Re-org of S4
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%root: main.tex
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\input{app_approx-alg-pseudo-code}
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%\input{app_approx-alg-pseudo-code}
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\subsection{Proof of Theorem \ref{lem:approx-alg}}\label{sec:proof-lem-approx-alg}
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Before proving~\Cref{lem:mon-samp}, we use it to argue our main result,~\Cref{lem:approx-alg}. The algorithm to prove~\Cref{lem:approx-alg} follows from the following observation. Given a query polynomial $\poly(\vct{X})=\polyf(\circuit)$ for circuit \circuit over $\bi$, we can exactly represent $\rpoly(\vct{X})$ as follows:
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\begin{equation}
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\label{eq:tilde-Q-bi}
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\rpoly\inparen{X_1,\dots,X_\numvar}=\hspace*{-1mm}\sum_{(\monom,\coef)\in \expansion{\circuit}} \hspace*{-2mm} \indicator{\monom\mod{\mathcal{B}}\not\equiv 0}\cdot \coef\cdot\hspace*{-2mm}\prod_{X_i\in \var\inparen{\monom}}\hspace*{-2mm} X_i
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\end{equation}
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Given the above, the algorithm is a sampling based algorithm for the above sum: we sample $(\monom,\coef)\in \expansion{\circuit}$ with probability proportional\footnote{We could have also uniformly sampled from $\expansion{\circuit}$ but this gives better parameters.} to $\abs{\coef}$ and compute $Y=\indicator{\monom\mod{\mathcal{B}}\not\equiv 0}\cdot \prod_{X_i\in \var\inparen{\monom}} p_i$. Taking $\numsamp$ samples and computing the average of $Y$ gives us our final estimate.
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The number of samples is computed by (see \Cref{app:subsec-th-mon-samp}):
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\begin{equation*}
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2\exp{\left(-\frac{\samplesize\error^2}{2}\right)}\leq \conf \implies\samplesize \geq \frac{2\log{\frac{2}{\conf}}}{\error^2}.
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\end{equation*}
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In order to prove~\Cref{lem:approx-alg}, we will need to argue the correctness of \approxq, which relies on the correctness of auxiliary algorithms \onepass and \sampmon.
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\begin{Lemma}\label{lem:one-pass}
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The $\onepass$ function completes in time:
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$$O\left(\size(\circuit) \cdot \multc{\log\left(\abs{\circuit(1\ldots, 1)}\right)}{\log{\size(\circuit}}\right)$$
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$\onepass$ guarantees two post-conditions: First, for each subcircuit $\vari{S}$ of $\circuit$, we have that $\vari{S}.\vari{partial}$ is set to $\abs{\vari{S}}(1,\ldots, 1)$. Second, when $\vari{S}.\type = \circplus$, \subcircuit.\lwght $= \frac{\abs{\subcircuit_\linput}(1,\ldots, 1)}{\abs{\subcircuit}(1,\ldots, 1)}$ and likewise for \subcircuit.\rwght.
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\end{Lemma}
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To prove correctness of~\Cref{alg:mon-sam}, we only use the following fact that follows from the above lemma: for the modified circuit ($\circuit_{\vari{mod}}$), $\circuit_{\vari{mod}}.\vari{partial}=\abs{\circuit}(1,\dots,1)$.
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\begin{Lemma}\label{lem:sample}
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The function $\sampmon$ completes in time
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$$O(\log{k} \cdot k \cdot \depth(\circuit)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log{\size(\circuit)}})$$
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where $k = \degree(\circuit)$. The function returns every $\left(\monom, sign(\coef)\right)$ for $(\monom, \coef)\in \expansion{\circuit}$ with probability $\frac{|\coef|}{\abs{\circuit}(1,\ldots, 1)}$.
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\end{Lemma}
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With the above two lemmas, we are ready to argue the following result (proof in~\Cref{sec:proofs-approx-alg}):
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\begin{Theorem}\label{lem:mon-samp}
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For any $\circuit$ with $\degree(poly(|\circuit|)) = k$, algorithm \ref{alg:mon-sam} outputs an estimate $\vari{acc}$ of $\rpoly(\prob_1,\ldots, \prob_\numvar)$ such that
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\[\probOf\left(\left|\vari{acc} - \rpoly(\prob_1,\ldots, \prob_\numvar)\right|> \error \cdot \abs{\circuit}(1,\ldots, 1)\right) \leq \conf,\]
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in $O\left(\left(\size(\circuit)+\frac{\log{\frac{1}{\conf}}}{\error^2} \cdot k \cdot\log{k} \cdot \depth(\circuit)\right)\cdot \multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log{\size(\circuit)}}\right)$ time.
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\end{Theorem}
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Before proving~\Cref{lem:mon-samp}, we use it to argue our main result,~\Cref{lem:approx-alg}.
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%The algorithm to prove~\Cref{lem:approx-alg} follows from the following observation. Given a query polynomial $\poly(\vct{X})=\polyf(\circuit)$ for circuit \circuit over $\bi$, we can exactly represent $\rpoly(\vct{X})$ as follows:
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%\begin{equation}
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%\label{eq:tilde-Q-bi}
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%\rpoly\inparen{X_1,\dots,X_\numvar}=\hspace*{-1mm}\sum_{(\monom,\coef)\in \expansion{\circuit}} \hspace*{-2mm} \indicator{\monom\mod{\mathcal{B}}\not\equiv 0}\cdot \coef\cdot\hspace*{-2mm}\prod_{X_i\in \var\inparen{\monom}}\hspace*{-2mm} X_i
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%\end{equation}
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%Given the above, the algorithm is a sampling based algorithm for the above sum: we sample $(\monom,\coef)\in \expansion{\circuit}$ with probability proportional\footnote{We could have also uniformly sampled from $\expansion{\circuit}$ but this gives better parameters.} to $\abs{\coef}$ and compute $Y=\indicator{\monom\mod{\mathcal{B}}\not\equiv 0}\cdot \prod_{X_i\in \var\inparen{\monom}} p_i$. Taking $\numsamp$ samples and computing the average of $Y$ gives us our final estimate.
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%The number of samples is computed by (see \Cref{app:subsec-th-mon-samp}):
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%\begin{equation*}
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%2\exp{\left(-\frac{\samplesize\error^2}{2}\right)}\leq \conf \implies\samplesize \geq \frac{2\log{\frac{2}{\conf}}}{\error^2}.
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%\end{equation*}
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\begin{proof}
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Set $\mathcal{E}=\approxq({\circuit}, (\prob_1,\dots,\prob_\numvar),$ $\conf, \error')$, where
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\[\error' = \error \cdot \frac{\rpoly(\prob_1,\ldots, \prob_\numvar)\cdot (1 - \gamma)}{\abs{{\circuit}}(1,\ldots, 1)},\]
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@ -109,45 +109,52 @@ we have $\abs{\circuit}(1,\ldots, 1)\le \size(\circuit)^{O(k)}.$
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Note that the above implies that with the assumption $\prob_0>0$ and $\gamma<1$ are absolute constants from \Cref{cor:approx-algo-const-p}, then the runtime there simplies to $O_k\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)^2\cdot \log{\frac{1}{\conf}}\right)$ for general circuits $\circuit$ and to $O_k\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)$ for the case when $\circuit$ satisfies the special conditions in~\Cref{lem:val-ub}. In~\Cref{app:proof-lem-val-ub} we argue that these conditions are very general and encompass many interesting scenarios.
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\subsection{Approximating $\rpoly$}
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\approxq (\cref{alg:mon-sam}) modifies \circuit with a call to \onepass. It then samples from $\circuit_{\vari{mod}}\numsamp$ times and uses that information to approximate $\rpoly$.
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The algorithm (\approxq) to prove~\Cref{lem:approx-alg} follows from the following observation. Given a query polynomial $\poly(\vct{X})=\polyf(\circuit)$ for circuit \circuit over $\bi$, we can exactly represent $\rpoly(\vct{X})$ as follows:
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\begin{equation}
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\label{eq:tilde-Q-bi}
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\rpoly\inparen{X_1,\dots,X_\numvar}=\hspace*{-1mm}\sum_{(\monom,\coef)\in \expansion{\circuit}} \hspace*{-2mm} \indicator{\monom\mod{\mathcal{B}}\not\equiv 0}\cdot \coef\cdot\hspace*{-2mm}\prod_{X_i\in \var\inparen{\monom}}\hspace*{-2mm} X_i
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\end{equation}
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Given the above, the algorithm is a sampling based algorithm for the above sum: we sample (via \sampmon) $(\monom,\coef)\in \expansion{\circuit}$ with probability proportional\footnote{We could have also uniformly sampled from $\expansion{\circuit}$ but this gives better parameters.} to $\abs{\coef}$ and compute $Y=\indicator{\monom\mod{\mathcal{B}}\not\equiv 0}\cdot \prod_{X_i\in \var\inparen{\monom}} p_i$. Taking $\numsamp$ samples and computing the average of $Y$ gives us our final estimate. \onepass is used to compute the sampling probabilities needed in \sampmon (details are in~\Cref{sec:proofs-approx-alg}).
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%\approxq (\cref{alg:mon-sam}) modifies \circuit with a call to \onepass. It then samples from $\circuit_{\vari{mod}}\numsamp$ times and uses that information to approximate $\rpoly$.
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\input{app_approx-alg-pseudo-code}
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%\subsubsection{Correctness}
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\subsubsection{Correctness}
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%In order to prove~\Cref{lem:approx-alg}, we will need to argue the correctness of \approxq, which relies on the correctness of auxiliary algorithms \onepass and \sampmon.
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In order to prove~\Cref{lem:approx-alg}, we will need to argue the correctness of \approxq, which relies on the correctness of auxiliary algorithms \onepass and \sampmon.
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%\begin{Lemma}\label{lem:one-pass}
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%The $\onepass$ function completes in time:
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%$$O\left(\size(\circuit) \cdot \multc{\log\left(\abs{\circuit(1\ldots, 1)}\right)}{\log{\size(\circuit}}\right)$$
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% $\onepass$ guarantees two post-conditions: First, for each subcircuit $\vari{S}$ of $\circuit$, we have that $\vari{S}.\vari{partial}$ is set to $\abs{\vari{S}}(1,\ldots, 1)$. Second, when $\vari{S}.\type = \circplus$, \subcircuit.\lwght $= \frac{\abs{\subcircuit_\linput}(1,\ldots, 1)}{\abs{\subcircuit}(1,\ldots, 1)}$ and likewise for \subcircuit.\rwght.
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%\end{Lemma}
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%To prove correctness of~\Cref{alg:mon-sam}, we only use the following fact that follows from the above lemma: for the modified circuit ($\circuit_{\vari{mod}}$), $\circuit_{\vari{mod}}.\vari{partial}=\abs{\circuit}(1,\dots,1)$.
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\begin{Lemma}\label{lem:one-pass}
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The $\onepass$ function completes in time:
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$$O\left(\size(\circuit) \cdot \multc{\log\left(\abs{\circuit(1\ldots, 1)}\right)}{\log{\size(\circuit}}\right)$$
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$\onepass$ guarantees two post-conditions: First, for each subcircuit $\vari{S}$ of $\circuit$, we have that $\vari{S}.\vari{partial}$ is set to $\abs{\vari{S}}(1,\ldots, 1)$. Second, when $\vari{S}.\type = \circplus$, \subcircuit.\lwght $= \frac{\abs{\subcircuit_\linput}(1,\ldots, 1)}{\abs{\subcircuit}(1,\ldots, 1)}$ and likewise for \subcircuit.\rwght.
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\end{Lemma}
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To prove correctness of~\Cref{alg:mon-sam}, we only use the following fact that follows from the above lemma: for the modified circuit ($\circuit_{\vari{mod}}$), $\circuit_{\vari{mod}}.\vari{partial}=\abs{\circuit}(1,\dots,1)$.
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%\begin{Lemma}\label{lem:sample}
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%The function $\sampmon$ completes in time
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%$$O(\log{k} \cdot k \cdot \depth(\circuit)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log{\size(\circuit)}})$$
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% where $k = \degree(\circuit)$. The function returns every $\left(\monom, sign(\coef)\right)$ for $(\monom, \coef)\in \expansion{\circuit}$ with probability $\frac{|\coef|}{\abs{\circuit}(1,\ldots, 1)}$.
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%\end{Lemma}
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\begin{Lemma}\label{lem:sample}
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The function $\sampmon$ completes in time
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$$O(\log{k} \cdot k \cdot \depth(\circuit)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log{\size(\circuit)}})$$
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where $k = \degree(\circuit)$. The function returns every $\left(\monom, sign(\coef)\right)$ for $(\monom, \coef)\in \expansion{\circuit}$ with probability $\frac{|\coef|}{\abs{\circuit}(1,\ldots, 1)}$.
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\end{Lemma}
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With the above two lemmas, we are ready to argue the following result (proof in~\Cref{sec:proofs-approx-alg}):
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\begin{Theorem}\label{lem:mon-samp}
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For any $\circuit$ with $\degree(poly(|\circuit|)) = k$, algorithm \ref{alg:mon-sam} outputs an estimate $\vari{acc}$ of $\rpoly(\prob_1,\ldots, \prob_\numvar)$ such that
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\[\probOf\left(\left|\vari{acc} - \rpoly(\prob_1,\ldots, \prob_\numvar)\right|> \error \cdot \abs{\circuit}(1,\ldots, 1)\right) \leq \conf,\]
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in $O\left(\left(\size(\circuit)+\frac{\log{\frac{1}{\conf}}}{\error^2} \cdot k \cdot\log{k} \cdot \depth(\circuit)\right)\cdot \multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log{\size(\circuit)}}\right)$ time.
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\end{Theorem}
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%With the above two lemmas, we are ready to argue the following result (proof in~\Cref{sec:proofs-approx-alg}):
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%\begin{Theorem}\label{lem:mon-samp}
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%For any $\circuit$ with $\degree(poly(|\circuit|)) = k$, algorithm \ref{alg:mon-sam} outputs an estimate $\vari{acc}$ of $\rpoly(\prob_1,\ldots, \prob_\numvar)$ such that
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%\[\probOf\left(\left|\vari{acc} - \rpoly(\prob_1,\ldots, \prob_\numvar)\right|> \error \cdot \abs{\circuit}(1,\ldots, 1)\right) \leq \conf,\]
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% in $O\left(\left(\size(\circuit)+\frac{\log{\frac{1}{\conf}}}{\error^2} \cdot k \cdot\log{k} \cdot \depth(\circuit)\right)\cdot \multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log{\size(\circuit)}}\right)$ time.
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%\end{Theorem}
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\subsection{\onepass\ Algorithm}
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\label{sec:onepass}
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%\subsection{\onepass\ Algorithm}
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%\label{sec:onepass}
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\noindent \onepass\ (Algorithm ~\ref{alg:one-pass-iter} in \Cref{sec:proofs-approx-alg}) iteratively visits each gate one time according to the topological ordering of \circuit annotating the \lwght, \rwght, and \prt variables of each node according to the definitions above. Lemma~\ref{lem:one-pass} is proved in~\Cref{sec:proofs-approx-alg}.
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%\noindent \onepass\ (Algorithm ~\ref{alg:one-pass-iter} in \Cref{sec:proofs-approx-alg}) iteratively visits each gate one time according to the topological ordering of \circuit annotating the \lwght, \rwght, and \prt variables of each node according to the definitions above. Lemma~\ref{lem:one-pass} is proved in~\Cref{sec:proofs-approx-alg}.
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\subsection{\sampmon\ Algorithm}
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\label{sec:samplemonomial}
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%\subsection{\sampmon\ Algorithm}
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%\label{sec:samplemonomial}
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A naive (slow) implementation of \sampmon\ would first compute $\expansion{\circuit}$ and then sample from it.
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Instead, \Cref{alg:sample} selects a monomial from $\expansion{\circuit}$ by top-down traversal of the input \circuit. More details on the traversal can be found in \cref{subsec:sampmon-remarks}.
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%A naive (slow) implementation of \sampmon\ would first compute $\expansion{\circuit}$ and then sample from it.
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%Instead, \Cref{alg:sample} selects a monomial from $\expansion{\circuit}$ by top-down traversal of the input \circuit. More details on the traversal can be found in \cref{subsec:sampmon-remarks}.
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%
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%$\sampmon$ is given in \Cref{alg:sample}, and a proof of its correctness (via \Cref{lem:sample}) is provided in \Cref{sec:proofs-approx-alg}.
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