(\ell\right)>0,\ k\in\mathcal{I},\
x_{j}^{\star}\left(\ell\right)=0,\ j\notin\mathcal{I}\\}$

where

$\mathcal{I}\subseteq\\{1,2,\dots,N\\},\ \mathcal{I}\neq\varnothing$

is the set of indexes of the non-zero entries of $x^{\star}$.

In each region $\mathcal{M}_{\mathcal{I}}$, a linear expression of the
solution of the optimization problem (3) can be obtained. Consider some
$\mathcal{I}$ and the associated region $\mathcal{M}_{\mathcal{I}}$. We assume
without loss of generality that the last $N-n^{\star}$ elements of $x^{\star}$
are null for all $\ell\in\mathcal{M}_{\mathcal{I}}$. From (40) and (43), one
has

$x_{j}^{\star}\left(\ell\right)=\begin{cases}\frac{\sum_{i=1}^{n^{\star}}\ell_{i}+E}{n^{\star}}-\ell_{j}&j\leqslant
n^{\star}\\\ 0&j>n^{\star}\end{cases}$ (44)

the Jacobian matrix of $x^{\star}\left(\ell\right)$ is

$\displaystyle\boldsymbol{H}\left(\ell\right)=$ (45)
$\displaystyle\left(\begin{array}[]{ccccccc}-1+\frac{1}{n^{\star}}&\frac{1}{n^{\star}}&\text{$\cdots$}&\frac{1}{n^{\star}}&0&\cdots&0\\\
\frac{1}{n^{\star}}&-1+\frac{1}{n^{\star}}&\cdots&\vdots&\vdots&&\vdots\\\
\vdots&\vdots&&\vdots&\vdots&&\vdots\\\
\frac{1}{n^{\star}}&\cdots&\cdots&-1+\frac{1}{n^{\star}}&0&\cdots&0\\\
0&\cdots&\cdots&0&0&\cdots&0\\\ \vdots&&&\vdots&\vdots&&\vdots\\\
0&\cdots&\cdots&0&0&\cdots&0\end{array}\right).$

Using (44) and the expression of $\boldsymbol{H}\left(\ell\right)$, one
obtains

$x^{\star}\left(\ell\right)=\boldsymbol{H}\left(\ell\right)\ell+b\left(\ell\right).$

###  A.3 Proof of Proposition 3

From (14), when $p\in\mathbb{N}^{+}$

$\displaystyle\frac{\partial\widehat{\Gamma}_{T}}{\partial\boldsymbol{B}}\left(\boldsymbol{B}\right)=-$
$\displaystyle\frac{1}{T}\sum_{i=1}^{T}2\left(u\left(x^{\star}\left(\ell^{\left(i\right)}\right);\ell^{\left(i\right)}\right)-u\left(x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right);\ell^{\left(i\right)}\right)\right)\frac{\partial\mathcal{G}_{i}}{\partial\boldsymbol{B}}$

where

$\displaystyle\mathcal{G}_{i}$
$\displaystyle=u\left(x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right);\ell^{\left(i\right)}\right)$
$\displaystyle=\left(\mathbf{1}^{\mathrm{T}}\cdot\underbrace{\left(x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right)+\ell^{\left(i\right)}\right)\odot\cdots\odot\left(x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right)+\ell^{\left(i\right)}\right)}_{p}\right)^{\frac{1}{p}}$

with $\mathbf{1}=\left[1,\dots,1\right]^{\mathrm{T}}$ the vector of $N$ ones.
Then

$\displaystyle d\mathcal{G}_{i}=$
$\displaystyle\frac{1}{p}\|x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right)+\ell^{\left(i\right)}\|_{p}^{1-p}*$
$\displaystyle
d\left(\mathbf{1}^{\mathrm{T}}\underbrace{\left(x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right)+\ell^{\left(i\right)}\right)\odot\cdots\odot\left(x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right)+\ell^{\left(i\right)}\right)}_{p}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{p}\|x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right)+\ell^{\left(i\right)}\|_{p}^{1-p}*p*\beta_{i}^{\mathrm{T}}d\left(x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right)+\ell^{\left(i\right)}\right)$
$\displaystyle=$
$\displaystyle\|x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right)+\ell^{\left(i\right)}\|_{p}^{1-p}\beta_{i}^{\mathrm{T}}d\left(\left(\boldsymbol{H}_{i}\boldsymbol{B}^{\mathrm{T}}\boldsymbol{B}+\boldsymbol{I}\right)\ell^{\left(i\right)}+b_{i}\right)$
$\displaystyle=$
$\displaystyle\|x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right)+\ell^{\left(i\right)}\|_{p}^{1-p}\beta_{i}^{\mathrm{T}}\left(\boldsymbol{H}_{i}d\boldsymbol{B}^{\mathrm{T}}\boldsymbol{B}\ell^{\left(i\right)}+\boldsymbol{H}_{i}\boldsymbol{B}^{\mathrm{T}}d\boldsymbol{B}\ell^{\left(i\right)}\right)$

where
$\boldsymbol{H}_{i}=\boldsymbol{H}\left(\widehat{\ell}^{\left(i\right)}\right)$,
$b_{i}=b\left(\widehat{\ell}^{\left(i\right)}\right)$, and

$\beta_{i}=\text{$\underbrace{\left(x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right)+\ell^{\left(i\right)}\right)\odot\cdots\odot\left(x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right)+\ell^{\left(i\right)}\right)}_{p-1}$}.$

As $d\mathcal{G}_{i}$ is a scalar, one has

$\displaystyle d\mathcal{G}_{i}=$
$\displaystyle\text{Tr}\left(d\mathcal{G}_{i}\right)$ $\displaystyle=$
$\displaystyle\|x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right)+\ell^{\left(i\right)}\|_{p}^{1-p}\left(\text{Tr}\left(\beta_{i}^{\mathrm{T}}\boldsymbol{H}_{i}d\boldsymbol{B}^{\mathrm{T}}\boldsymbol{B}\ell^{\left(i\right)}\right)+\text{Tr}\left(\beta_{i}^{\mathrm{T}}\boldsymbol{H}_{i}\boldsymbol{B}^{\mathrm{T}}d\boldsymbol{B}\ell^{\left(i\right)}\right)\right)$
$\displaystyle=$
$\displaystyle\|x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right)+\ell^{\left(i\right)}\|_{p}^{1-p}\left(\text{Tr}\left(\boldsymbol{B}\ell^{\left(i\right)}\beta_{i}^{\mathrm{T}}\boldsymbol{H}_{i}d\boldsymbol{B}^{\mathrm{T}}\right)+\text{Tr}\left(\ell^{\left(i\right)}\beta_{i}^{\mathrm{T}}\boldsymbol{H}_{i}\boldsymbol{B}^{\mathrm{T}}d\boldsymbol{B}\right)\right)$
$\displaystyle=$
$\displaystyle\|x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right)+\ell^{\left(i\right)}\|_{p}^{1-p}\text{Tr}\left(\left(\boldsymbol{B}\ell^{\left(i\right)}\beta_{i}^{\mathrm{T}}\boldsymbol{H}_{i}+\boldsymbol{B}\boldsymbol{H}_{i}^{\mathrm{T}}\beta_{i}\ell^{\left(i\right)T}\right)^{\mathrm{T}}d\boldsymbol{B}\right).$

Then, one can deduce

$\frac{\partial\mathcal{G}_{i}}{\partial\boldsymbol{B}}=\|x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right)+\ell^{\left(i\right)}\|_{p}^{1-p}\left(\boldsymbol{B}\ell^{\left(i\right)}\beta_{i}^{\mathrm{T}}\boldsymbol{H}_{i}+\boldsymbol{B}\boldsymbol{H}_{i}^{\mathrm{T}}\beta_{i}\ell^{\left(i\right)T}\right)$

and

$\frac{\partial\widehat{\Gamma}_{T}}{\partial\boldsymbol{B}}\left(\boldsymbol{B}\right)=-\frac{1}{T}\sum_{i=1}^{\mathrm{T}}C_{i}\left(\boldsymbol{B}\ell^{\left(i\right)}\beta_{i}^{\mathrm{T}}\boldsymbol{H}_{i}+\boldsymbol{B}\boldsymbol{H}_{i}^{\mathrm{T}}\beta_{i}\ell^{\left(i\right)T}\right)$
(46)

where

$\displaystyle C_{i}=2$
$\displaystyle\left(u\left(x^{\star}\left(\ell^{\left(i\right)}\right);\ell^{\left(i\right)}\right)-u\left(x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right);\ell^{\left(i\right)}\right)\right)||x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right)+\ell^{\left(i\right)}||_{p}^{1-p}.$
(47)

When $p\rightarrow+\infty$, from (14), we have

$\displaystyle\frac{\partial\widehat{\Gamma}_{T}}{\partial\boldsymbol{B}}\left(\boldsymbol{B}\right)=$
$\displaystyle-\frac{1}{T}\sum_{i=1}^{T}2\left(u\left(x^{\star}\left(\ell^{\left(i\right)}\right);\ell^{\left(i\right)}\right)-u\left(x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right);\ell^{\left(i\right)}\right)\right)\frac{\partial\mathcal{F}_{i}}{\partial\boldsymbol{B}}$

with

$\mathcal{F}_{i}=\max_{j}\left(\widehat{x}_{j}^{\left(i\right)}+\ell_{j}^{\left(i\right)}\right)$

and
$x^{\star}(\widehat{\ell}^{\left(i\right)})=\left[\widehat{x}_{1}^{\left(i\right)},\widehat{x}_{2}^{\left(i\right)},\cdots,\widehat{x}_{N}^{\left(i\right)}\right]^{\mathrm{T}}$.

Furthermore, we assume for $i$-th vector, the $k\left(i\right)$-th elements is
maximum. Then

$\displaystyle\mathcal{F_{\textmd{{i}}}}=$ $\displaystyle
s_{k\left(i\right)}^{\mathrm{T}}\left(x\left(\widehat{\ell}^{\left(i\right)}\right)+\ell^{\left(i\right)}\right)$
$\displaystyle=$ $\displaystyle
s_{k\left(i\right)}^{\mathrm{T}}\left(\left(\boldsymbol{H}_{i}\boldsymbol{B}^{\mathrm{T}}\boldsymbol{B}+\boldsymbol{I}\right)\ell^{\left(i\right)}+b_{i}\right)$

with

$s_{k\left(i\right)}=\left(\begin{array}[]{c}0_{(k\left(i\right)-1)\times
1}\\\ 1\\\ 0_{(N-k\left(i\right))\times 1}\end{array}\right).$

We have

$\displaystyle d\mathcal{F}_{\textmd{{i}}}$
$\displaystyle=s_{k\left(i\right)}^{\mathrm{T}}\left(\boldsymbol{H}_{i}d\boldsymbol{B}^{\mathrm{T}}\boldsymbol{B}\ell^{\left(i\right)}+\boldsymbol{H}_{i}\boldsymbol{B}^{\mathrm{T}}d\boldsymbol{B}\ell^{\left(i\right)}\right).$
$\begin{aligned}
\text{Tr}\left(d\mathcal{F}_{\textmd{{i}}}\right)&=\text{Tr}\left(\left(\boldsymbol{B}\ell^{\left(i\right)}s_{k\left(i\right)}^{\mathrm{T}}\boldsymbol{H}_{i}+\boldsymbol{B}\boldsymbol{H}_{i}^{\mathrm{T}}s_{k\left(i\right)}\ell^{\left(i\right)T}\right)^{\mathrm{T}}d\boldsymbol{B}\right)\end{aligned}.$

Consequently,

$\frac{\partial\mathcal{F}_{i}}{\partial\boldsymbol{B}}=\boldsymbol{B}\ell^{\left(i\right)}s_{k\left(i\right)}^{\mathrm{T}}\boldsymbol{H}_{i}+\boldsymbol{B}\boldsymbol{H}_{i}^{\mathrm{T}}s_{k\left(i\right)}\ell^{\left(i\right)T}$

and

$\frac{\partial\widehat{\Gamma}_{T}}{\partial\boldsymbol{B}}\left(\boldsymbol{B}\right)=-\frac{1}{T}\sum_{i=1}^{T}D_{i}\left(\boldsymbol{B}\ell^{\left(i\right)}s_{k\left(i\right)}^{\mathrm{T}}\boldsymbol{H}_{i}+\boldsymbol{B}\boldsymbol{H}_{i}^{\mathrm{T}}s_{k\left(i\right)}\ell^{\left(i\right)T}\right)$
(48)

with

$D_{i}=2\left(u\left(x^{\star}\left(\ell^{\left(i\right)}\right);\ell^{\left(i\right)}\right)-u\left(x^{\star}\left(\widehat{\ell}^{\left(i\right)}\right);\ell^{\left(i\right)}\right)\right).$
(49)

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