1]SAS LAB, DREXEL UNIVERSITY, PHILADELPHIA, PA 19104, USA 2]DEPARTMENT OF
MATHEMATICAL SCIENCES, MONTCLAIR STATE UNIVERSITY, MONTCLAIR, NJ 07043, USA
3]NONLINEAR SYSTEMS DYNAMICS SECTION, PLASMA PHYSICS DIVISION, CODE 6792, U.S.
NAVAL RESEARCH LAB, WASHINGTON, DC 20375, USA

KENNETH MALLORY
<EMAIL_ADDRESS>
# DISTRIBUTED ALLOCATION OF MOBILE SENSING SWARMS IN GYRE FLOWS

KENNETH MALLORY and M. ANI HSIEH  ERIC FORGOSTON  IRA B. SCHWARTZ  [  [  [

###### Abstract

We address the synthesis of distributed control policies to enable a swarm of
homogeneous mobile sensors to maintain a desired spatial distribution in a
geophysical flow environment, or workspace. In this article, we assume the
mobile sensors (or robots) have a “map” of the environment denoting the
locations of the Lagrangian coherent structures or LCS boundaries. Using this
information, we design agent-level hybrid control policies that leverage the
surrounding fluid dynamics and inherent environmental noise to enable the team
to maintain a desired distribution in the workspace. We discuss the stability
properties of the ensemble dynamics of the distributed control policies. Since
realistic quasi-geostrophic ocean models predict double-gyre flow solutions,
we use a wind-driven multi-gyre flow model to verify the feasibility of the
proposed distributed control strategy and compare the proposed control
strategy with a baseline deterministic allocation strategy. Lastly, we
validate the control strategy using actual flow data obtained by our coherent
structure experimental testbed.

Geophysical flows are naturally stochastic and aperiodic, yet exhibit coherent
structure. Coherent structures are of significant importance since knowledge
of them enables the prediction and estimation of the underlying geophysical
fluid dynamics. In realistic ocean flows, these time-dependent coherent
structures, or Lagrangian coherent structures (LCS), are similar to
separatrices that divide the flow into dynamically distinct regions, and are
essentially extensions of stable and unstable manifolds to general time-
dependent flows (Haller and Yuan, 2000). As such, they encode a great deal of
global information about the dynamics and transport of the fluidic
environment. For two-dimensional (2D) flows, ridges of locally maximal finite-
time Lyapunov exponent (FTLE) (Shadden et al., 2005) values correspond, to a
good approximation (though see (Haller, 2011)), to Lagrangian coherent
structures. Details regarding the derivation of the FTLE can be found in the
literature Haller (2000, 2001, 2002); Shadden et al. (2005); Lekien et al.
(2007); Branicki and Wiggins (2010).

Recent years have seen the use of autonomous underwater and surface vehicles
(AUVs and ASVs) for persistent monitoring of the ocean to study the dynamics
of various biological and physical phenomena, such as plankton assemblages
(Caron et al., 2008), temperature and salinity profiles (Lynch et al., 2008;
Wu and Zhang, 2011; Sydney and Paley, 2011), and the onset of harmful algae
blooms (Zhang et al., 2007; Chen et al., 2008; Das et al., 2011). These
studies have mostly focused on the deployment of single, or small numbers of,
AUVs working in conjunction with a few stationary sensors and ASVs. While data
collection strategies in these works are driven by the dynamics of the
processes they study, most existing works treat the effect of the surrounding
fluid as solely external disturbances (Das et al., 2011; Williams and
Sukhatme, 2012), largely because of our limited understanding of the
complexities of ocean dynamics. Recently, LCS have been shown to coincide with
optimal trajectories in the ocean which minimize the energy and the time
needed to traverse from one point to another (Inanc et al., 2005; Senatore and
Ross, 2008). And while recent works have begun to consider the dynamics of the
surrounding fluid in the development of fuel efficient navigation strategies
(Lolla et al., 2012; DeVries and Paley, 2011), they rely mostly on historical
ocean flow data and do not employ knowledge of LCS boundaries.

A drawback to operating both active and passive sensors in time-dependent and
stochastic environments like the ocean is that the sensors will escape from
their monitoring region of interest with some finite probability. This is
because the escape likelihood of any given sensor is not only a function of
the unstable environmental dynamics and inherent noise, but also the amount of
control effort available to the sensor. Since the LCS are inherently unstable
and denote regions of the flow where escape events occur with higher
probability (Forgoston et al., 2011), knowledge of the LCS are of paramount
importance in maintaining a sensor in a particular monitoring region.

In order to maintain stable patterns in unstable flows, the objective of this
work is to develop decentralized control policies for a team of autonomous
underwater vehicles (AUVs) and/or mobile sensing resources to maintain a
desired spatial distribution in a fluidic environment. Specifically, we devise
agent-level control policies which allow individual AUVs to leverage the
surrounding fluid dynamics and inherent environmental noise to navigate from
one dynamically distinct region to another in the workspace. While our agent-
level control policies are devised using a priori knowledge of
manifold/coherent structure locations within the region of interest, execution
of these control strategies by the individual robots is achieved using only
information that can be obtained via local sensing and local communication
with neighboring AUVs. As such, individual robots do not require information
on the global dynamics of the surrounding fluid. The result is a distributed
allocation strategy that minimizes the overall control-effort employed by the
team to maintain the desired spatial formation for environmental monitoring
applications.

While this problem can be formulated as a multi-task (MT), single-robot (SR),
time-extended assignment (TA) problem (Gerkey and Mataric, 2004), existing
approaches do not take into account the effects of fluid dynamics coupled with
the inherent environmental noise (Gerkey and Mataric, 2002; Dias et al., 2006;
Dahl et al., 2006; Hsieh et al., 2008; Berman et al., 2008). The novelty of
this work lies in the use of nonlinear dynamical systems tools and recent
results in LCS theory applied to collaborative robot tracking (Hsieh et al.,
2012) to synthesize distributed control policies that enables AUVs to maintain
a desired distribution in a fluidic environment.

The paper is structured as follows: We formulate the problem and outline key
assumptions in Section 1. The development of the distributed control strategy
is presented in Section 2 and its theoretical properties are analyzed in
Section 3. Section 4 presents our simulation methodology, results, and
discussion. We end with conclusions and directions for future work in Section
5.

##  1 Problem Formulation

Consider the deployment of $N$ mobile sensing resources (AUVs/ASVs) to monitor
$M$ regions in the ocean. The objective is to synthesize agent-level control
policies that will enable the team to autonomously maintain a desired
distribution across the $M$ regions in a dynamic and noisy fluidic
environment. We assume the following kinematic model for each AUV:

$\dot{\mathbf{q}}_{k}=\mathbf{u}_{k}+\mathbf{v}^{f}_{\mathbf{q}_{k}}\quad
k\in\\{1,\ldots,n\\},$ (1)

where $\mathbf{q}_{k}=[x_{k},\,y_{k},\,z_{k}]^{T}$ denotes the vehicle’s
position, $\mathbf{u}_{k}$ denotes the $3\times 1$ control input vector, and
$\mathbf{v}^{f}_{\mathbf{q}_{k}}$ denotes the velocity of the fluid
experienced/measured by the $k^{th}$ vehicle.

In this work, we limit our discussion to 2D planar flows and motions and thus
we assume $z_{k}$ is constant for all $k$. As such,
$\mathbf{v}^{f}_{\mathbf{q}_{k}}$ is a sample of a 2D vector field denoted by
$F(\mathbf{q})$ at $\mathbf{q}_{k}$ whose $z$ component is equal to zero,
i.e., $F_{z}(\mathbf{q})=0$, for all $\mathbf{q}$. Since realistic quasi-
geostrophic ocean models exhibit multi-gyre flow solutions, we assume
$F(\mathbf{q})$ is provided by the 2D wind-driven multi-gyre flow model given
by

$\displaystyle\dot{x}=-\pi A\sin(\pi\frac{f(x,t)}{s})\cos(\pi\frac{y}{s})-\mu
x+\eta_{1}(t),$ (2a) $\displaystyle\dot{y}=\pi
A\cos(\pi\frac{f(x,t)}{s})\sin(\pi\frac{y}{s})\frac{df}{dx}-\mu
y+\eta_{2}(t),$ (2b) $\displaystyle\dot{z}=0,$ (2c) $\displaystyle
f(x,t)=x+\varepsilon\sin(\pi\frac{x}{2s})\sin(\omega t+\psi).$ (2d)

When $\varepsilon=0$, the multi-gyre flow is time-independent, while for
$\varepsilon\neq 0$, the gyres undergo a periodic expansion and contraction in
the $x$ direction. In (2), $A$ approximately determines the amplitude of the
velocity vectors, $\omega/2\pi$ gives the oscillation frequency, $\varepsilon$
determines the amplitude of the left-right motion of the separatrix between
the gyres, $\psi$ is the phase, $\mu$ determines the dissipation, $s$ scales
the dimensions of the workspace, and $\eta_{i}(t)$ describes a stochastic
white noise with mean zero and standard deviation $\sigma=\sqrt{2I}$, for
noise intensity $I$. Figures 1 and 1 show the vector field of a two-gyre model
and the corresponding FTLE curves for the time-dependent case.

Figure 1: (a) Vector field and (b) FTLE field of the model given by (2) for
two gyres with $A=10$, $\mu=0.005$, $\varepsilon=0.1$, $\psi=0$, $I=0.01$, and
$s=50$. LCS are characterized by regions with maximum FTLE measures (denoted
by red). In 2D flows, regions with maximum FTLE measures correspond to 1D
curves.

Let ${\cal W}$ denote an obstacle-free workspace with flow dynamics given by
(2). We assume a tessellation of ${\cal W}$ such that the boundaries of each
cell roughly corresponds to the stable/unstable manifolds or LCS curves
quantified by maximum FTLE ridges as shown in Fig. 2. In general, it may be
unreasonable to expect small resource constrained autonomous vehicles to be
able to track the LCS locations in real time. However, LCS boundary locations
can be determined using historical data, ocean model data, e.g., data provided
by the Navy Coastal Ocean Model (NCOM) databases, and/or data obtained a
priori using LCS tracking strategies similar to (Hsieh et al., 2012). This
information can then be used to obtain an LCS-based cell decomposition of
${\cal W}$. Fig. 2 shows two manual cell decompositions of the workspace where
the cell boundaries roughly correspond to maximum FLTE ridges. In this work,
we assume the tessellation of ${\cal W}$ is given and do not address the
problem of automatic tessellation of the workspace to achieve a decomposition
where cell boundaries correspond to LCS curves.

A tessellation of the workspace along boundaries characterized by maximum FTLE
ridges makes sense since they separate regions within the flow field that
exhibit distinct dynamic behavior and denote regions in the flow field where
more escape events may occur probabilistically (Forgoston et al., 2011). In
the time-independent case, these boundaries correspond to stable and unstable
manifolds of saddle points in the system. The manifolds can also be
characterized by maximum FTLE ridges where the FTLE is computed based on a
backward (attracting structures) or forward (repelling structures) integration
in time. Since the manifolds demarcate the basin boundaries separating the
distinct dynamical regions, they are also regions that are uncertain with
respect to velocity vectors within a neighborhood of the manifold. Therefore,
switching between regions in neighborhoods of the manifold is influenced both
by deterministic uncertainty as well as stochasticity due to external noise.

Figure 2: Two examples of LCS-based cell decomposition of the region of
interest assuming a flow field given by (2). These cell decompositions were
performed manually. (a) A $4\times 4$ time-independent grid of gyres with
$A=0.5$, $\mu=0.005$, $\varepsilon=0$, $\psi=0$, $I=35$, and $s=20$. The
stable and unstable manifolds of each saddle point in the system is shown by
the black arrows. (b) An FTLE based cell decomposition for a time-dependent
double-gyre system with the same parameters as Fig. 1.

Given an FTLE-based cell decomposition of ${\cal W}$, let ${\cal G}=({\cal
V},{\cal E})$ denote an undirected graph whose vertex set ${\cal
V}=\\{V_{1},\ldots,V_{M}\\}$ represents the collection of FTLE-derived cells
in ${\cal W}$. An edge $e_{ij}$ exists in the set ${\cal E}$ if cells $V_{i}$
and $V_{j}$ share a physical boundary or are physically adjacent. In other
words, ${\cal G}$ serves as a roadmap for ${\cal W}$. For the case shown in
Fig. 2, adjacency of an interior cell is defined based on four neighborhoods.
Let $N_{i}$ denote the number of AUVs or mobile sensing resources/robots
within $V_{i}$. The objective is to synthesize agent-level control policies,
or $\mathbf{u}_{k}$, to achieve and maintain a desired distribution of the $N$
agents across the $M$ regions, denoted by
$\mathbf{\bar{N}}=[\bar{N}_{1},\ldots,\bar{N}_{M}]^{T}$, in an environment
whose dynamics are given by (2).

We assume that robots are given a map of the environment, ${\cal G}$, and
$\mathbf{\bar{N}}$. Since the tessellation of ${\cal W}$ is given, the LCS
locations corresponding to the boundaries of each $V_{i}$ is also known a
priori. Additionally, we assume robots co-located within the same $V_{i}$ have
the ability to communicate with each other. This makes sense since coherent
structures can act as transport barriers and prevent underwater acoustic wave
propagation (Wang et al., 2009; Rypina et al., 2011). Finally, we assume
individual robots have the ability to localize within the workspace, i.e.,
determine their own positions in the workspace. These assumptions are
necessary to enable the development of a prioritization scheme within each
$V_{i}$ based on an individual robot’s escape likelihoods in order to achieve
the desired allocation. The prioritization scheme will allow robots to
minimize the control effort expenditure as they move within the set ${\cal
V}$. We describe the methodology in the following section.

##  2 Methodology

We propose to leverage the environmental dynamics and the inherent
environmental noise to synthesize energy-efficient control policies for a team
of mobile sensing resources/robots to maintain the desired allocation in
${\cal W}$ at all times. We assume each robot has a map of the environment. In
our case, this translates to providing the robots the locations of LCS
boundaries that define each $V_{i}$ in ${\cal G}$. Since LCS curves separate
${\cal W}$ into regions with distinct flow dynamics, this becomes analogous to
providing autonomous ground or aerial vehicles a map of the environment which
is often obtained a priori. In a fluidic environment, the map consists of the
locations of the maximum FTLE ridges computed from data and refined,
potentially in real-time, using a strategy similar to the one found in (Hsieh
et al., 2012). Thus, we assume each robot has a map of the environment and has
the ability to determine the direction it is moving in within the global
coordinate frame, i.e., the ability to localize.

###  2.1 Controller Synthesis

Consider a team of $N$ robots initially distributed across $M$ gyres/cells.
Since the objective is to achieve a desired allocation of $\mathbf{\bar{N}}$
at all times, the proposed strategy will consist of two phases: an auction to
determine which robots within each $V_{i}$ should be tasked to leave/stay and
an actuation phase where robots execute the appropriate leave/stay controller.

####  2.1.1 Auction Phase

The purpose of the auction phase is to determine whether
$N_{i}(t)>\bar{N}_{i}$ and to assign the appropriate actuation strategy for
each robot within $V_{i}$. Let $Q_{i}$ denote an ordered set whose elements
provide robot identities that are arranged from highest escape likelihoods to
lowest escape likelihoods from $V_{i}$.

In general, to first order we assume a geometric measure whereby the escape
likelihood of any particle within $V_{i}$ increases as it approaches the
boundary of $V_{i}$, denoted as $\partial V_{i}$ (Forgoston et al., 2011).
Given ${\cal W}$, with dynamics given by (2), consider the case when
$\varepsilon=0$ and $I\neq 0$, i.e., the case when the fluid dynamics is time-
independent in the presence of noise. The boundaries between each $V_{i}$ are
given by the stable and unstable manifolds of the saddle points within ${\cal
W}$ as shown in Fig. 2. While there exists a stable attractor in each $V_{i}$
when $I=0$, the presence of noise means that robots originating in $V_{i}$
have a non-zero probability of landing in a neighboring gyre $V_{j}$ where
$e_{ij}\in{\cal E}$. Here, we assume that robots experience the same escape
likelihoods in each gyre/cell and assume that $P_{k}(\neg i|i)$, the
probability that a robot escapes from region $i$ to an adjacent region, can be
estimated based on a robot’s proximity to a cell boundary with some assumption
of the environmental noise profile (Forgoston et al., 2011).

Let $d(\mathbf{q}_{k},\partial V_{i})$ denote the distance between a robot $k$
located in $V_{i}$ and the boundary of $V_{i}$. We define the set
$Q_{i}=\\{k_{1},\ldots,k_{N_{i}}\\}$ such that $d(q_{k_{1}},\partial
V_{i})\leq d(q_{k_{2}},\partial V_{i})\leq\ldots\leq d(q_{N_{i}},\partial
V_{i})$. The set $Q_{i}$ provides the prioritization scheme for tasking robots
within $V_{i}$ to leave if $N_{i}(t)>\bar{N}_{i}$. The assumption is that
robots with higher escape likelihoods are more likely to be “pushed” out of
$V_{i}$ by the environment dynamics and will not have to exert as much control
effort when moving to another cell, minimizing the overall control effort
required by the team.

In general, a simple auction scheme can be used to determine $Q_{i}$ in a
distributed fashion by the robots in $V_{i}$ (Dias et al., 2006). If
$N_{i}(t)>\bar{N}_{i}$, then the first $N_{i}-\bar{N}_{i}$ elements of
$Q_{i}$, denoted by $Q_{i_{L}}\subset Q_{i}$, are tasked to leave $V_{i}$. The
number of robots in $V_{i}$ can be established in a distributed manner in a
similar fashion. The auction can be executed periodically at some frequency
$1/T_{a}$ where $T_{a}$ denotes the length of time between each auction and
should be greater than the relaxation time of the AUV/ASV dynamics.

####  2.1.2 Actuation Phase

For the actuation phase, individual robots execute their assigned controllers
depending on whether they were tasked to stay or leave during the auction
phase. As such, the individual robot control strategy is a hybrid control
policy consisting of three discrete states: a leave state, $U_{L}$, a stay
state, $U_{S}$, which is further distinguished into $U_{S_{A}}$ and
$U_{S_{P}}$. Robots who are tasked to leave will execute $U_{L}$ until they
have left $V_{i}$ or until they have been once again tasked to stay. Robots
who are tasked to stay will execute $U_{S_{P}}$ if $d(q_{k},\partial
V_{i})>d_{min}$ and $U_{S_{A}}$ otherwise. In other words, if a robot’s
distance to the cell boundary is below some minimum threshold distance
$d_{min}$, then the robot will actuate and move itself away from $\partial
V_{i}$. If a robot’s distance to $\partial V_{i}$ is above $d_{min}$, then the
robot will execute no control actions. Robots will execute $U_{S_{A}}$ until
they have reached a state where $d(q_{k},\partial V_{i})>d_{min}$ or until
they are tasked to leave at a later assignment round. Similarly, robots will
execute $U_{S_{P}}$ until either $d(q_{k},\partial V_{i})\leq d_{min}$ or they
are tasked to leave. The hybrid robot control policy is given by

$\displaystyle U_{L}(\mathbf{q}_{k})$ $\displaystyle=\bm{\omega}_{i}\times
c\frac{F(\mathbf{q}_{k})}{\|F(\mathbf{q}_{k})\|},$ (3a) $\displaystyle
U_{S_{A}}(\mathbf{q}_{k})$ $\displaystyle=-\bm{\omega}_{i}\times
c\frac{F(\mathbf{q}_{k})}{\|F(\mathbf{q}_{k})\|},$ (3b) $\displaystyle
U_{S_{P}}(\mathbf{q}_{k})$ $\displaystyle=0.$ (3c)

Here, $\mathbf{\omega}_{i}=[0,\,0,\,1]^{T}$ denotes counterclockwise rotation
with respect to the centroid of $V_{i}$, with clockwise rotation being denoted
by the negative and $c$ is a constant that sets the linear speed of the
robots. The hybrid control policy generates a control input perpendicular to
the velocity of the fluid as measured by robot $k$111The inertial velocity of
the fluid can be computed from the robot’s flow-relative velocity and
position. and pushes the robot towards $\partial V_{i}$ if $U_{L}$ is
selected, away from $\partial V_{i}$ if $U_{S_{A}}$ is selected, or results in
no control input if $U_{S_{P}}$ is selected The hybrid control policy is
summarized by Algorithm 1 and Fig. 3.

Algorithm 1 Auction Phase

1: if $ElapsedTime==T_{a}$ then

2: Determine $N_{i}(t)$ and $Q_{i}$

3: $\forall k\in Q_{i}$

4: if $N_{i}(t)>\bar{N}_{i}$ then

5: if $k\in Q_{L}$ then

6: $u_{k}\leftarrow U_{L}$

7: else

8: $u_{k}\leftarrow U_{S}$

9: end if

10: else

11: $u_{k}\leftarrow U_{S}$

12: end if

13: end if

In general, the Auction Phase is executed at a frequency of $1/T_{a}$ which
means robots also switch between controller states at a frequency of
$1/T_{a}$. To further reduce actuation efforts exerted by each robot, it is
possible to limit a robot’s actuation time to a period of time $T_{c}\leq
T_{a}$. Such a scheme may prolong the amount of time required for the team to
achieve the desired allocation, but may result in significant energy-
efficiency gains. We further analyze the proposed strategy in the following
sections.

Figure 3: Schematic of the single-robot hybrid robot control policy.

##  3 Analysis

In this section, we discuss the theoretical feasibility of the proposed
distributed allocation strategy. Instead of the traditional agent-based
analysis, we employ a macroscopic analysis of the proposed distributed control
strategy given by Algorithm 1 and (3). We first note that while the single
robot controller shown in Fig. 3 results in an agent-level stochastic control
policy, the ensemble dynamics of a team of $N$ robots each executing the same
hybrid control strategy can be modeled using a polynomial stochastic hybrid
system (pSHS). The advantage of this approach is that it allows the use of
moment closure techniques to model the time evolution of the distribution of
the team across the various cells. This, in turn, enables the analysis of the
stability of the closed-loop ensemble dynamics. The technique was previously
illustrated in (Mather and Hsieh, 2011). For completeness, we briefly
summarize the approach here and refer the interested reader to (Mather and
Hsieh, 2011) for further details.

The system state is given by
$\mathbf{N}(t)=\left[N_{1}(t),\ldots,N_{M}(t)\right]^{T}$. As the team
distributes across the $M$ regions, the rate in which robots leave a given
$V_{i}$ can be modeled using constant transition rates. For every edge
$e_{ij}\in{\cal E}$, we assign a constant $a_{ij}>0$ such that $a_{ij}$ gives
the transition probability per unit time for a robot from $V_{i}$ to land in
$V_{j}$. Different from Mather and Hsieh (2011), the $a_{ij}$s are a function
of the parameters $c$, $T_{c}$, and $T_{a}$ of the individual robot control
policy (3), the dynamics of the surrounding fluid, and the inherent noise in
the environment. Furthermore, $a_{ij}$ is a macroscopic description of the
system and thus a parameter of the ensemble dynamics rather than the agent-
based system. As such, the macroscopic analysis is a description of the
steady-state behavior of the system and becomes exact as $N$ approaches
infinity.

Given ${\cal G}$ and the set of $a_{ij}$s, we model the ensemble dynamics as a
set of transition rules of the form:

$\displaystyle N_{i}\xrightarrow{a_{ij}}N_{j}\quad\textrm{$\forall$
$e_{ij}\in{\cal E}$}.$ (4)

The above expression represents a stochastic transition rule with $a_{ij}$ as
the per unit transition rate and $N_{i}(t)$ and $N_{j}(t)$ as discrete random
variables. In the robotics setting, (4) implies that robots at $V_{i}$ will
move to $V_{j}$ with a rate of $a_{ij}N_{i}$. We assume the ensemble dynamics
is Markovian and note that in general $a_{ij}\neq a_{ji}$ and $a_{ij}$ encodes
the inverse of the average time a robot spends in $V_{i}$.

Given (4) and employing the extended generator we can obtain the following
description of the moment dynamics of the system:

$\displaystyle\tfrac{d}{dt}\mathrm{E}[\mathbf{N}]$
$\displaystyle=\mathbf{A}\mathrm{E}[\mathbf{N}]$ (5)

where $[\mathbf{A}]_{ij}=a_{ji}$ and
$[\mathbf{A}]_{ii}=-\sum_{(i,j)\in\cal{E}}a_{ij}$ (Mather and Hsieh, 2011). It
is important to note that $\mathbf{A}$ is a Markov process matrix and thus is
negative semidefinite. This, coupled with the conservation constraint
$\sum_{i}N_{i}=N$ leads to exponential stability of the system given by (5)
(Klavins, 2010).

In this work, we note that $a_{ij}$s can be determined experimentally after
the selection of the various parameters in the distributed control strategy.
While the $a_{ij}$s can be chosen to enable the team of robots to autonomously
maintain the desired steady-state distribution (Hsieh et al., 2008),
extraction of the control parameters from user specified transition rates is a
direction for future work. Thus, using the technique described by Mather and
Hsieh (2011), the following result can be stated for our current distributed
control strategy

###### Theorem 1

Given a team of $N$ robots with kinematics given by (1) and $\mathbf{v}_{f}$
given by (2), the distributed allocation strategy given by Algorithm 1 and
(3), at the ensemble level is stable and achieves the desired allocation
strategy.

For the details of the model development and the proof, we refer the
interested reader to (Mather and Hsieh, 2011).

##  4 Simulation Results

We validate the proposed control strategy described by Algorithm (1) and (3)
using three different flow fields:

  1. 1.

the time invariant wind driven multi-gyre model given by (2) with
$\varepsilon=0$,

  2. 2.

the time varying wind driven multi-gyre model given by (2) for a range of
$\omega\neq 0$ and $\varepsilon\neq 0$ values, and

  3. 3.

an experimentally generated flow field using different values of $T_{a}$ and
$c$ in (3).

We refer to each of these as Cases 1, 2, and 3 respectively. Two metrics are
used to compare the three cases. The first is the mean vehicle control effort
to indicate the energy expenditure of each robot. The second is the population
root mean square error (RMSE) of the resulting robot population distribution
with respect to the desired population. The RMSE is used to show effectiveness
of the control policy in achieving the desired distribution.

All cases assume a team of $N=500$ robots. The robots are randomly distributed
across the set of M gyres in ${\cal W}$. For the theoretical models, the
workspace ${\cal W}$ consists of a $4\times 4$ set of gyres, and each
$V_{i}\in{\cal V}$ corresponds to a gyre as shown in Fig. 2. We considered
three sets sets of desired distributions, namely a Ring formation, a Block
formation, and an L-shaped formation as shown in Fig. 4. The experimental flow
data had a set of $3\times 4$ regions. The inner two cells comprised ${\cal
W}$, while the complement, ${\cal W}^{C}$ consisted of the remaining cells.
This designation of cells helped to isolate the system from boundary effects,
and allowed the robots to escape the center gyres in all directions. The
desired pattern for this experimental data set was for all the agents to be
contained within a single cell. Each of the three cases was simulated a
minimum of five times and for a long enough period of time until steady-state
was reached.

Figure 4: Three desired distributions of the team of $N=500$ mobile sensing
resources/robots. (a) A Ring pattern formation, (b) a Block pattern formation,
and (c) an L-shaped pattern formation. Each box represents a gyre and the
number designates the desired number of robots contained within each gyre.

###  4.1 Case I: Time-Invariant Flows

For time-invariant flows, we assume $\varepsilon=0$, $A=0.5$, $s=20$,
$\mu=0.005$, and $I=35$ in (2). For the ring pattern, we consider the case
when the actuation was applied for $T_{c}=fT_{a}$ amount of time where
$f=0.1,0.2,\ldots,1.0$, and $T_{a}=10$. For the Block and L-Shape patterns, we
considered the cases when $T_{c}=0.5T_{a}$ and $T_{c}=T_{a}$. The final
population distribution of the team for the case with no controls and the
cases with controls for each of the patterns are shown in Fig. 5.

((a)) No Control

((b)) Ring

((c)) Block

((d)) L-Shape

Figure 5: Histogram of the final allocations in the time-invariant flow field
for the swarm of (a) passive robots exerting no controls and robots exerting
control forming the (b) Ring pattern with $T_{c}=0.8T_{a}$ at $t=450$, (c)
Block pattern with $T_{c}=T_{a}$ at $t=450$, and (d) L-shape pattern with
$T_{c}=0.5T_{a}$ at $t=450$.

We compared our results to a baseline deterministic allocation strategy where
the desired allocation is pre-computed and individual robots follow fixed
trajectories when navigating from one gyre to another. For this baseline case,
robots travel in straight lines at fixed speeds using a simple PID trajectory
follower and treat the surrounding fluid dynamics as an external disturbance
source. The RMSE results for all patterns are summarized in Table 1 and Fig.
6. The cumulative control effort per agent is shown in Fig. 7. From Fig. 6, we
see that our proposed control strategy performs comparable to the baseline
case especially when $T_{c}=T_{a}=10$ $sec$. In fact, even when $T_{c}<T_{a}$,
our proposed strategy achieves the desired distribution. The advantage of the
proposed approach lies in the significant energy gains when compared to the
baseline case, especially when $T_{c}<T_{a}$, as seen in Fig. 7. We omit the
cumulative control effort plots for the other cases since they are similar to
Fig. 7.

Figure 6: Comparison of the population RMSE in the time-varying flow for the
(a) Ring formation, (b) the Block formation, and (c) the L-shape formation for
different $T_{c}$, and for the PID control baseline controller in the Ring
case with time-invariant flows.  Figure 7: Comparison of the total control
effort for the Ring pattern for different $T_{c}$ with the baseline controller
for time-invariant flows.

In time-invariant flows, we note that for large enough $T_{c}$, our proposed
distributed control strategy performs comparable to the baseline controller
both in terms of steady-state error and convergence time. As $T_{c}$
decreases, less and less control effort is exerted and thus it becomes more
and more difficult for the team to achieve the desired allocation. This is
confirmed by both the RMSE results summarized in Table 1 and Fig. 6-6.
Furthermore, while the proposed control strategy does not beat the baseline
strategy as seen in Fig. 6, it does come extremely close to matching the
baseline strategy performance. while requiring much less control effort as
shown in Fig. 7 even at high duty cycles, i.e., when $T_{c}/T_{a}>0.5$.

More interestingly, we note that executing the proposed control strategy at
100% duty cycle, i.e., when $T_{c}=T_{a}$, in time-invariant flows did not
always result in better performance. This is true for the cases when
$T_{c}=0.5T_{a}=5$ for the Block and L-shaped patterns shown in Fig. 6 and 6.
In these cases, less control effort yielded improved performance. However,
further studies are required to determine the critical value of $T_{c}$ when
less control yields better overall performance. In time-invariant flows, our
proposed controller can more accurately match the desired pattern while using
approximately $20\%$ less effort when compared to the baseline controller.

###  4.2 Case II: Time-Varying Flows

For the time-varying, periodic flow, we assume $A=0.5$, $s=20$, $\mu=0.005$,
$I=35$, and $\psi=0$ in (2). Additionally, we considered the performance of
our control strategy for different values of $\omega$ and $\varepsilon$ with
$T_{a}=10$ and $T_{c}=8$ for the Ring formation and $T_{c}=5$ for the L-shaped
formation. In all these simulations, we use the FTLE ridges obtained for the
time-independent case to define the boundaries of each $V_{i}$. The final
population distribution of the team for the case with no controls and the
cases with controls for the Ring and L-shape patterns are shown in Fig. 8. The
final population RMSE for the cases with different $\omega$ and $\varepsilon$
values for the Ring and L-shape patterns are shown in Fig. 9. These figures
show the average of $10$ runs for each $\omega$ and $\varepsilon$ pair. In
each of these runs, the swarm of mobile sensors were initially randomly
distributed within the grid of $4\times 4$ cells. Finally, Fig. 10 shows the
population RMSE as a function of time for the Ring and L-shape patterns
respectively.

In time-varying, periodic flows we note that our proposed control strategy is
able to achieve the desired final allocation even at 80% duty cycle, i.e.,
$T_{c}=0.8T_{a}$. This is supported by the results shown in Fig. 9. In
particular, we note that the proposed control strategy performs quite well for
a range of $\omega$ and $\varepsilon$ parameters for both the Ring and L-shape
patterns. While the variation in final RMSE values for the Ring pattern is
significantly lower than the L-shape pattern, the variations in final RMSE
values for the L-shape are all within 10% of the total swarm population.

((a)) No Control

((b)) Ring

((c)) L-Shape

Figure 8: Histogram of the final allocations in periodic flows, with
parameters of $\omega=\frac{5*\pi}{40}$ and $\varepsilon=5$, for the swarm of
(a) passive robots exerting no controls and robots exerting control forming
the (b) Ring pattern with $T_{c}=0.8T_{a}$ at $t=450$, and (c) L-Shape pattern
with $T_{c}=0.5T_{a}$ at $t=450$.

((a)) Ring

((b)) L-Shape

Figure 9: Final population RMSE for different values of $\omega$ and
$\varepsilon$ for (a) the Ring formation and (b) the L-shaped formation.

Figure 10: Comparison of RMSE over time for select $\omega$ and $\varepsilon$
pairs for the (a) Ring and (b) L-shaped patterns in periodic flows.

###  4.3 Case III: Experimental Flows

Using our $0.6m\times 0.6m\times 0.3m$ experimental flow tank equipped with a
grid of $4x3$ set of driving cylinders, we generated a time-invariant multi-
gyre flow field to use in simulation. Particle image velocimetry (PIV) was
used to extract the surface flows at $7.5$ $Hz$ resulting in a $39\times 39$
grid of velocity measurements. The data was collected for a total of $60$
$sec$. Figure 11 shows the top view of our experimental testbed and the
resulting flow field obtained via PIV. Further details regarding the
experimental testbed can be found in (Michini et al., 2013). Using this data,
we simulated a swarm of $500$ mobile sensors executing the control strategy
given by (3).

Figure 11: (a) Experimental setup of flow tank with $12$ driven cylinders. (b)
Flow field for image (a) obtained via particle image velocimetry (PIV).

To determine the appropriate tessellation of the workspace, we used the LCS
ridges obtained for the temporal mean of the velocity field. This resulted in
the discretization of the space into a grid of $4\times 3$ cells. Each cell
corresponds to a single gyre as shown Fig. 12. The cells of primary concern
are the central pair and the remainder boundary cells were not used to avoid
boundary effects and to allow robots to escape the center gyres in all
directions. The robots were initially uniformly distributed across the two
center cells and all $500$ robots were tasked to stay within the upper center
cell. When no control effort is exerted by the robots, the final population
distribution achieved by the team is shown in Fig. 13. With controls, the
final population distribution is shown in Fig. 13. The control strategy was
applied assuming $T_{c}/T_{a}=0.8$. The final RMSE for different values of $c$
in (3) and $T_{a}$ is shown in Fig. 14 and RMSE as a function of time for
different values of $c$ and $T_{a}$ are shown in Fig. 14.

The results obtained using the experimental flow field shows that the proposed
control strategy has the potential to be effective in realistic flows.
However, the resulting performance will require good matching between the
amount of control effort a vehicle can realistically exert, the frequency in
which the auctions occur within a cell, and the time scales of the
environmental dynamics as shown in in Figs. 14 and 14. This is an area for
future investigation.

Figure 12: FTLE field for the temporal mean of the experimental velocity data.
The field is discretized into a grid of $4\times 3$ cells whose boundaries are
shown in black.

Figure 13: Population distribution for a swarm of $500$ mobile sensors over a
period of $60$ $sec$ (a) with no controls, i.e., passive, and (b) with
controls with $T_{c}=0.8T_{a}$.

Figure 14: (a) Final RMSE for different values of $c$ and $T_{a}$ using the
experimental flow field. $T_{c}/T_{a}=0.8$ is kept constant throughout. (b)
RMSE over time for select $c$ and $T_{a}$ parameters on an experimental flow
field. The duty cycle $T_{c}/T_{a}=0.8$ is kept constant throughout.

##  5 Conclusions and Future Outlook

In this work, we presented the development of a distributed hybrid control
strategy for a team of robots to maintain a desired spatial distribution in a
stochastic geophysical fluid environment. We assumed robots have a map of the
workspace which in the fluid setting is akin to having some estimate of the
global fluid dynamics. This can be achieved by knowing the locations of the
material lines within the flow field that separate regions with distinct
dynamics. Using this knowledge, we leverage the surrounding fluid dynamics and
inherent environmental noise to synthesize energy efficient control strategies
to achieve a distributed allocation of the team to specific regions in the
workspace. Our initial results show that using such a strategy can yield
similar performance as deterministic approaches that do not explicitly account
for the impact of the fluid dynamics while reducing the control effort
required by the team.

For future work we are interested in using actual ocean flow data to further
evaluate our distributed allocation strategy in the presence of jets and
eddies (Rogerson et al., 1999; Miller et al., 2002; Kuznetsov et al., 2002;
Mancho et al., 2008; Branicki et al., 2011; Mendoza and Mancho, 2012). We also
are interested in using more complicated flow models including a bounded
single-layer PDE ocean model (Forgoston et al., 2011), a multi-layer PDE ocean
model (Wang et al., 2009; Lolla et al., 2012), and realistic 2D and 3D
unbounded flow models provided by the Navy Coastal Ocean Model (NCOM)
database. Particularly, we are interested in extending our strategy to non-
periodic, time-varying flows. In addition, we are currently developing an
experimental testbed capable of generating complex 2D flows in a controlled
laboratory setting. The objective is to be able to evaluate the proposed
control strategy using experimentally generated flow field data whose dynamics
are similar to realistic ocean flows. Finally, since our proposed strategy
requires robots to have some estimate of the global fluid dynamics, another
immediate direction for future work is to determine how well one can estimate
the fluid dynamics given knowledge of the locations of Lagrangian coherent
structures (LCS) in the flow field.

###### Acknowledgements.

KM and MAH were supported by the Office of Naval Research (ONR) Award No.
N000141211019. EF was supported by the U.S. Naval Research Laboratory (NRL)
Award No. N0017310-2-C007. IBS was supported by ONR grant N0001412WX20083 and
the NRL Base Research Program N0001412WX30002. The authors additionally
acknowledge support by the ICMAT Severo Ochoa project SEV-2011-0087.

## References

  * Berman et al. (2008) Berman, S., Halasz, A., Hsieh, M. A., and Kumar, V.: Navigation-based Optimization of Stochastic Deployment Strategies for a Robot Swarm to Multiple Sites, in: Proc. of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, 2008.
  * Branicki and Wiggins (2010) Branicki, M. and Wiggins, S.: Finite-time Lagrangian transport analysis: Stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents, Nonlinear Proc. Geoph., 17, 1–36, 2010.
  * Branicki et al. (2011) Branicki, M., Mancho, A. M., and Wiggins, S.: A Lagrangian description of transport associated with a Front-Eddy interaction: application to data from the North-Western Mediterranean Sea, Physica D, 240, 282–304, 2011\.
  * Caron et al. (2008) Caron, D., Stauffer, B., Moorthi, S., Singh, A., Batalin, M., Graham, E., Hansen, M., Kaiser, W., Das, J., de Menezes Pereira, A., A. Dhariwal, B. Z., Oberg, C., and Sukhatme, G.: Macro- to fine-scale spatial and temporal distributions and dynamics of phytoplankton and their environmental driving forces in a small subalpine lake in southern California, USA, Journal of Limnology and Oceanography, 53, 2333–2349, 2008.
  * Chen et al. (2008) Chen, V., Batalin, M., Kaiser, W., and Sukhatme, G.: Towards Spatial and Semantic Mapping in Aquatic Environments, in: IEEE International Conference on Robotics and Automation, pp. 629–636, Pasadena, CA, 2008.
  * Dahl et al. (2006) Dahl, T. S., Mataric̀, M. J., and Sukhatme, G. S.: A machine learning method for improving task allocation in distributed multi-robot transportation, in: Understanding Complex Systems: Science Meets Technology, edited by Braha, D., Minai, A., and Bar-Yam, Y., pp. 307–337, Springer, Berlin, Germany, 2006.
  * Das et al. (2011) Das, J., Py, F., Maughan, T., O’Reilly, T., Messie, M., J. Ryan, G. S., and Rajan, K.: Simultaneous Tracking and Sampling of Dynamic Oceanographic Features with AUVs and Drifters, Submitted to International Journal of Robotics Research, 2011.
  * DeVries and Paley (2011) DeVries, L. and Paley, D. A.: Multi-vehicle control in a strong flowfield with application to hurricane sampling, Accepted for publication in the AIAA J. Guidance, Control, and Dynamics, 2011.
  * Dias et al. (2006) Dias, M. B., Zlot, R. M., Kalra, N., and Stentz, A. T.: Market-based multirobot coordination: a survey and analysis, Proceedings of the IEEE, 94, 1257–1270, 2006\.
  * Forgoston et al. (2011) Forgoston, E., Billings, L., Yecko, P., and Schwartz, I. B.: Set-based corral control in stochastic dynamical systems: Making almost invariant sets more invariant, Chaos, 21, 2011.
  * Gerkey and Mataric (2002) Gerkey, B. P. and Mataric, M. J.: Sold!: Auction methods for multi-robot control, IEEE Transactions on Robotics & Automation, 18, 758–768, 2002.
  * Gerkey and Mataric (2004) Gerkey, B. P. and Mataric, M. J.: A Formal Framework for the Study of Task Allocation in Multi-Robot Systems, International Journal of Robotics Research, 23, 939–954, 2004.
  * Haller (2000) Haller, G.: Finding finite-time invariant manifolds in two-dimensional velocity fields, Chaos, 10, 99–108, 2000.
  * Haller (2001) Haller, G.: Distinguished material surfaces and coherent structures in three-dimensional fluid flows, Physica D, 149, 248–277, 2001.
  * Haller (2002) Haller, G.: Lagrangian coherent structures from approximate velocity data, Phys. Fluids, 14, 1851–1861, 2002.
  * Haller (2011) Haller, G.: A variational theory of hyperbolic Lagrangian Coherent Structures, Physica D, 240, 574–598, 2011.
  * Haller and Yuan (2000) Haller, G. and Yuan, G.: Lagrangian coherent structures and mixing in two-dimensional turbulence, Phys. D, 147, 352–370, 10.1016/S0167-2789(00)00142-1, URL http://dl.acm.org/citation.cfm?id=366463.366505, 2000.
  * Hsieh et al. (2008) Hsieh, M. A., Halasz, A., Berman, S., and Kumar, V.: Biologically inspired redistribution of a swarm of robots among multiple sites, Swarm Intelligence, 2008\.
  * Hsieh et al. (2012) Hsieh, M. A., Forgoston, E., Mather, T. W., and Schwartz, I. B.: Robotic Manifold Tracking of Coherent Structures in Flows, in: in the Proc. of the IEEE International Conference on Robotics and Automation, Minneapolis, MN USA, 2012.
  * Inanc et al. (2005) Inanc, T., Shadden, S., and Marsden, J.: Optimal trajectory generation in ocean flows, in: American Control Conference, 2005. Proceedings of the 2005, pp. 674 – 679, 10.1109/ACC.2005.1470035, 2005.
  * Klavins (2010) Klavins, E.: Proportional-Integral Control of Stochastic Gene Regulatory Networks, in: Proc. of the 2010 IEEE Conf. on Decision and Control (CDC2010), Atlanta, GA USA, 2010.
  * Kuznetsov et al. (2002) Kuznetsov, L., Toner, M., Kirwan, A. D., and Jones, C.: Current and adjacent rings delineated by Lagrangian analysis of the near-surface flow, J. Mar. Res., 60, 405–429, 2002.
  * Lekien et al. (2007) Lekien, F., Shadden, S. C., and Marsden, J. E.: Lagrangian coherent structures in $n$-dimensional systems, J. Math. Phys., 48, 065 404, 2007.
  * Lolla et al. (2012) Lolla, T., Ueckermann, M. P., Haley, P., and Lermusiaux, P. F. J.: Path Planning in Time Dependent Flow Fields using Level Set Methods, in: in the Proc. IEEE International Conference on Robotics and Automation, Minneapolis, MN USA, 2012.
  * Lynch et al. (2008) Lynch, K. M., Schwartz, I. B. Yang, P., and Freeman, R. A.: Decentralized environmental modeling by mobile sensor networks, IEEE Trans. Robotics, 24, 710–724, 2008.
  * Mancho et al. (2008) Mancho, A. M., Hernández-García, E., Small, D., and Wiggins, S.: Lagrangian Transport through an Ocean Front in the Northwestern Mediterranean Sea, J. Phys. Oceanogr., 38, 1222–1237, 2008.
  * Mather and Hsieh (2011) Mather, T. W. and Hsieh, M. A.: Distributed Robot Ensemble Control for Deployment to Multiple Sites, in: 2011 Robotics: Science and Systems, Los Angeles, CA USA, 2011.
  * Mendoza and Mancho (2012) Mendoza, C. and Mancho, A. M.: The Lagrangian description of aperiodic flows: a case study of the Kuroshio Current, Nonlinear Proc. Geoph., 19, 449–472, 2012.
  * Michini et al. (2013) Michini, M., Mallory, K., Larkin, D., Hsieh, M. A., Forgoston, E., and Yecko, P. A.: An experimental testbed for multi-robot tracking of manifolds and coherent structures in flows, in: To appear at the 2013 ASME Dynamical Systems and Control Conference, 2013.
  * Miller et al. (2002) Miller, P. D.and Pratt, L. J., Helfrich, K., Jones, C., Kanth, L., and Choi, J.: Chaotic transport of mass and potential vorticity for an island recirculation, J. Phys. Oceanogr., 32, 80–102, 2002.
  * Rogerson et al. (1999) Rogerson, A. M., Miller, P. D., Pratt, L. J., and Jones, C.: Lagrangian motion and fluid exchange in a barotropic meandering jet, J. Phys. Oceanogr., 29, 2635–2655, 1999.
  * Rypina et al. (2011) Rypina, I. I., Scott, S., Pratt, L. J., and Brown, M. G.: Investigating the connection between trajectory complexities, Lagrangian coherent structures, and transport in the ocean, Nonlinear Processes in Geophysics, 18, 977–987, 2011.
  * Senatore and Ross (2008) Senatore, C. and Ross, S.: Fuel-efficient navigation in complex flows, in: American Control Conference, 2008, pp. 1244 –1248, 10.1109/ACC.2008.4586663, 2008.
  * Shadden et al. (2005) Shadden, S. C., Lekien, F., and Marsden, J. E.: Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Physica D: Nonlinear Phenomena, 212, 271 – 304, DOI: 10.1016/j.physd.2005.10.007, URL http://www.sciencedirect.com/science/article/pii/S0167278905004446, 2005\.
  * Sydney and Paley (2011) Sydney, N. and Paley, D. A.: Multi-vehicle control and optimization for spatiotemporal sampling, in: IEEE Conf. Decision and Control, pp. 5607–5612, Orlando, FL, 2011.
  * Wang et al. (2009) Wang, D., Lermusiaux, P. F., Haley, P. J., Eickstedt, D., Leslie, W. G., and Schmidt, H.: Acoustically focused adaptive sampling and on-board routing for marine rapid environmental assessment, Journal of Marine Systems, 78, 393–407, 2009.
  * Williams and Sukhatme (2012) Williams, R. and Sukhatme, G.: Probabilistic Spatial Mapping and Curve Tracking in Distributed Multi-Agent Systems, in: Submitted to IEEE International Conference on Robotics and Automation, Minneapolis, MN, 2012.
  * Wu and Zhang (2011) Wu, W. and Zhang, F.: Cooperative Exploration of Level Surfaces of Three Dimensional Scalar Fields, Automatica, the IFAC Journall, 47, 2044–2051, 2011\.
  * Zhang et al. (2007) Zhang, F., Fratantoni, D. M., Paley, D., Lund, J., and Leonard, N. E.: Control of Coordinated Patterns for Ocean Sampling, International Journal of Control, 80, 1186–1199, 2007.

$T_{c}$ | 2 | 5 | 8 | 9 | 10
---|---|---|---|---|---
Ring Pattern | 12.99 | 5.98 | 3.45 | 3.49 | 3.66
Block Pattern | - | 11.21 | - | - | 12.72
L Pattern | - | 30.09 | - | - | 30.45
Table 1: Summary of the RMSE for each simulation pattern at $t=450$ with the
time-invariant flow field. The RMSE for the Baseline case is 4.09.


# DBFed: Debiasing Federated Learning Framework based on Domain-Independent

Jiale Li  School of Software
Dalian University of Technology
Dalian, China
<EMAIL_ADDRESS>Zhixin Li  School of Computer Science
Fudan University
Shanghai, China
<EMAIL_ADDRESS>Yibo Wang  School of Software
Dalian University of Technology
Dalian, China
<EMAIL_ADDRESS>Yao Li  School of Software
Dalian University of Technology
Dalian, China
<EMAIL_ADDRESS>Lei Wang∗ * Lei Wang is corresponding author School of
Software
Dalian University of Technology
Dalian, China
<EMAIL_ADDRESS>
###### Abstract

As digital transformation continues, enterprises are generating, managing, and
storing vast amounts of data, while artificial intelligence technology is
rapidly advancing. However, it brings challenges in information security and
data security. Data security refers to the protection of digital information
from unauthorized access, damage, theft, etc. throughout its entire life
cycle. With the promulgation and implementation of data security laws and the
emphasis on data security and data privacy by organizations and users,
Privacy-preserving technology represented by federated learning has a wide
range of application scenarios. Federated learning is a distributed machine
learning computing framework that allows multiple subjects to train joint
models without sharing data to protect data privacy and solve the problem of
data islands. However, the data among multiple subjects are independent of
each other, and the data differences in quality may cause fairness issues in
federated learning modeling, such as data bias among multiple subjects,
resulting in biased and discriminatory models. Therefore, we propose DBFed, a
debiasing federated learning framework based on domain-independent, which
mitigates model bias by explicitly encoding sensitive attributes during
client-side training. This paper conducts experiments on three real datasets
and uses five evaluation metrics of accuracy and fairness to quantify the
effect of the model. Most metrics of DBFed exceed those of the other three
comparative methods, fully demonstrating the debiasing effect of DBFed.

###### Index Terms:

data security; federated learning; model fairness; information security

##  I Introduction

Artificial intelligence technology relies heavily on large amounts of data as
input, which is used for learning and training to recognize patterns, discover
rules, make decisions, and predict outcomes. This data can be structured, such
as table data in a database, or unstructured, such as images, text, and
speech. Due to the diversity and scale of data, AI technology requires
powerful computing capabilities and algorithm support to realize its
application value. Additionally, to protect personal privacy and data
security, reasonable restrictions and protections need to be placed on the use
of sensitive data. Therefore, as AI technology continues to evolve, ensuring
the quality and security of data in terms of acquisition, storage, sharing,
and utilization becomes an important issue. For enterprises, expanding data
sources, establishing a complete data lifecycle management system, and
adopting privacy-preserving computing technology can better utilize data
resources, improve the efficiency of mining and utilizing data value, and
better meet the needs of data security and privacy protection. Therefore, with
the development of artificial intelligence technology, how to ensure the
quality and security of data in terms of data acquisition, storage, sharing,
and utilization through privacy computing technology has become an important
frontier research topic. Many fields, such as finance, healthcare, and
communication[1], have extremely high requirements for data security. As a
result, data between different institutions often becomes a data island,
making it difficult to share and use it safely, resulting in ineffective
utilization of data value and hindering the development of artificial
intelligence technology. In order to protect data security, Privacy-preserving
computation technology represented by federated learning has been widely
applied in the field of artificial intelligence.

Privacy-preserving computation refers to a computing mode for computing and
data processing under the premise of protecting data privacy. It can encrypt,
share, calculate, and analyze data without exposing the original data, thereby
protecting personal privacy and business secrets. Federated learning is a
distributed machine learning technology that allows multiple participants to
jointly train a model, but each participant can only access local data,
thereby protecting data privacy. Federated learning can avoid collecting and
storing user data on a central server, which better protects user privacy.
However, the issue of fairness also becomes more important when using
federated learning. Fairness issues usually involve inequalities between
different participants, including data imbalance, computing resource
imbalance, capability differences, etc. Fairness issues in federated learning
can be grouped into the following categories:

####  I-1 Data Bias

The data distribution and characteristics of different participants may
differ, leading to the imbalanced performance of the model among different
participants. Factors such as gender, race, geographical location, etc. can
lead to the neglect or unfair treatment of certain clients’ datasets.

####  I-2 Model Bias

Since different participants have different data distributions and
characteristics, the model may be biased towards certain participants,
resulting in the unbalanced performance of the model, which will also affect
model fairness.

####  I-3 Imbalance of Computing Resources

Different participants may have different computing resources. Some
participants may have more computing power and storage resources to conduct
more iterative training locally, while others may only have limited iterative
training. This may result in some participants having better model performance
than others, leading to fairness issues.

####  I-4 Capability Differences

Different participants may have different abilities and professional
knowledge. Some participants may have more domain expertise and experience to
better understand and interpret the model results, while others may lack these
abilities, which may lead to issues with the interpretability and fairness of
the model.

Addressing bias issues in federated learning can improve the robustness and
generalization ability of federated learning models, as well as improve the
model’s coverage and service quality for different groups. Therefore, in order
to address these fairness issues, it is necessary to conduct relevant research
and develop appropriate algorithms and tools to promote fair federated
learning. This includes methods based on multi-party data joint learning, the
use of distributed privacy protection technology in federated learning, and
the development of appropriate metrics to evaluate the fairness of the model.

The main contributions of this paper are as follows:

  * •

This paper proposes a debiasing federated learning framework based on domain-
independent, which alleviates model bias by explicitly encoding sensitive
attributes during client training, effectively improving the fairness of deep
learning classification models in federated learning.

  * •

This paper conducts experiments on 3 real datasets, and uses 4 fairness
indicators to quantify the debiasing effect of the model for multi-
classification and multi-sensitive attribute tasks. The effectiveness of DBFed
was verified through experiments.

##  II Related Work

###  II-A Fedrated Learning

Federated learning is a distributed machine learning framework proposed by
McMahan et al.[2], which allows clients to collaborate with each other to
train machine learning models without exposing their own data. Nowadays,
federated learning has been applied to fields such as healthcare and finance.
Yang et al. [3] and others divided federal learning into horizontal federal
learning, vertical federal learning, and federal transfer learning according
to the distribution and alignment characteristics of data in the model
training process. In the horizontal federated learning framework, datasets
from different clients share the same sample space, but the samples are
different; In the vertical federated learning framework, datasets from
different clients have the same or partially identical sample IDs, but the
feature spaces of the datasets are not the same; In the federated transfer
learning framework, the sample ID and feature space of datasets on different
clients are different, but different clients have similar business scenarios.

In federated learning, the client protects local data privacy by passing model
gradients or weights instead of data sharing. However, Zhu et al. [4] showed
that gradient leakage attacks could achieve recovery of client datasets,
thereby compromising customer data privacy. After obtaining gradients, this
method randomly generates a pair of pseudo features and pseudo labels to
perform forward and backward propagation, After deriving the gradient,
optimize the pseudo features and pseudo labels by minimizing the distance
between the pseudo gradient and the real gradient, so that the pseudo data
continuously approaches the original data. Hanchi et al. [5] proposed an
adversarial generation network for generating random data to minimize the
distance between pseudo gradients and real gradients, thereby inferring the
raw data of federated learning clients. Melis et al. [6] have demonstrated
that malicious clients can deceive the global model by using multitasking
learning, allowing the model to learn more of its desired features and extract
more data information from other clients. In order to protect the privacy of
clients in the federated learning process, researchers applied homomorphic
encryption [7] and differential privacy [8] to federated learning. Homomorphic
encryption applies the encryption algorithm to the process of gradient
exchange. The value encrypted by the algorithm is decoded after addition and
multiplication, and the result is the same as that of decoding before the
operation. Therefore, it can effectively protect data privacy in the gradient
exchange process between the client and the server. But homomorphic encryption
reduces the computational efficiency of the federated learning framework.
Differential privacy effectively protects data privacy by adding noise to the
dataset or blurring certain features through generalization methods, making it
difficult for attackers to distinguish between samples and recover data.
However, due to modifications made to the dataset, differential privacy
technology usually needs to balance accuracy and security.

Many researchers have made trade-offs between security, computational
efficiency, and accuracy in federated learning, and have improved and
optimized the federated learning framework. The FedCG proposed by Wu et al.
[9]. utilizes conditional adversarial generation networks to achieve a high
level of privacy protection while maintaining the computational performance of
the model. FedCG has a private extractor and a public classifier on each
client, and in the process of weight aggregation, a client generator is used
instead of a public extractor. The client knowledge is aggregated through
knowledge distillation, protect and the privacy of extractor weights prevents
user information leakage while aggregating client knowledge. In the training
process of the client, the extractor and classifier are first trained to learn
the features of the local dataset, and the output distribution of the
extractor is closer to the generation distribution of the generator. Then, the
local adversarial generation network, namely the generator, and the
discriminator are trained separately from the local data to improve the
accuracy of the model. Zhu et al. [10] proposed a method to aggregate client
knowledge through knowledge distillation without using additional data. They
set up a lightweight generator on the server and used the learned knowledge as
induction bias to adjust local training. This method uses fewer communication
times and has good generalization ability.

###  II-B Model bias and debiasing

Although machine learning models have been applied to a wide range of life
scenarios such as face recognition and medical image analysis, some models
make decisions based on information such as race, gender, and nationality,
resulting in algorithmic bias. As Larson et al. [11] once pointed out, the
COMPAS system has a certain degree of racial biases. The research of Kohavi et
al. [12] shows that under the same circumstances, the deep neural network
usually predicts that male salaries are higher than females. Ashraf et al.
[13] pointed out that commercial gender classification systems developed by
Microsoft, Face++, and IBM have a high recognition error rate for dark-skinned
women. Some works have proposed solutions to algorithmic bias. Mehrabi et
al.[14]divided the solutions to algorithmic bias into three types:
preprocessing, in-process, and post-processing. Preprocessing technology
eliminates or alleviates algorithm bias by changing the training dataset; in-
process technology modifies the machine learning algorithm itself to eliminate
or reduce algorithm bias during training; postprocessing technology usually
refers that by regarding the trained model as back-box, the label output by
the black-box is recalculated according to a new function to eliminate or
reduce algorithm bias. In the field of community detection, Mehrabi et al.
[15] proposed a community detection method for nodes with low connection
attributes, which can alleviate the bias of low-degree nodes. In the field of
classifiers, Bilal et al. [16]proposed a fair constraint to prevent
classifiers from making predictions related to sensitive attributes in the
data, and Kamishima et al. [17] controlled classification accuracy and results
by adjusting regularization parameters and the trade-off between fairness.
This regularization method is applicable to any probabilistic discriminant
model prediction algorithm. In the field of language models, Bordia et al.
[18] proposed a regularization loss term for language models, which minimizes
the projection of the encoder-trained embeddings onto the gender-encoding
embedding subspace, effectively alleviating gender bias in language models. In
the field of causal inference, Lu et al. [19] proposed a framework for
discovering and eliminating bias for causal networks, capturing direct and
indirect discrimination through the causal effects of protected attributes on
decisions passed along different causal paths.

The problems of algorithm bias and model fairness also exist in the field of
federated learning. In recent years, some researchers have proposed some
debiasing methods under the framework of federated learning. Zhang et al.
[20]designed a reward mechanism to adjust the training model’s accuracy and
fairness, which drives fairness across all demographic groups and addresses
the challenges of limited information and limited coordination. The FairBatch
framework proposed by Roh et al. [21], while retaining the standard training
algorithm as an internal optimizer, incorporates an external optimizer to
equip the internal problem with additional features, implementing adaptively
choosing the mini-batch size, so that it will improve the fairness of the
model. This framework can significantly improve the fairness of any pre-
trained model through fine-tuning. Papadaki et al. [22] proposed an algorithm
for maximum and minimum fairness in federated learning, where the server
requires each client to explicitly share the performance of the model on each
race separately.

Figure 1: The training process of DBFed.

##  III Method

This chapter details the federated training process of DBFed. As shown in
Figure 1, in the learning process of DBFed, it is assumed that there is a
global server and clients to jointly train a deep neural network image
classification model, the server first initializes the global model and then
sends the weights to each client to initialize the local model of the client.
After receiving the model weights, the clients use the gradient descent
principle to update the model weights on the local dataset to minimize the
local loss function. After a certain number of rounds, the clients send the
local model weights to the server, and the server performs federated average
aggregation on the model weights of the client to obtain global model weights
and then distributes the global model weights to the client. The specific
process of client training and server aggregation will be introduced in detail
below.

###  III-A Client Domain-Independent Training

Inspired by the research of Wang [35] etc., this paper improves the fairness
of the model through domain-independent training, which encoding sensitive
attributes explicitly. For the problem of bias in deep learning classification
tasks, the predicted features are called target attributes, and the
potentially biased populations are called sensitive attributes. The fully
connected layer of the deep learning classification model sets $ND$-way
discriminant classifiers, where $N$ is the categories of target attribute, and
$D$ is the categories of the sensitive attribute. DBFed mitigates model bias
by explicitly encoding sensitive attribute information during training and
reducing the correlation between sensitive attributes and predicted attributes
during prediction. Assuming $f(\cdot)$ is a deep learning classification
model, there is $N\times D$ neuron in the last layer of the model, that is,
the number output value of the last layer of the model is $N\times D$,
recording $f_{z}(x,\theta)$ as the output result of the $z-th$ node of the
classification layer, where $x$ is the data sample, and $\theta$ is the model
weight. The output of the $ND$-way discriminant classifier can be passed
through the activation function:

$Softmax\left({{f}_{z}}(x;\theta)\right)=\frac{{{e}^{{{f}_{z}}(x;\theta)}}}{\sum\limits_{i=0}^{N}{{{e}^{{{f}_{i}}(x;\theta)}}}}$
(1)

The activated data results can be considered as probabilities, then the
probability that the predicted result of sample $x$ with sensitive feature $d$
is $y$ is:

$P(y|d,x)=Softmax(f_{(y+dN)}(x;\theta))$ (2)

For a data sample, according to the full probability formula, the prediction
result of the deep learning classifier can be calculated according to the
following formula

$\hat{y}=arg{\max\limits_{y}}P(y,x)=arg{\max\limits_{y}}\sum\limits_{d}^{G}{P}(y|d,x)P(d|x)$
(3)

where $P(d|x)$ is the probability of sensitive attribute d of the data sample
$x$, and $G$ is the set of sensitive attributes, then $|G|=D$. For a data
sample with a known sensitive attribute, the predicted value
$\hat{y}=arg\max\limits_{y}P(y|d,x)$. However, in order to achieve blind
review on the sensitive attribute, that is, to ignore the correlation between
predicted attributes and sensitive attributes during the prediction process,
as well as achieve Demographic Parity on sensitive attributes, that means, for
any $a,b\in G$, $P(a|x)=P(b|x)$, to guarantee the fairness between the
prediction different data samples of sensitive attributes and reduce the bias
of the algorithm. So in the prediction process, take $P(d|x)=\frac{1}{|G|}$.

For the training data sample x whose real value of the target attribute is $y$
and the sensitive attribute is $d$, its cross entropy can be calculated:

$L(x,y,d;\theta)=-logP(y|d,x)$ (4)

Therefore, using the gradient descent algorithm, the model weight update
formula of the client $k$ can be expressed as:

${{\theta}^{k}}={{\theta}^{k}}-\eta\nabla\ L\left(b;{{\theta}^{k}}\right)$ (5)

where $\eta$ is the learning rate and $b$ is a batch of training samples. In
the local training of the client, the local training data set is divided into
multiple batches, and each batch of data is selected for training in each
epoch. After multiple iterations of training, the local model weight data is
sent to the server.

Input: Communication round $R$, number of clients $K$, number of epoch $E$.
Training dataset of client $\\{\mathcal{X}_{k}\\}_{k=1}^{K}$, client local
model weights $\\{\theta^{k}\\}_{k=1}^{K}$. Global model weights
$\theta_{0}^{g}$. Learning rate $\eta$

1 Server initializes $\theta_{0}^{g}$;

2 foreach  _round $t$ from 1 to $R$_ do

3 Server sends $\theta_{t}^{g}$ to clients;

4 //Clients local domain-independent training;

5 foreach  _client $k$ from 1 to $K$ in parallel_ do

6 Client k receives $\theta_{t}^{g}$ ;

7 $\theta_{1}^{k}=\theta_{t}^{g}$;

8 foreach  _epoch $e$ from 1 to $E$_ do

9 forall  _$x,y,d\in\mathcal{X}_{k}$_ do

10 $\theta_{e}^{k}=\theta_{e}^{k}-\eta\nabla\ L(x,y,b;\theta_{e}^{k})$;

11

12 end forall

13 $\theta_{e+1}^{k}=\theta_{e}^{k}$;

14

15 end foreach

16 Client sends $k$$\theta_{E}^{k}$ to Server;

17

18 end foreach

19 //Server aggregates the weights;

20 Server executes:;

21 Server receives$\\{\theta_{E}^{k}\\}_{k=1}^{K}$;

22 $\theta_{t+1}^{g}=\sum_{k=1}^{K}{\frac{n_{k}}{n}\theta_{t}^{k}}$;

23

24 end foreach

Algorithm 1 Debiasing federated learning framework based on domain-independent

###  III-B Server Aggregation

After the server receives the weight of the client, it performs weight
aggregation through the FedAvg algorithm. The aggregation formula of the t+1
round global weight can be expressed as:

$\theta_{t+1}^{g}=\sum_{k=1}^{K}{\frac{n_{k}}{n}\theta_{t}^{k}}$ (6)

where $n_{k}$ is the number of samples in the local training data set of the
client k, and $n=\sum\limits^{K}\limits_{k=1}n_{k}$ is the sum of the number
of samples in the training data set of all clients.

TABLE I: Data Distribution and Setting Dataset | Dataset Size | Target Attribute | Sensitive Attribute
---|---|---|---
Training Set | Testing Set | Label |  | Number of
---
Categories
Label |  | Number of
---
Categories
CelebA | 162,770 | 19,962 | Smiling | 2 | Male | 2
FairFace | 86,744 | 10,954 | Age | 9 | Race | 7
UTKFace | 18,964 | 4,741 | Age | 9 | Race | 4

###  III-C Joint Training

In the training process of the federated learning debiasing framework, as
shown in Algorithm 1, the global model weights $\theta_{0}^{g}$ are first
initialized by the server, and then many rounds of communication are
performed. In each round, the server first sends the global model weights of
this round to each client, and then each client receives the global model
weights and performs local training in parallel. The client executes many
iterations for each local training. In the iterations, the batch data sets are
extracted in batches, and the local model weights are adjusted through
gradient descent to optimize the local model loss function. After the client
is trained locally, the client sends the local weights to the server. The
server receives the weights of all clients in this round and starts to
aggregate the weights of this round to obtain the global model weights of a
new round, finally ending the communication training of this round.

##  IV Experiment

###  IV-A Dataset

####  IV-A1 CelebA

CelebA dataset [23] is a face attribute dataset provided by Liu et al. It
contains 202,599 face pictures of 10,177 celebrity identities. The training
data set contains 162,770 pictures, the test dataset contains 19,962 pictures,
and each picture is marked with 40 features such as gender, hair color and
lips. This dataset is widely used in computer vision deep learning tasks.TABLE
LABEL:table1 shows the data distribution and settings of the dataset in this
paper. In the experiment, this paper chooses the ”Smiling” label as the target
attribute and uses the ”Male” label as the sensitive attribute to study the
bias of the model in the gender population when predicting smiles.

####  IV-A2 FairFace

The FairFace dataset [24] is a face image dataset proposed by Karkkainen et
al. It consists of 108,501 pictures, of which the training data set contains
86,744 pictures, and the verification data set contains 10,954 pictures. It
includes seven ethnic groups: White, Black, Indian, East Asian, Southeast
Asian, Middle Eastern, and Latin. Each facial image is labeled with race,
gender, and age, with age attributes classified into nine categories based on
age group. In the experiment, this paper chooses the ”Age” label as the target
attribute and uses the ”Race” label as the sensitive attribute to study the
bias of the deep learning model on the racial group when predicting age.

####  IV-A3 UTKFace

The UTKFace dataset [25] is a facial dataset with a long age span proposed by
Zhang et al., which contains over 20,000 images. In this paper, the dataset is
divided into a training dataset with a size of 18,964 and a testing dataset
with a size of 4,741 in a ratio of 80% and 20%. Each image in the dataset is
labeled with gender, age, and race. There are five types of race labels,
including white, black, Asian, Indian, and Others. This article uses images
from the first four races for experiments111The number of the last race
”Others” is too small and not properly labeled, which has a significant random
impact on the experiment., using the ”Race” label as a sensitive attribute.
The age tags in the image are divided into nine categories based on the age
groups of ”less than 2”, ”3-9”, ”10-19”, ”20-29”, ”30-39”, ”40-49”, ”50-59”,
”60-69”, and ”more than 70”, with the ”Age” label as the target attribute.

###  IV-B Evaluating Metrics

In order to quantify the actual debiasing effectiveness of deep learning
classification models, this paper selects one metric to measure classification
accuracy and four metrics to measure model fairness as evaluating metrics.

####  IV-B1 Accuracy

Accuracy refers to the probability that the model correctly classifies
predictive attributes, which can be calculated by the following formula:

$ACC=P\left(\hat{Y}=c\middle|\ Y=c\right)$ (7)

where $c\in C$ and $C$ are the set of the target attribute.

####  IV-B2 Skewed Error Ratio(SER)

The skewed error ratio is a metric that evaluates the maximum difference
between different sensitive attributes. It mainly represents the difference
between the sensitive attribute with the highest accuracy and the sensitive
attribute with the lowest accuracy. The larger the value, the greater the
difference in the algorithm’s discrimination accuracy for different races. The
formula can be expressed as:

$SER=\frac{\min\limits_{g\in G}{Error_{g}}}{\max\limits_{g\in G}{Error_{g}}}$
(8)

where $g$ is the sensitive attribute, $G$ is the set of sensitive attribute,
and $Error_{g}$ representing the classification error rate of images with
sensitive attribute $g$.

####  IV-B3 Equal of Opportunity(EO)

Equal of Opportunity is a metric that evaluates the equal discrimination
between different races, mainly indicating the equality of correctly
classified races. The larger the value, the greater the difference in the
probability of correctly classified races. Achieving equal opportunities
requires the model to have an equal true positive or false negative rate, and
the conditions for achieving equal opportunities can be expressed as: for all
$a,b\in G$, $P\left(\hat{Y}=1\middle|\
S=a,Y=1\right)=P\left(\hat{Y}=1\middle|\ S=b,Y=1\right)$.

Since this paper focuses on the situation of multiple categories of target
attribute and sensitive attribute, the calculation formula for Equality of
Opportunity is defined as the mean accuracy variance of each sensitive
attribute image in different target attribute images, as follows:

$EO=\frac{1}{\left|C\right|}\sum_{c\in C}{\mathop{var}\limits_{g\in
G}\left(P\left(\hat{Y}=c\middle|\ S=g,Y=c\right)\right)}$ (9)

where $var(\cdot)$ is variance calculation function.

####  IV-B4 Bias Amplification(BA)

Bias amplification is metric that evaluates the degree of inclination of
algorithm decisions towards specific types of target attributes, mainly
indicating the unfairness of the algorithm among the types of target
attributes. The larger the value, the more inclined the algorithm is towards
certain specific target attributes. Bias Amplification can be calculated using
the following formula:

$BA=\frac{1}{\left|C\right|}\sum_{c\in C}\frac{\max\limits_{g\in
G}{g_{c}}}{\sum\limits_{g\in G}\ g_{c}}-\frac{1}{\left|G\right|}$ (10)

####  IV-B5 Demographic Parity(DP)

Demographic Parity is a metric that evaluates the similarity of algorithm
decisions to different races, mainly indicating the degree of similarity of
algorithm decisions to different races. The larger the value, the greater the
difference in algorithm decisions for different populations. The condition
that the model meets Demographic Parity is that all $a,b\in G$, meet
$P\left(\hat{Y}=1\middle|\ S=a\right)=P\left(\hat{Y}=1\middle|\ S=b\right)$.
The calculation formula for Demography Parity is as follows:

$DP=\frac{1}{C}\sum\limits_{c\in C}{\mathop{var}\limits_{g\in
G}}\left(P\left(\hat{Y}=c\middle|S=g\right)\right)$ (11)

###  IV-C Comparative Experiment

####  IV-C1 Environment Settings

The experimental operating system is Linux 3.10.0, and the development
environment is Anaconda3, Python3.10.9, and Pycharm. The deep learning model
is mainly written based on the deep learning framework Pyorch 1.13.1 and
trained on NVIDIA A100.

####  IV-C2 Comparison Methods

This paper chooses the Federated Average (FedAvg) and local training algorithm
as the baseline and chooses Fair Federated Learning Model[26] (FairFed)
proposed by Ezzeldin et al. as state-of-the-art (SOTA). In local training
algorithms, each client only trains on the local dataset and does not
aggregate weights through communication.

####  IV-C3 Parameter settings

This paper uses Resnet34 [23] as the basic model for deep learning for
experiments. Resnet34 is a deep residual network with 34 convolutional layers,
which is easy to optimize and widely used in computer vision tasks. In the
experiment, the adaptive moment estimation (Adam)[27] optimizer was used to
implement the gradient descent principle, which can achieve high computational
efficiency in small memory. The learning rate of the optimizer is 0.0001, and
the weight decay is 0.0003. During the model training process. Five clients
are set up and randomly divided into equal datasets as their local training
dataset. The batch size is 128, and the client performs communication
aggregation after every three local training iterations. This paper mainly
uses three image datasets, CelebA, FairFace, and UTKFace for experiments. Each
image has three channels, R, G, and B, with values ranging from 0 to 255. All
images are uniformly adjusted to pixel size.

####  IV-C4 Results and Analysis

TABLE II shows the experiment results, the best-performing data for each
metric in each dataset are highlighted in bold, while the second-best data is
underlined. The experimental effect of DBFed on the CelebA data set is the
best. It outperforms other methods in terms of Equality of Opportunity and
Demographic Parity and ranks second in terms of Accuracy and Skewed Error
Ratio. This shows that DBFed has a good effect on model fairness while having
a high prediction accuracy, and effectively reduces the gender bias in smile
classification prediction. In the experiments on the FairFace dataset, DBFed
achieved the highest Accuracy, but the other four fairness metrics did not
significantly surpass other methods. This may be due to the lack of
significant unbiased effects caused by the large variety of sensitive
attributes in the dataset. In the experiment of UTKFace, this framework
outperformed other methods in terms of Skewed Error Ratio, Bias Amplification,
and population equality while achieving high accuracy, indicating excellent
results in model accuracy and fairness. Overall, DBFed performs well in both
model accuracy and fairness, effectively reducing population discrimination
when performing classification tasks.

TABLE II: Experimental result data | Metric | FedAvg | Local | DBFed | FairFed
---|---|---|---|---|---
CelebA | ACC$\uparrow$ | 0.9054 | 0.8968 | 0.9058 | 0.9071
SER$\downarrow$ | 1.2673 | 1.2265 | 1.2565 | 1.2607
EO$\downarrow$ | 0.00044 | 0.00048 | 0.00022 | 0.00033
BA$\downarrow$ | 0.1140 | 0.1133 | 0.1156 | 0.1138
DP$\downarrow$ | 0.01863 | 0.01855 | 0.01748 | 0.01832
FairFace | ACC$\uparrow$ | 0.5447 | 0.5360 | 0.5551 | 0.5485
SER$\downarrow$ | 1.1315 | 1.1424 | 1.1214 | 1.1130
EO$\downarrow$ | 0.00461 | 0.00483 | 0.00515 | 0.00460
BA$\downarrow$ | 0.0527 | 0.0556 | 0.0591 | 0.0546
DP$\downarrow$ | 0.00175 | 0.00190 | 0.00201 | 0.00186
UTKFace | ACC$\uparrow$ | 0.5670 | 0.5675 | 0.5727 | 0.5783
SER$\downarrow$ | 1.5561 | 1.6297 | 1.5242 | 1.668
EO$\downarrow$ | 0.00640 | 0.00830 | 0.00997 | 0.0089
BA$\downarrow$ | 0.3085 | 0.3109 | 0.3001 | 0.3063
DP$\downarrow$ | 0.0418 | 0.0420 | 0.0391 | 0.0422

##  V Conclusion

This paper proposes a debiasing federated learning framework based on domain-
independent that can predict classification without using sensitive attribute
labels, mitigate model bias model biases during federated learning, and
improve model fairness. This paper verifies the depolarization effect of DBFed
through experiments on three datasets and five evaluation metrics. In
addition, due to the need for sensitive attribute labels during the training
process, there are certain requirements for dataset annotation. The research
in this paper is a new attempt at model fairness under the federated learning
computing architecture, which can be applied to many scenarios that require
high data security, such as model fine-tuning and model unbinding in human
recognition or fund trading intelligent dialogue models in the financial
field. The next research direction will focus on exploring how to remove model
biases in federated learning processes without using sensitive attribute
labels and strive to perform better debiasing effects on more categories of
the sensitive attribute. based on domain-independent that can predict
classification without using sensitive attribute labels, mitigate model bias
model biases during federated learning, and improve model fairness. This paper
verifies the depolarization effect of DBFed through experiments on three
datasets and five evaluation metrics. In addition, due to the need for
sensitive attribute labels during the training process, there are certain
requirements for dataset annotation. The research in this paper is a new
attempt at model fairness under the federated learning computing architecture,
which can be applied to many scenarios that require high data security, such
as model fine-tuning and model unbinding in human recognition or fund trading
intelligent dialogue models in the financial field. The next research
direction will focus on exploring how to remove model biases in federated
learning processes without using sensitive attribute labels and strive to
perform better debiasing effects on more categories of the sensitive
attribute.

##  VI Acknowledgements

Joint Research and Development Project of Yangtze River Delta Region
Technology and Innovation Community(2022CSJGG0800).

## References

  * [1] X. Liu, X. Gao, J. Li, Y. Wang, C. Lin, Z. Qin, and L. Wang, “Uqcom: Underwater safety communication based on 3d blue-green qr arrays,” in _Proceedings of the SIGCOMM ’21 Poster and Demo Sessions_ , ser. SIGCOMM ’21. New York, NY, USA: Association for Computing Machinery, 2021, p. 36–38. [Online]. Available: https://doi.org/10.1145/3472716.3472859
  * [2] B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y. Arcas, “Communication-Efficient Learning of Deep Networks from Decentralized Data,” in _Proceedings of the 20th International Conference on Artificial Intelligence and Statistics_ , ser. Proceedings of Machine Learning Research, A. Singh and J. Zhu, Eds., vol. 54. PMLR, 20–22 Apr 2017, pp. 1273–1282. [Online]. Available: https://proceedings.mlr.press/v54/mcmahan17a.html
  * [3] Q. Yang, Y. Liu, T. Chen, and Y. Tong, “Federated machine learning: Concept and applications,” _ACM Trans. Intell. Syst. Technol._ , vol. 10, no. 2, jan 2019. [Online]. Available: https://doi.org/10.1145/3298981
  * [4] L. Zhu and S. Han, _Deep Leakage from Gradients_. Cham: Springer International Publishing, 2020, pp. 17–31. [Online]. Available: https://doi.org/10.1007/978-3-030-63076-8_2
  * [5] H. Ren, J. Deng, and X. Xie, “Grnn: Generative regression neural network—a data leakage attack for federated learning,” _ACM Trans. Intell. Syst. Technol._ , vol. 13, no. 4, may 2022. [Online]. Available: https://doi.org/10.1145/3510032
  * [6] L. Melis, C. Song, E. D. Cristofaro, and V. Shmatikov, “Inference attacks against collaborative learning,” 2018.
  * [7] L. T. Phong, Y. Aono, T. Hayashi, L. Wang, and S. Moriai, “Privacy-preserving deep learning via additively homomorphic encryption,” _IEEE Transactions on Information Forensics and Security_ , vol. 13, no. 5, pp. 1333–1345, 2018.
  * [8] C. Dwork, “Differential privacy: A survey of results,” in _Theory and Applications of Models of Computation_ , M. Agrawal, D. Du, Z. Duan, and A. Li, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2008, pp. 1–19.
  * [9] Y. Wu, Y. Kang, J. Luo, Y. He, L. Fan, R. Pan, and Q. Yang, “Fedcg: Leverage conditional gan for protecting privacy and maintaining competitive performance in federated learning,” in _Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence, IJCAI-22_ , L. D. Raedt, Ed. International Joint Conferences on Artificial Intelligence Organization, 7 2022, pp. 2334–2340, main Track. [Online]. Available: https://doi.org/10.24963/ijcai.2022/324
  * [10] Z. Zhu, J. Hong, and J. Zhou, “Data-free knowledge distillation for heterogeneous federated learning,” in _Proceedings of the 38th International Conference on Machine Learning_ , ser. Proceedings of Machine Learning Research, M. Meila and T. Zhang, Eds., vol. 139. PMLR, 18–24 Jul 2021, pp. 12 878–12 889. [Online]. Available: https://proceedings.mlr.press/v139/zhu21b.html
  * [11] K. Lauren, L. Jeff, M. Surya, and A. Julia. (2016) How we analyzed the compas recidivism algorithm. [Online]. Available: https://www.propublica.org/article/how-we-analyzed-the-compas-recidivism-algorithm
  * [12] R. Kohavi, “Scaling up the accuracy of naive-bayes classifiers: A decision-tree hybrid,” in _Proceedings of the Second International Conference on Knowledge Discovery and Data Mining_ , ser. KDD’96. AAAI Press, 1996, p. 202–207.
  * [13] A. Khalil, S. G. Ahmed, A. M. Khattak, and N. Al-Qirim, “Investigating bias in facial analysis systems: A systematic review,” _IEEE Access_ , vol. 8, pp. 130 751–130 761, 2020.
  * [14] N. Mehrabi, F. Morstatter, N. Saxena, K. Lerman, and A. Galstyan, “A survey on bias and fairness in machine learning,” _ACM Comput. Surv._ , vol. 54, no. 6, jul 2021. [Online]. Available: https://doi.org/10.1145/3457607
  * [15] N. Mehrabi, F. Morstatter, N. Peng, and A. Galstyan, “Debiasing community detection: The importance of lowly connected nodes,” in _2019 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM)_ , 2019, pp. 509–512.
  * [16] M. B. Zafar, I. Valera, M. G. Rogriguez, and K. P. Gummadi, “Fairness Constraints: Mechanisms for Fair Classification,” in _Proceedings of the 20th International Conference on Artificial Intelligence and Statistics_ , ser. Proceedings of Machine Learning Research, A. Singh and J. Zhu, Eds., vol. 54. PMLR, 20–22 Apr 2017, pp. 962–970. [Online]. Available: https://proceedings.mlr.press/v54/zafar17a.html
  * [17] T. Kamishima, S. Akaho, H. Asoh, and J. Sakuma, “Fairness-aware classifier with prejudice remover regularizer,” in _Proceedings of the 2012th European Conference on Machine Learning and Knowledge Discovery in Databases \- Volume Part II_ , ser. ECMLPKDD’12. Berlin, Heidelberg: Springer-Verlag, 2012.
  * [18] S. Bordia and S. R. Bowman, “Identifying and reducing gender bias in word-level language models,” in _North American Chapter of the Association for Computational Linguistics_ , 2019.
  * [19] L. Zhang, Y. Wu, and X. Wu, “A causal framework for discovering and removing direct and indirect discrimination,” in _Proceedings of the 26th International Joint Conference on Artificial Intelligence_ , ser. IJCAI’17. AAAI Press, 2017, p. 3929–3935.
  * [20] D. Y. Zhang, Z. Kou, and D. Wang, “Fairfl: A fair federated learning approach to reducing demographic bias in privacy-sensitive classification models,” in _2020 IEEE International Conference on Big Data (Big Data)_ , 2020, pp. 1051–1060.
  * [21] Y. Roh, K. Lee, S. E. Whang, and C. Suh, “Fairbatch: Batch selection for model fairness,” in _International Conference on Learning Representations_ , 2021\. [Online]. Available: https://openreview.net/forum?id=YNnpaAKeCfx
  * [22] Z. Wang, K. Qinami, I. C. Karakozis, K. Genova, P. Nair, K. Hata, and O. Russakovsky, “Towards fairness in visual recognition: Effective strategies for bias mitigation,” in _2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2020, pp. 8916–8925.
  * [23] Z. Liu, P. Luo, X. Wang, and X. Tang, “Deep learning face attributes in the wild,” in _2015 IEEE International Conference on Computer Vision (ICCV)_ , 2015, pp. 3730–3738.
  * [24] K. Kärkkäinen and J. Joo, “Fairface: Face attribute dataset for balanced race, gender, and age for bias measurement and mitigation,” in _2021 IEEE Winter Conference on Applications of Computer Vision (WACV)_ , 2021, pp. 1547–1557.
  * [25] Z. Zhang, Y. Song, and H. Qi, “Age progression/regression by conditional adversarial autoencoder,” in _2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2017, pp. 4352–4360.
  * [26] Y. H. Ezzeldin, S. Yan, C. He, E. Ferrara, and A. S. Avestimehr, “Fairfed: Enabling group fairness in federated learning,” _ArXiv_ , vol. abs/2110.00857, 2021.
  * [27] D. Kingma and J. Ba, “Adam: A method for stochastic optimization,” _International Conference on Learning Representations_ , 12 2014.