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Nissan Altima vs. Honda Civic
At a Glance: Altima vs. Civic
Trying to choose between the 2023 Nissan Altima and the 2023 Honda Civic? You've come to the right place! In this comparison, we at McGrath Nissan of Elgin will show you how these two models stack up against each other. Drivers near Elgin, Hoffman Estates, Barlett, and Streamwood should read on to see which model comes out on top!
Altima vs Civic
MAXIMUM HORSEPOWER
STANDARD TOUCH-SCREEN SIZE
7" | 7"
Select a category below to learn more about how the Nissan Altima compares to the Honda Accord.
Thanks to their lightweight frames and audacious engines, the Nissan Altima and the Honda Civic allow drivers to experience race car-like thrills in an everyday commuter. Which model will satisfy your hunger for power?
Both the 2023 Nissan Altima and the 2023 Honda Civic offer two engine options, but the Altima outshines the Civic at every turn. The Altima gives you the choice between a standard 188-horsepower 2.5L Direct Injection Gasoline (DIG) engine and an available 2.0L Variable Compression Turbo (VC-Turbo™) engine, which can make up to a whopping 248 horsepower.
The Honda's available turbocharged 4-cylinder engine only cranks out 180 horsepower, which pales in comparison to the Nissan's standard and available engine!
Stable Travels
Plus, the Nissan Altima can be configured with available Intelligent All-Wheel Drive. With this system, you'll be able to conquer anything on the road in front of you.
With the Honda Civic, you won't feel quite as confident in inclement weather with its standard and only drivetrain: a front-wheel drive system.
Comfort is a top priority, whether you're jetting off to the office, running errands, or setting out on a weekend adventure with the family. The Altima and the Civic both have plenty of comfort amenities, but which model provides ultimate convenience?
Room to Enjoy
The 2023 Nissan Altima and the 2023 Honda Civic both offer available leather seating surfaces, heated front seats, and ample legroom for all five passengers.
However, the Altima slightly edges out the Civic in terms of passenger volume. The Nissan offers 100.6 cubic feet of space, while the Honda has 99 cubic feet.
Cozy and Convenient Cabin
The Nissan Altima takes comfort and convenience to the next level, though, with a few extra available features such as:
Remote Engine Start System with Intelligent Climate Control
While the Honda Civic has Push Button Start as well, only the Nissan Altima offers Remote Engine Start, which makes sure you're getting into a preconditioned cabin to combat the weather conditions outside.
You love to drive. You love to explore new paths and travel to new locations. However, with unpredictable roads ahead, you'll want a little extra assurance behind the wheel. The standard safety driver assistance suites on these sedans will provide the protection you desire.
Active Safety
Both the 2023 Nissan Altima and the 2023 Honda Civic offer standard active safety suites. The Altima's Nissan Safety Shield 360® and the Civic's Honda Sensing® suite provide similar features, including Automatic Emergency Braking with Pedestrian Detection, Lane Departure Warning, Traffic Sign Recognition, and High Beam Assist.
Both sedans also offer Blind Spot Warning and Rear Cross Traffic Alert; however, these systems come standard on the Altima and are only available as options on the Civic.
Additional Protection
However, there are also a few extra features with the Nissan Altima that you won't find in the Honda Civic:
Intelligent Driver Alertnessmonitors your behavior and suggests a break when it senses that you're fatigued or distracted.
ProPILOT Assist helps you maintain a safe following distance from other vehicles.
The Altima also offers an available Intelligent Around View® Monitor, which provides a 360-degree view of your vehicle's surroundings. While the Civic offers an available Multi-Angle Rearview Camera, it has severely limited vision compared to the Altima.
As you can see, the 2023 Nissan Altima is the sedan for drivers near Elgin, Hoffman Estates, Barlett, and Streamwood. With its one-of-a-kind powertrain, high-end interior, and advanced technology features, the Altima goes above and beyond the call of duty.
Now that you've learned a bit more about what this model has to offer, it's time to experience it for yourself. Give us a call at McGrath Nissan of Elgin to schedule a test drive today!
All of the tools you need to move forward with experiencing a new 2023 Alitma for yourself are conveniently available right here online.
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text:
\section{Introduction}

In online learning problems, an agent sequentially makes
decisions and receives an associated reward. When this reward is
stochastic, the problem takes the form of a stochastic multi-armed
bandit problem \cite{bubeck2012regret}. However, stochastic bandits assume
stationary reward processes that are rarely the case in practice; they are too optimistic about the environment.
To deal with this shortcoming, one can model the rewards as being determined by
an adversary, which leads to a formulation known as non-stochastic or adversarial bandits
\cite{bubeck2012regret}. Naturally, the performance guarantees against
such a powerful adversary are much weaker than in the stochastic case, as
adversarial bandits are usually overly pessimistic about the environment.
Is there an alternative that lies in the gap between the two?

Multi-player bandits is a promising answer, which has seen a surge of interest in recent years
\cite{Liu2010,Vakili2013,Lai2008,Anandkumar2011,liu2019competing,magesh2019multi,Liu2013,Avner2014,sankararaman2019social,Evirgen2017,Cohen2017,Avner2016}.
One primary motivation for studying multi-player bandits is distributed resource allocation.
Examples include channels in communication networks, computation resources on servers, consumers and items, etc.
In most applications of interest, the reward of a player is a stochastic function of the decisions of other players that operate in the same environment.
Thinking of arms as resources, we see that while these players are not adversaries, conflicts still arise due to the players' preferences among the limited resources. To model that, we assign zero reward to players that choose the same arm.
The goal of multi-player bandit algorithms is to provide a distributed
way to learn how to share these resources optimally in an online manner. This is
useful in applications where agents (players) follow a standard or protocol,
like in wireless networks, autonomous vehicles, or with a team of robots.

A common network performance objective is the sum of rewards of the
players over time. As such, maximizing the sum of rewards has received the vast majority of the attention in the multi-player bandit literature
\cite{hanawal2018multi,Besson2018,tibrewal2019distributed,bistritz2018distributed,Kalathil2014,Nayyar2016,boursier2019sic,boursier2019practical}.
However, in the broader literature of network optimization, the sum
of rewards is only one possible objective.
One severe drawback of this objective is that it has no fairness guarantees.
As such, the maximal sum of rewards
assignment might starve some users. In many applications,
the designer wants to make sure that all users will enjoy at least
minimal target Quality of Service (QoS).

Ensuring fairness has recently been recognized by the machine learning community as a problem of
key importance. In addition to interest in fair classifiers
\cite{zemel2013learning}, fairness has been recognized as a major design parameter
in reinforcement learning and single-player bandits as well \cite{jabbari2017fairness,wei2015mixed,joseph2016fairness}. Our work addresses this major concern for the emerging field of multi-player bandits.

In the context of maximizing the sum of rewards, many works on multi-player
bandits have considered a model where all players have the same vector
of expected rewards \cite{Rosenski2016,boursier2019sic,alatur2019multi,bubeck2019non}. While being relevant in
some applications this model is not rich enough to study fairness, as then the worst off player is simply the one that was allocated the worst resource.
To study fairness a heterogeneous model is necessary, where players have different expected rewards
for the arms (a matrix of expected rewards).
In this case, a fair allocation may prevent
some players from getting their best arm in order to significantly
improve the allocation for less fortunate players.

Despite being a widely-applied objective in the broader resource allocation literature  \cite{radunovic2007unified,zehavi2013weighted,asadpour2010approximation}, max-min fairness in multi-player bandits has yet to be studied. Some bandit works have studied alternative objectives
that can potentially exhibit some level of fairness  \cite{darak2019multi,bar2019individual}.
In the networking literature, a celebrated notion of fairness is \textit{$\alpha$-fairness} \cite{mo2000fair_alpha} where $\alpha=1$ yields proportional fairness and $\alpha=2$ yields sum of utilities. While for constant $\alpha$, $\alpha$-fairness can be maximized in a similar manner to \cite{bistritz2018distributed}, the case of max-min fairness corresponds to $\alpha\rightarrow \infty$ and is fundamentally different.

Learning to play the max-min fairness allocation involves major technical
challenges that do not arise in the case of maximizing the sum of
rewards (or in the case of $\alpha$- fairness). The sum-rewards
optimal allocation is unique for ``almost all'' scenarios (randomizing the expected rewards).
This is not the case with max-min fairness, as there will typically be multiple optimal allocations.
This complicates
the distributed learning process, since players will have to agree on a specific
optimal allocation to play, which is difficult to do without communication.
Specifically, this rules out using similar techniques to those used
in \cite{bistritz2018distributed} to solve the sum of rewards case
under a similar multi-player bandit setting (i.e., matrix of expected
rewards, no communication, and a collision model).

Trivially, any matching where all players achieve reward greater than $\gamma$
has max-min value of at least $\gamma$. We observe that finding these matchings, called $\gamma$-matchings in this paper,
can be done via simple dynamics that introduce an absorbing
Markov chain with the $\gamma$-matchings as the absorbing states.
This insight allows for a more robust algorithm than that of \cite{bistritz2018distributed}
which relies on ergodic Markov chains that have an exploration
parameter $\varepsilon$ that has to be tuned.

In this work we provide an algorithm
that has provably order optimal regret up to a $\log\log T$ factor
(that can be arbitrarily improved to any factor that increases with the horizon
$T$). We adopt the challenging model with heterogeneous
arms (a matrix of expected rewards) and no communication between the
players. The only information players receive regarding their peers is
through the collisions that occur when two or more players pick the
same arm.

\subsection{Outline}

In Section 2 we formulate
our multi-player bandit problem of learning the max-min matching under the collision model with no communication
between players. In Section 3 we present our distributed max-min fairness algorithm and state our regret bound (proved in Section 8).
Section 4 analyzes the exploration phase of our algorithm. Section
5 analyzes the matching phase of our algorithm and bounds the probability of the exploitation error event. Section 6 presents simulation
results that corroborate our theoretical findings and demonstrate that
our algorithm learns the max-min optimal matching faster than our analytical
bounds suggest. Finally, Section 7 concludes the paper.

\section{Problem Formulation}
We consider a stochastic game played by a set of $N$ players $\mathcal{N}=\left\{ 1,...,N\right\} $
over a finite time horizon $T$. The strategy
space of each player is the set of $M$ arms with indices denoted
by $i,j\in\{1,...,M\}$. We assume that $M\geq N$, since otherwise the
max-min utility is trivially zero. The horizon $T$ is not known by any of the players, and is considered to be much larger than $M$ and $N$ since we assume that the game is played for a long time.
Let $t$ be the discrete turn index.  At each turn $t$, all players
simultaneously pick one arm each. The arm that player $n$ chooses
at turn $t$ is $a_{n}\left(t\right)$ and the strategy profile (vector of arms selected) at
turn $t$ is $\boldsymbol{a}\left(t\right)$. Players do not know
which arms the other players chose, and need not even know the number of players $N$.

Define the no-collision indicator of arm $i$ in strategy profile
$\boldsymbol{a}$ to be
\begin{equation}
\eta_{i}\left(\boldsymbol{a}\right)=
\begin{cases}
0 & \Bigl|\mathcal{N}_{i}\left(\boldsymbol{a}\right)\Bigr|>1\\
1 & otherwise.
\end{cases}
\label{eq:1}
\end{equation}
where $\mathcal{N}_{i}\left(\boldsymbol{a}\right)=\left\{ n\,|\,a_{n}=i\right\} $
is the set of players that chose arm $i$ in strategy profile $\boldsymbol{a}$.
The instantaneous utility of player $n$ at time $t$ with strategy profile $\boldsymbol{a}\left(t\right)$ is
\begin{equation}
\upsilon_{n}\left(\boldsymbol{a}\left(t\right)\right)=r_{n,a_{n}\left(t\right)}\left(t\right)\eta_{a_{n}\left(t\right)}\left(\boldsymbol{a}\left(t\right)\right)\label{eq:2}
\end{equation}
where $r_{n,a_{n}\left(t\right)}\left(t\right)$ is a random reward
which is assumed to have a continuous distribution on $\left[0,1\right]$.
The sequence of rewards of arm $i$ for player $n$, $\left\{ r_{n,i}\left(t\right)\right\} _{t=1}^T$,
 is i.i.d. with expectation
$\mu_{n,i}$.

An immediate motivation for the collision model above is channel allocation in wireless networks, where the transmission of one user creates interference for other users on the same channel and causes their transmission to fail. Since coordinating a large number of devices in a centralized manner is infeasible, distributed channel allocation algorithms are desirable in practice. In this context, our distributed algorithm learns over time how to assign the channels (arms) such that the maximal QoS guarantee is maintained for all users. However, the collision model is relevant to many other resource allocation scenarios where the resources are discrete items that cannot be shared.

Next we define the total expected regret for this problem. This is the expected regret that the cooperating but non-communicating players accumulate over time from not playing the optimal max-min allocation.
\begin{defn}
The total expected regret is defined as
\begin{equation}
R\left(T\right)=\sum_{t=1}^{T} \left(\gamma^{*} - \underset{n}{\min}\ \mathbb{E}\left\{ \upsilon_{n}\left(\boldsymbol{a}\left(t\right)\right)\right\}\right)
\label{eq:3}
\end{equation}
where $\gamma^{*}=\underset{\boldsymbol{a}}{\max}\ \underset{n}{\min }\ \mathbb{E}\left\{ \upsilon_{n}\left(\boldsymbol{a}\right)\right\}$.
The expectation is over the randomness of the rewards $\left\{ r_{n,i}\left(t\right)\right\} _{t}$, that dictate the random actions $\left\{ a_{n}\left(t\right)\right\}_{t}$.

\end{defn}

Note that replacing the minimum in \eqref{eq:3} with a sum over the players yields the regret for the sum-reward objective case, after redefining $\gamma^{*}$ to be the optimal sum-reward \cite{bistritz2018distributed,Kalathil2014,Nayyar2016,tibrewal2019distributed,boursier2019sic}.

Rewards with a continuous distribution are
natural in many applications (e.g., SNR in wireless networks). However,
this assumption is only used to argue that since the probability for
zero reward in a non-collision is zero, players can properly estimate their expected rewards. In the case
where the probability of receiving zero reward is not zero, we can assume
instead that each player can observe their no-collision indicator in addition
to their reward. This alternative assumption requires no modifications to our algorithm or analysis.
Observing one bit of feedback signifying whether other players chose
the same arm is significantly less demanding than observing the actions of other players.

According to the seminal work in \cite{Lai1985}, the optimal regret
of the single-player case is $O\left(\log T\right)$.
The next proposition shows that $\Omega\left(\log T\right)$ is a lower bound for our multi-player bandit case, since any multi-player bandit algorithm can be used as a single-player algorithm by simulating other players.
\begin{prop}
\label{Prop Lower Bound}The total expected regret as defined in \eqref{eq:3} of any algorithm is at least $\Omega\left(\log T\right)$.
\end{prop}
\begin{proof}
For $N=1$, the result directly follows from \cite{Lai1985}. Assume that for $N>1$ there is a policy that results in total expected regret better than
$\varOmega\left(\log T\right)$. Then any single player, denoted player $n$,
can simulate $N-1$ other players such that all their expected rewards are larger than her maximal expected reward.
Player $n$ can also generate
the other players' random rewards, that are independent of the actual rewards she
receives. Player $n$ also simulates the policies for other players,
and even knows when a collision occurred for herself and can assign
zero reward in that case.
In this scenario, the expected reward of player $n$ is the minimal expected reward among the non-colliding players.
This implies that $\gamma^{*}$ is the largest expected reward of player $n$.
Hence, in every turn $t$ without a collision, the $t$-th term in \eqref{eq:3} is equal to the $t$-th term of the single-player regret of player $n$.
If there is a collision in turn $t$, then the $t$-th term in \eqref{eq:3} is $\gamma^{*}$, which bounds from above the $t$-th term of the single-player regret of player $n$.
Thus, the total expected regret upper bounds the single-player regret of player $n$.
Hence, simulating $N-1$ fictitious players
is a valid single player algorithm that violates the
$\varOmega\left(\log T\right)$ bound, which is a contradiction. We
conclude that the $\varOmega\left(\log T\right)$ bound is also valid for $N>1$.
\end{proof}

\section{My Fair Bandit Algorithm}

In this section we describe our distributed multi-player bandit algorithm
that achieves near order optimal regret for the max-min fairness problem.
The key idea behind our algorithm is a global search parameter $\gamma$
that all players track together (with no communication required).
We define a $\gamma$-matching, which is a matching of players to
arms such that the expected reward of each player is at least $\gamma$.
\begin{defn}
An allocation of arms $\boldsymbol{a}$ is a $\gamma$-matching if and
only if $\underset{n}{\min}\ \mathbb{E}\left\{ \upsilon_{n}\left(\boldsymbol{a}\right)\right\}\geq\gamma$.
\end{defn}

Essentially, the players want to find the maximal $\gamma$ for which
there still exists a $\gamma$-matching. However, even for a given achievable
$\gamma$, distributedly converging to a $\gamma$-matching is a
challenge. Players do not know their expected rewards, and their coordination is extremely limited. To address
these issues we divide the unknown horizon of $T$ turns into epochs, one starting
immediately after the other. Each epoch is further divided into four
phases. In the $k$-th epoch we have:
\begin{enumerate}
\item \textbf{Exploration Phase} - this phase has a length of $\lceil c_{1}\log (k+1)\rceil$
turns for some $c_{1}\ge4$. It is used for estimating the expectation of the arms. As shown in Section \ref{sec:Exploration}, the exploration phase contributes $O\left(\log\log T\log T\right)$
to the total expected regret.
\item \textbf{Matching Phase} - this phase has a length of $\lceil c_{2}\log (k+1) \rceil$
turns for some $c_{2}\ge1$. In this phase, players attempt to converge
to a $\gamma_{k}$-matching, where each player plays an arm that
is at least as good as $\gamma_{k}$, up to the confidence intervals
of the exploration phase.
To find the matching,
the players follow distributed dynamics that induce an absorbing Markov
chain with the strategy profiles as states. The absorbing states of this
chain are the desired matchings. When the matching phase is long enough,
the probability that a matching exists but is not found is small.
If a matching does not exist, the matching phase naturally does not converge. As shown in Section \ref{sec:Matching}, the matching phase adds $O\left(\log\log T \log T\right)$
to the total expected regret.
\item \textbf{Consensus Phase} - this phase has a length of $M$ turns. The
goal of this phase is to let all players know whether the matching
phase ended with a matching, using the collisions for signaling. During this phase, every player who did not end the matching phase with a collision plays their matched arm, while the players that ended the matching phase
with a collision sequentially play all the arms for the next $M$ turns. If players deduce that they collectively converged
to a matching, they note this for future reference in the matching
indicator $S_k$. If the matching phase succeeded, the search parameter is updated
as $\gamma_{k+1}=\gamma_{k}+\varepsilon_{k}$. The step size $\varepsilon_{k}$
 is decreasing such that if even a slightly better
matching exists, it will eventually be found. However, it might be
that no $\gamma_{k}$-matching exists. Hence once in a while, with
decreasing frequency, the players reset $\gamma_{k+1}=0$ in order to
allow themselves to keep finding new matchings. This phase adds $O(M\log T)$ to the total expected
regret.
\item \textbf{Exploitation Phase} - this phase has a length of $\left\lceil c_{3}\left(\frac{4}{3}\right)^{k} \right\rceil$
turns for some $c_{3}\geq1$. During this phase, players play the best recently found matching $\boldsymbol{\tilde{\boldsymbol{a}}}_{k^{*}}$, where $k^{*}$ is the epoch within the last $\frac{k}{2}$ epochs with the largest $\gamma$ that resulted in a matching.
This phase adds a vanishing term (with $T$) to the total expected
regret since players eventually play an optimal matching with exponentially small error probability.
\end{enumerate}
\vspace{-.3cm}
The Fair Bandit Algorithm is detailed in Algorithm \ref{alg:FairBandits}.
Our main result is given next, and is proved in Section 8.
\begin{thm}[Main Theorem]
\label{Main Theorem}Assume that the rewards $\left\{ r_{n,i}\left(t\right)\right\} _{t}$
are independent in $n$ and i.i.d. with $t$, with continuous distributions
on $\left[0,1\right]$ with expectations $\left\{ \mu_{n,i}\right\} $.
Let $T$ be the finite deterministic horizon of the game, which is unknown to the players. Let
each player play according to Algorithm \ref{alg:FairBandits} with
any constants $c_{1},c_{2},c_{3}\geq4$. Then the total expected regret satisfies
\begin{align*}
R\left(T\right)
&\leq C_{0} +\left(M+2\left(c_{1}+c_{2}\right)\log\log_{\frac{4}{3}}\frac{T}{c_{3}}\right)\log_{\frac{4}{3}}\frac{T}{c_{3}}\\
&=O\left(\left(M+\log\log T\right)\log T\right) \numberthis \label{eq:4}
\end{align*}
where $C_{0}$ is a constant independent of $T$ and the $\log\log T$ can be improved to any increasing function of $T$ by changing the lengths of the exploration and matching phases.
\end{thm}
\vspace{-.1cm}
The purpose of the epoch structure is to address the main challenge arising from having multiple players: coordinating between players without communication. To this end, the players in our algorithm try together to find  a $\gamma$-matching, where $\gamma$ is a mutual parameter that they can all update simultaneously but independently, obviating the need for a central entity. Then the main measure of multi-player ``problem hardness'' is the absorption time $\Bar{\tau}$ of the matching Markov chain (see Lemma \ref{lem:Convergence Lemma}), which is unknown. To find a matching we then need a matching phase with increasing length (to eventually surpass $\Bar{\tau}$), taken to be of length $\lceil c_2\log (k+1) \rceil$ for the $k$-th phase, which alone contributes $O(\log\log T \log T)$ to the total expected regret.  Hence, the coordination challenge dominates the regret.

The additive constant $C_0$ (see \eqref{eq:23}) is essentially the total regret accumulated during the initial epochs when the confidence intervals were still not small enough compared to the gap $\Delta=\underset{n}{\min}\ \underset{i\neq j}{\min}\left|\mu_{n,i}-\mu_{n,j}\right|$ (formalized in \eqref{eq:12-1}) or when the length of the matching phase was not long enough compared to $\Bar{\tau}$ (formalized in \eqref{eq:15}). Hence, $C_0$ depends on  $\Delta$  and $\Bar{\tau}$. In a simplified scenario when $\Delta$  and $\Bar{\tau}$ are known or can be bounded, the lengths of the exploration and matching phase can be made constant  ($c_1$ and $c_2$) and the confidence intervals in \eqref{eq:10} can be set to $\frac{\Delta}{4}$.  Note that with constant length phases, the $\log\frac{T_{e}\left(k\right)}{5M}$ in \eqref{eq:11} would be replaced with $\frac{M}{\Delta^2}$ and the  $\log (\frac{k}{2}+1)$ in \eqref{eq:15} replaced with a constant. Then $c_1,c_2$ can be chosen such that $C_0=0$, by making the probability in \eqref{eq:11} vanish faster than $\left( \frac{3}{4} \right)^k$ and satisfying \eqref{eq:15}. This amounts to choosing $c_1=O\left(\frac{M}{\Delta^2}+M^2\right)$ and $c_2=O(\Bar{\tau})$, which makes our regret bound in \eqref{eq:4} become $O\left((\frac{M}{\Delta^{2}}+M^2+\Bar{\tau})\log T\right)$, since the $\log\log T $ becomes 1 with constant length phases. Nevertheless, this issue is mainly theoretical since in practice, it is easy to choose large enough $c_1,c_2$ such that $C_0$ is very small across various experiments, as can be seen in our simulations in Section \ref{sec:Simulations}.
\vspace{-.2cm}
\section{Exploration Phase \label{sec:Exploration}}
\vspace{-.1cm}
Over time, players receive stochastic rewards from different arms and
average them to estimate their expected reward for each arm. In each
epoch, only $\lceil c_{1}\log (k+1) \rceil$ turns are dedicated to exploration. However,
the estimation of the expected rewards uses all the previous exploration phases, so the
number of samples for estimation at epoch $k$ is $\Theta(k \log k) $. Since players
only have estimates of the expected rewards, they can never be
sure if a matching is a $\gamma$-matching. The purpose of the exploration
phase is to help the players become more confident over time that the matchings
they converge to in the matching phase are indeed $\gamma$-matchings.
\input{algorithm.tex}
In our exploration phase each player picks an arm uniformly at random.
This type of exploration phase is common in various multi-player bandit
algorithms \cite{Rosenski2016,bistritz2018distributed}. However, the nature of what the players
are trying to estimate is different. With a sum of rewards objective, players just need to improve over time the accuracy of the estimation of the expected rewards. With max-min fairness each player needs to make a hard (binary) decision whether a certain arm has expected reward above or below $\gamma$. After the confidence intervals become small enough, if the estimations do fall within their confidence intervals, players can be confident about this hard decision. Under this success event, where the confidence intervals are small enough,
a matching $\boldsymbol{a}$ is a $\gamma$-matching if all players observe that $\mu_{n,a_n}^{k}\geq\gamma-C_{n,a_n}^{k}$. The next lemma bounds the probability that this success event
does not occur, so the estimation for epoch $k$
failed.
\begin{lem}[Exploration Error Probability]
\label{lem: exploration} Let $\left\{ \mu_{n,i}^{k}\right\} $ be
the estimated reward expectations using all the exploration phases
up to epoch $k$, with confidence intervals $\left\{ C_{n,i}^{k}\right\} $.
Define the minimal gap by
\begin{equation}
\Delta \triangleq \underset{n}{\min}\ \underset{i\neq j}{\min}\left|\mu_{n,i}-\mu_{n,j}\right|.\label{eq:5}
\end{equation}
Define the $k$-th exploration error event as
\begin{equation}
E_{e,k}=\left\{ \exists n,i\,\bigg|\,\left|\mu_{n,i}^{k}-\mu_{n,i}\right|\geq C_{n,i}^{k}\textnormal{ or }C_{n,i}^{k}\ensuremath{\geq}\frac{\Delta}{4}\right\} .\label{eq:6}
\end{equation}
Then for all $k>k_0$ for a large enough constant $k_0$ we have
\begin{equation}
\P\left(E_{e,k}\right)\leq 3NMe^{-\frac{c_{1}}{6}k}.\label{eq:7}
\end{equation}
\end{lem}
\begin{proof}
After the $k$-th exploration phase, the estimation of the expected
rewards is based on $T_{e}\left(k\right)$ samples, and
\begin{equation}
T_{e}\left(k\right)\geq c_{1}\sum_{i=1}^{k}\log (i+1)\geq c_{1}\frac{k}{2}\log\frac{k}{2}.\label{eq:8}
\end{equation}
Let $A_{n,i}\left(t\right)$ be the indicator that is equal to
one if only player $n$ chose arm $i$ at time $t$.  Define $V_{n,i}$,
as the number of visits of player $n$ to arm $i$ with no collision
up to time $t$, and $V_m=\underset{n,i}{\min}\ V_{n,i}$.
The exploration phase consists of uniform and independent arm
choices, so
$\P\left(A_{n,i}\left(t\right)=1\right)=\frac{1}{M}\left(1-\frac{1}{M}\right)^{N-1}$.\label{eq:12}
We show that each player pulls each arm many times without collisions. Formally:
\begin{align}
\P\bigg(V_m<\frac{T_{e}\left(k\right)}{5M}\bigg)&= \P\left(\bigcup_{i=1}^{M}\bigcup_{n=1}^{N}\left\{ V_{n,i}<\frac{T_{e}\left(k\right)}{5M}\right\} \right)\nonumber\\
&\underset{\left(a\right)}{\leq} NM\P\left(V_{1,1}<\frac{T_{e}\left(k\right)}{5M}\right)\nonumber\\
&\underset{\left(b\right)}{\leq} NMe^{-2\frac{1}{M^{2}}\left(\left(1-\frac{1}{M}\right)^{N-1}-\frac{1}{5}\right)^{2}T_{e}\left(k\right)}\nonumber\\
&\underset{\left(c\right)}{\leq}NMe^{-\frac{1}{18M^{2}}T_{e}\left(k\right)}
\label{eq:9}
\end{align}
where (a) is a union bound, (b) is Hoeffding's inequality for
Bernoulli random variables and (c) follows since $M\geq N$ and $\left(1-\frac{1}{M}\right)^{M-1}-\frac{1}{5}\geq e^{-1}-\frac{1}{5}>\frac{1}{6}$.
By Hoeffding's inequality for random variables \cite{hoeffding1994probability} on $\left[0,1\right]$
\begin{align*}
&\P\left(\bigcup_{n=1}^{N}\bigcup_{i=1}^{M}\left\{ \left|\mu_{n,i}^{k}-\mu_{n,i}\right|\geq C_{n,i}^{k}\,\right\} \bigg|\,\left\{ V_{n,i}\right\} \right)\\&\hspace{.5cm}\leq\sum_{n=1}^{N}\sum_{i=1}^{M}2e^{-2V_{n,i}\left(C_{n,i}^{k}\right)^{2}}\underset{(a)}{\leq}2NMe^{-2\frac{MV_{m}}{\log V_{m}}}\numberthis \label{eq:10}
\end{align*}
where (a) uses $C_{n,i}^{k}=\sqrt{\frac{M}{\log V_{n,i}}}$ and $V_m\ge 3$. Now note that for all $k>k_0$ for a sufficiently large $k_0$:
\begin{equation}
\sqrt{\frac{M}{{\log\left(\frac{T_{e}(k)}{5M}\right)}}}\leq\sqrt{\frac{M}{{\log\left(\frac{c_{1}\frac{k}{2}\log\frac{k}{2}}{5M}\right)}}}<\frac{\Delta}{4}\label{eq:12-1}
\end{equation}

and therefore $V_m\geq\frac{T_{e}\left(k\right)}{5M}$ implies $\underset{n,i}{\max}\ C_{n,i}^{k}<\frac{\Delta}{4}$. Hence, given \eqref{eq:12-1} the event $\bigcup_{n=1}^{N}\bigcup_{i=1}^{M}\left\{ \left|\mu_{n,i}^{k}-\mu_{n,i}\right|\geq C_{n,i}^{k}\,\right\}$ coincides with $E_{e,k}$, so for all $k>k_0$:
\begin{align*}
    \P&\left(E_{e,k}\bigg|\ V_{m}\geq\frac{T_{e}\left(k\right)}{5M}\right)\\
    &= \P\left(\bigcup_{n=1}^{N}\bigcup_{i=1}^{M}\left\{ \left|\mu_{n,i}^{k}-\mu_{n,i}\right|\geq C_{n,i}^{k}\,\right\} \bigg|\,V_{m}\geq\frac{T_{e}\left(k\right)}{5M}\right)\\
    &\underset{\left(a\right)}{\leq}2NMe^{-2\frac{\frac{T_{e}\left(k\right)}{5}}{\log\frac{T_{e}\left(k\right)}{5M}}}\numberthis\label{eq:12-2}
\end{align*}
where (a) uses the law of total probability with respect to $\{V_{n,i}\}$  with Bayes's rule on $\{V_{m}\geq\frac{T_{e}\left(k\right)}{5M}\}$, using the bound in \eqref{eq:10}.
We conclude that for all $k>k_0$:
\begin{align*}
\P\big(E_{e,k}\big)
&=\P\left(E_{e,k}|\ V_m < \frac{T_{e}\left(k\right)}{5M}\right) \P\left(V_m<\frac{T_{e}\left(k\right)}{5M}\right) \\
&\hspace{.4cm}+\P\left(E_{e,k}|\ V_m\geq\frac{T_{e}\left(k\right)}{5M}\right) \P\left(V_{m}\geq\frac{T_{e}\left(k\right)}{5M}\right)\\
&\leq\P\left(V_m<\frac{T_{e}\left(k\right)}{5M}\right)
+\P\left(E_{e,k}|\ V_m\geq\frac{T_{e}\left(k\right)}{5M}\right)\\
&\underset{(a)}{\leq} NMe^{-\frac{T_{e}\left(k\right)}{18M^{2}}} + 2NMe^{-\frac{2T_{e}\left(k\right)}{5\log\frac{T_{e}\left(k\right)}{5M}}}\numberthis
\label{eq:11}
\end{align*}
where (a) uses \eqref{eq:9} and \eqref{eq:12-2}. Finally, \eqref{eq:7} follows by using  \eqref{eq:8} in \eqref{eq:11} for a sufficiently large $k$.
\end{proof}
\section{Matching Phase\label{sec:Matching}}
In this section we analyze the matching phase, where the goal is to distributedly
find $\gamma$-matchings based on the estimated expected rewards from
the exploration phase. We conclude by upper bounding
the probability that an optimal  $\gamma^{*}$-matching
is not played during the exploitation phase. During the matching phase,
the rewards of the arms are ignored, as only the binary decision of
whether an arm is better or worse than $\gamma$ matters. These binary
decisions induce the following bipartite graph between the $N$ players and $M$ arms:
\begin{defn}
Let $G_k$ be the bipartite graph where edge $\left(n,i\right)$
exists if and only if $\mu_{n,i}^{k}\geq\gamma_{k}-C_{n,i}^{k}$.
\end{defn}
During the $k$-th matching phase, players follow our dynamics to switch arms in order to
find a $\gamma_{k}$-matching in $G_{k}$. These $\gamma_{k}$-matchings (up to confidence intervals)
are absorbing states in the sense that players stop switching arms
if they are all playing a $\gamma_{k}$-matching. The dynamics of
the players induce the following Markov
chain:
\begin{defn}
\label{Def:MatchingDynamics}Define $\mathcal{E}_{n}^{k}=\left\{ i\,|\,\mu_{n,i}^{k}\geq\gamma_{k}-C_{n,i}^{k}\right\} $.
The transition into $\boldsymbol{a}\left(t+1\right)$
is dictated by the transition of each player $n$:
\begin{enumerate}
\item If $\eta_{a_{n}\left(t\right)}\left(\boldsymbol{a}\left(t\right)\right)=1$
then $a_{n}\left(t+1\right)=a_{n}\left(t\right)$ with probability
1.
\item If $\eta_{a_{n}\left(t\right)}\left(\boldsymbol{a}\left(t\right)\right)=0$
then $a_{n}\left(t+1\right)=i$ with probability $\frac{1}{\left|\mathcal{E}_{n}^{k}\right|}$
for all $i\in\mathcal{E}_{n}^{k}$.
\end{enumerate}
\end{defn}
Note that the matchings in $G_{k}$ are $\gamma_{k}$-matchings only when the confidence intervals are small enough.
Next we prove that if a matching exists in $G_{k}$, then
the matching phase will find it with a probability
that goes to one. However, we do not need this probability
to converge to one, but simply to exceed a large enough constant.
\begin{lem}
\label{Lem:MatchingTrial} Let $\mathcal{G}_{N,M}$ be the set of all
bipartite graphs with $N$ left vertices and $M$ right vertices that
have a matching of size $N$. Define the random variable $\tau(G,\boldsymbol{a}(0))$
as the first time the process of Definition \ref{Def:MatchingDynamics}, $\{\boldsymbol{a}(t)\}$,
constitutes a matching of size $N$, starting from $\boldsymbol{a}(0)$. Define

\begin{equation}
\Bar{\tau}=\max_{G\in\mathcal{G}_{N,M}, \boldsymbol{a}(0)}\mathbb{E}\left\{ \tau\left(G,\boldsymbol{a}(0)\right)\right\}
\end{equation}
If $G_{k}$ permits a matching then the $k$-th matching phase converges
to a matching with probability $p\geq 1-\frac{\Bar{\tau}}{\lceil c_{2}\log (k+1)\rceil}$.
\end{lem}
\begin{proof}
We start by noting that the process $\boldsymbol{a}\left(t\right)$
that evolves according to the dynamics in Definition \ref{Def:MatchingDynamics}
is a Markov chain. This follows since all transitions are a function
of $\boldsymbol{a}\left(t\right)$ alone, with no dependence on $\boldsymbol{a}\left(t-1\right),...,\boldsymbol{a}\left(0\right)$
given $\boldsymbol{a}\left(t\right)$. Let $\mathcal{\mathcal{M}}$ be a matching in $G_{k}$.
Define $\Phi_{\mathcal{M}}\left(\boldsymbol{a}\right)$ to be the number of players
that are playing in $\boldsymbol{a}$ the arm they are matched to
in $\mathcal{M}$. Observe the process $\Phi_{\mathcal{M}}\left(\boldsymbol{a}\left(t\right)\right)$.
If there are no colliding players, then $\boldsymbol{a}\left(t\right)$
is a matching (potentially different from $\mathcal{M}$) and no player will ever change
their chosen arm. Otherwise, for every collision, at least one of the colliding
players is not playing their arm in $\mathcal{M}$. There is a positive probability
that this player will pick their arm in $\mathcal{M}$ at random and all other players will
stay with the same arm. Hence, if $\boldsymbol{a}\left(t\right)$
is not a matching, then there is a positive probability that $\Phi_{\mathcal{M}}\left(\boldsymbol{a}\left(t+1\right)\right)=\Phi_{\mathcal{M}}\left(\boldsymbol{a}\left(t\right)\right)+1$.
We conclude that every non-matching $\boldsymbol{a}$
has a positive probability path to a matching, making $\boldsymbol{a}\left(t\right)$ an absorbing
Markov chain with the matchings as the absorbing states. By Markov's
inequality
\begin{align*}
\P\big(\tau&\left(G_{k},\boldsymbol{a}(0)\right)\geq \lceil c_{2}\log (k+1)\rceil\big) \numberthis
\\
& \hspace{.5cm} \leq \frac{\mathbb{E}\left\{ \tau\left(G_{k},\boldsymbol{a}(0)\right)\right\} }{\lceil c_{2}\log (k+1)\rceil}
\leq\frac{\Bar{\tau}}{\lceil c_{2}\log (k+1)\rceil}. \qedhere
\end{align*}\label{eq:13}
\end{proof}
Intriguingly, the Bernoulli trials stemming from trying to find a matching in $\{G_\ell\}$ over consecutive epochs are \textit{dependent}, as after enough successes, there will no longer be a matching in $G_\ell$, yielding success probability 0. The next Lemma shows that Hoeffding's inequality for binomial random variables still applies as long as there are few enough successes, such that there is still a matching in $G_k$.

\begin{lem} \label{lem:baharavineq}
Consider a sequence of i.i.d. Bernoulli random variables $X_{1},\hdots,X_L$ with success probability p (or at least $p$ for each trial).
For $x < Lp$, consider $S_x = \sum_{i=1}^L X_i\mathds{1} \{ \sum_{j<i} X_j < x \}$. Then
\begin{equation}
    \P(S_x < x) \le e^{ -2L\left(p-\frac{x}{L}\right)^2}\label{eq:14}.
\end{equation}
\end{lem}

\begin{proof}

If $S_x=m<x$ then $\sum_{i=1}^L X_i < x$, as otherwise the indicators in $S_x$ of the first $x$ indices $i$ where $X_i=1$ will be active, and so $S_x \ge x$, contradicting $S_x=m<x$. Therefore

\begin{align*}
    \P(S_x<x) &\le \P\left(\sum_{i=1}^L X_i <x\right) \le e^{ -2L\left(p-\frac{x}{L}\right)^2}. \qedhere
\end{align*}
\end{proof}
We conclude this section by proving the main Lemma used to prove Theorem
\ref{Main Theorem}. The idea of the proof is to show that if the
past $\frac{k}{2}$ exploration phases succeeded, and enough matching
trials succeeded, then a $\gamma^{*}$-matching was found
within the last $\frac{k}{2}$ matching phases. This ensures that a $\gamma^{*}$-matching is
played during the $k$-th exploitation phase.
\begin{lem}[Exploitation Error Probability]
\label{lem:Convergence Lemma}Define the $k$-th exploitation error
event $E_{k}$ as the event where the actions $\boldsymbol{\tilde{a}}_{k^{*}}$
played in the $k$-th exploitation phase are not a $\gamma^{*}$-matching.
Let $k_0$ be large enough such that for all $k>k_0$
\begin{equation}
\varepsilon_{\lceil\frac{k}{2}\rceil}<\frac{\Delta}{4}\textnormal{ and   }1-\frac{\Bar{\tau}}{\lceil c_{2}\log (\frac{k}{2}+1)\rceil} - \frac{1 + \log k}{k/6}\ge \frac{3}{\sqrt{10}} .\label{eq:15}
\end{equation}
Then for all $k>k_0$ we have
\begin{equation}
\P\left(E_{k}\right)\leq 7NMe^{-\frac{c_{1}}{12}k}+e^{-\frac{3k}{10}}.\label{eq:16}
\end{equation}
\end{lem}
\begin{proof}
Define $E_{c,\ell}$ as the event where a matching existed in $G_{\ell}$ and was not found in the $\ell$-th matching phase. From Lemma \ref{Lem:MatchingTrial}
we know that if there is a matching in $G_{\ell}$, then the $\ell$-th
trial has success probability at least $1-\frac{\Bar{\tau}}{\lceil c_{2}\log (\ell+1)\rceil}$.

Next we bound from below the number of trials we have between resets in order to find a $\gamma^{*}$ matching. We define $k_{w}\geq\left\lceil\frac{k}{2}\right\rceil$
as the first epoch since $\left\lceil\frac{k}{2}\right\rceil$ where a reset occurred (so
$\gamma_{k_{w}}=0$). In the worst case the algorithm resets in epoch $\left\lceil\frac{k}{2}\right\rceil-1$. Even still, the algorithm will reset again no later than
$k_{w}\leq \left\lceil\frac{k}{2}\right\rceil-1+ \left\lceil\frac{\left\lceil\frac{k}{2}\right\rceil-1}{3}\right\rceil \leq\frac{2}{3}k$.
The subsequent reset will then happen at $k_{w+1}$, where $k_{w+1}\leq\frac{2}{3}k+\left\lceil\frac{2}{9}k\right\rceil\leq\left\lceil\frac{8}{9}k\right\rceil<k$ for $k>9$.
We conclude that during the past $\frac{k}{2}$ epochs, there exists at least one full period (from reset to reset) with length at least
$\frac{k}{6}$.
Recall the definition of the $\ell$-th exploration error event $E_{e,\ell}$ in \eqref{eq:6}. Define the event $A_k=\bigcap_{\ell=\left\lceil \frac{k}{2}\right\rceil }^{k}\bar{E}_{e,\ell}$
for which $\bar{A}_k=\bigcup_{\ell=\left\lceil \frac{k}{2}\right\rceil }^{k}E_{e,\ell}$.
We define $\beta_{k_w}$ as the number of successful trials needed after reset $w$ to reach $\gamma_k \geq \gamma^{*}-\frac{\Delta}{4}$. Note that $\beta_{k_w} \leq 1 + \log k$ since no more than $1 + \log k$ steps of size $\varepsilon_{k_w} = \frac{1}{1+ \log k}$ are needed.
Then for all $k>k_{0}$
\begin{align*}
\P(E_{k}\,|\,A_k)
&\underset{\left(a\right)}{\leq} \P\left(\gamma_{k}<\gamma^{*}-\frac{\Delta}{4}\,\bigg|\,A_k\right) \\
&\underset{\left(b\right)}{\leq}
\P\left(\sum_{\ell=k_{w}}^{k_{w+1}}\mathds{1}\left\{ \bar{E}_{c,\ell}\right\} <\beta_{k_w}\,\bigg|\,A_k\right) \\
&\underset{\left(c\right)}{\leq}
e^{-2\left(1-\frac{\Bar{\tau}}{\lceil c_{2}\log (\frac{k}{2}+1)\rceil}-\frac{1 +\log k}{k_{w+1} - k_w}\right)^{2}\left(k_{w+1}-k_{w}\right)}\\
&\underset{\left(d\right)}{\leq}e^{-\frac{3k}{10}} \numberthis \label{eq:17}
\end{align*}
where (a) follows since given $\bigcap_{\ell=\left\lceil \frac{k}{2}\right\rceil }^{k}\bar{E}_{e,\ell}$,
if $\gamma_{k}\geq\gamma^{*}-\frac{\Delta}{4}$ then a $\gamma^{*}$-matching
was found before the $k$-th exploitation phase and $E_{k}$ did not
occur. This follows since at the last success at $\ell\leq k$ we must have then had that for
all $n$
\begin{align*}
    \mu_{n,a_{n}}&\geq\mu_{n,a_{n}}^{\ell}-C_{n,a_{n}}^{\ell}\geq\gamma_{\ell}-2C_{n,a_{n}}^{\ell}\\
    &\geq \gamma^{*}-\frac{\Delta}{4}-\varepsilon_{k_w}-2C_{n,a_{n}}^{\ell}>\gamma^{*}-\Delta\numberthis \label{eq:18}
\end{align*}
which can only happen if $\mu_{n,a_{n}}\geq\gamma^{*}$. Inequality (b) in \eqref{eq:17} follows by noting that the probability that $\max_{\lceil \frac{k}{2}\rceil\le \ell \le k} \gamma_\ell < \gamma^{*}-\frac{\Delta}{4}$
with a constant step size (between resets) $\varepsilon_{k_w}$ implies fewer than $\left\lceil \frac{\gamma^{*}-\frac{\Delta}{4}}{\varepsilon_{k_w}}\right\rceil $
successful trials between $k_w$ and $k_{w+1}$.
Given $A_k$, in any trial $\ell \in [k_w,k_{w+1}]$ such that there have been fewer than $\left\lceil \frac{\gamma^{*}-\frac{\Delta}{4}}{\varepsilon_{k_w}}\right\rceil $ successes in $[k_w,\ell)$, at least one matching will exist in $G_\ell$ (an optimal matching $a_{n}^{*}$), since
\begin{equation}
\mu_{n,a_{n}^{*}}^{\ell}\geq\mu_{n,a_{n}^{*}}-C_{n,a_{n}^{*}}^{\ell}\geq\gamma^{*}-C_{n,a_{n}^{*}}^{\ell}\underset{(1)}{>}\gamma_{\ell}-C_{n,a_{n}^{*}}^{\ell}\label{eq:19}
\end{equation}
where (1) follows since for all $k>k_{0}$, $\varepsilon_{k_w}$ is sufficiently small such that $\gamma_{\ell}\leq\gamma^{*}-\frac{\Delta}{4}+\varepsilon_{k_w}<\gamma^{*}$.
Inequality (c) in \eqref{eq:17} follows from Lemma \ref{lem:baharavineq} with $p\triangleq1-\frac{\Bar{\tau}}{\lceil c_{2}\log (\frac{k}{2}+1)\rceil }$.
Inequality (d) follows from $k_{w+1}-k_{w}\geq\frac{k}{6}$ and \eqref{eq:15}.
Finally, \eqref{eq:16} is obtained by:
\begin{align*}
\P\left(E_{k}\right)&= \P\left(E_{k}\,|\bar{A}_{k}\right) \P\left(\bar{A}_{k}\right)+ \P\left(E_{k}\,|A_{k}\right) \P\left(A_{k}\right)\\
&\underset{\left(a\right)}\leq \bigg( 3NM\sum_{\ell=\left\lceil \frac{k}{2}\right\rceil }^{k}e^{-\frac{c_{1}}{6} \ell}\bigg)+e^{-\frac{3k}{10}}\\
&\underset{\left(b\right)}\leq 3NMe^{-\frac{c_{1}}{12}k} \left(\frac{1-e^{-\frac{c_{1}}{12}k}}{1-e^{-\frac{c_{1}}{6}}}\right)+e^{-\frac{3k}{10}}\\
&\leq 7NMe^{-\frac{c_{1}}{12}k}+e^{-\frac{3k}{10}}\numberthis \label{eq:21}
\end{align*}
where (a) is a union bound of $\bar{A}_k=\bigcup_{\ell=\left\lceil \frac{k}{2}\right\rceil }^{k}E_{e,\ell}$ using Lemma \ref{lem: exploration} together with \eqref{eq:17}, and (b) is a geometric sum. \qedhere
\end{proof}

\section{Numerical Simulations} \label{sec:Simulations}
We simulated two multi-armed bandit games with the following expected rewards matrices:
$$
 U_1=
\begin{bmatrix}
\frac{1}{2} & \frac{9}{10} & \frac{1}{10} & \frac{1}{4}\\[1pt]
\frac{1}{4} & \frac{1}{2} & \frac{1}{4} & \frac{1}{10}\\[1pt]
\frac{1}{10}  & \frac{1}{4} & \frac{1}{2} & \frac{1}{2} \\[1pt]
\frac{1}{10} & \frac{9}{10} & \frac{1}{4} & \frac{1}{2}
\end{bmatrix}
$$
$$
U_2=\begin{bmatrix}
\frac{9}{10} & \frac{2}{5} & \frac{4}{5} & \frac{1}{10} & \frac{3}{10} & \frac{1}{20} & \frac{1}{5} & \frac{1}{10} & \frac{3}{10} & \frac{1}{5}\\[1pt]
\frac{4}{10} & \frac{3}{10} & \frac{3}{10} & \frac{1}{10} & \frac{1}{5} & \frac{3}{10} & \frac{2}{5} & \frac{2}{5} & \frac{3}{10} & \frac{2}{5}\\[1pt]
\frac{1}{10} & \frac{1}{20} & \frac{1}{10} & \frac{2}{5} & \frac{1}{10} & \frac{1}{5} & \frac{9}{10} & \frac{3}{10} & \frac{2}{5} & \frac{1}{10}\\[1pt]
\frac{1}{20} & \frac{1}{10} & \frac{9}{10} & \frac{1}{5} & \frac{9}{10} & \frac{3}{4} & \frac{1}{10} & \frac{9}{10} & \frac{1}{4} & \frac{1}{20}\\[1pt]
\frac{4}{5} & \frac{3}{10} & \frac{1}{10} & \frac{7}{10} & \frac{1}{10} & \frac{2}{5} & \frac{1}{20} & \frac{1}{5} & \frac{3}{4} & \frac{1}{20}\\[1pt]
\frac{2}{5} & \frac{1}{20} & \frac{3}{10} & \frac{7}{10} & \frac{1}{20} & \frac{1}{10} & \frac{1}{4} & \frac{3}{4} & \frac{3}{5} & \frac{1}{20}\\[1pt]
\frac{9}{10} & \frac{3}{10} & \frac{3}{10} & \frac{4}{5} & \frac{1}{10} & \frac{1}{4} & \frac{7}{10} & \frac{1}{20} & \frac{1}{5} & \frac{3}{10} \\[1pt]
\frac{3}{10} & \frac{1}{10} & \frac{2}{5}& \frac{1}{4} & \frac{1}{20} & \frac{9}{10} & \frac{1}{4} & \frac{1}{10} & \frac{1}{20} & \frac{2}{5} \\[1pt]
\frac{4}{5} & \frac{3}{4} & \frac{1}{10} & \frac{1}{5} & \frac{2}{5} & \frac{1}{20} & \frac{3}{10}  & \frac{1}{5} & \frac{1}{10} & \frac{1}{4} \\[1pt]
\frac{2}{5} & \frac{2}{5} & \frac{9}{10} & \frac{7}{10} & \frac{1}{4} & \frac{1}{5} & \frac{1}{20} & \frac{1}{10} & \frac{2}{5} & \frac{1}{4} \\[1pt]
\end{bmatrix}.
$$
Given expected rewards $\left\{ \mu_{n,i}\right\}$, the
rewards are generated as $r_{n,i}\left(t\right)=\mu_{n,i}+z_{n,i}\left(t\right)$
where $\left\{ z_{n,i}\left(t\right)\right\} $ are independent and
uniformly distributed on $\left[-0.05,0.05\right]$ for each $n,i$. The chosen
parameters were $c_{1}=1000$ and $c_{2}=2000$ and $c_{3}=4000$ for all experiments, and are chosen to ensure that the additive constant $C_0$ is small (since $k_0$ is small), as the exploration and matching phases are long enough from the beginning.

At the beginning of the $k$-th matching phase, each player played her action from the last exploitation phase if it is in  $\mathcal{E}_{n}^{k}$, or a random action from $\mathcal{E}_{n}^{k}$ otherwise. Although it has no effect on the theoretical bounds, it improved the performance in practice significantly. Another practical improvement was achieved by introducing a factor of 0.01 to the confidence intervals, which requires larger $c_1$ but does not affect the analysis otherwise. The step size sequence, that is updated only on resets, was chosen as $\varepsilon_k=\frac{0.2}{1+\log k}$.

In Fig. \ref{fig:totalRegretN4}, we present the total expected regret
versus time, averaged over 100 realizations, with $U_1$ as the expected reward matrix  ($N=4$). The shaded area denotes one standard deviation around the mean. This scenario has 24 matchings - 16
with minimal expected reward  $\frac{1}{10}$, 7 with $\frac{1}{4}$, and one optimal
matching with $\frac{1}{2}$.  It can be seen that
in all 100 experiments the players learned the max-min optimal matching by the end of the third epoch. This suggests that $k_{0}$ is much smaller than our theoretical bound. As expected, the regret scales (approximately) logarithmically. For comparison, the optimal sum of expected rewards for $U_1$ is 2.15, but the matching that achieves it has a minimal expected reward of $\frac{1}{4}$. Hence, a multi-player bandit algorithm that optimizes the expected sum of rewards will have regret $\Omega\left(\frac{T}{4}\right)$.

In Fig. \ref{fig:totalRegretN5}, we present the total expected regret
versus time, averaged over 100 realizations, with $U_2$ as the expected reward matrix ($N=10$). The shaded area denotes one standard deviation around the mean. In this scenario only 136 matchings out of the $10!=3628800$ are optimal with minimal utility of 0.4. There are 3798 matchings with 0.3, 16180 with 0.25, 62066 with 0.2,  785048 with 0.1 and 2761572 with 0.05. With more players and arms, $k_{0}$ is larger, but players still learn the max-min optimal matching by the sixth epoch. Again, the regret scales  (approximately) logarithmically as guaranteed by Theorem 1. For comparison, the optimal sum of expected rewards for $U_2$ is 7.35, but the matching that achieves it has a minimal expected reward of 0.3. Hence, a multi-player bandit algorithm that optimizes the expected sum of rewards will have regret $\Omega\left(\frac{T}{10}\right)$.
\begin{figure}[tbh]
\centering
\includegraphics[width=7cm,height=7cm,keepaspectratio]{Figures/N4FinalShade.pdf}
\vspace{-.5cm}
\caption{Total regret as a function of time, averaged
over 100 experiments and with $U_1$ ($N=4$).\label{fig:totalRegretN4}}
\end{figure}
\begin{figure}[tbh]
\vspace{-.5cm}
\centering
\includegraphics[width=7cm,height=7cm,keepaspectratio]{Figures/N10Shade.pdf}
\vspace{-.5cm}
\caption{Total  regret as a function of time, averaged
over 100 experiments and with $U_2$ ($N=10$).\label{fig:totalRegretN5}}
\vspace{-.5cm}
\end{figure}
\section{Conclusions and Future Work}
We studied a multi-player multi-armed bandit game where players cooperate to learn how to allocate arms, thought of as resources, so as to maximize the minimal expected reward received by any player. To allow for a meaningful notion of fairness, we employed the heterogeneous model where arms can have different expected rewards for each player.
Our algorithm operates in the restrictive setting of bandit feedback, where each player only observes the reward for the arm she played and cannot observe the actions or rewards of other players.
We proposed a novel fully distributed algorithm that achieves a near order optimal total expected regret of $O\left(\log \log T \log T\right)$, where $\log \log T$ can be improved to any increasing function of $T$.

It is still an open question whether a total expected regret of $O(\log T)$ is achievable in our scenario, when the problem parameters are unknown. Following our discussion on the additive constant $C_0$, an algorithm that achieves  $O(\log T)$ even with unknown parameters will expose the multiplicative factors of the $\log T$ in the regret bound, and their dependence on $\Delta$ and $M$. These factors are likely to be strongly affected by the time it takes the algorithm to find a matching. If some limited communication
is allowed, more sophisticated algorithms to distributedly compute
the matching are possible, based on gossip, message passing, or auctions \cite{bayati2008max, naparstek2016expected}. These algorithms do not need a consensus phase, eliminating the $M$ factor in \eqref{eq:4} and reducing the convergence time $\bar{\tau}$, but it is unclear if these approaches can achieve $O(\log T)$ regret.
Focusing on our setting, an interesting question is whether one can design a better distributed matching algorithm that can operate with no communication between players.

In some applications one is interested in guaranteeing a target QoS for each user that is ``good enough'' (see \cite{lai1984asymptotically,katz2020true} for the single-player case). This is a weaker requirement than max-min fairness between users. Hence, an interesting open question is whether better regret bounds for our multi-player bandit scenario can be obtained in this case.

\section{Proof of Theorem 1}
Let $K$ be the number of epochs that start within the $T$ turns.
Since
\begin{align
T&\geq\sum_{k=1}^{K-1}\left(c_{1}\log k+c_{2}\log k+M+c_{3}\left(\frac{4}{3}\right)^{k}\right) \nonumber\\
&\geq3c_{3}\left(\left(\frac{4}{3}\right)^{K}-\frac{4}{3}\right)
\end{align}
$K$ is upper bounded by $K\leq\log_{\frac{4}{3}}\left(\frac{T}{3c_{3}}+\frac{4}{3}\right)$.
Let $k_{0}$ be a constant epoch index that is large enough for
the bounds of Lemma \ref{lem: exploration}, Lemma \ref{lem:Convergence Lemma}, and inequality (c) in \eqref{eq:22} to hold. Intuitively,
this is the epoch after which the matching phase duration is long
enough, the step size  $\varepsilon_{k}$ is small enough, and the confidence intervals are sufficiently tight. Define $E_{k}$
as the event where a $\gamma^{*}$-matching is not played in the
$k$-th exploitation phase. We now bound the total expected regret
of epoch $k>k_{0}$, denoted by ${R}_{k}$:
\begin{align*}
    {R}_{k}&\leq M + \left(c_{1}+c_{2}\right)\log (k+1)+\P\left(E_{k}\right)c_{3}\left(\frac{4}{3}\right)^{k} +3\nonumber\\
    & \underset{\left(a\right)}{\leq} M + (c_{1}+c_{2})\log (k+1)+3\nonumber\\
    &\hspace{1cm} +\left(7NMe^{-\frac{c_{1}}{12}k}+e^{-\frac{3k}{10}}\right)c_{3}\left(\frac{4}{3}\right)^{k}\nonumber\\
    &\underset{\left(b\right)}{\leq} M+3+(c_{1}+c_{2})\log (k+1)+8NMc_{3}\beta^{k}\nonumber\\
    &\underset{\left(c\right)}{\leq}M+2(c_{1}+c_{2})\log k \numberthis \label{eq:22}
\end{align*}
where (a) uses Lemma \ref{lem:Convergence Lemma}, (b) follows for
some constant $\beta<1$ since $e^{-\frac{3}{10}}<\frac{3}{4}$ and
$c_{1}\geq4$ and (c) follows for $k>k_{0}$. We conclude that, for some additive constant $C_0$,
\begin{align*}
R(T)&=\sum_{k=1}^{K}{R}_{k} \underset{\left(a\right)}{\leq}MK+2\sum_{k=k_{0}+1}^{K}\left(c_{1}+c_{2}\right)\log k\\&\hspace{.5cm}+\sum_{k=1}^{k_{0}}\left(\left(c_{1}+c_{2}\right)\log (k+1)+c_{3}\left(\frac{4}{3}\right)^{k}+3\right)
\\
&{\leq} C_0+MK+ 2\left(c_{1}+c_{2}\right)K\log K \numberthis\label{eq:23}
\end{align*}
where (a) follows by completing the last epoch to a full epoch
which only increases $R(T)$, and by using \eqref{eq:22}. Then, we obtain \eqref{eq:4} by upper bounding $K\leq\log_{\frac{4}{3}}\left(\frac{T}{3c_{3}}+\frac{4}{3}\right)\leq\log_{\frac{4}{3}}\left(\frac{T}{c_{3}}\right)$, where the second inequality is only used to simplify the expression, and holds for all $T\geq2c_3$.
\vspace{-.25cm}
\section*{Acknowledgements}
\vspace{-.25cm}
The authors gratefully acknowledge funding from the Koret Foundation grant for Smart Cities and Digital Living, NSF GRFP, Alcatel-Lucent Stanford Graduate Fellowship, and ISF grant 1644/18.

\bibliographystyle{icml2020.bst}

redpajama_set_name: RedPajamaArXiv


text:
  <md-toolbar color="primary">

      <md-icon>home</md-icon>

    GitHyde

    <gh-user-menu></gh-user-menu>
  </md-toolbar>

    <router-outlet></router-outlet>

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text:
Q: How I can delete lines which have less than 11 numbers but more than 8 numbers in one line in notepad++ How I can delete lines which have less than 11 numbers but more than 8 numbers in one line in notepad++. The numbers are separeted from each other with letters or spaces, etc.

A: Your requirement says to remove lines having 9 or 10 digits, but not more or less than this.  You may try using lookaheads to handle this.  In regex mode, try finding the following pattern:
^(?!.*\d.*\d.*\d.*\d.*\d.*\d.*\d.*\d.*\d.*\d.*\d)(?=.*\d.*\d.*\d.*\d.*\d.*\d.*\d.*\d.*\d).*

Then just replace that with empty string (nothing).  Follow the demo below to see that the pattern correctly flags the appropriate lines.
Demo
Edit:
Here is another pattern you may use, without lookaheads, which is a bit easier on the eyes:
^\D*\d\D*\d\D*\d\D*\d\D*\d\D*\d\D*\d\D*\d\D*\d\D*\d?\D*$

This again says to match any line which contains either 9 or 10 digits, but not more or less than this.

A: *

*Ctrl+H

*Find what: ^(?:\D*\d){8}(?:\D*\d){0,3}(?:\R|$)

*Replace with: LEAVE EMPTY

*check Wrap around

*check Regular expression

*Replace all
Explanation:
^                   # beginning of line
  (?:\D*\d){8}      # non capture group, 0 or more NON digit and 1 digit, may appear 8 times
  (?:\D*\d){0,3}    # non capture group, 0 or more NON digit and 1 digit, may appear 0 upto 3 times
(?:\R|$)            # non capture group, linebreak or end of file

Given:
1234567
12345678
123456789
1234567890
12345678901
123456789012
a1b2c3d4e5f6g7
a1b2c3d4e5f6g7h8
a1b2c3d4e5f6g7h8i9
a1b2c3d4e5f6g7h8i9j0k1l2

Result for given example:
1234567
123456789012
a1b2c3d4e5f6g7
a1b2c3d4e5f6g7h8i9j0k1l2

Screen capture:

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text:
Mutiny Reflections
Before 1857 and Beyond
The Bengal Presidency
The North Western Provinces
In the Mofussil – May 1857
Etah, Aligarh, Bulandshahr and Secunderabad
Etawah and Mynpoorie
Hansi, Hissar and Sirsa
Rohilkhand
Shajahanpur,Muhamdi and Sitapur
Benares, Jaunpore and Allahabad
Jhansi and Nowgong
Cawnpore
Fatehpur,Banda,Nagode and Mirzapore
Fyzabad, Sultanpore and Salon
Secrora. Gonda and Bahraich
The Lucknow Residency
Lucknow Defenders
The Punjab
Ferozepore,Mian Mir, Sialkote and Simla
Jalandhar, Phillour, Ludhiana
Central India June-September 1857
Jottings
Category: Secrora. Gonda and Bahraich
Posted on June 15, 2022 December 6, 2022 by Eva Chatterji
"You are at Liberty to Consult Your Own Safety" – Secrora
Our Station from "Curry and Rice on Forty Plates" – Capt. Geo.F.Atkinson, 1858
Although this is not Secrora, many stations in India followed the same pattern.
There was hardly to be found a more charming station in Oudh than Secrora. Close to the Nepal Terai, it was considered a sportsman's dream – tigers and wild elephants abounded, and the jheels teemed with waterbirds. The jungles boasted a healthy population of antelopes and deer, while on the banks of the river Gogra, pig-sticking was the past time of the day. For an energetic young officer like Lieutenant John Bonham and his equally sports-happy companion, Captain John Stewart Tulloh, there was no better place to be.
Pig-sticking from "Curry and Rice on Forty Plates" – Capt. Geo.F.Atkinson, 1858
They had served together in the Lahore where Tulloh had taken the newly arrived Bonham under his wing – and with his influence secured him one of the most coveted berths to be had, an appointment to the fledgling Oudh Irregulars.
Tulloh and Bonham arrived in Secrora in early 1857 at the head of a battery of Oudh Irregular Artillery. Secrora already had a compliment of Irregulars – Daly's Horse (1st Oudh Irregular Cavalry) and 2nd Oudh Irregular Infantry. By the time Bonham arrived in Secrora however, Daly was well on his way to the Punjab. Taking his place was Captain Hamilton Forbes, while Captain G.W.Boileau commanded the infantry. An officer of considerable standing, Boileau commanded all the troops at the station, by virtue of seniority.
Several ladies had accompanied their husbands to Secrora and they were busy making this new station a happy community. On the 30th of January, 1857, Mrs. Daly wrote,
"Mr Jackson, (Colville Jackson) the acting Chief Commissioner, is coming- here in a few days with his two pretty nieces; there are five ladies in this station and two at Gonda,
15 miles off; it is decided that there are enough to get up a dance, and we are all highly busy now planning- how it shall be." Two years later, after her return to England, she noted: —" I remember well how gay and merry we were preparing for that party! The kutcherry, or court-room, which was to be the ball-room, being whitewashed and decorated; the stands of arms, banners, and garlands of flowers that adorned the walls, the tents to be pitched, and the supper devised and cooked…But then we knew nothing of coming horrors. All was gay and everyone cheerful, and the little ball, got up in a rude outstation, gave more pleasure than many a grand entertainment."
"Our Ball" from "Curry and Rice on Forty Plates" – Capt. Geo.F.Atkinson, 1858)
The coming horrors Mrs. Daly speaks of are of the deaths of Georgina Jackson, Mr. Loungueville Clarke, Mr. Bax and Mr. Charles Boileau – all who attended the ball that night, dancing so gaily. Mr. Boileau was the "favoured lover" of poor Georgina and his death would be the first shock the little station would receive. That came shortly after the ball.
Dacoits and Mr. Charles Boileau, March 1857
Family group of Dacoits (Robbers). India, ca. 1900, Image by © Bettmann/CORBIS
Dacoity was a serious problem in India, so much so that the EICo established the Thuggee and Dacoity Department in 1830 and then went even further to implement the Thuggee and Dacoity Suppression Acts of 1836-1848.
Dacoits were (and still are) bands of armed robbers or bandits and were even defined as "A member of a class of robbers in India and Burma who plunder in armed bands." This extended to include pirates who plied their trade on the Ganges, robbing and plundering boats. What pushed people to take up arms in such a manner is disputed, however, feudal exploitation which left people with little choice but to resort to banditry has been suggested as one factor, though as some of the gangs were also made up of high caste and wealthy people there are links in dacoity to tradition.
Thuggee is not the same as dacoity. Though the end goal was the same – robbery – thuggee was a practice from North and Central India involving not just robbery but ritualized murder and mutilation and could be more correctly defined as a cult. Colonel William Sleeman was very energetic in stamping out thuggee but the department which he worked for would continue in existence until 1904.
Anonymous Indian artist. Made for Capt. James Paton, Assistant to the British Resident at Lucknow, 1829-1840
Below is an idea of how dacoity affected just Bengal in the years 1852-1853. The various acts used to deal with it did not appear to be making much of a dent in the numbers. However, the punishment awaiting dacoits if caught might have caused many of them to raise their levels of violence as clemency could not be expected under the suppression acts.
Statement showing the number of Dacoities and attempts to commit Dacoity, ascertained to have
been committed in the Lower Provinces, during the years 1852 and 1853
In the vicinity of Secrora shortly before Bonham's arrival a band of dacoits, headed by one Fazl Ali, had started plying their trade in the district. Previously he had confined his depredations to the Lucknow area in defiance of the King of Oudh but following the annexation of the province, "His headquarters for years has been along the Tulsipur and Nandparah forest, and when pressed he has taken to Nepal. The Rani of Nandparah was constantly at war with the Raja of Tulsipur. Fazl Ali was occasionally on one side, occasionally on the other; but, wherever he was, that side had the prestige of success. After the annexation, he went into Nepal, but of late has returned to our territory, and some horridly bloody murders have been committed by him…"
Besides murder Fazl Ali had been plundering police posts and villages; his very presence was causing havoc in the district, and orders were sent out from Lucknow to apprehend the man.
Two men took it on themselves to put an end to Fazl Ali. Having been sighted close to Gonda or thereabouts, Mr. Lounguville Clarke took to the field with a company of the 10th Irregular Infantry stationed to find Fazl Ali. As his band was constantly on the move and the local population much disinclined to give any information as to their whereabouts, it took some time to locate the elusive dacoits. In the meantime, quite of his own volition, Mr. Charles Elliot Boileau, the Deputy Commissioner of Gonda, acting on his own intelligence, took some mounted police with him and set off in pursuit. He had written Henry Daly, first, however, informing him he was taking with him an escort of six men and one jemadar- Ram Singh – to find Fazl Ali – hardly enough men to engage a gang of violent dacoits. It would almost appear that Boileau was not confident he would actually find them, after all, no one else had until then.
Arriving at the village of Tulsipur, Boileau posted a few men sentry at the different outlets although Jemadar Ram Singh entreated Boileau to make a rush at the village – a surprise attack – but Boileau declined and entered the village with only Jemadar Ram Singh by his side. Much to their surprise in the village, they actually found Fazl Ali. What happened next is not clear- it would appear the 2 men tried to convince the dacoit they were a sizable force and he should surrender, while Fazl Singh himself made a show of asking for mercy, but pleasantries were short.
Fazl Ali suddenly shot Boileau in the chest and then in the ensuing panic, he took the opportunity to cut off Boileau's head and hang it from a tree. The killing of Boileau was described as such:
"Ram Singh…alone remained with him. Whether he meant to wait till Lieutenant Clarke, known to be near, came up with his party, or what, none knew…suddenly a volley was fired …Mr Boileau fell from his mare exclaiming, ' Oh Ram Singh! I am killed.' Another Sikh, Bhugwan Singh, came up; together they lifted the Sahib's body onto his horse; it fell off. Bhugwan Singh appears to have gone away. Ram Singh went in search of the other sowars to endeavour to persuade them to bring off poor Boileau's body; he could not induce them to venture. The firing was close and heavy, the Sahib dead, and they had but their swords. Still Henry is angry, and says they shall leave the regiment for deserting the body. Ram Singh returned alone and stood by the Sahib's body till he had four bullets in himself and his clothes; one in the wrist disabled him, but he made his escape. He went to Ghulam Mohi-ud-din, a resaldar who was at a village with a detachment not far off; they returned, but could not find the body, and Fazl Ali had gone off. Afterwards the body was found and sent into Lucknow for burial."
Henry Daly was not having it. He immediately called together 60 sowars of the 1st Oudh Irregulars and went off after Fazl Ali, joining up with Longueville Clarke.
"On our arrival at the scene of his depredations scarcely a word of information could we procure. However, with some sixty sowars I made a forced march to the end of the first ridge of hills where the Raptee jerks round the Valley of Nepal to flow into our territory. Here the hills sink into the plain, which is a dense forest. Goolerie, marked in the map, is a small space cut in the woods where the cow-herds bring their cattle to graze ; here we found clear and recent traces of the fugitive, but the same silence amongst the people; not from disaffection to me, but from the awe of him and the fear of his revenge. This was his old haunt; here there was no food for man or beast, but what we had brought with us. I attempted to cut him off. I dismounted thirty Sikhs and Hillmen, and sent them up the pass which leads us into Nepal. These men took with them their swords and carbines, and a little flour; their trip was vain. We marched to the Nepal border, and had a communication with their posts, very friendly. We have since this captured one of his followers, who tells us that, at Goolerie, we were within bowshot of Fazl Ali, who, until our arrival, did not know of the pursuit…"
After a "long and exciting chase", Fazl Ali and his band were run down but they did not go quietly. After an intense fight in which three sepoys were killed, Clarke and 5 of his men were wounded. Fazl Ali and most of his band were killed.
This was the work the Oudh Irregulars were meant for and they had shown themselves a hardy force. Daly could leave Secrora, confident his troops were worthy.
Unfortunately for Charles Boileau, although recognised in official reports as a young and promising civil servant full of dash and daring, he had lost his life due to a "misguided sense of duty" and "fell victim to his own unheeding gallantry." It was after all the duty the Oudh Irregulars were supposed to do and not a civilian, no matter how well-intentioned.
It was a lesson perhaps one civilian took to heart a little too earnestly, as we shall see.
Should a Mutiny Break Out – June 8th-10th, 1857
Up until the end of May, Secrora remained quiet. There was little to report; Tulloh had been sent to the hills on sick leave, so Bonham now commanded the artillery. The problems started when the first outbreak took place in Lucknow and was particularly noticeable among the men of G.W.Boileau's regiment, the 2nd Oudh Irregular Infantry.
Dissension took the familiar pattern – random acts of unaccountable incendiarism which was confined to the burning down of huts belonging to a regiment absent on detachment duty and the not-so-secretive, late-night meetings.
News of other uprisings continued to come in – but the horrifying news of the killings of Mr. Christian at Sitapore finally convinced the men it was time to send off the ladies and children to Lucknow. A message was dispatched to Sir Henry Lawrence of the intention.
Captain Forbes, who at the time was away at Lucknow on patrol duty was ordered to return to Secrora by Sir Henry and bring to Lucknow all the ladies and children in the Division. Another 4 civilians volunteered for the task of escorting the party. The officer's families were hastily summoned from nearby Gonda and on the 7th of June, seven women, four officers and 12 children along with the four volunteers left Secrora. They arrived in Lucknow safely on the 9th of June only to soon find themselves shut in at the Lucknow Residency during the siege.
With the women and children out of the way, Lieutenant Bonham and the others had to decide how best to protect themselves. "There was, indeed, very little to be done, but we determined to hold together as much as possible, and keeping a close watch on Boileau's men, to be prepared for flight should the necessity arise." There were however only 5 European officers left in Secrora now – the Commissioner Mr. Wingfield, Captain Boileau and his adjutant Lieutenant Hale, Assistant-Surgeon Kendell and Lieutenant Bonham. Boileau and Bonham, in order to be close their the lines opted to reside in Mr. Wingfield's house. Two staff sergeants from Bonham's regiment, and Bewsey and Miller and Sergeant Major Court from Boileau's were tasked with keeping watch during the night, to give the others timely notice should a disturbance occur. Hale and Kendell remained across the road in their own house.
Around midnight on the 8th of June Boileau and the others were awakened by the Sergeant on sentry duty, "rushing into the house, shouting that the Infantry had risen and were advancing on the bungalow…" With no time to lose, Boileau, Bonham and the Sergeant repaired with haste to the veranda – the Sergeant said he had seen sepoys sneaking up under cover of the low mud wall that surrounded Wingfield's house. Although they could not see anything, they could hear shouting in the Infantry Lines, convincing Bonham that his place was with his own men. Telling Boileau of his intentions, Bonham went with all speed down to the Battery, joined moments later by Boileau, Wingfield and the three sergeants. Bonham found the artillery at least were still manageable.
"My men behaved splendidly. They turned out at once, and taking the guns from the gun park, and running them out by hand, soon had them in position, bearing upon the Infantry lines. It was a bright moonlit night, and after some time, the disturbance having quieted down, I sent a mounted man down to reconnoitre, and when he returned and told us it was all quiet in that direction, we parked the guns and returned to the bungalow, thinking it must have been a false alarm."
Whatever the case may have been, it brought matters to a head.
The following morning, a few infantry sepoys came up to Wingfield's bungalow, demanding to see their commander, Captain Boileau,
"…and after taking him to task for going to the Artillery lines when he ought to have gone to his own men for protection, said that the guns had been brought out against them and that they had only escaped being murdered in their sleep by the refusal of the gunners to fire upon them."
In other words, the infantry believed wholeheartedly their own European officers had been plotting to murder them in their beds. Boileau had betrayed their confidence and as such, they no longer saw any reason to give loyalty when they were so clearly mistrusted by their own commander. It was a severe reprimand for Boileau – he tried to explain matters to the men but soon found his authority was lost. Finally, the sepoys dictated their own terms, demanding Boileau call a parade and hand out the contents of the magazine. It was a brash request and it worked. Boileau, perhaps hoping to save a little face and seeing it would be dangerous at this point to argue further, agreed.
A parade was ordered for that very evening, with Boileau and Hale in attendance but when they reached the lines they found the magazine had already been broken open and the men had taken possession of the ammunition.
After seeing to stable duties in his lines which had remained quiet, Bonham joined the others for dinner at Mr. Wingfield's house. Finding Boileau, Hale and Kendell sitting outside, he noticed suddenly that their host, Mr. Wingfield was missing. He had gone out for his evening ride and thinking him delayed, the men waited for him to return. Finally, they went in to dinner sans Wingfield and had barely started eating when a note arrived. "It was merely a few lines in pencil, to the effect that he did not like the state of affairs at Secrora, and seeing that Boileau's regiment was on the verge of mutiny, had gone out for his usual evening ride, and was on his way to Gonda…." Mr. Wingfield had left, without a word of warning to anyone, leaving everything standing as it was in his house.
Wingfield's own explanation, or rather excuse was no less shamefaced.
"Some old servants who had been with me ever since I had been in India, had that day and the previous one told me that some of them had been warned to quit me, or they might lose their lives; and now Captain Boileau came and told me he no longer commanded the troops, and that he was going on parade in compliance with the intention he had expressed to that effect, but did not expect to leave it alive." Frightened by the night's events, and by Boileau's statement, Wingfield turns to justification.
"So evident was it for some time past that the troops were fast hurrying into revolt, that I would have left Secrora, which was not a civil station, or my legitimate place or residence, for Gonda before, had not Captain Boileau urged me to remain, alledging my departure would shew want of confidence in them.." Mr. Wingfield should have been residing in Bayretch but a problem with housing had sent him to Secrora, so as he said, it was not his legitimate station, but that was hardly a good enough excuse.
"I now say that my remaining any longer would be imperilling my own life, and therefore taking the advantage of the habit of an evening ride mounted my horse, and rode over to Gonda…Sir Henry Lawrence had previously written to Captain Boileau and myself in these words – 'Should a mutiny break out or appear inevitable, you are liberty to consult your own safety.." In Wingfield's mind at least, the troops had thrown off their loyalty and the question was really, how long it would be before they murdered their officers. Wingfield for one was not going to wait and find out.
Chulye, Sahib!
As it was Hale and Boileau had both returned from the parade. The men had finally refused to turn in and after grossly insulting their officers, finished helping themselves to the contents of the magazine but besides behaving badly, left their officers standing, scared but alive. With this last act, the 2nd Oudh Irregular Infantry ceased to exist.
Demoralised, Hale and Boileau returned to Wingfield's house, to find the Commissioner had bolted.
Captain Boileau wasted no time in making his own feelings clear. With Wingfield gone, there was little point in them staying. He produced the letter from Sir Henry (already alluded to in Wingfield's excuse) and he considered it was time to take advantage of the permission the letter gave. Bonham, however, baulked. His men had behaved well and until now he could see no reason to abandon them and more importantly, the guns. He would stay in Secrora intending the next morning to march the battery to Lucknow.
While the discussions were going on, the Havildar Major of the Infantry came to the bungalow, wishing to see Boileau. Boileau stepped outside to meet him.
The Havildar informed Boileau that it had become known abroad in the lines that the officers were planning their escape; the infantry had no intention of letting them go. He then told Boileau to send a company of the infantry to Byram Ghat to prevent the officers from crossing, should there be any attempt made to get to Lucknow. Boileau conceded and ordered the company to march to Byram Ghat, assuring the Havildar that he would join them there.
Captain Boileau had already made up his mind though to follow Wingfield to Gonda, the opposite direction to Lucknow and then make his way to Bulrampore where the raja had promised him shelter. Arrangments were quickly made and that night, Boileau, Kendell and Hale left Secrora leaving Bonham and the three sergeants behind.
Bonham had already arranged to spend the night in the lines with his men. From the very beginning, he had seen the only way to keep any hold over them was quite simply, to trust them and as such, he spent his time now mixing with his men as much as possible and doing all he could to prevent them from communicating with the Infantry and Cavalry. Subadar Sheik Murdan Ali helped Bonham in this regard, a man Bonham had implicit confidence in. "He gave me the most trustworthy information of what was going on, not only in the Battery but also in the other regiments. I always had the highest opinion of him, and he remained perfectly faithful and true to the end."
In the artillery lines, Bonham found "a good spirit still prevailed." The guns were taken out and placed to cover the flanks of the stables and to command the road from the Infantry lines. Thus reassured, Bonham went to bed in his tent, telling the Subadar to call him if there was any trouble during the night.
In the morning Bonham awoke and to his surprise, the first thing he saw was the guns had been withdrawn and returned to the gun park. Upon asking what was the meaning of this, the Subadar informed the lieutenant that during the night the Artillery and Infantry soldiers had met and decided that "they would all act together – whatever one did, the other would do – which meant of course, that they were going to mutiny."
His hopes of saving his regiment and indeed the guns were at least for now, dashed. Hardly knowing what to do next, Bonham decided the best thing to do was carry on as usual. He went to the stables and found the stable duties were being carried on as usual. As he walked up and down the lines thinking of what he should do next, Bonham saw a group of men approaching. He sent the Subadar to find out who they were, and above all, what they wanted. They turned out to be men of Boileau's regiment and they wished an audience with Bonham.
"They were quite respectful in their manner, and after saluting me, began a tale of the manner in which they had been treated. They seemed very despondent and complained bitterly of having been deserted by their officers. The misbehaviour of the men, they said, was confined to a few budmashes (scoundrels), who, for their own purposes, wished to lead the others astray. They assured me that the majority of the regiment was perfectly loyal, and as they had now only me to look to, begged me to take command of the regiment."
Thinking perhaps that there was some truth to what they were saying, considering Boileau and Hale's behaviour, Bonham decided to try his luck. He told the men he was planning to take the guns to Lucknow; if any of the infantry wanted to prove their loyalty, that was their chance.
"They seemed greatly pleased at this, and assuring me they would do anything I ordered them, they asked me to go with them to their lines and tell the men, myself, I had taken command of them.
This was indeed coming to the point, but it seemed reasonable enough, and little as I liked the job, I at once consented. I had no sooner said the words than with wild shouts, of "Chulye, Sahib!" (Come on, Sahib!) they carried me off to their lines."
"A large crowd soon gathered round me. I told them I had at the request of the native officers to take command of the regiment, that I was going to march the battery to Lucknow at twelve o'clock, and was asking any of those who wished to go with me to come forward. They cried, "We'll all go! We'll all go!" They would go they said, to Delhi and fight the budmashes of mutineers, and wherever I went they would be with me, ar as they put it, they would be 'a full pace in front of me.'"
Confident now that if the Infantry marched, he would have no problems with his own men, Bonham prepared to march.
Or so he thought.
Just then the Subadar came up to Bonham to report that a large quantity of Government treasure had just arrived in Secrora. Bonham followed the usual procedure, escorting the treasure to the Infantry lines and handed it over in the normal way to the jemadar of the guard. Yet something happened and all of a sudden the mood changed and the very men who had but moments before professed their loyalty now crowded around the officer and began to abuse and threaten him.
Bonham quickly retreated back to the Battery and told the Subadar what had happened. Alarmed, the Subadar told Bonham he would go and see what the commotion was, and if nothing else, "his life would be taken before mine…"
His next shock came from the Quartermaster Sergeant of the 2nd Oudh Irregular Infantry. Hearing that the man was still in his house in the infantry lines, Bonham sent word that he had better repair to the Battery. The obstinant man came up to the line but refused to heed Bonham's arguments; his wife, the man said, was a local and besides he had a large quantity of property in his house that he refused to leave behind. In any case, seeing as he was married to an Indian woman, he had nothing to fear. He left Bonham to consult with his wife and was never seen again, killed by his own men, becoming the only British victim of the mutiny at Secrora.
Too hot to sit any longer in the tents, Bonham, Court, Bewsey and Miller found a little refuge under a large tree close to the battery quarter guard. As fate would have it on duty that day was Havildar Jehangir Bux, a man who had come over from the regular army and one in whom Bonham had no faith at all, "an ill-conditioned fellow…one of the worst men in Battery." While keeping an eye on the Havildar, the four men heard "the rattle of arms…and the ring of the ramrods as they drove home the bullets…told us they were loading their carabines…"With the Subadar still missing, Bonham decided to go down to the lines himself to find out what was going on.
There was no one there. In the stillness, Bonham guessed the men were in their huts but the whole place was deadly silent as if everyone was waiting for something to happen. He hurried back to the Battery.
He had barely arrived when the Assembly was sounded and the sepoys suddenly hurried down from their lines, collecting in excited groups between the stables and gun park. The guard marched in file to guns and stood at the ready. Bonham ran out amongst them, asking what had happened and why had they all turned out, to which the reply came, "We cannot tell, the bugle has sounded."
"Almost immediately a shout arose; "Tulungar ata hai! Tulungar ata hai!" (The Infantry are coming); and true enough we saw Boileau's regiment coming up in formation towards the Battery."
Determined to see the business through, no matter transpired, Bonham and the sergeants mounted their horses – which Bonham had had the sense to keep at the ready – and the four men rode out to meet the Infantry, shouting at them to not approach the Battery. Although it was rapidly becoming clear no one was listening to Bonham anymore, he still hoped at least some of his men would stay true.
Carrying spikes with which Bonham and the sergeants had intended to use to disable at least some of the guns and if nothing else to defend themselves with, Bonham returned to his guns. The guard turned on their lieutenant, threatening him with their carbines to stay away. A few men circled around Bonham and the sergeants, begging them to go. Seeing there was nothing for it, they left, the same men escorting them through the Battery.
As Bonham left, the Infantry came up and took possession of the guns he had tried so hard to save. The Subadar remained "true to the last" but as they were leaving the men from the Battery shouted for him to come back; Bonham ordered him to return to avoid any further trouble. He told him to escape when the opportunity presented itself and come to Lucknow. Sorrowful, the Subadar had some last words for Bonham, "Don't go to Bayram Ghat."
"We went off with what little dignity remained to us at a foot's pace, and had gone scarcely more than half a mile when we were overtaken by four of my men, who came with us and proved most useful on the way."
The question, however, was where to go?
Selections from the Records of the Bengal Government N° XVIII, Correspondence Relating to the Suppression of Dacoity in Bengal – Thos.Jones, Calcutta Gazette Office (1854)
Annals of the Indian Rebellion – Noah Alfred Chick (1859)
Kaye's and Malleson's History of the Indian Mutiny of 1857-58 Vol III Cabinet Edition, edited by Col. G.B.Malleson (1889)
Memoirs of General Sir Henry Dermot Daly – Major H. Daly, (1905)
Oudh in 1857. Some Memories of the Indian Mutiny – Colonel John Bonham (1924)
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redpajama_set_name: RedPajamaCommonCrawl


MINSK, 15 April (BelTA) – The reserve fund of the Belarusian president will provide financing to host 415 Syrian children at the national children's education and recuperation center Zubryonok in May-June 2019. Belarus President Aleksandr Lukashenko signed a corresponding executive order on 12 April, BelTA learned from the press service of the Belarusian leader.
The program of stay includes recreation and recuperation services, excursions, cultural, sports and mass events in Minsk and Belarusian regions.
Aleksandr Lukashenko also signed an executive order to finance the participation of 43 junior athletes from Syria in joint training camps at the Belarusian Olympic reserve school in April 2019. The program of stay includes sports and recuperation events, excursions and cultural events in Belarusian towns.
Belarus welcomes Syrian children as part of agreements to promote humanitarian cooperation between the two countries and in order to enhance Belarus' positive image in the world.


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> Wednesday December 14th is a night to remember for the WH Panther Wrestling team. Not only did they beat Quincy high school 34-32 in a very exciting match, they also achieved a milestone 500th victory in the process.
> But that wasn't all that happened tonight as the Panthers also unveiled a State Champions banner with four state champions in attendance. These included the first ever state champion for WH in Steve Wright, followed by Kevin Broderick, Nathan and Ben Dineen.
> Leading things off for the Panthers was Ben Cordingley with a pin in 4:06 in the 132 pound weight class. Joel McInnis followed with a tough 15-7 loss at 138 pounds. After several forfeits senior co-captain Jake Filicchia won a hard fought 13-4 victory over a quality Quincy wrestler winning by a score of 13-4. The match score at this point was WH 34 Quincy 22. Finishing things off for the Panthers was 120 pound freshmen Matt Rock. Matt fought extremely hard against a superior wrestler and loss his match by a 10-2 score enabling Whitman-Hanson to hold on for the final score of WH 34 Quincy 32. Coach "Rabbi" was extremely proud of all of his wrestlers tonight as they showed true grit and perseverance in every match. The Panthers are currently 2-2 on the young season and travel to Marshfield on Saturday for a Super Quad starting at 9:30am.
> Group picture of Whitman-Hanson Panthers after winning their 500 team victory Wednesday December 14, 2017.


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Dennis Kay Murray
Dennis Kay Murray of Wellsville Utah died Thursday February 27, 2020 in Logan, Utah. He died from complications of Pulmonary Fibrosis he battled for the last 2 years. Born on the 10th of November 1948 to Heber Kay & Mardene Murray. Dennis grew... View Obituary & Service Information
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