text:
\section{Introduction}
As one of the most fundamental problems in statistics and machine learning, (stochastic) Multi-Armed Bandits (MAB), and its general form, bandits learning, have already been studied for more than half a century, start from \cite{thompson1933likelihood,robbins1952some}. They find numerous applications in many areas such as medicine \cite{gutierrez2017multi}, finance \cite{shen2015portfolio}, social science \cite{nakayama2017nash} and clinical research \cite{press2009bandit}. The wide applications of bandits learning also present some new challenges to this method. Particularly, due to the existence of sensitive data  and their  distributed nature in many applications like recommendation system, biomedicine, and genomics, it is often challenging to preserve the privacy of such data as they are
extremely difficult to aggregate and learn from.

To preserve the privacy of these sensitive data, Differential Privacy (DP) \cite{dwork2006calibrating} has received a great deal of attentions and now has established itself as a de facto notation of privacy for data analysis. Over the past decade, differentially private bandits learning has been extensively studied from different perspectives including MAB~\cite{tossou2016algorithms,ren2020multi,sajed2019optimal,mishra2015nearly,kalogerias2020best}, combinatorial semi-bandits~\cite{chen2020locally}, and contextual bandits~\cite{shariff2018differentially,malekzadeh2019privacy,hannun2019privacy,zheng2020locally}. Additionally, \cite{wang2020global,dubey2020differentially,dubey2020private} recently investigated bandits learning in the federated/distributed setting.
\begin{table}[]
\begin{center}
\resizebox{\textwidth}{!}
\begin{tabular}{|c|l|l|l|}
\hline
Problem                              & Model          & Upper Bound                                                                                & Lower Bound                                                                                \\ \hline
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Heavy-tailed Reward\\ (Instance-dependent Bound)\end{tabular}}   & $\epsilon$-DP  & $ O\left(\frac{\log T}{\epsilon}\sum_{\Delta_a>0}{(\frac{1}{\Delta_a})^{\frac{1}{v}}}+\max_a\Delta_a\right)$     & $\Omega \left(\frac{\log T}{\epsilon}\sum_{\Delta_a>0}{(\frac{1}{\Delta_a})^{\frac{1}{v}}}\right) $                    \\ \cline{2-4}
                                     & $\epsilon$-LDP & ${O} \left(\frac{\log T}{\epsilon^2}\sum_{\Delta_a>0}{(\frac{1}{\Delta_a})^{\frac{1}{v}}}+\max_a\Delta_a\right)$ & $\Omega \left(\frac{\log T}{\epsilon^2}{(\frac{1}{\Delta})^{\frac{1}{v}}}\right)^* $                                                                                \\ \hline
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Bounded Reward\\ (Instance-dependent Bound)\end{tabular}}      & $\epsilon$-DP  & $O\left(\frac{K\log T}{\epsilon}+\sum_{\Delta_a>0} \frac{\log T}{\Delta_a}\right)$ \cite{sajed2019optimal}                                                                                &         $\Omega\left(\frac{K\log T}{\epsilon}+\sum_{\Delta_a>0} \frac{\log T}{\Delta_a}\right)$                              \cite{shariff2018differentially}                                                                                \\ \cline{2-4}
                                     & $\epsilon$-LDP &     $O\left(\frac{1}{\epsilon^2}\sum_{\Delta_a>0} \frac{\log T}{\Delta_a}+\Delta_a\right)$ \cite{ren2020multi}                                                                                &      $\Omega\left(\frac{1}{\epsilon^2}\sum_{\Delta_a>0} \frac{\log T}{\Delta_a}\right)$  \cite{ren2020multi}                                                                                \\ \hline

\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Heavy-tailed Reward\\ (Instance-independent Bound)\end{tabular}}     & $\epsilon$-DP  &     $O\left(\left(\frac{K\log T}{\epsilon}\right)^{\frac{v}{1+v}}T^{\frac{1}{1+v}}\right)$                                                                                &               ----                                                                                \\ \cline{2-4}
                                     & $\epsilon$-LDP &                                                                                $O\left(\left(\frac{K\log T}{\epsilon^2}\right)^{\frac{v}{1+v}}T^{\frac{1}{1+v}}\right)$      &         $\Omega\left(\left(\frac{K}{\epsilon^2}\right)^{\frac{v}{1+v}}T^{\frac{1}{1+v}}\right)$                                                                                \\ \hline

\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Bounded Reward\\ (Instance-independent Bound)\end{tabular}}     & $\epsilon$-DP  &     $O\left(\sqrt{KT\log T}+ \frac{K\log T}{\epsilon}\right)$     \cite{sajed2019optimal}                                                                                &                  $\Omega\left(\sqrt{KT}+ \frac{K\log T}{\epsilon}\right)$ \cite{sajed2019optimal}                                                                                \\ \cline{2-4}
                                     & $\epsilon$-LDP &      $O\left(\frac{\sqrt{KT\log T}}{\epsilon}\right)$ \cite{ren2020multi}                                                                                &          $\Omega(\frac{\sqrt{KT}}{\epsilon})$ \cite{basu2019differential}                                                                                \\ \hline
\end{tabular}
}
\vspace{2mm}
\caption{Summary of our contributions and comparison with the bounded reward distribution case. All the results are in the expected regret form. For the heavy-tailed reward distribution case, we assume the $(1+v)$-th moment of each reward distribution is bounded by $1$ for some known $v\in (0, 1]$. For the bounded reward distribution case, we assume the rewards are bounded by 1. Here $K$ is the number of arms, $T$ is the number of rounds, and $\Delta_a$ is the mean reward gap of arm $a$. * means that the lower bound only holds for the two-armed bandit case. }
\label{Table:1}
\end{center}
\vspace{-0.3in}
\end{table}

However, these problems are still not well-understood. For example, all of the previous results and methods need to assume that the rewards are sampled from some bounded (or sub-Gaussian) distributions to guarantee the DP property. However, such assumptions may not always hold when designing decision-making algorithms for real-world complex systems. In particular, previous papers have shown that the rewards or the interactions in such systems often lead to heavy-tailed and power law distributions \cite{dubey2019thompson}, such as modeling stock prices \cite{bradley2003financial}, preferential attachment in social networks \cite{mahanti2013tale}, and online behavior on websites \cite{kumar2010characterization}. Thus, new methods are needed  to deal with these heavy-tailed rewards in  private bandits learning.

To address the above issue, in this paper, we focus on the most fundamental bandits model, {\em i.e.,} MAB, with heavy-tailed rewards.  We conduct a comprehensive and the first study on MAB with heavy-tailed rewards in both central and local models of DP, where the reward distribution of each arm only has the $(1+v)$-th moment for some $v\in (0, 1]$. Our contributions are summarized as follows.
\begin{itemize}
    \item In the first part, we consider the problem in the central $\epsilon$-DP model. Specifically, we first propose a method which is based on a robust version of the Upper Confidence Bound (UCB) algorithm, and a new mechanism which could be seen as an adaptive version of the Tree-based mechanism  in \cite{dwork2010pan}. To further improve the result, we then develop a private and robust version of the Successive Elimination (SE) algorithm and show that compared with the first method we can improve by a factor of $\log^{1.5+\frac{1}{v}} T$ in the (expected) regret bound, where $T$ is the number of rounds and $v$ is a known constant in $(0, 1]$. Moreover, we also establish the lower bound and show that the instance-dependent regret bound of $ O\left(\frac{\log T}{\epsilon}\sum_{\Delta_a>0}{(\frac{1}{\Delta_a})^{\frac{1}{v}}}+\max_a\Delta_a\right)$ in our second algorithm is optimal (up to $\text{poly}(\log\log \frac{1}{\Delta_a})$ factors), where $\Delta_a$ is the mean gap defined in  Section \ref{sec:3.1}.
    \item In the second part of the paper, we study the problem in the $\epsilon$-LDP model. We first develop a LDP version of the SE algorithm which achieves an instance-dependent regret bound of ${O} \left(\frac{\log T}{\epsilon^2}\sum_{\Delta_a>0}{(\frac{1}{\Delta_a})^{\frac{1}{v}}}+\max_a\Delta_a\right)$, and an $\tilde{O}\left(\left(\frac{K}{\epsilon^2}\right)^{\frac{v}{1+v}}T^{\frac{1}{1+v}}\right)$ instance-independent bound. Then, we show that the above instance-dependent regret bound is optimal for two-armed bandits case and the instance-independent regret bound is near-optimal (up to $\text{poly}(\log T)$ factor). All of our results can also reveal the differences between the problem of private MAB with bounded rewards and heavy-tailed rewards (see Table \ref{Table:1} for details).  To achieve these (near) optimal results, we develop several new hard instances, mechanism and private robust estimators as byproducts, which could be used to other related problems, such as private contextual bandits \cite{shariff2018differentially} or private reinforcement learning \cite{vietri2020private}.
    \end{itemize}

\vspace{-0.1in}
\section{Related Work}
\vspace{-0.1in}
As mentioned earlier, there are enormous previous work on either MAB with bounded reward distributions in  the (local) DP model \cite{mishra2015nearly,tossou2016algorithms,basu2019differential,gajane2018corrupt,sajed2019optimal,shariff2018differentially,ren2020multi,vietri2020private,zheng2020locally} or MAB with heavy-tailed reward distributions \cite{bubeck2013bandits,lee2020optimal,yu2018pure,lattimore2017scale,agrawal2021regret,vakili2013deterministic,agrawal2020optimal}. However, to the best of our knowledge, MAB with heavy-tailed reward in the (local) DP model has not been studied before. In the following we only discuss the previous work that are the most close to ours.

In the previous studies of MAB with bounded rewards, to guarantee DP property, the most direct way is to modify the classical UCB algorithm \cite{mishra2015nearly,tossou2016algorithms,ren2020multi}. Our first algorithm is motivated by a robust version of the UCB algorithm in \cite{bubeck2013bandits}. However, there are several differences. First, unlike the non-private setting, in this paper we show that this approach could only achieve a near-optimal instance-dependent (expected) regret bound, due to the noises added by the Tree-based mechanism \cite{chan2011private}. Secondly, due to the noises we added, the parameters such as the thresholds are quite different with the non-private case. To achieve an improved regret bound, our  second algorithm is based on the Successive Elimination (SE) \cite{even2006action} algorithm, whose private version has been studied in \cite{sajed2019optimal} for the bounded reward case. In this paper we extend the method to the heavy-tailed case and the LDP model. We also show that  our algorithms are (near) optimal. For the lower bounds, previously many papers \cite{ren2020multi,basu2019differential,sajed2019optimal} established hard instances in either private MAB with bounded rewards or heavy-tailed MAB in the non-private case. However, these instances cannot be used to our problem and in this paper we build new hard instances.

Private and robust estimation has drawn much attention in recent years.   \cite{barber2014privacy} provided the first study on private mean estimation for distributions with bounded moment, which is extended by \cite{kamath2020private,liu2021robust,brunel2020propose} recently. However, all of them need to assume the underlying distribution has the second-order moment, while in this paper we only need to assume the reward distributions have the $(1+v)$-th  moment for some $v\in (0, 1]$. Moreover, all of these work only focus on the central DP model and offline setting and it is unknown whether they could be extended to the stream setting. Thus,  our problem is more general. Besides the mean estimation problem, recently \cite{wang2020differentially} studied  Differentially Private Stochastic Convex Optimization with heavy-tailed data. However, it still needs to assume the distribution of  gradient has second-order moment and cannot be used to the stream setting.
\vspace{-0.2in}
\section{Preliminaries}
\vspace{-0.1in}
\subsection{Stochastic Multi-Armed Bandits with Heavy-tailed Rewards} \label{sec:3.1}
In a stochastic multi-armed bandits (MAB) problem, there is a learner interacting with  environments sequentially over $T$ rounds. The learner is faced with a set of $K$ independent arms $\{1,\ldots, K\}$. In each round $t\in[T]$, the learner selects an arm $a_t \in[K]$ to pull and then obtains a reward $x_{t}$ drawn i.i.d. from a fixed but unknown probability distribution $\gamma_{a_t}$ associated with the chosen arm. We denote by $\mu_a$ the mean of each distribution $\gamma_a$ for $a\in [K]$, denote by  $\mu^*=\max_{a\in [K]}\mu_a$ the maximum, and define $\Delta_a:=\mu^*-\mu_a$ as the mean reward gap for arm $a$. The learner aims to maximize her/his expected cumulative reward over time, {\em i.e.,} to minimize the (expected) cumulative \textit{regret}, defined as
\begin{equation}
    \mathcal{R}_T:=T\mu^*-\mathbb{E}\left[\sum\limits_{t=1}^{T}{x_t}\right],
\end{equation}
where the expectation is taken with respect to all the randomness. In this paper, we consider a heavy-tailed setting where reward distributions have finite moments of order $1+v$ for some $v\in(0,1]$. Specifically, we assume that there is a constant $u>0$ such that for each reward distribution $\gamma_a$,
\begin{equation}\label{eq:2}
    \mathbb{E}_{X\sim \gamma_a} [|X|^{1+v}] \leq u.
\end{equation}
In this paper, we assume both $v$ and $u$ are known constants, that is for any constant $c$ we regard $c^\frac{1}{v}$ as a constant. Note that the assumption is commonly used in robust estimation \cite{catoni2012challenging} and MAB with heavy-tailed rewards \cite{bubeck2013bandits,lee2020optimal}.
\vspace{-0.1in}
\subsection{Differential Privacy}
\vspace{-0.1in}
We introduce the definition of differential privacy (DP) in the stream setting since rewards are released continually.
According to \cite{dwork2010pan}, for data streams there are two different settings, {\em i.e.,}  event-level setting
and user-level setting. In this paper we will focus on the event-level setting, {\em i.e.,} two data streams $\sigma$ and $\sigma^\prime$ are adjacent if they differ at exactly one timestep. Intuitively, an algorithm is differentially private  if it cannot be used to distinguish any two adjacent streams.
\begin{definition}[Differential Privacy \cite{dwork2010pan}]
   An algorithm $\mathcal{M}$ is $\epsilon$-differentially private (DP) if for any adjacent streams $\sigma$ and $\sigma^\prime$, and any measurable subset $\mathcal{O}$ of the output space of $\mathcal{M}$, we have
       $\mathbb{P}\left[\mathcal{M}(\sigma)\in\mathcal{O}\right]
       \le e^\epsilon\cdot\mathbb{P}\left[\mathcal{M}(\sigma^\prime)\in\mathcal{O}\right].$
\end{definition}

Compared with DP, Local Differential Privacy~(LDP) is a stronger notion of privacy. In LDP, each user's data is required to be perturbed to protect privacy before collection. Thus, there is no need to define corresponding stream version.

\begin{definition}[Local Differential Privacy]
   An algorithm $\mathcal{M}:\mathcal{X}\to\mathcal{Y}$ is said to be $\epsilon$-locally differentially private (LDP) if for any $x, x^\prime\in\mathcal{X}$, and any measurable subset $\mathcal{O}\subset\mathcal{Y}$, it holds that
$ \mathbb{P}\left[\mathcal{M}(x)\in\mathcal{O}\right]
       \le e^\epsilon\cdot\mathbb{P}\left[\mathcal{M}(x^\prime)\in\mathcal{O}\right].
 $
\end{definition}
In this paper, we will mainly use the Laplacian and an adaptive version of the Tree-based mechanism (see Section \ref{sec:central} for details), and the Parallel Composition Theorem to guarantee the DP property.
\begin{lemma}[Parallel Composition]
    Suppose there are $n$ $\epsilon$-differentially private mechanisms $\{\mathcal{M}_i\}_{i=1}^n$ and $n$ disjoint datasets denoted by $\{D_i\}_{i=1}^n$. Then for the algorithm which applies each $\mathcal{M}_i$ on the corresponding $D_i$, it is $\epsilon$-DP.
\end{lemma}
\begin{definition}[Laplacian Mechanism]\label{def:3}
		Given a function $f : \mathcal{X}^n\rightarrow \mathbb{R}^d$, the Laplacian Mechanism is defined as:
		$\mathcal{M}_L(D,f,\epsilon)=f(D)+ (Y_1, Y_2, \cdots, Y_d),$
		where $Y_i$ is i.i.d. drawn from a Laplacian Distribution $\text{Lap}(\frac{\Delta_1(f)}{\epsilon}),$ where $\Delta_1(f)$ is the $\ell_1$-sensitivity of the function $f$, {\em i.e.,}
		$\Delta_1(f)=\sup_{D\sim D'}||f(D)-f(D')||_1.$ For a parameter $\lambda$, the Laplacian distribution has the density function $\text{Lap}(\lambda) (x)=\frac{1}{2\lambda}\exp(-\frac{|x|}{\lambda})$.
		Laplacian Mechanism preserves $\epsilon$-DP.
\end{definition}

\vspace{-0.2in}
\section{Optimal Rates of Heavy-tailed MAB in the DP Model}\label{sec:central}
\vspace{-0.1in}
In this section, we will study the problem of designing $\epsilon$-DP algorithms for MAB with heavy-tailed rewards. Recall that, in the classical setting where the rewards follow some bounded distributions, the most commonly used approach is using the Tree-based mechanism to privately calculate the sum of rewards and then modify the Upper Confidence Bound (UCB) algorithm \cite{auer2002finite}, such as \cite{mishra2015nearly,tossou2016algorithms}. However, their methods cannot be directly generalized to the heavy-tailed setting, since now the reward is unbounded. Thus, the most direct approach is to first preproccess the rewards to make them bounded and then use the Tree-based mechanism and some UCB based algorithm.

To address MAB with heavy-tailed in the non-private case, \cite{bubeck2013bandits} presented a general near-optimal framework called robust-UCB by combining the UCB algorithm with several robust mean estimators. Specifically, the framework first truncates the rewards to some bounded value and then performs a robust version of UCB. Motivated by this we first show a most straightforward method based on the above non-private UCB algorithm~\cite{bubeck2013bandits}, see Algorithm \ref{Algo-UCB} for details.

The key idea of our algorithm is that, in the first $K$ rounds, we establish a tree instance $Tree_a$ for each arm $a\in[K]$ (step 4-9). After that, in the $t$-th round, when the arm $a_t$ is pulled, we truncate the newly generated reward by a certain range $B_{n_{a_{t}}}$ and insert the truncated reward to $Tree_a$ (step 10-15). Here we use a robust version of UCB to select the arm, where the sum of rewards is given by the Tree-based mechanism (since we only insert the truncated rewards, we can use the mechanism). We note that in the original Tree-based mechanism in \cite{chan2011private,dwork2010pan} each element in the data steam is bounded by a uniform constant $B$. However, here the bound $B_{n_{a_{t}}}$ is adaptive and non-decreasing. Thus, we need a finer tree mechanism. In the following we will discuss the adaptive version of the Tree-based mechanism based on \cite{chan2011private,DworkNRRV_STOC09}. By the same proof as in \cite{chan2011private,DworkNRRV_STOC09} we have Lemma \ref{lemma-tree}.

\begin{algorithm}[htb]
    \caption{(Adaptive) Tree-based Mechanism}
    \label{Algo-Tree}
    \begin{algorithmic}[1]
        \Require Time horizon $T$, privacy budget $\epsilon$, a stream $\sigma$.
        \State Initialize each p-sum $\alpha_i$ and noisy p-sum $\widehat{\alpha}_i$ to $0$.
        \State Let $\epsilon^\prime\gets\epsilon/\log T$.
        \For {$t\gets1,\cdots,T$}
            \State Express $t$ in binary form: $t=\sum_j {\rm Bin}_j(t)\cdot 2^j$.
            \State Let $i\gets\min\{j:{\rm Bin}_j(t)\neq 0\}$.
            \State $\alpha_i\gets\sum_{j<i}{\alpha_j}+\sigma(t)$.
            \For {$j\gets0,\cdots,i-1$}
                \State Set $\alpha_j\gets0$, $\widehat{\alpha}_j\gets0$.
            \EndFor
            \State $\widehat{\alpha}_i\gets\alpha_i+{\rm Lap}(2B_t/\epsilon^\prime)$.
            \State Output the estimate at time $t$: $\hat{S}(t)\gets\sum_{j:{\rm Bin}_j(t)=1}{\widehat{\alpha}_j}$.
        \EndFor
    \end{algorithmic}
\end{algorithm}

\begin{definition}[p-sum]\label{def-psum}
        A p-sum is a partial sum of consecutive data items. Let $1\le i\le j$. For a data stream $\sigma$ of length $T$, we use $\sigma(t)$ denotes the data item at time $t\in[T]$ and $\sum[i,j]:=\sum_{k=i}^j\sigma(k)$ to denote a partial sum involving data items $i$ through $j$. We use the notation $\alpha_i^t$ to denote the p-sum $\sum[t-2^i+1,t]$.
\end{definition}

\begin{lemma}[{(Adaptive) Tree-based Mechanism}]
\label{lemma-tree}
Given a stream $\sigma$ such that $\sigma(t)\in[-B_t, B_t]$ for $\forall t\in[T]$, where $B_t$ is non-decreasing with $t$, we want to privately and continually release the sum of the stream $S(t):=\sum\nolimits_{i=1}^{t}{\sigma(i)}$ for each $t\in [T]$. Tree-based Mechanism (Algorithm \ref{Algo-Tree}) outputs an estimation $\widehat{S}(t)$ for $S(t)$ at each $t\in[T]$ such that $\widehat{S}(t)$ preserves $\varepsilon$-differential privacy and guarantees the following noise bound with probability at least $1-\delta$ for any $\delta>0$,
\begin{equation}
    \left| \widehat{S}(t) - S(t) \right| \leq  \frac{2B_t}{\epsilon} \cdot\log^{1.5} T \cdot \log\frac{1}{\delta}.
\end{equation}
\end{lemma}

We can see that when $B_t=B$, Algorithm \ref{Algo-Tree} will be the same as the original one. The privacy guarantee of Algorithm~\ref{Algo-UCB} is presented in Theorem~\ref{theorem-privacy-ucb}.

\begin{algorithm}[!t]
    \caption{DP Robust Upper Confidence Bound}
    \label{Algo-UCB}
    \begin{algorithmic}[1]
        \Require Time horizon $T$, privacy budget $\epsilon$, $(1+v)$-th moment which is upper bounded by $u$.
        \State Create an empty tree $Tree_{a}$ with $T$ leaves for each arm $a\in[K]$.
        \State Initialize the pull number $n_a\gets 0$ for each arm $a\in[K]$.
        \State Denote $B_{n}$ as $(\frac{\epsilon un}{\log^{1.5}T})^{1/(1+v)}$ for any $n\in\mathbb{N}^+$.
        \For {$t\gets 1,\cdots,K$}
            \State Pull arm $t$ and observe a reward $x_t$.
            \State Update the pull number $n_t\gets n_t+1$.
            \State Truncate the reward by $\widetilde{x}_t\gets x_t\cdot\mathbbm{1}_{| x_t|\le B_{n_t}}$.
            \State Insert $\widetilde{x}_t$ into $Tree_t$.
        \EndFor
        \For {$t\gets K+1,\cdots,T$}
            \State Obtain total (sum) reward $S_a(t)$ from $Tree_a$ for each arm $a$ via the Tree-based mechanism Algorithm \ref{Algo-Tree}.
            \State Pull arm $a_t=\mathop{\arg\max}_{a} \frac{S_a(t)}{n_a}+18u^{\frac{1}{1+v}}
        (\frac{\log(2t^4)\log^{1.5+\frac{1}{v}}T}
        {n_a\epsilon}
        )^{\frac{v}{1+v}}$ and observe the reward $x_{t}$.
            \State Update the pull number $n_{a_t}\gets n_{a_t}+1$.
            \State Truncate the reward by $\widetilde{x}_{t}\gets x_{t}\cdot\mathbbm{1}_{|x_{t}|\le B_{n_{a_t}}}$.
            \State Insert $\widetilde{x}_t$ into $Tree_{a_t}$.
        \EndFor
    \end{algorithmic}
\end{algorithm}

\begin{theorem} \label{theorem-privacy-ucb}
   For any $\epsilon>0$,  Algorithm~\ref{Algo-UCB} is $\epsilon$-differentially private.

\end{theorem}

In fact, the $S_a(t)/n_a$ term in step 12, which is denoted by $\widehat{\mu}_{a}(n_a, t)$, could be seen as a robust and private estimator of the mean $\mu_a$ after total $n_a$ pulls of arm $a$ till time $t$. Our selection strategy in step 12 is based on the following estimation error between $\widehat{\mu}_{a}(n_a, t)$ and $\mu_a$, which is also a key lemma that will be used to bound the regret of Algorithm~\ref{Algo-UCB}.

\begin{lemma}\label{lemma-esterr}

    In Algorithm \ref{Algo-UCB}, for a fixed arm $a$ and $t$, we have the following estimation error with probability at least $1-t^{-4}$,
    \begin{equation}
        \widehat{\mu}_{a}(n_a, t)\le \mu_a+18u^{\frac{1}{1+v}}
        \left(\frac{\log(2t^4)\log^{1.5+\frac{1}{v}}T}
        {n_a\epsilon}
        \right)^{\frac{v}{1+v}}.
    \end{equation}

\end{lemma}

We have the following instance-dependent regret bound by following the proof of theorem 1 in \cite{auer2002finite}.
\begin{theorem}\label{thm:2} Under our assumptions, for any $0<\epsilon\leq 1$
    the instance-dependent expected regret of Algorithm~\ref{Algo-UCB} satisfies
    \begin{equation}
        \mathcal{R}_T\leq O \left(\sum_{a:\Delta_a>0}{\Big( \frac{\log^{2.5+\frac{1}{v}} T}{\epsilon}\Big(\frac{u}{\Delta_a}\Big)^{\frac{1}{v}}+\Delta_a\Big)}\right).
    \end{equation}
\end{theorem}
Compared with the non-private version of robust UCB in \cite{bubeck2013bandits}, we can see that the main difference is the threshold value $B_{n_{a_t}}=(\frac{\epsilon un_{a_t}}{\log^{1.5}T})^{\frac{1}{1+v}}$, where \cite{bubeck2013bandits} sets it as $(\frac{ un_{a_t}}{\log (t^2)})^{\frac{1}{1+v}}$. Informally speaking, this is caused by the fact that, due to the privacy, the number of efficient samples now becomes to $n\epsilon$. Specifically, due to privacy constraint, the estimation error could be decomposed into three parts: the bias, variance due to the truncation, and the noise we added. We can show that setting $B_{n_{a_t}}$ as the threshold could provide an improved bound of error. Compared with the $O(\sum_{a:\Delta_a>0}{[\log T(\frac{u}{\Delta_a})^{\frac{1}{v}}+\Delta_i]})$ optimal rate of the regret in the non-private version~\cite{bubeck2013bandits}, we can see that there is an additional factor of $\frac{\log^{1.5+\frac{1}{v}} T}{\epsilon}$ in the private case. In contrast, in the problem where the reward distributions are bounded, it has been showed that \cite{shariff2018differentially}, there is only an additional factor of $\frac{1}{\epsilon}$ compared with the non-private case. Motivated by this, a natural question arises is \emph{whether it is possible to further improve the regret}.  We answer this question affirmatively by designing an optimal algorithm, see Algorithm~\ref{Algo-DPSE} for details.

Our algorithm is based on the Successive Elimination (SE) algorithm in \cite{even2006action}, whose DP variant has been studied in \cite{sajed2019optimal}. Briefly speaking, we first set all the arms as viable options (step 1), then in each epoch we pull all the viable arms to get the same (private) confidence interval around the empirical average of each viable arm's reward (step 4-18). Finally we eliminate the arms with lower empirical rewards from the viable options if they are sub-optimal compared with other viable arms (step 20).

\begin{algorithm}[!t]
    \caption{DP Robust Successive Elimination}
    \label{Algo-DPSE}
    \begin{algorithmic}[1]
        \Require Confidence $\beta$, Privacy budget $\epsilon$, $1+v$-th moment which is upper bounded by $u$.
        \State Let $\mathcal{S}\gets\{1,\cdots,K\}$
        \State Initialize: $t\gets0$, $\tau\gets0$.
        \Repeat
            \State $\tau\gets \tau+1$.
            \State Set $\bar{\mu}_a=0$ for all $a\in\mathcal{S}$.
            \State Set $r\gets 0$, $\Delta_\tau\gets2^{-\tau}$, $ R_\tau\gets \left\lceil u^{\frac{1}{v}}
            (
            \frac
            {
                24^{(1+v)/v}
                \log
                    (
                        4|\mathcal{S} |\tau^2/\beta
                    )
            }
            {
                \epsilon\Delta_\tau^{(1+v)/v}
            }
            )+1 \right\rceil$.
            \State Set $B_\tau\gets
                    (
                    \frac
                    {
                        uR_\tau\epsilon
                    }
                    {
                        \log( 4|\mathcal{S}|\tau^2/\beta)
                    }
                    )^{1/(1+v)}$.
            \While {$r<R_\tau$}
                \State $r\gets r+1$.
                \For {$a\in\mathcal{S}$}
                    \State $t\gets t+1$.
                    \State Sample a reward $x_{a,r}$ for each arm $a\in\mathcal{S}$.
                \State $\widetilde{x}_{a,r}\gets x_{a,r}\cdot\mathbbm{1}_{\{|x_{a,r}|\le B_\tau\}}$.

                \EndFor
            \EndWhile
            \State For each $a\in \mathcal{S}$, compute $\bar{\mu}_a\gets(\sum\limits_{l=1}^{R_\tau}\widetilde{x}_{a,l})/R_\tau$.
            \State Set $err_\tau\gets u^{1/(1+v)}
            (\frac
            {\log(4|\mathcal{S}|\tau^2/\beta)}
            {R_\tau\epsilon})^{v/(1+v)}$.
            \State Set $\widetilde{\mu}_a\gets\bar{\mu}_a+{\rm Lap}(\frac{2B_\tau}{R_\tau\epsilon})$ for all $a\in\mathcal{S}$.
            \State Let $\widetilde{\mu}_{\rm max}\gets\max_{a\in\mathcal{S}}\widetilde{\mu}_a$.
            \State Remove all arms $a$ from $\mathcal{S}$ such that $\widetilde{\mu}_{\rm max}-\widetilde{\mu}_a>12err_\tau$.
        \Until {$|\mathcal{S}|=1$}
        \State Pull the arms in $\mathcal{S}$ in all remaining $T-t$ rounds.
    \end{algorithmic}
\end{algorithm}

\begin{theorem}\label{thm:3}
 For any $\epsilon>0$, Algorithm~\ref{Algo-DPSE} is $\epsilon$-differentially private.

\end{theorem}
\begin{remark}
As mentioned earlier, \cite{sajed2019optimal} also studied a DP variant of the SE algorithm. However, there are several differences between their result and ours. The algorithm in \cite{sajed2019optimal} is only for bounded reward distributions, while here we focus on the heavy-tailed ones. Due to the irregularity of rewards, we need to preprocess and shrink these rewards. Moreover, the forms of parameters are also more complicated than the bounded distributions case. Finally, in the later section, we also extend Algorithm~\ref{Algo-DPSE} to the local model and show its optimality.
\end{remark}
The following lemma claims that the number of rounds to pull each arm $a$ is at most $\tilde{O}(\frac{1}{\epsilon (\Delta_a)^\frac{1+v}{v}})$.
\begin{lemma}\label{lemma-pulltime}
    For any instance of the $K$-MAB problem, denote by $a^*$ its optimal arm, and by $\Delta_a$ the gap between the mean of arm $a^*$ and any sub-optimal arm $a\ne a^*$. Fix the time horizon $T$ and confidence level $\beta \in (0,1)$. Then, with probability at least $1-\beta$, in Algorithm~\ref{Algo-DPSE}, the total number of rounds to pull each sub-optimal arm $a\ne a^*$, denoted by $T_a$, is at most
    \begin{equation}
        \min \left\{T, O\left( \frac{u^{\frac{1}{1+v}}}{\epsilon(\Delta_a)^{\frac{1+v}{v}}}\left(\log(\frac{K}{\beta})+\log\log(\frac{1}{\Delta_a})\right)\right)\right\}.
    \end{equation}

\end{lemma}

\begin{theorem}\label{thm:4}
   If we set $\beta=\frac{1}{T}$ in Algorithm~\ref{Algo-DPSE}. Then for sufficiently large $T$,  the instance-dependent expected regret of Algorithm~\ref{Algo-DPSE} satisfies
    \begin{equation}
          \mathcal{R}_T\leq  {O} \left(\frac{u^{\frac{1}{1+v}}\log T}{\epsilon}\sum_{\Delta_a>0}{\Big(\frac{1}{\Delta_a}\Big)^{\frac{1}{v}}}+\max_a\Delta_a\right).
    \end{equation}
Moreover, the instance-independent expected regret of Algorithm~\ref{Algo-DPSE} satisfies
  \begin{equation}
     \mathcal{R}_T \leq  O\left(u^{\frac{v}{(1+v)^2}}\left(\frac{K\log T}{\epsilon}\right)^{\frac{v}{1+v}}T^{\frac{1}{1+v}}\right),
  \end{equation}
  where the ${O}(\cdot)$-notation omits $\log\log \frac{1}{\Delta_a}$ terms.

\end{theorem}
From Theorem \ref{thm:4} we can see that compared with the regret bound $O(\frac{\log^{2.5} T}{\epsilon}\sum_{\Delta_a>0}{(\frac{1}{\Delta_a})^{\frac{1}{v}}})$ in Theorem \ref{thm:2}, we achieve an improved bound of $O(\frac{\log T}{\epsilon}\sum_{\Delta_a>0}{(\frac{1}{\Delta_a})^{\frac{1}{v}}})$. Moreover, we also have an instance-independent regret bound. While in the bounded rewards case it has been shown that a DP variant of the SE algorithm is  optimal \cite{sajed2019optimal}, it is still unknown whether Algorithm \ref{Algo-DPSE} is optimal in the heavy-tailed case. In the following we study the lower bound of regret for  heavy-tailed MAB problem in the $\epsilon$-DP model. We start from the two-armed case and show that the dependency on the term of $\frac{1}{\epsilon}(\frac{1}{\Delta})^\frac{1}{v}$ is unavoidable in general.
Then we extend to the case with $K$ arms and show the instance-dependent regret bound presented in Theorem \ref{thm:4} is {\bf optimal}. The lower bound of the instance-independent regret is still unclear, and we leave it as an open problem.
\begin{theorem}\label{thm:6}
   There exists a heavy-tailed two-armed bandit problem with arm $2$ being sub-optimal, $u\le1$ in (\ref{eq:2}), and $\Delta:=\mu_1-\mu_2\in(0,\frac{1}{5})$. Such that for any $\epsilon$-DP algorithm $\mathcal{A}$ with expected regret at most $T^{\frac{3}{4}}$ for sufficiently large number of rounds $T$,\footnote{Note that we can replace $\frac{3}{4}$ to other constants. The same to other results.} we have
    \begin{equation}
        \mathcal{R}_T\ge\Omega\left(\frac{\log T}{\epsilon}(\frac{1}{\Delta})^{\frac{1}{v}}\right).
    \end{equation}

\end{theorem}
\begin{theorem}\label{thm:7}
    There exists a heavy-tailed $K$-armed bandit instance with $u\le1$ in (\ref{eq:2}), $\Delta\in(0,\frac{1}{5})$, such that for any $\epsilon$-DP algorithm $\mathcal{A}$ with expected regret at most $T^{\frac{3}{4}}$ for sufficiently large number of rounds $T$, we have
    \begin{equation}
        \mathcal{R}_T\ge\Omega \left(\frac{\log T}{\epsilon}\sum_{\Delta_a>0}{(\frac{1}{\Delta_a})^{\frac{1}{v}}}\right).
    \end{equation}

\end{theorem}
Below we will sketch the proof of Theorem \ref{thm:7}. Notably, previous  hard instances in the bounded awards case cannot provide tight lower bounds under our setting. Moreover, the hard instance in the non-private heavy-tailed MAB case \cite{bubeck2013bandits} is only for two-armed setting and cannot be generalized to $K$ arms. Therefore, it is necessary to construct new hard instance to prove Theorem \ref{thm:7}. Specifically, we design the following heavy-tailed MAB problem as a hard instance: the instance $\bar{P}$ where the distribution of each arm $a\in[K]$ is $\nu_a=\Big(1-\frac{s_a^{1+v}}{2}\Big)\delta_0+\frac{s_a^{1+v}}{2}\delta_{1/s_a},$
    with some $\frac{1}{2}\ge\mu_1\ge\cdots\ge\mu_K$ and $s_a=(2\mu_a)^{\frac{1}{v}}$. That is when $X\sim \nu_a$, $X=0$ with probability  $(1-\frac{s_a^{1+v}}{2})$, and $X=\frac{1}{s_a}$ with probability $\frac{s_a^{1+v}}{2}$. It is easy to verify for each $a\in[K]$ that $\mathbb{E}[\nu_a]=\mu_a$ and the $(1+v)$-th moment is bounded by 1.
    Now, we fix an arm $a\ne1$ and denote $E_a$ as the event that the arm $a$ is pulled at most $t_a:=\frac{\log T}{100\epsilon 4^{1/v}}(\frac{1}{\Delta_a})^\frac{1+v}{v}$ times. We will mainly show  that $\mathbb{P}_{\mathcal{A},\bar{P}}(E_a)\le\frac{1}{2K}$. To prove this, we consider another instance $\bar{Q}_a$ where the distribution of any arm $a^\prime\ne a$ remains unchanged, and the distribution of arm $a$ is $\nu_a^\prime=[ 1-(\frac{s_a^{1+v}}{2}+2\Delta_a\gamma_a)]\delta_0+(\frac{s_a^{1+v}}{2} )\delta_{1/s_a}+(2\Delta_a\gamma_a)\delta_{1/\gamma_a},$
    where $\gamma_a=(4\Delta_a)^{\frac{1}{v}}$. Note that when setting $\mu_a^{1+v}\le\frac{1}{6}$ and $\Delta_a^{1+v}\le\frac{1}{12}$, we have $\frac{s_a^{1+v}}{2}+2\Delta_a\gamma_a=2^{\frac{1}{v}}\mu_a^{\frac{1+v}{v}}+2\cdot4^{\frac{1}{v}}\cdot\Delta_a^{\frac{1+v}{v}}\le(\frac{1}{3})^{\frac{1}{v}}+2\cdot(\frac{1}{3})^{\frac{1}{v}}\le1,$
    thus the postulated $\nu_a^\prime$ is reasonable. It is also easy to verify that $\mathbb{E}[\nu_a^\prime]=\mu_a+2\Delta_a=\mu_1+\Delta_a \leq 1$ and the $(1+v)$-th moment  of $v_a^\prime$ is bounded by 1. With the coupling lemma in \cite{KarwaV_ITCS18}, we can prove that $\mathbb{P}_{\mathcal{A},\bar{P}}(E_a)\le\frac{1}{2K}$. Thus, take the union bound we can get with probability at least $\frac{1}{2}$, the regret of $\bar{P}$ for any private algorithm $\mathcal{A}$ is $\Omega(\frac{\log T}{\epsilon}\sum_{\Delta_a>0}{(\frac{1}{\Delta_a})^{\frac{1}{v}}})$.
\begin{remark}\label{remark2}
In the MAB with bounded rewards case, it has been shown that the optimal rate of the expected rate is $O(\frac{K\log T}{\epsilon}+\sum_{\Delta_a>0} \frac{\log T}{\Delta_a})$ \cite{sajed2019optimal}. Compared with the optimal rate $O(\frac{\log T}{\epsilon}\sum_{\Delta_a>0}{(\frac{1}{\Delta_a})^{\frac{1}{v}}})$ in the heavy-tailed case, we can see there is a huge difference. First, the dependency on $\frac{1}{\Delta_a}$ now becomes to $(\frac{1}{\Delta_a})^{\frac{1}{v}}$. Secondly, the price of privacy in the bounded rewards case is an additional term of $O(\frac{K \log T}{\epsilon})$ compared with the non-private rate, while in the heavy-tailed case, there is an additional factor of $\frac{1}{\epsilon}$ compared with the non-private one.
\end{remark}
\vspace{-0.1in}
\section{Optimal Rates of Heavy-tailed MAB in the LDP Model}
\vspace{-0.1in}
In this section, we will investigate upper and lower bounds for heavy-tailed MAB in the \emph{local} DP model. We start from the upper bounds. To design an $\epsilon$-LDP algorithm, most of the previous work on MAB with bounded rewards modifies the UCB algorithm and uses the Laplacian mechanism to guarantee the LDP property, such as \cite{chen2020locally,ren2020multi,zhou2020local}. Moreover, it has been shown that this type of method is near-optimal \cite{chen2020locally,ren2020multi}. Thus, for MAB with heavy-tailed rewards, a straightforward way is to modify the UCB-based method, {\em i.e.,} Algorithm \ref{Algo-UCB}. Specifically, each reward will be shrunken to a certain range and then add Laplacian noise,  then the algorithm uses the confidence bound on these perturbed rewards to pull an arm. However, unlike the bounded rewards case, this approach may cause enormous error. The reason is that, similar to Algorithm \ref{Algo-UCB}, here the threshold of each reward depends on $n_a$. That is the Laplcaian noise we added for each reward will also proportional to $n_a$. Thus, unlike the Tree-based mechanism in the central model, this LDP version of the UCB algorithm will introduce a huge amount of error to estimate the mean.

To provide a better utility, we propose an $\varepsilon$-LDP version of the SE algorithm, see Algorithm \ref{Algo-LDPSE} for details. The basic idea is similar to Algorithm \ref{Algo-DPSE}, where the algorithm now maintains (private) confidence interval for each arm via the perturbed rewards instead of the noisy average. However, compared with the above LDP version of the UCB algorithm, we can see that here the Laplacian noise added to each reward is \emph{independent} on the number of rounds  $n_a$, which could be much smaller than the noise in the LDP added to UCB method when $T$ is sufficiently large. The following theorems provide the privacy and utility guarantees of Algorithm~\ref{Algo-LDPSE}.

\begin{theorem}\label{DPofLDP}
For any $\epsilon>0$, Algorithm \ref{Algo-LDPSE} is $\varepsilon$-local differentially private.
\end{theorem}

\begin{theorem}\label{UpperBoundLDP}
   Set $\beta=\frac{1}{T}$ in Algorithm~\ref{Algo-LDPSE}.  Then, for sufficiently large $T$,  the instance-dependent expected regret of Algorithm~\ref{Algo-LDPSE} satisfies
    \begin{equation}
          \mathcal{R}_T \leq {O} \left(\frac{ u^{\frac{2}{v}}\log T}{\epsilon^2}\sum_{\Delta_a>0}{\Big(\frac{1}{\Delta_a}\Big)^{\frac{1}{v}}}+\max_a\Delta_a\right).
    \end{equation}
Moreover, the instance-independent expected regret of Algorithm~\ref{Algo-LDPSE} satisfies
  \begin{equation}
    \label{eq:instance-independent-local-upper}
     \mathcal{R}_T \leq  O\left(u^{\frac{2}{1+v}}\left(\frac{K\log T}{\epsilon^2}\right)^{\frac{v}{1+v}}T^{\frac{1}{1+v}}\right),
  \end{equation}
where {the ${O}(\cdot)$-notations omit $\log \log \frac{1}{\Delta_a}$ terms.}

\end{theorem}

\begin{algorithm}[!t]
    \caption{LDP Robust Successive Elimination}
    \label{Algo-LDPSE}
    \begin{algorithmic}[1]
        \Require Confidence $\beta$, privacy budget $\epsilon$, $1+v$-th moment which is upper bounded by $u$.
        \State Let $\mathcal{S}\gets\{1,\cdots,K\}$
        \State Initialize: $t\gets0$, $\tau\gets0$.
        \Repeat
            \State $\tau\gets \tau+1$.
            \State Set $\bar{\mu}_a=0$ for all $a\in\mathcal{S}$.
            \State Set $r\gets 0$, $\Delta_\tau\gets4^{-\tau}$, $R_\tau\gets  \left\lceil u^{\frac{2}{v}}
            (
            \frac
            {
                28^{2(1+v)/v}
                \log
                    (
                        8|\mathcal{S} |\tau^2/\beta
                    )
            }
            {
                \epsilon^2\Delta_\tau^{2(1+v)/v}
            }
            )+\log
                    (
                        8|\mathcal{S} |\tau^2/\beta
                    ) \right\rceil$.
            \State Set $B_\tau\gets
                    (
                    \frac
                    {
                        u\sqrt{R_\tau}\epsilon
                    }
                    {
                       \sqrt{ \log( 8|\mathcal{S}|\tau^2/\beta)}
                    }
                    )^{1/(1+v)}$.
            \While {$r<R_\tau$}
                \State $r\gets r+1$.
                \For {$a\in\mathcal{S}$}
                    \State $t\gets t+1$.
                    \State Sample a reward $x_{a,r}$ for each arm $a\in\mathcal{S}$.
                \State $\widetilde{x}_{a,r}\gets x_{a,r}\cdot\mathbbm{1}_{\{|x_{a,r}|\le B_\tau\}}$.
                \State $\widehat{x}_{a,r}\gets\widetilde{x}_{a,r}+{\rm Lap}(\frac{2B_\tau}{\epsilon})$

                \EndFor
            \EndWhile
             \State For each $a\in \mathcal{S}$, compute $\bar{\mu}_a\gets(\sum\limits_{l=1}^{R_\tau}\widehat{x}_{a,l})/R_\tau$.
            \State Set $err_\tau\gets u^{1/(1+v)}
            (\frac
            {\sqrt{\log(8|\mathcal{S}|\tau^2/\beta)}}
            {R_\tau\epsilon})^{v/(1+v)}$.
            \State Let $\bar{\mu}_{\rm max}\gets\max_{a\in\mathcal{S}}\bar{\mu}_i$.
            \State Remove all arms $a$ from $\mathcal{S}$ such that $\bar{\mu}_{\rm max}-\bar{\mu}_a>14err_\tau$.
        \Until {$|\mathcal{S}|=1$}
        \State Pull the arms in $\mathcal{S}$ in all remaining $T-t$ rounds.
    \end{algorithmic}
\end{algorithm}

In the following, we derive both instance-dependent and instance-independent lower bounds of heavy-tailed MAB in the $\epsilon$-LDP model.

\begin{theorem}\label{LDPInstDepend}
(Instance-dependent Lower Bound). There exists a $\Delta \in (0,\frac{1}{5})$ and a heavy-tailed two-armed bandit instance with $\mu_1-\mu_2=\Delta$ and each reward distribution has a $(1+v)$-th moment bounded by 1, such that for any $\epsilon$-LDP algorithm with regret $\leq o(T^\alpha)$ for any $\alpha > 0$, the regret satisfies $$
\liminf _{T \rightarrow \infty} \frac{\mathcal{R}_T}{\log T}\geq  \Omega \left(\frac{1}{\epsilon^2 \Delta^{\frac{1}{v}}} \right).
$$
\end{theorem}
\begin{remark}
Theorem \ref{LDPInstDepend} reveals that the term of $\frac{1}{\epsilon^2 \Delta^{\frac{1}{v}}}$ is unavoidable in the regret bound. As a result, the attained bound in Theorem \ref{UpperBoundLDP} is  optimal in the two-armed bandit case. Compared with the optimal rate $O(\frac{1}{\Delta^{\frac{1}{v}}})$ in the non-private case, we can see the price of privacy is an additional factor of $\frac{1}{\epsilon^2}$, which is similar to other MAB with bounded rewards problems  in the LDP model \cite{zhou2020local,ren2020multi}. However, it is noteworthy to emphasize that Theorem \ref{LDPInstDepend} only holds for two-armed bandits and cannot be extended to general $K$-armed case, which is different with Theorem~\ref{thm:7} in the central model. The main reason is that, to prove the lower bound, in the central we only need to focus on the TV distance; by contrast, in the local model it is necessary to consider the KL divergence. We leave the lower bound of the general $K$-armed bandits as future work.
\end{remark}
\begin{theorem}\label{LDPIndepLower}
(Instance-independent Lower Bound) There exists a heavy-tailed $K$-armed bandit instance with the $(1+v)$-th bounded moment of each reward distribution is bounded by $1$. Moreover, if $T$ is large enough, for any the $\epsilon$-LDP algorithm $A$, the expected regret must satisfies
$$\mathcal{R}_T \geq \Omega\left( \Big(\frac{K}{\epsilon^2}\Big)^{\frac{v}{1+v}} T^{\frac{1}{1+v}}\right).$$
\end{theorem}
\begin{remark}
Unlike the instance-dependent lower bound presented in Theorem \ref{LDPInstDepend}, here our instance-independent lower bound holds for the general $K$-armed case. Moreover, the upper bound~\eqref{eq:instance-independent-local-upper} of Algorithm \ref{Algo-LDPSE} is nearly optimal. However, compared with instance-independent lower bound, there is still a $\text{poly}(\log T)$ factor gap. We conjecture this factor could be removed by using some more advanced robust estimator, such as the estimator in \cite{lee2020optimal} and we will leave it as an open problem. For MAB with bounded rewards in the LDP model, \cite{basu2019differential} showes that its instance-dependent regret bound is always at least $\Omega(\frac{\sqrt{KT}}{\epsilon})$, {\em i.e.,} there is an additional factor of $\frac{1}{\epsilon}$ compared with the non-private case. However, for heavy-tailed MAB, compared with the lower bound of $\Omega(K^{\frac{v}{1+v}}T^{\frac{1}{1+v}})$ in the non-private case, from Theorem \ref{LDPIndepLower} we can observe that the difference is  a factor of $(\frac{1}{\epsilon^2})^{\frac{v}{1+v}}$. Thus, combining with Remark \ref{remark2}, we can conclude that heavy-tailed MAB and bounded MAB are quite different in both central and local differential privacy models.
\end{remark}

    \bibliographystyle{plain}

    \input{Main_context.bbl}

\newpage

redpajama_set_name: RedPajamaArXiv


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Today's recipe is pretty, pink, and delicious! If you like cheesecake (even a little), you are likely going to love this! This Strawberry Cheesecake Mousse is light and fluffy, sweet (but not overwhelming), and has a deliciously light cream cheese flavor. I've made several similar recipes in the past… first Cherries on a Cloud, then Candy Corn Cheesecake Mousse, then Strawberry Blueberry Mini Cheesecake Trifles.
I made a few little changes this time, adding fresh fruit puree to the mixture, (instead of just laying fruit on top as with some of the previous recipes) and also, I used real whipped cream (in place of cool whip). Both adjustments were deliciously successful. I'm sure as spring and summer roll in, I'll try other fruit purees as well.
I like the addition of the crust (the graham cracker crumble) for a little texture, but it is not necessary.
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Crush graham crackers in a plastic bag. Crush crackers until the mixture still has some pea sized pieces (you do not need it to be all fine crumbs).
Put cracker crumbs in a bowl and pour melted butter on top. Add 1 teaspoon sugar, and stir mixture until crumbs are moist.
Spread mixture into a baking dish (such as a 9"x9" dish) and bake at 350*F for about 12 minutes.
In a food processor, or blender, chop/blend fresh or frozen (and slighly thawed) strawberries. Process until the berries are chopped, juicy, and no piece is larger than a pea.
In the bowl of an electric mixer (or with a hand mixer), whip heavy cream until it begins to thicken. Add powdered sugar, and continue to whip until thick and holds stiff peaks. Move whipped cream to a separate bowl (if using bowl of elecric mixer).
Add 1-2 tablespoons of graham cracker crumble into the bottom of 6 serving dishes.
Spoon or pipe (with a large piping bag, or large ziplock bag with a corner cut off) strawberry cheesecake mixture into dishes.
Best when eaten the day it was made.
This recipe can be made sugar free by omiting the graham cracker crumble, and using splenda, (or similar) in place of the powdered sugar.
If desired, Cool whip may be used in place of the whipped cream (I would suggest about 1 or 1 1/2 cups).
I used full fat (regular) cream cheese. Reduced fat cream cheese has a higher water content, and I have not experimented with it for this recipe.
If using fresh strawberries, make sure they are nice and ripe. If ripe strawberries are not available, use frozen. I used frozen strawberries in this recipe, and fresh berries just for garnish.
If you use frozen strawberries, make sure they are just strawberries, with no additional sweetener.
If desired, you may want to add a drop of pink food coloring to add to the pink hue. I did not add any this time, and you can see the color is very pale pink.
This Strawberry Cheesecake Mousse would be perfect as a Valentine's treat, or delicious as a light spring or summer dessert.
Looking for more Valentine's ideas? Browse all my Valentine's Day posts, or find lots of sweet and pretty ideas from around the web on my Valentine's Day Pinterest board.
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I am one of those non-cheesecake people. Really, I think cream cheese should be savory. But when I saw this in my feed from you, I knew it would be beautiful….and it is! A little mascarpone may work out for me. So pretty on this rainy January day!
Oh wow! These looks heavenly!
These caught my eye right away. I am going to try them for my grandchildren. I love this site!!
So pretty… they look delicious, too!
Can't wait to try this! Love anything with strawberries and what a great desert for a pink or baby girl baby shower!
These look absolutely amazing! I love the fact that you added info on how to make it sugar free. Beautiful photos, too!
This looks delicious, I can't wait tot try this.
These look delicious Glory! And incredibly beautiful as always!
I love simple and tasty desserts like this!
I just made this and it's chilling in the fridge!
The filling is delicious–creamy, thick, slightly tart from the berries and sweet. I added 2 TB of Sour Cream for a more tangy end to the mousse.
I made these for Christmas Eve and they were easy to make and a big hit! I did add just one drop of red food coloring so they would be nice and pink. I meant to add a little extra whipped cream and a strawberry garnish on top but they were eaten too quickly!
You are my go-to gal for desserts when I have company as your recipes never fail me, I plan on making this dessert this weekend and have a question. If I make these ahead (probably 8 hours ahead) will the graham cracker crust get soggy? Even though the assembly would likely be quick, I would love to just take the finished dessert out of the fridge when the time comes. Thanks Glory for all your superb recipes!
Super Easy and DELICIOUS! I have made this several times and it is always a hit-Thank you for such a great recipe.
This was a hit with my friends as a dip with cookies and extra strawberries.


As a creative and forward thinking business, Neil Shacklocks are continually looking to improve its performance and the recruitment of talented staff is core to us achieving this. We realise our staff are our biggest asset and their contribution plays a key role in the success of the business.
We aim to provide our staff a safe, positive and friendly workplace, committed to diversity and equality, combined with competitive rates of pay.
Qualified electricians with experience of domestic new build or social housing projects.
Should you be interested in a career at Neil Shacklocks, please forward your CV to careers@shacklocks.com.
Neil Shacklocks are a committed supporter of apprenticeships. To register your interest and to be considered for future selection, please email a covering letter and CV to careers@shacklocks.com.
Neil Shacklocks are committed to Equality and Diversity and promotes a workplace that is free of discrimination.
Please note, due to the volume of applications it is not our policy to contact you unless selected for interview.


From the consumer to the CEO, variety is a natural part of business. Both B2B and B2C products come in different stripes to suit either people's preferences or certain situations they experience everyday. What's more, innovations and new technologies only add to the selection.
Naturally, these varieties have in impact in the B2B Marketing department. Different strategies sprout up in response to a new type of business-customer situation and create new tools for the job. A recent example is the way social media marketing helps content goes viral; quickly spreading it even beyond digital.


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Read more>> Q: What type of computer I need?
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text:
Q: Dimension analysis of de Broglie equations One form of one of the de Broglie's equations is this:
$\lambda = \frac{2\pi\hbar}{p}$
Units:
$\lambda = [m]$
$\hbar = [Js]$
$p = [\frac{kg m}{s}]$
$J=[Nm]$
How can one show with dimension analysis that the right side is equal to the left side, having unit meter?
EDIT: Sorry for having used the wrong units. But I actually had them right at some point when trying this out. But I didn't "see" the last step. If someone finds this post later and have that problem this might help them,
$[m]=\frac{[Js]}{\frac{kgm}{s}}=m\frac{Ns^2}{kgm}=m\frac{N}{kg\frac{m}{s^2}}=[m]$
This follows because N is the unit for force, and $kg\frac{m}{s^2}$ is force by Newton's laws.

A: You just need to use the definition of Joule (J).

A: You have the units of momentum wrong. It is $[\frac{\textrm{kg} \textrm{m}}{\textrm{s}}]$. From there it is just simple cancellation.

redpajama_set_name: RedPajamaStackExchange


This is a fascinating account of some of history's most inspiring women. Among the female firsts are Pharaoh Hatshepsut, a woman who ruled as king, not as queen; Christine de Pisan, Europe's first professional female writer; and Valentina Tereshkova, Russian astronaut and in 1963 the first woman in space.
Bringing to bear a commanding knowledge of Hebrew Scripture, Osherow presents a series of case studies of biblical heroines who engage in poetry and in song. The author investigates how the cultural requirement for feminine silence informs early modern readings of these biblical characters, and furthermore, how they were used to counteract cultural constraints on women's speech. The book's chapters focus on Miriam, Hannah, Deborah, and a feminized King David.
Through an examination of the individual life trajectories of youth who participated in an educational exchange project between Mozambique and East Germany, this book offers an alternative reading of Mozambique's socialist past and makes a significant contribution to studies of post-socialist transitions.
Posted on You Let Some Girl Beat You?
You Let Some Girl Beat You?
Between 1850 and 1950, at least 115 women were lynched by mobs in the United States. The majority of these women were black. This book examines the phenomenon of the lynching of women, a much more rare occurence than the lynching of men. Over the same hundred year period covered in this text, more than 1,000 white men were lynched, while thousands of black men were murdered by mobs. Of particular importance in this examination is the role of race in lynching, particularly the increase in the number of lynchings of black women as the century progressed. Details are provided—when available—in an attempt to shine a light on this form of deadly mob violence.
Madam President? The question is not if, but rather when the United States will elect a female president - but timing may be the only certainty involved in shattering this most visible glass ceiling in U.S. society. Who will be included in the field of candidates for Madam President, and why? How will she have to position herself for a viable run at the Oval Office? Once in office, will she encounter gender-based biases in her handling of military and foreign affairs? Will Madam President blend seamlessly with the long line of Mr. Presidents - or will the very nature of the presidency be irrevocably changed? Anticipating Madam President's insightful blend of analysis, profiles, and interviews with prominent political women illustrates the realities of women in the upper echelons of public life, as well as the challenges likely to face a woman in one of the world's most powerful political positions.
This book of conversations between Margaret R. Miles and Hiroko Sakomura compares the experiences of two women who grew up in different societies, with different educations, different professions, and different religious orientations. Reflecting on the different ways in which Japanese and American societies inhibited and enabled them, these two women share their struggles, difficulties, and achievements. All of this is set in the context of one of the most radical social movements in the history of the world, as women are gaining increments of equality with men in designing and administering the institutions of public life with opportunities, dangers, and rewards. This is a moment in which a critical mass of women want it all now, in the best sense of the phrase, seeking to preserve and reinterpret traditional values while exercising their capabilities and skills both in the home and in public life. This book is the memoir of two women's painful and joyful experiences in getting here from there.
Flags of the Fifty States is an indispensable historical reference and a fascinating, beautiful pictorial guide to the rich diversity of America's fifty states. It provides a colorful way to learn about how the United States grew and prospered and shaped itself and its banners over the years. With stunning, full-color reproductions of each flag, this book offers a thorough and eminently readable account of how and why each flag was designed, what the various symbols and figures in the flags mean, and how each flag evolved. Throughout are interesting facts and sidebars that answer such questions as why Alaska is represented by the Big Dipper, what significance the diamond has in the flag of Arkansas, which state features the Union Jack on its flag, and what impact the Civil War had on designs of the state flags of the South.
Herald it as the coming of age of woman power or shrug it off as mere symbolism to placate all and sundry - the fact is that the glass ceiling at the highest level has been shattered. Even so, with the passage of time the novelty of having a woman President is bound to wear off, and the sole benchmark to evaluate Pratibha Patil would be her performance as the first citizen of the country. A seasoned politician and proven Congress loyalist, Pratibha Patil has the unique distinction of having won every election that she has contested in the 34 years of her long political career. Till recently Governor of Rajasthan, she will need to muster all the skills honed in the rough and tumble of politics, in the professional arena of the courtroom, and in sports stadia to uphold the dignity and decorum of the hallowed office of the President of the country. This book traces the journey of President Pratibha Patil from the small town of Jalgaon to the majestic Rashtrapati Bhawan on Raisina Hill as she assumes office as the twelfth President of India, after one of the murkiest, no-holds barred presidential elections ever, making history in the sixtieth year of India's independence.
Examines the political status of women in the world's governments and challenges the view that women in the United States and other countries are breaking through traditional barriers to achieve unprecendented political power. The factors that exclude women from power are detailed.


We've all heard of the man. Now get ready for the restaurant.
English celebrity chef and restauranteur Jamie Oliver, who is known for his food TV shows and cookbooks, is bringing his largest restaurant chain – Jamie's Italian – to Taiwan next year.
According to William Lyon, the CEO of Big Cat Group, the company that first introduced Jamie's Italian to Hong Kong, Jamie's Italian will open in Taiwan in 2016, with Taipei being the possible location.
The first ever Jamie's Italian restaurant was opened by Oliver and Oliver's mentor, Italian chef and restauranteur Gennaro Contaldo, in Oxford in 2008.
The duo began with the idea of creating an accessible and affordable restaurant where customers can enjoy authentic, rustic dishes using tried and tested recipes with a current and Jamie twist.
Each restaurant is uniquely designed to incorporate not just the history of the building but also the personality of the city, creating "an environment where the focus is on exceptional food, good company and the relaxed conviviality of the 'Italian table'".
There are now fifty-one Jamie's Italian sating the hunger of food connoisseurs globally, with three stores in Asia – one in Singapore and two in Hong Kong.
From the looks of every Jamie's Italian opened so far, we can safely expect a lavish interior with open kitchens, where customers can observe chefs cook with fresh, sustainable ingredients in a casual setting.
What's more, seasonal menus and great authentic Italian food are some things we can look forward to.
Big Cat Group has the right of management of Jamie's Italian chain in Hong Kong and says it will also be looking to expand to China.
Hong Kong Jamie Oliver perhaps is the worst Jamie I tried over the world. Singapore and Dubai and Australia are all better. If the Taiwan one is going to run by the Hong Kong restaurant I will not try.
Totally agree. The Hong Kong Jamie Oliver is the one "creatively" localised the menu. I saw this Chinese New Year menu with fish and rice and Asian dessert. This is hilarious. I am a fan of Jamie as a chef and I feel when he cooks Asian cuisine, at least he is cooking decently. Not steam fish in an ITALIAN restaurant. I am sure when they run the Taiwan restaurant, they will serve Xiao Lung Bao on plank or oyster pizza which try to drive you nuts.
Jamie Oliver HK daily taking now is only half of opening. No need to book and can walk straight in and get table. Chief chef changed three times in 2 years. High staff turnover. Average staff tenure is less than a year. Look like the restaurant is in life support mode. I heard that they don't open new restaurant in Central. This is just how bad the business is running. They also lower food quality I believe they change food supplier for lower price, which can tell from taste.
Meantime at least three staff managed to drop or break something. One waiter dropped a bunch of the cups used for serving fries. Rather than send them back to be washed he just cooly stacked them up again. The staff appeared at best incompetent.
20 minutes later the fries appeared on the table (now been waming for 30 minutes) and they got sent back. Burgers appeared on the warming counter 10 minutes later (so 70 minutes after we ordered them) with staff apparently clueless as to what to do with them. I was so annoyed at this point that I went and got them myself. Not sure if they were the original burnt burgers as they'd been grilled to a cinder and were virtually inedible. No burger bun, all three were in stale ciabatta. Oh – and the spaghetti and meatballs had disappeared and took the staff 5 minutes to locate.
The manager didn't charge us. I would have refused to pay anyway. If Jamie Oliver knows his name is being put to this abomination he should be ashamed.
Just have lunch with my family of 4.
Have "hair" inside my kid's ordered dish, informed the staff and she just took it away without saying anything.
Glasses placed upside down ( I believe that 's the standatd way they set the table) but they should have ensure that the table must be clean in order to do this. As the rim of the glasses were stuck with some 'bread crumbs' or 'grated cheese stuff', who will dare to use it.
The so call " Fresh Crabmeat linguine" tasted extremely fishy. I have informed the staff there, again, they just removed it and then replied that " the taste is like that' and they have tried. Come on, if it was the truth, obviously they used DEAD crabmeat instead of what they called "fresh".
For sure will not ho there again!!!!
Compare to The Bread Street Kitchen in Central, price is more or less the same but the standard of food , the restaurant environment and the staffs are much better there then in Jamie's .
The HK team is surrounded by useless chap – bs CEO who run IKEA and KFC and what can you expect him to run Italian restaurant? he know nothing on F&B and 99.99 of his ideas are misfire. Chinese menu, breakfast, all flops. Poor investors sinking money into this company believing he can make a fortune. Poor Jamie letting him to ruin the name.


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Build 10F569 of Mac OS X 10.6.4 Seeded to Developers
Monday June 14, 2010 1:29 pm PDT by Eric Slivka
Apple today seeded Build 10F569 of Mac OS X 10.6.4 to developers. The new build comes six days after the last seed and continues to list no known issues with the update. As with the last several developers builds, testers are asked to once again focus their efforts on Graphics Drivers, SMB, USB, Voice Over, and VPN.
Seeding of Mac OS X 10.6.4 began in late April and today's build represents the eighth version pushed out to developers for testing.
A public release of Mac OS X 10.6.4 had been rumored to occur during last week's Worldwide Developers Conference, but Apple appears to still be putting the final touches on the next maintenance update for Mac OS X Snow Leopard. The most recent public version of Snow Leopard, Mac OS X 10.6.3, was released in late March.
[ 47 comments ]

redpajama_set_name: RedPajamaCommonCrawl


text:
Sidmouth Town, 3-1 down with four minutes left - win 4-3!
Published: 2:46 PM February 4, 2017 Updated: 6:54 PM November 7, 2020
- Credit: Archant
Sidmouth Town were involved in an extraordinary game when they visited Bovey Tracey to contest three league points.
The Vikings played against nine men for 63 minutes after Bovey Tracey lost one player to bad language directed to the match official and another for dangerous play!
Despite being two men short, the home side found themselves 3-1 up with just five minutes of the match to play, and trooped off the pitch beaten 4-3!
Town boss Josh Stunell said: "I have been in the game for some time, but I cannot ever recall a game quite as strange as this one. For much of the game we were poor and well below the levels we have shown in recent games – and indeed what I expect from this group of players."
He continued: "It was very hard for me at the end because, when your team has just come from behind to win in such fashion you want to praise the players, but in all honesty this was a woeful. We lacked both intensity and work rate in a woeful all-round show and we can – and will – do better."
Town fell behind in the early exchanges, but, within a minute were level with Tyler Wellman drilling the ball home from the edge of the area.
The home team then lost two of their number and yet Town fell 2-1 and then 3-1 down with both goals down to individual mistakes.
With four minutes of normal time to play Connor Swingler was felled in the box and Ben Farrington scored the resultant penalty.
Dan Norman then came off the bench to level things up and then, in the one and only minute of added time Sean Densham headed the winner at the far post.
Stunell said: "With 15 minutes to go I was in despair. We must have had 30 attempts to the three they had managed. For me the clock was clearly ticking and I genuinely wondered where on earth we'd go from this one if we lost to a 9-man side.
"Then the footballing gods delivered, we got the spot kick and quickly levelled.
"At 3-3 there was no thought of shut up shop and take the point. No, quite the opposite as we went straight back up their end to get a winner and duly got it!"
Despite the below-par show, the bottom line is that Town have now won three successive matches, scoring 14 goals in the process and that has seen confidence levels rise considerably at Manstone Lane.
This Saturday they are on the road again with a visit to a Torridgeside outfit that are big scorers themselves and netted five on their visit to East Devon earlier in the season.
Stunell says of the trip to North Devon: 2They [Torridgeside] have been scoring goals for fun and we have begun to fine the net on a regular basis so we ought to see a goal fest at Torridgeside on Saturday afternoon."


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Q: asp.net mvc storing user data how should I store user data in asp.net mvc? Let's say a user want to see 50 records per page. I wanted to save it in Session, but if I am doing it right, the Session resets every time a new controller is initialized. So where? A cookie?

A: Typically the session is not reset on controller initialization! Make sure you aren't clearing the session from code.
Anyway, storing this in session cause the record-limit to be reset quite often (depend on the session timeout param).
Consider storing this in the user's profile kept in database (will be used after log in), or in cookie (don't require login to be used). This will keep this setting forever - your users will appreciate that :)

A: Instead of using the built in ProfileProvider-system in ASP.NET, which you should only use i you want to persist user settings across multiple visits, you could instead put a the settingsdata in the session. Maybe wrapped in a serializable object.
Session is cleared if

*

*you clear it in your code

*the cookie storing the sessionid expires (depends on your settings i web.config) (if a
cookie expires during a session, it does not truly expire before the user closes all browser windows)

*if the application is restarted (unless you use sticky sessions (DB based sessions) in which case sessiondata persists through application restart)

A: Session does not reset when a new controller is initialized. But it does when you leave the application (your session ends) or the application is restarted. You should use Profile to store this kind of information.
See this:
http://msdn.microsoft.com/en-us/library/2y3fs9xs.aspx
http://www.odetocode.com/articles/440.aspx


text:
Q: Rational canonical form of some square of a matrix Let $$ B =  \begin{bmatrix} -2 & 0 & 0 \\ -1 & -4 & -1 \\ 2 & 4 & 0 \\ \end{bmatrix} $$
Then
$$ B^2 = A = \begin{bmatrix} 4 & 0 & 0 \\ 4 & 12 & 4 \\ -8 & -16 & -4 \\ \end{bmatrix} $$
But the square of rcf of B is not equal to rcf of A.
Why is that so ?
Thank you .

A: Because matrix multiplication isn't transitive.
Meaning: Suppose there exists $A, B, C \in M_{n\times n}$ such that :$ C = A\cdot B$.
 Does not mean that $C = B \cdot A$
For that you need to demand more properties.
Let $\hat{B} = \begin{bmatrix} 2 & 0 & 0 \\ -1 & -4 & -1 \\ -2 & 4 & 0 \\ \end{bmatrix}$. notice that $B \neq \hat{B} \ $, but:
$$ B^2 = \left(\begin{bmatrix} -2 & 0 & 0 \\ -1 & -4 & -1 \\ 2 & 4 & 0 \\ \end{bmatrix}\right)^{2}  =  \left(\begin{bmatrix} 2 & 0 & 0 \\ -1 & -4 & -1 \\ -2 & 4 & 0 \\ \end{bmatrix}\right)^{2}  = (\hat{B})^2 =  \begin{bmatrix} 4 & 0 & 0 \\ 4 & 12 & 4 \\ -8 & -16 & -4 \\ \end{bmatrix} = A $$
If $B = \sqrt{A} \ \Longrightarrow \ B^2 = A $, but in order to prove the other part you'd have to demand more properties for $A$ and $B$. ;


text:
Putren is een bestuurslaag in het regentschap Nganjuk van de provincie Oost-Java, Indonesië. Putren telt 5022 inwoners (volkstelling 2010).

Plaats in Oost-Java

redpajama_set_name: RedPajamaWikipedia


The architectural concept of the Soleta houses suggests balance, a close relationship with nature and by the assimilation of innovations and new technologies we watched to reduce the quantity in favour of quality and lifespan.
A human scale concept that offers everything needed for a long and full of joy life. Large and bright spaces, warm materials, clean lines and perfect proportions.
Through the automation options you can control in real time the temperature, natural and artificial lighting, ventilation, energy consumption and the security level of the house.
Soleta zeroEnergy can be defined by the following principles underlying this concept.
We selected premium materials for a premium product. Soleta houses target a segment of connoisseur customers who appreciate quality, good taste and durability of a house. They are made from 97% recycled materials, exclusively from Romania.
So we're a little nationalist but more ecologists - the amount of fuel consumed for the transport of raw materials to the factory is greatly diminished by shortening distances to the sources of supply.
By using high quality materials, the prefabrication of house components is reduced by up to 75% the amount of debris generated at the installation place, making a comparison with a classic construction.
A high-tech solution to reduce energy consumption in Soleta houses and thus to reduce the negative impact on the environment is to implement a computerized energy management system.
The classical solutions already known (high performance thermal insulation, window surfaces with two rooms and inert gas, avoiding the greenhouse effect by calculating shadows, using class A electrical appliances, etc) are already applied as a standard to all SOLETA projects.
Going forward, the SOLETA zeroEnergy concept means to fit the house with energy production systems from unconventional sources to ensure at least the own consumption of the house.
Water is Earth's most precious resource, currently over 25% of the world's fresh water is used for flushing toilets.
By equipping the Soleta houses with rainwater collection systems and microbiological treatment mini-stations, we are making a very big step towards efficient consumption of drinking water.
The classic solutions are also recommended: use of dual flush toilets or of low-flow shower heads.
A house with a healthy indoor environment is a happy house. Our concern for a healthy environment is found in the use of non-toxic building materials, water-based eco lakes, ventilation and air purification systems and rainwater recycling system.
Classic constructions are extremely inefficient in terms of the amount of residues or energy consumed during the site work.
With the new technologies applied, in manufacturing SOLETA houses are produced up to 75% less waste in the manufacturing process. During the assembly of houses, waste is not generated and, by reducing drastically the installation time, the energy consumption specific to this operation is reduced by up to 50%.


text:
Brother-in-law does THIS to stop sister-in-law's marriage, ends up in jail with 'Saali' and her lover
New Delhi: In a unique incident, two people climbed the 150-foot-high tower because they were madly in love with the same girl. One of them was none other than the girl's brother-in-law, who climbed the tower to prevent her sister-in-law's marriage to her lover, prompting the police to arrive.
What's the exact matter?
On Saturday, August 5, in Uttar Pradesh's Lalitpur, a boy climbed a 150-foot-high tower and insisted on marrying his girlfriend. Even after convincing the cops, he refused to come down. In such a situation, when his girlfriend agreed to marry him, he got down with great difficulty. However, the brother-in-law then committed an act that required all three of them to go to the police station.
When the matter of marriage with the lover came to light, the brother-in-law climbed the same 150-foot-high tower to end the marriage and declared that he would not return until the marriage was ended. After that, the police had to return to the scene, and seeing the gravity of the situation, they took the brother-in-law, sister-in-law, and lover to the police station.
According to the police report, a girl from Jhansi used to live at her brother-in-house law's in Lalitpur. During this time, she fell in love with a young man, but her brother-in-law and his family found out. So they began beating him and banning him. In such a situation, the girl became terrified and decided to leave her lover. But where was this lover supposed to put his trust? He climbed the tower to marry the girl, and in order to get him off the tower, the girlfriend agreed to the marriage condition, but his brother-in-law climbed the tower and ruined the whole thing.
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Bihar: Woman surviving on dialysis after hospital stole kidney on pretext of operating tumor
'Need girl with govt job, will give dowry': Man stands in middle of market with poster, goes viral
'Retired professor couple attacked on heads, killed with sharp object in Bihar,' suspects Police
Intoxicated man sets himself on fire after friend refuses to give phone back
Cities Pranjal Gupta - February 1, 2023


text:
Het seizoen 2012/2013 is het 116e jaar in het bestaan van de voetbalclub Heart of Midlothian FC uit Edinburgh. De club komt uit in de Scottish Premier League 12/13 en neemt ook deel aan de toernooien om de Europa League, Scottish League Cup en de Scottish Cup.

Selectie 2012/13

Transfers

Uitgeleend
  Nr. 11. Andrew Driver naar  Houston Dynamo
  Nr. 17. David Smith naar  Raith Rovers FC
  Nr. 27. Calum Tapping naar  Alloa Athletic FC
  Nr. 28. Brad McKay naar  Stenhousemuir FC
  Nr. 31. Jack Hamilton naar  Forfar Athletic FC

Vertrokken
  Craig Thomson
  David Obua
  Aaron Murdoch
  Chris Tobin
  Ryan Stewart
  Colin Hamilton
  Matthew Park
  Ian Black
  Adrian Mrowiec
  Stephen Elliot
  Gary Glen
  Jordan Morton
  Jason Thomson
  Chris Kane
  Suso Santana
  Craig Beattie
  Gary Graham
  Johnny Stewart
  David Templeton
  Peter Enckelman
  Ryan McGowan
  Callum Wyllie
  Danny Thomson
  Michael Thomson
  Ewan Saunderson

Aangetrokken
  Peter Enckelman
  Ryan Stevenson
  Liam Gordon
  Alan Combe

Gehuurd
  Danny Wilson
  Michael Ngoo

Zie ook
 Scottish Premier League 12/13
 Europa League
 Scottish League Cup
 Scottish Cup

Heart of Midlothian FC naar seizoen
Heart of Midlothian FC


text:
Tsibi Geva: SUBSTRATA
albertz benda is pleased to present Tsibi Geva: SUBSTRATA, on view from January 9 through February 15, 2020. The exhibition features a new body of work created over the past two years, defined by Geva's focus on deconstruction and abstraction to convey the emotional and cultural
Jan 9th – Feb 15th
New York, 515 W 26 StreetMap
Tsibi Geva
albertz benda is pleased to present Tsibi Geva: SUBSTRATA, on view from January 9 through February 15, 2020. The exhibition features a new body of work created over the past two years, defined by Geva's focus on deconstruction and abstraction to convey the emotional and cultural tenor of his surroundings.
Motifs of terrazzo tiles, stone walls, and latticework, which have been prevalent in Geva's paintings since the nineteen-nineties, are pushed to the forefront in these new works. These Mediterranean urban patterns, once abstracted, become a reflection of a collective subconscious, revealing a conflicted, restless landscape where borders are in a constant negotiation.
The various patterns are applied in layers that flow into one another, in entropic fashion. With expressive and spontaneous gestures, Geva collapses the distance between the precarious cultural amalgam of the present and the region's splintered history. Transcending familiar schemas and grids, Geva continuously constructs, erases, and builds from the found remnants of his process to produce paintings that oscillate between various compositional options within a single canvas.
ABOUT TSIBI GEVA
Born in 1951 at Kibbutz Ein Shemer, Israel, Geva lives and works in Tel Aviv, Israel and Brooklyn, New York. He received a BFA from the Beit Berl School of Arts, Israel and proceeded to study at the New York Studio School, New York.
Concurrent to the show at albertz benda, his work is the subject of a major exhibition at the Herzliya Museum of Art, Israel on view through February 1, 2020. Geva's work has been the subject of numerous exhibitions at international institutions including the Institute of Contemporary Art, Boston; The American University Museum, Washington, DC; MACRO Testaccio Museum, Rome; Mönchenhaus – Museum of Modern Art in Goslar, Germany, and at Israel's leading museums, including the Haifa Museum of Art; the Ashdod Art Museum, Monart Center; The Israel Museum, Jerusalem; and a retrospective at the Tel Aviv Museum of Art.
From 1990 to 2005 Geva was represented in New York by Annina Nosei Gallery. His work is found in the permanent collection of the Israel Museum, Jerusalem; The Museum of Modern Art, New York; The Rothfeld Collection, Washington DC; The Burger Collection, Zurich; The Jewish Museum, New York; and the Tel Aviv Museum of art among others. In 2015, Geva was chosen to represent Israel at the 2015 Venice Biennale.
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Once Upon A Time's Monomythic Fairy Tale
Posted on 4 November 2011 | Leave a comment
"I know the hero never believes at first. If he did, it wouldn't be a very good story."
Diving back into this blog after a number of monumental changes in my life (new program, new city, new work), I'm going back to an old interest of mine: the durability of the monomyth. The monomyth, also known as the hero's journey, was identified by Joseph Campbell as an underlying structure of the majority of Western folklore and myths. It follows the hero through the call to adventure, various trials, victory, and finally reintegration into society. I wrote my undergraduate thesis in part about the re-workings of this narrative structure in Jewish-American fiction, and part of my work on that project included looking at the variety of ways the hero's journey structure has appeared in recent popular culture. The best example of this narrative is, of course, Star Wars. George Lucas directly attributes much of the film's structure to Joseph Campbell's The Hero With A Thousand Faces. Star Wars succeeds–at least partially–because it wears its monomyth on its sleeve. While I appreciate the variety of cultural producers who have twisted and reworked the hero's journey over the years, there's value in the commitment to telling a familiar story well. Which brings me to Once Upon A Time.
Like, apparently, 12 million other American viewers, I tuned into Once Upon A Time's premiere two weeks ago. I was instantly enamored with the fairy-tales-in-our-world program, but I didn't quite know why. I had thought I'd be constantly comparing the show to Bill Willingham's brilliant comic series, Fables, and find it wanting, but while watching the pilot that thought rarely entered my mind. Perhaps it was the unbridled cheese, perhaps it was the fantasy, or perhaps it was the power of the Disney machine. I couldn't pin the appeal down beyond it being a fun ride. Then I watched last Sunday's episode in which the plot-device-in-child-form, Henry utters the quote that starts this post. He's practically citing Campbell, referencing the Call to Adventure and more explicitly, the hero's Refusal of the Call (here, only an initial refusal).
Once Upon A Time appears to be taking Campbell's conception of the hero of myth and applying it to the world of fairy tales. In The Hero With A Thousand Faces Campbell writes:
The composite hero of the monomyth is a personage of exceptional gifts. . . . He and/or the world in which he finds himself suffers from a symbolical deficiency. In fairy tales this may be as slight as the lack of a certain golden ring, whereas in apocalyptic vision, the physical and spiritual life of the whole earth can be represented as fallen, or on the point of falling into ruin. Typically, the hero of the fairy tale achieves a domestic, microcosmic triumph, and the hero of myth a world-historical macrocosmic triumph. (37-8)
In Once Upon A Time, a story explicitly and wholly intent on fairy tales deficient in their own symbolic history, the fairy tale hero and the mythical hero are interestingly conflated. Emma Swann, the protagonist pictured above (with the Star–Wars-echoing designation of "The Hope") is the child of Snow White and Prince Charming, destined to awake the town filled of fairy-tale figures from their ignorance of their own history. The macrocosm of the fairy-tale kingdom from where they came has been funneled into the microcosm of Storeybrook, Maine, the town Emma must save. In her domestic achievement, she, supposedly, will save an entire world. Her goal, in fact, is the reintegration of society itself (melding fairy-world and Storybrooke), and in doing so, reintegrating herself into the family she never knew she had (Snow White and Prince Charming).
With only two episodes aired so far, I can only speculate on the true influence of the monomythic structure on the show, but there certainly appears to be some intriguing affinities between the two. Emma's son, Henry, appears to be fulfilling dual roles as Caller to the hero and her supernatural guide (despite his key characteristic being that he is not in any fairy tales), telling her about her destiny and giving her the knowledge he believes she needs to fulfill her storybook role. When Emma crossed the First Threshold by deciding to stay in Storybrooke, it is a moment of re-invigoration more than self-annihilation (marked by the town clock, symbol of the characters being stuck out of time, moving for the first time in memory). The changes to the familiar structure are mild and necessary to keep the story fresh, but at its core, Once Upon A Time seems happy to embrace its fairy-tale hero's journey.


Mr. Hitchens, a prolific essayist and the author of "God Is Not Great: How Religion Poisons Everything," discovered in June 2010 that he had Stage 4 esophageal cancer. He has lately curtailed his once busy schedule of public appearances, but he made an exception for the Atheist Alliance — or "the Triple A," as he called it — partly because the occasion coincided almost to the day with his move 30 years ago from his native England to the United States. He was already in Houston, as it happened, because he had come here for treatment at the MD Anderson Cancer Center, where he has turned his 12th-floor room into a temporary library and headquarters.


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Q: Recommendations for anomaly detection I have a binary classifier. The classifier is trained on both numeric and categorical variables. In a given month, I will have new data coming in, comparable to 5% of the observation count of the training sample. It takes a long time to observe the true binary outcome of these incoming observations, but I know all the right-hand-side variables or features of the observation right away, classic out-of-sample classification problem.
I make predictions on these data using the binary classifier. I would like a means of classifying how well these observations are represented in the training dataset, taking account of both categorical and numeric features. Can anyone recommend a methodology that would yield a score for how well represented an observation is in the development data?
For example, say $X_1$ and $X_2$ are features. In the development data, these variables' magnitudes are usually inversely correlated. If new observations come in with atypical association between the two, positively correlated, this atypical association would contribute to the "anomaly score".

A: Restating your question as I understood it, you have a new subject n+1. Based on the new subject's covariates (predictor variables), you want to know whether there are a good number of subjects who are similar enough to the new subject within your training set of n subjects.
One approach to consider is using multivariate outlier detection methods. I'm partial to methods that use a Mahalanobis distance, which can handle continuous and categorical variables together - albeit with some issues that for your needs you can probably ignore. You can replace k level categorical variables with k-1 dummy variables in the covariate matrix. Before considering the new subject, it will be helpful to apply your detection method to your n training subjects alone. This will help you discover and think about special features of your training set, which may influence how you uniquely implement the method for your data. A final note, a subject being classified as an outlier doesn't guarantee the model will predict poorly for them. It just means your data is sparse in this area so the model's performance in this region has a lot of uncertainty.
There are many methods to consider, but these two are a good place to start.
(1) P. Filzmoser, R.G. Garrett, and C. Reimann. Multivariate outlier detection in exploration geochemistry. Computers & Geosciences, 31:579-587, 2005.
-- implemented in the R package mvoutlier.
(2) Andrea Cerioli. Multivariate outlier detection with high-breakdown estimators. Journal of the American Statistical Association, 105(489):147-156, 2010. doi: 10.1198/jasa.2009.tm09147
-- implemented in the R package CerioliOutlierDetection.