# Wasserstein Distributionally Robust Regret-Optimal Control under Partial
Observability
††thanks: The authors are affiliated with the Department of Electrical
Engineering at Caltech. Emails<EMAIL_ADDRESS>
Joudi Hajar  Taylan Kargin  Babak Hassibi

###### Abstract

This paper presents a framework for Wasserstein distributionally robust (DR)
regret-optimal (RO) control in the context of partially observable systems.
DR-RO control considers the regret in LQR cost between a causal and non-causal
controller and aims to minimize the worst-case regret over all disturbances
whose probability distribution is within a certain Wasserstein-2 ball of a
nominal distribution. Our work builds upon the full-information DR-RO problem
that was introduced and solved in Yan et al., 2023 [1], and extends it to
handle partial observability and measurement-feedback (MF). We solve the
finite horizon partially observable DR-RO and show that it reduces to a
tractable semi-definite program whose size is proportional to the time
horizon. Through simulations, the effectiveness and performance of the
framework are demonstrated, showcasing its practical relevance to real-world
control systems. The proposed approach enables robust control decisions,
enhances system performance in uncertain and partially observable
environments, and provides resilience against measurement noise and model
discrepancies.

###### Index Terms:

regret-optimal control, Wasserstein distance, partial observability,
distributionally robust control

##  I Introduction

Regret-optimal control [2, 3, 4, 5, 1], is a new approach in control theory
that focuses on minimizing the regret associated with control actions in
uncertain systems. The regret measures the cumulative difference between the
performance achieved by a causal control policy and the performance achieved
by an optimal policy that could have been chosen in hindsight. In regret-
optimal control, the worst-case regret over all $\ell_{2}$-norm-bounded
disturbance sequences is minimized.

Distributionally robust control [6, 7, 8, 1], on the other hand, addresses
uncertainty in system dynamics and disturbances by considering a set of
plausible probability distributions rather than relying on a single
distribution as in LQG control, or on a worst-case disturbance, such as in
$H_{\infty}$ or RO control. This approach seeks to find control policies that
perform well across all possible distributions within the uncertainty set,
thereby providing robustness against model uncertainties and ensuring system
performance in various scenarios. The size of the uncertainty set allows one
to control the amount of desired robustness so that, unlike $H_{\infty}$
controllers, say, the controller is not overly conservative. The uncertainty
set is most often taken to be the set of disturbances whose distributions are
within a given Wasserstein-2 distance of the nominal disturbance distribution.
The reason is that, for quadratic costs, the supremum of the expected cost
over a Wasserstein ball reduces to a tractable semi-definite program (SDP).

The current paper considers and extends the framework introduced in [1] that
applied distributionally robust (DR) control to the regret-optimal (RO)
setting. In the full-information finite-horizon setting, the authors of [1]
reduce the DR-RO problem to a tractable SDP. In this paper, we extend the
results of [1] to partially observable systems where, unlike the full-
information setting, the controller does not have access to the system state.
Instead, it only has access to partial information obtained through noisy
measurements. This is often called the measurement feedback (MF) problem. Of
course, the solution to the measurement feedback problem in LQG and
$H_{\infty}$ control is classical. The measurement-feedback setting for DR
control has been studied in [7], [9], and for RO control in [10].

In the finite-horizon case, we reduce the DR-RO control problem with
measurement feedback to an SDP similar to the full-information case studied in
[1]. Furthermore, we validate the effectiveness and performance of our
approach through simulations, showcasing its applicability in real-world
control systems.

The organization of the paper is as follows. In section II, we review the LQG
and regret optimal control formulation in the measurement-feedback setting. In
section III, we present the distributionally robust regret-optimal with
measurement feedback (DR-RO-MF) problem formulation, in section IV we
reformulate the problem as a tractable SDP, and in section V we show numerical
results for controlling the flight of a Boeing 747 [11].

##  II Preliminaries

###  II-A Notations

$\mathbb{R}$ denotes the set of real numbers, $\mathbb{N}$ is the set of
natural numbers, $\|\cdot\|$ is the 2-norm, $\mathbb{E}_{(\cdot)}$ is the
expectation over $(\cdot)$, $\mathcal{M}(\cdot)$ is the set of probability
distributions over $(\cdot)$ and $\operatorname{Tr}$ denotes the trace.

###  II-B A Linear Dynamical System

We consider the following state-space model of a discrete-time, linear time-
invariant (LTI) dynamical system:

$\displaystyle x_{t+1}$ $\displaystyle=Ax_{t}+Bu_{t}+w_{t},$ (1)
$\displaystyle y_{t}$ $\displaystyle=Cx_{t}+v_{t}.$

Here, $x_{t}\in\mathbb{R}^{n}$ represents the state of the system,
$u_{t}\in\mathbb{R}^{m}$ is the control input, $w_{t}\in\mathbb{R}^{n}$ is the
process noise, while $y_{t}\in\mathbb{R}^{p}$ represents the noisy state
measurements that the controller has access to, and $v_{t}\in\mathbb{R}^{p}$
is the measurement noise. The sequences {$w_{i}$} and {$v_{i}$} are considered
to be randomly distributed according to an unknown joint probability measure
$P$ which lies in a specified compact ambiguity set, $\cal P$. For simplicity,
we take $x_{0}$ to be zero.

In the rest of this paper, we adopt an operator form representation of the
system dynamics (1). To this end, assume a horizon of $N\in\mathbb{N}$, and
let us define

$x\coloneqq\left[\begin{array}[]{c}x_{0}\\\ x_{1}\\\ \vdots\\\
x_{N-1}\end{array}\right]\in{\mathbb{R}}^{Nn}~{}~{}~{},~{}~{}~{}u\coloneqq\left[\begin{array}[]{c}u_{0}\\\
u_{1}\\\ \vdots\\\ u_{N-1}\end{array}\right]\in{\mathbb{R}}^{Nm}$

and similarly for $y\in{\mathbb{R}}^{Np}$, $w\in{\mathbb{R}}^{Nn}$, and
$v\in{\mathbb{R}}^{Np}$. Using these definitions, we can represent the system
dynamics (1) equivalently in operator form as

$\displaystyle x$ $\displaystyle=Fu+Gw,$ (2) $\displaystyle y$
$\displaystyle=Ju+Lw+v,$

where $F\in{\mathbb{R}}^{Nn\times Nm}$, $G\in{\mathbb{R}}^{Nn\times Nn}$,
$J\in{\mathbb{R}}^{Np\times Nm}$, and $L\in{\mathbb{R}}^{Np\times Nn}$ are
strictly causal time-invariant operators (i.e, strictly lower triangular block
Toeplitz matrices) corresponding to the dynamics (1).

We consider the Linear-Quadratic Gaussian (LQG) cost given as

$J(u,w,v)\coloneqq x^{T}Qx+u^{T}Ru$ (3)

where $Q,R\succ 0$ are positive definite matrices of the appropriate
dimensions. In order to simplify the notation, we redefine $x$ and $u$ as
$x\leftarrow Q^{\frac{1}{2}}x$, and $u\leftarrow R^{\frac{1}{2}}u$, so that
(3) becomes

$J(u,w,v)=\|x\|^{2}+\|u\|^{2}.$ (4)

###  II-C Controller Design

We consider a linear controller that has only access to the measurements:

$u=Ky,\quad K\in\mathcal{K},$ (5)

where $\mathcal{K}\subseteq\mathbb{R}^{Nm\times Np}$ is the space of causal
(i.e., lower triangular) matrices. Then, the closed-loop state measurement
becomes

$y=(I-JK)^{-1}(Lw+v).$ (6)

As in [10], let

$E=K(I-JK)^{-1},$ (7)

be the Youla parametrization, so that

$K=(I+EJ)^{-1}E.$ (8)

The closed-loop LQG cost (4) can then be written as:

$J(K,w,v)=\begin{bmatrix}w^{T}&v^{T}\end{bmatrix}T_{K}^{T}T_{K}\begin{bmatrix}w\\\
v\end{bmatrix},$ (9)

where $T_{K}$ is the transfer operator associated with $K$ that maps the
disturbance sequences $\begin{bmatrix}w\\\ v\end{bmatrix}$ to the state and
control sequences $\begin{bmatrix}x\\\ u\end{bmatrix}$:

$T_{K}\coloneqq\begin{bmatrix}FEL+G&FE\\\ EL&E\end{bmatrix}.$ (10)

###  II-D Regret-Optimal Control with Measurement-Feedback

Given a noncausal controller $K_{0}\\!\in\\!\cal K$, we define the regret as:

$\displaystyle R(K,w,v)$ $\displaystyle\coloneqq J(K,w,v)-J(K_{0},w,v),$ (11)
$\displaystyle=\begin{bmatrix}w^{T}&v^{T}\end{bmatrix}(T_{K}^{T}T_{K}-T_{K_{0}}^{T}T_{K_{0}})\begin{bmatrix}w\\\
v\\\ \end{bmatrix},$ (12)

which measures the excess cost that a causal controller suffers by not knowing
the future. In other terms, regret is the difference between the cost
accumulated by a causal controller and the cost accumulated by a benchmark
noncausal controller that knows the complete disturbance trajectory. The
problem of minimizing regret in the measurement-feedback setting is referred
to as (RO-MF) and is formulated as:

$\inf_{K\in\mathcal{K}}\sup_{\begin{subarray}{c}w,v\end{subarray}}\frac{R(K,w,v)}{\|w\|^{2}+\|v\|^{2}},$
(13)

which is solved suboptimally by reducing it to a level-1 suboptimal Nehari
problem [10].

##  III Distributionally Robust Regret-Optimal Control

In this section, we introduce the distributionally robust regret-optimal (DR-
RO) control problem with measurement feedback, which we refer to as DR-RO-MF.

In this setting, the objective is to find a controller $K\\!\in\\!\mathcal{K}$
that minimizes the maximum expected regret among all joint probability
distributions of the disturbances in an ambiguity set $\cal P$. This can be
formulated formally as

$\inf_{K\in\mathcal{K}}\sup_{P\in\mathcal{P}}\mathbb{E}_{P}[R(K,w,v)],$ (14)

where the disturbances $\begin{bmatrix}w\\\ v\end{bmatrix}$ are distributed
according to $P\\!\in\\!\cal P$.

To solve this problem, we first need to characterize the ambiguity set
$\mathcal{P}$ and explicitly determine a benchmark noncausal controller
$K_{0}$. As in [1], we choose $\mathcal{P}$ to be the set of probability
distributions that are at a distance of at most $r>0$ to a nominal probability
distribution, $P_{0}\\!\in\\!{\mathcal{M}}({\mathbb{R}}^{N(n+p)})$. Here, the
distance is chosen to be the type-2 Wasserstein distance defined as [12]:

$W_{2}^{2}(P_{1},P_{2}):=\\!\inf_{\pi\in\Pi(P_{1},P_{2})}\
\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\|z_{1}\\!-\\!z_{2}\|^{2}\,\pi(dz_{1},dz_{2}),$

where the set $\Pi(P_{1},P_{2})$ comprises all joint distributions that have
marginal distributions $P_{1}$ and $P_{2}$. Then, $\mathcal{P}$ can be written
as:

$\mathcal{P}:=\\{P\in\mathcal{M}(\mathbb{R}^{N(n+p)})\,|\,W_{2}(P_{0},P)\leq
r\\}.$ (15)

Unlike the full-information case, we know from Theorem 1 in [10] that in the
measurement feedback case, there is no optimal noncausal controller that
dominates every other controller for every disturbance. Therefore, we will
choose $K_{0}$ as the optimal noncausal controller that minimizes the
Frobenius norm of $T_{K}$. Theorem 3 in [10] shows that such a controller can
be found as:

$K_{0}=(I+E_{0}J)^{-1}E_{0},$ (16)

where the associated operator, $T_{K_{0}}$ is:

$\displaystyle T_{K_{0}}=\begin{bmatrix}FE_{0}L+G&&FE_{0}\\\
E_{0}L&&E_{0}\end{bmatrix},$ (17)

with

$\displaystyle E_{0}$ $\displaystyle\coloneqq-T^{-1}F^{T}GL^{T}U^{-1},$ (18)
$\displaystyle T$ $\displaystyle\coloneqq I+F^{T}F,$ (19) $\displaystyle U$
$\displaystyle\coloneqq I+LL^{T}.$ (20)

##  IV Tractable Formulation

In this section, we introduce a tractable reformulation of the DR-RO-MF
control problem (14).

###  IV-A DR-RO-MF Control Problem

Defining

$\mathcal{C}_{K}\coloneqq T_{K}^{T}T_{K}-T_{K_{0}}^{T}T_{K_{0}},$ (21)

we can rewrite the DR-RO-MF control problem (14) as

$\inf_{K\in\mathcal{K}}\sup_{P\in\mathcal{P}}\mathbb{E}_{P}\left[\begin{bmatrix}w^{T}&v^{T}\end{bmatrix}\cal
C_{K}\begin{bmatrix}w\\\ v\\\ \end{bmatrix}\right].$ (22)

The following theorem gives the dual problem of inner maximization and
characterizes the worst-case distribution.

######  Theorem IV.1.

[adapted from Theorems 2 and 3 in [1]]. Suppose $P_{0}$ is absolutely
continuous with respect to the Lebesgue measure on $\mathbb{R}^{N}$ and
$\begin{bmatrix}w_{0}\\\ v_{0}\end{bmatrix}\sim P_{0}$. The optimization
problem:

$\sup_{P\in\mathcal{P}}\mathbb{E}_{P}\left[\begin{bmatrix}w^{T}&v^{T}\end{bmatrix}\cal
C_{K}\begin{bmatrix}w\\\ v\\\ \end{bmatrix}\right]$ (23)

where $\begin{bmatrix}w\\\ v\end{bmatrix}\sim P$ and
$\mathcal{C}_{K}\in\mathbb{S}^{N(n+p)}$, with
$\lambda_{max}(\mathcal{C}_{K})\neq 0$, has a finite solution and is
equivalent to the convex optimization problem:

$\inf_{\begin{subarray}{c}\gamma\geq 0,\\\ \gamma
I\succ\mathcal{C}_{K}\end{subarray}}\gamma(r^{2}-\operatorname{Tr}(M_{0}))+\gamma^{2}\operatorname{Tr}(M_{0}(\gamma
I-\mathcal{C}_{K})^{-1}),$ (24)

where $M_{0}:=\mathbb{E}_{P_{0}}\left[\begin{bmatrix}w\\\
v\end{bmatrix}\begin{bmatrix}w^{T}&v^{T}\end{bmatrix}\right]$. Furthermore,
the disturbance that achieves the worst-case regret is
$\begin{bmatrix}w^{\ast}\\\ v^{\ast}\end{bmatrix}\sim P^{\ast}$, where
$\begin{bmatrix}w^{\ast}\\\
v^{\ast}\end{bmatrix}=\gamma^{\ast}(\gamma^{\ast}I-\mathcal{{C}}_{K})^{-1}\begin{bmatrix}w_{0}\\\
v_{0}\end{bmatrix}$, and $\gamma^{\ast}$ is the optimal solution of (24),
which also satisfies the algebraic equation:

$\operatorname{Tr}((\gamma(\gamma I-\mathcal{C}_{K})^{-1}-I)^{2}M_{0})=r^{2}$
(25)

###### Proof.

The proof follows from Theorems 2 and 3 in [1] and is omitted for brevity
here. ∎

We highlight two remarks pertaining to the presented theorem.

Remark 1: Notice that the supremum of the quadratic cost depends on $P_{0}$
only though its covariance matrix $M_{0}$. Note further that as
$r\rightarrow\infty$, the optimal $\gamma$ reaches its smallest possible value
(since $r^{2}$ multiplies $\gamma$ in (24)). The smallest possible value that
$\gamma$ can take is simply the operator norm of ${\cal C}_{K}$, which means
that the DR-RO-MF controller approaches the regret-optimal controller as
$r\rightarrow\infty$.

Remark 2: Notice that the worst-case disturbance takes on a Gaussian
distribution when the nominal disturbance is Gaussian. This is not immediately
evident as the ambiguity set $\mathcal{P}$ contains non-Gaussian
distributions. Note further that the worst-case disturbance is correlated even
if the nominal distribution has white noise.

Assuming the covariance of the nominal distribution to be

$M_{0}=\mathbb{E}_{P_{0}}\left[\begin{bmatrix}w\\\
v\end{bmatrix}\begin{bmatrix}w^{T}&v^{T}\end{bmatrix}\right]=I.$ (26)

so that $\operatorname{Tr}(M_{0})=N(n+p)$, the optimization problem (22) can
be cast equivalently using Theorem IV.1 as

$\displaystyle\inf_{K\in\mathcal{K}}\inf_{\begin{subarray}{c}\gamma\geq
0\end{subarray}}\gamma(r^{2}-N(n+p))+\gamma^{2}\operatorname{Tr}((\gamma
I-\mathcal{C}_{K})^{-1})$ (27) $\displaystyle\text{s.t. }\begin{cases}&\gamma
I\succ\mathcal{C}_{K}\\\
&\mathcal{C}_{K}=T_{K}^{T}T_{K}-T_{K_{0}}^{T}T_{K_{0}}\\\ \end{cases}$

As in [10], define the unitary matrices $\Psi$ and $\Theta$:

$\displaystyle\Theta=\begin{bmatrix}S^{-\frac{1}{2}}&&0\\\
0&&T^{-\frac{T}{2}}\end{bmatrix}\begin{bmatrix}I&&-F\\\ F^{T}&&I\end{bmatrix}$
(28) $\displaystyle\Psi=\begin{bmatrix}I&&L^{T}\\\
-L&&I\end{bmatrix}\begin{bmatrix}V^{-\frac{1}{2}}&&-0\\\
0&&U^{-\frac{T}{2}}\end{bmatrix}$ (29)

where $T$ and $U$ are as in (19) and (20), and

$\displaystyle S=I+FF^{T}$ (30) $\displaystyle V=I+L^{T}L.$ (31)

and $S^{\frac{1}{2}}$, $T^{\frac{1}{2}}$, $U^{\frac{1}{2}}$, and
$V^{\frac{1}{2}}$ are (block) lower triangular matrices, such that
$S=S^{\frac{1}{2}}S^{\frac{T}{2}}$, $T=T^{\frac{T}{2}}T^{\frac{1}{2}}$,
$U=U^{\frac{1}{2}}U^{\frac{T}{2}}$, $V=V^{\frac{T}{2}}V^{\frac{1}{2}}$. Then,
the optimization problem (27) is equivalent to:

$\displaystyle\inf_{\begin{subarray}{c}K\in\mathcal{K},\\\ \gamma\geq 0,\\\
\gamma
I\succ\bar{\mathcal{C}}_{K}\end{subarray}}\gamma(r^{2}-N(n+p))+\gamma^{2}\operatorname{Tr}((\gamma
I-\bar{\mathcal{C}}_{K})^{-1})$ (32) $\displaystyle\text{s.t.
}\begin{cases}&\bar{\mathcal{C}}_{K}=(\Theta T_{K}\Psi)^{T}\Theta
T_{K}\Psi-(\Theta T_{K_{0}}\Psi)^{T}\Theta T_{K_{0}}\Psi\end{cases}$

which holds true since trace is invariant under unitary $\Theta$ and $\Psi$.
By introducing an auxiliary variable $X\succeq\gamma^{2}(\gamma
I-\bar{\mathcal{C}}_{K})^{-1}$ and leveraging the Schur complement theorem as
in [1], the problem (32) can be recast as

$\displaystyle\inf_{\begin{subarray}{c}K\in\mathcal{K},\\\ \gamma\geq 0,\\\
X\succeq 0\end{subarray}}\gamma(r^{2}-N(n+p))+\operatorname{Tr}(X)$ (33)
$\displaystyle\text{s.t. }\begin{cases}&\begin{bmatrix}X&\gamma I\\\ \gamma
I&\gamma I-\bar{\mathcal{C}}_{K}\end{bmatrix}\succeq 0\\\ &\gamma
I-\bar{\mathcal{C}}_{K}\succ 0\\\ &\bar{\mathcal{C}}_{K}\\!=\\!(\Theta
T_{K}\Psi)^{T}\Theta T_{K}\Psi-(\Theta T_{K_{0}}\Psi)^{T}\Theta
T_{K_{0}}\Psi\end{cases}$

In the following lemma, we establish some of the important identities that are
utilized to convert problem (33) to a tractable convex program.

######  Lemma IV.2.

[adapted from [10]]. The following statements hold:

  1. 1.

.

$\displaystyle\gamma I-\bar{\mathcal{C}}_{K}=\begin{bmatrix}\gamma I&&-PZ\\\
-Z^{T}P^{T}&&\gamma I-Z^{T}Z\end{bmatrix}$ (34)

where

$\displaystyle Z=T^{\frac{1}{2}}EU^{\frac{1}{2}}-W$ (35) $\displaystyle
W=-T^{-\frac{T}{2}}F^{T}GL^{T}U^{-\frac{T}{2}}$ (36) $\displaystyle
P=V^{-\frac{T}{2}}G^{T}FT^{-\frac{1}{2}}$ (37)

and $E$, $T$, $U$ and $V$ are as defined in 7, 19, 20 and 31 respectively.

  2. 2.

.

$\gamma I-\bar{\mathcal{C}}_{K}\succ
0\Leftrightarrow\|Y-W_{-,\gamma}\|_{2}\leq 1$ (38)

where

$\displaystyle\gamma^{-1}I+\gamma^{-2}P^{T}P=M_{\gamma}^{T}M_{\gamma}$ (39)
$\displaystyle
M_{\gamma}=\left(\gamma^{-1}I+\gamma^{-2}P^{T}P\right)^{\frac{1}{2}}$ (40)
$\displaystyle W_{\gamma}=M_{\gamma}W$ (41) $\displaystyle
Y=M_{\gamma}T^{\frac{1}{2}}EU^{\frac{1}{2}}-W_{+,\gamma}$ (42)

and $W_{+,\gamma}$ and $W_{-,\gamma}$ are the causal and strictly anticausal
parts of $W_{\gamma}$. Here, $M_{\gamma}$ is lower triangular, and positive-
definite.

  3. 3.

$Y$ is causal iff $E$ is causal, where $E$ can be found as follows:

$E=T^{-\frac{1}{2}}M_{\gamma}^{-1}(Y+W_{+,\gamma})U^{-\frac{1}{2}}$ (43)

  4. 4.

The condition in (38) is recognized as a level-1 suboptimal Nehari problem
that approximates a strictly anticausal matrix $W_{-,\gamma}$ by a causal
matrix $Y$.

###### Proof.

The proof follows from Theorem 4 in [10] and is omitted for brevity here. ∎

Using Lemma IV.2, problem (33) can be reformulated as a tractable optimization
program:

$\displaystyle\inf_{\begin{subarray}{c}Z,Y\in\mathcal{K},\\\ \gamma\geq 0,\\\
X\succeq 0\end{subarray}}\gamma(r^{2}-N(n+p))+\operatorname{Tr}(X)$ (44)
$\displaystyle\text{s.t. }\begin{cases}&\begin{bmatrix}X_{11}&X_{12}&\gamma
I&0\\\ X_{12}^{T}&X_{22}&0&\gamma I\\\ \gamma I&0&\gamma I&-PZ\\\ 0&\gamma
I&-Z^{T}P^{T}&\gamma I-Z^{T}Z\end{bmatrix}\succeq 0\\\
&\|Y-W_{-,\gamma}\|_{2}\leq 1\end{cases}$

$\displaystyle=\inf_{\begin{subarray}{c}Z,Y\in\mathcal{K},\\\ \gamma\geq 0,\\\
X\succeq 0\end{subarray}}\gamma(r^{2}-N(n+p))+\operatorname{Tr}(X)$ (45)
$\displaystyle\text{s.t. }\begin{cases}&\begin{bmatrix}X_{11}&X_{12}&\gamma
I&0&0\\\ X_{12}^{T}&X_{22}&0&\gamma I&0\\\ \gamma I&0&\gamma I&-PZ&0\\\
0&\gamma I&-Z^{T}P^{T}&\gamma I&Z^{T}\\\ 0&0&0&Z&I\end{bmatrix}\succeq 0\\\
&\|Y-W_{-,\gamma}\|_{2}\leq 1\end{cases}$

where the last step follows from the Schur complement. Using (35), (43), and

$H_{\gamma}=M_{\gamma}^{-1}W_{+,\gamma}-W$ (46)

we establish our main theorem.

######  Theorem IV.3 (Tractable Formulation of DR-RO-MF).

The distributionally robust regret-optimal control problem in the measurement
feedback setting (14) reads:

$\displaystyle\inf_{\begin{subarray}{c}Y\in\mathcal{K},\\\ \gamma\geq 0,\\\
X\succeq 0\end{subarray}}\gamma(r^{2}-N(n+p))+\operatorname{Tr}(X)$ (47)
$\displaystyle\text{s.t.
}\begin{cases}&\begin{bmatrix}X_{11}&X_{12}\\!&\\!\gamma I&0&0\\\
X_{12}^{T}&X_{22}\\!&\\!0&\gamma I&0\\\ \gamma I&0\\!&\\!\gamma I&-P(*)&0\\\
0&\gamma I\\!&\\!-(*)^{T}P^{T}&\gamma I&(*)^{T}\\\
0&0&0&(*)&I\end{bmatrix}\\!\succeq\\!0\\\ &(*)=M_{\gamma}^{-1}Y+H_{\gamma}\\\
&\begin{bmatrix}I&(Y-W_{-,\gamma})^{T}\\\ Y-W_{-,\gamma}&I\end{bmatrix}\succ
0\end{cases}$

The optimal controller $K^{\ast}$ is then obtained using (8) and (43).

###  IV-B Sub-Optimal Problem

For a given value of $\gamma$, problem (LABEL:eq::finalpb) can be simplified
into a tractable SDP. In practical implementations, we can solve problem
(LABEL:eq::finalpb) by optimizing the objective function with respect to the
variables $Y$ and $X$ while fixing $\gamma$, thus transforming the problem
into an SDP, which can be solved using standard convex optimization packages.
We then iteratively refine the value of $\gamma$ until it converges to the
optimal solution $\gamma^{*}$. This iterative process ensures that we obtain
the best possible value for $\gamma$ that minimizes the objective function in
problem (LABEL:eq::finalpb).

###  IV-C LQG and RO-MF Control Problems as Special Cases

Interestingly, LQG and RO control in the measurement feedback setting can be
recovered from the DR-RO-MF control by varying the radius $r$ which represents
the extent of uncertainty regarding the accuracy of the nominal distribution
in the ambiguity set. When $r\rightarrow 0$, the ambiguity set transforms into
a singular set comprising solely the nominal distribution. Consequently, the
problem simplifies into a stochastic optimal control problem under partial
observability:

$\inf_{K\in\mathcal{K}}\mathbb{E}_{P_{0}}[J(K,w,v)]$ (48)

As $r\rightarrow\infty$, the ambiguity set transforms into the set of any
disturbance generated adversarially and the optimal $\gamma$ reaches its
smallest possible value which is the operator norm of ${\cal C}_{K}$. This
means that the problem reduces to the RO-MF control problem which we discussed
in section II-D.

##  V Simulations

###  V-A Flight Control

We focus on the problem of controlling the longitudinal flight of a Boeing 747
which pertains to the linearized dynamics of the aircraft, as presented in
[11]. The linear dynamical system provided describes the aircraft’s dynamics
during level flight at an altitude of 7.57 miles and a speed of 593 miles per
hour, with a discretization interval of 0.1 second. The state variables of the
system encompass the aircraft’s velocity along the body axis, velocity
perpendicular to the body axis, angle between the body axis and the horizontal
plane, and angular velocity. The inputs to the system are the elevator angle
and thrust. The process noise accounts for variations caused by external wind
conditions. The discrete-time state space model is:

$\displaystyle A=\begin{bmatrix}0.9801&0.0003&-0.0980&0.0038\\\
-0.3868&0.9071&0.0471&-0.0008\\\ 0.1591&-0.0015&0.9691&0.0003\\\
-0.0198&0.0958&0.0021&1.000\end{bmatrix}$ $\displaystyle
B=\begin{bmatrix}-0.0001&0.0058\\\ 0.0296&0.0153\\\ 0.0012&-0.0908\\\
0.0015&0.0008\end{bmatrix},C=\begin{bmatrix}1&0&0&0\\\ 0&0&0&1\end{bmatrix}.$

We conduct all experiments using MATLAB, on a PC with an Intel Core i7-1065G7
processor and 16 GB of RAM. The optimization problems are solved using the CVX
package [13].

We limit the horizon to $N=10$. We take the nominal distribution $P_{0}$ to be
Gaussian with mean $\mu_{0}=0$ and covariance $\Sigma_{0}=I$, and we
investigate various values for the radius $r$, specifically:

$r\in\\{0,0.2,0.4,0.6,0.8,1,1.5,2,4,8,16,32,126\\}.$

For each value of $r$, we solve the sub-optimal problem described in section
IV-B, iterating over $\gamma$ until convergence to $\gamma^{*}$.

Figure 1: Controller costs for $r\in{0,0.2,0.4,0.6,0.8,1,1.5,2,4}$.
At $r=0$, the top-performing controllers are DR-RO-MF and LQG, exhibiting
regret costs of 5.4. They are followed by $\text{H}_{\infty}$ with a regret
cost of 5.9, and finally RO-MF with a regret cost of 13.8. The ranking of the
controllers based on regret costs is: DR-RO-MF=LQG=5.34 $<$
$\textbf{H}_{\infty}$=5.47 $<$ RO-MF=13.8.
As $r$ increases to 4, DR-RO-MF remains the best-performing controller with a
regret of 141. It is followed by RO-MF with a regret of 144,
$\text{H}_{\infty}$ with a regret of 154, and finally $\text{H}2$ with a
regret of 156. The ranking of the controllers at $r=4$ based on regret costs
is: DR-RO-MF=141 $<$ RO-MF=144 $<$ $\textbf{H}_{\infty}$=154 $<$ LQG=156.
Figure 2: Controller costs for $r\in{4,8,16,32,126}$.
At $r=8$, the best-performing controller is DR-RO-MF with a regret of 437,
which is closely comparable to the regret of the RO-MF controller of 438. They
are followed by $\text{H}_{\infty}$ with a regret of 499, and finally
$\text{H}2$ with a regret of 505. The ranking of controllers based on regret
costs is as follows: DR-RO-MF=437 $\lesssim$ RO-MF=438 $<$
$\textbf{H}_{\infty}$=499 $<$ LQG=505.
When $r$ increases to 126, which approximates the behavior of $r$ approaching
infinity, the order of the best-performing controllers remains unchanged: DR-
RO-MF=RO-MF=$8.33\times 10^{4}$ $<$ $\textbf{H}_{\infty}$=$9.50\times 10^{4}$
$<$ LQG=$9.57\times 10^{4}$. DR-RO-MF and RO-MF controllers exhibit similar
performance in this regime.

To assess the performance of the controller, we compute the worst-case
disturbance, which lies at a Wasserstein distance $r$ from $P_{0}$, as
discussed in theorem IV.1. Finally, we compare the regret cost of the DR-RO-MF
controller with that of the LQG, $H_{\infty}$ [14], and RO-MF [10] controllers
while considering the worst-case disturbance corresponding to the DR-RO-MF
controller. The results are shown in Figures 1 and 2.

The DR-RO-MF controller achieves the minimum cost under worst-case disturbance
conditions for any given value of $r$. When $r$ is sufficiently small (less
than 0.2), the cost of the DR-RO-MF controller closely approximates that of
the LQG controller (figure 1). Conversely, for sufficiently large values of
$r$ (greater than 8), the cost of the DR-RO-MF controller closely matches that
of the RO-MF controller (figure 2). These observations align with theoretical
findings as elaborated in section IV-C.

Furthermore, it is worth noting that for large values of $r$ (figure 2), the
LQG controller yields the poorest results. Conversely, for small values of $r$
(figure 1), the LQG controller performs on par with the DR-RO-MF controller,
emerging as the best choice, as mentioned earlier. This discrepancy is
expected since LQG control accounts only for disturbances drawn from the
nominal distribution, assuming uncorrelated noise. On the other hand, RO-MF
exhibits inferior performance when $r$ is small (figure 1), but gradually
becomes the top-performing controller alongside DR-RO-MF as $r$ increases.
This behavior arises from the fact that RO-MF is specifically designed for
sufficiently large $r$. Lastly, note that the $H_{\infty}$ cost lies between
the costs of the other controllers, interpolating their respective costs.

###  V-B Performance Under Adversarially Chosen Distribution

For any given causal controller $K_{c}$, an adversary can choose the worst-
case distribution of disturbances for a fixed $r$ as

$\displaystyle\arg\max_{P\in\mathcal{P}}\mathbb{E}_{P}R(K_{c},w,v)\eqqcolon
P_{c},$ (49)

where $R$ is the regret as in (11). Denoting by $K_{\text{DR-RO-MF}}$ the
optimal DR-RO-MF controller and by $P_{\text{DR-RO-MF}}$ the worst-case
(adversarial) distribution corresponding to $K_{\text{DR-RO-MF}}$, we have
that

$\displaystyle\mathbb{E}_{P_{c}}R(K_{c},w,v)$
$\displaystyle=\max_{P\in\mathcal{P}}\mathbb{E}_{P}R(K_{c},w,v),$ (50)
$\displaystyle\geq\min_{K\in\mathcal{K}}\max_{P\in\mathcal{P}}\mathbb{E}_{P}R(K,w,v),$
(51) $\displaystyle=\mathbb{E}_{P_{\text{DR-RO-MF}}}R(K_{\text{DR-RO-
MF}},w,v),$ (52) $\displaystyle\geq\mathbb{E}_{P_{c}}R(K_{\text{DR-RO-
MF}},w,v),$ (53)

where the first equality follows from (49) and the last inequality is due to
the fact that $P_{\text{DR-RO-MF}}$ is the worst-case distribution for
$K_{\text{DR-RO-MF}}$. In other words, DR-RO-MF controller is robust to
adversarial changes in distribution as it yields smaller expected regret
compared to any other causal controller $K_{c}$ when the disturbances are
sampled from the worst-case distribution $P_{c}$ corresponding to $K_{c}$.

The simulation results presented in Subsection V-A show that DR-RO-MF
outperforms RO-MF, $H_{\infty}$, and LQG (designed assuming disturbances are
sampled from $P_{0}$) controllers under the worst-case distribution of the DR-
RO-MF controller $P_{\text{DR-RO-MF}}$, i.e

$\displaystyle\mathbb{E}_{P_{\text{DR-RO-
MF}}}R(K_{c},w,v)\geq\mathbb{E}_{P_{\text{DR-RO-MF}}}R(K_{\text{DR-RO-
MF}},w,v).$ (54)

This directly implies that the theoretically expected inequality

$\displaystyle\mathbb{E}_{P_{c}}R(K_{c},w,v)\geq\mathbb{E}_{P_{c}}R(K_{\text{DR-
RO-MF}},w,v)$ (55)

is validated and positively exceeded following the inequalities (53) and

$\displaystyle\mathbb{E}_{P_{c}}R(K_{c},w,v)\geq\mathbb{E}_{P_{\text{DR-RO-
MF}}}R(K_{c},w,v).$ (56)

To further support our claims, we assess the performance of LQG and RO-MF
controllers by measuring the relative reduction in expected regret when DR-RO-
MF controller is utilized under the worst-case distributions corresponding to
LQG and RO-MF controllers, respectively:

$\frac{\mathbb{E}_{P_{c}}R(K_{c},w,v)-\mathbb{E}_{P_{c}}R(K_{\text{DR-RO-
MF}},w,v)}{\mathbb{E}_{P_{c}}R(K_{c},w,v)}\times 100,$ (57)

where $K_{c}$ is either LQG or RO-MF controller and $P_{c}$ is the
corresponding worst-case distribution. The results are shown in Table I for
$r\in\\{0.2,1,2,4,16,32\\}$.

$\mathbf{r}$ | 0.2 | 1 | 2 | 4 | 16 | 32
---|---|---|---|---|---|---
LQG(%) | 0.860 | 8.17 | 14.8 | 21.9 | 28.5 | 29.3
RO-MF(%) | 56.6 | 43.0 | 32.3 | 17.2 | 1.95 | 0.465
TABLE I: Relative difference in % (as in (57)) between the expected regret of
LQG/RO-MF and of DR-RO-MF controllers, under the worst-case disturbance of
LQG/RO-MF, respectively, as in (49) for different values of $r$

###  V-C Limitations

In our scenario with a relatively short planning horizon of $N=10$, the cost
reduction achieved by employing DR-RO-MF control, in comparison to traditional
controllers such as LQG and $H_{\infty}$, is moderate. However, it is
anticipated that this reduction would become more pronounced with the
utilization of a longer planning horizon. Unfortunately, in our experimental
setup, we were restricted to using $N=10$ due to computational limitations.
Solving semi-definite programs involving large matrices is computationally
inefficient, necessitating this constraint. In practice, this limitation can
be overcome by implementing the controller in a receding horizon fashion,
where the controller is updated every $x$ time steps.

##  VI Conclusion

In conclusion, this paper extended the distributionally robust approach to
regret-optimal control by incorporating the Wasserstein-2 distance [1] to
handle cases of limited observability. The proposed DR-RO-MF controller
demonstrated superior performance compared to classical controllers such as
LQG and $H_{\infty}$, as well as the RO-MF controller, in simulations of
flight control scenarios. The controller exhibits a unique interpolation
behavior between LQG and RO-MF, determined by the radius $r$ that quantifies
the uncertainty in the accuracy of the nominal distribution. As the time
horizon increases, solving the tractable SDP to which the solution reduces,
becomes more challenging, highlighting the practical need for a model
predictive control approach. Overall, the extended distributionally robust
approach presented in this paper holds promise for robust and effective
control in systems with limited observability.

## References

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ifaamas [AAMAS ’22]Proc. of the 21st International Conference on Autonomous
Agents and Multiagent Systems (AAMAS 2022)May 9–13, 2022 OnlineP. Faliszewski,
V. Mascardi, C. Pelachaud, M.E. Taylor (eds.) 2022 2022 University of
Groningen & University of Amsterdam The Netherlands

# Social Choice Around the Block:
On the Computational Social Choice of Blockchain

Davide Grossi<EMAIL_ADDRESS>
###### Abstract.

One of the most innovative aspects of blockchain technology consists in the
introduction of an incentive layer to regulate the behavior of distributed
protocols. The designer of a blockchain system faces therefore issues that are
akin to those relevant for the design of economic mechanisms, and faces them
in a computational setting. From this perspective the present paper argues for
the importance of computational social choice in blockchain research. It
identifies a few challenges at the interface of the two fields that illustrate
the strong potential for cross-fertilization between them.

###### Key words and phrases:

Blockchain; Computational Social Choice; Multi-agent Systems

##  1\. Introduction

A blockchain is a decentralized state machine, in its simplest form, a
decentralized ledger for financial transactions. The machine is controlled by
several distinct processes, called nodes or agents, and computes by packaging
state transitions (e.g., the order of transfering $x$ tokens from Alice’s
account to Bob’s) in batches, which are called blocks. Each block determines
the next global state of the machine (e.g., the next state of the ledger,
where Alice’s account has $x$ tokens less and Bob’s $x$ more). Each new block
is appended to the list of older blocks, thereby determining a growing append-
only list of global states of the machine—its computation history. Crucially,
each block points in a temper-proof way (via a cryptographic hash) to the
previous block to which it was appended, thereby enforcing an immutable
description of the history of the machine. The core of a blockchain is the
protocol that the nodes follow in order to agree on which transitions to
incorporate into the list, i.e., to achieve consensus on the state of the
machine.

#### Paper motivation

Blockchain was born as the backbone of the Bitcoin cryptocurrency. The
consensus protocol behind Bitcoin, known as Nakamoto consensus Nakamoto
(2008); Garay et al. (2015), showed that such a decentralized consensus on an
append-only list is possible even in large, open peer-to-peer networks. This
was a significant breakthrough with respect to existing approaches to
consensus, which worked specifically on systems of limited size with
controlled access. This breakthrough relied on one key insight: nodes in an
open system cannot be controlled, so their influence on consensus should be
kept at bay by linking it to the ownership of a non-monopolizable resource—in
the case of Bitcoin, computing power. In other words, nodes in an open network
cannot be controlled nor trusted, but can be incentivized. Nodes that
contribute to achieving consensus are rewarded with (they ‘mine’) tokens, that
is, units of currency. This is the key intuition behind the application to
consensus of the Proof-of-Work (PoW) technique (originally developed to thwart
email spamming Dwork and Naor (1992)), which has proven extremely robust,
maintaining the Bitcoin blockchain for over a decade (cf. for recent overviews
Narayanan et al. (2016); Wattenhofer (2017)). So, blockchain consensus is the
result of a rational response to incentives.

Traditionally, research in blockchain has focused mostly on the cryptographic
foundations and the distributed computing aspects (e.g., protocol correctness)
of the technology. At the same time, a game-theoretic perspective on
blockchain protocols has also been gaining attention: is behavior in
accordance with the protocol economically rational, in some precise
equilibrium-theoretic sense? In other words, are protocols strategy-proof?
This game-theoretic perspective has historically been marginal in distributed
computing Abraham et al. (2011), but has proven significant in blockchain. By
now, it has been extensively applied—including by researchers in the AAMAS
community—to Nakamoto consensus (e.g., Eyal (2015); Sapirstein et al. (2016);
Sompolinsky and Zohar (2018); Alkalay-Houlian and Shah (2019)), as well as to
other protocols (e.g., Abraham et al. (2016); Kiayias et al. (2017); Amoussou-
Guenou et al. (2020); Chen et al. (2021)). See Judmayer et al. (2021) for an
extensive recent overview. However, the economic issues that the designer of a
blockchain system faces go well beyond incentive-compatibility alone, and
reflect broad issues in the design of collective decision-making mechanisms,
such as forms of equity and fairness. This interface with the economic theory
of group decision-making, and specifically social choice theory, is the focus
of the present paper.

#### Paper contribution

The contribution of this paper consists in arguing how social choice theory,
and its algorithmically-focused branch—computational social choice—have an
important role to play in providing stronger foundations for the principled
development of blockchain technology. This paper sketches a number of research
challenges at the interface of computational social choice and blockchain. I
claim there exists now a perfect match between the state-of-the-art in
blockchain research on the one hand, and the state-of-the-art in on
computational social choice on the other. Blockchain offers a wealth of novel
questions that can push the boundaries of the existing body of results of
computational social choice, and in doing so contribute concrete solutions to
the challenges blockchain research itself currently faces.

To substantiate this claim, this paper reviews how key mechanisms that have a
long tradition within (computational) social choice find deployment in
blockchain systems: randomized mechanisms, voting mechanisms, and trust
mechanisms. In reviewing these mechanisms this paper identifies challenges
that their deployment in blockchain gives rise to, phrasing them in terms of
properties that, again, have occupied social choice theorists in the past:
fairness of lotteries, manipulation of voting, and false-name proofness of
trust mechanisms. Such an overview has no claim to be exhaustive and rather
aims at illustrating, by means of examples, the potential for cross-
fertilization between the two research areas. Given the nature of this paper I
will try to limit technical jargon to a minimum, use informal language as much
as possible and just convey the gist of my arguments without resorting to
explicit mathematical details.

##  2\. Randomization

###  2.1. Lotteries in Blockchain and Social Choice

Randomization has a long tradition in distributed computing as a way to bypass
impossibility results such as the so-called FLP impossibility theorem Fischer
et al. (1985).111The Fisher, Lynch, Paterson (FLP) impossibility result states
that in an asynchronous system no deterministic consensus protocol exists
which can tolerate even just one faulty node, where a faulty node is a node
which stops interacting with the protocol. Blockchain has further stressed the
importance of randomization. Lotteries are at the heart of the main approaches
to blockchain consensus such as proof-of-work (PoW, currently used in both
Bitcoin and Ethereum Buterin (2013)) and proof-of-stake (PoS, currently used
for instance in the Ouroboros protocols Daian et al. (2016); David et al.
(2017) and Algorand Gilad et al. (2017)). At a high level, and leaving network
latency issues aside, these protocols work as follows: nodes participate in a
distributed lottery; the winner appends a new block to the chain (or,
depending on the specific protocol, becomes part of a committee which will
then vote on the block to be appended) and receives a compensation in the
native currency—what is referred to as mining. Importantly, the chances of
winning this lottery depend on a resource that is assumed not to be
monopolizable—such as computational power (in PoW) or currency ownership (in
PoS). This makes participation to the lottery (directly or indirectly) costly
and prevents the manipulation of the lottery through identity forging (the so-
called Sybil attack Douceur (2002)). Based on this blueprint, several so-
called ‘proof-of-X’ (PoX) Bano et al. (2019) schemes have been proposed (e.g.,
proof-of-storage Miller et al. (2014)).

Like in distributed computing, also in social choice theory randomization is
an established route to circumvent fundamental impossibility theorems of the
deterministic social choice framework, such as the Gibbard-Satterthwaite
theorem Gibbard (1973); Satterthwaite (1975).222The theorem states that no
social choice function exists which is simultaneously non-manipulable and non-
dictatorial. In a famous later theorem Gibbard himself Gibbard (1977) showed
how randomization can provide a possibility result that is out of reach in the
deterministic setting: the random draw of one agent (the ‘dictator’, or
‘leader’) from the set of agents is the only decision mechanism that is
‘lottery’ strategy-proof (in the specific sense of stochastic dominance333I
provide a sketch of the definition here: $N$ is the set of agents; $A$ is the
set of alternatives; ${\bf P}=\langle
P_{1},\ldots,P_{|N|}\rangle\in\mathcal{P}$ is a profile of preferences over
$A$. A randomized decision rule is a function $r:\mathcal{P}\to\Delta(A)$,
that is, a function assigning a lottery over $A$ to each profile of
preferences. A lottery $p$ stochastically dominates a lottery $q$ for $i$ iff
$\sum_{x\in A:xP_{i}y}p(x)\geq\sum_{x\in A:xP_{i}y}q(y)$, for any $y\in A$. A
randomized decision rule is strategy-proof, w.r.t. stochastic dominance, if
for all $i\in N$ it never selects a lottery which is stochastically dominated
by another lottery for $i$. ) and ‘lottery’ efficient (in the sense of never
assigning positive probability to alternatives that are Pareto dominated).
From a blockchain perspective the theorem can be thought to offer a
justification for the use of randomization in PoW and PoS consensus: using a
lottery to select the next block in the chain is economically efficient and
elicits the true preferences of nodes about what block should be included in
the chain. A string of works have further extended Gibbard’s result (e.g.,
Gibbard (1977); Procaccia (2010); Aziz et al. (2013)) and randomization is
recognized as an important toolbox in the design of economic mechanisms, from
allocation to voting Brandt (2017). This body of theory offers a sophisticated
framework to understand important economic properties of the sort of lotteries
deployed in blockchain protocols. One such property is fairness.

###  2.2. Challenge: Long-Term Fairness

By linking winning chances to resource ownership and linking lottery outcomes
to currency allocation, blockchain protocols make participation resistant to
Sybil attacks (participation is costly) and at the same time monetarily
rewarding (’no-show’ is disincentivized). This scheme, however, has been shown
to induce centralizing effects on several blockchain systems (see, e.g.,
Kondor et al. (2014); Fanti et al. (2019)). Participants with more resources,
all other things being equal, have higher chances of being selected by the
lottery and thus accrue more resources in the long term. The resulting
allocation of resources invested in the system becomes therefore increasingly
unequal over time, and this inequality in resource ownership de facto biases
the lotteries upon which consensus is based. In blockchain, such bias
translates in a centralization effect, which makes the blockchain more
vulnerable to failures or attacks. So, randomized blockchain consensus
protocols need to implement, in the long term, allocations of winning chances
which are fair towards participating nodes in two somewhat opposing senses:
fair in the sense of being lotteries that are not too biased towards few
participants, in order to preserve decentralization; and at the same time fair
in the sense of being responsive, even proportional, to the resources invested
so that participation is suitably incentivized. The analysis of such a trade-
off is an inherently social choice-theoretic question: can randomized
mechanisms for blockchain be designed which achieve participation
incentivization and equitability at the same time, in the long-term?
Extensions of existing work in computational social choice may provide the
right stepping stone to approach this question, along the following lines.

#### Perpetual lotteries

First, while randomized mechanisms in social choice are normally studied in
one-shot interaction, fairness properties in blockchain should be
conceptualized in the context of indefinitely repeating interaction: a lottery
may satisfy forms of fairness in one-shot interaction (e.g., assigning winning
chances proportionally to invested effort, like in PoW and PoS lotteries) but
may fail fairness criteria in the long run (see Fanti et al. (2019)). This
suggests the study of randomized mechanisms within the recently developed
perpetual voting setting of Lackner (2020) (see also Freeman et al. (2017)). A
key feature of the perpetual voting framework that appears particularly
relevant for applications in blockchain is, in particular, the history-
dependence of decisions. In blockchain this dependence manifests itself via
the positive feedback between current and future winning chances. Functions
capturing this positive feedback could be approached axiomatically as well as
via computer simulations (see Wang et al. (2020) for recent work in this
direction) in order to provide a framework to understand long-term fairness
properties of blockchain protocols viewed as perpetual randomized mechanisms.

#### Contest-based lotteries

Second, work on randomized mechanisms in social choice normally assumes
uniform probabilities or is agnostic about the specific probability mass
functions defining the lotteries. Furthermore, when lotteries are iterated, as
in random serial dictatorships Brandt (2017), probability functions do not
vary over time. As noticed above, this is not the case in blockchain: an
agent’s chance to win equals its share of a total non-monopolizable resource.
But there is another crucial element of lotteries in blockchain. The winning
probabilities for $i$ can be defined as $P_{i}=\frac{f(e_{i})}{\sum_{j\in
N}f(e_{j})},$ where $e_{i}$ is $i$’s invested effort to acquire a non-
monopolizable resource (e.g., currency, computing power), and $f$ is a
function (here assumed unique for all agents for simplicity) mapping each
agent’s effort to its acquired amount of resource (see Dimitri (2017)). An
agent’s $i$ expected utility in this type of interaction is, therefore,
$P_{i}r_{i}-e_{i}.$ That is, $i$’s probability of winning times its reward
(e.g., block reward plus transaction fees in Nakamoto consensus) minus the
effort invested. In economics jargon this type of interaction is called a
contest Cournot (1838); Tullock (1980); Corchon (2007). Linking contest theory
to the make-up of the lotteries in randomized mechanisms would capture a
crucial aspect of perpetual randomization in blockchain, and offer a
parsimonious framework in which to study long-term fairness properties.

##  3\. Voting

###  3.1. Voting in Blockchain and Social Choice

Voting mechanisms have been at the heart of distributed computing since its
early days (e.g., Garcia-Molina (1982)). Still, almost no crossover has
happened between the social choice literature on the analysis of voting
mechanisms and the distributed computing literature (a recent exception is
Melnyk et al. (2018)). This is perhaps not surprising, as the role that voting
plays in the two traditions is fundamentally different. In distributed
computing, voting is a consensus mechanism producing agreement in contexts
where what matters is agreement itself, and not so much the option upon which
agreement settles. That is, the agents involved in consensus are assumed not
to be invested in any specific option, but to just aim at reaching consensus.
In social choice instead, voting is eminently a mechanism for preference
aggregation. Given that blockchains are open systems and that, therefore,
agents’ interests cannot be assumed to align, the social choice theoretic
perspective on voting becomes naturally relevant.

There are several applications of voting in blockchain, but fork resolution is
arguably the main one.444Forks are branches of the blockchain. When these are
due to network latency issues or, possibly, attacks—rather than deliberate
choices of developers (so-called hard forks)—the protocol is supposed to
resolve them. I will illustrate this application of voting with respect to a
specific protocol proposed for Ethereum called Casper Buterin and Griffith
(2017), although voting mechanisms for fork resolution are used in many
systems (e.g., the Spectre protocol Sompolinsky et al. (2017)). In a nutshell,
the aim of Casper is to use voting among a randomly selected committee of
agents (called validators) to resolve forks in the blockchain and guarantee
consensus on a canonical chain (the so-called finality property). Whenever a
fork occurs, agents vote on blocks occurring on different branches in the fork
(this can be thought of as voting on the nodes of the tree of possible
chains). Each agent’s vote is weighted by the agent’s stakes, that is, the
agent’s deposit in currency (so, a high deposit means greater voting power),
and a block is considered to belong to the canonical chain whenever a weighted
supermajority of $\frac{2}{3}$ votes for that block. This voting procedure
guarantees that the winning blocks identify a legitimate chain (i.e., without
forks), provided that the agents submitting individually consistent votes
(that is, not voting for blocks on different chains) own at least
$\frac{2}{3}$ of the total deposits.

###  3.2. Challenge: Manipulating Byzantine Voting

Voting procedures are vulnerable to manipulations of different kinds:
strategic voting Taylor (2005), lobbying Christian et al. (2007), bribery and
control Faliszewski and Rothe (2016), vote negotiation Grandi et al. (2019).
Given the incentives layer of blockchain protocols, all such forms of vote
manipulation are potentially relevant in view of the deployment of voting
mechanisms. For example, returning to Casper, although the protocol can be
shown to be robust against a share of Byzantine voters worth $\frac{1}{3}$ of
the total deposit in stakes, to what extent is the protocol robust against
strategic forms of vote manipulation of the types mentioned above? More
generally: to what extent should voting mechanisms in blockchain consensus be
robust against forms of vote manipulation?

Given that voting in blockchain systems is conducted by computational
processes, the computational complexity approach to robustness against
manipulations appears especially natural: robustness to manipulations pursued
through computational intractability Bartholdi et al. (1989); Faliszewski and
Rothe (2016). In this perspective, one might argue more specifically that
voting mechanisms for blockchain should satisfy two properties: have a
tractable winner determination problem; have intractable manipulation problems
with respect to forms of manipulation that, given the application, may be
considered relevant. Results from the above mentioned literature offer an
obvious starting point, but should be extended in order to incorporate the
possibility—which is fundamental in the distributed computing perspective—of
Byzantine agents, in a way akin to what has been pursued in applications of
game theory to distributed computing (see, for instance, the so-called
Byzantine-Altruistic-Rational fault-tolerance models Clement et al. (2007)).
In particular, the decision problems themselves concerning the existence of
manipulation strategies would need to be adjusted to this setting, by allowing
for the possibility of shares of the agents population to be Byzantine.

##  4\. Trust

###  4.1. Trust in Blockchain and Social Choice

Among the key limitations of PoW are its high energy demand,555The estimated
annual energy consumption of the Bitcoin network is 194.95 TWh, roughly
comparable to the annual energy consumption of a country like Thailand.
Source: digiconomist.net (last accessed on 19.11.21). high latency and low
transactions throughput Bonneau et al. (2015). An influential approach trying
to address these limitations has been proposed by blockchain systems like
Stellar Mazierès (2016) and Ripple Schwartz et al. (2014)666They represent,
respectively, the 7th and 24th cryptocurrency in terms of market
capitalization (in the order of hundreds of millions of dollars). Source:
coinmarketcap.com (last accessed on 19.11.21). or TrustChain Otte et al.
(2017). The approach of these systems to safeguard consensus against Sybil
attacks is not based on lotteries but on the idea of leveraging existing
‘real-world’ trust relations in order to select the agents that may
participate in consensus. In other words, the system remains open, but new
participants are admitted only if trusted by existing ones. According to the
proponents of these systems, this should make it possible then to use well-
established consensus approaches proper of permissioned (i.e., closed)
systems, like Byzantine fault-tolerant consensus (BFT, Lamport et al. (1982);
Castro and Liskov (1999)). Nodes participate in consensus, but only in as much
as they are trusted by others. This controls participation to the consensus
process by restricting access. It does so, however, in a way that is in
principle open and decentralized.

Abstractly, the above systems work on the basis of an underlying trust network
that links nodes according to who trusts whom. At any given time, each honest
node broadcasts a truthful opinion about the state of the next block (e.g.,
whether a given block should be included or not) to the nodes that listen to
it. But the system may contain Byzantine nodes, and such nodes may reveal
different opinions to different honest nodes. A consensus protocol running on
such a trust network can be viewed as a discrete time dynamical system
generating a stream of opinion vectors of nodes. The final vector should be
such that all honest nodes hold the same opinion, that is, the system does not
fork—the so-called ‘safety’ property.

At least three strands of research in computational social choice appear
relevant for the above class of systems. The first one is the axiomatic and
computational analysis of trust and reputation mechanisms (e.g., Ghosh et al.
(2007); Seuken et al. (2010); Tennenholtz and Zohar (2013)). The second one is
the analysis of influence and power in structured groups and social networks
(e.g., Hu and Shapley (2003b, a); Grabisch and Rusinowska (2011)). The third
one is the analysis Sybil-proofness, or false-name proofness (e.g., Conitzer
and Yokoo (2010); Todo et al. (2011); Waggoner et al. (2012)). In what follows
I will outline how especially the latter two strands of research can offer
insights into the fundamental tradeoffs between decentralization, safety and
sybil-proofness that the above systems need to handle.

###  4.2. Challenge: Decentralized Sybil-proof Trust

Understanding how to deploy BFT consensus in a permissionless setting is
explicitly recognized as an open problem in distributed computing Cachin and
Vukolic (2017); Vukolic (2015). Yet, little academic research exists on the
trust-based approach to consensus used by systems like Ripple and Stellar.
Computational social choice could provide a fruitful framework from which to
study this approach: Can trust systems for blockchain consensus be designed
that maintain safety (i.e., absence of forks) are decentralized and, at the
same time, Sybil-proof?

#### Decentralization & safety

Trust involves a form of influence: a node’s voting decision on whether to
validate a block depends on the decisions of the nodes it trusts and ‘listens
to’. At the same time, safety demands that no two honest nodes express
dissenting (finalized) votes and this in turn imposes structural properties on
the trust structure: if we do not want forks to occur, then trust structures
should satisfy specific properties (see Mazierès (2016); Bracciali et al.
(2020)). But how decentralized a system really is when such properties hold is
unclear, because some nodes may end up accruing disproportionate influence in
the network. De facto centralization is a well-known phenomenon in PoW and PoS
blockchains (see the above discussion about randomized mechanisms) but a
comparable understanding for consensus protocols based on trust mechanisms is
still lacking. In particular it is unclear how to even quantify influence on
the consensus process in such systems, although some proposals have been put
forth using the theory of power indices Bracciali et al. (2020).

#### Sybil-proofness of trust mechanisms

If consensus is based on trust systems like the ones sketched above, what kind
of Sybil-proofness guarantees can be achieved? Intuitively, trust relations
should make the system harder to access for Sybils as they would be able to
exercise influence on consensus only if trusted by honest nodes. But can this
argument be made exact?

PoW and PoS protocols implement a form of costly identity. They make
participation to consensus costly by making it dependent on the investment in
a resource (e.g., computing power). In a way nodes purchase identities on a
continuum and proportionally to their ownership of a non-monopolizable
resource. Costly identities have been investigated also in social choice and
mechanism design as one possible approach to achieve the so-called false-name-
proofness of a mechanism. A mechanism is said to be false-name-proof if no
agent participating in it would benefit by using more than one identifier to
interact with the mechanism. A number of routes to enforce false-name-
proofness have been investigated in this literature and costly identities have
been studied also in the context of voting Wagman and Conitzer
(2008).777Intuitively, a voting mechanism is then false-name-proof if the cost
of casting additional votes always exceeds the gains an agent would obtain by
doing so. In this context, while creating Sybils may be costless, creating
trustworthy ones can be made costly. The issue then translates into
understanding the tradeoffs that manipulators face between identifiers’ costs
and their payoffs measured in terms of expected influence on the consensus
process.

##  5\. Conclusions

This paper has highlighted how the applications to blockchain of mechanisms
such as lotteries, voting or trust systems give rise to challenges that have a
natural social choice dimension. These challenges all revolve around
understanding fundamental tradeoffs, often in the long run, among properties
that are crucial to the correct behavior of blockchain consensus protocols and
are linked to the incentive dimensions of such protocols. In pursuing such
problems from a computational social choice perspective, one would need to
adapt standard concepts, definitions and techniques to the blockchain setting.
This adaptation opens up promising lines of research and this paper has tried
to illustrate some such lines, which appear promising to the author.

What covered, however, should not be considered exhaustive. In particular,
this paper has focused on the problem of consensus, but looking more broadly
at the blockchain field one finds many more points of contact with social
choice, some of which have already been identified in the literature. For
example, voting theory and coalitional games have been applied to the problem
of the algorithmic governance of hard forks in blockchain Abramowitz et al.
(2021), and epistemic social choice has been applied to the so-called oracle
problem Goel et al. (2020); Cai et al. (2020), that is, how to reliably link
blockchain records to real-world events. For all these reasons I believe there
exists now a deep cross-fertilization potential between the two areas of
research, and one that would greatly benefit both.

I thank the anonymous reviewers for their helpful comments.

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# Improving accuracy of field work calculations for the analysis of the
electron acceleration mechanisms in near-critical density plasma

Diana Gorlova  Faculty of Physics, Lomonosov Moscow State University,
Leninskye gory 1, 119991, Moscow, Russia  Institute for Nuclear Research of
the Russian Academy of Sciences, prospekt 60-letiya Oktyabrya 7a, 117312,
Moscow, Russia  Ivan Tsymbalov  Faculty of Physics, Lomonosov Moscow State
University, Leninskye gory 1, 119991, Moscow, Russia  Institute for Nuclear
Research of the Russian Academy of Sciences, prospekt 60-letiya Oktyabrya 7a,
117312, Moscow, Russia  Konstantin Ivanov  Faculty of Physics, Lomonosov
Moscow State University, Leninskye gory 1, 119991, Moscow, Russia  Ekaterina
Starodubtseva  Faculty of Physics, Lomonosov Moscow State University,
Leninskye gory 1, 119991, Moscow, Russia  Andrei Savelev  Faculty of Physics,
Lomonosov Moscow State University, Leninskye gory 1, 119991, Moscow, Russia

###### Abstract

It is demonstrated that the inherent structure of Boris pusher may result in
the accumulation of errors in numerical integration when estimating the works
of electric fields on particles. This, in turn, leads to an incorrect
estimations of particle acceleration mechanisms. A method of error correction,
reducing error to the acceptable levels of $<10\%$, is implemented. Numerical
integration error in question is most pronounced in scenarios where
substantial number of electrons are injected into the accelerating structure
through wave breaking of plasma waves i.e. in near-critical density plasma.

###### keywords:

laser-plasma acceleration, electron acceleration, PIC simulation, near-
critical density plasma, wave breaking, electric field work

##  1 INTRODUCTION

Modern laser systems generate pulses of short femtosecond duration with peak
powers up to several petawatts [1]. When focused, this allows one to achieve
relativistic intensities ($10^{18}-10^{21}$ W/cm2 for Ti:Sa laser systems)
[2]. The interaction of radiation of relativistic intensity with target atoms
forms a laser plasma, which, can be used to efficiently accelerate electrons
and ions [3]. The most commonly discussed electron acceleration mechanisms in
relativistic laser plasma are laser wakefield acceleration (LWFA [4]) and
direct laser acceleration (DLA [5]), the first one resulting in electron beams
with high energies (up to several GeV) and the second one - with high charge
(up to $\mu$C [6]).

However, currently there is an interest in studying so-called ”hybrid”
acceleration regimes, where several electron acceleration mechanisms are
present, for example LWFA to DLA regime [7] or DLA to VLA (vacuum laser
acceleration) [8]. There, in order to correctly establish acceleration
mechanisms, an analysis of particle-in-cell (PIC) simulation results should be
conducted; specifically one needs to calculate the work of various electric
field components along the trajectories of test particles. Electron energy
gain attributed to different components is explicitly related to various
electron acceleration mechanisms (longitudinal field $E_{x}$ \- LWFA
acceleration, etc.). In this work, this procedure will be examined in more
detail, and it will be shown that due to the inherent structure of the
numerical scheme used in the PIC code an accumulation of integration errors
occurs. A method of correcting this error will be proposed and implemented.

##  2 NUMERICAL SIMULATION OF ELECTRON ACCELERATION

###  2.1 Simulation Setup

In the present work we used SMILEI PIC code to conduct the simulations [9].
The calculations to establish the mechanisms of electron acceleration and the
parameters of the electron beam were performed in 3D, while the simulations
for electron beam deflection on the transverse gradient of electron
concentration [10] were carried out in 2D3V mode due to the need for a large
spatial domain. The problem under investigation is common for both simulation
regimes, at it arises from type of electron’s trajectory. Thus here, we will
only highlight some common simulation parameters.

In all conducted calculations, a Gaussian laser pulse with a normalized vector
potential of $a_{0}=1.5$ with a full width at half maximum (FWHM) duration of
$\tau\textsubscript{FWHM}=50$ fs was focused to a spot with a diameter of
$d\textsubscript{FWHM}=4$ $\mu$m. This corresponded to a vacuum intensity of
approximately $I\approx 5\cdot 10^{18}$ W/cm², reflecting experimental
conditions. At the beginning of the simulation, the target consisted of
neutral carbon atoms, and their density profile was taken from the results of
hydrodynamic modeling ($n_{e}\approx 0.05-0.2n_{cr}$, detailed target
parameters can be found in [11, 10]). In 3D simulations, 1 particle per cell
was used due to their high computational complexity. For 2D simulations, 9
particles per cell were employed. Note, that 1 particle per cell for ions
corresponds to 4-6 particles per cell for the electrons due to ionization.

For subsequent analysis, during the simulations, the distributions of fields
$E_{x},E_{y},E_{z},B_{z}$, electron and ion concentrations $n_{e}$ and
$n_{i}$, electron spatial and energy distributions, as well as trajectories of
individual particles were recorded. Present work concerns problems of
trajectory analysis specifically. All post-processing was carried out using
Python scripts. In the discussion commonly used normalized quantities will be
employed: spatial dimensions in units of the central wavelength of the laser
pulse $\lambda$, time moments in units of $\tau=\lambda/c$, electron $n_{e}$
and ion $n_{i}$ concentrations in units of the critical electron concentration
$n_{cr}$, field strengths $B$ and $E$ in units of the normalized vector
potential $a_{0}=\frac{eE}{\omega_{L}mc}=\frac{eB}{\omega_{L}m_{e}}$, where
$e,m$ are the charge and mass of the electron, kinetic energies of electrons
$E$ in units of the Lorentz factor $E=(\gamma-1)mc^{2}$, and field works $W$
in units of $mc^{2}\approx 0.5$ MeV. Grid step sizes along the corresponding
coordinates will be denoted as $\Delta x,\Delta y,\Delta z,\Delta\tau$.

###  2.2 Calculation of the Work of Fields on an Electron Along its Trajectory

Full results on the acceleration process analysis will be published elsewhere,
here we will discuss problem with work calculation that may arise during PIC
simulation results analysis. As mentioned earlier, the most straightforward
way to establish particle (here - electrons) acceleration mechanisms in laser-
plasma acceleration setups is through the analysis of the works of various
components of the electric field along the trajectories of electrons that
acquired a significant momentum at the end of the calculation. Here only the
electrons with final $p_{x}>5$ were analyzed. The work at each step
$t=N\Delta\tau$ is calculated using the formula:

$W_{j}=-e\int_{0}^{t}E_{j}v_{j}dt$ (1)

where $j=x,y,z$ represents the projection onto the corresponding axis. During
the analysis of numerical simulation results, the sum of works may deviate
from the particle’s energy, i.e.:

$W_{x}+W_{y}+W_{z}\neq\gamma+1$ (2)

ant the discrepancy can be significant ($\gg\gamma$). This inconsistency is
primarily caused by the fact that in the commonly used Boris solver [12] (and
many other solvers as well), the moments in time at which the electric fields
$E_{x,y,z}$ (solving Maxwell’s equations) and particle velocities $v_{x,y,z}$
(solving equations of motion) are determined are shifted by $\Delta\tau/2$.
Therefore, the output data of the numerical simulation will also be determined
at time points shifted by $\Delta\tau/2$, i.e., they will have a phase shift
$\delta=\frac{\omega\Delta\tau}{2}$.

Let’s illustrate why this phase shift will lead to errors in calculating the
work done by fields using the formula (1). Consider the motion of a single
electron in the field of a plane wave as the first approximation of the
interaction between the front of the laser pulse and electrons (i.e., when
plasma waves have not yet formed). It can be shown that:

$E\propto cos(\omega t)\quad v\propto sin(\omega t)\quad\rightarrow\quad
Ev\propto sin(2\omega t)\quad\rightarrow\quad\int_{0}^{\tau}Evdt\propto
cos(2\omega\tau)=0$ (3)

i.e., the average work done by the field over a period is zero. Now, let’s
introduce an additional phase shift $\delta$ between the field and the
electron velocity, i.e.,

$\displaystyle E\propto\cos(\omega t)\quad v\propto\sin(\omega
t+\delta)\quad\rightarrow\quad Ev\propto\sin(2\omega t+\delta)+\sin(\delta)$
(4a)
$\displaystyle\int_{0}^{\tau}Evdt\propto\frac{\cos(2\omega\tau+\delta)-\cos(\delta)}{2\omega}+\tau\sin(\delta)=\boxed{\tau\sin(\delta)\neq
0}$ (4b)

i.e., the average work done by the field over a period is now not zero but
depends on the period $\tau$ and the phase shift $\delta$. Obviously, when
integrating over a time interval containing many periods, this constant
component will accumulate linearly. Additionally, since the laser pulse is
Gaussian, the field intensity $E$ rapidly increases at its front, further
amplifying the error (Fig. 1, curve $W[i]$). Currently, an embedded method for
calculating field works is being developed in the SMILEI code, which, at the
moment, also leads to error accumulation (Fig. 1, curve $W[i]$ SMILEI).

Figure 1: The field works $W=W_{x}+W_{y}+W_{z}$, calculated directly from the
values of $E$ and $v$ along the trajectory (yellow curve $W[i]$), with the
field $E$ shifted during post-processing (blue curve $W[i-1/2]$), obtained
from the beta version of the embedded SMILEI module (green curve $W[i]$
SMILEI), as well as $\gamma-1$ for a test particle as a function of time.

Note that this issue is of concern only in the initial stages of particle
acceleration. During the acceleration process and as the particle gains
kinetic energy, the frequency of the electromagnetic field in the particle’s
co-moving frame will experience a Doppler shift into the red region. This
means that both $\lambda$ and $\tau$ will significantly increase, causing the
error accumulation to become negligible: $\delta\rightarrow
0,\tau\sin(\delta)\rightarrow const\ll\gamma$. Therefore, in scenarios where
electron acceleration occurs to high energies ($\gamma\gtrapprox 20$, i.e.
laser wakefield acceleration), the relative error may be small.

This issue is of big importance, however, for the acceleration regimes,
concerned with generating electron beams with high charge and relatively low
energies, i.e. most commonly - DLA in high density plasma ($n_{e}\gtrapprox
0.1n_{cr}$), as in our case. In such interaction regimes two injection
mechanisms are present: wave breaking of plasma waves (usually of backward
stimulated Raman scattering (BSRS) origin) and ionization injection. In
ionization injection, an electron, ”breaking away” from an ion, immediately
enters a region of strong field ($a_{0}>1$) and acquires a significant
longitudinal velocity ($v_{x}/c>0.5$, Fig. 2a). In injection due to the wave
breaking of plasma waves electron initially undergoes oscillations near the
equilibrium position in the plasma wave ($v_{x}/c\approx 0$, Fig. 2b) and
acquires a significant longitudinal velocity only after the wave breaking.
This prolonged electron’ oscillation near the equilibrium position is what
leads to fast error accumulation during trajectory analysis, i.e. error in
work analysis is of importance for the interaction regimes where injection
through the wave breaking is heavily present.

Figure 2: Characteristic trajectories of electrons (red lines) and the
magnetic field component of the laser pulse $B_{z}$ (red-blue) for electrons
undergoing ionization injection (a) and injection due to the wave breaking of
plasma waves (b).

In the considered acceleration regime, the error accumulated at the pulse
front routinely reaches $\frac{\mid W-(\gamma-1)\mid}{\gamma}\gg 1$, leading
to an incorrect estimation of the field component works. The most
straightforward and evident way to address this was the correction of the
phase shift. This was done by linearly interpolating fields from time moments
$t=n\Delta\tau$ to $t^{{}^{\prime}}=(n-1/2)\Delta\tau$, where $n=0..N$ \- time
steps in the grid. For this, the fields $E_{x,y,z}$ were recorded at each time
step $n$, and the values used for calculating works at this step were:

$E_{j}[n-1/2]=\frac{E_{j}[n]+E_{j}[n-1]}{2}$ (5)

where $j=x,y,z$. The calculation of the work with the phase shift correction
is shown in Fig. 1 (curve $W[i-1/2]$). It can be observed that the discrepancy
between $W$ and $(\gamma-1)$ is still present (likely due to interpolation
inaccuracies), but the error stops accumulating over time.

###  2.3 Comparison of Results for a Large Sample of Test Particles

Figure 3: The fields work $W=W_{x}+W_{y}+W_{z}$, calculated directly from the
values of $E$ and $v$ (a, $W[i]$), $\gamma-1$ (b) and the divergence between
them (c) for a sample of thousand of test particles as a function of time. All
units are normalized in $mc^{2}$.  Figure 4: The fields work
$W=W_{x}+W_{y}+W_{z}$, calculated from the values of $E$ shifted during post-
processing (a, $W[i-1/2]$), $\gamma-1$ (b) and the divergence between them (c)
for a sample of thousand of test particles (same as Fig.3) as a function of
time. All units are normalized in $mc^{2}$.

With further acceleration of the test particles, the relative error
$\frac{\mid W-(\gamma-1)\mid}{\gamma}$ decreases, and for most accelerated
particles, it is less than $10\%$ by the end of the simulation. The results of
a works calculation for a large sample of test particles are presented in Fig.
3 and Fig. 4. It can be observed that in the absence of correction in the
field work calculation (Fig. 3), the divergence between the particle kinetic
energy and the sum of the field works at the end of the simulation is
significant and can reach up to $W[i]-(\gamma-1)\approx-5$ (this value can be
both negative and positive depending on the initial $\delta$, i.e. determined
by solver and code properties). There is also some non-physical behavior
present: at the beginning stages of acceleration $W[i]<0$, while $\gamma-1\geq
0$ condition is always met. The use of the algorithm of error correction
described above leads to a significant reduction in this divergence (see Fig.
4). For most test particles at the end of the simulation the condition $\mid
W[i-1/2]-(\gamma-1)\mid<1$ is met, and non-physical behavior does not arise.
Further improvements in the fields interpolation scheme may reduce the
discrepancy (2) to the required level, we found the $10\%$ level satisfactory
for further analysis.

##  3 CONCLUSIONS

It has been demonstrated that the inherent structure of Boris pusher, commonly
used in PIC codes, may result in the accumulation of errors in numerical
integration when estimating the works of electric fields on particles. This,
in turn, leads to an incorrect estimations of particle acceleration
mechanisms. A method of error correction based on fields interpolation during
post-processing has been proposed and successfully implemented, reducing
$\mid\frac{W}{\gamma-1}\mid$ to the acceptable levels of $<10\%$. This
divergence effect is most pronounced in scenarios where the electron
trajectory involves a significant amount of oscillations near the equilibrium
position, i.e. when substantial number of electrons are injected into the
accelerating structure through wave breaking of plasma waves (usually in near-
critical density plasma). Nevertheless, the results demonstrated in this work
are general as they are caused by the structure of the Boris numerical scheme
itself.

###### Acknowledgements.

This work was supported by RSCF Grant №22-79-10087. D.G. and E.S. acknowledge
the Foundation for Theoretical Research ‘BASIS’ for financial support.

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