Update on Overleaf.

app_approx-alg-analysis.tex

@@ -115,7 +115,9 @@ Applying this bound in the runtime bound in \Cref{lem:approx-alg} gives the firs
 \begin{proof}
 The circuit \circuit' is built from \circuit in the following manner.  For each input gate $\gate_i$ with $\gate_i.\val = X_\tup$, replace $\gate_i$ with the circuit \subcircuit encoding the sum $\sum_{j = 1}^\bound j\cdot X_{\tup, j}$.  We argue that \circuit' is a valid circuit by the following facts.  Let $\pdb = \inparen{\worlds, \bpd}$ be the original \abbrCTIDB \circuit was generated from.  Then, by~\Cref{prop:ctidb-reduct} there exists a \abbrOneBIDB $\pdb' = \inparen{\onebidbworlds{\tupset'}, \bpd'}$, with $\tupset' = \inset{\intup{\tup, j}~|~\tup\in\tupset, j\in\pbox{\bound}}$, from which the conversion from \circuit to \circuit' follows.  Both $\polyf\inparen{\circuit}$ and $\polyf\inparen{\circuit'}$ have the same expected multiplicity since (by~\Cref{prop:ctidb-reduct}) the distributions $\bpd$ and $\bpd'$ are equivalent and each $j\cdot\worldvec'_{\tup, j} = \worldvec_\tup$ for $\worldvec'\in\inset{0, 1}^{\bound\numvar}$ and $\worldvec\in\worlds$.  Finally, note that because there exists a (sub) circuit encoding $\sum_{j = 1}^\bound j\cdot X_{\tup, j}$ that is a \emph{balanced} binary tree, the above conversion implies the claimed size and depth bounds of the lemma.
 
-Next we argue the claim on $\gamma\inparen{\circuit'}$.  Consider the list of expanded monomials $\expansion{\circuit}$ for \abbrCTIDB circuit \circuit.  Let \monom be an arbitrary monomial such that the set of variables in \monom is $\encMon = X_{\tup_1}^{d_1},\ldots,X_{\tup_\ell}^{d_\ell}$ with $\ell$ variables.  Then \monom yields the set of monomials $\vari{E}_\monom\inparen{\circuit'}=\inset{j_1^{d_1}\cdot X_{\tup, j_1}^{d_1}\times\cdots\times j_\ell^{d_\ell}\cdot X_{\tup, j_\ell}^{d_\ell}}_{j_1,\ldots, j_\ell \in \pbox{0, \bound}}$ in $\expansion{\circuit'}$.  Recall that a cancellation occurs when we have a monomial \monom' such that there exists $\tup\neq\tup'$ in the same block $\block$ where variables $X_\tup, X_{\tup'}$ are in the set of variables $\encMon'$ of \monom'.  Observe that cancellations can only occur for each $X_{\tup}^{d_\tup}\in \encMon$, where the expansion $\inparen{\sum_{j = 1}^\bound j\cdot X_{\tup, j}}^{d_\tup}$ represents the monomial $X_\tup^{d_\tup}$ in $\tupset'$.  Consider the number of cancellations for $\inparen{\sum_{j = 1}^\bound j\cdot X_{\tup, j}^{d_\tup}}^{d_\ell}$.  Then $\gamma \leq 1 - \bound^{d_\tup - 1}$, since for each element in the set of cross products $\inset{\bigtimes_{i\in\pbox{d_\tup}, j_i\in\pbox{\bound}}X_{\tup, j_i}}$ there are \emph{exactly} $\bound$ surviving elements with $j_1=\cdots=j_{d_\tup}$, i.e. $X_j^{d_\tup}$ for each $j\in\pbox{\bound}$.  The rest of the $\inparen{\bound}^{d_\tup-1}$ cross terms cancel.  Regarding the whole monomial \monom', it is the case that the proportion of non-cancellations across each $X_\tup^{d_\tup}\in\encMon'$ multiply because non-cancelling terms for $X_\tup$ can only be joined with non-cancelling terms of $X_{\tup'}^{d_{\tup'}}\in\encMon'$ for $\tup\neq\tup'$.  This then yields the fraction of cancelled monomials $1 - \prod_{i = 1}^{\ell}\bound^{d_i - 1}\leq \gamma \leq 1 - \bound^{-\inparen{k - 1}}$ where the inequalities take into account the fact that $\sum_{i = 1}^\ell d_i \leq k$.
+Next we argue the claim on $\gamma\inparen{\circuit'}$.  Consider the list of expanded monomials $\expansion{\circuit}$ for \abbrCTIDB circuit \circuit.  Let 
+ $\encMon = X_{\tup_1}^{d_1},\ldots,X_{\tup_\ell}^{d_\ell}$ be an arbitrary monomial with $\ell$ variables.  Then \monom yields the set of monomials $\vari{E}_\monom\inparen{\circuit'}=\inset{j_1^{d_1}\cdot X_{\tup, j_1}^{d_1}\times\cdots\times j_\ell^{d_\ell}\cdot X_{\tup, j_\ell}^{d_\ell}}_{j_1,\ldots, j_\ell \in \pbox{\bound}}$ in $\expansion{\circuit'}$.  Recall that a cancellation occurs when we have a monomial \monom' such that there exists $\tup\neq\tup'$ in the same block $\block$ where variables $X_\tup, X_{\tup'}$ are in the set of variables $\encMon'$ of \monom'.  Observe that cancellations can only occur for each $X_{\tup}^{d_\tup}\in \encMon$, where the expansion $\inparen{\sum_{j = 1}^\bound j\cdot X_{\tup, j}}^{d_\tup}$ represents the monomial $X_\tup^{d_\tup}$ in $\tupset'$.  Consider the number of cancellations for $\inparen{\sum_{j = 1}^\bound j\cdot X_{\tup, j}}^{d_t}$.  Then $\gamma \leq 1 - \bound^{d_\tup - 1}$, since 
+ for each element in the set of cross products $\inset{\bigtimes_{i\in\pbox{d_\tup}, j_i\in\pbox{\bound}}X_{\tup, j_i}}$ there are \emph{exactly} $\bound$ surviving elements with $j_1=\cdots=j_{d_\tup}=j$, i.e. $X_{t,j}^{d_\tup}$ for each $j\in\pbox{\bound}$.  The rest of the $\bound^{d_\tup}-c$ cross terms cancel.  Regarding the whole monomial \monom', it is the case that the proportion of non-cancellations across each $X_\tup^{d_\tup}\in\encMon'$ multiply because non-cancelling terms for $X_\tup$ can only be joined with non-cancelling terms of $X_{\tup'}^{d_{\tup'}}\in\encMon'$ for $\tup\neq\tup'$.  This then yields the fraction of cancelled monomials $\gamma\le 1 - \prod_{i = 1}^{\ell}\bound^{d_i - 1} \leq 1 - \bound^{-\inparen{k - 1}}$ where the inequalities take into account the fact that $\sum_{i = 1}^\ell d_i \leq k$.
 
 Since this is true for arbitrary \monom, the bound follows for $\polyf\inparen{\circuit'}$.
 \end{proof}

approx_alg.tex

@@ -115,7 +115,7 @@ as well as for all three queries of the PDBench \abbrBIDB benchmark (see \Cref{a
 
 \begin{Lemma}
 \label{lem:ctidb-gamma}
-Given $\raPlus$ query $\query$ and \abbrCTIDB $\pdb$, let \circuit be the circuit computed by $\query\inparen{\tupset}$.  Then, for the reduced \abbrOneBIDB $\pdb'$ there exists an equivalent circuit \circuit' obtained from $\query\inparen{\tupset'}$, such that $\gamma\inparen{\circuit'}\leq 1 - \inparen{\bound}^{-\inparen{k-1}}$ with $\size\inparen{\circuit'} \leq \size\inparen{\circuit} + \bigO{\numvar\bound}$ 
+Given $\raPlus$ query $\query$ and \abbrCTIDB $\pdb$, let \circuit be the circuit computed by $\query\inparen{\tupset}$.  Then, for the reduced \abbrOneBIDB $\pdb'$ there exists an equivalent circuit \circuit' obtained from $\query\inparen{\tupset'}$, such that $\gamma\inparen{\circuit'}\leq 1 - \bound^{-\inparen{k-1}}$ with $\size\inparen{\circuit'} \leq \size\inparen{\circuit} + \bigO{\numvar\bound}$ 
  and $\depth\inparen{\circuit'} = \depth\inparen{\circuit} + \bigO{\log{\bound}}$.
 \end{Lemma}
 

introduction.tex

@@ -226,7 +226,9 @@ Substituting $\vct{\prob}$ for $\vct{X}$,
 \end{align*}
 \end{footnotesize}
 If we assume that all probability values are at least $p_0>0$, then given access to $\refpoly{1, }^2\inparen{\vct{\prob}} - 4\prob_A^2\prob_{X_1}\prob_{X_2}\prob_B^2$
-we get that $\refpoly{1, }^2\inparen{\vct{\prob}} - 4\prob_A^2\prob_{X_1}\prob_{X_2}\prob_B^2$ is in the range $\left(\inparen{p_0}^3\cdot\inparen{\rpoly^2_1\vct{\prob}}, \rpoly_1^2\inparen{\vct{\prob}}\right]$.  We can simulate sampling from $\refpoly{1, }^2\inparen{\vct{X}}$ by sampling monomials from $\refpoly{1, }^2$ while ignoring any samples $A^2X_1X_2B^2$.  Note however, that this is \emph{not a tight approximation}.  
+we get that $\refpoly{1, }^2\inparen{\vct{\prob}} - 4\prob_A^2\prob_{X_1}\prob_{X_2}\prob_B^2$ is in the range $\left(\inparen{p_0}^3\cdot\inparen{\rpoly^2_1\vct{\prob}}, \rpoly_1^2\inparen{\vct{\prob}}\right]$.  
+%We can simulate sampling from $\refpoly{1, }^2\inparen{\vct{X}}$ by sampling monomials from $\refpoly{1, }^2$ while ignoring any samples $A^2X_1X_2B^2$.  
+Note however, that this is \emph{not a tight approximation}.  
 In~\cref{sec:algo} we demonstrate that a $(1\pm\epsilon)$ (multiplicative) approximation with competitive performance is achievable.
 To get an $(1\pm \epsilon)$-multiplicative approximation and solve~\Cref{prob:intro-stmt}, using \circuit we uniformly sample monomials from the equivalent \abbrSMB representation of $\poly$ (without materializing the \abbrSMB representation) and `adjust' their contribution to $\widetilde{\poly}\left(\cdot\right)$.