text:
\section{Supplementary Material}
\label{sec:appendix}

\subsection{Proof of Supporting Lemmas}

\subsubsection{Proof of Lemma \ref{lemma:breg}}
We refer to \cite{nemirovski2009robust} for a proof.

\subsection{Proof of Lemma \ref{lemma:prop9}}
The proof of this result is almost identical to the proof of Proposition 9 in \cite{lan2016algorithms}. We present this here for completeness. For any $s \leq t \leq N$, from Lemma \ref{lemma:breg} we have,
\begin{align}
\label{eq:lemma21}
B_{\psi}(x, x_{t+1}) \leq B_{\psi}(x, x_t) + \gamma_t \langle h_t, x - x_t \rangle + \frac{\gamma_t^2}{2} \norm{h_t}_{\psi^*}^2.
\end{align}
Note that, if $t \in \mathcal{B}$, $\langle h_t, x_t - x \rangle = \langle F'(x_t, \xi_{0,t}), x_t - x \rangle$ and if $t \in \mathcal{N}_j$, then due to convexity of $G_j$,
\begin{align*}
\langle h_t, x_t - x \rangle &= \langle G_j'(x_t, \xi_{j,t}), x_t - x \rangle \geq G_j(x_t, \xi_{j,t}) -  G_j(x, \xi_{j,t}).
\end{align*}
Moreover, we have
\begin{align}
\label{eq:fourth term}
\norm{h_t}_{\psi^*}^2  = \norm{F'(x_t, \xi_{0,t})}_{\psi^*}^2  \one_{\{t \in \mathcal{B}\}} + \sum_{j=1}^m \norm{G'(x_t, \xi_{j,t})}_{\psi^*}^2 \one_{\{t \in \mathcal{N}_j\}}
\end{align}
Thus we get,
\begin{align}
\label{eq:fundamental inequality}
B_{\psi}(x, x_{t+1}) &\leq B_{\psi}(x, x_t) - \gamma_t \langle F'(x_t, \xi_{0,t}), x_t - x \rangle ~\one_{\{t \in \mathcal{B}\}} \\
&\qquad - \sum_{j=1}^m\gamma_t \big(G_j(x_t, \xi_{j,t}) -  G_j(x, \xi_{j,t})\big)~\one_{\{t \in \mathcal{N}_j\}} \\
& \qquad + \frac{\gamma_t^2}{2}\left(\norm{F'(x_t, \xi_{0,t})}_{\psi^*}^2  \one_{\{t \in \mathcal{B}\}} + \sum_{j=1}^m \norm{G'(x_t, \xi_{j,t})}_{\psi^*}^2 \one_{\{t \in \mathcal{N}_j\}} \right) \qquad \text{a.s.}
\end{align}
Adding up from $t = s$ to $N$ we get,
\begin{align*}
B_{\psi}(x, x_{N+1}) &\leq B_{\psi}(x, x_{s}) - \sum_{t = s}^N \gamma_t \langle F'(x_t, \xi_{0,t}), x_t - x \rangle ~\one_{\{t \in \mathcal{B}\}}\\
& - \sum_{t = s}^N\sum_{j=1}^m\gamma_t \big(G_j(x_t, \xi_{j,t}) -  G_j(x, \xi_{j,t})\big)~\one_{\{t \in \mathcal{N}_j\}}\\
&+ \sum_{t=s}^N \frac{\gamma_t^2}{2}\left(\norm{F'(x_t, \xi_{0,t})}_{\psi^*}^2  \one_{\{t \in \mathcal{B}\}} + \sum_{j=1}^m \norm{G'(x_t, \xi_{j,t})}_{\psi^*}^2 \one_{\{t \in \mathcal{N}_j\}} \right)  \qquad \text{a.s.}
\end{align*}
Rearranging the terms and using $ B_{\psi}(x, x_{N+1}) \geq 0$ we arrive at the required inequality, hence proving the result.
\qed

\subsubsection{Proof of Lemma \ref{lemma:prop9b}}
By taking a conditional expectation given $x_t$ and ${\boldsymbol{\tilde{\xi}}}$ on both sides of \eqref{eq:fundamental inequality} and then adding up from $t = s$ to $N$ we get,
\begin{align*}
\sum_{t = s}^N {\mathbb E} &\left[\gamma_t \langle F'(x_t, \xi_{0,t}), x_t - x \rangle~\one_{\{t \in \mathcal{B}\}} \mid x_t, {\boldsymbol{\tilde{\xi}}} \right] +  \nonumber\\
& \sum_{j=1}^m \sum_{t = s}^N {\mathbb E}\left[ \gamma_t \big(G_j(x_t, \xi_{j,t}) -  G_j(x, \xi_{j,t})\big)~\one_{\{t \in \mathcal{N}_j\}} \mid x_t, {\boldsymbol{\tilde{\xi}}} \right] \nonumber \\
& \leq {\mathbb E} [B_{\psi}(x, x_{s}) \bigm| x_t, {\boldsymbol{\tilde{\xi}}} ] + \sum_{t = s}^N \frac{\gamma_t^2}{2}  {\mathbb E} \left[ \norm{F'(x_t, \xi_{0,t})}_{\psi^*}^2 ~ \one_{\{t \in \mathcal{B}\}}  \bigm| x_t,  {\boldsymbol{\tilde{\xi}}} \right] + \nonumber \\
 & \qquad \sum_{t = s}^N \sum_{j=1}^m  \frac{\gamma_t^2}{2}  {\mathbb E} \left[ \norm{G'(x_t, \xi_{j,t})}_{\psi^*}^2~ \one_{\{t \in \mathcal{N}_j\}} \bigm| x_t , {\boldsymbol{\tilde{\xi}}}\right]
\end{align*}
Note that given $x_t$, the random variables $\one_{\{t \in \mathcal{B}\}}$ and $\one_{\{t \in \mathcal{N}_j\}}$ (depending on
$\boldsymbol{\tilde{\xi}} :=  \{ \{ \tilde{\xi}_{j,\ell}\}_{\ell = 1}^L\}_{j=1}^m$) are independent of ${\boldsymbol{\xi}}_t$. Hence we get,
\begin{align}
\label{eq:second term}
{\mathbb E} \bigg[ \langle F'(x_t, \xi_{0,t}), x_t - x \rangle ~\one_{\{t \in \mathcal{B}\}} \biggm| x_t, {\boldsymbol{\tilde{\xi}}} \bigg] = &~ {\mathbb E} \big[ \langle F'(x_t, \xi_{0,t}), x_t - x \rangle \bigm| x_t \big] ~ \one_{\{t \in \mathcal{B}\}} \nonumber \\
=&~ \langle f'(x_t), x_t - x \rangle \one_{\{t \in \mathcal{B}\}},
\end{align}
and
\begin{align}
\label{eq:third term}
{\mathbb E} \bigg[ \big(G_j(x_t, \xi_{j,t}) -  G_j(x, \xi_{j,t}) \big)~\one_{\{t \in \mathcal{N}_j\}} \biggm| x_t, {\boldsymbol{\tilde{\xi}}} \bigg]= &~ {\mathbb E} \big[G_j(x_t, \xi_{j,t}) -  G_j(x, \xi_{j,t}) \bigm| x_t \big] \one_{\{t \in \mathcal{N}_j\}} \nonumber \\
=&~ (g(x_t) - g(x))  ~ \one_{\{t \in \mathcal{N}_j\}}.
\end{align}
Similarly, we have
\begin{align}
\label{eq:fourth term}
&\sum_{t = s}^N \frac{\gamma_t^2}{2}  {\mathbb E} \left[ \norm{F'(x_t, \xi_{0,t})}_{\psi^*}^2 ~ \one_{\{t \in \mathcal{B}\}} \bigm| x_t,   {\boldsymbol{\tilde{\xi}}} \right] +  \sum_{t = s}^N \sum_{j=1}^m  \frac{\gamma_t^2}{2}  {\mathbb E} \left[ \norm{G'(x_t, \xi_{j,t})}_{\psi^*}^2~ \one_{\{t \in \mathcal{N}_j\}} \bigm| x_t , {\boldsymbol{\tilde{\xi}}}\right]  \nonumber \\
& = \sum_{t = s}^N \frac{\gamma_t^2}{2}  {\mathbb E} \big[ \norm{F'(x_t, \xi_{0,t})}_{\psi^*}^2 \bigm| x_t \big] ~ \one_{\{t \in \mathcal{B}\}} + \sum_{t = s}^N \sum_{j=1}^m  \frac{\gamma_t^2}{2}  {\mathbb E} \big[ \norm{G'(x_t, \xi_{j,t})}_{\psi^*}^2 \bigm| x_t \big] ~ \one_{\{t \in \mathcal{N}_j\}}  \nonumber \\
& \leq \sum_{t = s}^N \frac{\gamma_t^2}{2} M_F^2 ~ \one_{\{t \in \mathcal{B}\}} + \sum_{t = s}^N \sum_{j=0}^m \frac{\gamma_t^2}{2} M_{G_j}^2  \one_{\{t \in \mathcal{N}_j\}}  \leq M^2 \sum_{t = s}^N \frac{\gamma_t^2}{2}
\end{align}
where the first inequality follows from Assumption \ref{assump:norm} and the second inequality follows from the definition of $M$. By combining \eqref{eq:second term}, \eqref{eq:third term} and \eqref{eq:fourth term} and by taking a conditional expectation over $x_t$ given ${\boldsymbol{\tilde{\xi}}}$, we obtain the desired inequality, hence proving the result.

\subsubsection{Proof of Lemma \ref{lemma:conc}}
The proof follows from Lemma 7 of \cite{lan2016algorithms}.
Note that $\zeta_t$ conditional on ${\boldsymbol{\tilde{\xi}}}$ is a deterministic function of $\xi_1, \ldots, \xi_{t-1}$, where $\xi_t = (\xi_{1,t}, \ldots, \xi_{m,t})$ is the sample drawn at $t$-th iteration. Now,
${\mathbb E}(\zeta_t | {\boldsymbol{\tilde{\xi}}}) = 0$ and from Assumption \ref{assump:lighttail}, ${\mathbb E}(\exp(\zeta_t^2 / \gamma_t^2 \sigma^2) | {\boldsymbol{\tilde{\xi}}}) \leq \exp(1)$.
Thus, now from the result of Lemma 7 in \cite{nemirovski2009robust} we have for all $\lambda > 0$,
\begin{align*}
P\left(\sum_{t=s}^{N} \zeta_t > \lambda \sqrt{\sum_{t=s}^N \gamma_t^2 \sigma^2} \;\;\Bigg| \;\;{\boldsymbol{\tilde{\xi}}} \right) \leq \exp\bigg(-\frac{\lambda^2}{3}\bigg).
\end{align*}
\qed

\subsubsection{Proof of Lemma \ref{lemma:donsker}}
Fix $j \in \{1,\ldots,m\}$. We begin by defining the \emph{bracketing number} $N_{[]}(\epsilon, \mathcal{F}_j, L_2(P))$ of the function class $\mathcal{F}_j = \{G_j(x, \cdot) : x \in \mathcal{X} \}$ with respect to the $L_2(P)$ norm defined as $||G_j(x, \cdot)||_{L_2(P)}^2 = {\mathbb E}[G_j(x, \xi)^2]$. The set of all functions in $\mathcal{F}$ satisfying $||G_j(x, \xi) - G_j(y, \xi)||_{L_2(P)} < \epsilon$ is an \emph{$\epsilon$-bracket} with boundaries $G_j(x, \xi)$ and $G_j(y, \xi)$. Then the bracketing number $N_{[]}(\epsilon, \mathcal{F}_j, L_2(P))$ is defined as the minimum number of $\epsilon$-bracket required to cover $\mathcal{F}_j$.

Next, we formally define the Donsker property for $\mathcal{F}_j$ and its relation with the bracketing number. The function class $\mathcal{F}_j$ is said to be $P$-Donsker if $\hat{G}(x)$ converges in distribution to a Gaussian random element (called Brownian bridge) in the space $\ell^{\infty}(\mathcal{X})$ of all bounded functions $h: \mathcal{X} \rightarrow \mathbb{R}$, equipped with the sup norm $||h||_{\sup} = \sup_{x} |h(x)|$. A sufficient condition for a function class $\mathcal{F}$ to be Donsker $P$-Donsker is given by
\begin{align}
\label{eq:suff cond}
N_{[]}(\epsilon, \mathcal{F}, L_r(P) = \mathcal{O}\bigg( \frac{1}{\epsilon^d} \bigg),~~\text{for some $d > 0$}.
\end{align}
We note that \eqref{eq:suff cond} holds for $\mathcal{F} = \mathcal{F}_j$ with $d = dimension(\mathcal{X})$. This follows from Assumption \ref{assump:donsker}, Theorem 2.7.11 of \cite{VaartWellner96} and the fact that $\mathcal{X}$ is a compact set. Therefore, the first part of Lemma \ref{lemma:donsker} follows from the continuous mapping theorem, since $h(x, \xi) = \sup_{x \in \mathcal{X}}|G_j(x, \xi) - g(x)|$ is a continuous function for each $\xi$.

The second part of Lemma \ref{lemma:donsker} follows from \eqref{eq:suff cond} and Theorem 2.14.2 of \cite{VaartWellner96}. \qed

\subsubsection{Proof of Lemma \ref{lemma:fkg}}
We refer to \cite{FKG71} for a proof.

\subsubsection{Proof of Lemma \ref{lemma:fkg application}}
Let $T = \{s, \ldots, N\}$ and let $\Omega(T)$ be the set of all subsets of $T$. We define a partial ordering on $\Omega(T)$ of as follows: $B_1 \preceq B_2$ if and only if $B_1 = \{t_1,\ldots,t_{k_1}\}$ and $B_2 = \{t_1,\ldots,t_{k_2}\}$ where $s \leq t_1 < t_2 < \cdots < t_{k_1} < \cdots < t_{k_2} \leq N$. We define a probability distribution on $\Omega(T)$ as $\mu(A) : = P(\mathcal{B} = A)$. Then it is easy to verify that $(\Omega(T), \vee, \wedge)$ is a finite distributive lattice for
\[
A\vee B := \mathop{\mathrm{argmax}} \{\mu(A), \mu(B)\} ~~\text{and}~~ A\wedge B := \mathop{\mathrm{argmin}} \{\mu(A), \mu(B)\}.
\]
Therefore, we have $P(A\vee B)~P(A\wedge B) = P(A) P(B)$. Finally, note that both $h_1(B) := {\mathbb E} [ \sum_{t \in \mathcal{B}} \gamma_t (f(x_t) - f(x^*)) \mid \mathcal{B} = B]$ and $h_2(B) := -{\mathbb E} [ 1 /(\sum_{t \in \mathcal{B}} \gamma_t) \mid \mathcal{B} = B]$ are increasing functions in $\Omega(T)$ as $f(x) - f(x^*) \geq 0$ for all $x \in \mathcal{X}$ and $\gamma_t > 0$ for all $t$. Thus Lemma \ref{lemma:fkg} follows directly from Lemma \ref{lemma:fkg}. \qed

\subsection{Experimental Setup}

To initialize, we choose $x_1 = 0.5 \times \one_d$, a $d$-dimensional vector where each co-ordinate is 0.5
and we fix $d = 100$. Moreover we choose $\{\eta_j\}_t = M^2/\sqrt{N}$ and $\{\gamma\}_t  = D_\mathcal{X}/(M\sqrt{N})$. We choose $\psi(x) = x^Tx$ and $M = 10$. We first convert this into a minimization problem and we pick, $\mu_0 = -0.8\times\one_d$ and $\Sigma_0 = \mathbf{I}_{d\times d}$.

\subsubsection{Feasible Configuration:}
We begin with a feasible configuration. We choose, $\mu_j = \mu \times \one_d$,  $\Sigma_j = \sigma^2 \mathbf{I}_{d\times d}$ for $j = 1, \ldots, m$. We run both variants of the MCSA algorithm along with the state-of-the-art \cite{yu2017online} for difference choices of $(\mu,\sigma^2)$.  Specifically, we choose $\mu = \{-0.2, -0.001\}$ and $\sigma^2 = \{0.01, 2.5, 5\}$. Note that $\mu = -0.2$ it is a relatively easier problem since the feasible region is easier to obtain as compared to $\mu = -0.001$.

In all of these settings, the optimal solution is $f(x^*) = 80$. Each of these six setting are repeated 100 times and we plot the mean function decay $f(x_t) -  f(x^*)$ across the iterations. We also add a 95\% confidence interval from the repeated experiments. Throughout we are considering the corresponding minimization problem corresponding to \eqref{eq:example_problem}.

\subsubsection{Infeasible Configuration:}
Note that as long as $\mu < 0$, the problem in \eqref{eq:example_problem} is a feasible problem. To test infeasibility, we choose $\mu = 0.2 \times \one_d$ and $\sigma^2 = 1$. Furthermore we increase the number of iterations $N = 10000$. In this cases, both the algorithms failed to converge as expected. The plot is for MCSA shown in Figure \ref{fig:infeasible}.

\begin{figure} [ht!]
\centering
\includegraphics[scale=0.35]{plots/DecayRate-infeas1}
\caption{Function decay for an infeasible problem.}
\label{fig:infeasible}
\vspace{-0.3cm}
\end{figure}

\section{Related Works}
\vspace{-0.3cm}
\section{Discussion}
\label{sec:discussion}
\vspace{-0.3cm}
In this paper, we have described a solution to a stochastic optimization problem specifically containing multiple expectation constraints. We introduce the MCSA algorithm, prove its optimal convergence by a careful consideration of the dependent structure. This work also plugs in several gaps in the proof of \cite{lan2016algorithms}. We compare an online version of our algorithm to the state-of-the-art and empirically show some instances where it performs better. As a future step, we are interested in extending the algorithm to more general problems, especially the cases where the expectation constraints are replaced by probability. Being able to solve such general problems will have a high impact in many research and scientific domains.

\section*{Acknowledgement}
We thank Abhishek Chakrabortty for many helpful discussions on empirical processes and Donsker's theorem.
\section{Experiments}
\label{sec:experiments}
\vspace{-0.3cm}
In this section, we describe simulated experiment
to showcase the efficacy of our algorithm. Throughout this section, we compare our algorithm to the state-of-the-art algorithm in online convex optimization with stochastic constraints \cite{yu2017online}. We focus our experiments on a real-world problem which motivated our work.
\vspace{-0.2cm}
\subsection{Personalized Parameter Selection in Large-Scale Social Network}
\vspace{-0.3cm}
Most social networks do extensive experimentation to identify which parameter gives rise to the biggest online metric gains. However, in many cases choosing a single global parameter is not very prudent. That being said, experimenting to identify member level parameter is an extremely challenging problem. A good middle pathway lies in identifying cohorts of members and estimating the optimal parameter for each cohort. \cite{tu2019prophet} tries to solve this problem by framing this problem as \eqref{eq:main_problem}.

Let us focus on the minimum gap parameter in the ranking ads problem. Let us assume that the parameter can take $K$ possible  values and there are $M + 1$ metrics we are interested in. One primary metric (revenue) and $M$ guardrail metrics (ads click-through-rate, organic click-through-rate, etc). Specifically, we can estimate the causal effect when treatment $k \in K$ is applied to a cohort for each of these $M+1$ metrics. This effect is a random variable which is usually distributed as a Gaussian with some mean and variance. Our aim is to identify the optimal allocation $x^*$ which maximizes the expected revenue. Formally, let $x_{ik}$ denote the probability of of assigning the $k$-th treatment to the $i$-th cohort and let $U_{ik}^j$ denote the metric lift in the $j$-th metric when $k$-th treatment is assignment to $i$-th cohort.
\vspace{-0.1cm}
Thus we want to solve,
\begin{equation}
\label{eq:example_main}
\begin{aligned}
& \underset{x}{\text{Maximize}} &  \sum_{ik} x_{ik}U_{ik}^0\\
& \text{subject to}
&  \sum_{ik}x_{ik}U_{ik}^j \leq c_j &\qquad \text{ for } j = 1, \ldots, M. \\
&& 0 \leq x_{ik} \leq 1 &\qquad \text{ for all } i,k \qquad \text{ and } \qquad \sum_{k} x_{ik} = 1 \;\;\; \text{ for all } i.
\end{aligned}
\end{equation}
\vspace{-0.4cm}

Note that each $U_{ik}^j$ is a random variable distributed usually as a Gaussian distribution with some mean and variance. Hence this problem can be translated to the following:
\vspace{-0.1cm}
\begin{equation}
\label{eq:example_problem}
\begin{aligned}
& \underset{x}{\text{Maximize}} & & f(x) := {\mathbb E}_{ \xi_0} \left(x^T\xi_0\right)\\
\vspace{-0.5cm}
& \text{subject to}
& & g_j(x) :=  {\mathbb E}_{ \xi_j} \left(x^T \xi_j - c_j\right) \leq 0\qquad \text{ for } j = 1, \ldots, M, \qquad \text{ and } \qquad x \in \mathcal{X}
\end{aligned}
\end{equation}
where $\mathcal{X}$ are the set of non-stochastic constraints. For simplicity in our simulation setting we choose, $\mathcal{X} = [0,1]^d$, and $c_j = 0$ for all $j$. Further, we assume that each $\xi_j \sim N_d(\mu, \Sigma)$ for $j = 0, \ldots, M$. Note that, this also satisfies Assumptions \ref{assump:norm} and \ref{assump:donsker}. All parameter settings are pushed to the supplementary for brevity.

We run the algorithm using different configurations of the mean and variance.
Although, we fix ${\boldsymbol{\tilde{\xi}}}$ in our algorithm to get $\hat{G}_{j}$ in \eqref{eq:Gfunc1}, it can very easily be changed to an online version, by choosing
$\hat{G}_{j}(x_t) := \frac{1}{t} \sum_{\ell = 1}^t G_j(x_t, \tilde{\xi}_{j,\ell})$.
We call this the MCSA-online algorithm and show the results to ensure a fair comparison with Yu et. al. (2017) \cite{yu2017online}. The results are shown in Figure \ref{fig:main}.
\vspace{-0.2cm}
\begin{figure*}[!th]
\centering
\includegraphics[width=\linewidth]{plots/decay_final}
\caption{\label{fig:main} This shows the function decay vs iterations for six different simulation settings corresponding to $\mu \in \{-0.2, -0.001\}$ and $\sigma^2 \in \{0.01, 2.5, 5\}$. The top and the bottom rows correspond to an easy and a more difficult feasibility domain respectively, while the columns depict an increase in variance we move towards the right. The dotted lines denote the 95\% confidence interval using 100 repeats of each configuration.}
\end{figure*}

From Figure \ref{fig:main}, we observe that in a high signal-to-noise ratio setting all algorithms work well. For the settings with low signal-to-noise ratios, MCSA and MCSA-online outperform the state-of-the-art. For a harder problem setting, a more clear evidence can be seen that our algorithm performs better than \cite{yu2017online}. We have also tried a similar exercise with different dimensions $d$ and different number of constraints $M$ and in each case, we observed a similar result.

\section{Introduction}
\label{sec:introduction}
\vspace{-0.3cm}
In this paper we focus on a stochastic optimization problem with multiple expectation constraints. Specifically, we are interested in solving a problem of the following form:
\begin{equation}
\label{eq:main_problem}
\begin{aligned}
& \underset{x}{\text{min}} & & f(x) := {\mathbb E}_{ \xi_0} \left(F(x, \xi_0)\right)\\
& \text{subject to}
& & g_j(x) :=  {\mathbb E}_{ \xi_j} \left(G_j(x, \xi_j)\right) \leq 0\qquad \text{ for } j = 1, \ldots, m. \\
&&& x \in \mathcal{X}
\end{aligned}
\end{equation}
where $\mathcal{X} \subseteq {\mathbb R}^n$ is a convex compact set, $\xi_j$ are random variables
for $j = 0, \ldots, m$.
$F(\cdot), G_j(\cdot)$ for $j = 1, \ldots, m$ are closed convex functions with respect to $x$ for a.e. $\xi_j \in \mathcal{P}_j$ for $j = 0, \ldots, m$.

There are several applications of the above problem formulation \eqref{eq:main_problem} especially, in fields such as control theory \cite{primbs2009stochastic}, management science \cite{charnes1959chance}, finance \cite{rockafellar2000optimization}, etc.
Our specific motivation comes from a problem arising from a large-scale social network platform. Any such platform does extensive experimentation to identify which parameters should be applied to certain members to get the most amount of metric gains. An example of such an influential parameter is the gap between successive ads on a newsfeed product (described in details in Section \ref{sec:experiments}).

Traditionally, stochastic optimization routines were solved either via sample average approximation (SAA) \cite{kleywegt2002sample, shapiro2003monte, wang2008sample} or via stochastic approximation (SA) \cite{robbins1951stochastic}.  In SAA, each function $g_j$ for $j = 1, \ldots, m$ are approximated by a sample average $\hat{g}_j(x) = \sum_{i=1}^n G_j(x, \xi_{j,i}) /n$ and then solved via traditional means. The SAA solution is computationally challenging, may not be applicable to the online setting and the approximation might lead to an infeasible problem. In SA, the algorithm follows the usual gradient descent algorithm by using the stochastic gradient rather than $f'(x)$ to solve \eqref{eq:main_problem} \cite{benveniste2012adaptive, spall2005introduction}. The SA solution requires a projection onto the domain specified by $\{g_j(x) \leq 0\}$, which may not be possible when we have the expectation formulation.

There have been several papers showing improvement over the original SA method especially for strongly convex problems \cite{polyak1992acceleration}, and for general class of non-smooth convex stochastic programming problems \cite{nemirovski2009robust}. However, these methods may not be directly applicable to the case where each $g_j$ is an expectation constraint. Very recently \cite{yu2017online} developed a method of stochastic online optimization with general stochastic constraints by generalizing Zinkevich's online convex optimization. Their approach is much more general and as a result may be not the optimal approach to solving this specific problem. For more related works, we refer to \cite{lan2016algorithms} and \cite{yu2017online} and the references therein.

Lan et. al. (2016) \cite{lan2016algorithms} introduced the cooperative stochastic approximation (CSA) algorithm to solve problem \eqref{eq:main_problem} with $m=1$. In this paper, we extend their algorithm to multiple expectation constraints and also close several gaps in their proof of optimal convergence. Specifically, we prove that an optimal point $\hat{\mathbf{x}}$ satisfying
\begin{align*}
{\mathbb E}(f(\hat{\mathbf{x}}) - f(\mathbf{x}^*)) \leq c/\sqrt{N} \text{ and }  {\mathbb E}(g_j(\hat{\mathbf{x}}))  \leq C/\sqrt{N}  \;\; \forall j \in \{1,\ldots, m\}
\end{align*}
can be obtained with $N$ steps of our algorithm. These rates are optimal due to the lower bound results following from \cite{agarwal2009information}. We primarily focus on the theoretical analysis of the algorithm in this paper. We use a completely novel proof technique and overcome several gaps in the proof of the \cite{lan2016algorithms} which we highlight in Section \ref{sec:convergence}. Furthermore, we run experiments on simulated data to show that this algorithm empirically outperforms the algorithm in \cite{yu2017online}. For more practical results, we refer the reader to \cite{tu2019prophet}.

The rest of the paper is organized as follows. In Section \ref{sec:algo}, We introduce the multiple cooperative stochastic approximation (MCSA) algorithm for solving the problem stated in \eqref{eq:main_problem}. In Section \ref{sec:convergence}, we prove the convergence guarantee of our algorithm. We discuss some empirical results in Section \ref{sec:experiments} before concluding with a discussion in Section \ref{sec:discussion}.

\section{Convergence Analysis}
\label{sec:convergence}
\vspace{-0.3cm}
In this section, we study the convergence of the MCSA algorithm as described in Section \ref{sec:mcsa}. Specifically, we show that if we run $N$ iterations in Algorithm \ref{algo:mcsa}, then expected error and expected violation of the constraints is $\mathcal{O}(1/\sqrt{N})$. Note that the presence of interrelated stochastic components of the MCSA algorithm poses a novel challenge in deriving the rate of convergence of MCSA. In particular, at every step MCSA makes a random decision for choosing $h_t$ depending on $x_t$ and $\{\tilde{\xi}_{j, \ell} : j = 1, \ldots, m,~ \ell = 1,\ldots, L \}$, while $x_t$ is a random variable depends on $x_{t-1}$, $\{\xi_{j,t} : j = 0,\ldots, m \}$ and a similar random decision made at the previous step for generating $x_{t-1}$. A careful consideration of this intertwined stochasticity reveals several gaps in the convergence analysis of the CSA algorithm as presented in \cite{lan2016algorithms}. For this reason, we refrained from deriving the rate of convergence of MCSA by extending the convergence analysis of \cite{lan2016algorithms}. Our rigorous convergence analysis is the main contribution of this paper and this also reinforces the $\mathcal{O}(1/\sqrt{N})$ convergence rate of the CSA algorithm \cite{lan2016algorithms}.

\textbf{Proof Overview}: We begin with the identification of a sufficient condition on the threshold parameters $\{\eta_{j,t}\}, \{\gamma_t\}$ such that the solution $\hat{\mathbf{x}}$ is well-defined and on average consists of at least $cN$ $x_t$'s for some $c \in (0, 1]$ (Theorem \ref{theorem:welldefined}). In order to show that result, we use a concentration bound (Lemma \ref{lemma:conc}) as well as the properties of a Donsker class of functions (Lemma \ref{lemma:donsker}). Finally, we prove the $\mathcal{O}(1/\sqrt{N})$ convergence rate of the MCSA algorithm in Theorem \ref{thm:main1} using Theorem \ref{theorem:welldefined}, an application of the FKG inequality (Lemma \ref{lemma:fkg application}) and Lemma \ref{lemma:donsker}. We use the FKG inequality to untangle the dependency between the random variables $\{x_t : t \in \mathcal{B}\}$ and the random set $\mathcal{B}$.

\vspace{-0.2cm}
\subsection{Supporting Results}
\label{sec:supporting}
\vspace{-0.2cm}
\textbf{Bregman Divergence:} We begin by stating our first lemma connecting Bregman divergence and the proximal projection function \eqref{eq:prox_proj}. All proofs of lemmas in this sub-section are pushed to the supplementary material for brevity.
\begin{lemma}
\label{lemma:breg}
For any $x,z \in \mathcal{X}$ and $y \in {\mathbb R}^n$, we have $B_{\psi}\left( z, P_x(y) \right) \leq B_{\psi}( z, x ) + y^T(z - x) + \frac{1}{2}\norm{y}_{{\psi}^*}^2 $
where $\norm{\cdot}_{{\psi}^*}$ is the dual norm of $\norm{\cdot}_{\psi}$.
\end{lemma}
The following result follows from Lemma \ref{lemma:breg} and a careful analysis of Algorithm \ref{algo:mcsa}.
\begin{lemma}
\label{lemma:prop9}
For any $s \in T$, we have
\vspace{-0.2cm}
\begin{align}
\label{eq:prop9}
\sum_{t \in \mathcal{B}}^N \gamma_t \langle F'(x_t, \xi_{0,t}), x_t - x \rangle &+ \sum_{j=1}^m \sum_{t \in \mathcal{N}_j}\gamma_t \big(G_j(x_t, \xi_{j,t}) -  G_j(x, \xi_{j,t})\big) \leq B_{\psi}(x, x_{s}) \nonumber \\
&+ \sum_{t \in \mathcal{B}} \frac{\gamma_t^2}{2} \norm{F'(x_t, \xi_{0,t})}_{\psi^*}^2 +  \sum_{j=1}^m \sum_{t \in \mathcal{N}_j}\norm{G'(x_t, \xi_{j,t})}_{\psi^*}^2 \;\; \text{a.s.}
\end{align}
\end{lemma}
We bound the right hand side of \eqref{eq:prop9} in Lemma \ref{lemma:prop9} by making the following assumption.
\begin{assumption}
\label{assump:norm}
For any $x \in \mathcal{X}$, the following holds
\begin{align*}
{\mathbb E}( \norm{F'(x, \xi_0)}_{{\psi}^*}^2)  \leq M_F^2 \qquad \text{ and } \qquad
{\mathbb E}( \norm{G'_j(x, \xi_j)}_{{\psi}^*}^2)  \leq M_{G_j}^2 \;\; \forall \; j \in \{1, \ldots, m\}.
\end{align*}
\end{assumption}
\begin{lemma}
\label{lemma:prop9b}
Under Assumption \ref{assump:norm}, for any $s \in T$, we have
\vspace{-0.2cm}
\begin{align}
\label{eq:prop9b}
~\sum_{t = s}^N \sum_{j=1}^m \gamma_t~{\mathbb E}\bigg[\big(g_j(x_t) -  g_j(x) \big) ~\one_{\{t \in \mathcal{N}_j\}} \biggm| {\boldsymbol{\tilde{\xi}}} \bigg] ~&+~ \sum_{t = s}^N  \gamma_t ~{\mathbb E} \bigg[ \langle f'(x_t), x_t - x \rangle ~\one_{\{t \in \mathcal{B}\}} \biggm| {\boldsymbol{\tilde{\xi}}} \bigg] \nonumber \\
& \leq ~ {\mathbb E} \bigg[ B_{\psi}(x, x_{s}) \bigg| {\boldsymbol{\tilde{\xi}}} \bigg]  ~+~ \frac{M^2}{2}~ \sum_{t = s}^N\gamma_t^2,
\end{align}
where $M^2 := \max \{M_F^2, M_{G_1}^2, \ldots, M_{G_m}^2\}$, ${\boldsymbol{\tilde{\xi}}} :=  \{ \{ \tilde{\xi}_{j,\ell}\}_{\ell = 1}^L\}_{j=1}^m$ and $\mathcal{N}_j \subseteq \{s, \ldots, N\}$ denotes the random set of indices for which $j^* = j$ in Algorithm \ref{algo:mcsa}.
\end{lemma}

\textbf{Concentration Bounds:} We use a concentration result to achieve the optimal convergence. Towards that, we first define,
\vspace{-0.2cm}
\begin{align*}
\zeta_t = \sum_{j=1}^m \gamma_t \left( g_j(x_t)\one_{\{t \in \mathcal{N}_j\}} -  {\mathbb E} \left[ g_j(x_t) ~\one_{\{t \in \mathcal{N}_j\}} \bigg| {\boldsymbol{\tilde{\xi}}} \right]\right).
\end{align*}
Now, we assume that the distribution of $\zeta_t$ has a light-tail.
\begin{assumption}
\label{assump:lighttail}
There exists a $\sigma^2$ such that ${\mathbb E}[ \zeta_t^2 / (\gamma_t^2 \sigma^2) ~ | ~ {\boldsymbol{\tilde{\xi}}} ] \leq  \exp(1).$
\end{assumption}
\begin{lemma}
\label{lemma:conc}
Under Assumption \ref{assump:lighttail}, for any $\lambda > 0$,
$P\big(\sum_{t=s}^{N} \zeta_t > \lambda \sqrt{\sum_{t=s}^N \gamma_t^2 \sigma^2} \;\big|\;{\boldsymbol{\tilde{\xi}}} \big) \leq \exp\big(-\frac{\lambda^2}{3}\big).$
\end{lemma}

\textbf{Donsker Class:} Recall that we approximate $g_j(x_t)$ by $\hat{G}_{j}(x_t)$ in Algorithm \ref{algo:mcsa}. Although the central limit theorem guarantees the convergence of $\sqrt{L}(\hat{G}_{j}(x) - g_j(x))$ to a zero mean Gaussian distribution for each $x$, it does not ensure a ``uniform convergence'' for all $x \in \mathcal{X}$ as in Lemma \ref{lemma:donsker}. To achieve that we show $\mathcal{F}_j = \{ G_j(x, \cdot): x \in \mathcal{X} \}$ is a \emph{Donsker class} \cite{VaartWellner96} under the following assumption.

\begin{assumption}
\label{assump:donsker}
For each $j = 1, \ldots, m$, the class of functions $\mathcal{F}_j = \{ G_j(x, \cdot): x \in \mathcal{X} \}$ satisfies the following Lipschitz condition:
\vspace{-0.2cm}
\[
|G_j(x, \xi) - G_j(y, \xi)| \leq |x - y|~ \phi(\xi),
\]
for all $x,y \in \mathcal{X}$ and for some function $\phi$ satisfying ${\mathbb E}[\phi(\xi)^2] < \infty$.
\end{assumption}
The Lipschitz condition in Assumption \ref{assump:donsker} ensures $G_j(\cdot, \xi)$ is sufficiently smooth for each $\xi$. It is easy to see that Assumption \ref{assump:donsker} holds if the derivative $G_j'(x, \xi)$ is uniformly bounded by a constant for all $\xi$. We use Assumption \ref{assump:donsker} to prove the following lemma.

\begin{lemma}
\label{lemma:donsker}
Under Assumption \ref{assump:donsker},
\vspace{-0.2cm}
\[
\sup_{x \in \mathcal{X}}\sqrt{L}\left|\hat{G}_{j}(x) - g_j(x)\right| \overset{d}{\longrightarrow} \sup_{x \in \mathcal{X}} \left|\mathbb{G}(x) \right| ~~\text{and}~~ {\mathbb E}\bigg(  \sup_{x \in \mathcal{X}}\left| \hat{G}_{j}(x) - g_j(x) \right| \bigg) = \mathcal{O}(1 / \sqrt{L}),
\]
where $\mathbb{G}(x)$ is a zero mean Gaussian process with covariance function ${\mathbb E}[\mathbb{G}(x)\mathbb{G}(y)] = {\mathbb E}[G(x,\xi)G(y,\xi)] - {\mathbb E}[G(x,\xi)] {\mathbb E}[G(y,\xi)]$.
\end{lemma}

\textbf{FKG Inequality:}
The Fortuin-Kasteleyn-Ginibre (FKG) \cite{FKG71} inequality asserts positive correlation between two increasing functions on a finite partially ordered set.
\begin{lemma}
\label{lemma:fkg}
Let $L$ be partially ordered set such that $(L, \vee, \wedge)$ be a finite distributive lattice. Further, let $\mu$ be a probability measure on $L$ such that $\mu(\ell_1\wedge \ell_2) \mu(\ell_1\vee \ell_2) \geq \mu(\ell_1) \mu(\ell_2)$. If both $h_1$ and $h_2$ are increasing functions on $L$ with respect to the partial ordering of $L$, then
\vspace{-0.2cm}
\[
\sum_{\ell \in L} h_1(\ell)h_2(\ell) \mu(\ell) \geq \bigg(\sum_{\ell \in L} h_1(\ell) \mu(\ell) \bigg) \bigg(\sum_{\ell \in L} h_2(\ell) \mu(\ell) \bigg).
\]
\end{lemma}
The following result follows from Lemma \ref{lemma:fkg} and the fact that $f(x_t) - f(x^*) \geq 0$.
\begin{lemma}
\label{lemma:fkg application}
Let $\{\gamma_t\}$, $\{x_t\}$ be as in Algorithm \ref{algo:mcsa} and let $x^* := \underset{x}{\mathop{\mathrm{argmin}}} f(x)$. Then
\vspace{-0.2cm}
\[
{\mathbb E}\bigg[\frac{\sum_{t \in \mathcal{B}} \gamma_t (f(x_t) - f(x^*))}{\sum_{t \in \mathcal{B}} \gamma_t}\bigg] \leq {\mathbb E}\bigg[\sum_{t \in \mathcal{B}} \gamma_t (f(x_t) - f(x^*))\bigg]~{\mathbb E}\bigg[\frac{1}{\sum_{t \in \mathcal{B}} \gamma_t}\bigg].
\]
\end{lemma}
\vspace{-0.3cm}
\subsection{Main Result}
\label{sec:main_result}
\vspace{-0.3cm}
First, we present a sufficient condition under which ${\mathbb E}[|\mathcal{B}|/N] \geq c$. We define,
\vspace{-0.2cm}
\begin{align}
\label{eq:tau_def}
\tau := D_{\mathcal{X}}^2 +   \frac{M^2}{2}  \sum_{t = s}^N \gamma_t^2,
\end{align}
where $D_\mathcal{X} = \sqrt{\max_{x, z \in \mathcal{X}} B_{\psi}(z, x)}$ denotes the diameter of $\mathcal{X}$.

\begin{theorem}
\label{theorem:welldefined}
Let $\{\gamma_t\}$, $\eta_{j,t}$, $\{x_t\}$ be as in Algorithm \ref{algo:mcsa} and let $x^* := \underset{x}{\mathop{\mathrm{argmin}}} f(x)$. Let
\vspace{-0.3cm}
\begin{align}
\label{eq:welldefined}
\gamma_t = \frac{\sqrt{2}K_1}{\sqrt{N}} ~~\text{and}~~ \eta_{j,t} = \frac{\sqrt{2}K_2}{\sqrt{N}} ~~\text{for all $t$},
\end{align}
for some sufficiently large constant $K_1$ and $K_2$. Then one of the following two conditions hold:
\vspace{-0.2cm}
\begin{enumerate}
\item $P\left(\frac{|\mathcal{B}|}{N} > \frac{1}{2} - \frac{s-1}{N} - \frac{\tau}{2K_1K_2}\right) > \frac{1}{4}$ which implies ${\mathbb E}(|\mathcal{B}| / N) > c$ for some $c \in (0,1]$
\vspace{-0.2cm}
\item $\mathcal{B} \neq \emptyset$ and  $\sum_{t = s}^N  \gamma_t \langle f'(x_t), x_t - x^* \rangle ~\one_{\{t \in \mathcal{B}\}}  \leq 0$ almost surely.
\end{enumerate}
\end{theorem}
\vspace{-0.3cm}
\begin{proof}
We show that if the second condition does not hold then the first condition must hold. Note that if the second condition does not hold,
\vspace{-0.2cm}
\begin{align*}
&\sum_{t = s}^N  \gamma_t \langle f'(x_t), x_t - x^* \rangle ~\one_{\{t \in \mathcal{B}\}}  \geq 0 \qquad
\Rightarrow \qquad \sum_{t = s}^N  \gamma_t ~{\mathbb E} \bigg[ \langle f'(x_t), x_t - x^* \rangle ~\one_{\{t \in \mathcal{B}\}} \biggm| {\boldsymbol{\tilde{\xi}}} \bigg] \geq 0,
\end{align*}
where the equality holds if $\mathcal{B} = \emptyset$. Thus it follows from Lemma \ref{lemma:prop9b} with $x = x^*$ that
\vspace{-0.2cm}
\begin{align*}
~\sum_{t = s}^N \sum_{j=1}^m \gamma_t~{\mathbb E} \bigg[\big(g_j(x_t) -  g_j(x^*) \big) ~\one_{\{t \in \mathcal{N}_j\}} \biggm| {\boldsymbol{\tilde{\xi}}} \bigg] \leq ~ {\mathbb E} \bigg[ B_{\psi}(x^*, x_{s}) \bigg| {\boldsymbol{\tilde{\xi}}} \bigg]  ~+~ \frac{M^2}{2}  \sum_{t = s}^N \gamma_t^2.
\end{align*}
Moreover, since $g_j(x^*) \leq 0$, $B_{\psi}(x^*, x_s) \leq D_{\mathcal{X}}^2$, we have
\vspace{-0.1cm}
\begin{align}
\label{eq:lemma41}
\sum_{t = s}^N \sum_{j=1}^m \gamma_t {\mathbb E} \left[ g_j(x_t) ~\one_{\{t \in \mathcal{N}_j\}} \bigg| {\boldsymbol{\tilde{\xi}}} \right] \leq \tau
\end{align}
where $\tau$ is defined in \eqref{eq:tau_def}. Note that, for $t \in \mathcal{N}_j$, we have $\hat{G}_{j}(x_t) > \eta_{j,t}$. Thus we have,
\vspace{-0.1cm}
\begin{align}
\label{eq:lemma42}
 \sum_{t = s}^N \sum_{j=1}^m \gamma_t \bigg( \hat{G}_{j}(x_t) \one_{\{t \in \mathcal{N}_j\}} - {\mathbb E} \left[ g_j(x_t) ~\one_{\{t \in \mathcal{N}_j\}} \bigg| {\boldsymbol{\tilde{\xi}}} \right] \bigg) > \sum_{t = s}^N \sum_{j=1}^m \gamma_t \eta_{j,t} \one_{\{t \in \mathcal{N}_j\}} - \tau.
\end{align}
Now consider the following two sets:
\vspace{-0.2cm}
\begin{align}
A_1 &:= \bigg\{\sum_{t = s}^N \sum_{j=1}^m \gamma_t (\hat{G}_{j}(x_t) - g_j(x_t)) \one_{\{t \in \mathcal{N}_j\}} \leq \frac{1}{2} \sum_{t = s}^N \sum_{j=1}^m \gamma_t \eta_{j,t} \one_{\{t \in \mathcal{N}_j\}} \bigg\}; \\
A_2 &:= \bigg\{\sum_{t = s}^N \sum_{j=1}^m \gamma_t \left( g_j(x_t)\one_{\{t \in \mathcal{N}_j\}} -  {\mathbb E} \left[ g_j(x_t) ~\one_{\{t \in \mathcal{N}_j\}} \bigg| {\boldsymbol{\tilde{\xi}}} \right]\right) \leq \frac{K_1K_2 - \tau}{2} \bigg\}.
\end{align}

On the set $A_1 \cap A_2$, we have
\vspace{-0.2cm}
\begin{align}\label{eq:intersection}
 \sum_{t = s}^N \sum_{j=1}^m \gamma_t \big( \hat{G}_{j}(x_t) \one_{\{t \in \mathcal{N}_j\}} - {\mathbb E} \big[ g_j(x_t) ~\one_{\{t \in \mathcal{N}_j\}} \big| {\boldsymbol{\tilde{\xi}}} \big] \big) \leq \frac{1}{2}\sum_{t = s}^N \sum_{j=1}^m \gamma_t \eta_{j,t} \one_{\{t \in \mathcal{N}_j\}} + \frac{K_1K_2 - \tau}{2}.
\end{align}
By combining \eqref{eq:welldefined}, \eqref{eq:lemma42} and \eqref{eq:intersection}, we get
\vspace{-0.2cm}
\begin{align*}
\frac{2K_1K_2(N-s+1 - |\mathcal{B}|)}{N} \leq \sum_{t = s}^N \sum_{j=1}^m \gamma_t \eta_{j,t} \one_{\{t \in \mathcal{N}_j\}} < K_1K_2 + \tau.
\end{align*}
This implies on the set $A_1 \cap A_2$, we have $\frac{|\mathcal{B}|}{N} > \frac{1}{2} - \frac{s-1}{N} - \frac{\tau}{2K_1K_2}$. Thus, we get,
\vspace{-0.2cm}
\begin{align*}
P\left(\frac{|\mathcal{B}|}{N} > \frac{1}{2} - \frac{s-1}{N} - \frac{\tau}{2K_1K_2}\right) & \geq P(A_1 \cap A_2) \geq 1 - P(A_1^c) - P(A_2^c).
\end{align*}
Next, we derive upper bounds for $P(A_1^c)$ and $P(A_2^c)$. From straightforward calculations, it follows that
\vspace{-0.2cm}
\begin{align}
\label{eq:firstbound}
P(A_1^c) & \leq P\bigg(\sum_{t = s}^N \sum_{j=1}^m \left|\gamma_t (\hat{G}_{j}(x_t) - g_j(x_t)) \one_{\{t \in \mathcal{N}_j\}}\right| > \sum_{t=s}^N \sum_{j=1}^m \frac{K_1K_2}{N} \bigg) \nonumber \\
& \leq \sum_{j=1}^m P\bigg( \sup_{x \in \mathcal{X}} \left| \sqrt{L} (\hat{G}_{j}(x) - g_j(x))\right| >   \sqrt{\frac{LK_2^2}{2N}} \bigg) < \frac{1}{2}.
\end{align}
where the last inequality follows from Lemma \ref{lemma:donsker} for sufficiently large $K_2$, since we have chosen $L = \Omega(N)$. Now using Lemma \ref{lemma:conc} and choosing $\lambda = (K_1K_2 - \tau)/\sqrt{4 \sum_{t=s}^N \gamma_t^2 \sigma^2}$, we have
\vspace{-0.1cm}
\begin{align}
\label{eq:secondbound}
P(A_2^c) & \leq \exp \left( -\frac{N (K_1K_2 - \tau)^2}{12 \sigma^2 (N-s+1) K_1^2} \right) < \frac{1}{4},
\end{align}
where the last inequality follows from choosing suitably large $K_1, K_2$. Thus, using \eqref{eq:firstbound} and \eqref{eq:secondbound} we get
\vspace{-0.4cm}
\begin{align*}
P\left(\frac{|\mathcal{B}|}{N} > \frac{1}{2} - \frac{s-1}{N} - \frac{\tau}{2K_1K_2}\right) > \frac{1}{4}.
\end{align*}
\vspace{-0.2cm}
\end{proof}
\begin{theorem}
\label{thm:main1}
Let $\{\gamma_t\}$, $\eta_{j,t}$, $\{x_t\}$, $x^*$ be as in Theorem \ref{theorem:welldefined}. Then under Assumptions \ref{assump:norm} and \ref{assump:donsker}, we have
\vspace{-0.1cm}
\begin{align}
\label{eq:rate_eq}
{\mathbb E}[f(\hat{\mathbf{x}}) - f(x^*)] = \mathcal{O}\bigg(\frac{1}{\sqrt{N}}\bigg) \;\;\text{ and } \;\;{\mathbb E}[ g_j(\hat{\mathbf{x}})] = \mathcal{O}\bigg(\frac{1}{\sqrt{L}} + \frac{1}{\sqrt{N}}\bigg) \;\; \forall j.
\end{align}
\end{theorem}

\begin{proof}
First observe that the second condition of Theorem \ref{theorem:welldefined} implies that our Algorithm has already converged. Specifically, if $\mathcal{B} \neq \emptyset$ our algorithm is well-defined and we have,
\vspace{-0.2cm}
\begin{align}
\label{eq:rightbound1}
0 &\geq \sum_{t = s}^N  \gamma_t \langle f'(x_t), x_t - x^* \rangle ~\one_{\{t \in \mathcal{B}\}} \geq  \sum_{t = s}^N \gamma_t (f(x_t) - f(x^*)) ~\one_{\{t \in \mathcal{B}\}} \nonumber \\
& \geq \bigg( \sum_{t = s}^N  \gamma_t \one_{\{t \in \mathcal{B}\}} \bigg) (f(\hat{\mathbf{x}}) - f(x^*))
\end{align}
where we have used the successively used the convexity of $f$. Since $\mathcal{B} \neq \emptyset$, we get $f(\hat{\mathbf{x}}) - f(x^*) \leq 0$ and hence the algorithm has already converged, i.e.\ $\hat{\mathbf{x}} = x^*$. In this case, we have ${\mathbb E}[f(\hat{\mathbf{x}}) - f(x^*)] = 0$ and ${\mathbb E}[ g_j(\hat{\mathbf{x}})] = {\mathbb E}[ g_j(x^*)] \leq 0$. Thus, either our algorithm has already converged or the first condition of Theorem \ref{theorem:welldefined} holds. From the first condition of Theorem \ref{theorem:welldefined} we have ${\mathbb E}(|\mathcal{B}| / N) > c$ for sufficiently large $N$ and for some $c > 0$. Now using convexity of $g_j$ and the fact that $\hat{G}_{j}(x_t) \leq \eta_{j,t}$ for $t \in \mathcal{B}$, we have
\vspace{-0.2cm}
\begin{align*}
{\mathbb E}( g_j(\hat{\mathbf{x}}) ) & \leq {\mathbb E} \left[ \frac{\sum_{t=s}^N \gamma_t g_j(x_t) \one_{\{t \in \mathcal{B}\}}}{ \sum_{t = s}^N \gamma_t \one_{\{t \in \mathcal{B}\}}}\right] \\
&= {\mathbb E} \left[ \frac{\sum_{t=s}^N \gamma_t (g_j(x_t) - \hat{G}_{j}(x_t))\one_{\{t \in \mathcal{B}\}}}{ \sum_{t = s}^N \gamma_t \one_{\{t \in \mathcal{B}\}}}\right] + {\mathbb E} \left[ \frac{\sum_{t=s}^N \gamma_t \hat{G}_{j}(x_t)\one_{\{t \in \mathcal{B}\}}}{ \sum_{t = s}^N \gamma_t \one_{\{t \in \mathcal{B}\}}}\right] \\
& \leq {\mathbb E} \left[ \sup_{x} (g_j(x) - \hat{G}_{j}(x))\right] +  \frac{\sqrt{2}K_2}{\sqrt{N}} = \mathcal{O}\bigg(\frac{1}{\sqrt{L}} + \frac{1}{\sqrt{N}}\bigg),
\end{align*}
where the we have used Lemma \ref{lemma:donsker} to get the last equality. Similarly, using convexity of $f$
\begin{align}
\label{eq:first_bound}
{\mathbb E} \left[ f(\hat{\mathbf{x}}) - f(x^*)\right] &\leq {\mathbb E} \left[ \frac{\sum_{t=s}^N \gamma_t (f(x_t) - f(x^*))\one_{\{t \in \mathcal{B}\}}}{ \sum_{t = s}^N \gamma_t \one_{\{t \in \mathcal{B}\}}}\right] \nonumber \\
& \leq {\mathbb E} \left[ \sum_{t=s}^N \gamma_t (f(x_t) - f(x^*))\one_{\{t \in \mathcal{B}\}}\right] {\mathbb E}\left[\frac{1}{ \sum_{t = s}^N \gamma_t \one_{\{t \in \mathcal{B}\}}}\right]
\end{align}
where the last inequality follows from Lemma \ref{lemma:fkg application}. Note that applying Jensen's inequality for the concave function $1/x$ we have,
\vspace{-0.2cm}
\begin{align}
\label{eq:jensen}
{\mathbb E}\left[\frac{1}{ \sum_{t = s}^N \gamma_t \one_{\{t \in \mathcal{B}\}}}\right] \leq \frac{1}{{\mathbb E} \left[ \sum_{t = s}^N \gamma_t \one_{\{t \in \mathcal{B}\}} \right] } \leq \frac{\sqrt{N}}{\sqrt{2}K_1 N{\mathbb E}\left[ \frac{| \mathcal{B}|}{N}\right]} \leq \frac{1}{(\sqrt{2}cK_1)\sqrt{N}}.
\end{align}
where we have used the definition of $\gamma_t$ and the fact that ${\mathbb E}(|\mathcal{B}| / N) > c$. Moreover, using the fact that both $x_t$ and $\one_{\{t \in \mathcal{B}\}}$ do not depend on $\xi_{0,t}$ we have,
\vspace{-0.2cm}
\begin{align}
\label{eq:first_term}
{\mathbb E} &\bigg[ \sum_{t=s}^N  \gamma_t (f(x_t) - f(x^*))\one_{\{t \in \mathcal{B}\}}\bigg] =
\sum_{t=s}^N \gamma_t {\mathbb E} \left[  (F(x_t, \xi_{0,t}) - F(x^*, \xi_{0,t}))\one_{\{t \in \mathcal{B}\}}\right]
\nonumber \\
& \leq \sum_{t=s}^N \gamma_t {\mathbb E} \left[\langle F'(x_t, \xi_{0,t}), x_t - x^* \rangle\one_{\{t \in \mathcal{B}\}}\right]
\nonumber \\
& \leq {\mathbb E}(B_{\psi}(x^*, x_{s})) - \sum_{t = s}^N \sum_{j=1}^m  {\mathbb E} \left[\gamma_t \big(G_j(x_t, \xi_{j,t}) -  G_j(x^*, \xi_{j,t})\big) \one_{\{t \in \mathcal{N}_j\}}\right] + \frac{M^2}{2}  \sum_{t = s}^N \gamma_t^2 \nonumber \\
& \leq \tau - \sum_{t = s}^N \sum_{j=1}^m \gamma_t~{\mathbb E}\big[ g_j(x_t) ~\one_{\{t \in \mathcal{N}_j\}} \big]
\end{align}
where the first inequality follows from the convexity of $F(\cdot, \xi_{j,t})$, the second inequality follows from Lemma \ref{lemma:prop9} with $x = x^*$ and the definition of $M$ given in Lemma \ref{lemma:prop9b} and the third inequality follows the definition of $\tau$ from \eqref{eq:tau_def} and the fact that $g_j(x^*) \leq 0$. By plugging in \eqref{eq:jensen} and \eqref{eq:first_term} into \eqref{eq:first_bound} we get,
\vspace{-0.2cm}
\begin{align*}
{\mathbb E} \left[ f(\hat{\mathbf{x}}) - f(x^*)\right] &\leq \frac{1}{(\sqrt{2}cK_1)\sqrt{N}} \bigg(\tau - \sum_{t = s}^N \sum_{j=1}^m \gamma_t~{\mathbb E}\big[ g_j(x_t) ~\one_{\{t \in \mathcal{N}_j\}} \big]  \bigg)
\end{align*}
Finally,
\vspace{-0.2cm}
\begin{align*}
{\mathbb E}\left[ g_j(x_t) ~\one_{\{t \in \mathcal{N}_j\}} \right]  &= {\mathbb E}\left[ (g_j(x_t) - \hat{G}_{j}(x_t))~\one_{\{t \in \mathcal{N}_j\}} \right] + {\mathbb E}\left[ \hat{G}_{j}(x_t)~\one_{\{t \in \mathcal{N}_j\}} \right] \\
& \geq {\mathbb E}\left[ (g_j(x_t) - \hat{G}_{j}(x_t))~\one_{\{t \in \mathcal{N}_j\}} \right] + \eta_{j,t} \\
& \geq -{\mathbb E}\left[ \sup_{x} \left|\hat{G}_{j}(x) - g_j(x)\right| \right] + \frac{\sqrt{2}K_2}{\sqrt{N}} \geq -\frac{C_2}{\sqrt{L}}~~\text{for some $C_2 > 0$,}
\end{align*}
where the last inequality follows from the second part of Lemma \ref{lemma:donsker} and the fact that $K_2 > 0$.
Thus, we have
\vspace{-0.4cm}
\begin{align*}
{\mathbb E} \left[ f(\hat{\mathbf{x}}) - f(x^*)\right] &\leq \frac{1}{(\sqrt{2}cK_1)\sqrt{N}}  \bigg(\tau + \sum_{t = s}^N \sum_{j=1}^m  \frac{C_2\gamma_t}{\sqrt{L}} \bigg) =  \mathcal{O}\bigg( \frac{1}{\sqrt{N}} \bigg),
\end{align*}
where the last equality follows from the fact that $\gamma_t = \mathcal{O}(1/\sqrt{N})$.
\end{proof}

\section{Multiple Cooperative Stochastic Approximation (MCSA)}
\label{sec:algo}
\vspace{-0.3cm}
\label{sec:mcsa}
We begin with the definition of a projection operator. Let $\psi : \mathcal{X} \rightarrow {\mathbb R}$ be a 1-strongly convex proximal function, i.e., we have
$\psi(y) \geq \psi(x) + \langle \nabla \psi(x), y - x \rangle + \frac{1}{2} \norm{x-y}_{}^2.$
The Bregman divergence \cite{duchi2011adaptive} associated with a strongly convex and differentiable function $\psi$ is defined as
$B_{\psi}(x, y) := \psi(x) - \psi(y) - \langle \nabla \psi(y), x - y \rangle.$
Note that due to the strong convexity of $\psi$ we have $B_{\psi}(y, x) \geq \frac{1}{2} \norm{x-y}_{2}^2 \;\; \forall x,y  \in \mathcal{X}. $

\begin{definition}\label{definition: projection}
Based on the function $\psi$ we define the proximal projection function $P_{x, \mathcal{X}}^{\psi}: {\mathbb R}^n \rightarrow {\mathbb R}^n$ as
\vspace{-0.3cm}
\begin{align}
\label{eq:prox_proj}
P_{x, \mathcal{X}}^{\psi}(\cdot) := \mathop{\mathrm{argmin}}_{z \in \mathcal{X}} \left\{ \langle \cdot, z \rangle + B_{\psi}(z,x) \right\}.
\end{align}
We fix $\psi$ and in the rest of the paper, we denote the proximal projection function by $P_x(\cdot)$.
\end{definition}

We assume that the objective function $f(x)$ and each constraint function $g_j(x)$ in \eqref{eq:main_problem} are well-defined, finite valued, continuous and convex functions for $x \in \mathcal{X}$. Similar to the CSA algorithm \cite{lan2016algorithms}, at each iterate $t > 0$, we move along the subgradient direction $f'(x_t)$ if all $g_j(x_t) \leq \eta_{j,t}$ for all $j$ and for the tolerance sequence $\{\eta_j\}_t > 0$. Otherwise we move along a randomly chosen subgradient direction $g'_{j^*}(x_t)$ where $j^*$ is chosen randomly from the set $\left\{j : g_j(x_t) > \eta_{j,t} \right\}$. This modification to the CSA algorithm allows us to work with multiple constraints. At each stage, we move along the chosen direction with a stepsize $\gamma_t$.
We will show in the Sections \ref{sec:main_result} how we can choose the tolerances $\{\eta_{j,t}\}, \{\gamma_t\}$ so that we can achieve the optimal convergence rates.

Since we do not have access to the exact functions $f', g_j'$ and $g_j$ for $j = 1,\ldots, m$, we use an approximation $\hat{G}_j$ for $g_j$ and use the stochastic gradients $G_j'( x_t, \xi_{j,t})$ and $F'(x_t, \xi_{0,t})$ for $g_j'(x_t)$ and $f'(x_t)$ respectively, where $\xi_{j,t}$ are i.i.d. observations that we get to observe from the distribution of $\xi_j$ at iteration $t$. We run $N$ steps of our algorithm and choose our final solution $\hat{\mathbf{x}}$ as the mean over a set of indices that is defined by
\begin{align}
\label{eq:Bdef}
\mathcal{B} = \left\{ s \leq t \leq N  :  \hat{G}_{j}(x_t) \leq \eta_{j,t} \; \forall j \in \{1, \ldots, m\} \right\}
\end{align}
Here $s$ denotes a burn-in period.
Note that we need to get an approximation of $g_j(x_t)$ for every stage $t$ for all $j$. A consistent estimate of $g_j(x_t)$ is given by
\vspace{-0.2cm}
\begin{align}
\label{eq:Gfunc1}
\hat{G}_{j}(x_t) := \frac{1}{L} \sum_{\ell = 1}^L G_j(x_t, \tilde{\xi}_{j,\ell}),
\end{align}
where $\tilde{\xi}_{j,\ell}$ for $\ell = 1, \ldots, L$ are i.i.d.\ observations from the distribution of $\xi_j$. Throughout this paper we assume that $L$ grows in the same rate as $N$. The full algorithm is  written out as Algorithm \ref{algo:mcsa}.

\begin{algorithm}[!h]
\caption{Multiple Cooperative Stochastic Approximation}\label{algo:mcsa}
\begin{algorithmic}[1]
\State Input : Initial $x_1 \in \mathcal{X}$, Tolerances $\{\eta_j\}_t, \{\gamma\}_t$, The number of iterations $N$
\For{ $t = 1, \ldots, N$ }
\State Estimate $\hat{G}_{j}(x_t)$ for all $j \in 1,\ldots, m$ using \eqref{eq:Gfunc1}.
\If{$\hat{G}_{j}(x_t) \leq \eta_{j,t} \text{ for all } j$}
\State Set $h_t = F'(x_t, \xi_{0,t})$
\Else
\State Randomly select $j^*$ from $\{j : \hat{G}_{j}(x_t) > \eta_{j,t}\}$
\State Set $h_t = G_{j^*}'(x_t, \xi_{j^*,t})$
\EndIf
\State Compute $x_{t+1} = P_{x_t}(\gamma_t h_t)$
\EndFor
\State Define $\mathcal{B} = \left\{ s \leq t \leq N  : \hat{G}_{j}(x_t) \leq \eta_{j,t} \; \forall j \in \{1, \ldots, m\} \right\}$
\State \Return $\hat\mathbf{x} := \frac{\sum_{t \in \mathcal{B}} x_t \gamma_t}{\sum_{t \in \mathcal{B}} \gamma_t}$
\end{algorithmic}
\end{algorithm}


IN 2019 the maritime terminal at the port of Naples will resume hosting the same number of cruise passengers it had in 2016, following a downturn suffered in 2017. Looking ahead, the company that manages the terminal plans to build a new berth for latest-generation ships (over 300m), and a hotel within the port, for home port passengers and crews. And, meanwhile, road access improvements are proceeding on target, following a six-year period in which the port was under special administration, effectively freezing every major work project. The analysis comes from Tommaso Cognolato, CEO at Terminal Napoli SpA, the operator that has the concession for the terminal, jointly (and equally) with Royal Caribbean, Costa Crociere, Marinvest and Alilauro, and with MSC Cruises (which has a smaller share).
Are the existing infrastructures sufficient for your needs?
Have you got any investments planned?
The terminal is managed by some of the major players. Does this disadvantage the other companies?
So, you're also generating business for the airport?


We shake up the industry as a leading example of innovative practices within the world of fashion. We deepen your consumer experience and welcome you on our aesthetic journey.
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We design with a conscious mind. Every single wardrobe piece is effortless, timeless, and resilient. Top-notch natural materials are chosen to create the perfect garment. We aim to find a balance between running our company and doing our part for a better environmental and social footprint. To uphold our principles, we only work with carefully selected factories whom we consider as part of the Aesthetic Stories Family.
At Aesthetic Stories we make sure you pay a fair price for each and every product. We want to go beyond traditional retail practices. So that everyone can possess a pretty piece with a unique story.


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Safin Hasan is 22 years old lad who recently qualified UPSC CSE with Rank 570 and GPSC Class 1 with rank 34, both in the first attempt. His success story is full of challenges and inspiration. On the day of the exam when he was going towards exam center, he met with an accident and his knee ligament got completely torn. He had injuries on his left hand and head. The only thing he noticed was his right hand which was safe and hence he decided to go to the exam center. After giving that paper (GS3), he went for the primary checkup and gave next five papers in that situation.
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Legal Bites: Did you integrate your Prelims or Mains preparation or was it separate in the changing scenario?
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Legal Bites: What were the sources of information for general reading? How did you come to know that which sources of reading materials are standard?
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Legal Bites: What is your advice on notes-making for other aspirants?
It depends on the individual. I maintained class notes and referred monthly magazines of Current Affairs for quick revision. One should make notes of what he/she reads from the internet as it is difficult to revise it frequently from internet.
Legal Bites: How helpful is one's academic background in his/her success?
Mr. Safin Hasan: I am an electronics engineer and it helped me to understand the latest development in IT and general science. Same is the case with Commerce graduate or History student. If one is genuinely interested in graduation subject, he/she can choose it as optional.
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Legal Bites: Civil Services Exam process is quite strenuous. It requires long hours of constant study. How did you maintain your tempo and what did you do to break the monotony of preparation?
Mr. Safin Hasan: As it was my first attempt, I knew I had to put extra efforts and hence I was always ready for it. I used to take off from studies every 15 days or 20 days and refreshed my mind so as to start next day with full of energy. My performance in the tests was also an encouragement for me. And yes, it's always your inner motivation which keeps you working for this exam.
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Mr. Safin Hasan: My advice would be, just ask yourself why you want to be a civil servant? If you get a convincing and strong reason for it, nothing would stop you from reaching the destination. This exam process is part of the job training where your dedication, patience, honesty and many other qualities are tested which makes you a successful candidate. Have faith in yourself and work hard… success is yours.


My name is Gregory S. Taylor, Attorney at Law, and I have been practicing law since 1974. In 1991, I founded a private practice law firm in Ada, Oklahoma. My primary areas of practice are estate planning, probate & estate Administration, real estate law, and energy law.
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Vayu, a Ystrategies portfolio company, has launched its wind energy optimization cloud software, and also announced that its renaming from WindWISDEM.
Vayu CEO Jim Kiles said: "We've achieved our goal of creating a cloud computing service that can optimize any wind farm and produce an optimization report for any wind service stakeholder in the world, including wind farm operators, OEMs and investors.
Vayu today also announced the addition of August Kiles as Lead Product Engineer responsible for managing the development of all VayuAI products and services. Kiles' background is in operations research and financial engineering.


In 2011 (HB 2693) and 2012 (HB 4260), Delegate Fleischauer was the lead sponsor of autism legislation that increased funding and insurance coverage for children with autism spectrum disorder. Six million dollars was added to the state budget to cover increased costs in Medicaid, the Children's Health Insurance Program (CHIP) and the Public Employees Insurance (PEIA). These programs and most private insurance companies operating in West Virginia are now required to provide coverage for behavioral therapy - up to $30,000 per year for the first three years, and up to $24,000 thereafter. This coverage is available under these insurance plans to children with autism from the date of diagnosis to age 18, so long as a physician certifies that the treatment is medically necessary. With intensive early treatment, most children with autism can become productive members of society. This was a huge heart-warming win for children and families.


text:
Home / Culture / Film, Theatre & TV
Saswata Chatterjee to act in Indo-Danish film
10 February, 2018 13:29:45
Rakasree Basu has always been a fan of Saswata Chatterjee. Be it as Tutti Futti in Jagga Jasoos, or Bob Biswas in Shobor, Chatterjee has always been Basu's favourite actor. So no wonder when this Mumbai based producer, decided to make a children's movie, instead of taking a leading actor from Bollywood, she decided to return to Kolkata, to her roots. Basu spent her formative years in Kolkata before shifting base to Mumbai. But she believes in a strong Bengali connect and hence her earlier project for a Swedish TV show was with another Tollywood actor Joy Sengupta.
PROFESSOR SHONKU IS HERE!
Basu believes there are several talents in Tollywood's backyard and hence she tries to add a touch of Bengal to all her projects. The Indo-Danish international project is titled Iqbal and the Indian Jewel. The earlier Iqbal films had a huge audience in Denmark and are very popular. Saswata's role has a lot of action. Well-known cinematographer Martin Top Jacobsen has been roped in and the film is based on a novel by Manu Sareen, a former parliamentarian of Denmark.
The man, the actor, and my Soumitra 'kaku'
Covid protocol in place, Kolkata welcomes 160 films from 42 nations
Kidderpore girls punch their way to freedom and power

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Bad Bargain is an original novel based on the U. Having sealed the Hellmouth, the Scooby Gang do not realise that anything is odd when things to be sold at the first annual band fund-raising rummage sale are stored in the school basement, which is directly above the Hellmouth. The rummage sale begins, and the items on sale seem to be having an unexpected effect on those that buy them.
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Teresa Southwick discovered her love for the written word because she was lazy. In a high school history class she was given a list of possible projects and she chose to do an imaginary diary of Marie Antoinette since it seemed to require the least amount of work. But she soon realized that to come up with any plausible personal entries for poor Marie she needed to know a little something about the woman.
Research was required. After all, Teresa sincerely wanted to pass the class. Nowadays, she finds that knowing as much as she can about her characters is more fun than it is work. She is the author of 20 books, four of them historicals for which she had to do research. She s happy to say laziness played no part in the creative process and no brain cells were harmed in the writing of those books.
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Stacy Connelly dreamed of publishing books since she was a kid writing about a girl and her horse.
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Publication date. December 26, Media type, Print (paperback). Pages, ISBN · · OCLC · Bad Bargain is an original novel based on the U.S. television series Buffy the Vampire Slayer.
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In the 1960's, WENGER created a new type of professional knife for butchers, under the trade name SWIBO.
Since then, WENGER has consistently evolved and improved the line, and today it has become a standard of reference in this demanding product field, acknowledged throughout the world.
Ergonomic concept
Maximum safety
Totally hygienic
SWIBO knife handles are made of genuine Grilon (Swiss Nylon Quality). Consequently they are unbreakable, heatproof to 150 degrees Celsius, and do not absorb any grease or dirt. They are ageless and hygienic. They are safe because of their security yellow colour with their hand-adapted shape with safety guards and non-slip surface.
Swibo boning knives made by the makers of the original Swiss army knife.
Swibo bread knives made by the makers of the original swiss army knife.
Butchers Knives
Swibo butchers knives made by the makers of the original swiss army knife.
Swibo chopper made by the makers of the original swiss army knife.
Swibo cooks knives made by the makers of the original swiss army knife.
Swibo fillet knives made by the makers of the original swiss army knife.
Fish Knife
Swibo fish knife made by the makers of the original swiss army knife.
Multi Purpose Knife
Swibo multi purpose knife made by the makers of the original swiss army knife.
Swibo palette knives made by the makers of the original swiss army knife.
Swibo parer knives made by the makers of the original swiss army knife.
Swibo skinning knives made by the makers of the original swiss army knife.
Swibo slicing knives made by the makers of the original swiss army knife.
Swibo vegetable knives made by the makers of the original swiss army knife.
Swibo carving forks made by the makers of the original swiss army knife.
Swibo peelers made by the makers of the original swiss army knife.
Swibo sharpening steels made by the makers of the original swiss army knife.


In the event of a natural disaster there are guidelines on how to handle Mold Remediation electronic equipment that has been exposed to water damage through flooding, firefighting activities, hurricanes and tornadoes. This information is designed for use by electronic suppliers, installers, inspectors, and users of specialized electronic products. Electronic equipment that was exposed to water can be extremely dangerous if allowed to reenergize without proper reconditioning or replacement.
Reductions in integrity of electrical insulation due to excessive moisture or Mold Remediation that has lodged in the equipment components, and other factors, can damage electronic equipment due to affecting the ability of the equipment to perform its intended function. Damage to electronic equipment may also result from flood waters that are contaminated with chemicals, sewage, oil, and other debris that will affect and jeopardize the integrity and performance of the equipment. Ocean water and salt spray could be particularly damaging because of the corrosive and conductive nature of the salt water residue.
Distributors of electronic equipment shall not use any inventory that has been subjected to water damage. Mold Remediation damaged inventory should not be sold to resellers that might put him and place the equipment back into the market. This could lead to water damaged equipment still being operated and creating a hazard to individuals or property. To contact the manufacturer, a working knowledge of electronic systems and of the damaged equipment in question is required to evaluate damage caused by contact with water. The original manufacturer of the electronic equipment should be contacted if any questions should arise or specific recommendations and specifications are needed. In many cases, total replacement will be necessary.
After an initial consultation with the manufacturer, some larger types of electronic equipment may be refurbished by properly trained personnel. The ability to refurbish the electronic equipment could vary with the nature of the electronic equipment function, the degree of Mold Remediation, the age of the equipment, and the length of time the equipment was exposed to the water damage. Attempts to recondition electronic equipment without consulting the manufacturer could result in additional hazards because of the use of improper cleaning agents, which can further Mold Remediation damage the equipment.
Electronic stereo equipment usually involves several switches and low-voltage protective components such as molded case circuit breakers and fuses, within assemblies such as enclosures, panel boards, and complex switchboards. These assemblies could be connected to electronic stereo systems using various other wiring methods. The Mold Remediation protective components that are critical to the safe operation of many distribution circuits. Their ability to protect these circuits will be adversely affected by exposure to mold and water and to the minerals and dust particles which may be present in the water. In molded case stereo circuit breakers and switches, such exposure could affect the overall operation of the mechanism through corrosion, also through the presence of foreign particles, and through the removal of lubricants.


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E-Piim to start building EUR 100 mln dairy processing plant in Paide in summer
TALLINN – Estonian dairy cooperative E-Piim is preparing for the establishment of a 100 million euro dairy processing plant in Paide, for which final coordination with the Agricultural Registers and Information Board (PRIA) and signatures to construction and technology contracts are needed.
E-Piim received 15 million euros' worth of support from PRIA in 2017 and last year, it was reported that the world's biggest wholesaler of dairy products, Dutch company Interfood Holding BV, invested 15 million euros in the dairy processing plant of E-Piim and as a result it acquired a 25 percent holding in E-Piim.
In July, the city government of Paide issued a building permit for a property of 14 hectares for the construction of the dairy processing plant, two thirds of which is to be covered with bank loans.
While construction company Nordecon told the stock exchange last week that the tender of Nordecon AS was declared successful in the procurement for the construction of the new dairy of E-Piim, the cost of which without VAT is 29 million euros, construction cannot start before the technology contracts have been signed and the endeavor has been coordinated with PRIA.
Aare Annus, production project manager at E-Piim, told BNS that as, when it comes to the plant, everything revolves around technology, a box will essentially be built around equipment. According to Annus, the construction will be timed so that inputs will be purchased first and then a construction contract and other smaller procurement contracts will be concluded, for example, to establish communications.
"With every large projects, things do not go quickly. We are waiting for PRIA's final answer, which should arrive around March 20 -- according to the regulation, the measure included two stages, when E-Piim was deemed the winner in July 2017, there was a year and a half to prepare all the documentation for PRIA, including a building permit, detailed plan, procurements carried out and other smaller documents," Annus said.
"Everything was submitted to PRIA before Christmas and, according to the regulation, they now have 60 working days to review the materials, if there are questions or correction, these must be handled and the final answer should come around March 20," Annus said.
The revenue of the dairy cooperative E-Piim increased 24.7 percent to 56.3 million euros in 2017, while profit rose from 58,000 euros in 2016 to 1.3 million euros in 2017. The cooperative sold 14.7 percent of its production to outside the European Union. For example, in cooperation with Denmark's largest organic milk cooperative Thise, high value add demineralized whey powder was sold to the Chinese market, and in cooperation with Interfood, export to Asia should increase.
According to Annus, the contract with Interfood gave banks the certainty to fund the dairy processing plant construction. If necessary, the Dutch company, which is among the two largest producers in Europe, will buy the entire production.
The main production of the new plant will include cheese and powders as E-Piim has longstanding experience in those product groups.
According to Nordecon, construction activities will start in the summer of 2019, and the new dairy will be completed in the fall of 2021. The new dairy will be built in Paide and the complex will consist of a two-storey administrative building, one-storey production space and a seven-storey section for the milk powder production. Total useful space of the complex is 30,589 square meters.


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Virginia Beach will discuss moving its Confederate monument on Thursday
By Alissa Skelton
Virginia Beach's Confederate monument on the site of the former Princess Anne Courthouse at the municipal center. As seen Friday, June 12, 2020. (Stephen M. Katz)
Virginia Beach — The future of the Virginia Beach Confederate monument may be determined Thursday night at the Convention Center.
The Virginia Beach City Council will listen to public comment starting at 7 p.m. Attendees should wear masks and follow physical distancing guidelines to reduce the spread of COVID-19.
The 27-foot monument has been located near the intersection of Princess Anne and North Landing roads at the city's Municipal Center since 1905. Like the majority of Confederate statues in South Hampton Roads, Virginia Beach's is placed outside a former site where slaves were once sold. Activists in Virginia and across the nation have said the statues celebrate racism and white supremacy and have demanded the structures be removed.
The city has been exploring its options for the statue for months. To date, however, no action has been taken.
If the council members can reach an agreement, they could vote tonight on whether to relocate the statue to an alternative location.
After a vote to relocate takes place, state law requires the city to invite other interested parties to submit relocation proposals. It must be offered to a museum, historical society, battlefield or government under the new state legislation, the city has said.
Council members will make the final relocation decision.
"Obviously we want to get this resolved for a lot of reasons as soon as possible," Mayor Bobby Dyer said.
The city covered the statue on June 12 and placed barriers around it. Tom Leahy, the former city manager, said this was done to protect people and the structure after a man was injured when a Confederate statue was torn down during a June 10 protest in Portsmouth. Norfolk removed its downtown monument in June.
All who are interested in speaking during the meeting are encouraged to sign up before 5 p.m. by calling 757-385-4303 or emailing ABarnes@vbgov.com. Commenters may speak during the meeting from home via video conferencing.
Alissa Skelton, 757-222-5155, alissa.skelton@pilotonline.com.


Who is Micah Robinson? in today's world of formulas and followers he is genuinely something new. As a singer, He is a rare combination of polish and passion. As a writer, he maintains a pleasant simplicity that somehow still reconnects you with your deepest thoughts. As an artist, he presents a style that is fresh and forward, while still carrying a vintage flare and familiarity that makes you feel like you've been listening to him for years. He is more than any single a statement can say. Micah Robinson is an experience, and a name you want to know.


KARACHI: Uber and Careem have reached an agreement for Uber to acquire Careem for $3.1 billion, consisting of $1.7 billion in convertible notes and $1.4 billion in cash.
The acquisition of Careem is subject to applicable regulatory approvals. The transaction is expected to close in Q1 2020. Uber will acquire all of Careem's mobility, delivery, and payments businesses across the greater Middle East region, ranging from Morocco to Pakistan, with major markets including Egypt, Jordan, Pakistan, Saudi Arabia, and the United Arab Emirates, according to a news release issued to PPI Karachi on Tuesday.
said Uber CEO Dara Khosrowshahi.
said Careem CEO and co-founder, Mudassir Sheikha.


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Esther, Obert & Grace's stories
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School fees in the rural areas cost £30 per year – in the 9 primary schools and 7 ECD centres we work in that includes a daily meal and highly trained teachers. If you would like to donate towards keeping rural area children in school then please click here.
Township Kids Sponsorship
In one of Zimbabwe's largest townships, just outside Harare we offer sponsorship opportunities to a few children who live in extreme poverty. Township life can be brutal for these children – living squashed up in tin shacks with up to 11 others sharing scraps of food, whilst open sewers and rubbish bring severe health conditions and school is but a distant dream. In many cases young girls end up pregnant and trapped in the cycle of poverty.
Sponsorship includes school fees, uniforms, books, mahewu porridge, food hampers and access to basic medical treatment. It costs £25/month for individual sponsorship or £12.50/month for shared sponsorship. We ask the children to communicate their appreciation and encourage them to write as often as they feel comfortable doing so. It is vital to note that the protection and welfare of the children is our priority.
Please read Bridget's story here if you are interested in sponsoring a township child, according to the costs and specifics mentioned above. Please kindly contact sharon@zimbabweschildren.org or continue to our donate page here. Thank you!
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We have one main goal – giving education and hope back to the children of Zimbabwe.
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