Todos for appendix and S4

approx_alg.tex

@@ -192,8 +192,12 @@ The algorithm to prove~\Cref{lem:approx-alg} follows from the following observat
 Given the above, the algorithm is a sampling based algorithm for the above sum: we sample $(v,c)\in \expandtree{\etree}$ with probability proportional\footnote{We could have also uniformly sampled from $\expandtree{\etree}$ but this gives better parameters.}
 %\AH{Regarding the footnote, is there really a difference?  I \emph{suppose} technically, but in this case they are \emph{effectively} the same.  Just wondering.} 
 %\AR{Yes, there is! If we used uniform distribution then in our bounds we will have a parameter that depends on the largest $\abs{coef}$, which e.g. could be dependent on $n$. But with the weighted probability distribution, we avoid paying this price. Though I guess perhaps we can say for the kinds of queries we consider thhese coefficients are all constants?}
-to $\abs{c}$ and compute $Y=\indicator{\monom\mod{\mathcal{B}}\not\equiv 0}\cdot \prod_{X_i\in \var\inparen{v}} p_i$. Taking enough samples and computing the average of $Y$ gives us our final estimate. Algorithm~\ref{alg:mon-sam} has the details.
-\OK{Even if the proof is offloaded to the appendix, it would be useful to state the formula for $N$ (line 4 of \Cref{alg:mon-sam}), along with a pointer to the appendix.}
+to $\abs{c}$ and compute $Y=\indicator{\monom\mod{\mathcal{B}}\not\equiv 0}\cdot \prod_{X_i\in \var\inparen{v}} p_i$. Taking $\numsamp$ samples and computing the average of $Y$ gives us our final estimate.  Algorithm~\ref{alg:mon-sam} has the details for $\approxq$.  The derivation of $\numsamp$ (\Cref{alg:mon-sam-global2}) can be found in~\Cref{app:subsec-th-mon-samp}, and from those results, one can further see that
+\begin{equation*}
+2\exp{\left(-\frac{\samplesize\error^2}{2}\right)}\leq \conf \implies\samplesize \geq \frac{2\log{\frac{2}{\conf}}}{\error^2}.
+%\exp{\left(-\frac{\samplesize\error^2}{2}\right)}\leq \frac{\conf}{2}\\
+%\frac{\samplesize\error^2}{2}\geq \log{\frac{2}{\conf}}\\
+\end{equation*}
 
 %We state the approximation algorithm in terms of a $\bi$.
 %\subsubsection{Description}

hardness-app.tex

@@ -379,7 +379,7 @@ The claim on the runtime follows since
 which completes the proof.
 
 We now return to the proof of~\Cref{lem:mon-samp}:
-\subsection{Proof of Theorem \ref{lem:mon-samp}}
+\subsection{Proof of Theorem \ref{lem:mon-samp}}\label{app:subsec-th-mon-samp}
 Consider now the random variables $\randvar_1,\dots,\randvar_\numvar$, where each $\randvar_i$ is the value of $\vari{Y}_{\vari{i}}$ after~\Cref{alg:mon-sam-product} is executed. In particular, note that we have
 \[Y_i= \onesymbol\inparen{\monom\mod{\mathcal{B}}\not\equiv 0}\cdot \prod_{X_i\in \var\inparen{v}} p_i,\]
 where the indicator variable handles the check in~\Cref{alg:check-duplicate-block}