text:
\section{Introduction}
In modern distributed computing paradigms such as federated learning (FL), a group of agents typically interact with a parameter server to train a common statistical model. A major bottleneck in such settings is the network communication cost of uploading (potentially high-dimensional) models and gradient vectors to the server. Motivated by this emerging concern, several works draw on ideas from quantization theory \cite{seide,strom, wen,qsgd,khirirat,DIANA,ADIANA,horvathq,fedpaq,Fedcomgate}, sparsification \cite{emp1,emp2,alistarhsparse,stichsparse,reddySignSGD,reddystich,beznosikov,lin_comp,mitraNIPS21}, and rate-distortion theory \cite{mitchell} to design communication-efficient algorithms that achieve a desired level of precision while exchanging as few bits as possible. This rich body of work contributes significantly to the study of static optimization problems under communication constraints. Nonetheless,  there remains a considerable gap in our understanding of similar questions when it comes to sequential decision-making under uncertainty (e.g., bandit problems and reinforcement learning). Our primary goal in this paper is to bridge the above gap.

A common abstraction for analyzing optimization under limited communication is one where a worker agent transmits quantized gradients to a server over a finite bit-rate communication channel \cite{mayekar,gandikota,kostina}. Inspired by this model, for our problem of interest, we introduce and study a new linear stochastic bandit formulation comprising of an agent connected to a decision-making entity (server) by a noiseless communication channel of finite capacity $B$; see Fig.~\ref{fig:Model}. The agent interacts with an environment and observes noisy rewards that depend linearly on an unknown parameter vector $\theta_* \in \mathbb{R}^d$. It then encodes and transmits finite-precision estimates of  $\theta_*$ to the server. Based on these estimates, the role of the server is to play a sequence of actions that maximizes the sum of rewards accrued over a time horizon $T$ - a performance metric captured by cumulative regret.\footnote{A formal description of our setup is provided in Section \ref{sec:model}.} Notably, the agent can only transmit encoded estimates of the parameter vector, but not the rewards themselves. The reason for this is twofold. First, our formulation is motivated by the popular federated learning framework \cite{konevcny} where due to privacy concerns, agents exchange their local models with the server instead of their raw observations. In our setup, the parameter vector is the model and the rewards are the observations. Second, our goal is to build a theory that is eventually applicable to multi-agent decision-making. For such settings, it is more natural for the server to perform fusion on the agents' local models instead of directly fusing observations that may belong to different signal spaces.

The main technical challenge in our setup arises from the fact that the channel from the agent to the server introduces additional uncertainty into the decision-making process. Unless accounted for carefully, the instantaneous encoding errors resulting from such uncertainty can accumulate over time and lead to sub-optimal regret bounds. Given this challenge, the central question we investigate is the following.

\vspace{1mm}
\textit{Under what conditions on the channel capacity $B$ can we achieve the order-optimal regret bound $\tilde{O}(d\sqrt{T})$?}\footnote{When the channel has infinite capacity, i.e., when $B=\infty$, $\tilde{O}(d\sqrt{T})$ regret is known to be optimal; see Chapter 24 of \cite{tor} for further details on this topic.}

\vspace{1mm}
In this work, we rigorously answer the above question via a set of algorithmic and theoretical contributions discussed below.

$\bullet$ \textbf{Algorithmic Contributions.} For the setting of interest, we develop a novel framework for statistical decision-making under communication constraints. Our approach hinges on two main components. The first is an adaptive quantization mechanism that encodes the change (\textit{innovation})  in successive estimates of $\theta_*$ at the agent. The main intuition here is that with high probability, the gap between successive model estimates shrinks over time; as a result, the innovation signals are contained in balls of progressively smaller radii. Thus, roughly speaking, to achieve the same precision, it takes fewer bits to encode the innovation signals as compared to the model estimates (that can be of a much larger magnitude). A key feature of our encoding scheme is that the dynamic quantizer ranges are designed based on statistical concentration bounds specific to the stochastic process we study. As such, our encoding scheme is novel,  and differs significantly from standard quantization approaches for optimization.

The second integral component of our framework is the decision-making policy at the server that comprises of two phases: (i) a pure exploration phase that facilitates the estimation of $\theta_*$, and (ii) an information-constrained exploration-exploitation phase. Specifically, in the latter phase, actions are taken based on certain  ``inflated" confidence sets that are carefully constructed: the radii of such sets need to be large enough to account for the errors induced by compression. At the same time, the compression errors need to gradually shrink to ensure that taking decisions based on ``inflated" confidence sets does not lead to sub-optimal regret bounds. Thus, the design of the  encoding scheme at the agent is tightly coupled with the  decision-making policy at the server. Notably, the construction of the confidence sets is a key  algorithmic contribution of our work that sets it apart from communication-constrained static optimization where a decision-making component is absent. We refer to our overall scheme as the Information-Constrained \texttt{LinUCB} algorithm (\texttt{IC-LinUCB}).

$\bullet $ \textbf{Theoretical Contributions.} Our first main result (Theorem \ref{thm:ICLinUCB}) reveals that with a channel capacity $B=O(d)$ bits, \texttt{\texttt{IC-LinUCB}} guarantees a regret bound of $\tilde{O}(d\sqrt{T})$. The main implication of this result is that one can achieve order-optimal regret guarantees  with a bit-rate that is independent of the horizon $T$, and that depends only on the dimension $d$ of the unknown model $\theta_*$. As far as we are aware, this is the first result of its kind for linear stochastic bandits, and complements similar results for stochastic optimization: the authors in \cite{mayekar} recently showed that with $d$-dimensional quantized gradients, a bit-rate of $\tilde{O}(d)$ bits is sufficient for achieving the optimal optimization convergence rate.  On the technical front, we note that the proof of Theorem \ref{thm:ICLinUCB} is non-trivial, and relies on some key intermediate ideas that we outline in Section \ref{sec:guarantees}.

To demonstrate the generality of our approach, we significantly extend our analysis to the generalized linear bandit setting that accounts for non-linear observation models \cite{GLM1,GLM2}. Once again, we establish that with a bit-rate of $O(d)$ bits, one can achieve optimal regret bounds; see Theorem \ref{thm:ICGLMUCB} for a formal statement of the result.

Finally, we ask: \textit{When the action sets have additional structure, can we exploit such structure to achieve optimal performance with fewer than $O(d)$ bits?} To answer this question, we study a special case of the linear bandit problem where the actions are the standard orthonormal basis vectors. This setting corresponds to the multi-armed bandit (MAB) problem with a finite number of arms \cite{auer}. For this setting, we prove that with a bit-rate $B=1$, one can achieve both gap-dependent (Theorem \ref{thm:ICUCB}) and gap-independent (Theorem \ref{thm:gap_ind}) regret bounds matching those of the celebrated upper-confidence bound (\texttt{UCB}) algorithm.

Overall, we envision that the algorithmic and analytical insights from this work will pave the way for studying more complex statistical decision-making problems in distributed and multi-agent settings under channel capacity constraints.

\textbf{Further Related Work.} Our formulation is inspired by the classical work \cite{tatikonda} that studies the problem of stabilizing a linear time-invariant dynamical system over a bit-constrained channel. There, as in our setup, the estimation module (sensor) is separated from the decision-making module (controller) by the channel. Aside from the fact that we study a fundamentally different problem, our work departs from \cite{tatikonda} in that our setup is inherently stochastic, while the authors in \cite{tatikonda} consider a fully deterministic setting. In particular, while the state estimates encoded in \cite{tatikonda} are deterministic, the model parameter estimates that we encode in our setting are high-dimensional random vectors.

Our work is naturally related to the seminal papers on linear stochastic bandits \cite{dani,abbasi} that introduce and analyze the \texttt{LinUCB} algorithm. The results in this paper extend those in \cite{dani,abbasi} to the communication-constrained setting that we study. In the context of multi-agent bandits \cite{landgren1,landgren2,shahrampour,kolla,dMAB3,sankararaman,martinez,dubeyICML20,dubeyNIPS20,lalitha20,chawla1,chawla2,ghoshbandits,agarwal21,FedMAB1,FedMAB3}, a body of work focuses on achieving benefits of collaboration while minimizing the number of communication rounds \cite{dMAB3, dubeyNIPS20, chawla1, agarwal21}. The main goal of these papers is to achieve desirable performance while minimizing the \textit{frequency} of communication. Our focus is orthogonal - that of studying the impact of finite-precision communication channels on the performance of bandit algorithms. As a result, our problem formulation, algorithmic techniques, and theoretical results differ considerably from the above strand of literature.

\textbf{Notation}. Given two scalars $a$ and $b$, we use $a \vee b$ and $a \wedge b$ to represent $\max\{a,b\}$ and $\min\{a,b\}$, respectively.  For any positive integer $n$, we use $[n]$ to denote the set of integers $\{1, \ldots, n\}$. We use $\mathcal{B}_d(0,1)$ and $\mathbb{S}^{d-1}$ to represent the $d$-dimensional Euclidean ball and the $d$-dimensional Euclidean sphere, respectively, of unit radius centered at the origin. Given a matrix $A$, we use $\lambda_{\max}(A)$ and $\lambda_{\min}(A)$ to represent the largest and smallest eigenvalues, respectively, of $A$. Moreover, we use $A'$ to denote the transpose of $A$. Given two symmetric positive semi-definite matrices $A$ and $B$, we use $B \preccurlyeq A$ to imply that $A-B$ is positive semi-definite.

\newpage
\section{Model and Problem Formulation}
\label{sec:model}

\begin{figure}[t]
\begin{center}
\begin{tikzpicture}
[->,shorten >=1pt,scale=.75,inner sep=1pt, minimum size=12pt, auto=center, node distance=3cm,
  thick, node/.style={circle, draw=black, thick},]
\tikzstyle{block1} = [rectangle, draw, fill=red!10,
    text width=8em, text centered, rounded corners, minimum height=0.8cm, minimum width=1cm];
\node (rect) at (0,0) (A) [draw, rounded corners, minimum width= 1.5cm, minimum height=1cm] {Agent};
\node (rect) at (6,0) (S) [draw, rounded corners, minimum width= 1.5cm, minimum height=1cm] {Server};
\coordinate (a1) at (1, 0.4);
\coordinate (s1) at (5,0.4);
\draw[->,thick]
 (s1) -- node[pos=.5,above]{action $a_t$} (a1);

\coordinate (a2) at (1,-0.4);
\coordinate (s2) at (5,-0.4);
\draw[->,thick]
 (a2) -- node[pos=.5,below] {$B$-bit symbol $\sigma_t$}  node[pos=.5,above] {Channel} (s2);

\node at (0,-1.4) {Observe $y_t$};
\node at (0,-2) {and encode $\theta_*$};
\node at (6, -1.4) {Decode $\theta_*$};
\node at (6, -2) {and play $a_{t+1}$};
\end{tikzpicture}
\end{center}
\caption{At each round $t$, the action $a_t$ played by the server is sent to the agent without any loss of information. The agent then observes a reward $y_t$ as per Eq. \eqref{eqn:obs_model}, encodes an estimate of the model $\theta_*$, and transmits the encoded symbol $\sigma_t$ back to the server under the $B$-bits per round channel constraint. The server performs decoding and plays the next action $a_{t+1}$.}
\label{fig:Model}
\end{figure}

We study a setting comprising of an agent and a decision-maker (server) separated by a noiseless communication channel of finite capacity; see Fig.~\ref{fig:Model}. Based on all the information acquired by the server up to time-step $t-1$, it chooses an action $a_t \in \mathcal{A}_t$ at time $t$, where $\mathcal{A}_t \subset \mathbb{R}^d$ is the feasible decision set at time $t$. The agent then receives a reward (observation) according to the following model:
\begin{equation}
    y_t= \langle \theta_*, a_t \rangle + \eta_t,
\label{eqn:obs_model}
\end{equation}
where $\{\eta_t\}$ is a sequence of i.i.d. $1$-subgaussian noise random variables.  Here, $\theta_*$ is an unknown parameter that belongs to a known compact set $\Theta \subset \mathbb{R}^d$; for each $\theta \in \Theta$, it holds that ${\Vert \theta \Vert}_2 \leq M$, where $M \geq 1$. Our performance measure of interest is the following regret metric $R_T$:
\begin{equation}
    R_T = \mathbb{E} \left[\sum_{t=1}^{T} \max_{a\in\mathcal{A}_t} \langle \theta_*, a-a_t \rangle \right],
\label{eqn:regret}
    \end{equation}
where $T$ is the time horizon. The goal of the server is to play a sequence of actions such that ${R}_T$ grows sub-linearly in $T$. When there is no loss of information from the agent to the server (i.e., in the absence of the  channel), it is well known that one can achieve  $\tilde{O}\left(d\sqrt{T}\right)$ regret by following the popular \texttt{LinUCB} algorithm \cite{abbasi}. Our \textbf{goal} in this work is to develop an algorithm that achieves the same performance subject to communication constraints that we describe next.

\textbf{Communication constraints.} To capture communication constraints, we assume that the channel from the agent to the server has a finite capacity of $B$ bits. Thus, at each time-step, the channel can transmit without error one of $2^B$ symbols denoted by $\sigma \in \Sigma$, where $\vert \Sigma \vert = 2^B$. As explained and motivated in the introduction,  we impose an additional information constraint that the agent can only transmit encoded estimates of the unknown model parameter $\theta_*$, but not the rewards themselves. We note here that encoding a high-dimensional model estimate is much more challenging than encoding a scalar-valued reward.

In section \ref{sec:guarantees}, we will establish that with $B=O(d)$ bits, one can ensure that  $R_T = \tilde{O}\left(d\sqrt{T}\right)$. Arriving at this result is however  quite  non-trivial, and requires overcoming certain key technical challenges that we outline next.

\textbf{Challenges.} In the standard linear stochastic bandit formulation, the chief difficulty lies in taking decisions that incur low regret despite statistical uncertainty concerning the unknown parameter $\theta_*$. In our setting, such uncertainty is accentuated by the loss of information incurred over the finite-capacity channel. Unless the server explicitly accounts for this additional source of error in its decision-making process, it can end up taking sub-optimal actions that generate low rewards. Moreover, since our problem is of an inherently sequential nature, the effect of ``poor" actions coupled with channel-induced errors can pile up over time, resulting in the agent-server pair suffering linear regret. The above discussion highlights the challenge in decision-making.

In terms of communication, one natural idea to encode the parameter $\theta_*$ could be to uniformly quantize the set $\Theta$ at each time-step, since $\theta_* \in \Theta$. To ensure that the sum of the instantaneous encoding errors do not grow linearly with the horizon $T$, such errors need to scale inversely with $T$. However, to achieve such a precision with a  \textit{non-adaptive} encoding scheme that always encodes the entire set $\Theta$, the channel capacity $B$, in turn, needs to scale with $T$.
This is highly undesirable since the horizon-length $T$ can be arbitrarily long. To sum up, the design of a joint encoding-decoding  and decision-making strategy that achieves order-optimal regret with a horizon-independent channel capacity is not at all obvious a priori. Nonetheless, we will develop such a strategy in the next section. For now, we lay down certain standard technical assumptions that will be made throughout the paper.

\begin{assumption} \label{ass:actions}
The following hold: \begin{enumerate}
    \item[(i)] $\max_{t\in [T]} \sup_{a,b \in \mathcal{A}_t} \langle \theta_*, a-b \rangle \leq 1.$
    \item[(ii)] ${\Vert a \Vert}_2 \leq L$, $\forall a \in  \bigcup_{t=1}^{T} \mathcal{A}_t$. \item[(iii)] At each time-step $t\in[T]$, the decision set $\mathcal{A}_t$ contains the unit sphere $\mathbb{S}^{d-1}$.
\end{enumerate}
\end{assumption}
While assumptions (i) and (ii) are typical in the literature on linear stochastic bandits \cite{tor}, assumption (iii) is also quite standard and has been used in various different contexts \cite{amani,yangICLR}. Without loss of generality, we assume that $L \geq 1$; furthermore, we assume that the horizon is long-enough relative to the dimension of the model:  $T \geq d^2$.

\section{Information-Constrained Optimism in the Face of Uncertainty}
\label{sec:lin_algo}

In this section, we will develop our proposed algorithm (Algorithm \ref{algo:ICLUCB}) called Information Constrained \texttt{LinUCB}  (\texttt{IC-LinUCB}) that comprises of two phases. Phase I is a pure exploration phase where the server picks i.i.d. actions from the uniform distribution over the unit sphere; such actions are feasible owing to Assumption \ref{ass:actions}-(iii). During this phase which lasts for $\bar{T}+1$ time-steps, the only transmission from the agent to the server takes place at time-step $\bar{T}+1$. The purpose of the pure exploration phase and the choice of the parameter $\bar{T}$ will be explained shortly. During each time-step of Phase II, the agent employs an \textit{adaptive} encoding strategy (outlined in Algorithm~\ref{algo:Encoder}) to transmit information about the unknown parameter $\theta_*$ to the server. Based on this information, the server takes decisions by constructing an ``inflated" confidence that accounts for encoding errors. We now describe in detail the two key ingredients of $\texttt{IC-LinUCB}$: (i) the adaptive encoding strategy at the agent, and (ii) the decision-making rule at the server.

$\bullet$ \textbf{Adaptive Encoding at Agent.} To describe the encoder, we will require the notion of an $\epsilon$-net \cite{versh}.

\begin{definition} (\textbf{$\epsilon$-net}). Consider a subset $\mathcal{K} \subset \mathbb{R}^{d}$ and let $\epsilon > 0$. A subset $\mathcal{N} \subseteq \mathcal{K}$ is called an $\epsilon$-net of $\mathcal{K}$ if every point in $\mathcal{K}$ is within a distance of $\epsilon$ of some point of $\mathcal{N}$, i.e.,
$$ \forall x \in \mathcal{K}, \exists x_0 \in \mathcal{N}: {\Vert x-x_0 \Vert}_2 \leq \epsilon. $$
Equivalently, $\mathcal{N}$ is an $\epsilon$-net of $\mathcal{K}$ if and only if $\mathcal{K}$ can be covered by balls with centers in $\mathcal{N}$ and radii $\epsilon$.
\end{definition}
Next, consider the least-squares estimate $\hat{\theta}^{(a)}_t$ maintained by the agent:
\begin{equation}
\hat{\theta}^{(a)}_t=V^{-1}_t \sum_{s=1}^{t}a_{s}y_{s}, \hspace{2mm} \textrm{where} \hspace{2mm}
    V_t = \lambda I_d + \sum_{s=1}^{t}a_s a'_s
\label{eqn:least_square}
\end{equation}
is the covariance matrix at time-step $t$.\footnote{We use $x'$ to denote the transpose of a vector $x$.}  Here,  $\lambda >0$ is a scalar regularization parameter. Let $\hat{\theta}^{(s)}_t$ be the estimate of $\theta_*$ maintained by the server; $\hat{\theta}^{(s)}_t$ is initialized from any arbitrary vector in $\Theta$ at time-step $\bar{T}+1$. The choice of this initial vector is known to both the agent and the server.

\textbf{Main Ideas.} The key ideas guiding our encoding strategy are as follows. Once the agent has acquired  sufficiently many observations, the gap $\hat{\theta}^{(a)}_{t}-\hat{\theta}^{(a)}_{t-1}$ between successive estimates will start shrinking due to the pure exploration phase; see Remark \ref{rem:Pureexp}.  Thus, at this stage, if the gap  $\hat{\theta}^{(a)}_{t-1} - \hat{\theta}^{(s)}_{t-1}$ is not too large, then the gap $e_t=\hat{\theta}^{(a)}_{t} - \hat{\theta}^{(s)}_{t-1}$ should not be too large either. In other words, eventually, a new observation $y_t$ will not cause the agent's estimate of $\theta_*$ to deviate drastically from the estimate of $\theta_*$ held by the server. Intuitively, it thus makes sense to encode and transmit only the \textit{new information} about $\theta_*$ contained in $y_t$, i.e., the ``innovation" signal $e_t$ (as opposed to encoding $\hat{\theta}^{(a)}_{t}$). However, given the stochastic nature of our setup, $e_t$ is a random variable. Thus, encoding $e_t$ poses the technical hurdle of characterizing the region containing $e_t$ with high probability. To this end, in Lemma \ref{lemma:enc_dec} of Section \ref{sec:guarantees}, we establish that with high probability, $\forall t \geq \bar{T}+1$, $e_t \in \mathcal{B}_d(0,p_t)$, where $p_t$ is the radius of the ball containing the innovation $e_t$. Our encoding strategy is adaptive since it requires dynamically updating the radius $p_t$ (as per Eq. \eqref{eqn:quantizer_eqs}) based on statistical concentration bounds specific to our problem.
\begin{figure}[t]
\centering
  \includegraphics[width=0.75\linewidth]{cover.png}
  \caption{Illustration of the  encoding technique in Algorithm   \ref{algo:Encoder}. The agent computes the innovation signal $e_t$ that belongs to $\mathcal{B}_d(0,p_t)$ with high probability. An $\epsilon p_t$-net of $\mathcal{B}_d(0,p_t)$ is constructed, and the center $\tilde{e}_t$ of the ball containing $e_t$ is decoded by the server.}
  \label{fig:covering}
\end{figure}

\begin{algorithm}[H]
\caption{Adaptive Encoding at the Agent}
\label{algo:Encoder}
 \begin{algorithmic}[1]
\Statex \hspace{-5mm} \textbf{Input Parameters:} $\hat{\theta}^{(s)}_{\bar{T}}$ is any arbitrary vector in $\Theta$; $q_{\bar{T}}=10M$; and $f(T)=\frac{3}{5L}\sqrt{\frac{\beta_T}{T\log(dLT)}}$.
\For {$t\in \{\bar{T}+1, \ldots, T\}$}
\State Observe $y_t$; compute $\hat{\theta}^{(a)}_t=V^{-1}_t \sum_{s=1}^{t}a_{s}y_{s}$ and innovation $e_t=\hat{\theta}^{(a)}_t-\hat{\theta}^{(s)}_{t-1}$.
\State Encode $e_t$ by constructing an $\epsilon p_t$-net of $\mathcal{B}_d(0,p_t)$, where
\begin{equation}
  q_{t}=\epsilon \left( q_{t-1} + f(T)\right); \hspace{2mm}  p_t=q_t+f(T).
\label{eqn:quantizer_eqs}
\end{equation}
\EndFor
\end{algorithmic}
 \end{algorithm}

\textbf{Summary of Encoding Strategy.} The overall encoding technique in Algorithm \ref{algo:Encoder} can be summarized as follows. At each time-step $t \geq \bar{T}+1$, the agent observes $y_t$, computes $\hat{\theta}^{(a)}_t$ as per \eqref{eqn:least_square}, and then evaluates the innovation signal $e_t=\hat{\theta}^{(a)}_t-\hat{\theta}^{(s)}_{t-1}$. Given that $e_t \in \mathcal{B}_d(0,p_t)$ with high probability (as justified by Lemma \ref{lemma:enc_dec}),  the region  $\mathcal{B}_d(0,p_t)$ is covered by balls of radius $\epsilon p_t$, where $\epsilon\in (0,1)$ is a pre-decided constant, i.e., the agent constructs an $\epsilon p_t$-net of $\mathcal{B}_d(0,p_t)$.\footnote{For a discussion on constructing such coverings, see \cite{dumer,verger} and the references therein.} The agent then determines the ball $e_t$ falls into, and transmits the symbol $\sigma\in\Sigma$ corresponding to that ball.\footnote{In case $e_t$ lands on the boundary of more than one ball, it is assigned the label/symbol of any one of those balls based on a fixed priority rule.} If $e_t$ falls outside $\mathcal{B}_d(0,p_t)$, the agent transmits a special symbol to indicate an overflow. We succinctly represent the entire operation described above by a dynamic encoder map $\mathcal{E}_t$ that takes as input $e_t$  and generates as output the symbolic encoding $\sigma_t \in \Sigma$ that is transmitted to the server.

\textbf{Decoding at Server.} As a basic requirement for correct decoding, we assume that the server is aware of the encoding operation at the agent. Moreover, note that the sequences $\{p_t\}$ and $\{q_t\}$  defined in Eq. \eqref{eqn:quantizer_eqs} are deterministic, and can be computed by the server at its end. Thus, at any time-step $t\geq \bar{T}+1$, the server is aware of the region $\mathcal{B}_d(0,p_t)$ being encoded. Upon receiving $\sigma_t$, the server can thus correctly determine the center $\tilde{e}_t$ of the ball containing  $e_t$. We represent the above decoding operation at time $t$ by the decoder map $\mathcal{D}_t$ that takes as input $\sigma_t$ and outputs $\tilde{e}_t$. Having decoded the innovation signal, the server computes an estimate $\hat{\theta}^{(s)}_t$ of $\theta_*$ as per line 7 of Algo. \ref{algo:ICLUCB}. The agent computes $\hat{\theta}^{(s)}_t$ on its end as well in order to evaluate the innovation signal at time $t+1$; see line 2 of Algo. \ref{algo:Encoder}. This concludes the description of the encoding-decoding operation, a pictorial illustration of which is presented in Figure \ref{fig:covering}.

Till now, we have only described how to transit information about $\theta_*$ from the agent to the server over a finite-capacity channel. However, the key question that remains unanswered is the following: \textit{How should the server take decisions that yield low cumulative regret while accounting for the additional uncertainty introduced by the channel?} We now turn to answering this question.

\begin{algorithm}[t]
\caption{Information Constrained \texttt{LinUCB}  (\texttt{IC-LinUCB})}
\label{algo:ICLUCB}
 \begin{algorithmic}[1]
\Statex \hspace{-5mm} \textbf{Input Parameters:} $\bar{T}=\ceil{10 L^2d\sqrt{T}\log(dLT)}$.
\Statex \hspace{-5mm} \textbf{Phase I:} \textit{Pure Exploration}
\For {$t\in \{1, \ldots, \bar{T}+1\}$}
\State Server plays $a_t \sim  \textrm{Unif}(\mathbb{S}^{d-1})$.
\State Agent receives reward $y_t$ as per \eqref{eqn:obs_model} and computes estimate  $\hat{\theta}^{(a)}_t=V^{-1}_t \sum_{s=1}^{t}a_{s}y_{s}$.
\EndFor
\State Agent encodes $e_{\bar{T}+1}=\hat{\theta}^{(a)}_{\bar{T}+1}-\hat{\theta}^{(s)}_{\bar{T}}$ as per Algo. \ref{algo:Encoder}, and transmits $\sigma_{\bar{T}+1}=\mathcal{E}_{\bar{T}+1}\left(e_{\bar{T}+1}\right)$.
\Statex \hrulefill
\Statex \hspace{-5mm} \textbf{Phase II:} \textit{Information-Constrained Exploration-Exploitation}
 \For {$t\in \{\bar{T}+2, \ldots, T\}$}
\State Server decodes $\tilde{e}_{t-1} =\mathcal{D}_{t-1}(\sigma_{t-1})$, and generates $\hat{\theta}^{(s)}_{t-1}=\hat{\theta}^{(s)}_{t-2}+\tilde{e}_{t-1}$.
\State Server constructs confidence ellipsoid:
\begin{equation}
    \mathcal{C}^{(s)}_t = \{\theta\in\mathbb{R}^d: {\Vert \theta - \hat{\theta}^{(s)}_{t-1} \Vert}_{V_{t-1}} \leq \sqrt{\beta_T} + \textcolor{black}{\left(\sqrt{\lambda+(t-1)L^2}\right) q_t\}},
\label{eqn:dec_conf_int}
\end{equation}
where  $V_{t-1}$, $q_t$, and $\sqrt{\beta_T}$ are given by \eqref{eqn:least_square}, \eqref{eqn:quantizer_eqs}, and \eqref{eqn:conf_radius}, respectively.
\State Server plays action $a_t = \argmax_{a\in\mathcal{A}_t} \max_{\theta\in\mathcal{C}^{(s)}_t} \langle \theta, a \rangle$.
\State Agent receives reward $y_t$ as per \eqref{eqn:obs_model} and computes estimate  $\hat{\theta}^{(a)}_t=V^{-1}_t \sum_{s=1}^{t}a_{s}y_{s}$.
\State Agent encodes the innovation $e_t=\hat{\theta}^{(a)}_t-\hat{\theta}^{(s)}_{t-1}$ as per Algo. \ref{algo:Encoder}, and transmits $\sigma_t = \mathcal{E}_t(e_t)$.
\EndFor
\end{algorithmic}
 \end{algorithm}

$\bullet$ \textbf{Decision-Making at the Server.} When there is no loss of information over the channel, i.e., when $\hat{\theta}^{(s)}_t=\hat{\theta}^{(a)}_t$, the celebrated \texttt{LinUCB} algorithm relies on the principle of \textit{optimism in the face of uncertainty.} Specifically, at each time-step, an ellipsoidal confidence set is constructed that contains $\theta_*$ with high-probability. The learner then acts optimistically by playing an action that yields the highest reward over all possible values of $\theta$ in the confidence set. While  our approach builds on the same high-level principle, it  relies crucially on the construction of a new ``inflated" ellipsoidal confidence set:
\begin{equation}
    \mathcal{C}^{(s)}_t = \{\theta\in\mathbb{R}^d: {\Vert \theta - \hat{\theta}^{(s)}_{t-1} \Vert}_{V_{t-1}} \leq \sqrt{\beta_T} + \textcolor{myblue}{\left(\sqrt{\lambda+(t-1)L^2}\right) q_t\}},
\end{equation}
where
\begin{equation}
    \sqrt{\beta_T} = \sqrt{\lambda}M + \sqrt{2\log\left(\frac{1}{\delta}\right)+d\log\left(\frac{d\lambda+TL^2}{d\lambda}\right)}.
\label{eqn:conf_radius}
\end{equation}
Here, recall that $M \geq 1$ is such that for each $\theta\in\Theta$, ${\Vert \theta \Vert}_2 \leq M$; $\delta\in(0,1)$ is a constant to be picked later.  Notably, when $\hat{\theta}^{(s)}_{t-1}=\hat{\theta}^{(a)}_{t-1}$, and $q_t=0$, $\mathcal{C}^{(s)}_t$ reduces to the confidence set in \texttt{LinUCB}. The inflation in the radius of the confidence set (relative to \texttt{LinUCB}) carefully accounts for the quantization errors resulting from the finite capacity of the channel. Our main technical contribution in this regard is to establish that $\forall t\geq \bar{T}+2$, $\theta_* \in \mathcal{C}^{(s)}_t$ with high probability; see Lemma \ref{lemma:dec_conf_region} in Section \ref{sec:guarantees}. This result, in turn,  justifies the optimistic decision-making rule of \texttt{IC-LinUCB} in line 9 of Algorithm  \ref{algo:ICLUCB}. During the pure exploration phase, the server simply samples actions independently from the uniform distribution over $\mathbb{S}^{d-1}$, i.e., $a_t \sim  \textrm{Unif}(\mathbb{S}^{d-1}), \forall t\in[\bar{T}+1]$.\footnote{To be more precise, a random variable $Z$ is uniformly distributed on $\mathbb{S}^{d-1}$ if, for every Borel subset $\mathcal{K} \subset \mathbb{S}^{d-1}$, the probability $\mathbb{P}(Z \in \mathcal{K})$ equals the ratio of the $(d-1)$-dimensional areas of $\mathcal{K}$ and $\mathbb{S}^{d-1}$.} At every time-step $t\in[T]$, the action $a_t$ decided upon by the server is passed down to the agent without any loss of information.

We summarize below the essential steps of our approach.
\begin{itemize}
\item \textbf{Bound Gap between Successive Model Estimates.} Bound the gap $\hat{\theta}^{(a)}_{t}-\hat{\theta}^{(a)}_{t-1}$ between successive estimates of $\theta_*$ (at the agent), and argue that this gap is eventually small due to the pure exploration phase; see Remark \ref{rem:Pureexp}.

\item \textbf{Characterize Encoding Region.} Using the bound from the above step, encode the innovation signal $e_t=\hat{\theta}^{(a)}_{t}-\hat{\theta}^{(s)}_{t-1}$ by characterizing the region that contains $e_t$ with high-probability.

\item \textbf{Construct Inflated Confidence Set.} Construct a confidence set that (i) is centered at the server's estimate of $\theta_*$, namely $\hat{\theta}^{(s)}_t$, (ii) accounts for encoding errors, and (iii) contains $\theta_*$ with high-probability. Play actions optimistically w.r.t. this confidence set.
\end{itemize}

In Section \ref{sec:gen_lin}, we will demonstrate that the above ideas are applicable to parameterized observation models well beyond the specific linear model considered in Eq. \eqref{eqn:obs_model}. As such, our approach provides a general recipe for decision-making under information constraints. Before moving on to the performance guarantees of \texttt{IC-LinUCB}, a couple of important remarks are in order.

\begin{remark} \label{rem:Pureexp}  (\textbf{Need for Pure Exploration Phase}) Note that $\forall t\geq \bar{T}+1$, $\lambda_{min}(V_t) \geq \lambda_{min}(V_{\bar{T}})$. Moreover, for $\bar{T}$ as chosen in Algorithm  \ref{algo:ICLUCB}, the pure exploration phase ensures that $\lambda_{min}(V_{\bar{T}})$ is bounded below by a linear function of $\bar{T}$.  The above two facts collectively imply that the gap between consecutive estimates of $\theta_*$ (at the agent) is eventually small - a key requirement for our encoding strategy. This point is made precise in Lemma \ref{lemma:iterate_bound} of Appendix \ref{app:ThmICLinUCB}. The duration $\bar{T}$ of the pure exploration phase is chosen carefully to ensure that the term $\left(\sqrt{\lambda+(t-1)L^2}\right) q_t$ in $\mathcal{C}^{(s)}_t$ is eventually $O\left(\sqrt{\beta_T}\right)$. In other words, such a choice of $\bar{T}$ enables us to preserve order-optimal regret despite taking actions based on an inflated confidence set.
\end{remark}

\begin{remark} (\textbf{Horizon-Independent Channel Capacity}) In our encoding technique, although the radii of the balls used to cover the encoding region shrink over time, the radius of the ball being encoded shrinks commensurately. This is precisely what enables us to achieve order-optimal regret with a channel capacity $B=O(d)$ bits that is independent of the time-horizon $T$. Roughly speaking, the main intuition here is that to achieve the same level of encoding precision, it takes fewer bits to encode the innovation $e_t$ as compared to the model $\theta_*$. This is because while $e_t$ resides (with high probability) in a ball of progressively shrinking radius, $\theta_*$ belongs to the set $\Theta$ of fixed radius.
\end{remark}
\section{Analysis of the \texttt{IC-LinUCB} Algorithm}
\label{sec:guarantees}
Our main result concerning the performance of the \texttt{IC-LinUCB} algorithm is as follows.

\begin{theorem} (\textbf{Regret of \texttt{IC-LinUCB}}) \label{thm:ICLinUCB}
Suppose Assumption \ref{ass:actions} holds, and let the channel capacity satisfy $B \geq 6d$. Then, with $\epsilon=1/2$ and $\delta=1/T$ in Algorithm \ref{algo:ICLUCB}, the \texttt{IC-LinUCB} algorithm guarantees:
\begin{equation}
    R_T = O\left(L^2 d \sqrt{T} \log(dLT)\right) = \tilde{O}\left(d\sqrt{T}\right).
\end{equation}
\end{theorem}

\textbf{Discussion.} We note that for the \texttt{IC-LinUCB} algorithm, the dependence of the regret on $d$ and $T$ exactly matches that of \texttt{LinUCB}. Thus, our work is the first to establish that with a horizon-independent channel capacity of $O(d)$ bits, one can achieve the same performance as when the channel has infinite capacity. Thus, Theorem \ref{thm:ICLinUCB} can be seen as an extension of the results in \cite{dani,abbasi} to the communication-constrained setting of interest in this work.
Interestingly, \cite{mayekar} recently showed that for stochastic optimization with $d$-dimensional quantized gradients, a bit-rate of $\Omega(d)$ is necessary for achieving the optimal convergence rate of $O(1/\sqrt{T})$, where $T$ is the number of iterations. We conjecture that to achieve order-optimal regret, a similar lower-bound of $B=\Omega(d)$ will hold for our setup as well; we leave verifying this as future work.

We prove Theorem \ref{thm:ICLinUCB} in Appendix \ref{app:ThmICLinUCB}. In what follows, we briefly outline the key technical steps in the proof.

\textbf{Outline of the proof.} We start by constructing an appropriate ``clean" event $\mathcal{G}$ of measure at least $1-5/T$, and condition on this event throughout the subsequent analysis. There are three main steps in the proof of Theorem \ref{thm:ICLinUCB}, and we describe them below.

$\bullet$ \textbf{Step 1.} On the clean event $\mathcal{G}$, we argue that the gap between successive model estimates at the agent is eventually small. More precisely, in Lemma  \ref{lemma:iterate_bound} of Appendix \ref{app:ThmICLinUCB}, we establish that
$$ {\Vert \hat{\theta}^{(a)}_{t+1}-\hat{\theta}^{(a)}_{t}\Vert}_2 \leq f(T), \forall t \geq \bar{T}, $$
where $f(T)$ is as defined in the input parameters of Algorithm \ref{algo:Encoder}. The proof of Lemma \ref{lemma:iterate_bound} in turn relies on the fact that with high probability,
$$ \lambda_{\min}(V_t) \geq 5 L^2 \sqrt{T} \log(dLT), \forall t\geq \bar{T}. $$
The above claim is established in Lemma \ref{lemma:eigen_cov} of Appendix \ref{app:ThmICLinUCB} by appealing to the Matrix Bernstein inequality \cite[Theorem 5.4.1]{versh}.
\vspace{2mm}

$\bullet$ \textbf{Step 2.} The next key result justifies the encoding strategy in Algorithm \ref{algo:Encoder}.\footnote{We note that results of a similar conceptual flavor are established in \cite{tatikonda} and \cite{kostina} in the context of stabilization of an LTI system, and optimization, respectively. While the results in these papers pertain to deterministic settings, Lemma \ref{lemma:enc_dec} carefully exploits statistical concentration bounds specific to the stochastic process we study.}

\begin{lemma} (\textbf{Encoding Region}) The following is true with probability at least $1-5/T$:
$$ e_t \in \mathcal{B}_{d}(0,p_t), \forall t \in \{\bar{T}+1, \ldots, T\},$$
where $e_t$ is the innovation in line 2 of Algorithm \ref{algo:Encoder}, and $p_t$ is as defined in Eq.  \eqref{eqn:quantizer_eqs}.
\label{lemma:enc_dec}
\end{lemma}

The above result tells us that the innovation random variable $e_t$ always falls within the desired encoding region on the event $\mathcal{G}$, i.e., with high probability, there is never any overflow. It is easy to argue that $B=O(d)$ bits suffice to construct an $\epsilon p_t$ net of $\mathcal{B}_d(0,p_t)$.
\vspace{2mm}

$\bullet$ \textbf{Step 3.} It remains to justify the choice of the confidence set $\mathcal{C}^{(s)}_t$ in Eq. \eqref{eqn:dec_conf_int}. This is achieved in the following lemma.

\begin{lemma} (\textbf{Confidence Region at Server}) With probability at least $1-5/T$,  the following is true: $\theta_* \in \mathcal{C}^{(s)}_t, \forall t \in \{\bar{T}+2, \ldots, T\}$, where $\mathcal{C}^{(s)}_t$ is the confidence set defined in Eq. \eqref{eqn:dec_conf_int}.
Moreover, $\forall t \geq \bar{T}+\tilde{T}$, we have
\begin{equation}
   \left(\sqrt{\lambda+(t-1)L^2}\right) q_t \leq 4 \sqrt{\frac{\beta_T}{\log(dLT)}},
\end{equation}
where
\begin{equation}
    \tilde{T} =  \ceil*{\frac{\log\left(\frac{10M}{f(T)}\right)}{\log(2)}} \vee 2 = O\left(\log(dLT)\right).
\end{equation}
\label{lemma:dec_conf_region}
\end{lemma}

The above result implies that the inflated confidence set (that accounts for encoding errors) eventually contains the true parameter $\theta_*$ with high probability. At the same time, the quantization error $q_t$ decays fast enough to ensure that the radius of the confidence set is eventually $O(\sqrt{\beta_T})$ - exactly as in the \texttt{LinUCB} algorithm. In other words, our approach ensures that the impact of the quantization error on decision-making vanishes over time.

Equipped with Lemma's \ref{lemma:enc_dec} and \ref{lemma:dec_conf_region}, the rest of the proof of Theorem \ref{thm:ICLinUCB} follows fairly standard arguments. We fill in these detailed arguments in Appendix \ref{app:ThmICLinUCB}.
\newpage
\section{Extension to Generalized Linear Models}
\label{sec:gen_lin}
The main goal of this section is to demonstrate the generality of the algorithmic approach developed in Section \ref{sec:lin_algo}. To do so, we will now consider an observation model where the rewards are no longer assumed to be linear functions of the parameter. Rather, they satisfy the following relationship \cite{GLM1,GLM2}:
\begin{equation}
    y_t=\mu\left(\langle \theta^*, a_t \rangle \right)+\eta_t,
\label{eqn:gen_obs_model}
\end{equation}
where $\mu:\mathbb{R} \rightarrow \mathbb{R}$ is a continuously differentiable function typically referred to as the (inverse) \textit{link} function, and $\{\eta_t\}$ is a 1-subgaussian noise process as before. Our goal now will be to control the following notion of regret:
\begin{equation}
\bar{R}_T=\mathbb{E} \left[\sum_{t=1}^{T} \left( \max_{a\in\mathcal{A}_t} \mu(\langle \theta_*, a \rangle) - \mu(\langle \theta_*, a_t \rangle) \right)  \right].
\label{eqn:GLM_regret}
\end{equation}
In \cite{GLM1}, the authors noted the following main difficulty in analyzing the above non-linear model relative to the linear setting: for the Generalized Linear Model (GLM) in \eqref{eqn:obs_model}, the relevant confidence regions may have a more complicated geometry in the parameter space than simple ellipsoids. To overcome this challenge, the \texttt{GLM-UCB} algorithm is developed in \cite{GLM1} by focusing on the reward space.

For our setting, however, recall that the agent is not allowed to transmit the reward $y_t$ (neither directly, nor in an encoded form) to the server, thereby further adding to the technical complexity. This raises the following questions. (i) How should the agent encode an estimate of $\theta_*$ based on the GLM in Eq. \eqref{eqn:gen_obs_model}? (ii) How should the server design the  confidence region for decision-making? (iii) Will $B=O(d)$ bits continue to suffice for achieving order-optimal regret? In what follows, we will answer the last question in the affirmative by appropriately adapting the general recipe outlined at the end of Section \ref{sec:lin_algo}.

To get started, we make the following standard assumption on the inverse link function \cite{GLM1,GLM2}.

\begin{assumption} \label{ass:link} The function $\mu:\mathbb{R} \rightarrow \mathbb{R}$ is continuously differentiable, Lipschitz with constant $k_2$, and such that $$k_1=\min\{1,\inf_{\theta\in\Theta, a \in \bigcup_{t\in[T]}\mathcal{A}_t} \dot{\mu}(\langle \theta, a \rangle) \} > 0.$$
Here, $\dot{\mu}(\cdot)$ is used to represent the derivative of $\mu(\cdot)$.
\end{assumption}

Without loss of generality, we will assume $k_2 \geq 1$. Next, let us define
\begin{equation}
    g_t(\theta)=\lambda \theta + \sum_{s=1}^{t}\mu(\langle \theta, a_s \rangle) a_s.
\label{eqn:g_func}
\end{equation}

We are now ready to describe the Information-Constrained \texttt{GLMUCB} algorithm  (\texttt{IC-GLMUCB}).

\noindent \textbf{Description of \texttt{IC-GLMUCB}}. The \texttt{IC-GLMUCB} algorithm shares the same structure as \texttt{IC-LinUCB}, but has some crucial algorithmic differences that we next outline. First, instead of computing the least-squares estimate as in Eq. \eqref{eqn:least_square}, the agent now computes an estimate $\hat{\theta}^{(a)}_t$ of $\theta^*$ by solving the following equation:
\begin{equation}
    g_t(\hat{\theta}^{(a)}_t) = \sum_{s=1}^{t} y_s a_s.
\label{eqn:non_lin_LS}
\end{equation}
It then encodes the innovation signal $e_t=\hat{\theta}^{(a)}_t-\hat{\theta}^{(s)}_{t-1}$ exactly as in line 3 of Algo. \ref{algo:Encoder}, with $f(T)$ in Eq. \eqref{eqn:quantizer_eqs} replaced by
\begin{equation}
    \bar{f}(T) = \frac{3}{5L} \sqrt{ \frac{\beta_T}{k_1 k_2 T\log(dLT)}},
\end{equation}
and $q_{\bar{T}}$ set to $\left(4+6/\sqrt{k_1 k_2}\right)M$. The server performs decoding exactly as in line 7 of Algorithm \ref{algo:ICLUCB}, and computes $\hat{\theta}^{(s)}_t$. The main distinction relative to \texttt{IC-LinUCB} is in the construction of the confidence region at the server. The server constructs the following confidence set:
\begin{equation}
    \mathcal{\bar{C}}^{(s)}_t =  \{\theta\in \Theta: \mathcal{H}_{t-1}(\theta) \leq \sqrt{\beta_T} + k_2\textcolor{black}{\left(\sqrt{\lambda+(t-1)L^2}\right) q_t\}}, \hspace{1mm} \textrm{where} \hspace{1mm}  \mathcal{H}_t(\theta) \triangleq {\Vert g_t(\theta)-g_t(\hat{\theta}^{(s)}_t) \Vert}_{V^{-1}_t}.
\label{eqn:GLM_conf_region}
\end{equation}
The server then acts optimistically w.r.t. the above confidence set:
\begin{equation}
    a_t = \argmax_{a\in\mathcal{A}_t} \max_{\theta\in\mathcal{\bar{C}}^{(s)}_t} \langle \theta, a \rangle, \forall t \in \{\bar{T}+2,\ldots,T\},
\end{equation}
with $\bar{T}$ now set to
$$ \bar{T}=\ceil*{10 \frac{k_2}{k_1}  L^2d\sqrt{T}\log(dLT)}. $$

For completeness, we provide the detailed steps of the \texttt{IC-GLMUCB} algorithm in Appendix \ref{app:GLM_analysis}. To analyze the performance of the \texttt{IC-GLMUCB} algorithm, we will make the following standard assumption that is in the same spirit as Assumption \ref{ass:actions}(i):
$$ \sup_{a,b \in \bigcup_{t=1}^{T} \mathcal{A}_t}  \mu(\langle \theta_*,a \rangle) - \mu(\langle \theta_*,b \rangle) \leq 1. $$

We now state the main result of this section.
\begin{theorem} (\textbf{Regret of \texttt{IC-GLMUCB}}) \label{thm:ICGLMUCB}
Suppose Assumptions \ref{ass:actions}(ii)-(iii) hold along with Assumption \ref{ass:link}. Moreover, let the channel capacity satisfy $B \geq 6d$. Then, the \texttt{IC-GLMUCB} algorithm guarantees:
\begin{equation}
    \bar{R}_T = O\left( { \left(\frac{k_2}{k_1}\right)}^{3/2} L^2 d \sqrt{T} \log(dLT) \right) = \tilde{O}\left(d\sqrt{T}\right).
\end{equation}
\end{theorem}

For the proof of Theorem \ref{thm:ICGLMUCB}, see Appendix \ref{app:GLM_analysis}.

\textbf{Discussion.} The above result significantly generalizes Theorem \ref{thm:ICLinUCB}, and reveals that the \texttt{IC-GLMUCB} algorithm yields the same regret bounds as \texttt{IC-LinUCB} (up to constants) under identical requirements on the channel capacity as before. Thus, the main takeaway here is that our approach can readily accommodate general non-linear observation models as well (albeit with a slightly more involved analysis).
\newpage
\section{One Bit Channel Capacity is Sufficient for the Multi-Armed Bandit Problem}
\label{sec:unstructred}
In Sections \ref{sec:guarantees} and \ref{sec:gen_lin}, we have seen that for a $d$-dimensional model, $O(d)$ bits suffice to achieve order-optimal regret. In this section, we investigate whether one can achieve similar order-optimal regret bounds with fewer bits when the set of feasible actions has additional structure. We will show that this is indeed the case for a particular setting of interest when $\mathcal{A}_t =\{e_1, \ldots, e_d\}, \forall t\in [T]$, where ${(e_i)}_i$ are the standard orthonormal unit vectors. This setting represents the popular unstructured multi-armed bandit problem with a finite number of arms.  Our main insight is the following: playing action/arm $i$ only reveals information about the $i$-th component of $\theta_*$, denoted by $\theta_i$, and hence, when the $i$-th action is played, it makes sense for the agent to encode and transmit the innovation related to only $\theta_i$. In other words, the above intuition suggests that encoding a scalar innovation signal (as opposed to a $d$-dimensional innovation vector) should suffice for the specific setting under consideration. In what follows, we formalize this reasoning.

To get started, let us note that the optimal action $a_*$ is the unit vector corresponding to the largest component of $\theta_*$. Without loss of generality, let this component be $\theta_1$, i.e., $\theta_1 = \max_{i\in[d]} \theta_i$. We thus have $a_*=e_1$. Let us denote by $\hat{\theta}^{(a)}_{i,k}$ (resp., $\hat{\theta}^{(s)}_{i,k}$) the estimate of $\theta_i$ at the agent (resp., at the server) after arm $i$ has been played $k$ times. We now develop an information-constrained variant of the celebrated upper confidence bound algorithm that we call \texttt{IC-UCB}.

\noindent \textbf{Description of \texttt{IC-UCB}}. Let $\gamma=1/2^B$ where $B$ is the channel capacity, and define the following sequences for $k\geq 1$:
\begin{equation}
    p_{k+1}=\gamma p_k + 2 f_k; \hspace{2mm} q_{k}=\gamma p_k; \hspace{2mm} f_k= 2 \sqrt{\frac{\log T}{k}},
\label{eqn:UCB_quant_eq}
\end{equation}
where $p_1=m+f_1$, and $m \geq 1$ is such that $\max_{i\in[d]} |\theta_i| \leq m$. Suppose the action at time $t$ is $a_t=e_i$. The agent first updates its estimate of $\theta_i$:
\begin{equation}
    \hat{\theta}^{(a)}_{i,n_i(t)}= \frac{1}{n_i(t)} \sum_{k=1}^{n_i(t)} y_{i,k},
\label{eqn:run_sum}
\end{equation}
where $y_{i,k}$ is the agent's observation when the $i$-th arm is played the $k$-th time, and $n_i(t)$ is the number of times arm $i$ is played up to (and including) time-step $t$. It then computes the \textit{scalar} innovation $e_{i,n_i(t)}=\hat{\theta}^{(a)}_{i,n_i(t)}-\hat{\theta}^{(s)}_{i,n_i(t)-1}$, where $\hat{\theta}^{(s)}_{i,0}=0, \forall i\in[d]$. If $e_{i,n_i(t)}$ falls in the interval $\mathcal{Z}_{i,t}=[-p_{n_i(t)}, p_{n_i(t)}]$, then $\mathcal{Z}_{i,t}$ is partitioned uniformly into $2^B$ bins, and the symbol $\sigma_t \in \Sigma$ encoding the bin containing $e_{i,n_i(t)}$ is transmitted to the server. The server then decodes the center $\tilde{e}_{i,n_i(t)}$ of that bin, and computes $$\hat{\theta}^{(s)}_{i,n_i(t)}=\hat{\theta}^{(s)}_{i,n_i(t)-1}+\tilde{e}_{i,n_i(t)}.$$

If $e_{i,n_i(t)} \notin \mathcal{Z}_{i,t} $, then there is no transmission from the agent to the server. As for decision-making, each arm is first played once by the server. Subsequently, the action chosen by the server at time-step $t+1$ is the one that maximizes the following index:
\begin{equation}
    \texttt{IC-UCB}_i(t)=\hat{\theta}^{(s)}_{i,n_i(t)}+q_{n_i(t)}+f_{n_i(t)},
\label{eqn:index}
\end{equation}
where $q_{n_i(t)}$ and $f_{n_i(t)}$ are as in Eq. \eqref{eqn:UCB_quant_eq}.

Let us define by $\Delta_i=\theta_1 - \theta_i$ the sub-optimality gap of arm $i$. To present our results in a clean way, we will focus on the particularly important case where the sub-optimality gaps are small: $\Delta_i \in (0,1], \forall i \in [d] \setminus\{1\}.$ Our results can be easily generalized to arbitrary values of the sub-optimality gaps. For the setting considered in this section, it is easy to verify that the regret $R_T$ in Eq. \eqref{eqn:regret} simplifies to
$$ R_T= \sum_{i=1}^{d}\Delta_i \mathbb{E}[n_i(T)]. $$
Our main result concerning the regret bound of \texttt{IC-UCB} is as follows.
\begin{theorem}
\label{thm:ICUCB}
(\textbf{Regret of \texttt{IC-UCB}}) Suppose the channel capacity is at least $1$ bit, i.e., $B \geq 1$. The \texttt{IC-UCB} algorithm then guarantees:
$$ R_T \leq  5 \sum_{i=1}^{d} \Delta_i + \sum_{i=1}^{d} O\left(\frac{\log (mT)}{\Delta_i}\right). $$
\end{theorem}

We can also establish the following gap-independent bound.
\begin{theorem}
\label{thm:gap_ind}
(\textbf{Gap-independent regret of \texttt{IC-UCB}}) Suppose the channel capacity satisfies $B \geq 1$. The \texttt{IC-UCB} algorithm then guarantees:
$$ R_T \leq  5 \sum_{i=1}^{d} \Delta_i + O\left(\sqrt{dT \log(mT)}\right). $$
\end{theorem}

\textbf{Discussion:} Our bounds above match those of the \texttt{UCB} algorithm, revealing that for the MAB problem, one can achieve order-optimal bounds with a bit-rate of \textit{just $1$ bit}. The main takeaway here is that when the action sets have more structure, there is hope for achieving optimal performance with a channel capacity of fewer than $O(d)$ bits. As future work, it would be interesting to see if one can draw similar conclusions for other types of common action sets.

\section{Conclusion and Future Directions}
We introduced and studied a new linear stochastic bandit problem subject to communication channel constraints. We developed a general algorithmic framework comprising of an adaptive compression mechanism, and a decision-making rule that explicitly accounts for encoding errors. We then showed how this framework leads to order-optimal regret bounds for (i) the linear bandit setting, (ii) the generalized linear bandit setting, and (iii) the MAB problem, with horizon-independent bit-rates that depend only on the dimension of the unknown model parameter. Our work opens up several interesting avenues of research; we discuss some of them below.

First, an immediate question that remains unresolved is whether $O(d)$ bits is indeed necessary for achieving order-optimal regret for the linear bandit problem. Answering this question would require establishing algorithm-independent lower bounds for our setup - this is quite non-trivial, and is the subject of our ongoing work. Second, while our focus was on minimizing the number of bits needed to achieve optimal regret, one may ask the following alternate question: Given a fixed communication budget, what is the best cumulative regret bound one can hope for? It would be particularly interesting to ascertain the minimal communication budget (as a function of the model parameter dimension) needed to achieve sublinear regret. Third, we plan to extend the ideas and results in our paper to more complex distributed/federated settings involving multiple agents. Finally, our goal is to explore whether the techniques developed in this paper are applicable to Markov decision processes in the context of reinforcement learning.
\bibliographystyle{unsrt}

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With rent increases in Australia's capital cities experiencing their slowest growth rate in nearly 20 years and residential vacancies continuing to surge, there's no denying tenants are in the driving seat at the moment.
Whatcan landlords do under these challenging conditions predicted to persist for some time?
The cyclical nature of the property market dictates that this current tenants' market will not last and momentum will eventually shift. But until conditions change, there are several things landlords can do to maximise rental opportunities and preserve their investment. Denial, ignorance or blind optimism are not options. If landlords do not accept the reality of the current market situation and take positive action, they could end up putting their investment at risk.
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				© Alex Moriarty

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Wow, tonight I got hammered by so-called pollution spammers for the first time since I put up this blog (I've had 2-3 spam comments before)! I got 4958 comments which all were spam, fortunately WordPress caught every single one of them.
If you want to learn more about pollution spamming, there's a good article over at The Net is Dead. Basically what this spam does is NOT link to the spammers' sites, but rather to innocent sites like news.linux.com (very often targeted) or any other 'regular' site like Yahoo! The objective is to pollute anti-spam filters and blacklists.


text:
Southwest Water Buys Riverview Sewer System, Plant for $22.5M
Southwest Water Co. recently announced the acquisition of the Riverview sewer system and related treatment plant from the Shelby County Governmental Utility Services Corp. of Alabama for $22.5 million.
"This acquisition represents an excellent opportunity for Southwest Water," said Mark A. Swatek, chief executive officer and chairman of the company. "The system currently services over 12,000 residents and is located in a high population-growth area that serves a major commercial corridor. Moreover, it is adjacent to our Shelby County wastewater system which will allow us to realize synergies by combining the operations of the two systems."
Southwest Water Co. provides water production, treatment and distribution; wastewater collection and treatment; utility billing and collection; utility infrastructure construction management; and public works services. The company owns regulated public utilities and also serves cities, utility districts and private companies under contract.
'Historic Investment' of $23M for Puerto Rico Water Infrastructure Projects Green-Lit
Final Fifth Drinking Water Contaminant Candidate List Published by EPA
Long Island Drinking Water Projects Get Grant Funding
Clean Water Projects Funded by Port of Long Beach
Texas Water Foundation Announces New Strategic Plan
How Industrial Facilities Can Transfer from Water Use to Water Efficiency

redpajama_set_name: RedPajamaCommonCrawl


Three graduates from Polytechnic Institute, who ended up in the same graduating class at Howard University Medical School, returned as the three commencement speakers.
When the Orioles make their selection in the amateur draft tomorrow, they should consider picking one of agent Scott Boras' clients. The move could pay benefits down the line.
DS NightHead to the Sidebar at 9 p.m. to play games with fellow Nintendo DS fans. Free. Call 410-659-4130.
A Garden of GizmosPort Discovery's new exhibit includes an interactive garden and more. Admission is $10.75. 410-727-8120 or portdiscovery.org.
U.S. Air Guitar ChampionshipsWannabe rockers battle it out at the 9:30 Club in D.C. Admission is $10. 202-393-0930 or 930.com.
House of PayneNew Tyler Perry series makes its debut at 9 p.m. on TBS.
Senior Exit ExhibitionArt students graduating from University of Maryland, Baltimore County have their works on display. Free. Call 410-455-3188.
As this year's tanning season begins, dermatologists expect many will ignore a decade of warnings about the risks posed by overexposure to UV light from sunlight and tanning booths.
The Columbia Festival of the Arts turns 20.
Reviews of the caper sequel Ocean's Thirteen - starring Brad Pitt and George Clooney - and the animated comedy Surf's Up.
Prowling the streets of Fells Point, Kevin Cowherd joins the fun as a group of adults revisit an elementary school favorite: the scavenger hunt.
1Thousands of dead fish, algae foul harbor airThousand of fish die when oxygen is depleted near the water's surface.
2Bad luck constant companion of Ravens' CodyInjury-plagued linebacker misses mini-camp with a knee injury.
3Death on the streetsHomicide rate makes Baltimore the second-most perilous city in the nation.
4O's fade again in late inningsBullpen meltdown in eighth extends team's losing streak to four.
5Car accident is a hard lesson in honestyWhen someone slammed into Charlie Lusco's Dodge Caravan, he did everything right; others didn't.


text:
Q: Как получить индекс элемента в jquery коллекции? На странице в разных частях есть несколько элементов, при клике по кнопкам, мне нужно получить индекс родителя. Из всей коллекции подобных.
попробовал метод индекс, но не подходит.
Кликнете по 5 элементу в списке, его индекс в коллекции 4, но в консоль выводится 5...

let $elements = $('[data-js="myel"]'),
  $tyk = $elements.find('button');

$tyk.click((e) => {
  let $current = $(e.target);
  $element = $current.closest('[data-js="myel"]');
  console.log($element.index());
});
.banner {
  display: block;
  position: relative;
  font-family: sans-serif;
  padding: 40px;
  background: #5eba7d;
  margin: 10px 0;
}

.el {
  display: block;
  position: relative;
  margin: 4px 0;
  background: #ccc;
  padding: 3px 15px;
  cursor: pointer;
}

    Первый элемент
    <button>Тык</button>

    второй элемент
    <button>Тык</button>

    трейтий элемент
    <button>Тык</button>

    четвертый элемент
    <button>Тык</button>

Какой то баннер

    пятый элемент
    <button>Тык</button>

    шестой элемент
    <button>Тык</button>

    седьмой элемент
    <button>Тык</button>

    восьмой элемент
    <button>Тык</button>

A: Просто укажите селектор для элементов, чей индекс нужно получить:

let $elements = $('[data-js="myel"]');

$elements.click((e) => {
  let $current = $(e.target);
  console.log($current.index('[data-js="myel"]'));
});
.banner {
  display: block;
  position: relative;
  font-family: sans-serif;
  padding: 40px;
  background: #5eba7d;
  margin: 10px 0;
}

.el {
  display: block;
  position: relative;
  margin: 4px 0;
  background: #ccc;
  padding: 3px 15px;
  cursor: pointer;
}

Первый элемент

второй элемент

трейтий элемент

четвертый элемент

Какой то баннер

пятый элемент

шестой элемент

седьмой элемент

восьмой элемент

redpajama_set_name: RedPajamaStackExchange


Mediamobile's V-Traffic platform is as flexible as you need – collecting and disseminating traffic information in any format over any distribution channel, from radio (RDS-TMC or digital radio) through mobile data (TPEG over IP) to the Internet (XML).
Mediamobile delivers solutions tailored to our customers' needs. No matter which mapping system you prefer, we're compatible with all of them – Nokia (Navteq), TomTom (TeleAtlas), Bing, Google Maps and many others.
Most of the time, V-Traffic information is based on TMC location referencing tables. This makes it easy to geolocate events happening on each country's main road network. TMC location tables are specific to each country and certified by TISA. Mediamobile has either created and/or helps maintain the TMC tables for all the countries in which we operate.
Updating maps is a costly business. Fortunately, the latest methods for encoding location references like Agora-C and OpenLR make it possible to identify and pinpoint locations or events on any type of map. Mediamobile can deliver data encoded in either of these formats, as you prefer.


Something Special Home Consignment & Antique Mart serves Semmes, AL and is located in the 36575-7406 ZIP code.
Something special is unique because of the type of items that are there. I have decorated my office starting from the floor to the top of my desk. I have become an avid customer.


Business customers are the most burdensome group for hotel service, mainly due to high requirements.
As we know, a hotel stay for a business client is tantamount to the performance of professional duties, more importantly, these are guests who regularly visit hotel facilities, so they have a comparison of hotel services and quality of service.
The requirements of business clients are usually specific and very high - while private guests easily accept a minimalist room, the business client expects at least a substitute for the house, i.e. the reduction of empty space and the provision of the room with an appropriate TV set. In addition, a desk and WiFi Internet access should be mandatory in the hotel room.
Most business clients expect an alternative between a standard room, apartment and studio.
Hotels focused on business guests, striving to attract regular customers, should have a well-equipped restaurant, which will encourage customers to take advantage of the offer. It is therefore necessary to take care of a wide range of meals, while taking into account the individual needs of allergy sufferers, as well as individual fashions and culinary tastes of guests. The optimal solution is international cuisine with a regional accent. The restaurant should be complemented by a cafe.
Another determinant of the hotel focused on business guests are the conference rooms, preferably a few. It is the hotel staff's responsibility to organize all events in the conference rooms.
Business guests are by definition active people, so hotels should take care of additional attractions such as a bar or swimming pool, a salt cave or wellness treatments is also a good idea.
An important issue for business guests is also the attractive location of the hotel, providing the opportunity to escape from the everyday hustle and bustle and bustle of the city.
It is also worth remembering that according to estimates, business customers generate 1/3 of all turnover in the domestic hotel industry.
Company lunches also prove to be an important factor - this is mainly due to the fact that company lunches are tax deductible under the law. We are talking about meals during which important business matters are dealt with. Company lunches are an attractive alternative to standard business trips.
In order to meet the needs of business customers, many hotels in the country offer conference rooms and restaurants adapted to serve business customers and their guests. Business lunches turn out to be particularly important for foreign guests for whom this solution is a standard.
It is also worth taking care of an attractive entertainment package, especially when these are guests who visited the city for the first time - in this case, however, it is necessary to take into account their individual schedule and the ability to plan time on their own, in which case business lunch turns out to be the optimal solution.
It should also be remembered that an attractive place to stay is not enough to fully satisfy business customers, it is important especially when an agenda is implemented during a meeting. It is the hotel's duty to provide spectacular decoration and tasty dishes.


Now held at West Salisbury Elementary, Camp Pinehurst is a summer camp with an emphasis on special needs youth. Campers work on socialization skills, physical activities, team building and special activities. There will be a fun candy theme for 2019.
Each participant is separated into age/ability settings to maximize their enjoyment and success in the program. The program is run by highly trained staff with years of experience in the field.
An open house will be held Tuesday, June 26, from noon-2 p.m.
Time(s): Monday-Thursday, 9 a.m.-3 p.m.
Additional Details: Transportation is available to Wicomico County residents.
Payment in full is due up front by the deadline date of May 24, 2019, for the entire service period that covers the range of June 26, 2019, to August 1, 2019.


A set of plugins for enhancing emailing within WordPress sites.
A plugin for synchronizing your WordPress readers with a list of subscribers in MailChimp easily. It is in beta release, but it's still pretty handy. It is an add-on for MailChimp for WordPress plugin, so to use it, it needs MailChimp for WordPress v2.2.3. It allows you to add users to a list, see data changes, delete users, manual sync, and more.


text:
Key Labs
Position: Home Home > Research > Department of Computer Science and Technology > Content
Institute of Computer Architecture
The Institute of Computer Architecture is also the Microprocessor Research and Development Center (MPRC) of Peking University. Combining with both computer science and microelectronics research of Peking University, the institute is committed to the research and design of microprocessors with an architecture based on an internally developed instruction set architecture and independent intellectual properties.
The institute faculty includes two professors, three associate professors, four other faculty members, and several international experts serving as guest professors. The institute has completed more than 30 research projects, with achievements including the first microprocessor hardware-software co-design platform in China, UniCore16 /32 microprocessor and supporting software system, heterogeneous multicore PKUnity SoC, PKUnity Network Computer, PKUnity Super-K netbook, PKUnity Nephoputer, etc.
The institute has pioneered development of microprocessor (CPU) and computer technology indigenous to China in the past decade. The low-cost computer systems designed by the institute has been lauded by the Chinese government as innovative contributions to the national information industry. Representative of the remarkable progress of Chinese computer technology, such low-cost computer systems have been widely used in all provinces throughout China.
Main research areas include:
(1) Modern microprocessor
(2) Multicore and SoC
(3) Hardware-Software co-design


text:
Boutique Obsession: Adeline
By Abby Gregory	• Published at 9:49 pm on July 6, 2011
Our latest shopping obsession in Dallas is the new boutique, Adeline. Specializing in contemporary clothing and home accessories, Adeline offers cool selections from labels like Vintage Havana and AnM. But the best part - nearly everything in their University Park store is priced under $150.
Adeline's owner, Graeme Gehring, posseses a luxe yet laid-back style, and though she's just recently graduated from the University of Oklahoma, her taste and trend-spotting abilities are well beyond her years. Turns out, she's learned from the best; her father, Randall Gehring, owns Enchanted Lighting, located just around the corner from Adeline.
"Retail sort of runs in the family. So, I was somewhat familiar with the process. When I was younger, I would go to the home markets with my Dad and try to convince him to buy accessories and bags to sell in his shop," says Graeme of her boutique background.
Some of our top picks on Adeline's well-stocked racks include unstructured dresses by Savannah Rose and Hazel. Other things we loved? The tunic tops by Ocean Drive and relaxed, dip-dyed tees by Freeloader. Unique clutches by Jesslyn Blake are also among our current favorites.
The hot shop, located at 4412 Lovers Lane, is open Monday through Saturday from 11 a.m. to 7 p.m. We also suggest you connect with Adeline via Facebook to stay on top of special sales and new arrivals.


If you would like to hold a Gold CSCS card to become a supervisor, you will need a better understanding of health and safety than most of the other works on the construction site.
On a worksite being a supervisor has some responsibilities, and you will have to make safety-critical decisions every day, and this will be a massive part of your job.
To get hold of your gold CSCS card, you will first have to pass the CITB Health, Safety & Environment Test. This test will consist of 50 multiple choice questions, you will need to get to the pass make which is 47 out of 50. These questions will have 38 questions of which will test you on health, safety and environmental issues and the other 12 questions will be behavioural case study questions. These will test your on how you should act on a construction site to keep yourself and others safe at all times.
Supervisor's taking their test soon can make use of this free CSCS mock test for supervisors, there are 50 questions in total, all multiple choice questions and some will need more than one answer. Good luck and let us know how you got on in the comments below.
1. When the construction (design and management) regulations 2015 apply, what must be in place before construction work begins?
2. Under the construction (design and management regulations) 2015, where would you find the agreement for managing health and safety for the project you are working on?
3. Under the construction (design and management) regulations 2015, what must be provided before construction work starts, and then maintained until the end of the project?
4. Under the construction design and management regulations 2015, which two of the following must you ensure workers have received before they start working on site?
5. How long must you keep inspection records under the construction *design and management) regulations 2015?
6. What is the purpose of the health and safety file that is handed to the client at the end of the project?
7. Which piece of equipment is used with a cable avoidance tool to detect cables?
8. In the colour coding of electrical power supplies on site, what voltage does a blue plug represent?
9. On the site electrical distribution system, which colour plug indicates a 400 volt supply?
10. Why must a RCD be used in conjunction with 230 volt electrical equipment?
11. How could a site worker check if the RCD through which a 230 volt hand tool is connected to the supply is working correctly?
12. Electric welding is about to start on your site. What should be used to protect passers-by from getting arc eye?
13. When should an escape route or routes ensure safe access to?
14. Which of the following tasks would you expect the Appointed Person for first aid to carry out if someone is injured?
15. What does the proactive monitoring of health and safety procedures involved?
16. Why may a young person be more at risk of having an accident?
17. In which regulations will you find the details of the welfare facilities that must be provided on a construction site?
18. Which two of the following factors must be considered when providing first aid facilities on site?
19. What is the purpose of using a permit to work system?
20. What is the best way for a supervisor or manager to make sure that the operatives doing a job have fully understood a method statement?
21. When is it advisable to take precautions to prevent people, plant or materials falling into an excavation?
22. Which of the following precautions should be taken to prevent a dumper that is tipping material into an excavation from falling into it?
23. When planning possible work in a confined space, what should be the first consideration?
24. Which of the following is a significant hazard when excavating alongside a building or structure?
25. What danger is created by excessive oxygen in a confined space?
26. On a scaffold, what must be the largest unprotected gap between any guard rail, toeboard, barrier or other similar means of protection?
27. Which of the following is a fall-arrest system?
28. Under what requirements of the work at heights regulations, what must be the minimum width of a working platform be?
29. For a fixed ladder, what is the maximum vertical height that may be climbed before an intermediate landing place is required?
30. What should you do if you noticed that operating working above a safety net are dropping off cuts of material and other debris into the net?
31. When must a competent person routinely inspect a working platform?
32. What must edge protection be designed to do?
33. The beaufort scale is important when working at height externally. What does it measure?
34. Why is it dangerous to use inflatable air bags for fall assist if they are too big for the area to be protected?
35. If a scaffold is not complete, which of the following actions should be taken by the supervisor?
36. Ideally, where should a safety net be rigged?
37. What is the main danger of leaving someone who has Fallen suspended in a harness for too long?
38. When should guard rails be fitted to a working platform?
39. A design feature of some airbags used for full assist is a controlled leak rate. If you are using these, what must you ensure about the inflation pump?
40. What is your least reliable source of information when assessing the level of vibration from a power tool?
41. Following a scaffold inspection under the work at height regulations, how soon must a report be given to the person on whose behalf the inspection was made?
42. What is the minimum height of the main guard rail on a scaffold?
43. How should cylinders containing LPG be stored on site?
44. Where should LPG cylinders be positioned when supplying an appliance in a site cabin?
45. What should be the capacity of a spillage Bund around a fuel storage tank, in addition to the volume of the tank?
46. Which of the following should be the first consideration if you need to use a hazardous substance?
47. Before allowing a lift operation to be carried out, where should you ensure that the sequence of operations is recorded to enable a safe lift?
48. Where must the number of people who may be carried in a passenger hoist on site be displayed?
49. What is the significance of a weekly or daily personal noise exposure of 87 decibels?
50. The heavy load has been delivered to the site. What is the first thing that should be considered before trying to move it?
Does this test include the case study questions?
My cscs test is at xmss could u send info to help me revise and pass.


As an online trading trader, it is possible to trade on a wide array of assets. Some newbies to the industry are under the misconception that online trading is all about currency trading. In reality, it is possible to pick specific company stocks on which to trade. In the automobile industry, it is common to see online trading traders investing in BMW, Fiat, Hyundai, and the stock we are going to look at now, Daimler AG.
Daimler AG was formed when Mercedes Benz merged with Chrysler Corporation with the intention of forming a transatlantic car producer. The merger failed and Chrysler was sold off. The company now makes cars, vans, and trucks under the Mercedes-Benz, Smart, Freightliner and Maybach brands.
Daimler AG stock can be found on the Frankfurt Stock Exchange under DAI, and is also featured in the DAX index. Online trading traders can use news and research on the company to either trade directly on the stock or contribute to their research on trades on the DAX index.
The world has changed significantly since the credit crunch of 2008. With rising fuel prices being felt more keenly than ever the public demand for more fuel efficiency is driving manufacturers to adapt, innovate and release new products. The competitive nature of the industry and these new product launches provide a suitable environment for online trading traders to seek out profits.
The company is, by sales volume the 13th largest car. and second largest van/truck manufacturer in the world. With increased economic prosperity in the world, Daimler AG trading will be well positioned to benefit from their quality brand image. Daimler AG stock can be an excellent additional asset to add to your research and trading portfolio.
Are you looking for how to trade Daimler AG in the stock market?
All the Daimler AG online trading brokers reviews you can find here are tested, verified and approved by our experts. That is why you can be sure that after having chosen any of Daimler AG online trading brokers from our list you will get an excellent trading experience. Moreover, you can easily find online videos together with other various education materials almost in every broker. We hope that online trading will become a new interesting and captivating activity that will bring you high payouts.


text:
Q: why we need EndInvoke() to return value of asynchronous call in delegate? i am having a bit trouble in understanding it that BeginInvoke() method call in a delegate when asynchronously calls methods, after completion of it's tasks why can't BeginInvoke() itself return the return value from the method, why we require EndInvoke() to query BeginInvoke() and return the return value from it.
i know this may be not a good query, but my confusion is real :-)
Addition: i can call a method asynchronously using DelagateName.BeginInvoke(parameters), now when the matching method in the delegate(suppose it returns a value) finishes it's work and returns a value, why i need DelegateName.EndInvoke() to get the returned value? why can't the first call itself can return value on method completion.

A: When BeginInvoke returns, the method hasn't finished - that's the whole point of it being asynchronous. So BeginInvoke can only give you a token representing the "promise" of a result at some point - and you use that promise to get the actual result using EndInvoke, usually in a callback.
In .NET 4 this is more pleasantly encapsulated using Task and Task<T> of course, but generics weren't part of .NET 1, hence the somewhat more convoluted approach for delegates.

A: You only need the BeginInvoke and matching EndInvoke if you are wanting to call the delegate asynchronously i.e. it won't block the current thread. This is useful for long running methods or methods that call processes that you don't control e.g. across application boundaries.
From what you describe you may be better off using the Invoke method which will return the value within the same method - and on the same thread i.e. synchronously. Of course your thread could be blocked by a long running process in this instance. It depends what your are trying to do.


우리는 중국에서 나일론 노트북 백 팩 제조 업체 및 공급 업체 / 공장 전문화되어 있습니다. 나일론 노트북 백 팩 중 하나 인 Shenzhen Olinb Bags Co., Ltd. 중국 유명 브랜드 중 하나 인 저렴한 가격 / 저렴한 가격으로 고품질의 나일론 노트북 백 팩 도매업.
Wholesale 나일론 노트북 백 팩 from China, Need to find cheap 나일론 노트북 백 팩 as low price but leading manufacturers. Just find high-quality brands on 나일론 노트북 백 팩 produce factory, You can also feedback about what you want, start saving and explore our 나일론 노트북 백 팩, We'll reply you in fastest.


text:
Who Was the Woman who Stood Up to Joe Paterno?
By Tom Shakely
Editors Note: Vicky Triponey is the former Vice President of Student Affairs at Penn State who has recently been in the news for her criticisms of the way Joe Paterno handled the discipline of football players. The author of this community post is the founder of the now inactive SafeguardOldState.com, which was often critical of Triponey's practices at Penn State.
Despite today's revisionist history, Vicky Triponey was not a crusader against the Penn State administration's entrenched culture of secrecy. On the contrary, she was its enforcer, and virtually her first act upon assuming her position was to threaten its most vocal critic using the power of the purse.
Long before there was a Louis Freeh report to castigate the Trustees or a Penn Staters for Responsible Stewardship to call them to account, there was Radio Free Penn State, a lone voice that dared to suggest that Graham Spanier held too much authority, his administration was too centrally organized and vertically integrated, the University trustees were too disengaged (and thus complicit by default), and that the crucial checks and balances of shared governance had all but evaporated within the University. The campus radio station aired these ideas for more than a decade before it became fashionable. Dr. Triponey inserted herself into that conversation to silence these dangerous, radical, and unacceptable sentiments using the best means at her disposal — denial of University funding.
"The first thing to go will be Radio Free Penn State."
So many at Penn State recall Vicky Triponey's time here so unfavorably not because of that which she considers her greatest failure (disciplining football players), but rather for what she regarded as her many "accomplishments." Bizarrely, for a woman in charge of stewarding student life, she chose to centralize and cloister power in the very ways she now suggests she opposed in speaking with national media.
Today, she recalls herself as the victim of consolidated power. In practice, she helped wipe away the last remnants of an older system that was designed to prevent it. Likely the most remembered aspect of Dr. Triponey's achievements will be her dismantling of student shared governance. Triponey empowered the student trustee, not only with a coveted title, but also money and resources, to carry out her plan to dissolve the student government, going so far as to attempt a campus-wide referendum justified with dozens of allegedly forged petitions, all offered in identical pink handwriting. For decades, the validity of such documents had been adjudicated by a student supreme court, a body Vicky Triponey had already stripped of its authority. Students at Penn State's peer institutions in the Big Ten recoiled at this naked power grab, taking the extraordinary step of refusing to recognize the flaccid "advocacy" group she had installed to replace the student self-government. The last five years of campus government reform efforts have been a testament to the dysfunctional, personality-dependent system she sought to impose.
Vicky Triponey's achievements in other areas include the increase and control of student activities and facilities fees that she created an administratively-led board to oversee, erasing a traditional student role of self-governance that in previous eras led to students' creation of the fraternity homes of State College and, in fact, the very students' Hetzel Union Building.
An insular "Cabinet of Student Leaders," and a clubbish approach to governing meant you were either an ally or an enemy — something many community stakeholders discovered too late in an environment less collaborative than caustic, and less open than agenda-driven. Success often could be measured by proximity to power.
Vicky Triponey's personality was graceful, but her demeanor imperious in her dealings with many students during her three busy years, and Safeguard Old State grew as perhaps the most visible symbol for student displeasure with her approach to leadership. It came to represent a unified student opposition that arose in response to an adversarial situation that was not of its making.
Vicky Triponey seemed to create her own enemies (and many are sharing their own recollections) and did so almost uniformly through her attempts to "modernize" student life by, perversely, curtailing authentic student leadership roles.
There is also something that no one will find among Safeguard Old State's chronicles and criticisms of Vicky Triponey or other primary source materials from the time: any knowledge or writing of her now-public clashes with Coach Joe Paterno. No one we knew had any knowledge of her disputes within the University administration, and the record should reinforce that her own troubles with students resulted neither from her clashes with Joe Paterno nor her attitude toward athletics. Student displeasure with Vicky Triponey flowed truly from experience, and the aggregate of frustration, disenfranchisement, and disregard so many student leaders felt in facing the results of her agenda.
These are recollections brought back to mind after five years as a result of Vicky Triponey's decision to speak to the media about herself in the wake of a tragic pedophilia scandal.
I have not really thought of Vicky Triponey in these five years since her dismissal — and God-willing, this will be the last time I ever write about her — but evidently she still thinks about us.
We of the Nittany Valley are at the dawn of a new era in the life of our town and University. We are faced with building up, with recovering the best aspects of our legacy with an eye toward our history and experience.
So if it's true that she was The Woman Who Stood Up to Joe Paterno, she was also The Woman Who Towered Over Student Life. To avoid history repeating, it's worth a moment to recall the full scope and complete portrait of her time in our little Nittany Valley.
Tom Shakely
Other posts by Tom
On The All-American Rathskeller
"All things considered, I'm cautiously optimistic that the Herlocher's local purchase of the Foster Building will achieve some degree of good conservation, although it's a tragedy for the distinctiveness of State College to lose the Skeller in the process."
Vision For Penn State Greeks
How Penn Staters Can Encounter Their Heritage
https://onwardstate.com/2012/07/18/who-was-the-woman-who-stood-up-to-joe-paterno/


text:
What car(s) did you hate at launch but have grown to like as time has passed? Or vice versa
Thread starter MWF
Your Brown Banana for Scale
East-Timor, USA
206 HP N/A washing machine
Blind_Io said:
I'd say the same about the current generation of Civic. I really didn't like the look when they first came out, but I saw one the other day next to a new Corolla and thought, "These two are competitors?" The Honda looks so much better, no matter how many fake vents Toyota puts on the Corolla.
I'm glad you said that, they vastly improved the civic in recent times (compared to the 2010 one that I had).
The new corolla looks terribad
Blind_Io
"Be The Match" Registered
06 Nissan XTerra Off Road, 00 VFR800, 07 ST1300
The exception is their models with all the go-fast on them; I still hate the styling on those, they look like a doodle where the kid didn't know when to stop drawing.
AlexGRFan97
Ferrari 360 Modena: Hated it when I first found out it replaced the 355 which I loved in NFS3 and all of Clarkson's early home videos. Now it's one of my favourite Ferrari designs of all time. Still nowhere near as good as its predecessors though.
Lamborghini Murcielago: Same as above, replace 355 with Diablo. Hell, I even like the later Aventadors now.
Lancia Stratos Concept: Aside from the puppy-dog headlights, this is what Lancia should be making instead of continuing to nail its own coffin.
Nissan GT-R: I remember praying in school that the next Skyline wouldn't look like the 2001 Proto version. They improved it a lot with the revised 2005 concept, but when it launched I was still underwhelmed. Now though? To put it frankly, this car has aged like wine. Even as late as 2016/2017 this car looked refreshingly new, and I can't wait to see what they'll do with the R36.
Noble M600: I thought this would've been the car to kill the underdog brand from Richard Noble, whose past cars looked like demonic Mondeos. Though it isn't as high-profile as their efforts were in the mid '00s, for me the M600 is easily one of the best looking cars of the decade.
Porsche 997: The earlier models, at least. Phase II still isn't that appealing to me.
Subaru Impreza GE/GH - I just thought the hatchback Impreza looked horrifically wrong the first time I saw it. When they brought back the sedan version I was a lot happier to say the least.
TVR Griffith: I remember hating this the second I saw it, and I'm still not a fan of the fish/special student front fascia. All this model's doing is making up for 13 years' of lost time, and the interior's familiar to anyone who's ever seen the inside of a past TVR. Here's hoping TVR becomes a household name once more.
Vauxhall VXR8: I loved the Monaro in Top Gear, and I similarly loved "driving" it in Need For Speed: Most Wanted. Then its successor came out, and what was once a sleek, fearsome coupé had bloated into a brutish, blocky sedan with a wing. Nowadays I love both equally, and wish that one day Holden is brought back from the dead, same with Aussie Ford.
JimCorrigan
HMCS Velvet Glove; The Last Samurai
I should add the current gen S550 Mustang (pre and post refresh) to the list. I remember thinking it looked disjointed front to back when it was released, but as time wore on, I think it's probably the prettiest Mustang since the first gen.
NecroJoe
San Francisco area, CA, USA
2015 Mazda 3 S GT, 2015 VW e-Golf
JimCorrigan said:
I'm with you there. When it was released, there was a lot of "Ford Fusion" front, and Mustang+Aston rear". I think it out grew that impression even before the upgrade, and I'm regularly pleasantly surprised whenever I get to see a newer one in the wild. On Saturday, I parked next to a nice dark red that looked fantastic.
MWF
Now needs wood
MWF HQ, Ukadia
MX-5 1.8i Indiana SE, update pending
I still don't like the 360 and I've liked very few Ferraris since. Some have been OK-ish but I can't think of one that makes me dribble or want to bite the back of my hand.
Crazyjeeper
NickGyver
F150, CJ7, Mustang GT, Tenere, Griso, GL1000
NecroJoe said:
Same. I went from meh to buying one.
CrzRsn
Mr. Fedewhatever
Motor City, Michigan
13 Ford Mustang GT, 17 Ford Fiesta ST
I still don't really like the 2015-2017 cars (except the GT350). I think the facelift looks a lot better.
Very Odd Looking Vehicular Object
4x4 diesel barge
Tram13 said:
I never hated the Škoda Superb, nor any of the large saloons, but I just thought of them as boring cars. I always liked small and nimble cars, or cars that were otherwise rated as fun by the likes of Evo Magazine. However, when I had a chance to drive a first-gen Škoda Superb 2.0 TDI, I was addicted. The space, the torque, the comfort, the leather seats (it was a higher-end model as it had an umbrella in one of its doors)...
Sure, the interior looked outdated even by the standards of its time and very much VW-ish, and I still can't wrap my head around why'd VW choose the Audi PL45 platform (i.e. one with a longitudinally placed engine in an font-wheel drive car) for the Passat (which this car essentially is), nor do I have a need for such a car, but it broadened the spectre of cars that I find interesting.
The 2002 Škoda Superb looks and feels like a stretched Passat because... that's what it is. It was developed by VW for China, because the Chinese like rear legroom and pretty much every sedan on the market can be had as a long wheel base version. VAG brought it back home and slapped a Skoda badge on it, basically. The Skoda buyers liked it and now we're up to generation 3.
And it wasn't the first Passat with a longitudinal engine. They've historically gone back and forth with longitudinal and and transverse layouts.
The longitudinal and FWD layout with the halfshafts coming out of the gearbox is an old DKW design which Audi inherited and apparently decided to keep using forever. It made sense with a tiny little 3-cylinder 2-stroke in there, but it's pretty ridiculous how far forward the engines are on the five-cylinder models from the 80's.
MWF said:
Not even the 458?
Nope. It's better than some recent efforts but it's not what I would call particularly pretty. Some of their bespoke creations from the last few years however are very handsome - take a look at the SP38 Deborah, F12 TRS or the SP12EC.


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