# Asynchronous Distributed Optimization via ADMM
with Efficient Communication

Apostolos I. Rikos, Wei Jiang, Themistoklis Charalambous, and Karl H.
Johansson Apostolos I. Rikos is with the Department of Electrical and Computer
Engineering, Division of Systems Engineering, Boston University, Boston, MA
02215, US. E-mail<EMAIL_ADDRESS>Jiang resides in Hong Kong, China. Email:
<EMAIL_ADDRESS>Charalambous is with the Department of Electrical and
Computer Engineering, School of Engineering, University of Cyprus, 1678
Nicosia, Cyprus. He is also with the Department of Electrical Engineering and
Automation, School of Electrical Engineering, Aalto University, Espoo,
Finland. Email<EMAIL_ADDRESS>K. H. Johansson is with
the Division of Decision and Control Systems, KTH Royal Institute of
Technology, SE-100 44 Stockholm, Sweden. He is also affiliated with Digital
Futures. E-mail<EMAIL_ADDRESS>of this work was supported by the Knut and
Alice Wallenberg Foundation, the Swedish Research Council, and the Swedish
Foundation for Strategic Research. The work of T. Charalambous was partly
supported by the European Research Council (ERC) Consolidator Grant MINERVA
(Grant agreement No. 101044629).

###### Abstract

In this paper, we focus on an asynchronous distributed optimization problem.
In our problem, each node is endowed with a convex local cost function, and is
able to communicate with its neighbors over a directed communication network.
Furthermore, we assume that the communication channels between nodes have
limited bandwidth, and each node suffers from processing delays. We present a
distributed algorithm which combines the Alternating Direction Method of
Multipliers (ADMM) strategy with a finite time quantized averaging algorithm.
In our proposed algorithm, nodes exchange quantized valued messages and
operate in an asynchronous fashion. More specifically, during every iteration
of our algorithm each node (i) solves a local convex optimization problem (for
the one of its primal variables), and (ii) utilizes a finite-time quantized
averaging algorithm to obtain the value of the second primal variable (since
the cost function for the second primal variable is not decomposable). We show
that our algorithm converges to the optimal solution at a rate of $O(1/k)$
(where $k$ is the number of time steps) for the case where the local cost
function of every node is convex and not-necessarily differentiable. Finally,
we demonstrate the operational advantages of our algorithm against other
algorithms from the literature.

##  I Introduction

The problem of distributed optimization has received extensive attention in
recent years. Due to the rise of large-scale machine learning [1], control
[2], and other data-driven applications [3], there is a growing need to solve
optimization problems that involve massive amounts of data. Solving these
problems in a centralized way is proven to be infeasible since it is difficult
or impossible to store and process large amounts of data on a single node.

Distributed optimization is a method that distributes data across multiple
nodes. Each node performs computations on its stored data and collaborates
with others to solve the optimization problem collectively. This approach
optimizes a global objective function by combining each node’s local objective
function and coordinating with the network. The advantage is reducing
computational and storage requirements for individual nodes. However, frequent
communication with neighboring nodes is necessary to update optimization
variables. This can become a bottleneck with increasing nodes or data. To
address this issue, recent attention from the scientific community focuses on
developing optimization algorithms with efficient communication. This leads to
enhancements on scalability and operational efficiency, while mitigating
issues like network congestion, latency, and bandwidth limitations.

Existing Literature. Most works in the literature assume that nodes can
process and exchange real values. This may result in communication overhead,
especially for algorithms requiring frequent and complex communication (see,
e.g., [4, 5, 6, 7, 8, 9, 10]). In practical applications, nodes must exchange
quantized messages to efficiently utilize network resources like energy and
processing power. For this reason, recent research focuses on communication-
efficient algorithms (e.g., [11, 12, 7, 13, 6, 14, 15]), but they often assume
perfectly synchronized nodes or bidirectional communication, limiting their
applicability. Addressing communication overhead remains a key challenge,
necessitating the development of communication-efficient algorithms that can
operate over directed networks asynchronously. Therefore, continued research
in this area is crucial to overcoming this bottleneck and enhancing the
performance of distributed optimization methods.

Main Contributions. Existing algorithms in the literature often assume that
nodes can exchange precise values of their optimization variables and operate
synchronously. However, transmitting exact values (often irrational numbers)
necessitates an infinite number of bits and becomes infeasible. Moreover,
synchronizing nodes within a distributed network involves costly protocols,
time-consuming to execute. In this paper, we present a distributed
optimization algorithm, which aims to address these challenges. More
specifically, we make the following contributions.
A. We present a distributed optimization algorithm that leverages the
advantages of the ADMM optimization strategy and operates over a directed
communication graph. Our algorithm allows nodes to operate in an asynchronous
fashion, and enables efficient communication as nodes communicate with
quantized messages; see Algorithm 1.
B. We prove that our algorithm converges to the optimal solution at a rate of
$O(1/k)$ even for non-differentiable and convex local cost functions (as it is
the case for similar algorithms with real-valued states). This rate is
justified in our simulations in which our algorithm exhibits comparable
performance with real-valued communication algorithms while guaranteeing
efficient (quantized) communication among nodes; see Section VI. Furthermore,
we show that the optimal solution is calculated within an error bound that
depends on the quantization level; see Theorem 1.

##  II NOTATION AND PRELIMINARIES

Notation. The sets of real, rational, integer and natural numbers are denoted
by $\mathds{R},\mathds{Q},\mathds{Z}$ and $\mathds{N}$, respectively. The
symbol $\mathds{Z}_{\geq 0}$ ($\mathds{Z}_{>0}$) denotes the set of
nonnegative (positive) integer numbers. The symbol $\mathds{R}_{\geq 0}$
($\mathds{R}_{>0}$) denotes the set of nonnegative (positive) real numbers.
The symbol $\mathds{R}^{n}_{\geq 0}$ denotes the nonnegative orthant of the
$n$-dimensional real space $\mathds{R}^{n}$. Matrices are denoted with capital
letters (e.g., $A$), and vectors with small letters (e.g., $x$). The transpose
of matrix $A$ and vector $x$ are denoted as $A^{\top}$, $x^{\top}$,
respectively. For any real number $a\in\mathds{R}$, the floor $\lfloor
a\rfloor$ denotes the greatest integer less than or equal to $a$ while the
ceiling $\lceil a\rceil$ denotes the least integer greater than or equal to
$a$. For any matrix $A\in\mathds{R}^{n\times n}$, the $a_{ij}$ denotes the
entry in row $i$ and column $j$. By $\mathds{1}$ and $\mathds{I}$ we denote
the all-ones vector and the identity matrix of appropriate dimensions,
respectively. By $\|\cdot\|$, we denote the Euclidean norm of a vector.

Graph Theory. The communication network is captured by a directed graph
(digraph) defined as $\mathcal{G}=(\mathcal{V},\mathcal{E})$. This digraph
consists of $n$ ($n\geq 2$) agents communicating only with their immediate
neighbors, and is static (i.e., it does not change over time). In
$\mathcal{G}$, the set of nodes is denoted as
$\mathcal{V}=\\{v_{1},v_{2},...,v_{n}\\}$, and the set of edges as
$\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}\cup\\{(v_{i},v_{i})\ |\
v_{i}\in\mathcal{V}\\}$ (note that each agent has also a virtual self-edge).
The cardinality of the sets of nodes, edges are denoted as $|\mathcal{V}|=N$,
$|\mathcal{E}|=m$, respectively. A directed edge from node $v_{i}$ to node
$v_{l}$ is denoted by $(v_{l},v_{i})\in\mathcal{E}$, and captures the fact
that node $v_{l}$ can receive information from node $v_{i}$ at time step $k$
(but not the other way around). The subset of nodes that can directly transmit
information to node $v_{i}$ is called the set of in-neighbors of $v_{i}$ and
is represented by
$\mathcal{N}_{i}^{-}=\\{v_{j}\in\mathcal{V}\;|\;(v_{i},v_{j})\in\mathcal{E}\\}$.
The subset of nodes that can directly receive information from node $v_{i}$ is
called the set of out-neighbors of $v_{i}$ and is represented by
$\mathcal{N}_{i}^{+}=\\{v_{l}\in\mathcal{V}\;|\;(v_{l},v_{i})\in\mathcal{E}\\}$.
The in-degree, and out-degree of $v_{j}$ are denoted by
$\mathcal{D}_{i}^{-}=|\mathcal{N}_{i}^{-}|$,
$\mathcal{D}_{i}^{+}=|\mathcal{N}_{i}^{+}|$, respectively. The diameter $D$ of
a digraph is the longest shortest path between any two nodes
$v_{l},v_{i}\in\mathcal{V}$. A directed path from $v_{i}$ to $v_{l}$ of length
$t$ exists if we can find a sequence of agents $i\equiv
l_{0},l_{1},\dots,l_{t}\equiv l$ such that
$(l_{\tau+1},l_{\tau})\in\mathcal{E}$ for $\tau=0,1,\dots,t-1$. A digraph is
strongly connected if there exists a directed path from every node $v_{i}$ to
every node $v_{l}$, for every $v_{i},v_{l}\in\mathcal{V}$.

ADMM Algorithm. The standard ADMM algorithm [16] is designed to solve the
following problem:

$\displaystyle\min_{x\in\mathds{R}^{p},z\in\mathds{R}^{m}}~{}$ $\displaystyle
f(x)+g(x),$ (1) s.t. $\displaystyle Ax+Bz=c,$

where $A\in\mathds{R}^{q\times p}$, $B\in\mathds{R}^{q\times m}$ and
$c\in\mathds{R}^{q}$ (for $q,p,m\in\mathds{N}$). In order to solve (1), the
augmented Lagrangian is:

$\displaystyle L_{\rho}(x,z,\lambda)=f(x)+g(x)$
$\displaystyle+\lambda(Ax+Bz-c)$
$\displaystyle+\frac{\rho}{2}\|Ax+Bz-c\|^{2},$ (2)

where $\lambda\in\mathds{R}$ is the Lagrange multiplier, and
$\rho\in\mathds{R}$ is the positive penalty parameter. The primary variables
$x$, $z$ and the Lagrangian multiplier $\lambda$ are initialized as
$[x,z,\lambda]^{\top}=[x^{[0]},z^{[0]},\lambda^{[0]}]^{\top}$. Then, during
every ADMM time step, the $x$, $z$ and $\lambda$ are updated as:

$\displaystyle x^{[k+1]}=$
$\displaystyle\operatorname*{arg\,min}_{x\in\mathds{R}^{p}}L_{\rho}(x,z^{[k]},\lambda^{[k]}),$
(3) $\displaystyle z^{[k+1]}=$
$\displaystyle\operatorname*{arg\,min}_{z\in\mathds{R}^{m}}L_{\rho}(x^{[k+1]},z,\lambda^{[k]}),$
(4) $\displaystyle\lambda^{[k+1]}=$
$\displaystyle\lambda^{[k]}+\rho(Ax^{[k+1]}+Bz^{[k+1]}-c),$ (5)

where $\rho$ in (5) is the penalty parameter from (II).

Asymmetric Quantizers. Quantization is a strategy that lessens the number of
bits needed to represent information. This reduces the required communication
bandwidth and increases power and computation efficiency. Quantization is
mainly used to describe communication constraints and imperfect information
exchanges between nodes [17]. The three main types of quantizers are (i)
asymmetric, (ii) uniform, and (iii) logarithmic. In this paper, we rely on
asymmetric quantizers to reduce the required communication bandwidth. Note
that the results of this paper are transferable to other quantizer types
(e.g., logarithmic or uniform). Asymmetric quantizers are defined as

$q_{\Delta}^{a}(\xi)=\Bigl{\lfloor}\frac{\xi}{\Delta}\Bigr{\rfloor},$ (6)

where $\Delta\in\mathds{Q}$ is the quantization level, $\xi\in\mathds{R}$ is
the value to be quantized, and $q_{\Delta}^{a}(\xi)\in\mathds{Q}$ is the
quantized version of $\xi$ with quantization level $\Delta$ (note that the
superscript “$a$” indicates that the quantizer is asymmetric).

##  III Problem Formulation

Problem Statement. Let us consider a distributed network modeled as a digraph
$\mathcal{G}=(\mathcal{V},\mathcal{E})$ with $n=|\mathcal{V}|$ nodes. In our
network $\mathcal{G}$, we assume that the communication channels among nodes
have limited bandwidth. Each node $v_{i}$ is endowed with a scalar local cost
function $f_{i}(x):\mathds{R}^{p}\mapsto\mathds{R}$ only known to node
$v_{i}$. In this paper we aim to develop a distributed algorithm which allows
nodes to cooperatively solve the following optimization problem

$\displaystyle\min_{x\in\mathds{R}^{p}}~{}$
$\displaystyle\sum_{i=1}^{n}f_{i}(x),$ (7)

where $x\in\mathds{R}^{p}$ is the global optimization variable (or common
decision variable). We will solve (7) via the distributed ADMM strategy.
Furthermore, in our solution we guarantee efficient communication between
nodes (due to communication channels of limited bandwidth in the network).

Modification of the Optimization Problem. In order to solve (7) via the ADMM
and guarantee efficient communication between nodes, we introduce (i) the
variable $x_{i}$ for every node $v_{i}$, (ii) the constraint
$|x_{i}-x_{j}|\leq\epsilon$ for every $v_{i},v_{j}\in\mathcal{V}$ (where
$\epsilon\in\mathds{R}$ is an error tolerance which is predefined), and (iii)
the constraint that nodes communicate with quantized values. The second
constraint is introduced to allow an asynchronous implementation of the
distributed ADMM strategy, and the third constraint to guarantee efficient
communication between nodes. Considering the aforementioned constraints (i),
(ii) and (iii), (7) becomes:

$\displaystyle\min_{x_{i}}~{}$
$\displaystyle\sum_{i=1}^{n}f_{i}(x_{i}),i=1,...,n$ (8) s.t.
$\displaystyle|x_{i}-x_{j}|\leq\epsilon,\forall v_{i},v_{j}\in\mathcal{V},$
(9) nodes communicate with quantized values. (10)

Let us now define a closed nonempty convex set $\mathcal{C}$ as

$\mathcal{C}=\left\\{\begin{bmatrix}x_{1}^{\mbox{\tiny T}}&x_{2}^{\mbox{\tiny
T}}&\ldots&x_{n}^{\mbox{\tiny T}}\end{bmatrix}^{\mbox{\tiny
T}}\in\mathds{R}^{np}\,:\,\|x_{i}-x_{j}\|\leq\epsilon\right\\}.$ (11)

Furthermore, denote $X\coloneqq\begin{bmatrix}x_{1}^{\mbox{\tiny
T}}&x_{2}^{\mbox{\tiny T}}&\ldots&x_{n}^{\mbox{\tiny
T}}\end{bmatrix}^{\mbox{\tiny T}}$ and its copy variable
$z\in\mathds{R}^{np}$. This means that $\eqref{admm_prob_min_probForm_2}$ and
(11) become

$X=z,\,\forall z\in\mathcal{C}.$ (12)

Now let us define the indicator function $g(z)$ of set $\mathcal{C}$ as

$g(z)=\left\\{\begin{array}[]{l}\begin{aligned}
&0,\quad\text{if}\,z\in\mathcal{C},\\\ &\infty,\,\
\text{otherwise}.\end{aligned}\end{array}\right.$ (13)

Incorporating (12) and (13) into (8), we have that (8) becomes

$\displaystyle\min_{z,x_{i}}$
$\displaystyle\left\\{\sum_{i=1}^{n}f_{i}(x_{i})+g(z)\right\\},i=1,\ldots,n$
(14) s.t. $\displaystyle X-z=0,\,\forall z\in\mathcal{C},$ nodes communicate
with quantized values.

As a result, in this paper, we aim to design a distributed algorithm that
solves (14) via the distributed ADMM strategy.

##  IV Preliminaries on Distributed Coordination

We now present a definition of asynchrony (borrowed from [8]) that defines the
operation of nodes in the network. Furthermore, we present a distributed
coordination algorithm that operates with quantized values and is necessary
for our subsequent development.

###  IV-A Definition of Asynchronous Operation

During their optimization operation, nodes aim to coordinate in an
asynchronous fashion. Specifically, let us assume that the iterations for the
optimization operation start at time step $t(0)\in\mathds{R}_{+}$.
Furthermore, we assume that one (or more) nodes transmit values to their out-
neighbors at a set of time instances
$\mathcal{T}=\\{t(1),t(2),t(3),\ldots\\}$. During the nodes’ asynchronous
operation, a message that is received at time step $t(\eta_{1})$ from node
$v_{i}$, is processed at time step $t(\eta_{2})$ where $\eta_{2}>\eta_{1}$.
This means that the message received at time step $t(\eta_{1})$ suffers from a
processing delay of $t(\eta_{2})-t(\eta_{1})$ time steps. An example of how
processing delays affecting transmissions is shown in Fig. 1 (that is borrowed
from [8]).

Figure 1: Example of how processing and transmission delays affect the
operation of nodes $v_{1}$, $v_{2}$, $v_{3}$. Blue dots indicate the
iterations and blue arrows indicate the transmissions. Transmissions occur at
time steps $t_{i}(\eta)$, and $t_{i}(\eta+1)-t_{i}(\eta)$ is the processing
delay, where $i\in\\{1,2,3\\}$, $\eta\in\mathds{Z}_{\geq 0}$. The time
difference from the blue dot to the blue arrow is the transmission delay [8].

Note here that the nodes states at time step $t(\eta)$ are indexed by $\eta$.
This means that the state of node $v_{i}$ at time step $t(\eta)$ is denoted as
$x_{i}^{\eta}\in\mathds{R}^{p}$.

We now present the following assumption which is necessary for the
asynchronous operation of every node.

###### Assumption 1

The number of time steps required for a node $v_{i}$ to process the
information received from its in-neighbors is upper bounded by
$\mathcal{B}\in\mathds{N}$. Furthermore, the actual time (in seconds) required
for a node $v_{i}$ to process the information received from its in-neighbors
is upper bounded by $T\in\mathds{R}_{\geq 0}$.

Assumption 1 states that there exists a finite number of steps $\mathcal{B}$
before which all nodes have updated their states and proceed to perform
transmissions to their neighboring nodes. The upper bound $\mathcal{B}$ is
translated to an upper bound of $T$ in actual time (in seconds). This is
mainly because it is not possible for nodes to count the number of time steps
elapsed in the network (and understand when $\mathcal{B}$ time steps have
passed. The value $T$ can be counted by each node individually.

###  IV-B Asynchronous $\max$/$\min$ \- Consensus

In asynchronous max/min consensus (see [18]), the update rule for every node
$v_{i}\in\mathcal{V}$ is:

$\displaystyle
x_{i}^{[k+\theta_{i}^{[k]}]}=\max_{v_{j}\in\mathcal{N}_{i}^{-}\cup\\{v_{i}\\}}\\{x_{j}^{[k+\theta_{ij}^{[k]}]}\\},$
(15)

where $\theta_{i}^{[k]}$ is the update instance of node $v_{i}$,
$x_{j}^{[k+\theta_{ij}^{[k]}]}$ are the states of the in-neighbors
$v_{j}\in\mathcal{N}_{i}^{-}\cup\\{v_{i}\\}$ during the time instant of
$v_{i}$’s update, $\theta_{ij}^{[k]}$ are the asynchronous state updates of
the in-neighbors of node $v_{i}$ that occur between two consecutive updates of
node $v_{i}$’s state. The asynchronous max/min consensus in (15) converges to
the maximum value among all nodes in a finite number of steps $s^{\prime}\leq
D\mathcal{B}$ (see [18]), where $D$ is the diameter of the network, and
$\mathcal{B}$ is the upper bound on the number of time steps required for a
node $v_{j}$ to process the information received from its in-neighbors.

##  V Distributed Asynchronous Optimization via ADMM with Efficient
Communication

In this section we present a distributed algorithm which solves problem (14).
Before presenting the operation of the proposed algorithm, we analyze the ADMM
operation over the problem (14).

In (14), let us denote $F(X)\coloneqq\sum_{i=1}^{n}f_{i}(x_{i})$. This means
that the Lagrangian function is equal to

$L(X,z,\lambda)=F(X)+g(z)+\lambda^{\mbox{\tiny T}}(X-z),$ (16)

where $\lambda\in\mathds{R}^{np}$ is the Lagrange multiplier. We now make the
following assumptions to solve the problem (14).

###### Assumption 2

Every cost function $f_{i}:\mathds{R}^{p}\rightarrow\mathds{R}$ is closed,
proper and convex.

###### Assumption 3

The Lagrangian $L(X,z,\lambda)$ has a saddle point. This means that there
exists $(X^{*},z^{*},\lambda^{*})$, for which

$L(X^{*},z^{*},\lambda)\leq L(X^{*},z^{*},\lambda^{*})\leq
L(X,z,\lambda^{*}),$ (17)

for all $X\in\mathds{R}^{np}$, $z\in\mathds{R}^{np}$, and
$\lambda\in\mathds{R}^{np}$.

Assumption 2 means that the local cost function $f_{i}$ of every node $v_{i}$
can be non-differentiable (see [19]). Furthermore, Assumptions 2 and 3 mean
that $L(X,z,\lambda^{*})$ is convex in $(X,z)$ and $(X^{*},z^{*})$ is a
solution to problem (14) (see [19, 20]). Note that this is also based on the
definition of $g(z)$ in (13). Note here that our results extend naturally to
strongly convex cost functions, since strong convexity implies convexity.

Let us now focus on the Lagrangian of the problem in (14). At time step $k$,
the augmented Lagrangian of (14) is

$\displaystyle L_{\rho}$ $\displaystyle(X^{[k]},z^{[k]},\lambda^{[k]})$ (18)
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}f_{i}(x_{i}^{[k]})+g(z^{[k]})+{\lambda^{[k]}}^{\mbox{\tiny
T}}(X^{[k]}-z^{[k]})$ $\displaystyle+$
$\displaystyle\frac{\rho}{2}\|X^{[k]}-z^{[k]}\|^{2}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\left(f_{i}(x_{i}^{[k]})+{\lambda_{i}^{[k]}}^{\mbox{\tiny
T}}(x_{i}^{[k]}-z_{i}^{[k]})+\frac{\rho}{2}\|x_{i}^{[k]}-z_{i}^{[k]}\|^{2}\right)$
$\displaystyle+$ $\displaystyle g(z^{[k]}),$

where $z_{i}\in\mathds{R}^{p}$ is the $i^{th}$ element of vector $z$. In
(3)–(5) we ignore terms that are independent of the optimization variables
such as $x_{i},z$ for node $v_{i}$. This means that (3)–(5) become:

$\displaystyle x_{i}^{[k+1]}=$
$\displaystyle\operatorname*{argmin}_{x_{i}}f_{i}(x_{i})+{\lambda_{i}^{[k]}}^{\mbox{\tiny
T}}x_{i}+\frac{\rho}{2}\|x_{i}-z_{i}^{[k]}\|^{2},$ (19) $\displaystyle
z^{[k+1]}=$
$\displaystyle\operatorname*{argmin}_{z}g(z)+{\lambda^{[k]}}^{\mbox{\tiny
T}}(X^{[k+1]}-z)+\frac{\rho}{2}\|X^{[k+1]}-z\|^{2}$ $\displaystyle=$
$\displaystyle\operatorname*{argmin}_{z}g(z)+\frac{\rho}{2}\|X^{[k+1]}-z+\frac{1}{\rho}\lambda^{[k]}\|^{2},$
(20) $\displaystyle\lambda_{i}^{[k+1]}=$
$\displaystyle\lambda_{i}^{[k]}+\rho(x_{i}^{[k+1]}-z_{i}^{[k+1]}).$ (21)

Note that for (20) we use the identity $2a^{T}b+b^{2}=(a+b)^{2}-a^{2}$ for
$a=\lambda^{[k]}/\rho$ and $b=X^{[k+1]}-z$.

Equations (19), (21) can be executed independently by node $v_{i}$ in a
parallel fashion. Specifically, node $v_{i}$ can solve (19) for
$x_{i}^{[k+1]}$ by a classical method (e.g., a proximity operator [19, Section
4]), and implement trivially (21) for $\lambda_{i}^{[k+1]}$.

In (13), $g(z)$ is the indicator function of the closed nonempty convex set
$\mathcal{C}$. This means that (20) becomes

$\displaystyle z^{[k+1]}=\Pi_{\mathcal{C}}(X^{[k+1]}+\lambda^{[k]}/\rho),$
(22)

where $\Pi_{\mathcal{C}}$ is the projection (in the Euclidean norm) onto
$\mathcal{C}$. It is important to note here that the elements of $z$ (i.e.,
$z_{1},z_{2},\ldots,z_{n}$) should belong into the set $\mathcal{C}$ in finite
time. This is due to the definition of $g(z)$ in (13). Specifically, if the
elements of $z$ do not belong in $\mathcal{C}$ then $g(z)=\infty$ (thus (20)
cannot be executed). Therefore, we need to adjust the values of the elements
of $z$ so that they belong in the set $\mathcal{C}$ in finite time (i.e., we
need to set the elements of $z$ such that $\|z_{i}-z_{j}\|\leq\epsilon,\forall
v_{i},v_{j}\in\mathcal{V}$). Note that if $z_{i}-z_{j}=0,\forall
v_{i},v_{j}\in\mathcal{V}$, then every node $v_{i}\in\mathcal{V}$ has reached
consensus. Specifically, we can have that in finite time the state $z_{i}$
becomes

$\displaystyle z_{i}=\frac{1}{n}\sum_{l=1}^{n}z_{l}^{[0]},\forall
v_{i}\in\mathcal{V},$ (23)

where $z_{l}^{[0]}=x_{l}^{[k+1]}+\lambda_{l}^{[k]}/\rho$. Furthermore,
$\|z_{i}-z_{j}\|\leq\epsilon,\forall v_{i},v_{j}\in\mathcal{V}$ means that

$\displaystyle
z_{i}\in[\frac{1}{n}\sum_{l=1}^{n}z_{l}^{[0]}-\frac{\epsilon}{2},\frac{1}{n}\sum_{l=1}^{n}z_{l}^{[0]}+\frac{\epsilon}{2}],\forall
v_{i}\in\mathcal{V},$ (24)

where $z_{l}^{[0]}=x_{l}^{[k+1]}+\lambda_{l}^{[k]}/\rho$. This means that for
every node $v_{i}$, $z_{i}$ enters a circle with its center at
$\frac{1}{n}\sum_{l=1}^{n}(x_{l}^{[k+1]}+\lambda_{l}^{[k]}/\rho)$ and its
radius as $\epsilon/2$. Finally, from (14), we have that each node in the
network needs to communicate with its neighbors in an efficient manner. For
this reason, we aim to allow each node $v_{i}$ coordinate with its neighboring
nodes by exchanging quantized values in order to fulfil (24).

###  V-A Distributed Optimization Algorithm

We now present our distributed optimization algorithm. The algorithm is
detailed below as Algorithm 1 and allows each node in the network to solve the
problem presented in (14). The operation of the proposed algorithm is based on
two parts. During these parts, each node $v_{i}$ (i) calculates
$x_{i}^{[k+1]}$, $z_{i}^{[k+1]}$, $\lambda_{i}^{[k+1]}$ according to (19)–(21)
(see Algorithm 1), and (ii) coordinates with other nodes in a communication
efficient manner in order to calculate $z_{i}^{[k+1]}$ that belongs in
$\mathcal{C}$ in (11) (see Algorithm 2). Note that Algorithm 2 is a finite
time coordination algorithm with quantized communication and is executed as a
step of Algorithm 1.

Note that during Algorithm 1, nodes operate in an asynchronous fashion.
Synchronous operation requires synchronization among nodes or the existence of
a global clock so that all nodes to agree on their update time. In our
setting, asynchronous operation arises when each node (i) starts calculating
$x_{i}^{[k+1]}$, $z_{i}^{[k+1]}$, $\lambda_{i}^{[k+1]}$ according to (19)–(21)
in Algorithm 1, and (ii) calculates $z_{i}^{[k+1]}$ that belongs in
$\mathcal{C}$ in (11) in Algorithm 2. This can be achieved by making the
internal clocks of all nodes have similar pacing, which will allow them to
execute the optimization step at roughly the same time [21]. Furthermore,
making the internal clocks of all nodes have similar pacing does not mean that
we have to synchronize the clocks of the nodes (or their time-zones). Note
that this is a common procedure in most modern computers as the clock pacing
specification is defined within the Advanced Configuration and Power Interface
(ACPI) specification [22].

We now make the following assumption which is important for the operation of
our algorithm.

###### Assumption 4

The diameter $D$ (or an upper bound) is known to every node $v_{i}$ in the
network.

Assumption 4 is necessary so that each node $v_{i}$ is able to determine
whether calculation of $z_{i}$ that belongs in $\mathcal{C}$ in (11) has been
achieved in a distributed manner. We now present the details of Algorithm 1.

Algorithm 1 QuAsyADMM - Quantized Asynchronous ADMM

Input: Strongly connected $\mathcal{G}=(\mathcal{V},\mathcal{E})$, parameter
$\rho$, diameter $D$, error tolerance $\epsilon\in\mathds{Q}$, upper bound on
processing delays $\mathcal{B}$. Assumptions 1, 2, 3, 4 hold. $k_{\text{max}}$
(ADMM maximum number of iterations).
Initialization: Each node $v_{i}\in\mathcal{V}$ sets randomly
$x^{[0]},z^{[0]},\lambda^{[0]}$, and sets $\Delta=\epsilon/3$.
Iteration: For $k=0,1,2,\dots,k_{\text{max}}$, each node $v_{i}\in\mathcal{V}$
does the following:

  * 1)

Calculate $x_{i}^{[k+1]}$ via (19);

  * 2)

Calculate $z_{i}^{[k+1]}$ = Algorithm
2($x_{i}^{[k+1]}+\lambda_{i}^{[k]}/\rho,D,\Delta,\mathcal{B}$);

  * 3)

Calculate $\lambda_{i}^{[k+1]}$ via (21).

Output: Each node $v_{i}\in\mathcal{V}$ calculates $x_{i}^{*}$ which solves
problem (14) in Section III.

Algorithm 2 QuAsAvCo - Quantized Asynchronous Average Consensus

Input: $x_{i}^{[k+1]}+\lambda_{i}^{[k]}/\rho,D,\Delta,\mathcal{B}$.
Initialization: Each node $v_{i}\in\mathcal{V}$ does the following:

  * $1)$

Assigns probability $b_{li}$ to each out-neigbor
$v_{l}\in\mathcal{N}^{+}_{i}\cup\\{v_{i}\\}$, as follows

$\displaystyle
b_{li}=\left\\{\begin{array}[]{ll}\frac{1}{1+\mathcal{D}_{i}^{+}},&\mbox{if
$l=i$ or $v_{l}\in\mathcal{N}_{i}^{+}$,}\\\ 0,&\mbox{if $l\neq i$ and
$v_{l}\notin\mathcal{N}_{i}^{+}$;}\end{array}\right.$

  * $2)$

$\text{flag}_{i}=0$, $\xi_{i}=2$, $y_{i}=2\
q_{\Delta}^{a}(x_{i}^{[k+1]}+\lambda_{i}^{[k]}/\rho)$ (see (6));

Iteration: For $\eta=1,2,\dots$, each node $v_{i}\in\mathcal{V}$, does:

  * $1)$

if $\eta\mod(D\mathcal{B})=1$ then sets $M_{i}=\lceil y_{i}/\xi_{i}\rceil$,
$m_{i}=\lfloor y_{i}/\xi_{i}\rfloor$;

  * $2)$

broadcasts $M_{i}$, $m_{i}$ to every $v_{l}\in\mathcal{N}_{i}^{+}$; receives
$M_{j}$, $m_{j}$ from every $v_{j}\in\mathcal{N}_{i}^{-}$; sets
$M_{i}=\max_{v_{j}\in\mathcal{N}_{i}^{-}\cup\\{v_{i}\\}}M_{j}$,
$m_{i}=\min_{v_{j}\in\mathcal{N}_{i}^{-}\cup\\{v_{i}\\}}m_{j}$;

  * $3)$

sets $d_{i}=\xi_{i}$;

  * $4)$

while $d_{i}>1$ do

    * $4.1)$

$c_{i}^{[\eta]}=\lfloor y_{i}\ /\ \xi_{i}\rfloor$;

    * $4.2)$

sets $y_{i}=y_{i}-c_{i}^{[\eta]}$, $\xi_{i}=\xi_{i}-1$, and $d_{i}=d_{i}-1$;

    * $4.3)$

transmits $c_{i}^{[\eta]}$ to randomly chosen out-neighbor
$v_{l}\in\mathcal{N}^{+}_{i}\cup\\{v_{i}\\}$ according to $b_{li}$;

    * $4.4)$

receives $c_{j}^{[\eta]}$ from $v_{j}\in\mathcal{N}_{i}^{-}$ and sets

$\displaystyle y_{i}$
$\displaystyle=y_{i}+\sum_{j=1}^{n}\sum_{r=0}^{\mathcal{B}}w^{[r]}_{\eta-r,ij}\
c^{[\eta-r]}_{j}\ ,$ (25) $\displaystyle\xi_{i}$
$\displaystyle=\xi_{i}+\sum_{j=1}^{n}\sum_{r=0}^{\mathcal{B}}w^{[r]}_{\eta-r,ij}\
,$ (26)

where $w^{[r]}_{\eta-r,ij}=1$ when the processing time of node $v_{i}$ is
equal to $r$ at time step $\eta-r$, so that node $v_{i}$ receives
$c^{[\eta]}_{i}$, $1$ from $v_{j}$ at time step $\eta$ (otherwise
$w^{[r]}_{\eta-r,ij}=0$ and $v_{i}$ receives no message at time step $\eta$
from $v_{j}$);

  * $5)$

if $\eta\mod(D\mathcal{B})=0$ and $M_{i}-m_{i}\leq 1$ then sets
$z_{i}^{[k+1]}=m_{i}\Delta$ and stops operation.

Output: $z_{i}^{[k+1]}$.

Note here that Algorithm 1 shares similarities with [8]. However, Algorithm 1
is adjusted to guarantee efficient communication between nodes by allowing
them to exchange quatized valued messages (this characteristic is not present
in [8]). More specifically, the intuition of Algorithm 1 is the following.
Each node $v_{i}$ aims to solve the problem presented in (14) via the ADMM
strategy. Initially, each node chooses a suitable quantization level $\Delta$
so that the constraints of (14) are fulfilled (as explained later in Remark
1). During each time step $k$ each node $v_{i}$ calculates $x_{i}^{[k+1]}$ via
(19). Then, each node $v_{i}$ executes Algorithm 2 in order to calculate
$z_{i}^{[k+1]}$ which belongs in the set $\mathcal{C}$ (i.e.,
$z_{i}^{[k+1]}\in\mathcal{C},\forall v_{i}\in\mathcal{V}$). Finally, each node
$v_{i}$ uses $x_{i}^{[k+1]}$ and the result of Algorithm 2 in order to
calculate $\lambda_{i}^{[k+1]}$ via (21). Algorithm 2 is a distributed
coordination algorithm which allows each node to calculate the quantized
average of each node’s initial state. The main characteristic of Algorithm 2
is that it combines (i) (asymmetric) quantization, (ii) quantized averaging,
and (iii) a stopping strategy. In Algorithm 2, initially each node $v_{i}$
uses an asymmetric quantizer to quantize its state; see Initialization-step
$2$. Then, at each time step $\eta$ each node $v_{i}$:

  * •

splits the $y_{i}$ into $\xi_{i}$ equal pieces (the value of some pieces might
be greater than others by one); see Iteration-steps $4.1$, $4.2$,

  * •

transmits each piece to a randomly selected out-neighbor or to itself; see
Iteration-step $4.3$,

  * •

receives the pieces transmitted from its in-neighbors, sums them with $y_{i}$
and $\xi_{i}$, and repeats the operation; see Iteration-step $4.4$.

Finally, every $D\mathcal{B}$ time steps, each node $v_{i}$ performs in
parallel a max-consensus and a min-consensus operation; see Iteration-steps
$1$ and $2$. If the results of the max-consensus and min-consensus have a
difference less or equal to one (see Iteration-steps $5$), each node $v_{i}$
(i) scales the solution according to the quantization level, (ii) stops the
operation of Algorithm 2, (iii) uses the value $z_{i}^{[k+1]}$ to continue the
operation of Algorithm 1. Note that Algorithm 2 converges in finite number of
steps according to [23, Theorem $1$], since QuAsAvCo has similar structure to
that in [23].

###### Remark 1

It is important to note here that during the initialization of Algorithm 1,
the error tolerance $\epsilon$ is chosen to be a rational number (i.e.,
$\epsilon\in\mathds{Q}$). This is not a limitation for the ADMM optimization
process in Algorithm 1. The real-valued $\epsilon$ can be chosen such that it
can be represented as a rational value. Furthermore, this choice facilitates
the operation of Algorithm 2. Specifically, a rational value for $\epsilon$
facilitates the choice of a suitable quantization level $\Delta$ (since
$\Delta=\epsilon/3$). During the execution of Algorithm 2 nodes quantize their
states, thus an error $e_{q_{1}}\leq\Delta$ is imposed to every state. Then,
Algorithm 2 converges to the quantized average thus, the final states of the
nodes have an error $e_{q_{2}}\leq\Delta$. This means that after executing
Algorithm 2, we have $|z_{i}-z_{j}|\leq 2\Delta<\epsilon$, and thus we have
$z_{i}^{[k+1]}\in\mathcal{C}$ in (11), $\forall v_{i}\in\mathcal{V}$. For this
reason, any choice of $\Delta$ for which $\Delta<\epsilon/2$ is suitable for
the operation of our algorithm for a given error tolerance $\epsilon$.

###### Remark 2

In practical applications, nodes do not know the value of $\mathcal{B}$.
However, $B$ time-steps (which is its upper bound) is guaranteed to be
executed within $T$ seconds (see Assumption 1). As noted previously,
consistent pacing of each node’s clock ensures that the check for convergence
at each node will happen at roughly the same time (see [21]). Therefore, at
every $DT$ seconds, each node checks whether Algorithm 2 can be terminated.

###  V-B Convergence of Algorithm 1

We now analyze the convergence time of Algorithm 1 via the following theorem.
Our theorem is inspired from [8] but is adjusted to the quantized nature of
Algorithm 1. However, due to space limitations we omit the proof (we will
include it at an extended version of our paper).

###### Theorem 1

Let us consider a strongly connected digraph
$\mathcal{G}=(\mathcal{V},\mathcal{E})$. Each node $v_{i}\in\mathcal{V}$, is
endowed with a scalar local cost function
$f_{i}(x):\mathds{R}^{p}\mapsto\mathds{R}$, and Assumptions 1-4 hold.
Furthermore, every node $v_{i}$ has knowledge of a parameter $\rho$, the
network diameter $D$, an error tolerance $\epsilon\in\mathds{Q}$, and an upper
bound on processing delays $\mathcal{B}$. During the operation of Algorithm 1,
let us consider the variables $\\{X^{[k]},z^{[k]},\lambda^{[k]}\\}$, where
$X^{[k]}=[{x_{1}^{[k]}}^{\mbox{\tiny T}},{x_{2}^{[k]}}^{\mbox{\tiny
T}},\ldots,{x_{n}^{[k]}}^{\mbox{\tiny T}}]^{\mbox{\tiny T}}$ and
$\lambda^{[k]}=[{\lambda_{1}^{[k]}}^{\mbox{\tiny
T}},{\lambda_{2}^{[k]}}^{\mbox{\tiny
T}},\ldots,{\lambda_{n}^{[k]}}^{\mbox{\tiny T}}]^{\mbox{\tiny T}}$; then,
define
$\bar{X}^{[k]}=\frac{1}{k}\sum_{s=0}^{k-1}X^{[s+1]},\bar{z}^{[k]}=\frac{1}{k}\sum_{s=0}^{k-1}z^{[s+1]}$.
During the operation of Algorithm 1 we have

$\displaystyle 0$ $\displaystyle\leq
L(\bar{X}^{[k]},\bar{z}^{[k]},\lambda^{*})-L(X^{*},z^{*},\lambda^{*})$ (27)
$\displaystyle\leq\frac{1}{k}\left(\frac{1}{2\rho}\|\lambda^{*}-\lambda^{[0]}\|^{2}+\frac{\rho}{2}\|X^{*}-z^{[0]}\|^{2}\right)+\mathcal{O}(2\Delta\sqrt{n}),$

for every time step $k$, where $\Delta$ is the quantization level for
calculating $z_{i}\in\mathcal{C}$ in (11) during the operation of Algorithm 2.

It is important to note that in Theorem 1 we focus on the convergence of the
optimization steps, i.e., the steps executed during the operation of Algorithm
1. Due to the operation of Algorithm 2 we have that in (27) an additional term
$\mathcal{O}(2\Delta\sqrt{n})$ appears. This term (as will be seen later in
Section VI) affects the precision according to which the optimal solution is
calculated. However, we can adjust Algorithm 2 to operate with a dynamically
refined quantization level $\Delta$. For example, we can initially set
$\Delta=\epsilon/3$ (where $\epsilon\in\mathds{Q}$). Then, execute Algorithm 2
during every time step $k$ with quantization level
$\Delta^{\prime}=\frac{\Delta}{10(k+1)}$. Since we have
$\frac{\Delta}{10(k+1)}<\frac{\Delta}{10(k)}$ for every $k$, then, Algorithm 2
will lead to a reduction of the error on the optimal solution that depends on
the quantization level (i.e., the term $\mathcal{O}(2\Delta\sqrt{n})$ in (27)
will be reduced after every execution of Algorithm 2). However, please note
that this analysis is outside of the scope of this paper and will be
considered in an extended version.

##  VI Simulation Results

In this section, we present simulation results in order to demonstrate the
operation of Algorithm 1 and its advantages. Furthermore, we compare Algorithm
1 against existing algorithms and emphasize on the introduced improvements.

In Fig. 2, we focus on a network comprised of $100$ nodes modelled as a
directed graph. Each node $v_{i}$ is endowed with a scalar local cost function
$f_{i}(x)=0.5x^{\top}P_{i}x+q_{i}^{\top}x+r_{i}$. This cost function is
quadratic and convex. Furthermore, for $f_{i}(x)$ we have that (i) $P_{i}$ was
initialized as the square of a randomly generated symmetric matrix $A_{i}$
(ensuring it is is positive definite), (ii) $q_{i}$ is initialized as the
negation of the product of the transpose of $A_{i}$ and a randomly generated
vector $b_{i}$ (i.e., it is a linear term), (iii) and $r_{i}$ is initialized
as half of the squared norm of the randomly generated vector $b_{i}$ (i.e., it
is a scalar constant). We execute Algorithm 1 and we show how the nodes’
states converge to the optimal solution for $\epsilon=0.03,0.003,0.0003$, and
$\Delta=0.01,0.001,0.0001$, respectively. We plot the error $e^{[k]}$ defined
as

$e^{[k]}=\frac{\sqrt{\sum_{j=1}^{n}(x_{j}^{[k]}-x^{*})^{2}}}{\sqrt{\sum_{j=1}^{n}(x_{j}^{[0]}-x^{*})^{2}}},$
(28)

where $x^{*}$ is the optimal solution of the optimization problem in
$\eqref{objective_function_no_quant}$. Note that from Remark 1, we have that
any $\Delta<\epsilon/2$ is suitable for the operation of Algorithm 1 for a
given $\epsilon$. In Fig. 2, we execute Algorithm 1 for $\Delta=\epsilon/3$.
We can see that Algorithm 1 converges to the optimal solution for the three
different values of $\epsilon$. However, Algorithm 1 is able to approximate
the optimal solution with precision that depends on the quantization level
(i.e., during Algorithm 1, nodes are able to calculate a neighborhood of the
optimal solution). Reducing the quantization level $\Delta$ allows calculation
of the optimal solution with higher precision. Furthermore, we can see that
after calculating the optimal solution our algorithm exhibits an oscillatory
behavior due to quantized communication. This means quantized communication
introduces nonlinearities to the consensus calculation which in turn affect
the values of other parameters such as $x$ and $z$, and $\lambda$ (see
iteration steps $1$, $2$, $3$), and for this reason we have this oscillatory
behavior. Finally, we can see that Algorithm 1 exhibits comparable performance
with [24] (which is plotted until optimization step $14$) until the
neighborhood of the optimal solution is calculated. However, in [24] nodes are
able to exchange real-valued messages. Specifically, in [24] nodes are
required to form the Hankel matrix and perform additional computations when
the matrix loses rank. This requires nodes to exchange the exact values of
their states. Therefore, the main advantage of Algorithm 1 compared to [24],
is that it exhibits comparable performance while guaranteeing efficient
(quantized) communication among nodes.

Figure 2: Comparison of Algorithm 1 with [24] over a directed graph comprised
of $100$ nodes for $\epsilon=0.03,0.003,0.0003$, and
$\Delta=0.01,0.001,0.0001$, respectively.

##  VII Conclusions and Future Directions

In this paper, we presented an asynchronous distributed optimization algorithm
which combines the Alternating Direction Method of Multipliers (ADMM) strategy
with a finite time quantized averaging algorithm. We showed that our proposed
algorithm is able to calculate the optimal solution while operating over
directed communication networks in an asynchronous fashion, and guaranteeing
efficient (quantized) communication between nodes. We analyzed the operation
of our algorithm and showed that it converges to a neighborhood of the optimal
solution (that depends on the quantization level) at a rate of $O(1/k)$.
Finally, we demonstrated the operation of our algorithm and compared it
against other algorithms from the literature.

In the future, we aim to enhance the operation of our algorithm to avoid the
oscillatory behavior after calculating the optimal solution. Furthermore, we
plan to develop strategies that allow calculation of the exact optimal
solution while guaranteeing efficient communication among nodes. Finally, we
will focus on designing efficient communication strategies for non-convex
distributed optimization problems.

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# Gender Differences in Class Participation in Online versus In-Person Core CS
Courses

Madison Brigham<EMAIL_ADDRESS>Department of Computer
ScienceUniversity of California, DavisDavisCaliforniaUSA95616 and  Joël
Porquet-Lupine  0000-0003-4634-2877<EMAIL_ADDRESS>Department of
Computer ScienceUniversity of California, DavisDavisCaliforniaUSA95616

###### Abstract.

The COVID-19 pandemic significantly altered how post-secondary students
receive their education. Namely, the transition from an in-person to an online
class format changed how students interact with their instructors and their
classmates. In this paper, we use student participation scores from two core
computer science classes across ten in-person and three online quarters at a
public research university to analyze whether the shift to primarily
asynchronous online learning has impacted the gender gap in student
participation scores and students’ attitudes towards themselves and their
peers. We observe a shift on the online class forum: in in-person classes,
males score higher on average and dominate the top scores while in online
classes, male and female students participate at approximately the same rate
classwide. To understand what might be driving changes in participation
behavior, we analyze survey responses from over a quarter of the students
enrolled in the online classes. While we find that students of both genders
tend to compare themselves to their peers less when classes are online, we
also find that this trend is much more accentuated for females than males.
This data suggests that observed female participation habits in typical in-
person classes are not inherent gender differences, but rather, a product of
the environment. Therefore, it is critical the community investigates the root
causes of these behavioral differences, and experiments with ways to mitigate
them, before we soon return to an in-person format.

Computer science education; Gender; Class participation

††copyright: rightsretained††ccs: Social and professional topics Computer
science education

## Forewords

Despite our best efforts, this manuscript was rejected twice from top CS
Education conferences (ACM SIGCSE TS and ACM ITiCSE) in 2022. This version
incorporates modifications from the latest round of feedback we received
(limitations section, mention of non-binary responses, etc.). Unfortunately,
it was never resubmitted as Maddii graduated at the end of 2022, and the
rationale for studying the effects of the pandemic was beginning to diminish.
We are making the decision to submit to arXiv as we strongly believe that this
work is valuable and should reach its intended audience, even if not via peer-
reviewed venues.

##  1\. Introduction

The onset of the COVID-19 pandemic in early 2020 drastically affected the
delivery of post-secondary education. Across the US, colleges and universities
rapidly moved their in-person instruction to an online format, which remained
the norm at many institutions throughout the 2020 to 2021 academic year. This
widespread shift to online learning subsequently led to a decline in in-person
lectures, where classroom climate is typically established.

As explained by Barker and Garvin-Doxas (Barker and Garvin-Doxas, 2004;
Garvin-Doxas and Barker, 2004), classroom climate describes the communication
and interaction patterns that shape the perceptions of students in a class. A
defensive classroom climate is characterized by judgement, apathy, and
superiority. This tends to result in students isolating from one another,
feeling the need to compete with each other, and seeing experience-level as
what constitutes intelligence. This kind of culture can have an especially
harmful impact on students in underrepresented groups, such as female
students, whose enrollment in computer science (CS) programs has steadily
declined since the mid-1980s (U.S. Department of Education, 2019).

In this paper, we investigate whether male and female students’ attitudes and
participation habits have changed with classes transitioning from in-person to
online. Does the shift from an in-person to an online class format impact the
gender gap in student participation scores? And has this shift also impacted
students’ attitudes towards themselves and their peers?

To measure this, we compared 1,860 student participation scores from ten in-
person offerings of two core CS classes with 547 student participation scores
from three –primarily asynchronous– online offerings of the same classes, as
explained in section 3. In section 4, we compare the scores received by male
and female students in the two main participation categories, which remained
consistent across class formats: participation on the class forum and survey
completion. In section 5, we evaluate students’ perceptions of how their
participation and view of themselves and their peers has or has not changed
following the transition online.

##  2\. Related work

The effects of the shift from in-person to online classes at the onset of the
COVID-19 pandemic remain relatively unstudied, as the pandemic only began a
little over a year and a half ago. With that being said, a limited body of
research has already been conducted in this area. Work by Mooney and Becker
(Mooney and Becker, 2021) found that male and female students who did not
consider themselves to be part of a CS minority experienced a decline in their
sense of belonging over the course of the pandemic. Conversely, female
students who considered themselves to be part of a minority saw an increase in
their sense of belonging during this time period. Work by Lewis et al. (Lewis
et al., 2021) found that students reported their stress levels and perception
of class difficulty in online classes to be similar to or lower than that of
in-person classes, likely due in part to the increased accessibility of
materials, the redesign of courses, and the increased flexibility in grading
policies. Our research aims to further the understanding of the effects of the
shift to online schooling. As the transition to online schooling subsequently
changed how students interact with each other, their instructors, and the
class material, we want to study potential changes in class participation and
classroom climate, both well-established areas of research in terms of pre-
pandemic schooling, under the lens of this shift.

Class participation in in-person classes can provide insight into the
experience of students in CS. Of particular interest is the experience of
female students, who are underrepresented in CS at the post-secondary level
(U.S. Department of Education, 2019). Work by Alvarado et al. (Alvarado et
al., 2017) found that female students were less comfortable asking questions
in lecture than males. Similarly, work by Brigham and Porquet-Lupine (Brigham
and Porquet-Lupine, 2021) found that the top class participation scorers in
lecture and on class forums were disproportionately male. Previous work has
found that female students are more likely than males to ask and answer
questions anonymously on the class forum (Thinnyun et al., 2021; Sobel et al.,
2016).

Past research has found low self-confidence among female CS students
(Rosenstein et al., 2020). Research by Winter et al. found that female
students commonly felt as though they were falling behind their peers, and
expressed concerns about their experience gaps with prior coding (Winter et
al., 2021). Even when quantitative ability is controlled for, female students
express lower self-confidence with computing than their male counterparts
(Alvarado et al., 2017; Beyer et al., 2003). Work by Treu and Skinner suggests
that low self-confidence in female students can be attributed to negative
stereotypes, subtle bias from instructors, and a lack of female role models
(Treu and Skinner, 2002). Work by Cohoon also found that lower presence of
female faculty was correlated with higher female attrition rates (Cohoon,
2001).

To combat this, efforts have been made to improve the self-perception of
females in CS. Research by Fisk et al. found that incorporating encouraging
emails from the instructor to increase the self-perception of top-performing
female students seemed to increase female intent to continue in CS (Fisk et
al., 2021). Research by Krause-Levy et al. found that the design of open
online computing classes that normalized struggle was correlated with higher
female enrollment and higher completion rates for both male and female
students (Krause-Levy et al., 2021). Efforts have also been made to expose
students to more female role models in the field (Alvarado et al., 2012;
Roberts et al., 2002), mitigate harmful side effects from experience gaps
between students starting out in CS (Alvarado et al., 2012; Barker et al.,
2009; Roberts et al., 2002), and combat the stereotype that CS is isolated
from other disciplines (Alvarado et al., 2012; Rich et al., 2004).

##  3\. Methodology

This study111Both the class data analysis and student survey in this study
were reviewed by our university’s Institutional Review Board. looks at in-
person and online offerings of two core classes that all CS majors at the
university are required to take: Data Structures and Algorithms (hereafter
referred to as CS3) and Operating Systems (hereafter referred to as CSOS).
Both of these quarter-long classes are concept-focused and programming-heavy.

CS3 is the third and final class in the introductory CS series, and students
typically take it towards the end of their freshman year or beginning of their
sophomore year. For the in-person offerings of this course, we analyzed 333
students’ grades across two quarters. Of these students, 76.88% were male and
23.12% were female. For the online offerings of CS3, we analyzed 153 students’
grades across one quarter. Of these students, 76.47% were male and 23.53% were
female. CSOS is an upper division course that students typically take towards
the end of their junior year or beginning of their senior year. For the in-
person offerings of this course, we analyzed 1,504 students’ grades across
eight quarters. Of these students, 75.40% were male and 24.60% were female.
For the online offerings of CSOS, we analyzed 392 students’ grades across two
quarters. Of these students, 74.74% were male and 25.26% were female.

All classes were taught by the same male instructor, who collected the in-
person class data from January 2017 to March 2020, and the online class data
from April 2020 to March 2021. Based on our university’s registrar data, which
uses a binary gender classification, students were classified as either male
or female222We acknowledge that this is an oversimplified gender spectrum..
The 0.01% of students (25 of the 2,407 students) whose gender was unknown by
the registrar were omitted from our data analysis.

A student’s participation score in CS3 and CSOS could typically earn them up
to 3% extra credit. In the in-person offerings of these classes, this score
was determined by three categories: lecture, forum, and survey. For the online
offerings of these classes, where lectures occurred asynchronously, the
categories that determined a student’s participation score slightly shifted:
instead of earning points for participating in lecture, a student could earn
points by attending synchronous office hours with either the instructor or
TAs. In this study, we focus on the two categories that remain consistent
across class formats: forum and survey.

##### Forum participation

In both the in-person and online classes, a student earned points for forum
participation by posting on Piazza, an online class forum where students can
interact with one another in real time. Students can post questions, answer
other students’ questions in a designated answer box on each post, or leave
follow-up questions or comments in the discussion section at the bottom of
each post. Each student’s forum score was determined by these three metrics,
which are tracked by Piazza, with answers earning the most points, then
questions, then follow-ups. Students are able to post anonymously to
classmates, but not to the instructor.

##### Survey participation

In both the in-person and online classes, a student earned points for survey
participation by completing class evaluations throughout the quarter. These
include the official university course evaluation, as well as a mid-quarter
and end-of-the-quarter survey created by the instructor. Surveys provide
opportunities for students to evaluate and leave feedback on the course, and
students’ responses to these surveys are seen only by the instructor.

##### Student input

To help us explain the trends we observed in the online offerings of CS3 and
CSOS, we developed a short survey to collect student input. This student
survey was sent in February 2021 to the 533 students who had taken or were
currently enrolled in CS3 online or CSOS online, so that we could capture
responses specifically from the students composing our online class data set.
Participation in the survey was optional, and student responses were
completely anonymous. We received 138 complete responses, giving us a response
rate of 25.89%. Of the students who responded, 71.01% identified as male,
27.54% identified as female, and 1.45% identified as non-binary. This leaves
the gender ratio of the survey respondents close to that of the students
composing the online class data set, suggesting that our survey responses are
representative of our target population as a whole. In this study, we
primarily focus on male and female responses only because a sample size of two
non-binary CS students is not sufficient to draw general conclusions about
this gender category.

##  4\. Participation scores

We began by comparing scores received in the two participation categories that
remained consistent across class formats: the forum and survey participation.
For all statistical tests, we used an alpha level of .05.

###  4.1. Forum

For CS3 in-person, the average male forum participation score (M = 23.75, SD =
32.79) was 1.69 times the average female forum participation score (M = 14.03,
SD = 20.73), t(331) = 3.11, p = .002. For CSOS in-person, the average male
forum participation score (M = 18.44, SD = 26.70) was 1.54 times the average
female forum participation score (M = 11.97, SD = 18.70), t(1502) = 5.16, p ¡
.001. Figure 1 shows a visual comparison of these averages on the left.

In the online offerings of these courses, the forum participation imbalance
disappeared. For CS3 online, there is no longer a significant difference
between the average male forum participation score (M = 13.93, SD = 21.99) and
average female forum participation score (M = 14.44, SD = 23.45), t(151) =
-0.12, p = .908. For CSOS online, there is also no significant difference
between the average male forum participation score (M = 21.09, SD = 28.92) and
average female forum participation score (M = 22.37, SD = 27.29), t(390) =
-0.40, p = .691. Figure 1 shows a visual comparison of these averages on the
right.

Figure 1. Average forum participation scores

While it is evident that the forum averages are much closer in the online
classes than the in-person ones, we were curious about what was driving this
change. In other words, were a couple of female outliers pulling up the
average for everyone, or had there been shifts in behavior classwide? In order
to measure this, we considered the distribution of participation scorers.

Decile distributions of forum participation scorers, like Figures 2 and 3,
help us visualize how male and female students rank in comparison to one
another classwide, and therefore provide insight into this shift in forum
averages. When we combine the participation scores from all in-person or
online quarters of a class, and rank them from highest to lowest, the
distribution shows what percentage of male and female students scored in the
top 10% of scorers, the second decile of scorers, and so on down to the bottom
10% of scorers. If male and female students’ scores were distributed evenly,
each decile of scorers would contain 10% of the female students and 10% of the
male students.

In the in-person offerings of CS3 and CSOS, which are shown in Figure 2, we
found a disproportionately low percentage of female students scoring in the
top deciles of scorers, and a disproportionately high percentage of them
scoring in the mid to low deciles.

In the online classes, which are shown in Figure 3, we see a different
pattern. For both CS3 and CSOS online, male and female students participated
on the forum at approximately the same rate classwide. In other words, the
proportion of male to female students who scored in each decile is
approximately equal to the proportion of males to females enrolled in each
class. The dented nature of the CS3 female line can be attributed to small
sample size. The 35 females enrolled in CS3 online were as evenly distributed
as possible, however, a difference of one student per decile (e.g., 3 instead
of 4) creates a misleadingly pronounced bend to the line.

Figure 2. In-person distribution of forum scorers Figure 3. Online
distribution of forum scorers

Moving online seems to have closed the gender gap in average forum scores.
Furthermore, the average female forum score is not being pulled up by a
handful of female students; instead, forum participation scores of male and
female students are more evenly distributed classwide.

###  4.2. Survey

For CS3 in-person, the average female survey participation score (M = 81.17,
SD = 32.48) was 1.22 times the average male survey participation score (M =
66.41, SD = 37.89), t(331) = -3.36, p = .001. For CSOS in-person, the average
female survey participation score (M = 79.79, SD = 31.47) was 1.06 times the
average male survey participation score (M = 75.43, SD = 34.69), t(1502) =
-2.25, p = .025. Figure 4 shows a visual comparison of these averages on the
left.

In the online offerings of these courses, survey participation has mostly
stayed the same. For CS3 online, the average female survey participation score
(M = 84.26, SD = 28.16) is 1.21 times the average male survey participation
score (M = 69.80, SD = 38.90), t(151) = -2.45, p = .017. For CSOS online,
while there is no longer a statistically significant difference between the
average male survey participation score (M = 79.98, SD = 30.86) and average
female survey participation score (M = 84.85, SD = 25.32), t(390) = -1.56, p =
.120, this may have to do with the smaller sample size of CSOS online compared
to CSOS in-person. Figure 4 shows a visual comparison of these averages on the
right.

Figure 4. Average survey participation scores

The decile distributions of survey participation scorers, shown in Figures 5
and 6, reveal a similar pattern across class formats. In both the in-person
and online offerings of CS3 and CSOS, we consistently see a slightly higher
percentage of female students scoring in the top and middle deciles, and a
noticeably lower percentage of them scoring in the bottom deciles.

Figure 5. In-person distribution of survey scorers Figure 6. Online
distribution of survey scorers

Based on these averages and distributions, survey participation patterns of
male and female students appear to be relatively unchanged, despite the shift
from an in-person to an online class format: the data suggests that in both
in-person and online offerings of CS3 and CSOS, female students were more
likely to complete the class surveys than their male counterparts.

##  5\. Student survey responses

While analyzing class data can help us identify trends, collecting student
input is vital in explaining them. We wanted to know why globally, females are
participating on the forum at the same rate as males now that classes are
online, and whether this change in participation habits can be explained in
terms of student attitudes. To help us accomplish this, we developed a short
student survey.

###  5.1. Survey details

In our survey, we wanted to assess two things: whether students believed that
their forum participation had changed since going online (Question 1), and
whether they believed that their attitudes had shifted as well. We decided to
capture the student attitudes which may have impacted participation on the
forum by asking whether the shift from an in-person to an online class format
had impacted students’ confidence in their ability to succeed in their CS
classes (Question 2), or the amount that they compared themselves to their
peers (Question 3).

Question 1: Which statement best describes your forum participation in in-
person versus online CS classes? When classes are online, I participate ___ on
the forum.

Question 2: Which statement best describes your confidence in your own ability
to succeed in your CS classes? When classes are online, I feel ___ confident
in my CS abilities.

Question 3: Which statement best describes your attitudes towards your
classmates? When classes are online, I compare myself to my classmates ___.

Students selected their response to each question from a five-point Likert
scale: significantly more, slightly more, the same, slightly less, or
significantly less. Students could also respond to an optional free response
following each question, if the student wished to explain their answer.

###  5.2. Survey results

####  5.2.1. Question 1

We see in Figure 7 that for Question 1, a higher proportion of males reported
having the impression of participating more on the forum in online classes,
compared to the female respondents. A higher proportion of females reported
having the impression of participating less on the forum in online classes, or
participating the same as in in-person.

Figure 7. Perceived online forum participation by gender

Conversely, our analysis of class data suggests that female students are
participating more on the forum now that classes are online, or perhaps that
males are participating less. But it seems that the students’ perception, at
least that of the respondents, is the opposite. Of course, it is possible that
we have self-selection bias in the population of students who chose to
complete our survey. But these responses suggest that female students may have
a more modest view of their participation on the forum.

In the free responses to this question, we observe two major trends. The first
trend was widely reported by both male and female respondents: now that
classes are online, many students are using the forum more because they see it
as the only means of communication. As a male student put it, ”since
everything is now online, this provides more incentive to engage with
professors and TAs through these platforms such as Discord, Piazza, and
CampusWire.” Another male student mentioned that ”[the] inability to talk to
classmates in person has caused [them] to participate slightly more on the
forum.” A female student commented that ”since everything is virtual, the only
way to communicate with others is through the internet so …class forums are
very useful and an essential way to feel slightly more connected with other
students when classes are online.”

On the other hand, the second trend was widespread only among female
respondents: in online classes, some female students use the forum more
because of increased comfort and confidence that comes from their anonymity in
the class. One male student did respond saying that ”in person, [they are] too
shy to talk to people, [and that] being online helps a bit.” However, this was
only one of the 24 male students who left free responses explaining why their
forum participation had increased, and therefore not enough to establish a
trend. In comparison, of the eight females who left free responses explaining
why their forum participation had increased, half of them referenced this
reasoning. One female student replied, ”[they] feel more comfortable while
people cannot see [them]” and another mentioned, ”online classes allow [them]
to be anonymous while participating.”

There is a bit of ambiguity here, as anonymous online forum participation is
not unique to online classes, and was already an option when the classes we
are considering were in-person. These responses do seem to suggest that some
female students have felt increased anonymity now that there is no in-person
component to the class, like in-person lecture, and that this overall increase
in anonymity is leading to their increase in forum participation.

####  5.2.2. Question 2

We see in Figure 8 that for Question 2, both the male and female responses
were rather normally distributed. However, a smaller proportion of female
respondents reported feeling that their confidence levels had been unaffected
by the transition. Instead, a higher proportion of females reported the shift
to have either positively or negatively affected their confidence levels, in
comparison to the male respondents.

Figure 8. Self-confidence by gender

In the free responses to this question, we observe three major trends. The
first trend, shared across all respondents, is that with online classes, many
students are less confident in their ability to succeed due to a lack of
support and motivation. One male student commented that ”[they] have less
opportunities to ask other classmates or professors for help and advice[, and
that they] also feel significantly less engaged when a class is online rather
than in person.” Another male student said, ”[they] don’t have a network of
people to rely on in the same way [as in in-person classes].” A female student
reported ”not learning as effectively as [they] would in an in person class
because it is much more difficult to stay motivated online.” Another female
commented that ”since there is no in person interaction, it makes [them] feel
as if [they’re] alone and don’t have the support needed.”

On the other hand, the second trend across both male and female students was
that in online classes, some students are more confident in their ability to
succeed due to the increased accessibility of class material. A male student
made several points, saying that ”[they] appreciate how [they] can take
classes and watch lectures at [their] own pace. When things are in person,
there is only one opportunity to listen to a lecture which limits the
possibility for a student to ask clarifying questions. The online content is
typically more condensed and easier to access.” A female student said that for
them, ”it’s more convenient to watch lectures in [their] own timeline and
[having] the ability to pause and take better notes [has] really boosted
[their] level of understanding and therefore [their] level of confidence in
the class.”

The third trend, which was unique to female respondents, was that some females
feel an increased sense of confidence in online classes because there is less
opportunity for them to compare themselves to their classmates. No male
respondents echoed this sentiment. One female student mentioned that they
”feel more comfortable when [they] don’t see all the students, who are
younger, smarter and better than [themselves]. [In online classes, they
aren’t] scared [of being] judged anymore.” Another female mentioned that
”something that used to lower [their] confidence in [their] cs abilities
[was], during in person classes, when someone would raise their hand and ask
questions that [they] had absolutely no idea about.”

In this last trend, we notice that female respondents already bring up
comparison to peers without being prompted. However, it is not until our next
question, Question 3, that we explicitly asked students to consider the amount
that they compared themselves to their classmates. This supports our initial
intuition that students’ attitudes towards their peers may have been a
significant driver of the behavioral changes we observed on the forum.

####  5.2.3. Question 3

We see in Figure 9 that for Question 3, a higher proportion of female
respondents reported comparing themselves to their peers less now that classes
were online.

Figure 9. Attitude towards peers by gender

In the free responses to this question, we observe three major trends. With
classes online, many male and female students compare themselves to their
peers less because not seeing them has created a feeling of indifference. Or,
as one male student put it, ”[they’re] all just Zoom names to me now :)”.
Another male student explained, ”it’s harder for [them] to see how prepared
[or] confident [their classmates] are, and [that they generally] think less
about them because [they] don’t see them personally each and every single day
now.” A female student commented that ”with classes online, [they’re] not
surrounded by [their] classmates everyday so [they’re] less likely to take
note of the differences [or] compare [themselves] with them.”

The second trend is the first trend unique to male students: in online
classes, some males report comparing themselves to their classmates more,
because there is more room to speculate and make assumptions about their
progress. This reasoning was not echoed among female respondents. One male
explained that ”since [they] feel less engaged in class, [they] worry that
[they are] not learning as much information as other students and thus feel
less prepared for quizzes, exams, and assignments when compared to other[s].”
Another commented that ”[they] don’t get to see [their classmates] in person
so [they] construct a version of them in [their] head that may or may not be
true and is only based on what [they] see online.” Another male student
mentioned they ”feel that at home, [they] are able to slack off more while
[their] peers are able to get ahead in assignments, projects, and
extracurriculars.”

Conversely, the final trend was unique to female respondents: in online
classes, some females compare themselves to their peers less because they feel
more comfortable not interacting with them in an in-person lecture
environment. This was not echoed by any male respondents. One female commented
that ”being in in-person lectures can make the class seem more intimidating
[and] intense. You don’t experience that environment when lectures are
online.” Another female explained that ”with in-person classes, there is more
opportunity for participation. Those who ask questions are usually more
outspoken and more knowledgeable than [they are], so [they] feel more self-
conscious of [their] own abilities and knowledge. These questions [in lecture]
are sometimes very specific and maybe even unrelated to the class. With the
forum, the questions are usually more focused and [they] do not feel as self-
conscious when answering them.”

This is an interesting difference between male and female students. It seems
like for some female students, seeing and interacting with classmates in
person makes them question their own abilities, whereas for some male
students, this interaction actually makes them more confident. Therefore, the
in-person lecture component of CS classes may be more favorable to some male
students, and less favorable to some female students, due to the class
format’s effect on students’ self-perception. This is telling of what kind of
student benefits from the classroom climate of CS classes. As classroom
climate can impact a student’s self-perception, this could explain why in in-
person classes, we even see reduced female participation in spaces outside of
the in-person components, namely, on the forum.

####  5.2.4. Non-binary responses

While the small sample size of non-binary students prevents us from drawing
statistically significant conclusions, it is still interesting to consider
this group’s responses. The two non-binary students responded neutrally to all
Likert scale questions; they reported that the shift to online classes had no
impact on their forum participation, confidence in their CS abilities, or
comparison to their classmates. Neither of the students left free responses.

##  6\. Limitations

We obtained official gender identification information for each student’s
course grade from the university registrar, which uses a binary gender
classification. To avoid an oversimplified gender spectrum in future studies,
students could self-report their gender identity.

Naturally, the duration of the pandemic limited the size of our online class
dataset. Analyzing data from online classes taught before the pandemic would
allow us to gauge the consistency of the observed trends.

The same male instructor taught all analyzed classes. Future research could
investigate whether we see consistent trends in similar classes taught under
different instructors.

Alongside the shift from in-person to online, the analyzed classes also moved
from a synchronous to a primarily asynchronous format. This secondary shift in
format may also impact how students of different gender identities participate
in class.

##  7\. Conclusion

As almost all aspects of CS3 and CSOS remained consistent between the in-
person and online offerings (e.g., the same professor, the same class
structure), this allows us to isolate and study the potential impacts of the
only few but significant changes: the transition from an in-person to online
delivery of lectures, discussions, and office hours, and the transition from
synchronous to asynchronous lectures. Our data analysis finds a correlation
between the shift from an in-person to a primarily asynchronous online class
format and the closure of the gender gap in forum participation class-wide,
which, with the removal of lecture participation, is now the most public form
of participating. These observations differ from existing research findings on
participation in in-person classes, which found that the most public forms of
participation are largely male-dominated (Alvarado et al., 2017; Brigham and
Porquet-Lupine, 2021; Sobel et al., 2016).

Student survey responses suggest that female students are less likely to
compare themselves to their peers in online classes, potentially contributing
to increased female participation on the forum. Conversely, our collected
responses suggest that male students are more likely than females to compare
themselves to their peers when they do not see them in person. This
observation suggests that the classroom climate in in-person settings could
benefit male students.

The solution to alleviating the class participation gender gap in CS classes
is not to remain online forever. However, these observed differences in
participation between in-person and online classes can help us better pinpoint
what we need to address to mitigate the gap when we return to an in-person
format. This data suggests that observed female participation habits in
typical in-person classes are not inherent gender differences but rather a
product of the environment. Future work should explore the causes of these
behavioral differences (i.e., why some females report feeling more confident
online than in person, why some males report the opposite) and experiment with
ways to ensure in-person computer science classes promote equity for students
of all gender identities.

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