$O(n^{2})$
+$f$
+$n$
+$G(v)$
+$s_{o}\oplus s_{a}\in\mathbb{V}^{n+m}$
+$Z\in\mathbb{R}^{m\times d_{\text{token}}}$
+$E_{\psi}(s)$
+$\displaystyle=F^{i}(E_{\psi}(s_{o})\oplus\text{Proj}_{\psi}(Z)).$
+$\displaystyle\cos(v,v_{t}^{image})+\lambda\cos(v,v_{t}^{text})$
+$\cos(\psi_{i},\psi_{j})$
+${}^{4}$
+$v_{t}^{text}=F^{t}(E_{\psi}(s^{\prime}))$
+${}^{*}$
+$\displaystyle\text{argmax}_{Z}$
+$\rightarrow$
+$\mathcal{A}(x,t,s_{o})$
+$\displaystyle=F^{i}(E_{\psi}(s_{o}\oplus s_{a}))$
+${}^{1}$
+$\text{Proj}_{\psi}(Z)_{i}=Z_{i}+\text{sg}(\psi_{j}-Z_{i})$
+$x_{t}$
+$500\times 20=10000$
+$w_{i},w_{j}$
+$v_{t}^{image}\leftarrow F^{i}(x_{t})$
+$m=4$
+$s_{a}=E_{\psi}^{-1}(\text{Proj}_{\psi}(Z))$
+${}^{5}$
+$Z_{i}$
+${}^{1,*}$
+$\text{Proj}_{\psi}(Z)$
+$s$
+$\displaystyle\text{argmax}_{s_{a}}$
+$t$
+$s^{\prime}\leftarrow$
+$v_{t}^{image}$
+$5\times 4\times 100=2000$
+${}^{1,2}$
+$\psi\in\mathbb{R}^{|\mathbb{V}|\times d_{\text{token}}}$
+$bestloss\leftarrow\mathcal{L},bestZ\leftarrow Z$
+$G$
+$\lambda=0$
+$\text{Proj}_{\psi}:\mathbb{R}^{m\times d_{\text{token}}}\rightarrow\mathbb{R}^% +{m\times d_{\text{token}}}$
+$i\leftarrow 1$
+$s\in\mathbb{V}^{*}$
+$\displaystyle\text{argmax}_{s_{a}}\mathbb{E}_{x\sim G(F^{t}(E_{\psi}(s_{o}% +\oplus s_{a})))}\mathcal{A}(x,t,s_{o})~{},$
+$\displaystyle\cos(v,v_{t}^{image})+\lambda\cos(v,v_{t}^{text}),$
+$\cos(a,b)=\frac{a^{T}b}{\|a\|\|b\|}$
+$\eta$
+$512\times 512$
+$x$
+$E_{\psi}(s_{o}\oplus s_{a})=E_{\psi}(s_{o})\oplus E_{\psi}(s_{a})$
+$N$
+$bestloss>\mathcal{L}$
+$v_{t}^{image}=F^{i}(x_{t})$
+$d_{\text{emb}}$
+$\displaystyle\text{argmax}_{s_{a}}\cos(F^{i}(E_{\psi}(s_{o}\oplus s_{a})),v_{t% +}).$
+$s^{\prime}=$
+${}^{3,*}$
+$Z\leftarrow Z-\eta\nabla_{Z}\mathcal{L}$
+$100$
+$s_{a}$
+$s_{o}\oplus s_{a}$
+$m$
+$v$
+$\displaystyle\text{s.t.}\quad v=F^{i}(E_{\psi}(s_{o}\oplus s_{a})),$
+$\mathbb{V}=\{w_{1},w_{2},\cdots,w_{|\mathbb{V}|}\}$
+$F^{i}$
+$\psi$
+$\displaystyle\text{s.t.}\quad v$
+$s_{o}$
+$F^{t}$
+${}^{2}$
+$\oplus$
+$E_{\psi}(s)_{i}=\psi_{j}$
+$5\times 4=20$
+$3\times 100$
+${}^{3}$
+$v\leftarrow F^{t}(E_{\psi}(s_{o})\oplus\text{Proj}_{\psi}(Z))$
+$\mathcal{L}=-\cos(v,v_{t}^{image})-\lambda\cos(v,v_{t}^{text})$
+$s_{o}\in\mathbb{V}^{n}$
+$s_{a}\leftarrow E_{\psi}^{-1}(\text{Proj}_{\psi}(bestZ))$
+$bestloss\leftarrow\infty,bestZ\leftarrow Z$
+$\displaystyle=F^{i}(E_{\psi}(s_{o}\oplus E_{\psi}^{-1}(\text{Proj}_{\psi}(Z))))$
+$t\in\mathbb{V}$
+$Z$
+$(\cdot)$
+$x\sim G(v)$
+$d_{\text{token}}$
+$s_{a}\in\mathbb{V}^{m}$
+$v_{t}$
+$\lambda$
+$\mathbb{V}$
+$w_{j}=s_{i}$
+$t\in\mathcal{V}$
+$x\sim G(F^{t}(E_{\psi}(s)))$
+$E_{\psi}$
+$j=\text{argmin}_{j^{\prime}}\|\psi_{j^{\prime}}-Z_{i}\|_{2}^{2}$
+$|s|\times d_{\text{token}}$
+$\displaystyle\text{argmax}_{v_{t}}\mathbb{E}_{x\sim G(v_{t})}\mathcal{A}(x,t,s% +_{o})~{}.$
+$E_{L}\cup E_{R}$
+$E_{L}=\{(u,w)|(u,w)\in E,w\neq v\}$
+ + + diff --git a/htmls/output_mathjax_example_10.html b/htmls/output_mathjax_example_10.html new file mode 100644 index 0000000000000000000000000000000000000000..b6906530e1c93fd43e4562b311c75d277a044804 --- /dev/null +++ b/htmls/output_mathjax_example_10.html @@ -0,0 +1,137 @@ + + + +$0.01$
+$\underset{\pm 0.10}{2.15}$
+$(\operatorname{\bm{\theta}}_{\text{client}}^{(t)}=\operatorname{\bm{\theta}}_{% +\text{client}}^{(0)}$
+$r=64$
+$297.78$
+$\mathbf{0.43}$
+$\operatorname{\mathbf{v}}_{i}\in\mathcal{V}$
+$0.76$
+$\underset{\pm 0.66}{45.89}$
+$0$
+$\rho$
+$0.26$
+$0.95$
+$p\approx 8.69\times 10^{-8}$
+$0.69$
+$47.32$
+$\mathbf{0.69}$
+$2.30$
+$\mathbf{A}\in\mathbb{R}^{d\times r}$
+$\operatorname{\bm{\theta}}_{\text{client}}^{(t)}$
+$500$
+$\operatorname{\mathbf{v}}_{i}$
+$0.78$
+$308$
+$\mathbf{0.36}$
+$-\frac{1}{|\mathcal{D}^{(t)}_{\bigtriangledown}|}\sum_{\operatorname{\mathbf{d% +}}^{(t)}\in\mathcal{D}^{(t)}_{\bigtriangledown}}\log p_{(\cdot)}\big{(}% +\operatorname{\mathbf{d}}^{(t)}\big{|}\operatorname{\bm{\theta}}_{(\cdot)}^{(t% +)}),$
+$\operatorname{\mathbf{pr}}_{\text{client}}$
+$p$
+$0.66$
+$2.65$
+$\operatorname{\bm{\theta}}_{\text{client}}^{(0)}$
+$25$
+$277.25$
+$\%$
+$57.16$
+$0.74$
+$0.77$
+$\tau=0.5$
+$256$
+$4.3$
+$\operatorname{\bm{\theta}}_{\text{agent}}^{(0)}$
+$0.90$
+$0.80$
+$4$
+$0.8$
+$9000$
+$\operatorname{\bm{\theta}}_{\text{client}}$
+$F_{1}$
+$\mathbf{55.25}$
+$0.60$
+$\operatorname{\mathbf{v}}_{\text{next}}\in\text{Children}(\operatorname{% +\mathbf{v}}_{i})$
+$\displaystyle\geq bx_{j}^{\prime}+\sum_{i>j}x_{i}^{\prime}+(b-1)\sum_{i>j}x_{i}$
+$x_{i}^{\prime}\geq x_{i}$
+$A_{1}$
+$\displaystyle(b^{k}-b^{k-1}-(b-1)^{k}+b^{k-1})x_{1}$
+$(u,w)$
+$C_{1},C_{2}$
+$P:(u,v)\cup P^{\prime}$
+$e_{>i}$
+$(u_{1},u_{2},\dots,u_{k},v)$
+$\displaystyle=\frac{(b-1)^{k-1}}{b^{k}-(b-1)^{k}}x.$
+$(v,z)\in N(u)\times N(u)$
+$1\leq i\leq k$
+$x_{\tau+1},\dots,x_{k}\mapsto\textsf{Chunk-Shortest-Edge}(k-\tau,x-\delta)$
+ $\sum_{i}x_{i} $\displaystyle=bx+c(v\to t)-c(u,w)-c(w\to t)=\delta+(b-1)x$ $p(e_{k}^{O})<\beta$ $i\geq\tau$ $\displaystyle=\begin{cases*}c(u,y)+cost[y,y,i]&if $(y,z)\in\mathcal{P}^{\prime%
+}(u,y)$\\
+\infty&otherwise\end{cases*}$ $y^{*}\leq\frac{x}{k-1}$ $x-\delta$ $p(e;b_{i})$ $\displaystyle=\frac{b^{k-1}}{b^{k-1}-(b-1)^{k-1}}\left(\frac{b^{k}-(b-1)^{k-1}%
+(b-1+1)}{b^{k}-(b-1)^{k}}\right)x+c(v\to t)$ $(s,s_{1})$ $i+j\leq k$ $\displaystyle=\left(\frac{b^{k-1}}{b^{k-1}-(b-1)^{k-1}}-1\right)x$ $\begin{array}[]{ll}p(e_{\tau})&=bx_{\tau}+c(u_{\tau+1}\to t)\\
+&=bx_{\tau}+\sum_{i=\tau+1}^{k}x_{i}+c(v\to t)\hfill\mbox{(shortest path from
+$u_{\tau+1}$ follows the chunking)}\\
+&=bx_{\tau}+x-\sum_{i=1}^{\tau}x_{i}+c(v\to t)\\
+&=b\cdot\frac{\delta}{\tau}-\tau\cdot\frac{\delta}{\tau}+x+c(v\to t)\hfill%
+\mbox{(substituing $x_{i}=\delta/\tau$ for $i\leq\tau$)}\\
+&=\frac{b\delta}{\tau}+c(u,w)+c(w\to t)\hfill\mbox{(since $\delta=x+c(v\to t)-%
+c(u,w)-c(w\to t)$).}\end{array}$ $(u_{3},z)$ $\displaystyle\frac{(b-1)(b^{k-1}-(b-1)^{k-1})+b^{k-1}}{b^{k-1}-(b-1)^{k-1}}x_{1}$ $\sum_{l>i}x_{l}\leq x$ $\displaystyle(b-1)x_{1}+x$ $c^{n}$ $\delta/k$ $p(e_{i})=bx_{i}+c(u,w)+c(w\to t)$ $\alpha=\beta$ $p(e_{i};b_{1})\leq\alpha_{u}^{(j)}$ $p(e_{i}^{C})=bx_{i}^{C}+c(u_{i}^{C}\to t)$ $x_{i}-x_{i}^{\prime}\geq 0$ $p(e_{i})=\beta^{\prime}$ $\displaystyle\min_{(v,z)\in\mathcal{P}(u,y)}\min(C_{1}(u,v,y),C_{2}(u,y,z),C_{%
+3}(u,v,y,z)).$ $c(u,v)$ $\beta^{*}=p(e_{i})$ $p(e_{j};b_{1})\leq\alpha_{u}^{(1)}$ $\displaystyle=\frac{x}{1-\left(\frac{b-1}{b}\right)^{k-\tau+1}}+c(v\to t)-c(w%
+\to t)-c(u,w).$ $x_{1}$ $O(|E|^{2}k^{3}\log k+|V|)$ $\beta\leftarrow\frac{x-\delta}{z_{\tau}}+c(v\to t)$ $min\_bottleneck\leftarrow\min(\alpha,\beta)$ $\delta/\tau$ $(y,z)$ $x_{t+1},\ldots x_{t+\ell}$ $O(|{\cal X}|^{2L})$ $\forall i=1,\ldots,L$ $1000$ ${\cal M}_{b}$ $(X^{\tau},Z^{L-\tau})\sim{\cal M}_{b}^{L}.$ $(x^{i},y^{j})$ $x^{L}$ $\forall x\in{\cal X}$ $\forall x^{i_{0}}\in{\cal X}^{i_{0}}$ $X^{L}\sim{\cal M}_{s}^{L}$ $\mathbb{E}_{X^{L}\sim{\cal M}_{s}^{L}}[\tau]$ $(X^{L},Y^{L})$ $\ell=L$ $Z^{L-\tau}$ $x^{i}\in{\cal X}^{i}$ ${\cal M}_{b}(\cdot|x^{n})=\text{Ber}(q),$ $\forall\ell\leq L$ $\tau,{\cal M}_{b,{\rm next}}$ $L-1$ $\beta(X^{L},Y^{L})=\tau$ $\displaystyle=\sum_{\ell=0}^{L}\ell\cdot\Pr{\left({\tau=\ell}\right)}=\sum_{%
+\ell=1}^{L}\sum_{x^{\ell}\in{\cal X}^{\ell}}\prod_{i=1}^{\ell}\min\{{\cal M}_{%
+b}(x_{i}\mid x^{i-1}),{\cal M}_{s}(x_{i}\mid x^{i-1})\},$ $\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}\sum_{\ell\leq L}\sum_{x^{%
+\ell}}\min\left\{\text{Pr}\left(X^{1:\ell}=x^{\ell}\right),\text{Pr}\left(Y^{1%
+:\ell}=x^{\ell}\right)\right\}$ $\displaystyle=\Pr{\left({X^{i_{0}+1}=x^{i_{0}+1},\tau\geq i_{0}+1}\right)}+\Pr%
+{\left({Y^{i_{0}}=x^{i_{0}},\tau\leq i_{0}}\right)}\cdot p_{\rm res}^{x^{i_{0}%
+}}(x_{i_{0}+1})$ $Y_{\tau+1}\neq X_{\tau+1}$ $\displaystyle=\sum_{x^{\ell+1}\in{\cal X}^{\ell+1}}\prod_{i=1}^{\ell}\min\{{%
+\cal M}_{b}(x_{i}\mid x^{i-1}),{\cal M}_{s}(x_{i}\mid x^{i-1})\}({\cal M}_{s}(%
+x_{\ell+1}\mid x^{\ell})-\min\{{\cal M}_{b}(x_{\ell+1}\mid x^{\ell}),{\cal M}_%
+{s}(x_{\ell+1}\mid x^{\ell})\})$ ${\cal M}_{s}(\cdot\mid X^{i-1}),{\cal M}_{b}(\cdot\mid X^{i-1}),\forall i=0,%
+\ldots,L$ $L=8$ $\displaystyle\min\{{\cal M}_{s}(x^{\ell}),{\cal M}_{b}(x^{\ell})\}=\min\{\prod%
+_{i=1}^{\ell}{\cal M}_{s}(x_{i}\mid x^{i-1}),{\cal M}_{b}(x_{i}\mid x^{i-1})\},$ ${\cal M}^{*}(\cdot\mid x^{t})$ $L-\beta(X^{L},Y^{L})$ $X_{1},\ldots,X_{L}\sim{\cal M}_{s}^{L}(\cdot)$ $\forall y^{L-\tau}\in{\cal X}^{{L-\tau}}$ $\Pr{\left({\tau=L}\right)}=\sum_{x^{L}\in{\cal X}^{L}}\prod_{i=1}^{L}\min\{{%
+\cal M}_{b}(x_{i}\mid x^{i-1}),{\cal M}_{s}(x_{i}\mid x^{i-1})\}.$ $\eta_{i}\sim U(0,1)$ $\Pi({\cal M}_{s}^{L},{\cal M}_{b}^{L})$ $Y\sim{\cal M}_{b}$ $L=4$ $\displaystyle=\Pr{\left({X^{\ell_{0}-1}=x^{\ell_{0}-1},\tau\geq\ell_{0}}\right%
+)}+\Pr{\left({X^{\ell_{0}-1}=x^{\ell_{0}-1},\tau=\ell_{0}-1}\right)}$ $\displaystyle=\Pr{\left({Y^{i_{0}+1}=x^{i_{0}+1},\tau\geq i_{0}+1}\right)}+\Pr%
+{\left({Y^{i_{0}+1}=x^{i_{0}+1},\tau\leq i_{0}}\right)}$ $L=1$ $\ell=\ell_{0}-1$ $\displaystyle p_{\rm rej}(x^{i})$ $\displaystyle{:=}\sum_{x}{\left({{\cal M}_{b}(x^{i},x)-{\cal M}_{s}(x^{i},x)}%
+\right)}_{+},$ $\displaystyle\;\;\;\Pr{\left({\tau=\ell}\right)}$ $\displaystyle=\sum_{\ell=1}^{L}\Pr{\left({\tau\geq\ell}\right)}=\sum_{\ell=1}^%
+{L}\sum_{x^{\ell}\in{\cal X}^{\ell}}\Pr{\left({X^{\ell}=x^{\ell},\tau\geq\ell}%
+\right)}$ $\displaystyle=\min\{\sum_{x\in{\cal X}}\min\{{\cal M}_{b}(x^{\ell_{0}-1},x),{%
+\cal M}_{s}(x^{\ell_{0}-1},x)\}+p_{\rm remain}(x^{\ell_{0}-1}),$ $\ell>1$ $\forall\ell\leq L-1,x^{\ell}\in{\cal X}^{\ell}$ $x^{t}\in{\cal X}^{t}$ $\Pr{\left({X^{\ell}=x^{\ell},\tau\geq\ell}\right)}=\min\{{\cal M}_{b}(x^{\ell}%
+),{\cal M}_{s}(x^{\ell})\}.$ $\displaystyle{:=}\sum_{x}{\left({{\cal M}_{s}(x^{i},x)-{\cal M}_{b}(x^{i},x)}%
+\right)}_{+}.$ $Y^{L}=(X^{\tau},Z^{L-\tau})$ $\displaystyle\sum_{y^{L}}\pi(x^{L},y^{L})$ $(x)_{+}=\max\{x,0\}.$ ${\cal M}_{b}(\cdot\mid X^{\tau},Y)$ $\displaystyle={\cal M}_{s}(x^{L})\cdot\min\left\{1,\frac{{\cal M}_{b}(x^{L})}{%
+{\cal M}_{s}(x^{L})}\right\}$ ${\cal M}^{L}_{s}(x^{L})$ $(x_{i},\ldots,x_{j})$ $0\leq\tau\leq L$ $\pi\in\Pi({\cal M}^{L}_{s},{\cal M}^{L}_{b})$ $\displaystyle=\sum_{\ell\leq L}\sum_{x^{\ell}}\text{Pr}_{X^{L},Y^{L}}\left(X^{%
+1:\ell}=Y^{1:\ell}=x^{\ell},\beta(X^{L},Y^{L})\geq\ell\right)$ $X^{L}$ $Y_{\tau+1}$ $x^{L}\in{\cal X}^{L}$ $\textsc{Verify}_{\pi}$ ${\cal M}_{b,{\rm next}}^{L-\tau}$ $\tau\leq L$ $\min\{{\cal M}_{b}(x^{L}),{\cal M}_{s}(x^{L})\}$ $\displaystyle=\prod_{i=1}^{\ell}\min\{1,\frac{{\cal M}_{b}(x_{i}\mid x^{i-1})}%
+{{\cal M}_{s}(x_{i}\mid x^{i-1})}\}{\left({1-\min\{1,\frac{{\cal M}_{b}(x_{%
+\ell+1}\mid x^{\ell})}{{\cal M}_{s}(x_{\ell+1}\mid x^{\ell})}\}}\right)}.$ $x^{t}$ $X_{i}\sim{\cal M}_{s}(\cdot\mid X^{i-1}).$ $\tau=L-i$ ${\rm E.O.S}\in X^{\tau}$ $\displaystyle=\sum_{x^{L}\in{\cal X}^{L}}\Pr{\left({\tau=\ell,X^{L}=x^{L}}%
+\right)}$ $\pi_{\rm token}$ $\displaystyle=\sum_{x^{\ell+1}\in{\cal X}^{\ell+1}}\Pr{\left({\tau=\ell,X^{%
+\ell+1}=x^{\ell+1}}\right)}$ $Z\sim{\cal M}_{b,{\rm next}}(\cdot)$ $X^{\tau}$ $(a)$ $\ell $\displaystyle=\Pr{\left({X^{\ell_{0}-1}=x^{\ell_{0}-1},\tau\leq\ell_{0}-1}%
+\right)}\cdot\Pr{\left({X^{\ell_{0}-1}\text{ is accepted.}}\right)}$ $Y_{1}=X_{1}$ $\displaystyle=\min\left\{{\cal M}_{b}(x^{\ell_{0}-1}),{\cal M}_{s}(x^{\ell_{0}%
+-1})\right\},$ $\displaystyle=\text{Pr}_{\pi}{\left({\beta{{\color[rgb]{0,0,0}=\ell}}\mid X^{L%
+}}\right)}{:=}{{\color[rgb]{0,0,0}\frac{\sum_{\ell:\beta(X^{L},Y^{L})=\ell}\pi%
+(X^{L},Y^{L})}{\pi(X^{L})}}},$ $Y_{2}$ $\displaystyle\max_{\pi\in\Pi({\cal M}_{s}^{L},{\cal M}_{b}^{L})}\mathbb{E}_{X^%
+{L},Y^{L}\sim\pi}\left[\beta(X^{L},Y^{L})\right]\leq\sum_{\tau=1}^{L}\sum_{x^{%
+\tau}\in{\cal X}^{\tau}}\min\{{\cal M}_{s}(x^{\ell}),{\cal M}_{b}(x^{\ell})\}$ $\forall x^{n}$ $\eta_{0}\leq\min\left\{\frac{{\cal M}_{b}(X^{L})}{{\cal M}_{s}(X^{L})},1\right\}$ $O(|{\cal X}|^{L})$ $Z^{L-\tau}\sim{\cal M}_{b,{\rm next}}$ $L=6$ ${\cal M}_{b}(Y_{3}\mid Y_{1},y)$ $Y^{L}=(X^{\tau},Z^{i-\tau})$ $\displaystyle=\sum_{\ell\leq L}\sum_{x^{\ell}}\min\{{\cal M}_{s}^{\ell}(x^{%
+\ell}),{\cal M}_{b}^{\ell}(x^{\ell})\}.$ $\displaystyle=\Pr{\left({X^{L}=x^{L}}\right)}\Pr{\left({\tau=L\mid X^{L}=x^{L}%
+}\right)}$ $p_{\rm res}(x)=\frac{{\left({{\cal M}_{b}(x\mid X^{\tau})-{\cal M}_{s}(x\mid X%
+^{\tau})}\right)}_{+}}{\sum_{x^{\prime}}{\left({{\cal M}_{b}(x^{\prime}\mid X^%
+{\tau})-{\cal M}_{s}(x^{\prime}\mid X^{\tau})}\right)}_{+}},$ ${\cal M}_{s}=\text{Ber}(1)$ $\frac{{\left({{\cal M}_{b}(x^{L})-{\cal M}_{s}(x^{L})}\right)}_{+}}{\sum_{x^{L%
+}}{\left({{\cal M}_{b}(x^{L})-{\cal M}_{s}(x^{L})}\right)}_{+}}.$ $i=0,1,\ldots,L$ $M(T)_{\mu\nu}$ $\gamma_{1}:1\succ 0$ $(V_{j},V_{i})$ $\Delta(N,N^{k,n}_{s})=2^{n-k-1}$ $|Pa(N_{s},V_{n})|=k$ $Pa(N_{s},V_{n})\cap Pa(N_{s^{\prime}},V_{n})=\emptyset$ $\binom{2c+1}{c+1}=\binom{2c+1}{c}=\frac{2c+1}{c+1}\binom{2c}{c}=\frac{4c+2}{c+%
+1}\binom{2c-1}{c}$ $2^{k-k^{\prime}+1}-2$ $k^{\prime}=0$ $(2^{n}-2^{n-k})\sum_{k^{\prime}=0}^{k}\binom{k}{k^{\prime}}\binom{n-k-1}{k-k^{%
+\prime}}\geq 3\cdot 2^{n-2}\binom{n-1}{k}\,.$ $V\in Pa(N_{s},V_{n})$ $N_{1},\ldots,N_{t}$ $N^{a}$ $(o,o^{\prime})$ $(T^{n-1,n})_{n\geq 3}$ $T^{n-1,n}$ $o^{\prime}$ $\binom{2d}{d}\leq 2^{2d-1}$ $\mathcal{V^{\prime}}\subseteq\mathcal{V}\setminus{V_{n}}$ $o\succ o^{\prime}$ $\sum_{1\leq s\leq t}|\operatorname{CPT}(N_{s},V_{n})|$ $2^{2k-k^{\prime}}$ $o^{\prime}\succ o$ $s\in\{1,\ldots,\binom{n-1}{k}2^{k}$ $n^{\prime}=\max_{1\leq s\leq t}|Pa(N_{s},V_{n})|$ $T_{\varepsilon}$ $N_{\varepsilon}$ $\binom{2d+2}{d+1}\leq 2^{2d+1}$ $o[V_{j}]=o^{\prime}[V_{j}]=\gamma[V_{j}]$ $CPT(N_{s},V_{n})$ $\operatorname{Inst}(P)$ $|\tau|$ $V_{4}$ $Pa(N_{1},V_{3})$ $000$ $\Leftarrow c\binom{2c-1}{c}\leq(2c+3)\cdot 2^{2c-4}$ $(c+1)\binom{2c+1}{c+1}\leq(c+1)\cdot 2^{2c-1}+\binom{2c}{c}$ $2^{k}-2^{k-k^{\prime}}$ $\gamma:0\succ 1$ $\Leftarrow c\binom{2c-1}{c}\leq(c+1)\cdot 2^{2c-3}$ $4c\binom{2c-1}{c}+2\cdot\binom{2c-1}{c}\leq(c+1)\cdot 2^{2c-1}+2\cdot\binom{2c%
+-1}{c}$ $f_{T^{k,n}}(N^{k,n}_{s})=(2^{n}-2^{n-k})\sum_{k^{\prime}=0}^{k}\binom{k}{k^{%
+\prime}}\binom{n-k-1}{k-k^{\prime}}$ $\mathcal{F}_{bad}=(T_{n})_{n\in\mathbb{N}}$ $o[\{V_{1},V_{2}\}]$ $2\cdot\binom{2c-1}{c}$ $Pa(N^{k,n}_{s},V_{n})\cup P$ $T_{n}=(N^{n}_{1},\ldots,N^{n}_{n-1})$ $s\in\{1,\ldots,t\}$ $N^{k,n}_{2}$ $\Leftarrow 2^{n-2c}\cdot c\binom{2c-1}{c}\leq(2c+2)\cdot 2^{n-4}$ $2^{|\mathcal{U}|}\prod_{s=1}^{t}2^{p_{s}-1}$ $2^{n-k-1}$ $\displaystyle\ \ +2^{k}-2^{k-k^{\prime}}\binom{k}{k^{\prime}}\binom{n-k-1}{k^{%
+\prime}}2^{n-k}\big{]}$ $|\mathcal{V}^{\prime}|=k$ $f_{T}(N)$ $f_{T}(N^{a})>f_{T}(N)$ $p_{s}=|Pa(N_{s},V_{n})|$ $\operatorname{Inst}(\mathcal{V}^{\prime})$ $\gamma:1\succ 0$ $P\setminus Pa(N^{k,n}_{s},V_{n})$ $\Delta(N^{k,n}_{s},N^{k,n}_{s^{\prime}})=2^{n-k}-2^{n-2k}$ $\kappa\leq t-\kappa$ $2^{k-k^{\prime}}$ $\operatorname{CPT}(N^{a},V_{n})$ $O(\sum_{1\leq s\leq t}|CPT(N_{s},V_{n})|)$ $f_{T}(N^{a})\leq f_{T}(N)$ $\Delta(N,N_{s})$ $O(2^{|P|}\cdot\sum_{1\leq s\leq t}|CPT(N_{s},V_{n})|)$ $V=\{V_{1},V_{2},V_{3}\}$ $T^{2,3}$ $N_{i}\in T$ $\sum_{\kappa=0}^{\lfloor\frac{t}{2}\rfloor}\kappa\cdot\binom{t}{\kappa}$ $\mathcal{V}^{\prime}$ $freq_{M}(0\succ 1)=t\cdot 2^{n-1}-freq_{M}(1\succ 0)$ $Pa(N^{k,n}_{s},V_{n})\cap P$ $s\neq s^{\prime}$ $2^{n-t-1}$ $\frac{2\cdot(2d+1)}{d+1}$ $2^{n-k}$ $f_{T^{\prime}}(N)=\begin{cases}2\cdot\sum_{\kappa=0}^{c-1}\kappa\binom{2c-1}{%
+\kappa}&\text{if }t=2c-1\\
+2\cdot\sum_{\kappa=0}^{c-1}\kappa\binom{2c}{\kappa}+c\binom{2c}{c}&\text{if }t%
+=2c\\
+\end{cases}$ $N_{4}$ $\Leftarrow c\binom{2c}{c}\leq(2c+3)\cdot 2^{2c-3}$ $freq_{M}(1\succ 0)=\sum\nolimits_{0\leq\mu<2^{n-1}}\sum\nolimits_{1\leq\nu\leq
+t%
+}M_{\mu\nu}$ $\mathcal{F}_{bad}$ $f_{T}(N_{s})=\min\{f_{T}(N_{s^{\prime}})\mid 1\leq s^{\prime}\leq t\}\leq\frac%
+{4}{3}f_{T}(N)$ $N^{k,n}_{s}$ $2^{k}\binom{n-k-1}{k}$ $2^{n-2}$ $\Delta(N^{k,n}_{s},N^{k,n}_{s^{\prime}})$ $freq_{M}(1\succ 0)$ $o[V_{i}]=0$ $f_{T^{k,n}}(N)=2^{n-1}\binom{n-1}{k}$ $f_{T^{k,n}}(N^{k,n}_{s})$ $\{V_{1},V_{2}\}$ $(T^{k,n})_{k,n}$ $1\leq s\leq t=2c$ $\Leftarrow 2^{n-2c-1}\cdot c\binom{2c}{c}\leq(2c+3)\cdot 2^{n-4}$ $1\leq k^{\prime}\leq k-1$ $P:=\bigcup_{1\leq s\leq t}Pa(N_{s},V_{n})$ $t=n-1$ $0.058\scriptscriptstyle\pm\scriptstyle.010$ $0.027\scriptscriptstyle\pm\scriptstyle.004$ $\mu_{c}^{\mathcal{D}_{\text{test}}}$ $0.146\scriptscriptstyle\pm\scriptstyle.002$ $0.124\scriptscriptstyle\pm\scriptstyle.006$ $0.038\scriptscriptstyle\pm\scriptstyle.003$ $0.243\scriptscriptstyle\pm\scriptstyle.002$ $0.338\scriptscriptstyle\pm\scriptstyle.005$ $\mathbf{0.043\scriptscriptstyle\pm\scriptstyle.000}$ $\mathbf{0.010\scriptscriptstyle\pm\scriptstyle.002}$ $1769$ $\mathbf{0.077\scriptscriptstyle\pm\scriptstyle.005}$ $0.045\scriptscriptstyle\pm\scriptstyle.004$ $0.152\scriptscriptstyle\pm\scriptstyle.001$ $\mathbf{0.042\scriptscriptstyle\pm\scriptstyle.003}$ $0.025\scriptscriptstyle\pm\scriptstyle.000$ $0.081\scriptscriptstyle\pm\scriptstyle.007$ $0.066\scriptscriptstyle\pm\scriptstyle.005$ $0.013\scriptscriptstyle\pm\scriptstyle.006$ $0.781\scriptscriptstyle\pm\scriptstyle.003$ $0.021\scriptscriptstyle\pm\scriptstyle.003$ $0.012\scriptscriptstyle\pm\scriptstyle.000$ $0.135\scriptscriptstyle\pm\scriptstyle.014$ $0.228\scriptscriptstyle\pm\scriptstyle.007$ $0.069\scriptscriptstyle\pm\scriptstyle.035$ $0.228\scriptscriptstyle\pm\scriptstyle.001$ $0.065\scriptscriptstyle\pm\scriptstyle.012$ $0.078\scriptscriptstyle\pm\scriptstyle.007$ $\mathcal{D}_{\text{Test}}$ $0.153\scriptscriptstyle\pm\scriptstyle.002$ $0.077\scriptscriptstyle\pm\scriptstyle.010$ $0.332\scriptscriptstyle\pm\scriptstyle.001$ $0.360\scriptscriptstyle\pm\scriptstyle.004$ $0.083\scriptscriptstyle\pm\scriptstyle.002$ $\mathbf{0.033\scriptscriptstyle\pm\scriptstyle.003}$ $0.081\scriptscriptstyle\pm\scriptstyle.001$ $0.045\scriptscriptstyle\pm\scriptstyle.001$ $0.046\scriptscriptstyle\pm\scriptstyle.002$ $0.018\scriptscriptstyle\pm\scriptstyle.007$ $\mathbf{0.051\scriptscriptstyle\pm\scriptstyle.024}$ $\mathbf{0.041\scriptscriptstyle\pm\scriptstyle.003}$ $0.028\scriptscriptstyle\pm\scriptstyle.006$ $0.042\scriptscriptstyle\pm\scriptstyle.000$ $0.149\scriptscriptstyle\pm\scriptstyle.001$ $0.409\scriptscriptstyle\pm\scriptstyle.005$ $0.065\scriptscriptstyle\pm\scriptstyle.001$ $0.108\scriptscriptstyle\pm\scriptstyle.003$ $0.028\scriptscriptstyle\pm\scriptstyle.009$ $0.133\scriptscriptstyle\pm\scriptstyle.007$ $0.166\scriptscriptstyle\pm\scriptstyle.003$ $\mathbf{0.034\scriptscriptstyle\pm\scriptstyle.001}$ $\mathbf{0.007\scriptscriptstyle\pm\scriptstyle.001}$ $0.033\scriptscriptstyle\pm\scriptstyle.006$ $0.117\scriptscriptstyle\pm\scriptstyle.002$ $0.083\scriptscriptstyle\pm\scriptstyle.024$ $0.052\scriptscriptstyle\pm\scriptstyle.001$ $0.321\scriptscriptstyle\pm\scriptstyle.002$ $0.058\scriptscriptstyle\pm\scriptstyle.004$ $\mathcal{SRE}$ $\mathcal{SR}$ $\mathbf{PP}(p_{i})=\frac{1}{Length(\overline{p^{e}_{i}\Phi(p^{e}_{i})})+dist^{%
+M(X)}(center,\Phi(p^{e}_{i})},$ $Crosswise$ $Sb_{i}$ $\mathbf{V}_{1},\mathbf{V}_{2},\mathbf{V}_{3},\mathbf{V}_{4}$ $\mathbf{X}\subset\mathbb{R}^{2}$ $p_{1},p_{2}\leftarrow$ $M(\mathbf{X})$ $E_{area}$ $\mathbf{I}=\{I_{i}\},i=0,\dots,N_{I}$ $\mathbf{A}^{c}=\{ac_{k}\}$ $center=\operatorname*{arg\,max}_{m\in M(X)}\;CR(v_{i},v_{j})\;\mid v_{i},v_{j}%
+\in\pi(m),$ $\pi(z)=\{z\in\partial\mathbf{X}:\|z-x\|=D(\mathbf{X})(z)\}$ $p^{\prime}(x^{\prime},y^{\prime})=\zeta(p(x,y)),$ $\mathbf{K_{i}}$ $p(x,y)\in a_{k}$ $p_{triangle}$ $D(\mathbf{X})(z)=\inf_{x\in\partial\mathbf{X}}\|z-x\|,$ $4^{S-1}$ $Sb_{i}^{*}$ $M(\mathbf{X})=\{z\in\mathbf{X}:\lvert\pi(z)\rvert>1\}.$ $P_{\mathbf{X}}$ $x\in\pi(z)$ $D(\mathbf{X}):\mathbb{R}^{2}\mapsto\mathbb{R}$ $\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{H}_{3},\mathbf{H}_{4}$ $z\in\mathbb{R}^{2}$ $\zeta(.)$ $4^{1}\cdot 8/2$ $Crosswise(z)$ $\mathbf{D_{\mathbf{T}}}$ $E_{area}=\sum_{i=1}^{S_{T}}Area(Sb_{i}^{*})$ $\mathbf{\mathcal{O}}^{*}=\operatorname*{arg\,max}_{\mathbf{D_{i}},\mathbf{K_{i%
+}}}\;E_{area},$ $[\;\;]$ $\mathbf{A}^{i}=\{a_{k}\},k=1,\dots,8$ $\Phi\big{(}z\big{)}$ $M_{o}=\dfrac{P_{o}}{P_{\mathbf{X}}},$ $\tau_{p}=0.75$ $Axial$ $M_{n}=\dfrac{1}{N}\sum_{i}\|(L_{i}-L_{ci})\|,$ $Sb^{*}$ $Sb_{i}=[bx_{1},by_{1},bx_{2},by_{2}]$ $\mathfrak{L}_{T}$ $\tau_{e}=2$ $ct\leftarrow G.centroid$ $O(4^{n})$ $dividing\_line\leftarrow$ $\tau_{e}=3$ $4^{3}\cdot n/4$ $\mathfrak{L}_{\mathbf{T}}$ $E_{area}=\sum_{i=1}^{S_{T}}(Area(Sb_{i}^{*})\cdot p_{triangle}(\mathbf{G})),$ $N_{I}>N_{p}$ $N_{I}>>N_{p}$ $\mathfrak{R}_{\mathbf{T}}$ $\mathbf{G_{\mathbf{T}}}$ $\mathbf{S}_{i}=\big{[}N_{I}\cdot\dfrac{Area(p_{i})}{Area(\mathbf{X})}\big{]},$ $slope\leftarrow Axial(ct)$ $Sb$ $N_{c}=N_{I}$ $CR(v_{i},v_{j})=dist^{B}(v_{i},v_{j})-Length(\overline{v_{i}v_{j}}),$ $\Phi(p^{e}_{i})$ $\mathbf{\mathcal{O}}^{*}$ $Axial(z)$ $\|\;\;\|$ $\mathbf{G}_{T}$ $dist^{B}$ $\mathbf{G}\leftarrow p_{2}$ $\bigcup_{i}S_{i}$ $\mathbb{R}^{2}\setminus\mathbf{X}$ $\mathbf{\mathcal{O}}$ $z\in\mathbf{X}$ $M_{a}=\dfrac{|\bigcup_{i}S_{i}|}{P_{\mathbf{X}}},$ $\tau_{e}=1$ $slope$ $dividing\_line$ $p_{triangle}(polygon)=\begin{cases}0.8&polygon\text{ is a triangle}\\
+1.0&\text{otherwise.}\end{cases}$ $8/2$ $R^{m}_{i}$ $S_{i}-1$ $p_{1},p_{2}\leftarrow G$ $dist^{M(X)}$ $S_{T}-1$ $4^{1}\cdot n/2$ $M_{c}=\dfrac{P_{w}}{P_{\mathbf{X}}},$ $\gamma^{u}$ $4^{2^{\tau_{e}}-1}\cdot n/2^{\tau_{e}}$ $\Phi(z,M(\mathbf{X}))=\operatorname*{arg\,min}_{m\in M(\mathbf{X})}\|z-m\|$ $ac_{k}$ $L_{ci}$ $M_{s}=1-\dfrac{|\bigcup_{i}S_{i}|}{\sum_{i}|S_{i}|}.$ $\mathfrak{R}_{T}$ $slope\leftarrow Crosswise(ct)$ $p^{\prime}\in ac_{k}$ $\mathbf{G}\leftarrow p_{1}$ $\mathbf{P}=\{p_{1},\dots,p_{N_{p}}\}$ $p^{e}_{i}$ $Sb=[bx_{1},by_{1},bx_{2},by_{2}]$ $\mathcal{T}_{t}^{(d)}(m)$ $\mathcal{T}_{r}^{(d)}(m)=\lVert(\mathcal{T}^{(d)}(s,m))_{s\in\mathcal{M},s\neq
+t%
+}\rVert_{2}$ $\mathcal{T}_{t}^{(d)}(m)=\lVert(\mathcal{T}^{(d)}(m,t))_{t\in\mathcal{M},t\neq
+s%
+}\rVert_{2}$ $\alpha=\frac{2}{255}$ $\displaystyle x^{t+1}=\prod_{x+\mathcal{B}}(x^{t}+\alpha\text{sgn}(\nabla_{x}L%
+(\theta,x,y)))$ $\mathcal{T}_{r}^{(d)}(m)$ $\mathcal{T}^{(d)}(s,t)$ $\mathcal{A}_{s}^{(d)}=\{(x,y)\}$ $\mathcal{A}_{s}^{(d)}$ $(s,t)\in\mathcal{M}^{2}$ $(7,308+4,542+7,517)*6=116,202$ $\mathcal{T}^{(d)}(s,t)=\frac{|\{(x,y)\in\mathcal{A}_{s}^{(d)};\hat{y}_{t}\neq y%
+\}|}{|\mathcal{A}_{s}^{(d)}|}$ $N_{e}=100$ $l^{\eta}_{i,j\in B}$ $BNM$ $L^{\textit{AttnMask}}_{DM}=L_{DM}(\mathcal{M}\odot x,\mathcal{M}\odot\tilde{x}),$ $[v^{*},\ldots,v^{\&}]:=\operatorname*{arg\,min}_{\mathcal{V}}E_{x_{0},{%
+\epsilon}\sim N(0,I)}\\
+\|{\epsilon}-{\epsilon}_{\theta}(x_{t},t,[c_{\theta}(y),v^{*},\ldots,v^{\&}]\|%
+^{2}$ $V=f_{V}(v)$ $v^{\eta}_{i}$ $[y,p^{*}]$ $v^{\eta}_{j}$ $L_{PromptCL}$ $\overline{M}^{p}=1/T\sum_{t=1}^{T}M_{t}^{p}$ $\{v^{*},\ldots,v^{\&}\}$ $v^{*}=c_{\theta}(p^{*})$ $L_{DM}=L_{DM}(x,\tilde{x}):=E_{x_{0},{\epsilon}\sim N(0,I),t\sim\text{Uniform}%
+(1,T)}\|{\epsilon}-{\epsilon}_{\theta}(x_{t},t,c_{\phi}(p))\|^{2},$ $6100$ $L=L^{\textit{AttnMask}}_{DM}+\gamma L_{PromptCL}^{adj},$ $M=\text{Softmax}(QK^{T}/\sqrt{d})$ $step=1,\ldots,S$ $\{v^{\&}\}$ $(0.2,0.0005)$ $\mathcal{V}=[v^{*},\ldots,v^{\&}]$ $(p^{*},v^{*})$ $(\mathcal{P},\mathcal{V})$ ${\bm{\epsilon}}\sim\mathcal{N}(\mathbf{0},\textbf{I})$ $L_{PromptCL}^{adj}$ $v=c_{\phi}(p)$ $\eta\in MN$ $L^{\textit{AttnMask}}_{DM}$ $(0.3,0.00075)$ $Q=f_{Q}(z)$ $\mathcal{P}=[p^{*},\ldots,{p}^{\&}]$ $\mathcal{M}=\bigcup_{p\in\mathcal{P}}B(M^{p})$ $\{v^{*}\}$ $K=f_{K}(v)$ $sim(v_{i},v_{j})=v_{i}^{T}.v_{j}/||v_{i}||||v_{j}||$ $B(M^{p}):=\{1\text{ if }M^{p}>k,0\text{ otherwise}\}$ $v^{*}:=\operatorname*{arg\,min}_{v}E_{x_{0},{\epsilon}\sim N(0,I)}\\
+\|{\epsilon}-{\epsilon}_{\theta}(x_{t},t,[c_{\theta}(y),v^{*}]\|^{2}$ $\{v_{b}^{n}\}_{b=1}^{B},_{n=1}^{N}$ $\begin{gathered}L_{PromptCL}=\frac{1}{N}\frac{1}{B}\sum_{\eta=1}^{N}\sum_{i=1}%
+^{B}{l^{\eta}_{i,j\in B}},\qquad L_{PromptCL}^{adj}=\frac{1}{NM}\frac{1}{B}%
+\sum_{\eta=1}^{NM}\sum_{i=1}^{B}{l^{\eta}_{i,j\in B}}\end{gathered}$ $(\tau,\gamma)$ $[v^{*},\ldots,v^{\&}]=[c_{\theta}(p^{*}),\ldots,c_{\theta}(p^{\&})]$ $[y,p^{*},\ldots,p^{\&}]$ $c_{\phi}$ $l^{\eta}_{i,j\in B}=-log(\frac{exp(sim(v^{\eta}_{i},v^{\eta}_{j}))/\tau}{\sum_%
+{\eta=1}^{N}\sum_{j=1,j\neq{i}}^{B}exp(sim(v^{\eta}_{i},v^{\eta}_{j})/\tau)})$ $\eta\in N$ $\tau=10^{-4}$ $J^{\tau}(\mathbf{\chi}^{k+1},T)$ $q_{1}=1,\ q_{2}=100$ $d=0.1l$ $\tilde{\chi}_{s}$ $\Phi_{h}=(q_{1}-q_{2})G_{\tau}\ast(T_{h}-T^{\ast}_{h})+\gamma\sqrt{\frac{\pi}{%
+\tau}}G_{\tau}\ast(1-2\chi_{h})+(\kappa_{1}-\kappa_{2})G_{\tau}\ast(\frac{\xi}%
+{2}\nabla T_{h}\cdot\nabla T_{h}+\nabla T_{h}\cdot\nabla T_{h}^{\ast}).$ $\kappa_{1}=5,\ 10,\ 15$ $\tau=3\times 10^{-5}$ $\Omega\in\mathbb{R}^{d}\ (d=2,3)$ $\mathbf{\chi}_{h}$ $\int_{\Omega}\kappa(\chi)\nabla T\cdot\nabla T\ d\textbf{x}$ $y=1/2$ $\Omega_{1}\subset\Omega$ $(\ref{ad})$ $\frac{q_{1}}{q_{2}}$ $z=0,\ y=1/2$ $\Phi_{h}$ $T^{\ast}_{h}$ $J^{\tau}(\chi,T)=\int_{\Omega}q(\chi)T\ d\textbf{x}+\frac{\xi}{2}\int_{\Omega}%
+\kappa(\chi)\nabla T\cdot\nabla T\ d\textbf{x}+\gamma\sqrt{\frac{\pi}{\tau}}%
+\int_{\Omega}\chi G_{\tau}\ast(1-\chi)\ d\textbf{x},$ $\displaystyle\ \ \ \ +\int_{\Omega}\kappa(\chi)\nabla T^{k}\cdot\nabla T^{*k}%
+\ d\textbf{x}-\int_{\Omega}q(\chi)T^{*k}\ d\textbf{x}.$ $\displaystyle\int_{\Omega}\chi G_{\tau}\ast(1-\chi)\ d\textbf{x}=$ $J^{\tau}(\tilde{\chi}_{s},\tilde{T}_{s})>J^{\tau}(\chi^{k},T^{k})$ $\kappa_{1}=10,\ \kappa_{2}=1,\ q_{1}=1,\ q_{2}=100,\ \tau=1\times 10^{-4}$ $\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}\leq\sigma_{1},\chi^{k}(p)=0%
+\big{\}},$ $\tilde{T}_{s}$ $\kappa_{1}=40,\ \kappa_{2}=1$ $\arg\min_{\chi\in\mathcal{B}}\tilde{J}^{\tau,k}(\chi)\in\arg\min_{\chi\in%
+\mathcal{H}}\tilde{J}^{\tau,k}(\chi).$ $\tilde{B}\subset B$ $A_{1}=\{p\in A:\phi_{A}(p)\leq\sigma_{1}\}\ \ \ \ B_{1}=\{p\in B:\phi_{B}(p)%
+\leq\sigma_{2}\}$ $H^{1}_{\Gamma_{D}}(\Omega)=\{v\in H^{1}(\Omega)\ |\ v|_{\Gamma_{D}}=0\}$ $(\ref{and})$ $\displaystyle\left\{\begin{aligned} -\nabla\cdot(\kappa(\chi^{k})\nabla T)-q(%
+\chi^{k})&=0,\ \ &&\rm in\ \ \Omega,\\
+T&=0,\ \ &&\rm on\ \ \Gamma_{D},\\
+\kappa(\chi^{k})\nabla T\cdot\mathbf{n}&=0,\ \ &&\rm on\ \ \Gamma_{N},\end{%
+aligned}\right.$ $\displaystyle\mathcal{L}^{\tau,k}_{\chi^{k}}(\chi)$ $\sigma_{1}=\tilde{\phi}_{A}^{k}(n_{s})$ $tol>0$ $\displaystyle\int_{\Omega}(q(\chi^{k})T^{k})\ d\textbf{x}+\frac{\xi}{2}\int_{%
+\Omega}\kappa(\chi^{k})\nabla T^{k}\cdot\nabla T^{k}\ d\textbf{x}+\gamma\sqrt{%
+\frac{\pi}{\tau}}\int_{\Omega}\chi^{k}G_{\tau}\ast(1-\chi^{k})\ d\textbf{x}.$ $T\in Q(\chi)$ $\frac{\xi}{2}\int_{\Omega}\kappa(\chi)\nabla T\cdot\nabla T\ d\textbf{x}$ $G_{\tau}=\frac{1}{(4\pi\tau)^{d/2}}\exp\left(-\frac{|\textbf{x}|^{2}}{4\tau}\right)$ $\int_{\Omega}\chi^{k+1}\ d\textbf{x}=V_{0}$ $\chi^{k+1}=\tilde{\chi}_{s}$ $\mathbb{R}^{d}\setminus\overline{\Omega}$ $T^{*k}$ $\displaystyle\tilde{B}_{1}$ $\Omega_{1}\in\Omega$ $\Gamma\colon=\Gamma_{D}\cup\Gamma_{N}$ $\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}>\sigma,\chi^{k}(p)=1\big{\}}.$ $\displaystyle\chi^{k+1}=\begin{cases}1&\ \textrm{if}\ \Phi^{k}\leq\sigma,\\
+0&\ \textrm{otherwise}.\end{cases}$ $\frac{\delta\tilde{J}^{\tau}}{\delta T}=0,\ \ \ \frac{\delta\tilde{J}^{\tau}}{%
+\delta T^{*}}=0.$ $\chi^{k+1}=\arg\min_{\chi\in\mathcal{B}}\tilde{J}^{\tau,k}(\chi)$ $\tilde{\chi}_{s}=\chi_{A_{1}}+\chi^{k}-\chi_{B_{1}}$ $\kappa_{1},\kappa_{2},q_{1},q_{2}$ $T^{\ast}_{h}\in V_{h}^{0}$ $\Phi^{k}=(q_{1}-q_{2})G_{\tau}\ast(T^{k}-T^{*k})+\gamma\sqrt{\frac{\pi}{\tau}}%
+G_{\tau}\ast(1-2\chi^{k})+(k_{1}-k_{2})G_{\tau}\ast(\frac{\xi}{2}\nabla T^{k}%
+\cdot\nabla T^{k}+\nabla T^{k}\cdot\nabla T^{\ast k})$ $\displaystyle\tilde{J}^{\tau,k}(\chi)$ $\displaystyle\int_{\Omega}\left[G_{\tau/2}\ast\chi\right]\left[G_{\tau/2}\ast(%
+1-\chi)\right]\ d\textbf{x}$ $\kappa_{1}=10,\ \kappa_{2}=1,\ q_{1}=1,\ q_{2}=100$ $\kappa_{1}=40,\kappa_{2}=1$ $\int_{\Omega}\chi\ d\textbf{x}=V_{0}$ $T^{k+1}$ $\chi^{0}$ $\tilde{\phi}_{A}^{k}$ $\displaystyle\tilde{A}_{2}$ $\chi^{k+1}=Proj_{[0,1]}\left(\chi^{k}-s\left.\frac{\delta J^{\tau}}{\delta\chi%
+}\right|_{\chi^{k}}\right),$ $\kappa(\chi)$ $0.1l$ $\displaystyle\chi^{k+1}=\chi_{A}+\chi^{k}-\chi_{B}.$ $J^{\tau}(\chi^{k+1},T^{k+1})\leq J^{\tau}(\chi^{k+1},T^{k}),$ $\kappa_{2}=1$ $\displaystyle\kappa(\chi)=\kappa_{1}G_{\tau}\ast\chi+\kappa_{2}G_{\tau}\ast(1-%
+\chi),$ $\tilde{J}^{\tau}(\chi,T,T^{*})=J^{\tau}(\chi,T)+\int_{\Omega}(-\nabla\cdot(%
+\kappa(\chi)\nabla T)-q(\chi))\cdot T^{*}d\textbf{x}.$ $q_{2}=100$ $\chi^{1},T^{1},T^{*1},\chi^{2},T^{2},T^{*2},\cdots,\chi^{k},T^{k},T^{*k},\cdots$ $\tilde{J}^{\tau}(\mathbf{\chi}^{k+1},T^{k})\leq\tilde{J}^{\tau}(\mathbf{\chi}^%
+{k},T^{k}),$ $J^{\tau}(\chi,T)$ $\displaystyle\left\{\begin{aligned} \nabla\cdot(\kappa(\chi^{k})\nabla T^{*})&%
+=q(\chi^{k})-\xi(\nabla\cdot(\kappa(\chi^{k})\nabla T)),\ \ &&\rm in\ \ \Omega%
+,\\
+T^{*}&=0,\ \ &&\rm on\ \ \Gamma_{D},\\
+\kappa(\chi^{k})\nabla T^{*}\cdot\mathbf{n}&=0,\ \ &&\rm on\ \ \Gamma_{N}\end{%
+aligned}\right.$ $J^{\tau}(\mathbf{\chi}^{k+1},T^{k+1})\leq J^{\tau}(\mathbf{\chi}^{k},T^{k})$ $\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}>\sigma_{2},\chi^{k}(p)=1\big{%
+\}},$ $\chi^{k+1}\in\mathcal{B}$ $\|\chi^{k+1}-\chi^{k}\|_{2}>tol$ $\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}\in(\sigma_{1},\sigma),\chi^{k}%
+(p)=0\big{\}},$ $\kappa=\kappa_{1}\chi_{\Omega_{1}}+\kappa_{2}\chi_{\Omega_{2}}$ $\displaystyle\ J^{\tau}(\mathbf{\chi}^{k+1},T^{k+1})\leq J^{\tau}(\mathbf{\chi%
+}^{k},T^{k}).$ $\xi=1\times 10^{-5}$ $600\times 600$ $P_{1}(K)$ $T^{k},T^{*k}$ $\int_{\Omega}\chi^{k+1}(\textbf{x})d\textbf{x}=V_{0}$ $J(\chi,T)$ $\int_{\Omega}\chi G_{\tau}\ast(1-\chi)\ d\textbf{x}.$ $n_{s}=N-\lfloor N*\theta^{s}\rfloor$ $\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}\leq\sigma,\chi^{k}(p)=0\big{\}},$ $\kappa_{1}=5,\ 10,\ 20$ $k_{1}/k_{2}$ $\kappa_{1}=20,\ \kappa_{2}=1$ $\chi^{k+1}=\arg\min_{\chi\in B}\tilde{J}^{\tau,k}(\chi).$ $\chi^{k}$ $\chi^{\ast}\in\mathcal{B}$ $\mathcal{H}:=\big{\{}\chi\in BV(\Omega)\ |\ \chi(\textbf{x})\in[0,1],\int_{%
+\Omega}\chi\ d\textbf{x}=V_{0}\big{\}}.$ $\displaystyle=\int_{\Omega}q(\chi)T^{k}\ d\textbf{x}+\frac{\xi}{2}\int_{\Omega%
+}\kappa(\chi)\nabla T^{k}\cdot\nabla T^{k}\ d\textbf{x}+\gamma\sqrt{\frac{\pi}%
+{\tau}}\int_{\Omega}\chi G_{\tau}\ast(1-\chi)\ d\textbf{x}$ $\xi=1\times 10^{-7}$ $\tilde{B}_{2}$ $\displaystyle=(q(\chi_{h}),\varphi_{h}),\ \ \forall\varphi_{h}\in V_{h}^{0},$ $J^{\tau}(\mathbf{\chi}^{k+1},T^{k})\leq J^{\tau}(\mathbf{\chi}^{k},T^{k})$ $q_{1}=1,\ 20,\ 80$ $q(\chi)$ $\displaystyle\chi(\textbf{x}):=\left\{\begin{aligned} &1,\ \ \ \ \textrm{if}\ %
+\textbf{x}\in\ \Omega_{1},\\
+&0,\ \ \ \ \textrm{otherwise}.\end{aligned}\right.$ $\begin{cases}&\min\limits_{\mathbf{\chi}\in\mathcal{B}}J^{\tau}(\mathbf{\chi},%
+T),\\
+&\textrm{s.t.}\ T\in Q(\chi),\end{cases}$ $\tau=1\times 10^{-4}$ $\ \tau>0$ $\chi^{k+1}=\arg\min_{\chi\in\mathcal{H}}\mathcal{L}^{\tau,k}_{r^{k}}(\chi)=%
+\arg\min_{\chi\in\mathcal{H}}\int_{\Omega}\chi\Phi^{k}\ d\textbf{x},$ $\chi_{1}.$ $\displaystyle\chi^{k+1}=\chi_{\tilde{A}_{1}}+\chi^{k}-\chi_{\tilde{B}_{1}}.$ $Q(\chi)$ $\displaystyle\chi^{k+1}=$ $\tilde{J}^{\tau,k}(\chi)\approx\tilde{J}^{\tau,k}(\chi^{k})+\mathcal{L}^{\tau,%
+k}_{\chi^{k}}(\chi-\chi^{k}),$ $\chi^{k+1}$ $\xi=1e-7$ $\displaystyle\tilde{A}_{1}$ $\Phi^{k}=(q_{1}-q_{2})G_{\tau}\ast(T^{k}-T^{*k})+\gamma\sqrt{\frac{\pi}{\tau}}%
+G_{\tau}\ast(1-2\chi^{k})+(k_{1}-k_{2})G_{\tau}\ast(\frac{\xi}{2}\nabla T^{k}%
+\cdot\nabla T^{k}+\nabla T^{k}\cdot\nabla T^{\ast k}).$ $\chi^{k+1}(x)=\begin{cases}1\ \ \textrm{if}\ \Phi^{k}(x)\leq\sigma,\\
+0\ \ \textrm{otherwise,}\end{cases}$ $\begin{cases}\dfrac{\delta(\mathcal{L}^{\tau,k}_{\chi^{k}}(\chi)-\sigma(\int_{%
+\Omega}\chi\ d\textbf{x}-V_{0}))}{\delta\chi}(x)=\Phi^{k}-\sigma\leq 0\ \ \ %
+\textrm{if}\ \ \chi(x)=1,\\
+\dfrac{\delta(\mathcal{L}^{\tau,k}_{\chi^{k}}(\chi)-\sigma(\int_{\Omega}\chi\ %
+d\textbf{x}-V_{0}))}{\delta\chi}(x)=\Phi^{k}-\sigma\geq 0\ \ \ \textrm{if}\ \ %
+\chi(x)=0,\\
+\dfrac{d(\mathcal{L}^{\tau,k}_{\chi^{k}}(\chi)-\sigma(\int_{\Omega}\chi\ d%
+\textbf{x}-V_{0}))}{d\sigma}=\int_{\Omega}\chi\ d\textbf{x}-V_{0}=0.\end{cases}$ $T_{h}\in V_{h}^{0}$ $\displaystyle\left\{\begin{aligned} \nabla\cdot(\kappa\nabla T)+q&=0,\ \ &&\rm
+in%
+\ \Omega,\\
+(\kappa\nabla T)\cdot\mathbf{n}&=0,\ \ &&\rm on\ \Gamma_{N},\\
+T&=0,\ \ &&\rm on\ \Gamma_{D},\\
+|\Omega_{1}|&=\beta|\Omega|,\ \ &&\rm with\ a\ fixed\ parameter\ \beta\in(0,1)%
+,\end{aligned}\right.$ $\beta=0.1,\ 0.2,\ 0.3$ $\int_{\Omega}\chi\ d\textbf{x}$ $\tilde{\phi}_{B}^{k}$ $\displaystyle\int_{\Omega}\chi G_{\tau/2}\ast\left[G_{\tau/2}\ast(1-\chi)%
+\right]\ d\textbf{x}$ $\chi_{\Omega_{i}}$ $\kappa_{1}=40,\ \kappa_{2}=1,\gamma=20$ $\displaystyle\left\{\begin{aligned} -\nabla\cdot(\kappa(\chi)\nabla T)-q(\chi)%
+&=0,\ \ &&\rm in\ \ \Omega,\\
+T&=0,\ \ &&\rm on\ \ \Gamma_{D},\\
+\kappa(\chi)\nabla T\cdot\mathbf{n}&=0,\ \ &&\rm on\ \ \Gamma_{N},\end{aligned%
+}\right.$ $\chi_{h}\in\mathcal{B}_{h}$ $\displaystyle J^{\tau}(\mathbf{\chi}^{k},T^{k})=$ $\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}\in(\sigma,\sigma_{2}),\chi^{k}%
+(p)=1\big{\}},$ $\gamma=12,\ 15,\ 50$ $|\Gamma|\approx\sqrt{\frac{\pi}{\tau}}\int_{\Omega}\chi G_{\tau}*(1-\chi)\ d%
+\textbf{x},$ $J^{\tau}(\mathbf{\chi}^{k},T^{k})$ $\tilde{A}\subset A$ $q=q_{1}\chi_{\Omega_{1}}+q_{2}\chi_{\Omega_{2}}$ $J^{\tau}(\chi^{k+1},T^{k+1})\leq J^{\tau}(\chi^{k},T^{k}).$ $J^{\tau}(\tilde{\chi}_{s},\tilde{T}_{s})$ $\displaystyle\tilde{B}_{2}$ $q_{1}/q_{2}$ $\chi^{k+1}=\arg\min_{\chi\in\mathcal{\mathcal{H}}}\tilde{J}^{\tau,k}(\chi),$ $\tilde{J}^{\tau}(\chi,T^{k},T^{*k})$ $BV(\Omega)$ $G_{\tau}\ast$ $\gamma=15$ $\kappa_{1}=10,\ \kappa_{2}=1$ $60\times 60\times 60$ $\kappa_{1},\ \kappa_{2},\ q_{1},\ q_{2},\ \tau,\ \gamma,\ \xi$ $\frac{\kappa_{1}}{\kappa_{2}}$ $\displaystyle q(\chi)=q_{1}G_{\tau}\ast\chi+q_{2}G_{\tau}\ast(1-\chi).$ ${0}\times[0.45,0.55]$ $\tilde{J}^{\tau,k}(\chi)$ $\displaystyle=\int_{\Omega}\chi\Phi^{k}\ d\textbf{x},$ $\mathcal{B}:=\big{\{}\chi\in BV(\Omega)\ |\ \chi(\textbf{x})=\{0,1\},\int_{%
+\Omega}\chi\ d\textbf{x}=V_{0}\big{\}},$ $\displaystyle=(q(\chi_{h}),\varphi_{h})+\xi(\kappa(\chi_{h})\nabla T_{h},%
+\nabla\varphi_{h}),\ \ \forall\varphi_{h}\in V_{h}^{0}.$ $\mathbf{\chi}$ $\displaystyle V_{h}=\{v\in H^{1}(\Omega)\ |\ v\in P_{1}(K),\forall K\in%
+\mathcal{T}_{h}\},\quad V_{h}^{0}=V_{h}\cap H^{1}_{\Gamma_{D}}(\Omega),$ $\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}>\sigma,\chi^{k}(p)=1\big{\}},$ $\displaystyle(\kappa(\chi_{h})\nabla T_{h},\nabla\varphi_{h})$ $\tau=0.35\times 10^{-4}$ $\displaystyle\left\{\begin{aligned} \nabla\cdot(\kappa(\chi)\nabla T^{*})&=q(%
+\chi)-\xi(\nabla\cdot(\kappa(\chi)\nabla T)),\ \ &&\rm in\ \ \Omega,\\
+T^{*}&=0,\ \ &&\rm on\ \ \Gamma_{D},\\
+\kappa(\chi)\nabla T^{*}\cdot\mathbf{n}&=0,\ \ &&\rm on\ \ \Gamma_{N}.\end{%
+aligned}\right.$ $\displaystyle-(\kappa(\chi_{h})\nabla T^{\ast}_{h},\nabla\varphi_{h})$ $\sigma_{2}=\tilde{\phi}_{B}^{k}(n_{s})$ $\chi^{1},\chi^{2},\cdots,\chi^{k+1},\cdots$ $\chi^{k},~{}T^{k},~{}T^{*k}$ $q_{1}=1,\ 40,\ 80$ $\displaystyle\int_{\Omega}\left[G_{\tau/2}\ast\chi\right]\left[1-G_{\tau/2}%
+\ast\chi\right]\ d\textbf{x},$ $G_{\tau/2}\ast$ $Proj_{[0,1]}(v)=\begin{cases}v&{\rm if}\ v\in[0,1],\\
+0&{\rm if}\ v<0,\\
+1&{\rm if}\ v>1.\end{cases}$ $\displaystyle\arg\min_{\chi\in\mathcal{B}}\tilde{J}^{\tau,k}(\chi)$ $\kappa_{1}=40$ $\Omega_{2}=\Omega\setminus\Omega_{1}\in\Omega$ $\kappa_{1}/\kappa_{2}$ $V_{0}=\beta|\Omega|$ $J^{\tau}(\mathbf{\chi}^{k},T)$ $\min_{(\Omega_{1},T)}J(\Omega_{1},T)=\int_{\Omega}qTd\textbf{x}+\gamma|\Gamma|,$ $\displaystyle=\mathbb{E}_{q(Z_{1/N:1}|Z_{0})}\left[\log\frac{q(Z_{1}|Z_{0})}{p%
+_{v_{\theta}}(Z_{1})p_{v_{\theta}}(Z_{0}|Z_{1/N})}\right]+\sum_{i=1}^{N}%
+\mathbb{E}_{\hat{Z}_{i/N}\sim\int p_{v_{\theta}}(Z_{i/N}|Z_{1})q(Z_{1}|Z_{0})%
+dZ_{1}}$ $Z_{7/8}$ $\displaystyle\operatorname*{arg\,min}_{v_{\theta}}\mathbb{E}_{(Z_{1},Z_{0},t)}%
+||g_{v_{\theta}}(Z_{t},t)-(\alpha_{t}[\log(b)-\log(a)]X^{S})||_{2}^{2}.$ $\alpha_{1}X^{S}$ $\forall Z_{1-i/K}$ $q(\cdot|\cdot)$ $\displaystyle\mathcal{L}_{\textrm{OFM-KT}}=\mathbb{E}_{(X^{S},Y)}[\frac{1}{N}%
+\sum_{i=0}^{N-1}$ $\min L(g^{S}(X^{S}),g^{T}(X^{T}))$ $\displaystyle\mathcal{L}_{\textrm{FM-KT}}=\mathbb{E}_{(X^{S},X^{T})}\frac{1}{N%
+}\sum_{i=1}^{N}||\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}-g_{v_{\theta}}(Z_{1-i/%
+N},1-i/N))/-\nabla_{t}\sigma_{t})-X^{T}||_{2}^{2},$ $p_{v_{\theta}}(\cdot|\cdot)$ $\displaystyle=\operatorname*{arg\,min}_{v_{\theta}}\mathbb{E}_{(Z_{1},Z_{0},t)%
+}||g_{v_{\theta}}(Z_{t},t)-(\nabla_{t}\alpha_{t}X^{S}+\nabla_{t}\sigma_{t}X^{T%
+})||_{2}^{2}.$ $\left\lfloor{\beta_{d}}B\right\rfloor$ $\displaystyle Z_{t}=\alpha_{t}X^{S}+\sigma_{t}X^{T},\ s.t.\ \lim_{t\rightarrow
+0%
+}\alpha_{t}=0,\lim_{t\rightarrow 0}\sigma_{t}=1,\lim_{t\rightarrow 1}\sigma_{t%
+}=0.$ $\displaystyle\operatorname*{arg\,min}_{v_{\theta}}\mathbb{E}_{(Z_{1},Z_{0},t)}%
+||g_{v_{\theta}}(Z_{t},t)-((\frac{1}{2}a(1-t)+\frac{1}{2}b)\alpha_{t}X^{S}-%
+\frac{\alpha_{t}}{\sqrt{1-\alpha_{t}^{2}}}\alpha_{t}(\frac{1}{2}a(1-t)+\frac{1%
+}{2}b)X^{T})||_{2}^{2}.$ $g^{S}(\cdot)$ $\displaystyle+\underbrace{L(\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}-g_{v_{%
+\theta}}(Z_{1-i/N},1-i/N))/-\nabla_{t}\sigma_{t}),Y)}_{\textrm{match the %
+ground truth label}}].$ $\{X_{t}\}_{t}$ $\displaystyle=\mathcal{E}(Z_{1-i/K})[1-(1/K)\psi(1-i/K)]+(1/K)\mathcal{K}(1-i/%
+K).$ $Z_{1-i/N}=Z_{1-(i-1)/N}-g_{v_{\theta}}(Z_{1-(i-1)/N},1-(i-1)/N)/N$ $\mathcal{E}(Z_{1-i/K})$ $\hat{Z}_{1}\sim\pi_{1}$ $\displaystyle\mathcal{L}_{\textrm{FM-KT}^{\Theta}}=$ $Z_{1}=\alpha_{1}X^{S}$ $\frac{\sigma_{t}-\sigma_{t-\Delta t}}{t-\Delta t}$ $\displaystyle-\log p_{v_{\theta}}(Z_{0})\leq\mathbb{E}_{q(Z_{1/N:1}|Z_{0})}%
+\left[\log\frac{q(Z_{1}|Z_{0})}{p_{v_{\theta}}(Z_{1})p_{v_{\theta}}(Z_{0}|Z_{1%
+/N})}+\sum_{i=1}^{N}\log\frac{q(Z_{(i-1)/N}|Z_{i/N},Z_{0})}{p_{v_{\theta}}(Z_{%
+(i-1)/N}|Z_{i/N})}\right].$ $\sigma_{t}\in\mathbb{R}^{+}$ ${}_{\textrm{L}}$ $\displaystyle\left[||q(Z_{(i-1)/N}|Z_{i/N},Z_{0})-p_{v_{\theta}}(Z_{(i-1)/N}|%
+\hat{Z}_{i/N})||_{2}^{2}\right],\quad s.t.\quad\textrm{Law}(Z_{i/N})\stackrel{%
+{\scriptstyle\sim}}{{=}}\textrm{Law}(\hat{Z}_{i/N}).$ $\mathcal{L}_{\textrm{FM-KT}}$ $X_{t}=tX^{S}+(1-t)X^{T}$ $\displaystyle Z_{1-(i+1)/K}$ $(X^{S},X^{T},Y)$ $\mathit{a=0.02}$ $\displaystyle\quad\quad-g_{v_{\theta}}(Z_{1-(i-1)/N},1-(i-1)/N)/N,\quad s.t.%
+\quad i\geq 1.$ $\displaystyle\mathcal{E}(Z_{1-2/K})=\mathcal{E}(Z_{1-1/K})(1-(1/K)\psi(1-1/K))%
++(1/K)\mathcal{K}(1-1/K)$ $\{Z_{1-i/K}\}_{i=1}^{K}$ $Z_{i/N}$ $\displaystyle,X^{T})+\underbrace{L(\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}-g_{v%
+_{\theta}}(Z_{1-i/N},1-i/N))/-\nabla_{t}\sigma_{t}),Y)}_{\textrm{match the %
+ground truth label (optional)}}],$ $\frac{d\hat{Z}_{t}}{dt}=-g^{*}_{v_{\theta}}(\hat{Z}_{t},t)$ $Z_{1-(i-1)/K}$ $\sigma_{t}=1,\ s.t.\quad a=0.02,b=100$ $\{dZ_{t-\Delta_{t}},dZ_{t-2\Delta_{t}},\cdots,dZ_{s}\}$ $\mathit{b=0.1}$ $\textrm{Law}(Z_{i/N})\stackrel{{\scriptstyle\sim}}{{=}}\textrm{Law}(\hat{Z}_{i%
+/N})$ $g_{v_{\theta}}(Z_{t},t)$ $\displaystyle\mathcal{E}(Z_{0})=(1/K)\left[\sum_{i=0}^{K-1}\mathcal{K}(1-i/K)%
+\right]+(1/K^{2})\left[\sum_{j=1}^{K-1}\left[\psi(1-j/K)\left(\sum_{i=0}^{j-1}%
+\mathcal{K}(1-i/K)\right)\right]\right]+\mathcal{O}(1/K^{3}).$ $\displaystyle\textrm{the sampling process:}\quad Z_{1-i/N}=Z_{1-(i-1)/N}-g_{v_%
+{\theta}}(Z_{1-(i-1)/N},1-(i-1)/N)/N,\quad s.t.\quad i\geq 1,$ $\mathbb{E}_{\hat{Z}_{i/N},Z_{1},Z_{0}}D_{\mathrm{KL}}(q(Z_{(i-1)/N}|Z_{i/N},Z_%
+{0})||p_{v_{\theta}}(Z_{(i-1)/N}|\hat{Z}_{i/N}))$ $Z_{1-i/K}=X_{1-i/K}+\mathcal{E}(Z_{1-i/K})$ $Z_{1-i/K}$ $Z_{1-i/N}\!=\!Z_{1\!-\!(i\!-\!1)/N}\!-\!g_{v_{\theta}}(Z_{1\!-\!(i\!-\!1)/N},1%
+\!-\!(i\!-\!1)/N)/N$ $\displaystyle=Z_{1-i/K}-(1/K)g_{v_{\theta}}(X_{1-i/K}+\mathcal{E}(Z_{1-i/K}),1%
+-i/K)$ ${}_{\textrm{M}}$ $(X^{S},X^{T})\in(\mathbb{R}^{d},\mathbb{R}^{d})$ $\displaystyle\approx X_{1-i/K}+\mathcal{E}(Z_{1-i/K})-(1/K)\left[g_{v_{\theta}%
+}(X_{1-i/K},1-i/K)+\mathcal{E}(Z_{1-i/K})\nabla_{X_{t}}g_{v_{\theta}}(X_{1-i/K%
+},1-i/K)\right],$ $X^{S}$ $\log p_{v_{\theta}}(Z_{0})\geq-\mathbb{E}_{q(Z_{1/N:1}|Z_{0})}[\log\frac{q(Z_{%
+1}|Z_{0})}{p_{v_{\theta}}(Z_{1})p_{v_{\theta}}(Z_{0}|Z_{1/N})}+\sum_{i=1}^{N}%
+\log\frac{q(Z_{(i-1)/N}|Z_{i/N},Z_{0})}{p_{v_{\theta}}(Z_{(i-1)/N}|Z_{i/N})}]$ $\displaystyle\mathbb{E}[L(\mathcal{T}_{\textrm{vanilla}}(X^{S}),\mathcal{T}(Z_%
+{0}))+\alpha^{\Theta}L(\mathcal{T}_{\textrm{vanilla}}(X^{S}),Y)]+\mathcal{L}_{%
+\textrm{FM-KT}},$ $\mathcal{L}_{\textrm{OFM-KT}}$ $B\!-\!\left\lfloor{\beta_{d}}B\right\rfloor$ $\lim_{t\rightarrow 1}\nabla_{t}\alpha_{t}=+\infty$ $p_{v_{\theta}}(\cdot|Z_{i/N})$ $X^{T}\left[0:B\!-\!\left\lfloor{\beta_{d}}B\right\rfloor\right]$ $X^{T}\left[0:B\!-\!\left\lfloor{\beta_{d}}B\right\rfloor\right]=\textbf{{%
+shuffle}}\left(X^{T}\left[0:B\!-\!\left\lfloor{\beta_{d}}B\right\rfloor\right]%
+\right).$ $\mathcal{E}(Z_{1-(i+1)/K})=\mathcal{E}(Z_{1-i/K})[1-(1/K)\psi(1-i/K)]+(1/K)%
+\mathcal{K}(1-i/K)$ $\left[\frac{dX_{t}}{dt}-g_{v_{\theta}}(\mathcal{H}(t),t)\right]$ $Z_{1}=\alpha_{1}X_{S}$ $\mathit{b=100}$ $Z_{t}+\int_{t}^{s}g_{v_{\theta}}(Z_{\tau},\tau)d\tau$ $\displaystyle-\log p_{v_{\theta}}(Z_{0})\leq\mathbb{E}_{q(Z_{1/N:1}|Z_{0})}%
+\left[\log\frac{q(Z_{1}|Z_{0})}{p_{v_{\theta}}(Z_{1})p_{v_{\theta}}(Z_{0}|Z_{1%
+/N})}+\sum_{i=1}^{N}\log\frac{q(Z_{(i-1)/N}|Z_{i/N},Z_{0})}{p_{v_{\theta}}(Z_{%
+(i-1)/N}|Z_{i/N})}\right]$ $\{\hat{Z}_{i/N}\}_{i}$ $\mathcal{K}(t)$ $\displaystyle\operatorname*{arg\,min}_{v_{\theta}}\mathbb{E}_{(Z_{1},Z_{0},t)}%
+||g_{v_{\theta}}(Z_{t},t)-\nabla_{t}Z_{t}||_{2}^{2}$ $\nabla_{t}\sigma_{t}$ $\nabla_{t}\sigma_{t}\equiv 0$ $\displaystyle=Z_{1-i/K}-(1/K)g_{v_{\theta}}(Z_{1-i/K},1-i/K)$ $g^{T}(\cdot)$ ${}_{\textrm{50}}$ $\displaystyle\operatorname*{arg\,min}_{v_{\theta}}\mathbb{E}_{(Z_{1},Z_{0},t)}%
+||g_{v_{\theta}}(Z_{t},t)-(X^{S}-X^{T})||_{2}^{2}.$ $\rho_{t}(Z):\mathbb{R}^{d}\times[0,1]\rightarrow\mathbb{R}^{d}$ $\psi(t)=\nabla_{X_{t}}g_{v_{\theta}}(X_{t},t)$ $\textrm{Law}(Z_{1})\stackrel{{\scriptstyle\sim}}{{=}}\textrm{Law}(\hat{Z}_{1})$ $\hat{Z}_{0}\sim\pi_{0}$ $B-\left\lfloor{\beta_{d}}B\right\rfloor$ $\alpha_{t}=a(\frac{b}{a})^{t}$ $\displaystyle\approx\mathbb{E}_{q(Z_{1/N:1}|Z_{0})}\left[\log\frac{q(Z_{1}|Z_{%
+0})}{p_{v_{\theta}}(Z_{1})p_{v_{\theta}}(Z_{0}|Z_{1/N})}\right]+\sum_{i=1}^{N}%
+\mathbb{E}_{\hat{Z}_{i/N}\sim\int p_{v_{\theta}}(Z_{i/N}|Z_{1})q(Z_{1}|Z_{0})%
+dZ_{1}}$ $\displaystyle\mathcal{L}_{\textrm{FM-KT++}}=\mathbb{E}_{(X^{S},X^{T},Y)}\frac{%
+1}{N}\sum_{i=0}^{N-1}L(\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}-g_{v_{\theta}}(Z%
+_{1-i/N},1-i/N))/-\nabla_{t}\sigma_{t}),X^{T})$ $\{Z_{1-i/N}\}_{i}$ $\displaystyle=X_{1-i/K}+\mathcal{E}(Z_{1-i/K})-(1/K)g_{v_{\theta}}(X_{1-i/K}+%
+\mathcal{E}(Z_{1-i/K}),1-i/K)$ $\mathcal{K}(t)\geq 0$ $\mathcal{L}_{\textrm{guided}}$ $\displaystyle\left[D_{\mathrm{KL}}(q(Z_{(i-1)/N}|Z_{i/N},Z_{0})||p_{v_{\theta}%
+}(Z_{(i-1)/N}|\hat{Z}_{i/N}))\right],\quad s.t.\quad\textrm{Law}(Z_{i/N})%
+\stackrel{{\scriptstyle\sim}}{{=}}\textrm{Law}(\hat{Z}_{i/N})$ $\displaystyle\quad\underbrace{L(\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}-g_{v_{%
+\theta}}(Z_{1-i/N},1-i/N))/-\nabla_{t}\sigma_{t}),Z_{0})}_{\textrm{the Online %
+KD loss}}$ $\displaystyle\approx X_{1-(i+1)/K}+\mathcal{E}(Z_{1-i/K})+(1/K)\mathcal{K}(1-i%
+/K)-(1/K)\mathcal{E}(Z_{1-i/K})\psi(1-i/K)$ ${}_{\textrm{S}}$ $\displaystyle Z_{1-(i+1)/K}-X_{1-(i+1)/K}$ $Z_{1-(i+1)/K}\!=\!Z_{1-i/K}-g_{v_{\theta}}(Z_{1-{i/K}},1-i/K)dt$ $\frac{\alpha_{t}-\alpha_{t-\Delta t}}{t-\Delta t}$ $\nabla_{t}\alpha_{t}$ $\mathcal{E}(Z_{1-i/K})\geq 0$ $X^{T}\in\mathbb{R}^{B\times C\times H\times W}$ $\displaystyle\operatorname*{arg\,min}_{v_{\theta}}$ $g^{*}_{v_{\theta}}(\cdot)$ $dZ_{t}$ $i\!\geq\!1$ $\{\textrm{Law}(Z_{i/N})\stackrel{{\scriptstyle\sim}}{{=}}\textrm{Law}(\hat{Z}_%
+{i/N})\}_{i=0}^{N-1}$ $\displaystyle=X_{1-(i+1)/K}+\mathcal{E}(Z_{1-i/K})[1-(1/K)\psi(1-i/K)]+(1/K)%
+\mathcal{K}(1-i/K)$ $\sigma(t)=1-0.1t$ $\rho_{0}(X^{T})=\rho_{1}(X^{S})+\int_{\rho_{1}(Z)}^{\rho_{0}(Z)}\partial\rho_{%
+t}(Z)$ $L(\cdot,\cdot)$ $\displaystyle\int_{0}^{1}\mathbb{E}[||\partial\rho_{t}(Z)/\partial t-g_{v_{%
+\theta}}(Z_{t},t)||]dt.$ $g_{v_{\theta}}(\cdot)$ ${}_{\textrm{75}}$ $\hat{Z}_{0}$ ${Z}_{i/N}$ $\displaystyle=\mathbb{E}_{q(Z_{1/N:1}|Z_{0})}\left[\log\frac{q(Z_{1}|Z_{0})}{p%
+_{v_{\theta}}(Z_{1})p_{v_{\theta}}(Z_{0}|Z_{1/N})}\right]+\sum_{i=1}^{N}%
+\mathbb{E}_{q(Z_{i/N}|Z_{0})}\mathbb{E}_{q(Z_{(i-1)/N}|Z_{i/N},Z_{0})}\left[%
+\log\frac{q(Z_{(i-1)/N}|Z_{i/N},Z_{0})}{p_{v_{\theta}}(Z_{(i-1)/N}|Z_{i/N})}\right]$ $\displaystyle\mathcal{L}_{\textrm{FM-KT}}\!=\!\mathbb{E}[\frac{1}{N}\sum_{i=0}%
+^{N-1}\!L(\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}\!\!-\!\!g_{v_{\theta}}(Z_{1\!%
+-\!i/N},1-i/N))/\!-\!\nabla_{t}\sigma_{t})$ $\sigma_{t}=\sqrt{1-\alpha_{t}^{2}},\ s..t.\quad a=19.9,b=0.1$ $\displaystyle=X_{1-i/K}+\mathcal{E}(Z_{1-i/K})-(1/K)\left[g_{v_{\theta}}(X_{1-%
+i/K},1-i/K)+\mathcal{E}(Z_{1-i/K})\psi(1-i/K)\right],$ $\hat{Z}_{i/N}$ $\alpha^{\Theta}$ $\textrm{Law}(Z_{(i-1)/N})\stackrel{{\scriptstyle\sim}}{{=}}\textrm{Law}(\hat{Z%
+}_{(i-1)/N})$ $\displaystyle+\underbrace{L(\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}-g_{v_{%
+\theta}}(Z_{1-i/N},1-i/N))/-\nabla_{t}\sigma_{t}),Y)}_{\textrm{match the %
+ground truth label (optional)}},$ $\alpha_{t}=\textrm{exp}(-\frac{1}{4}a(1-t)^{2}-\frac{1}{2}b(1-t))$ $\lim_{t\rightarrow 1}\alpha_{t}=1$ $\{Z_{t}\}_{t}$ $\sigma(t)=1$ $X^{S}-X^{T}=\frac{dX_{t}}{dt}$ $\displaystyle\mathcal{E}(Z_{1-1/K})=(1/K)\mathcal{K}(1)$ $\sigma_{t}=\sqrt{1-\alpha_{t}^{2}}$ $\mathit{a=19.9}$ $\displaystyle\textrm{where}\quad Z_{1-i/N}=Z_{1-(i-1)/N}$ $\mathcal{H}(t)=\operatorname*{arg\,sup}_{X_{t}}\{||\frac{dX_{t}}{dt}-g_{v_{%
+\theta}}(X_{t},t)||_{2}^{2}\}$ $||\frac{dX_{t}}{dt}-g_{v_{\theta}}(X_{t},t)||_{2}^{2}$ $t\sim\mathcal{U}[0,1]$ $\mathcal{T}_{\textrm{vanilla}}(\cdot)$ $\sim 10^{6}\times 10^{4}=10^{10}$ $\sim 1\,\mathrm{KB}$ $\displaystyle-~{}NCC(\mathcal{X}_{\mathrm{fx}},\mathcal{X}_{\mathrm{wp,n}})+%
+\lambda\sum_{p\in\Omega}||\nabla\varphi(p)||^{2}$ $\varphi=\sum_{i=0}^{n}\varphi_{i}$ $\mathcal{X}_{\mathrm{wp}}$ $w_{k,i}=|m(i\in\omega)|^{g}$ $\mathcal{X}_{\mathrm{mv}}$ $0.7\times 0.7\times 3.0$ $\mathcal{X}_{\mathrm{fx}}$ $0.837\pm 0.021$ $1.2\times 1.2\times 3.0$ $\displaystyle\begin{split}\mathcal{X}_{\mathrm{wp,n}}=\mathcal{X}_{\mathrm{mv}%
+}\circ\sum_{i=0}^{n}\varphi_{i}.\end{split}$ $\varphi:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ $\mathcal{X}_{\mathrm{wp,1}}$ $\mathcal{X}_{\mathrm{wp,n}}$ $0.926\pm 0.012$ $0.847\pm 0.008$ $\centering\mathcal{X}_{\mathrm{wp}}=\mathcal{X}_{\mathrm{mv}}\circ\varphi%
+\approx\mathcal{X}_{\mathrm{fx}}.\@add@centering$ $\mathcal{L}_{\mathrm{sim}}$ $0.866\pm 0.020$ $0.915\pm 0.012$ $230\times 230$ $\mathcal{X}_{\mathrm{wp,0}}$ $0.811\pm 0.023$ $0.807\pm 0.23$ $290\times 250$ $lr_{\mathrm{epoch}}=3\cdot 10^{-4}\cdot e^{-3\mathrm{epoch}/500}$ $H\times W\times L$ $0.920\pm 0.014$ $\%|J_{\phi}|\leq 0$ $\%|J_{\varphi}|<0$ $\mathcal{L}_{\mathrm{smooth}}$ $\displaystyle~{}\mathcal{L}_{\mathrm{sim}}+\lambda\mathcal{L}_{\mathrm{smooth}}$ $0.866\pm 0.013$ $94.65\%$ $\langle search\rangle$ $P(\hat{a}=\langle search\rangle|\theta^{\prime},q)$ $LM_{\theta^{\prime}}$ $LM_{\theta^{\prime}}:Q\mapsto\Omega\cup\{\langle search\rangle\}$ $LM_{\theta}:Q\mapsto\Omega$ $\psi\circ LM_{\theta}:Q\mapsto\Omega\cup\{\langle search\rangle\}$ $alpha=32$ $\underset{\theta}{\text{argmin}}\prod_{q\in Q}\left[P(\hat{a}=\langle search%
+\rangle|\theta^{\prime},q)+\lambda P(\hat{a}\notin A|\theta^{\prime},q)\right]$ $(\langle search\rangle)$ $7e-5$ $LM_{\theta}$ $\psi(LM_{\theta}(q))=\begin{cases}\mathds{1}(\hat{a}),&\text{if }\hat{a}\in A%
+\\
+\langle search\rangle,&\text{otherwise}\end{cases}$ $P(\hat{a}\notin A|\theta^{\prime},q)$ $733.5$ $TR^{(1)},TR^{(2)},\cdots,TR^{(m)}$ $\mathcal{V}_{\mathrm{target}}$ $\hat{P}_{v}$ $\mathcal{V}_{target}$ $\operatorname*{arg\,min}_{\theta}\sum_{G}\mathcal{L}(G,\theta).$ $p^{(1)},p^{(2)},\cdots,p^{(m)}$ $q^{(1)},q^{(2)},\cdots,q^{(n)}$ $\hat{\mathcal{V}}_{target}$ $\in TR^{i}$ $\displaystyle-\frac{1}{|\mathcal{V}|}\sum_{v\notin\mathcal{V}_{\textrm{target}%
+}}\log P(\hat{P}_{v}=0|\mathcal{G},\theta)$ $\displaystyle\mathcal{L}(G,\theta)=$ $\displaystyle-\frac{1}{|\mathcal{V}|}\sum_{v\in\mathcal{V}_{\textrm{target}}}%
+\log P(\hat{P}_{v}=1|\mathcal{G},\theta)$ $1,2,\cdots,m$ $TR^{i}$ $N=1923$ $p^{(1)}_{ref},p^{(2)}_{ref},\cdots,p^{(m)}_{ref}$ $\mathcal{V}_{\mathrm{target}}\subset\mathcal{V}$ $[0.1,0.01,0.001,0.0001]$ $\mathrm{EE}_{y}$ $\left[\mathcal{D}_{t},\mathcal{D}_{r}\right]$ $\left[\mathcal{D}_{r},\mathcal{H}_{r}\right]$ $W=F\cdot S$ $\mathcal{L}_{GPs}=\mathcal{L}_{G}+w_{P}\mathcal{L}_{P}+w_{S}\mathcal{L}_{s}$ $\eta_{t}>6$ $\left[\mathcal{D}_{t},\mathcal{H}_{t}\right]$ $\eta_{min}<6$ $X_{syn}$ $\displaystyle l_{3}+l_{2} $\delta S_{out}$ $F\cdot S=\tau\cdot\theta$ $\mathrm{EE}_{x}$ $\left[\mathcal{D}_{t},\mathcal{H}_{t}\right]=\left[\{d_{t,1},d_{t,2},...,d_{t,%
+i}\},\{\eta_{t,1},\eta_{t,2},...,\eta_{t,i}\}\right]$ $\mathcal{L}_{P}=\frac{1}{N}\sum^{N}_{i=1}\left[(d_{r,i}-d_{t,i})^{2}+(\eta_{r,%
+i}-\eta_{t,i})^{2}\right]$ $\left[\mathcal{H}_{t},\mathcal{H}_{r}\right]$ $|F_{in}\cdot\delta S_{in}|=|F_{out}\cdot\delta S_{out}|$ $\eta=\left|\frac{\delta S_{out}}{\delta\theta_{in}}\right|$ $(\mathrm{x}_{i},\mathrm{x}_{j})$ $i=100$ $d_{t}=1.0,\eta_{t}=2.0$ $d_{r},\eta_{r}$ $\delta\theta_{in}$ $|\tau_{in}\cdot\delta\theta_{in}|=|F_{out}\cdot\delta S_{out}|$ $\mathcal{L}_{s}=-\frac{1}{N}\sum^{N}_{i=1}\min{D\left(\mathrm{x}_{i},\mathrm{x%
+}_{j}\right)}$ $|W_{in}|=|W_{out}|$ $\left[\mathcal{D}_{r},\mathcal{H}_{r}\right]=\left[\{d_{r,1},d_{r,2},...,d_{r,%
+i}\},\{\eta_{r,1},\eta_{r,2},...,\eta_{r,i}\}\right]$ $y\left[i,j\right]=\sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}h\left[m,n%
+\right]\cdot x\left[i-m,j-n\right]$ $0-\infty$ $\mathcal{M}=\{M_{i}|0\leq i $\mathcal{M}=\{M_{i|0\leq i $\{1e^{-5},2e^{-5}\}$ $M=10,50,70,100$ $R_{t+1,A_{t},B_{t}}$ $\displaystyle\mathbb{E}_{0}\left[\sqrt{\sum_{a}\left(\tilde{C}^{+}_{T,i}\right%
+)^{2}}\right]\leqslant\mathbb{E}_{0}\left[\sqrt{\sum_{t=1}^{T}\sum_{a}\left(%
+\tilde{\Re}_{t,a}\right)^{2}}\right]\leqslant\sqrt{\sum_{t=1}^{T}\frac{2(1+%
+\sigma_{w}^{2})\left|\mathcal{A}\right|^{2}}{\gamma_{t}}}$ $\displaystyle w_{t}(a,b)^{2}=\beta_{t}I_{t}(\theta;R_{t+1,A_{t},B_{t}}\>|\>A_{%
+t}=a,B_{t}=b)=\frac{\beta^{\prime}_{t}\log(1+\sigma_{w}^{-2}\sigma^{2}_{t}(a,b%
+))}{\log(1+\sigma_{w}^{-2})}\geqslant\beta^{\prime}_{t}\sigma^{2}_{t}(a,b).$ $\displaystyle+C\sum_{t=0}^{T}\mathbb{P}(\neg\mathcal{E}^{c}_{t}(\tilde{R},U^{%
+\prime},A_{t},B_{t})\cup\neg\mathcal{E}^{c}_{t}(f_{\theta},B_{t}))$ $\mathbb{P}(\max_{i}y_{ti}\leqslant\frac{t}{t+\sigma_{n}^{2}}(1-\Delta)+\sqrt{%
+\frac{2\sigma_{n}^{2}\log(M/\delta_{1})}{t+\sigma_{n}^{2}}})\geqslant 1-\delta%
+_{1}$ $\displaystyle\Re_{\operatorname{full}}(a;T,\operatorname{adv},\tilde{R})$ $\displaystyle\leqslant\sum_{a}\left({\tilde{C}_{T-1,a}^{+}}\right)^{2}+\sum_{i%
+}(\tilde{\Re}_{T,a})^{2}$ $\displaystyle{\tilde{C}_{T,a}}={\tilde{C}_{T-1,a}}+\tilde{\Re}_{T,a},$ $\displaystyle\mathbb{E}\left[\max_{a\in\mathcal{A}}\sum_{t=0}^{T-1}\mathbb{E}%
+\left[\operatorname{pess}_{t+1}(a)\mid\theta\right]\right]$ $\pi=\pi^{\operatorname{adv-est}}$ $\phi(a,b)=P_{a}(\theta)/(P_{n_{0}}+P_{b}\mathds{1}(a=b))$ $\Re_{full}(T,\operatorname{adv},\tilde{R})$ $\displaystyle\mathbf{k}_{t}(a,b)$ $\Re_{t1}$ $\displaystyle\mathbb{P}(\tilde{R}_{t1}\geqslant\tilde{R}_{t2})$ $\displaystyle\Re_{\operatorname{full}}(T,{\operatorname{adv}},(r_{t})_{t})=%
+\max_{a\in\mathcal{A}}\mathbb{E}\left[\sum_{t=0}^{T-1}r_{t}(a)-r_{t}(A_{t})\right]$ $c(\Delta,\sigma_{n})>0$ $GP(0,k((a,b),(a^{\prime},b^{\prime})))$ $X_{2}=X_{1}$ $\mathcal{N}(0.5,2.0)$ $r_{t}(a)=f_{\theta}(a,B_{t})$ $\tilde{f}_{t+1}(a,b)$ $U=(\mu_{t}(a,b)+\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b):t\in\mathbb{N}).$ $\displaystyle\geqslant\sum_{t=1}^{\infty}\log\mathbb{P}(N_{t}\leqslant 0)$ $\sqrt{2\log\mathcal{A}\mathcal{B}\sqrt{T}}$ $\displaystyle\Re_{t}(a)-\mathbb{E}_{t}\left[\tilde{\Re}_{t,a}\right]$ $\displaystyle\mathbf{R}_{t}$ $\displaystyle=\mathbb{E}\left[\sum_{t=0}^{T-1}\tilde{R}_{t+1}(a)-\tilde{R}_{t+%
+1}(A_{t})\right].$ $f_{1},f_{2},f_{3}\in\mathcal{F}$ $\displaystyle\mathbb{P}(\omega_{t}|\Omega_{t-1})$ $\frac{x}{\log(1+x)}$ $X_{2}=[0,1]$ $\displaystyle\sum_{t=0}^{T-1}\mathbb{E}\left[f_{\theta}(a,B_{t})-f_{\theta}(A_%
+{t},B_{t})\>|\>\theta\right]$ $\pi=\pi^{\operatorname{adv-OTS}}$ $r^{i}(a^{i},a^{-i})$ $\displaystyle\geqslant(1-\delta_{1})\mathbb{P}(\max_{i}x_{ti}\geqslant\frac{t}%
+{t+\sigma_{n}^{2}}(1-\Delta)+\sqrt{\frac{2\sigma_{n}^{2}\log(M/\delta_{1})}{t+%
+\sigma_{n}^{2}}}\mid\epsilon)$ $\begin{cases}x_{ti}&\sim\mathcal{N}_{i}(0,1),\quad i=1,\ldots,M\\
+y_{ti}&\sim\mathcal{N}_{i}\left(\frac{t}{t+\sigma_{n}^{2}}(1-\Delta),\frac{%
+\sigma_{n}^{2}}{\sigma_{n}^{2}+t}\right),\quad i=1,\ldots,M\end{cases}$ $(10^{-1})$ $\displaystyle=\mathbb{E}\left[\tilde{R}_{t+1}(a)-\tilde{R}_{t+1}(A_{t})\mid%
+\theta\right],$ $\neg{\mathcal{E}}$ $N(\mu_{t}(a,b),\sigma_{t}(a,b))$ $f_{\theta}(a,b)$ $\tilde{R}=(\tilde{R}_{1},\ldots,\tilde{R}_{t+1},\ldots)$ $U=(U_{t}\>|\>t\in\mathbb{N})$ $A^{*}=\operatorname*{arg\,max}_{a\in\mathcal{A}}\sum_{t=0}^{T-1}\mathbb{E}%
+\left[R_{t+1,a,B_{t}}\>|\>\theta\right]$ $reg_{1}=\tilde{R}_{1}-\tilde{R}_{1}^{T}X_{1}\cdot\mathbf{1},$ $\mathcal{O}(d\log T)$ $(a^{i},a^{-i})$ $[a^{i}]_{e}=U^{i}$ $0.15(\sum_{i=1}^{N}[a^{i}]_{e}/C_{e})^{4}$ $\displaystyle=\mathbb{P}(\sum_{k=2}^{t}m_{k}\leqslant 0|\Omega_{t-1})\left(%
+\mathbb{P}(\Omega_{t-1})+\mathbb{P}(\bar{\Omega}_{t-1})\right)$ $(A_{t},B_{t})$ $[a^{i}]_{e}$ $\displaystyle\Re_{\operatorname{full}}(T,{\operatorname{adv}})$ $\displaystyle\left\langle\tilde{C}_{t-1}^{+},\tilde{\Re}_{t}\right\rangle$ $W_{t+1}=Y_{t+1,A_{t},B_{t}}-f_{\theta}(A_{t},B_{t})$ $\displaystyle\Re(T,\pi^{\operatorname{alg}},\pi^{B})=\mathbb{E}\left[\Re(T,\pi%
+^{\operatorname{alg}},\pi^{B},\theta)\right]$ $\displaystyle=\mathbb{E}\left[\sum_{t}I_{t}(\theta;Z_{t})\right]=\sum_{t=0}^{T%
+-1}I(\theta;Z_{t}\>|\>Z_{0},\ldots,Z_{t-1})$ $[0,1]^{\mathcal{A}}$ $\displaystyle I_{t}\left(\theta;R_{t+1,A_{t},B_{t}}\mid A_{t}=a,B_{t}=b\right)%
+=\frac{1}{2}\log\left(1+\frac{\sigma^{2}_{t}(a,b)}{\sigma_{w}^{2}}\right)$ $W_{t}=Y_{t+1,A_{t},B_{t}}-f_{\theta}(A_{t},B_{t})$ $\displaystyle\quad+C\sum_{t=0}^{T-1}\mathbb{P}(\neg\mathcal{E}^{c}_{t}(\tilde{%
+R},U^{\prime},A_{t},B_{t}))+2\mathbb{P}(\neg\mathcal{E}^{c}_{t}(f_{\theta},B_{%
+t}))+\mathbb{P}\left(\neg{\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})}\right)$ $t_{e}(u)=c_{e}(1+0.5(\frac{u}{C_{e}})^{4}),$ $k(x,x)\leqslant 1$ $\displaystyle\geqslant 1-\delta_{1}-(1-\delta_{1})\left(1-\frac{f}{\sqrt{2\pi}%
+(f^{2}+1)e^{f^{2}/2}}\right)^{M}$ $\displaystyle\leqslant\mathds{1}_{\mathcal{E}^{c}_{t}(\tilde{R},U^{\prime},A_{%
+t},B_{t})\cap\mathcal{E}^{c}_{t}(f_{\theta},B_{t})}(U^{\prime}_{t}(A_{t},B_{t}%
+)-L_{t}(A_{t},B_{t}))$ $\mathcal{O}((\log T)^{d+1})$ $\displaystyle=\underbrace{\mathbb{E}\left[\tilde{R}_{t+1}(a)-\tilde{R}_{t+1}(A%
+_{t})\>|\>\theta\right]}_{(I)}+\underbrace{\mathbb{E}\left[f_{\theta}({a,B_{t}%
+})-\tilde{R}_{t+1}(a)\>|\>\theta\right]}_{(II)}+\underbrace{\mathbb{E}\left[%
+\tilde{R}_{t+1}(A_{t})-f_{\theta}(A_{t},B_{t})\>|\>\theta\right]}_{(III)}$ $\Re^{*}(T,\text{IWE-Hedge})=\mathcal{O}(\sqrt{T\mathcal{A}\log\mathcal{A}}).$ $x_{t}^{-i}$ $\Omega_{t-1}$ $\beta_{t}=\frac{2\beta^{\prime}_{t}}{\log(1+\sigma_{w}^{-2})}.$ $\Re_{t}=[0.5m_{1}+\sum_{k=2}^{t}m_{k},-0.5m_{1}]$ $\{z^{j}_{t+1}\}_{j\in[M]}$ $\mu(a,b)=\mathbb{E}\left[f_{\theta}(a,b)\right]$ $\log(M\sqrt{T})/\log(1+\sigma_{w}^{-2})+2\beta_{T}=\mathcal{O}(\log\mathcal{A}%
+T+\log T+\log\log\mathcal{A}T).$ $(M=10,20,30)$ $\Re_{\operatorname{adv}}(a;T,\text{Hedge},\tilde{R})=\mathcal{O}(2c\sqrt{T\log%
+\mathcal{A}})$ $R_{t+1,A_{t},B_{t}}=\theta_{A_{t},B_{t}}$ $a^{+}=\max\{a,0\}$ $o=f_{\theta}(a)+w$ $\mathbb{P}\left(\neg{\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})}\right)$ $B_{0:T}$ $\displaystyle\mathbb{E}\left[\sum_{t=0}^{T-1}I_{t}(\theta;A_{t},B_{t},R_{t+1,A%
+_{t},B_{t}})\right]$ $\displaystyle\gamma_{T}:=\max_{A_{0:T},B_{0:T}}I(\theta;A_{0},B_{0},\ldots,A_{%
+T-1},B_{T-1})$ $f_{\theta}(a,b)=\phi(a,b)^{\top}\theta$ $\Re_{t}\in\mathbb{R}^{\mathcal{A}}$ $\displaystyle=R_{t+1,A_{t},B_{t}}\left(\frac{\tilde{C}_{t-1,A_{t}}^{+}}{X_{t,A%
+_{t}}}-\frac{\hat{X}_{t,A_{t}}}{{X}_{t,A_{t}}}\sum_{a}\tilde{C}_{t-1,a}^{+}\right)$ $\tilde{R}^{\operatorname{est}}=(\tilde{R}_{t+1},t=0,1,\ldots)$ $\tilde{R}_{t}=\left[\tilde{R}_{t1},\tilde{R}_{t2}\right]$ $L=(L_{t}\>|\>t\in\mathbb{N})$ $\mathbb{E}_{t}\left[R_{t,a,b}^{2}\right]:=\mathbb{E}_{t}\left[(f_{\theta]}(a,b%
+)+W_{t+1})^{2}\right]=\mathbb{E}_{t}\left[f_{\theta]}(a,b)^{2}+W_{t+1}^{2}%
+\right]\leqslant 1+\sigma_{w}^{2}$ $\epsilon_{t}\sim\mathcal{N}(0,0.1)$ $\tilde{R}^{\operatorname{est}}$ $N_{t}=\sum\limits_{k=2}^{t}m_{k}\sim\mathcal{N}\left(-\sum\limits_{k=2}^{t}%
+\frac{k}{k+\sigma_{n}^{2}}(1-\Delta),\ \sum\limits_{k=2}^{t}(1+\frac{\sigma_{n%
+}^{2}}{k+\sigma_{n}^{2}})\right)\triangleq\mathcal{N}(\mu_{t},\sigma_{t}^{2}).$ $\displaystyle=\log\prod_{t=1}^{\infty}\mathbb{P}(\omega_{t}|\Omega_{t-1})=\sum%
+_{t=1}^{\infty}\log\mathbb{P}(\omega_{t}|\Omega_{t-1})$ $\displaystyle=\sum_{t=1}^{\infty}\log\Phi\left(\frac{\sum_{k=2}^{t}\frac{k}{k+%
+\sigma_{n}^{2}}(1-\Delta)}{\sqrt{\sum_{k=2}^{t}(1+\frac{\sigma_{n}^{2}}{k+%
+\sigma_{n}^{2}})}}\right)$ $\Re_{\operatorname{full}}(a;T,\operatorname{adv},\tilde{R})$ $\displaystyle\mathbb{E}_{0}\left[\sum_{t=1}^{T}\sum_{a}\left(\tilde{\Re}_{t,a}%
+\right)^{2}\right]$ $\tilde{R}_{t}(2nd)$ $X_{1}=[0.5,0.5]$ $\mathcal{E}(c)$ $\displaystyle=\mathbb{E}\left[\max_{a\in\mathcal{A}}\sum_{t=0}^{T-1}\mathbb{E}%
+\left[R_{t+1,a,B_{t}}-R_{t+1,A_{t},B_{t}}\mid\theta\right]\right]$ $\textbf{r}_{t}=(f_{\theta}(a,B_{t}))_{a\in\mathcal{A}}$ $m_{1}<0$ ${N}(\mu_{p},\Sigma_{p})$ $\mathcal{O}\big{(}\sqrt{T\mathcal{A}}+\sqrt{\gamma_{T}\beta T}\big{)}$ $c^{\prime}=-0.62$ $(b^{i},b^{-i})$ $U^{\prime}_{t}\geqslant L_{t}$ $r^{i}(a^{i},a^{-i})=-\ell^{i}(a^{i},a^{-i})$ $H_{t+1}=(H_{t},A_{t},B_{t},R_{t+1,A_{t},B_{t}})$ $\mathcal{O}\left(\left(\sqrt{\log\mathcal{A}}+\sqrt{\log(\mathcal{A}T)\log(T)^%
+{d+1}}\right)\sqrt{T}\right)$ $\displaystyle\beta_{t}I_{t}(\theta;R_{t+1,A_{t},B_{t}}\>|\>A_{t}=a,B_{t}=b)%
+\geqslant\beta^{\prime}_{t}\sigma^{2}_{t}(a,b).$ $\mathcal{E}^{c}_{t}(\tilde{R},U^{\prime},A_{t},B_{t}):=\{\tilde{R}_{t+1}(A_{t}%
+)\leqslant U^{\prime}_{t}(A_{t},B_{t})\}.$ $X_{1}=\hat{X}_{1}$ $k_{\rm L}(\cdot,\cdot)$ $(a+b)^{+}\leqslant(a^{+}+b)^{+}\leqslant\left|a^{+}+b\right|.$ $\tilde{R}_{t+1}$ $\displaystyle\Re^{*}(T,\pi,\theta)$ $\displaystyle=\sum_{a}(\gamma_{t}\hat{X}_{t,a}-\gamma_{t}/\mathcal{A})\sum_{b}%
+Y_{t,b}f_{\theta}(a,b)$ $\sqrt{\gamma_{T}}$ $\displaystyle\mathbb{P}(\left|f_{\theta}(a,b)-{\mu_{t}(a,b)}\right|\geqslant%
+\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b),\forall a\in\mathcal{A}\mid H_{t})%
+\leqslant 2\mathcal{A}\exp(-\beta^{\prime}_{t}/2),$ $\displaystyle=\sum_{a}\frac{\mathbb{E}_{t}\left[R_{t+1,a,B_{t}}^{2}\right]}{X_%
+{t,a}}+\sum_{a}\frac{\hat{X}_{t,a}}{{X}_{t,a}}\mathbb{E}_{t}\left[R_{t+1,A_{t}%
+,B_{t}}^{2}\right]\left(\left|\mathcal{A}\right|\hat{X}_{t,a}-2\right)$ $\tilde{R}^{\operatorname{est}}=(\tilde{R}_{t},t\in\mathbb{Z}_{++})$ $\displaystyle\leqslant\mathbb{E}_{0}\left[\tilde{C}_{T,a}^{+}\right]=\mathbb{E%
+}_{0}\left[\sqrt{(\tilde{C}_{T,a}^{+})^{2}}\right]\leqslant\mathbb{E}_{0}\left%
+[\sqrt{\sum_{a}\left(\tilde{C}_{T,a}^{+}\right)^{2}}\right],$ $M_{1}=M_{2}=\ldots=M_{T}=M=\mathcal{O}(\log\mathcal{A}T)$ $\sigma(H_{t},A_{t},R_{t+1,A_{t},B_{t}})$ $\mathcal{R}:\mathbb{R}\mapsto[0,1]$ $\displaystyle\mathbb{E}_{0}\left[\sqrt{\sum_{a}\left(\tilde{C}^{+}_{T,i}\right%
+)^{2}}\right]\leqslant\mathbb{E}_{0}\left[\sqrt{\sum_{t=1}^{T}\sum_{a}\left(%
+\tilde{\Re}_{t,a}\right)^{2}}\right]$ $H_{t},B_{t}$ $\tilde{R}_{t2}$ $\displaystyle\leqslant\min_{t>0}\frac{\exp(\sigma^{2}t^{2}/2)}{\exp(tc)}=\exp(%
+-c^{2}/\sigma^{2})$ $\displaystyle\mu_{t+1}=\Sigma_{t+1}\left(\Sigma_{t}^{-1}\mu_{t}+\frac{R_{t+1,A%
+_{t},B_{t}}}{\sigma_{w}^{2}}\phi(A_{t},B_{t})\right)$ $\displaystyle\mathbb{P}(X-\mu\geqslant c)$ $\pi_{t}(H_{t})$ $v\in\mathbb{R}^{\mathcal{A}}$ $\sigma_{p}\leqslant 1$ $\hat{X}_{t+1}$ $\tilde{R}_{t+1}(a)\in[0,C]$ $\tilde{R}_{t}=\left[z_{t},\frac{t}{t+\sigma_{n}^{2}}(1-\Delta)+\sqrt{\frac{%
+\sigma_{n}^{2}}{\sigma_{n}^{2}+t}}z^{\prime}_{t}\right],$ $H_{t},A_{t},B_{t}$ $\displaystyle\leqslant(1+\sigma_{w}^{2})\left(\sum_{a}\frac{1}{{X}_{t,a}}+\sum%
+_{a}\frac{\hat{X}_{t,a}}{{X}_{t,a}}\left(\left|\mathcal{A}\right|\hat{X}_{t,a}%
+-2\right)\right)$ $f(x)=\theta^{\top}x,\theta\sim N(0,\sigma_{0}I)$ $\mathbb{E}_{t}\left[\cdot\right]=\mathbb{E}\left[\cdot\mid H_{t},\theta\right]$ $\bm{a}=(a^{i},a^{-i})$ $\displaystyle\geqslant\sum_{t=1}^{\infty}\left(-\frac{1}{\sqrt{2\pi}f_{t}(%
+\Delta,\sigma_{n})e^{f_{t}^{2}(\Delta,\sigma_{n})/2}}\right)$ $\displaystyle V^{*}=\max_{P\in\mathcal{D}(\mathcal{A})}\min_{Q\in\mathcal{D}(%
+\mathcal{B})}\mathbb{E}_{A\sim P,B\sim Q}\left[f_{\theta}(A,B)\right],$ $\mathbb{P}(f_{\theta}\in\mathcal{F}_{t})\geqslant 1-2\mathcal{A}\exp(-\beta^{%
+\prime}_{t}/2).$ $\operatorname{clip}_{[-c,c]}(x)\geqslant\min(x,c)$ $0.01\sqrt{2\log\mathcal{A}\mathcal{B}\sqrt{T}}$ $\sum_{t=0}^{T-1}\mathbb{E}\left[R_{t+1,A_{t},B_{t}}\>|\>\theta\right]$ $\text{SINR}(a,b;\theta)=\phi(a,b)^{T}\theta$ $\delta=1/\sqrt{t}$ $\displaystyle\lim_{t\to\infty}\log\mathbb{P}(\Omega_{t})$ $\text{Regret}^{i}(T)=\frac{1}{T}\max_{a\in\Delta^{\mathcal{D}(\mathcal{A}^{i})%
+}}\mathbb{E}\left[\sum_{t=1}^{T}\phi\left(a,x_{t}^{-i}\right)-\phi\left(x_{t}^%
+{i},x_{t}^{-i}\right)\right],$ $\displaystyle\leqslant\Re_{\operatorname{full}}(T,\operatorname{adv})+\sqrt{%
+\beta I(\theta;H_{T})T},$ $\displaystyle\mathbb{E}_{0}\left[\sum_{t=1}^{T}\Re_{t}(a)\right]\leqslant 2^{4%
+/3}(1+\sigma_{w}^{2})^{1/3}\left|\mathcal{A}\right|^{2/3}T^{2/3}.$ $g_{t}(\cdot):\mathbb{R}_{+}^{\mathcal{A}}\times\mathbb{R}_{+}^{\mathcal{A}}%
+\mapsto\mathbb{R}_{+}^{\mathcal{A}}$ $m_{t}=z_{t}-\frac{t}{t+\sigma_{n}^{2}}(1-\Delta)-\sqrt{\frac{\sigma_{n}^{2}}{t%
++\sigma_{n}^{2}}}z^{\prime}_{t}=z_{t}-\sqrt{\frac{\sigma_{n}^{2}}{t+\sigma_{n}%
+^{2}}}z^{\prime}_{t}-\frac{t}{t+\sigma_{n}^{2}}(1-\Delta)$ $f_{t}(\Delta,\sigma_{n})=\frac{(1-\Delta)\left(t-\sigma_{n}^{2}\ln{(t+\sigma_{%
+n}^{2})}+2\sigma_{n}^{2}\ln{\sigma_{n}}-1/(\sigma_{n}^{2}+1)\right)}{\sqrt{t+%
+\sigma_{n}^{2}\ln{(t+\sigma_{n}^{2})}-2\sigma_{n}^{2}\ln{\sigma_{n}}-(\sigma_{%
+n}^{2}+2)/(\sigma_{n}^{2}+1)}},$ $\displaystyle\leqslant\sum_{a}\left|\gamma_{t}\hat{X}_{t,a}-\gamma_{t}/%
+\mathcal{A}\right|$ $\tilde{R}_{t1}\geqslant\tilde{R}_{t2}$ $(\gamma_{t})_{t\geqslant 0}$ $\displaystyle\mathbb{P}(\mathcal{E}_{t}(\tilde{R},U,B_{t})\mid H_{t},B_{t})$ $\phi(a,b)\in\mathbb{R}^{d}$ $\sigma^{2}_{t}(a,b)=k((a,b),(a,b))-\mathbf{k}_{t}((a,b))^{\top}(\mathbf{K}_{t}%
++\sigma^{2}{\bm{I}}_{t})\mathbf{k}_{t}(a,b)$ $\tilde{C}^{+}_{t-1,a}=0$ $\mathcal{A}_{J}=\mathcal{F}$ $(r_{t})_{t}$ $\displaystyle\geqslant 1-\delta_{1}-(1-\delta_{1})\exp{\left(\frac{-Mf}{\sqrt{%
+2\pi}(f^{2}+1)e^{f^{2}/2}}\right)}$ $\displaystyle(U^{\prime}_{t}(A_{t},B_{t})-L_{t}(A_{t},B_{t}))$ $Y_{t}=\max_{y}X_{t}^{\top}\theta y$ $X_{t,a}$ $U=(\mu_{t}(a,b)+\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b):t\in\mathbb{N}),\quad
+U%
+^{\prime}=(\mu_{t}(a,b)+\sqrt{2\log(M\sqrt{t})}\sigma_{t}(a,b):t\in\mathbb{N}).$ $(f_{\theta}(a,b):(a,b)\in\mathcal{A}\times\mathcal{B})$ $\mathcal{O}\big{(}\sqrt{T\mathcal{A}}\big{)}$ $a^{i},b^{i}\in\mathcal{A}^{i}$ $\displaystyle=\left(\frac{\sqrt{2\log(M\sqrt{t})}}{\sqrt{\beta^{\prime}_{t}}}+%
+1\right)\sqrt{\beta_{t}I_{t}(\theta;A_{t},B_{t},R_{t+1,A_{t},B_{t}})}$ $\displaystyle\geqslant\sum_{t=1}^{\infty}\log\left(1-\frac{1}{\sqrt{2\pi}f_{t}%
+(\Delta,\sigma_{n})e^{f_{t}^{2}(\Delta,\sigma_{n})/2}}\right)$ $\Delta,\sigma_{n}$ $\displaystyle\leqslant\left(\frac{\sqrt{2\log(M\sqrt{t})}}{\sqrt{\beta^{\prime%
+}_{t}}}+1\right)\sqrt{\beta_{t}I_{t}(\theta;R_{t+1,A_{t},B_{t}}\>|\>A_{t},B_{t%
+})}$ $\left|\mathcal{A}\right|$ $k(x,x^{\prime})=\exp(-(2l^{2})^{-1}\left\|x-x^{\prime}\right\|^{2}s)$ $\Phi(\beta^{\prime}_{t})^{M}=1/\sqrt{t}$ $\tilde{R}_{t}\in[0,1]^{\mathcal{A}}$ $m_{t}=\tilde{R}_{t1}-\tilde{R}_{t2}$ $\overline{x}_{T}=\frac{1}{T}\sum_{t=1}^{T}x_{t},\quad\overline{y}_{T}=\frac{1}%
+{T}\sum_{t=1}^{T}y_{t}.$ $k((a,b),(a^{\prime},b^{\prime}))$ $\left\langle\tilde{C}_{t-1}^{+},\tilde{\Re}_{t}\right\rangle\leqslant 0$ $N(0,\sigma_{w}^{2})$ $\displaystyle=\mathbb{E}_{0}\left[\sum_{t=1}^{T}R_{t+1,A_{t},B_{t}}^{2}\left(%
+\frac{1}{X_{t,A_{t}}^{2}}-\frac{2\hat{X}_{t,A_{t}}}{X_{t,A_{t}}^{2}}+\left|%
+\mathcal{A}\right|\frac{\hat{X}_{t,A_{t}}^{2}}{X_{t,A_{t}}^{2}}\right)\right]$ $\displaystyle\tilde{\Re}_{t,a}=\frac{\mathbb{I}_{A_{t}=a}R_{t+1,A_{t},B_{t}}}{%
+X_{t,a}}-R_{t+1,A_{t},B_{t}}\frac{\hat{X}_{t,A_{t}}}{X_{t,A_{t}}}$ $(X_{a})_{a\in\mathcal{A}}$ $Y_{t}=\min_{y}X_{t}^{\top}\theta y$ $\log(\text{average regret})\propto(1/2)\log(M+N)$ $H_{t}=\left(A_{0},B_{0},Y_{1,A_{0},B_{0}},\ldots,A_{t-1},B_{t-1},Y_{t,A_{t-1},%
+B_{t-1}}\right)$ $U^{\prime}\geqslant U\geqslant L$ $\mathcal{O}(T^{d(d+1)/(2\nu+d(d+1))}(\log T))$ $\mathcal{O}\big{(}\sqrt{T\mathcal{A}\log\mathcal{A}}\big{)}$ $\displaystyle:=\operatorname{clip}_{[0,1]}\left(\tilde{f}_{t+1}^{\operatorname%
+{OTS}}(a,B_{t})\right).$ $n_{t}(a,b)$ $\tilde{\Re}_{t}$ $\Phi(x)\leqslant 1-\frac{x}{\sqrt{2\pi}(x^{2}+1)e^{x^{2}/2}}$ $\displaystyle=\sum_{t=1}^{\infty}\log\Phi(-\frac{\mu_{t}}{\sigma_{t}})$ $g_{t}:\Delta^{\mathcal{A}}\times[0,1]^{\mathcal{A}}\mapsto\Delta^{\mathcal{A}}$ $c=0.54$ $\tilde{\mathcal{O}}(\sqrt{MN/T})$ $X_{t+1}=g_{t}(X_{t},(f_{\theta}(a,B_{t}))_{a\in\mathcal{A}})$ $\displaystyle=\sum_{a}(\hat{X}_{t,a}-X_{t,a})\sum_{b}Y_{t,b}f_{\theta}(a,b)$ $\mathbb{P}(\neg\mathcal{E}^{c}_{t}(f_{\theta},B_{t}))$ $Z_{t+1}$ $\displaystyle\geqslant(1-\delta_{1})\mathbb{P}(\max_{i}x_{i}\geqslant\max_{i}y%
+_{i}\mid\epsilon)$ $X_{2}=[1,0]$ $\mathcal{O}\big{(}\sqrt{T\log\mathcal{A}}\big{)}$ $\displaystyle\mathcal{F}_{t}:=\left\{f_{\theta}:\left|f_{\theta}(a,B_{t})-\mu_%
+{t}(a,B_{t})\right|\leqslant\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,B_{t}),%
+\forall a\in\mathcal{A}\right\}$ $w\sim N\left(0,\sigma_{w}^{2}\right)$ $\frac{\sqrt{2\log M\sqrt{t}}}{\sqrt{\beta^{\prime}_{t}}}=\sqrt{\frac{2\log M%
+\sqrt{t}}{\log\mathcal{A}\sqrt{t}}}$ $G_{J}$ $\displaystyle\tilde{R}_{t+1}(A_{t})-f_{\theta}(A_{t},B_{t})\leqslant(\mathds{1%
+}_{\mathcal{E}^{c}_{t}(\tilde{R},U^{\prime},A_{t},B_{t})})(U^{\prime}_{t}(A_{t%
+},B_{t})-f_{\theta}(A_{t},B_{t}))+C(1-\mathds{1}_{\mathcal{E}^{c}_{t}(\tilde{R%
+},U^{\prime},A_{t},B_{t})}).$ $\displaystyle=\sum_{t=1}^{\infty}\log\mathbb{P}(\mu_{t}+\sigma_{t}Z\leqslant 0%
+),\quad Z\sim\mathcal{N}(0,1)$ $\text{regret-matching}^{+}(\text{RM}^{+})$ $\displaystyle f_{\theta}(a,B_{t})-f_{\theta}(A_{t},B_{t})=\underbrace{\tilde{R%
+}_{t+1}(a)-\tilde{R}_{t+1}(A_{t})}_{(I)\leavevmode\nobreak\ \operatorname{adv}%
+_{t+1}(a)}+\underbrace{f_{\theta}(a,B_{t})-\tilde{R}_{t+1}(a)}_{(II)%
+\leavevmode\nobreak\ \operatorname{pess}_{t+1}(a)}+\underbrace{\tilde{R}_{t+1}%
+(A_{t})-f_{\theta}(A_{t},B_{t})}_{(III)\operatorname{est}_{t+1}}$ $\sum_{a}\tilde{C}_{t-1,a}^{+}\leqslant 0$ $\displaystyle f_{\theta}(a,B_{t})-\tilde{R}_{t+1}(a)\leqslant C(1-\mathds{1}_{%
+\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})\cap\mathcal{E}^{c}_{t}(f_{\theta},B_{t}%
+)}),\quad\forall a\in\mathcal{A}.$ $\displaystyle\leqslant\sqrt{T\left(8\log(M\sqrt{T})/\log(1+\sigma_{w}^{-2})+2%
+\beta_{T}\right)I(\theta;H_{T})}$ $\mathbb{P}(\tilde{R}_{t1}\geqslant\tilde{R}_{t2})$ $\displaystyle\Re(T,\pi^{\operatorname{alg}},\pi^{B})$ $\displaystyle=\mathbb{P}(0.5m_{1}+\sum_{k=2}^{t}m_{k}\leqslant 0|\Omega_{t-1})$ $(H_{t},A_{t},B_{t},\theta)$ $\beta=\mathcal{O}(\log\mathcal{A}T)$ $\sigma_{n}=0.1$ $[-c,c]^{\mathcal{A}}$ $\displaystyle\geqslant\mathbb{P}(\sum_{k=2}^{t}m_{k}\leqslant 0|\Omega_{t-1})$ $\lim_{t\to\infty}\mathbb{P}(\Omega_{t})\geqslant c>0,$ $\displaystyle=\mathbb{P}\left(\max_{j\in[M]}z^{j}_{t+1}\geqslant\sqrt{\beta^{%
+\prime}_{t}}\right)$ $\sigma_{t}(a,b)=\left\|\phi(a,b)\right\|_{\Sigma_{t}}$ $\mathbb{P}(\Omega_{t})\geqslant\lim\limits_{t\to\infty}\mathbb{P}(\Omega_{t})\geqslant
+c$ $\displaystyle\operatorname{NashRegret}_{t}=\mathbb{E}\left[V^{*}-R_{t+1,A_{t},%
+B_{t}}\right]$ $\tilde{R}_{t}\in\mathbb{R}^{\mathcal{A}}$ $\displaystyle\geqslant 1-\delta_{1}-(1-\delta_{1})\Phi^{M}\left(\frac{t}{t+%
+\sigma_{n}^{2}}(1-\Delta)+\sqrt{\frac{2\sigma_{n}^{2}\log(M/\delta_{1})}{t+%
+\sigma_{n}^{2}}}\right)$ $I(\theta;H_{T})=I(\theta;A_{0},B_{0},\ldots,A_{T-1},B_{T-1})\leqslant\gamma_{T}.$ $\displaystyle\mathbb{P}(\left|f_{\theta}(a,b)-{\mu_{t}(a,b)}\right|\geqslant%
+\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b)\mid H_{t})$ $r=(\sqrt{2\nu}/l)\left\|x-x^{\prime}\right\|$ $(a,b),(a^{\prime},b^{\prime})\in\mathcal{A}\times\mathcal{B}$ $\mu_{t}(a,b)$ $\sum_{a}\left({\tilde{C}_{T,a}^{+}}\right)^{2}\leqslant\sum_{t=1}^{T}\sum_{a}%
+\left(\tilde{\Re}_{t,a}\right)^{2}$ $\tilde{R}_{t}$ $\displaystyle\tilde{f}^{\operatorname{OTS}}_{t+1}(a,B_{t}):=(\max_{j\in[M]}z_{%
+t+1}^{j})\cdot\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,B_{t})+\mu_{t}(a,B_{t})%
+\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \tilde{R}_{t+1}(a)=%
+\operatorname{clip}_{[-c,c]}\left(\tilde{f}_{t+1}^{\operatorname{OTS}}(a,B_{t}%
+)\right),\forall a\in\mathcal{A}.$ $\displaystyle\tilde{R}^{\operatorname{OTS}}_{t+1}(a)$ $\displaystyle=R_{t+1,A_{t},B_{t}}\left(\frac{\tilde{C}_{t-1,A_{t}}^{+}}{X_{t,A%
+_{t}}}-\frac{\tilde{C}_{t-1,A_{t}}^{+}/\sum_{a}\tilde{C}_{t-1,a}^{+}}{{X}_{t,A%
+_{t}}}\sum_{a}\tilde{C}_{t-1,a}^{+}\right)=0$ $\gamma=\sqrt[3]{((1+\sigma_{w}^{2})\left|\mathcal{A}\right|^{2})/{2T}}$ $\tilde{R}_{t+1}=E(H_{t+1},Z_{t+1})\in\mathbb{R}^{\mathcal{A}}.$ $\displaystyle\leqslant\Re_{\operatorname{full}}(T,\operatorname{adv},\tilde{R}%
+)+\sum_{t=0}^{T-1}\mathbb{E}\left[(U^{\prime}_{t}(A_{t},B_{t})-L_{t}(A_{t},B_{%
+t}))\right]$ $\displaystyle\leqslant\sum_{t=0}^{T-1}\mathbb{E}\left[(U^{\prime}_{t}(A_{t},B_%
+{t})-L_{t}(A_{t},B_{t}))\right]$ $\displaystyle=\mathbb{E}\left[f_{\theta}({a,B_{t}})-\tilde{R}_{t+1}(a)\mid%
+\theta\right],$ $(III)$ $\displaystyle f_{\theta}(a,B_{t})-\tilde{R}_{t+1}(a)\leqslant\mathds{1}_{%
+\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})}(f_{\theta}(a,B_{t})-U_{t}(a,B_{t}))+C(%
+1-\mathds{1}_{{\mathcal{E}}^{o}_{t}(\tilde{R},U,B_{t})}).$ $\Re(T)\geqslant 2\mathbb{P}(\Omega_{T})\Delta\cdot T.$ $\displaystyle\mathbb{E}\left[R_{t+1,a,B_{t}}-R_{t+1,A_{t},B_{t}}\>|\>\theta%
+\right]=(I)+(II)+(III)$ $reg_{t}$ $\displaystyle=I(\theta;Z_{0},\ldots,Z_{T-1})=I(\theta;H_{T})$ $w\in\mathbb{R}_{+}$ $\mathbb{P}(\Omega_{t})$ $\displaystyle\leqslant\left(\frac{\sqrt{2\log(M\sqrt{t})}}{\sqrt{\beta^{\prime%
+}_{t}}}+1\right)\sqrt{\beta^{\prime}_{t}}\sigma_{t}(A_{t},B_{t})$ $\Re_{\operatorname{full}}(a;T,\text{RM},\tilde{R})=\mathcal{O}(2c\sqrt{T%
+\mathcal{A}})$ $\sigma^{2}_{t}(a,b)\leqslant k((a,b),(a,b))\leqslant 1$ $\displaystyle\mathbb{P}(\neg\mathcal{E}^{c}_{t}(\tilde{R},U^{\prime},A_{t},B_{%
+t}))\leqslant\frac{1}{\sqrt{t}}.$ $\displaystyle\leqslant 2\exp\left(-\frac{\beta^{\prime}_{t}}{2}\right)$ $\mathbb{P}(f_{\theta}(a,B_{t})\geqslant\tilde{R}_{t+1}(a)\mid\theta)\leqslant%
+\mathcal{O}(1/\sqrt{T}).$ $\displaystyle=\mathbb{P}(\max_{i}x_{i}\geqslant\max_{i}y_{i})$ $\sqrt{2\log\mathcal{A}\sqrt{T}},0.2\sqrt{2\log\mathcal{A}\mathcal{B}\sqrt{T}},%
+0.05\sqrt{2\log\mathcal{A}\mathcal{B}\sqrt{T}}$ $\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})$ $\mathcal{O}\big{(}\sqrt{T\log\mathcal{A}}+\sqrt{\gamma_{T}\beta T}\big{)}$ $\displaystyle\sim N(\mu_{t}(a,B_{t}),\sigma_{t}(a,B_{t})),j\in[M_{t+1}]$ $\ell^{i}(a^{i},a^{-i})=\sum_{e\in\mathcal{E}}[a^{i}]_{e}t_{e}([a^{i}]_{e}+[g(a%
+^{-i})]_{e}).$ $a^{i}\in\mathcal{A}^{i}\subset\mathbb{R}^{|\mathcal{E}|}$ $\beta=\log(\mathcal{A}T)$ $K^{i}((a^{i},a^{-i}),(b^{i},b^{-i}))=k_{\rm L}(a^{i},b^{i})k_{\rm P}(a^{i}+g(a%
+^{-i}),b^{i}+g(b^{-i})),$ $\tilde{R}_{t2}=\max\limits_{i}y_{ti}$ $\beta_{t}=\sqrt{2\log{\mathcal{A}\sqrt{t}}}$ $\tilde{R}^{OTS}_{t+1}(1st)$ $\log(\text{average regret})\propto\frac{1}{2}\log(MN)$ $\tilde{C}_{t,a}=\sum_{s=0}^{t}\tilde{\Re}_{s,a}$ $\mathcal{O}(\sqrt{T\mathcal{A}\log\mathcal{A}})$ $X_{t+1}=g_{t}(X_{t},r_{t})$ $min(x,c)$ $\pi^{\operatorname{alg}}=(\pi_{t})_{t\in\mathbb{N}}$ $a^{-i},b^{-i}\in\mathcal{A}^{-i}$ $\displaystyle\begin{cases}\textrm{Hedge:}\ &g_{t,a}(X_{t},r_{t})=X_{t,a}\exp%
+\left(\eta_{t}r_{t}(a)\right),\\
+\textrm{RM:}\ &g_{t,a}(X_{t},r_{t})=\max\left(0,\sum\limits_{s=0}^{t}r_{t}(a)-%
+r_{t}(A_{s})\right),\end{cases}$ $\displaystyle\hat{X}_{t+1,a}=\begin{cases}{\tilde{C}^{+}_{t,a}}/{\sum_{a\in%
+\mathcal{A}}\tilde{C}_{t,a}^{+}},&\text{if }\sum_{a\in\mathcal{A}}\tilde{C}_{t%
+,a}^{+}>0,\\
+\text{arbitrary vector on simplex, e.g. }1/\mathcal{A},&\text{otherwise}\end{cases}$ $\sigma_{t}^{2}(a,b)\leqslant\frac{1}{\log(1+\sigma_{w}^{-2})}\log(1+\sigma_{w}%
+^{-2}\sigma^{2}_{t}(a,b)).$ $\displaystyle\left((a+b)^{+}\right)^{2}\leqslant(a^{+})^{2}+2(a^{+})b+b^{2}$ $[a^{i}]_{e}+[g(a^{-i})]_{e}$ $\displaystyle\mathbb{E}\left[\sum_{t=0}^{T-1}U^{\prime}_{t}(A_{t},B_{t})-L_{t}%
+(A_{t},B_{t})\right]$ $\displaystyle=\mathbb{E}\left[\tilde{R}_{t+1}(A_{t})-f_{\theta}(A_{t},B_{t})%
+\mid\theta\right].$ $\Sigma_{p}=\sigma_{p}I$ $\mathbb{P}(\Omega_{t})=\mathbb{P}(\omega_{1})\mathbb{P}(\omega_{2}|\Omega_{1})%
+\ldots\mathbb{P}(\omega_{t}|\Omega_{t-1})$ $\displaystyle=[R_{1,A_{0},B_{0}},\ldots,R_{t,A_{t-1},B_{t-1}}]^{\top}$ $\displaystyle\geqslant\mathbb{P}(\sum_{k=2}^{t}m_{k}\leqslant 0|\Omega_{t-1})%
+\mathbb{P}(\Omega_{t-1})+\mathbb{P}(\sum_{k=2}^{t}m_{k}\leqslant 0|\bar{\Omega%
+}_{t-1})\mathbb{P}(\bar{\Omega}_{t-1})$ $\displaystyle=\Re_{\operatorname{full}}(T,\operatorname{adv},\tilde{R})+%
+\mathbb{E}\left[\max_{a\in\mathcal{A}}\sum_{t=0}^{T-1}\mathbb{E}\left[%
+\operatorname{pess}_{t+1}(a)\mid\theta\right]\right]+\sum_{t=0}^{T-1}\mathbb{E%
+}\left[\operatorname{est}_{t+1}\right]$ $\operatorname{NashRegret}_{t}$ $\theta=\begin{bmatrix}1&1-\Delta\\
+1-\Delta&1\end{bmatrix},$ $\displaystyle\operatorname{NashRegret}(T)=\sum_{t=1}^{T}\operatorname{%
+NashRegret}_{t}$ $I(\theta;R_{t+1,A_{t},B_{t}}|H_{t})$ $f_{\theta}(a,b)=\theta(a,b)$ $P_{X}(i)={X_{i}}/\sum_{i\in\mathcal{A}}X_{i}$ $\displaystyle\sigma_{t}(a,b)=\sqrt{\frac{\sigma_{w}^{2}}{\sigma_{w}^{2}/\sigma%
+^{2}_{p}(a,b)+n_{t}(a,b)}}$ $KL(\overline{x}_{T},x^{)}$ $\Re_{\operatorname{full}}$ $k((a,b),(a,b))=\phi(a,b)^{\top}\Sigma_{p}\phi(a,b)$ $\{\tilde{R}_{t}:t\in\mathbb{Z}_{++}\}$ $\displaystyle\mathbb{E}\left[\tilde{\Re}_{t,a}\mid\theta,H_{t}\right]=\mathbb{%
+E}\left[f_{\theta}(a,B_{t})\mid\theta,H_{t}\right]-\sum_{a}\hat{X}_{t,a}%
+\mathbb{E}\left[f_{\theta}(a,B_{t})\mid\theta,H_{t}\right]$ $\displaystyle\tilde{f}^{\operatorname{OTS}}_{t+1}(a,B_{t})$ $I(\theta;H_{T})$ $\mathbb{P}\left(\eta_{j}\leqslant w\right)=\mathbb{P}({\eta_{j}}/{\sigma}%
+\leqslant{w}/{\sigma})=\Phi({w}/{\sigma}).$ $\displaystyle\Re(T)\geqslant 2\mathbb{P}(\Omega_{T})\Delta\cdot T$ $\phi(a,b)$ $\mathcal{E}_{t}(\tilde{R},U,B_{t}):=\{\tilde{R}_{t+1}(a)\geqslant U_{t}(a,B_{t%
+}),\forall a\in\mathcal{A}\}.$ $\displaystyle\stackrel{{\scriptstyle(ii)}}{{\geqslant}}\mathbb{P}\left(\left(%
+\max_{j\in[M]}z^{j}_{t+1}\right)\sigma_{t}(a,B_{t})\geqslant\sqrt{\beta^{%
+\prime}_{t}}\sigma_{t}(a,B_{t}),\forall a\in\mathcal{A}\>|\>H_{t},B_{t}\right)$ $\phi(a,b)=e_{a,b}$ $\displaystyle\mathbb{E}_{0}\left[\tilde{C}_{T,a}\right]$ $f(t,\Delta,\sigma_{n})=\frac{t}{t+\sigma_{n}^{2}}(1-\Delta)+\sqrt{\frac{2%
+\sigma_{n}^{2}\log(M/\delta_{1})}{t+\sigma_{n}^{2}}}$ $\mathcal{A}_{R}=\mathcal{F}\times\mathcal{F}\times\mathcal{F}$ $B_{t}\sim P_{Y_{t}}$ $\mathbb{P}(\neg\mathcal{E}^{c}_{t}(\tilde{R},U^{\prime},A_{t},B_{t}))$ $\displaystyle\Re_{\operatorname{full}}(T,{\operatorname{adv}},(r_{t})_{t})=%
+\max_{a\in\mathcal{A}}\mathbb{E}\left[\sum_{t=0}^{T-1}r_{t}(a)-r_{t}(A_{t})%
+\right].$ $\tilde{R}_{t}(1nd)$ $\lim_{t\to\infty}\log\mathbb{P}(\Omega_{t})\geqslant\log c^{\prime}>-\infty$ $0\leqslant L_{t}\leqslant U_{t}\leqslant C$ $\sigma_{n},M$ $0\leqslant{\hat{X}_{t,a}}/{X}_{t,a}\leqslant 1/(1-\gamma_{t})$ $\displaystyle\geqslant 1-\delta_{1}-(1-\delta_{1})\Phi^{M}(f)$ $X_{t+1,a}\propto X_{t,a}\exp(\eta_{t}\tilde{R}_{t+1}(a))$ $\displaystyle\mathbb{P}(\left|f_{\theta}(a,B_{t})-{\mu_{t}(a,B_{t})}\right|%
+\geqslant\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b),\forall a\in\mathcal{A}\mid H%
+_{t})\leqslant 2\mathcal{A}\exp(-\beta^{\prime}_{t}/2).$ $X_{2}\propto\max(reg_{1},0)$ $f_{\theta}(a,b)=\theta_{a,b}$ $\displaystyle\leqslant(1+\sigma_{w}^{2})\left(\sum_{a}\frac{\left|\mathcal{A}%
+\right|}{\gamma_{t}}+\min(\frac{\left|\mathcal{A}\right|}{\gamma_{t}},\frac{%
+\left|\mathcal{A}\right|}{1-\gamma_{t}})(\left|\mathcal{A}\right|-2)\right)%
+\leqslant\frac{2(1+\sigma_{w}^{2})\left|\mathcal{A}\right|^{2}}{\gamma_{t}}$ $\displaystyle\tilde{f}^{\operatorname{TS},j}_{t+1}(a,B_{t})$ $B_{0:T}=(B_{0}=b_{0},B_{1}=b_{1},\ldots,B_{T-1}=b_{T-1})$ $\mathbb{P}(\sum_{k=2}^{t}m_{k}\leqslant 0|\Omega_{t-1})\geqslant\mathbb{P}(%
+\sum_{k=2}^{t}m_{k}\leqslant 0|\bar{\Omega}_{t-1})$ $(W_{t}:t\in\mathbb{Z}_{++})$ $\displaystyle\tilde{h}(a)=\mathbb{I}_{A=a}\frac{h(A)}{X_{a}},\forall a\in%
+\mathcal{A}$ $\displaystyle\mathbb{E}_{t}\left[R_{t+1,A_{t},B_{t}}^{2}\left(\frac{1}{X_{t,A_%
+{t}}^{2}}-\frac{2\hat{X}_{t,A_{t}}}{X_{t,A_{t}}^{2}}+\left|\mathcal{A}\right|%
+\frac{\hat{X}_{t,A_{t}}^{2}}{X_{t,A_{t}}^{2}}\right)\right]$ $(10$ $f_{\theta}(a,B_{t})$ $\displaystyle\Re^{*}(T,\pi,\theta)\leqslant\Re_{\operatorname{full}}(T,%
+\operatorname{adv},\tilde{R}^{\operatorname{est}})+\sqrt{\log(\mathcal{A}T)I(%
+\theta;H_{T})T}$ $\displaystyle I(\theta;o)=H(o)-H(o\mid\theta)=\frac{1}{2}\log 2\pi e(\sigma(a)%
+^{2}+\sigma_{w}^{2})-\frac{1}{2}\log 2\pi e\sigma_{w}^{2}=\frac{1}{2}\log(1+%
+\sigma_{w}^{-2}\sigma(a))$ $\tilde{r}t=A{ij}+\epsilon_{t}$ $\operatorname{adv}$ $M=\frac{\log(\mathcal{A}\sqrt{T})}{\log\frac{1}{\Phi(\beta^{\prime}_{t})}},%
+\beta^{\prime}_{t}=2\log\mathcal{A}\sqrt{T}$ $\tilde{R}_{t+1,A_{t},B_{t}}$ $X_{t}=[0,1],\forall t\geqslant 2$ $\mathcal{O}\big{(}T^{2/3}\mathcal{A}^{2/3}\big{)}$ $\phi:\mathcal{A}\times\mathcal{B}\mapsto\mathbb{R}^{d}$ $\mathbb{P}\left(\max_{j\in[M]}\eta_{j}\leqslant\sqrt{2\sigma^{2}\log(M/\delta)%
+}\right)\geqslant 1-\delta.$ $(II)$ $k(x,x^{\prime})=(2^{1-\nu}/\Gamma(\nu))r^{\nu}B_{\nu}(r)$ $\mathcal{F}=\{f_{1},f_{2},f_{3}\}$ $\displaystyle=C\sum_{t=0}^{T-1}\mathbb{P}\left(\neg{\mathcal{E}^{o}_{t}(\tilde%
+{R},U,B_{t})}\cup\neg{\mathcal{E}^{c}_{t}(f_{\theta},B_{t})}\right)$ $|\phi(a,b)|\leqslant 1$ $N=528$ $reg_{t}=\tilde{R}_{t}-X_{t}^{T}\tilde{R}_{t}\cdot\mathbf{1}=[m_{t},0],\quad%
+\forall t\geqslant 2$ $R_{t+1,A_{t},B_{t}}=\mathcal{R}(Y_{t+1,A_{t},B_{t}})$ $A_{t}\sim P_{X_{t}}$ $X_{t+1}=g_{t}(X_{t},\tilde{R}_{t+1}).$ $a^{-i}\in\mathcal{A}^{-i}$ $w_{t}(a,b)$ $\Re^{*}(T,\text{IWE-RM})=\mathcal{O}(T^{2/3}\mathcal{A}^{2/3}).$ $(H_{t},\theta,A_{t},B_{t})$ $\mathbb{E}\left[\operatorname{pess}_{t+1}\>|\>\theta\right]=\mathbb{E}\left[%
+\operatorname{est}_{t+1}\>|\>\theta\right]=0$ $\displaystyle\mathbb{E}\left[R_{t+1,a,B_{t}}-R_{t+1,A_{t},B_{t}}\>|\>\theta%
+\right]=\mathbb{E}\left[f_{\theta}(a,B_{t})-f_{\theta}(A_{t},B_{t})\>|\>\theta\right]$ $\mathbb{P}(\neg\mathcal{E}^{c}_{t}(f_{\theta},B_{t}))\leqslant 2\mathcal{A}%
+\exp(-\beta_{t}^{\prime}/2)=2/\sqrt{t}$ $\min\{f_{T}(N_{s})\mid 1\leq s\leq t\}<2f_{T}(N)$ $\gamma^{\prime}[V]=\gamma^{\prime\prime}[V]$ $N^{k,n}_{s^{\prime}}$ $\mathcal{V}\setminus(\mathcal{V}^{\prime}\cup\{V_{n}\})$ $V_{n-1}$ $Pa(N,V_{i})$ $Pa(N^{k,n}_{s},V_{n})$ $\Leftarrow(2c-1)\cdot 2^{n-2}-2^{n-2c}\cdot c\binom{2c-1}{c}\geq(6c-6)\cdot 2^%
+{n-4}$ $1\leq s\leq t=2c-1$ $(2^{k}-1)\cdot 1=2^{k}-1$ $1\succ 0$ $0\leq k^{\prime}\leq k$ $\mathcal{V}^{\prime}\subseteq\mathcal{V}\setminus\{V_{n}\}$ $\Leftarrow(8c-4)\cdot 2^{n-4}-2^{n-2c}\cdot c\binom{2c-1}{c}\geq(6c-6)\cdot 2^%
+{n-4}$ $\mathcal{V}^{\prime}\subseteq\mathcal{V}$ $0\leq\kappa\leq\lfloor\frac{t}{2}\rfloor$ $\mathcal{V}=\{V_{1},\ldots,V_{n}\}$ $M(T^{\prime})$ $2c\cdot 2^{n-2}-2^{n-2c-1}\cdot c\binom{2c}{c}\geq\frac{3}{4}(2c-2)\cdot 2^{n-2}$ $o[V_{i}]=b$ $\mathcal{V}^{\prime},\mathcal{V}^{\prime\prime}\subseteq\mathcal{V}$ $Pa(N,V_{n})=P_{N}\subseteq P$ $k\in\{2,\ldots,n-1\}$ $1\leq s^{\prime}\leq\binom{n-1}{k}2^{k}$ $\mathcal{V}\setminus{V_{n}}$ $001$ $P\subseteq\{V_{1},\ldots,V_{n-1}\}$ $o,o^{\prime}$ $2^{k-k^{\prime}}-1$ $I=\{1,\ldots,m\}$ $2^{k+1}-2$ $\{V_{1},\ldots,V_{n-1}\}$ $T_{\varepsilon}=(N^{\varepsilon}_{1},\ldots,N^{\varepsilon}_{t_{\varepsilon}})$ $\Delta(N^{k,n}_{s},N^{k,n}_{s^{\prime}})=2^{n-k}$ $4c\binom{2c-1}{c}\leq(c+1)\cdot 2^{2c-1}\,.$ $(s,o,o^{\prime})$ $1\leq s\leq t$ $1\leq s\leq\binom{n-1}{k}2^{k}$ $o[V]=o^{\prime}[V]=\gamma[V]$ $(c+1)\binom{2c+1}{c+1}\leq(c+2)\cdot 2^{2c-1}$ $f_{T}(N_{s})$ $\operatorname{CPT}(N_{s},V_{n})$ $|Pa(N^{k,n}_{s},V_{n})\cap Pa(N^{k,n}_{s^{\prime}},V_{n})|=:k^{\prime}$ $s^{\prime}\neq s$ $2^{k-k^{\prime}+1}$ $T^{k,n}$ $\mathcal{F}_{t\in O(1)}$ $f_{T}(N)=$ $f_{T}(N)=\begin{cases}t\cdot 2^{n-2}-2^{n-t-1}\cdot c\binom{2c-1}{c}&\text{if %
+}t=2c-1\\
+t\cdot 2^{n-2}-2^{n-t-1}\cdot c\binom{2c}{c}&\text{if }t=2c\\
+\end{cases}$ $2^{k}-1$ $\gamma:b\succ b^{\prime}$ $f_{T^{k,n}}(N^{k,n}_{s})=\sum_{s^{\prime}\neq s}\Delta(N^{k,n}_{s},N^{k,n}_{s^%
+{\prime}})$ $Pa(N^{a},V_{n})\subseteq P$ $\displaystyle\sum_{k^{\prime}=1}^{k-1}\big{[}2^{k-k^{\prime}}\binom{k}{k^{%
+\prime}}\binom{n-k-1}{k^{\prime}}(2^{n-k}-2^{n-2k+k^{\prime}})$ $c\binom{2c-1}{c}\leq(c+1)\cdot 2^{2c-3}=(2c+2)\cdot 2^{2c-4}$ $|\mathcal{V}^{\prime}|$ $001,101$ $\Leftarrow 8c\cdot 2^{n-4}-2^{n-2c-1}\cdot c\binom{2c}{c}\geq(6c-3)\cdot 2^{n-4}$ $Pa(N^{k,n}_{s},V_{n})\cap P=k^{\prime}$ $c\binom{2c-1}{c}\leq(c+1)\cdot 2^{2c-3}\,,$ $Pa(N^{a},V_{n})\subseteq Pa(N_{s},V_{n})$ $\displaystyle(2^{k}-1)\cdot 2^{n-k}+2^{k}\binom{n-k-1}{k}(2^{n-k}-2^{n-2k})$ $f_{T^{k,n}}(N^{k,n}_{s})\geq(3/2)f_{T^{k,n}}(N)$ $Pa(N,V_{n})\subseteq P$ $t-\kappa$ $\mathcal{V}=\{V_{1},\ldots,V_{n-1}\}$ $P\in\{Pa(N_{s},V_{n})\mid 1\leq s\leq t\}$ $\gamma_{2}:1\succ 0$ $\Leftarrow c\binom{2c-1}{c}\leq(2c+2)\cdot 2^{2c-4}$ $V_{i}\in\mathcal{V}^{\prime}$ $o^{\prime}[V_{i}]=1$ $\gamma\in\operatorname{Inst}(\mathcal{V}^{\prime})$ $2^{p_{s}-1}$ $\frac{2\cdot(2d+1)}{d+1}<4$ $Pa(N,V_{i})=\emptyset$ $f_{T_{n}}(N^{*})$ $\gamma_{2}\in\operatorname{Inst}(P)$ $2\leq k\leq n-1$ $Pa(N_{s},V_{n})$ $\gamma[V_{i}]$ $freq_{M^{\prime}}(0\succ 1)\leq freq_{M^{\prime}}(1\succ 0)$ $CPT(N_{i},V_{3})$ $\Delta(N^{k,n}_{s},N^{k,n}_{s^{\prime}}=2^{n-k}$ $\gamma^{\prime}\in\operatorname{Inst}(P)\setminus\{\gamma\}$ $T^{k,n}=(N^{k,n}_{1},\ldots,N^{k,n}_{t})$ $V\in\mathcal{V}^{\prime}\cap\mathcal{V}^{\prime\prime}$ $\gamma\in\operatorname{Inst}(Pa(N_{s},V_{n}))$ $(o^{\prime},o)$ $V\in\mathcal{V}^{\prime}$ $\gamma_{1}=00$ $Pa(N^{k,n}_{s^{\prime}},V_{n})$ $f_{T^{n-1,n}}(N^{n-1,n}_{s})>(2-\varepsilon)f_{T^{n-1,n}}(N)$ $Inst(\mathcal{V}^{\prime})$ $\gamma_{1}\in\operatorname{Inst}(Pa(N^{k,n}_{s},V_{n}))$ $|P|\leq\max\{|Pa(N_{s},V_{n})\mid 1\leq i\leq t\}$ $1\leq s\leq 2^{n-1}$ $T=(N_{1},\ldots,N_{t})$ $\Leftarrow 2c\cdot 2^{n-2}-2^{n-2c-1}\cdot c\binom{2c}{c}\geq(6c-3)\cdot 2^{n-4}$ $\{b,b^{\prime}\}=\{0,1\}$ $\gamma^{\prime}\in\operatorname{Inst}(\mathcal{V}^{\prime})$ $\tilde{\mathcal{O}}(\sqrt{(M+N)/T})$ $\text{SINR}(a,b;\theta)$ $\sqrt{2\log\mathcal{A}\sqrt{T}}$ $\operatorname{NashRegret}(T)$ $\displaystyle\geqslant\sum_{t=1}^{\infty}\log\Phi(f_{t}(\Delta,\sigma_{n}))$ $(\Delta,\sigma_{n}^{2})$ $\displaystyle\tilde{R}_{t+1}(a)=1-\frac{\mathbb{I}_{A_{t}=a}(1-R_{t+1,A_{t},B_%
+{t}})}{X_{t,a}}$ $w_{t}(a,b)=\sqrt{\beta_{t}I_{t}\left(\theta;R_{t+1,A_{t},B_{t}}\mid A_{t}=a,B_%
+{t}=b\right)}\quad\text{ with }\quad\beta_{t}=\frac{2\beta^{\prime}_{t}}{\log(%
+1+\sigma_{w}^{-2})}.$ $\mu_{t}(a,b)=\mathbf{k}_{t}((a,b))^{\top}(\mathbf{K}_{t}+\sigma^{2}{\bm{I}}_{t%
+})^{-1}\mathbf{R}_{t}$ $\eta_{1},\eta_{2},\ldots,\eta_{M}$ $z_{t},z^{\prime}_{t}\sim\mathcal{N}(0,1)$ $\theta\in\mathbb{R}^{\mathcal{A}\times\mathcal{B}}$ $\beta_{t}=2\beta^{\prime}_{t}/\log(1+\sigma_{w}^{-2})$ $\Re_{\operatorname{full}}(T,{\operatorname{Hedge}})=\mathcal{O}(\sqrt{T\log%
+\mathcal{A}})$ $\displaystyle:=\max_{j\in[M_{t+1}]}\tilde{f}^{\operatorname{TS},j}_{t+1}(a,B_{%
+t}),$ $\displaystyle\leqslant\max_{a\in\mathcal{A}}\Re_{\operatorname{full}}(a;T,%
+\operatorname{adv},\tilde{R})+\mathbb{E}\left[\max_{a\in\mathcal{A}}\sum_{t=0}%
+^{T-1}\mathbb{E}\left[\operatorname{pess}_{t+1}(a)\mid\theta\right]\right]+%
+\sum_{t=0}^{T-1}\mathbb{E}\left[\operatorname{est}_{t+1}\right]$ $\tilde{r}_{t}=A_{ij}+\epsilon_{t}$ $N(\mu_{p},\Sigma_{p})$ $M=\frac{\log(\sqrt{t})}{\log\frac{1}{\Phi(\sqrt{\beta^{\prime}_{t}})}}$ $\displaystyle=\mathbb{P}(\sum_{k=2}^{t}m_{k}\leqslant 0)$ $\displaystyle\mathbb{E}\left[\sum_{t=0}^{T-1}\operatorname{est}_{t+1}\right]$ $\displaystyle\mathbb{E}_{t}\left[\tilde{R}_{t+1}(a)\right]=1-\mathbb{E}_{t}%
+\left[\mathbb{I}_{A_{t}=a}\frac{1-R_{t+1,a,B_{t}}}{X_{t,a}}\right]=1-\mathbb{E%
+}_{t}\left[\mathbb{I}_{A_{t}=a}\right]\frac{1-\mathbb{E}_{t}\left[f_{\theta}(a%
+,B_{t})\right]}{X_{t,a}}=\mathbb{E}_{t}\left[f_{\theta}(a,B_{t})\right].$ $\sigma_{t}(a,b)$ $\displaystyle=k((A_{i},B_{i}),(A_{j},B_{j}))$ $\Re(T,\pi^{\operatorname{alg}},B_{0:T},\theta)$ $\displaystyle\mathbb{E}_{0}\left[\sum_{t=1}^{T}\Re_{t}(a)\right]=\mathbb{E}_{0%
+}\left[\sum_{t=1}^{T}\mathbb{E}_{t}\left[\tilde{\Re}_{t,a}\right]+\sum_{t=1}^{%
+T}2\gamma_{t}\right]=\mathbb{E}_{0}\left[\tilde{C}_{T,a}+\sum_{t=1}^{T}2\gamma%
+_{t}\right]\leqslant\sqrt{\sum_{t=1}^{T}\frac{2(1+\sigma_{w}^{2})\left|%
+\mathcal{A}\right|^{2}}{\gamma_{t}}}+2\gamma_{t}$ $\displaystyle=\mathbb{P}(\tilde{R}_{t+1}(a)\geqslant U_{t}(a,B_{t}),\forall a%
+\in\mathcal{A}\mid H_{t},B_{t})$ $\Re_{t}(a):=\mathbb{E}\left[R_{t+1,a,B_{t}}-R_{t+1,A_{t},B_{t}}\mid\theta,H_{t%
+}\right]=\mathbb{E}\left[f_{\theta}(a,B_{t})\mid\theta,H_{t}\right]-\sum_{a}{X%
+}_{t,a}\mathbb{E}\left[f_{\theta}(a,B_{t})\mid\theta,H_{t}\right]$ $\displaystyle\sum_{t=0}^{T-1}\mathbb{E}\left[R_{t+1,A^{*},B_{t}}-R_{t+1,A_{t},%
+B_{t}}\>|\>\theta\right],$ $\displaystyle\leqslant\Re_{\operatorname{full}}(a;T,\operatorname{adv},\tilde{%
+R})+\sum_{t=0}^{T-1}\mathbb{E}\left[\operatorname{pess}_{t+1}(a)\mid\theta%
+\right]+\sum_{t=0}^{T-1}\mathbb{E}\left[\operatorname{est}_{t+1}\>|\>\theta\right]$ $\mathbb{E}\left[\tilde{R}_{t+1}(a)\>|\>H_{t},\theta\right]=\mathbb{E}\left[f_{%
+\theta}(a,B_{t})\>|\>H_{t},\theta\right]=\mathbb{E}\left[g_{\theta}(e_{a},Y_{t%
+})\>|\>H_{t},\theta\right]$ $\displaystyle\leqslant\mathbb{E}\left[\sum_{t=0}^{T-1}\sqrt{\left(\sqrt{2\log(%
+M\sqrt{t})/\beta^{\prime}_{t}}+1\right)^{2}\beta_{t}I_{t}(\theta;A_{t},B_{t},R%
+_{t+1,A_{t},B_{t}})}\right]$ $\Re_{\operatorname{full}}(T,\text{RM},\tilde{R}^{\operatorname{est}})=\mathcal%
+{O}(\sqrt{T\mathcal{A}})$ $(a=b)$ $\mathbb{P}(\Omega_{t})\geqslant c$ $\beta^{\prime}_{t}=2\log\mathcal{A}\sqrt{t}.$ $\displaystyle\Re^{*}(T,\pi^{\operatorname{alg}})=\sup_{B_{0:T}\in\mathcal{B}^{%
+T}}\Re(T,\pi^{\operatorname{alg}},B_{0:T})=o(T),$ $\mathbf{K}_{t}$ $\tilde{R}^{OTS}_{t+1}(1st)>\tilde{R}^{OTS}_{t+1}(2nd)$ $\displaystyle\leqslant\mathbb{E}\left[\max_{a\in\mathcal{A}}\sum_{t=0}^{T-1}%
+\mathbb{E}\left[C(1-\mathds{1}_{\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})\cap%
+\mathcal{E}^{c}_{t}(f_{\theta},B_{t})})\mid\theta\right]\right]$ $\nu\rightarrow\infty$ $\displaystyle\quad+C\left(1-\mathds{1}_{\mathcal{E}^{c}_{t}(\tilde{R},U,A_{t},%
+B_{t})}\mathds{1}_{\mathcal{E}^{c}_{t}(f_{\theta},B_{t})}\right)$ $\displaystyle X_{t,a}=(1-\gamma_{t})\hat{X}_{t,a}+\gamma_{t}(1/\mathcal{A}),%
+\forall a\in\mathcal{A}$ $\textrm{Hedge:}\ g_{t,a}(X_{t},r_{t})=X_{t,a}\exp(\eta_{t}r_{t}(a)),\quad%
+\textrm{RM:}\ g_{t,a}(X_{t},r_{t})=\max\left(0,\sum_{s=0}^{t}r_{t}(a)-r_{t}(A_%
+{s})\right).$ $\displaystyle\mathcal{E}_{t}(f_{\theta},B_{t}):=\{\forall a\in\mathcal{A},f_{%
+\theta}(a,B_{t})\in[L_{t}(a,B_{t}),U_{t}(a,B_{t})]\}.$ $reg_{1}$ $f_{\theta}(a,b)\mid H_{t}$ $[a^{i}]_{e}=0$ $\displaystyle=\mathbb{E}_{0}\left[\sum_{t=1}^{T}\sum_{a}R_{t+1,A_{t},B_{t}}^{2%
+}\left(\left(\frac{\mathbb{I}_{A_{t}=a}}{X_{t,A_{t}}}\right)^{2}-\frac{2\hat{X%
+}_{t,A_{t}}}{X_{t,A_{t}}^{2}}\mathbb{I}_{A_{t}=a}+\frac{\hat{X}_{t,A_{t}}^{2}}%
+{X_{t,A_{t}}^{2}}\right)\right]$ $\tilde{R}_{t1}=\max\limits_{i}x_{ti}$ $\displaystyle\mathbb{P}(\neg\mathcal{E}_{t}(\tilde{R},U,B_{t}))\leqslant\frac{%
+1}{\sqrt{t}}.$ $\displaystyle\mathbf{K}_{t}(i,j)$ $\nu>1$ $\tilde{R}_{t+1}(a)$ $\tilde{R}:=\{\tilde{R}_{t}:t\in\mathbb{Z}_{++}\}$ $\mathbb{P}(A_{t}\in\cdot\>|\>\pi_{t})=\mathbb{P}(A_{t}\in\cdot\>|\>H_{t})=\pi_%
+{t}(\cdot)$ $\displaystyle\begin{cases}\textrm{UCB:}&\tilde{f}_{t+1}(a,B_{t})\>|\>H_{t+1}=%
+\mu_{t}(a,B_{t})+\beta_{t}\sigma_{t}(a,B_{t}),\\
+&\tilde{R}_{t+1}(a)=\tilde{f}_{t+1}(a,B_{t})\wedge 1,\forall a\in\mathcal{A}.%
+\\
+\textrm{TS:}&\tilde{f}_{t+1}(a,B_{t})\>|\>H_{t+1}\sim N(\mu_{t}(a,B_{t}),%
+\sigma_{t}(a,B_{t})),\\
+&\tilde{R}_{t+1}(a)=\tilde{f}_{t+1}(a,B_{t})\wedge 1,\forall a\in\mathcal{A}.%
+\end{cases}$ $\Sigma_{0}=\Sigma_{p}$ $\displaystyle\leqslant(1+\sigma_{w}^{2})\left(\sum_{a}\frac{1}{X_{t,a}}+\sum_{%
+a}\frac{\hat{X}_{t,a}}{{X}_{t,a}}\left(\left|\mathcal{A}\right|-2\right)\right)$ $\mathds{1}(a=b)$ $\operatorname{est}$ $\mathbf{k}_{t}((a,b))$ $\mathcal{E}_{t}(\tilde{R},U^{\prime},A_{t},B_{t})$ $\displaystyle\leqslant\sum_{a}\left(\left|\gamma_{t}\hat{X}_{t,a}\right|+\left%
+|\gamma_{t}/\mathcal{A}\right|\right)=2\gamma_{t}$ $\mathcal{B}={1,\ldots,\left|\mathcal{B}\right|}$ $M=\frac{\log(\sqrt{t})}{\log\frac{1}{\Phi(\sqrt{\beta^{\prime}_{t}})}}.$ $\tilde{R}_{t+1}(a)=\min(\tilde{f}_{t+1}(a,B_{t}),1)$ $\displaystyle\mathbb{E}_{0}\left[\sum_{t=1}^{T}\Re_{t}(a)\right]\leqslant\sqrt%
+{T}\sqrt{\frac{2(1+\sigma_{w}^{2})\left|\mathcal{A}\right|^{2}}{\gamma}}+2%
+\gamma T,$ $\displaystyle\mathbb{P}\left(\max_{j\in[M]}\eta_{j}\geqslant w\right)=1-\left[%
+\Phi\left(\frac{w}{\sigma}\right)\right]^{M}$ $Z_{t}=(A_{t},B_{t},R_{t+1,A_{t},B_{t}}))$ $(r_{t})_{t\in[T]}\in[0,1]^{\mathcal{A}\times T}$ $\displaystyle\stackrel{{\scriptstyle(iii)}}{{=}}1-\Phi\left(\sqrt{\beta_{t}^{%
+\prime}}\right)^{M}.$ $\displaystyle>-\infty,$ $KL(\overline{y}_{T},y^{)}$ $\displaystyle\tilde{R}_{t+1}(a)-f_{\theta}(a,B_{t})=\mathds{1}_{\mathcal{E}^{o%
+}_{t}(\tilde{R},U,B_{t})}(\tilde{R}_{t+1}(a)-f_{\theta}(a,B_{t}))+(1-\mathds{1%
+}_{{\mathcal{E}}^{o}_{t}(\tilde{R},U,B_{t})})(\tilde{R}_{t+1}(a)-f_{\theta}(a,%
+B_{t}))$ $\mathcal{A}={1,\ldots,\left|\mathcal{A}\right|}$ $\Re^{*}(T,\text{UCB-Hedge})=\mathcal{O}(\sqrt{T\log\mathcal{A}}+\sqrt{\gamma_{%
+T}\beta T}),\ \Re^{*}(T,\text{UCB-RM})=\mathcal{O}(\sqrt{T\mathcal{A}}+\sqrt{%
+\gamma_{T}\beta T}).$ $\Re_{\operatorname{full}}(T,\text{Hedge},\tilde{R}^{\operatorname{est}})=%
+\mathcal{O}(\sqrt{T\log\mathcal{A}})$ $U=(\mu_{t}(a,b)+\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b):t\in\mathbb{N}),\quad
+L%
+=(\mu_{t}(a,b)-\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b):t\in\mathbb{N}).$ $c(\Delta,\sigma_{n})\approx 0.54$ $k_{\rm P}(\cdot,\cdot)$ $\sigma_{n}>0$ $k((a,b),(a^{\prime},b^{\prime}))=\mathbb{E}\left[(f_{\theta}(a,b)-\mu(a,b))(f_%
+{\theta}(a^{\prime},b^{\prime})-\mu(a^{\prime},b^{\prime}))\right]$ $\displaystyle=[k((A_{0},B_{0}),(a,b)),\ldots,k((A_{t-1},B_{t-1}),(a,b))]^{\top}$ $\sigma_{t}^{2}(a,b)$ $\displaystyle\leqslant\sum_{a}\left(\left({\tilde{C}_{T-1,a}^{+}}\right)^{2}+2%
+{\tilde{C}_{T-1,a}^{+}}{\tilde{\Re}_{T,a}}+\left(\tilde{\Re}_{T,a}\right)^{2}\right)$ $A\in\mathbb{R}^{10\times 5}$ $\displaystyle\tilde{R}_{t+1}(A_{t})-f_{\theta}(A_{t},B_{t})$ $g(a^{-i})=\sum_{j\neq i}a^{j}$ $\gamma_{t}=\gamma$ $P_{a}(\theta)$ $\displaystyle\sum_{a}\left({\tilde{C}_{T,a}^{+}}\right)^{2}$ $\sum\limits_{k=1}^{t}\frac{1}{k+a}\leqslant\int_{0}^{t}\frac{1}{k+a}dk$ $y^{*}=\operatorname*{arg\,min}\limits_{y\in\Delta}y^{T}(Ax),$ $\displaystyle\mathbb{P}(\max_{j\in[M]}\eta_{j}\geqslant w)=1-\mathbb{P}(\max_{%
+j\in[M]}\eta_{j}\leqslant w)=1-\mathbb{P}(\forall j\in[M],\eta_{j}\leqslant w)%
+=1-\left[\Phi\left(\frac{w}{\sigma}\right)\right]^{M}.$ $m_{1}\leqslant 0$ $\tilde{R}_{t+1}\in[0,C]$ $\mathcal{D}(\mathcal{A}^{i})$ $\displaystyle\sum_{t=1}^{T}\frac{1}{\sqrt{t}}=2\sum_{t=1}^{T}\frac{t-(t-1)}{%
+\sqrt{t}+\sqrt{t}}\leqslant 2\sum_{t=1}^{T}\frac{t-(t-1)}{\sqrt{t}+\sqrt{t-1}}%
+=2\sum_{t=1}^{T}(\sqrt{t}-\sqrt{t-1})=2\sqrt{T}$ $50)$ $\mu_{0}=\mu_{p}$ $\text{Gap}(x,y)=\max_{\left(x^{\prime},y^{\prime}\right)\in\Delta}\mathbb{E}%
+\left[\phi\left(x^{\prime},y\right)-\phi\left(x,y^{\prime}\right)\right]$ $Y_{t+1,A_{t},B_{t}}=f_{\theta}(A_{t},B_{t})+\eta_{t},$ $(1-x)^{M}\leqslant e^{-Mx}$ $\theta(a,b)\in\mathbb{R}$ $\Re_{\operatorname{full}}(T,{\operatorname{Hedge}},(r_{t})_{t})=\mathcal{O}(%
+\sqrt{T\log\mathcal{A}}),\ \Re_{\operatorname{full}}(T,{\operatorname{RM}},(r_%
+{t})_{t})=\mathcal{O}(\sqrt{T\mathcal{A}}).$ $f_{t}(\Delta,\sigma_{n}^{2})$ $\displaystyle=\max_{(r_{t})_{t}}\Re_{\operatorname{full}}(T,{\operatorname{adv%
+}},(r_{t})_{t}).$ $I(\theta;o)=\frac{1}{2}\log\left(1+{\sigma_{w}^{-2}\sigma(a)}\right)$ $(\overline{x}_{T},\overline{y}_{T})$ $N\left(0,\sigma^{2}\right)$ $v^{+}=(v_{a}^{+})_{a\in\mathcal{A}}$ $\displaystyle\leqslant\mathcal{O}\left(\sqrt{T\log(\mathcal{A}T)I(\theta;H_{T}%
+)}\right).$ $\pi=(\pi_{t})_{t\in\mathbb{N}}$ $\mathbb{I}_{A_{t}=a}R_{t+1,A_{t},B_{t}}=\mathbb{I}_{A_{t}=a}R_{t+1,a,B_{t}}$ $\displaystyle=\mathbb{E}_{0}\left[\sum_{t=1}^{T}\mathbb{E}_{t}\left[R_{t+1,A_{%
+t},B_{t}}^{2}\left(\frac{1}{X_{t,A_{t}}^{2}}-\frac{2\hat{X}_{t,A_{t}}}{X_{t,A_%
+{t}}^{2}}+\left|\mathcal{A}\right|\frac{\hat{X}_{t,A_{t}}^{2}}{X_{t,A_{t}}^{2}%
+}\right)\right]\right],$ $M=5,10,20,50,70,100$ $\displaystyle=\mathbb{E}_{0}\left[\sum_{t=1}^{T}\sum_{a}\left(\frac{\mathbb{I}%
+_{A_{t}=a}R_{t+1,A_{t},B_{t}}}{X_{t,a}}-R_{t+1,A_{t},B_{t}}\frac{\hat{X}_{t,A_%
+{t}}}{X_{t,A_{t}}}\right)^{2}\right]$ $c\geqslant 0$ $k(x,x^{\prime})=x^{\top}x^{\prime}$ $U^{\prime},U,L$ $\Re_{\operatorname{full}}(T,{\operatorname{RM}})=\mathcal{O}(\sqrt{T\mathcal{A%
+}})$ $\displaystyle\geqslant\sum_{t=1}^{\infty}\log\Phi\left(\frac{(1-\Delta)\left(t%
+-\sigma_{n}^{2}\ln{(t+\sigma_{n}^{2})}+2\sigma_{n}^{2}\ln{\sigma_{n}}-1/(%
+\sigma_{n}^{2}+1)\right)}{\sqrt{t+\sigma_{n}^{2}\ln{(t+\sigma_{n}^{2})}-2%
+\sigma_{n}^{2}\ln{\sigma_{n}}-(\sigma_{n}^{2}+2)/(\sigma_{n}^{2}+1)}}\right)$ $A\in\mathcal{R}^{10\times 5}$ $U^{\prime}=(\mu_{t}(a,b)+\sqrt{2\log(M\sqrt{t})}\sigma_{t}(a,b):t\in\mathbb{N}).$ $\tilde{R}^{OTS}_{t+1}(2nd)$ $\displaystyle\leqslant\sqrt{\mathbb{E}\left[\sum_{t=0}^{T-1}\left(\sqrt{2\log(%
+M\sqrt{t})/\beta^{\prime}_{t}}+1\right)^{2}\beta_{t}\right]}\sqrt{\mathbb{E}%
+\left[\sum_{t=0}^{T-1}I_{t}(\theta;A_{t},B_{t},R_{t+1,A_{t},B_{t}})\right]}$ $(P^{*},Q^{*})$ $\pi_{\operatorname{alg}}$ $\pm\tilde{r}_{t}$ $\displaystyle\Sigma_{t+1}=\left(\Sigma_{t}^{-1}+\frac{1}{\sigma_{w}^{2}}\phi(A%
+_{t},B_{t})\phi(A_{t},B_{t})^{\top}\right)^{-1}$ $\mathbb{P}(\Omega_{t})\geqslant c(\Delta,\sigma_{n})$ $\mathcal{E}^{c}_{t}(f_{\theta},B_{t})$ $\sum_{t=0}^{T-1}\mathbb{E}\left[(U^{\prime}_{t}(A_{t},B_{t})-L_{t}(A_{t},B_{t}%
+))\right]$ $\mu_{t}(a,b)=\phi(a,b)^{\top}\mu_{t}$ $c=e^{c^{\prime}}$ $\displaystyle\mathbb{E}\left[\tilde{h}(a)\right]=\mathbb{E}\left[\mathbb{I}_{A%
+=a}\right]h(a)/X_{a}=h(a).$ $\displaystyle\stackrel{{\scriptstyle(i)}}{{\geqslant}}\mathbb{P}(\tilde{R}_{t+%
+1}(a)\geqslant\min\{\mu_{t}(a,B_{t})+\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,B_{%
+t}),c\},\forall a\in\mathcal{A}\>|\>H_{t},B_{t})$ $\displaystyle=\sum_{t=1}^{\infty}\log\mathbb{P}(Z\leqslant-\frac{\mu_{t}}{%
+\sigma_{t}})$ $f_{\theta}(a)\sim N(\mu(a),\sigma(a))$ $\displaystyle=\mathbb{E}\left[\sum_{t=0}^{T-1}\mathbb{E}\left[C(1-\mathds{1}_{%
+\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})\cap\mathcal{E}^{c}_{t}(f_{\theta},B_{t}%
+)})\right]\right]$ $31.9\pm 1.6$ $46.4\pm 0.8$ $\text{Uniform}[0.05,0.1]$ $71.6\%\rightarrow 78.9\%$ $65.2\pm 1.1$ $\phi:\mathcal{X}\rightarrow\mathbb{R}^{k}$ $93.8\pm 0.3$ $b\in{\bm{b}}$ ${\rho^{\prime}},{\alpha^{\prime}},{\beta^{\prime}}$ $\mathcal{A}_{\text{ft}}(x^{\prime}\mid x)=1$ $\frac{{\beta}}{{\gamma}}$ $\displaystyle\mathcal{L}_{\text{pretrain}}(\phi)=-2\cdot$ ${\alpha^{\prime}}$ $67.84\pm 0.70$ $32.1\pm 0.8$ $31.2\%\rightarrow 31.9\%$ $50.6\pm 1.3$ $91.4\pm 0.9$ $\displaystyle\mathcal{A}_{\text{prop}}({x^{\prime}}\mid x)=\begin{cases}{\rho^%
+{\prime}}&x={x^{\prime}}\\
+{\alpha^{\prime}}&\{x^{\prime},x\}\in\{\{1,3\},\{3,5\},\{5,7\},\{2,4\},\{4,6\}%
+,\{6,8\},\{1,7\},\{2,8\}\\
+{\beta^{\prime}}&\{{x^{\prime}},x\}\in\{\{1,2\},\{3,4\},\{5,6\},\{7,8\}\}\\
+{\gamma^{\prime}}&\{{x^{\prime}},x\}\in\{\{1,4\},\{2,3\},\{3,6\},\{4,5\},\{5,8%
+\},\{6,7\},\{1,8\},\{2,7\}\}\\
+\end{cases}.$ $30.4\%\rightarrow 34.5\%$ $\text{21M}\rightarrow\text{69M}$ $27.9\pm 0.5$ $y_{x}\in\mathbb{R}^{k}$ $p^{*}(\cdot\mid x)$ $\text{loguniform}(0.95z,\;\text{min}(1.5(1+z)-1,\;5z))$ ${\bm{X}}^{\prime}=\{\tilde{{\bm{X}}^{\prime}}_{:,j}+\epsilon_{j}\}_{j=1}^{W},{%
+\bm{X}}^{\prime}_{\text{err}}=\left\{\sqrt{\tilde{{\bm{X}}^{\prime}}_{\text{%
+err},:,j}^{2}+\epsilon_{j}^{2}}\right\}_{j=1}^{W}.$ $z=z^{\prime}$ $46.4\%\rightarrow 50.6\%$ $\{F(t_{i},w_{j})\}_{i=1,j=1}^{T,W}$ ${\rho^{\prime}}>\max\{{\alpha^{\prime}},{\beta^{\prime}}\}$ $\mathbf{51.7\pm 0.8}$ $65.2\%\rightarrow 91.4\%$ $64.5\pm 1.2$ $i\in\{1,...,T\},j\in\{1,...,W\}$ $0.320\pm 0.009$ $92.3\pm 0.7$ $\mathcal{T}=\{3,4,5,6,7,8\}$ $0.304\pm 0.010$ $\mathcal{A}_{\text{ft}}(x^{\prime}\mid x)=\sum_{z^{\prime}}T(x^{\prime}\mid x,%
+z^{\prime})\hat{p_{T}}(z^{\prime}\mid z)$ $67.54\pm 0.32$ $\mathbf{79.90\pm 0.60}$ $\{F^{\prime}_{\text{err}}(t_{\text{new},i},w_{\text{new},i})\}_{i=1,j=1}^{T,W}$ $\widehat{f}_{\text{erm}}\in\operatorname*{arg\,min}_{f}\mathcal{L}_{\text{ERM}%
+}(f)$ $\{F(t_{i},b_{j})\}_{i=1,j=1}^{T,W},\{F_{\text{err}}(t_{i},b_{j})\}_{i=1,j=1}^{%
+T,W}$ $\mathbf{0.247\pm 0.005}$ ${\bm{X}}^{\prime}_{\text{err}}$ $\mathcal{L}_{0-1}(\widehat{f})=0$ $T(x^{\prime}|x,z^{\prime})$ $((y_{1},d_{1}),(y_{2},d_{2}))$ $x\in\{1,3,5,7\}$ $46.4\%\rightarrow 48.5\%$ $\displaystyle\mathcal{A}_{\text{pre}}({x^{\prime}}\mid x)=\begin{cases}{\rho^{%
+\prime}}&x={x^{\prime}}\\
+{\alpha^{\prime}}&\{x^{\prime},x\}\in\{\{1,4\},\{3,5\},\{5,7\},\{2,5\},\{4,6\}%
+,\{6,8\},\{1,8\},\{2,7\}\\
+{\beta^{\prime}}&\{{x^{\prime}},x\}\in\{\{1,2\},\{3,4\},\{5,6\},\{7,8\}\}\\
+{\gamma^{\prime}}&\{{x^{\prime}},x\}\in\{\{1,3\},\{2,4\},\{3,6\},\{4,5\},\{5,8%
+\},\{6,7\},\{1,7\},\{2,8\}\}\\
+\end{cases}.$ $y_{x}=1$ $0.27\rightarrow 0.25$ $96.7\pm 0.0$ $48.5\pm 3.2$ $\widehat{\phi}=\operatorname*{arg\,min}_{\phi}\mathcal{L}_{\text{pretrain}}(\phi)$ $46.4\%\rightarrow 47.5\%$ $d_{-}$ $0.277\pm 0.004$ $77.72\pm 0.59$ $\mathcal{T}=\{x\in\mathcal{T}:d_{x}=2\}$ $0.274\pm 0.016$ $\displaystyle\mathcal{L}_{\text{ERM}}(f)=\mathbb{E}_{x\sim P_{S},x^{\prime}%
+\sim\mathcal{A}_{\text{ft}}(\cdot\mid x)}[\ell(f(x^{\prime}),y_{x})].$ $\mathbf{36.9\pm 0.7}$ $\mathcal{A}_{\text{ft}}$ $34.5\pm 1.4$ $71.6\%\rightarrow 68.8\%$ ${\beta}>{\gamma}$ $\mathcal{L}_{0-1}(\widehat{f}_{\text{erm}})=1/3$ $\mathbf{98.5\pm 0.0}$ $\text{SNR}(x,x_{\text{err}})=\frac{|x|}{x_{\text{err}}}$ $30.4\%\rightarrow 32.1\%$ $x\in\{2,4,6,8\}$ $\text{SNR}({\bm{X}}^{\prime}_{i,j},{\bm{X}}^{\prime}_{\text{err},i,j})\geq 5$ $\displaystyle\mathcal{A}_{\text{ft}}({x^{\prime}}\mid x)=\begin{cases}1&\{{x^{%
+\prime}},x\}\in\{1,4\},\{2,3\}\\
+1&x={x^{\prime}}\text{ and }x\notin\{1,2\}\\
+0&\text{otherwise}\end{cases}$ $61.26\pm 1.10$ $\widehat{f}_{\text{erm}}$ $x,{x^{\prime}}$ $\text{loss}_{\text{ft}}:\mathbb{R}^{n}\times\mathcal{Y}\rightarrow\mathbb{R}$ $86.1\pm 1.3$ $\tilde{x^{\prime}}$ $96.7\%\rightarrow 98.5\%$ $P_{U}=\beta P_{S}+(1-\beta)P_{T}$ $\min\{{\alpha^{\prime}},{\beta^{\prime}}\}>{\gamma^{\prime}}$ $F^{\prime},F^{\prime}_{\text{err}}$ $z^{\prime}\sim\text{loguniform}(0.95z_{\text{orig}},\;\text{min}(1.5(1+z_{%
+\text{orig}})-1,\;5z_{\text{orig}}))$ $90.5\pm 0.4$ $\widehat{f}(x)=\operatorname*{arg\,max}_{i\in[r]}(\widehat{B}\widehat{\phi}(x)%
+)_{i}$ $92.3\pm 0.2$ ${x^{\prime}}\in\mathcal{X}$ $65.2\rightarrow 91.4$ $\tilde{{\bm{X}}^{\prime}}_{\text{err}}=10^{0.4(d(z^{\prime})-d(z_{\text{orig}}%
+))}\{F^{\prime}_{\text{err}}(t_{\text{new},i},w_{\text{new},j})\}_{i=1,j=1}^{T%
+,W},$ $\alpha>\gamma+\beta$ $\displaystyle\mathcal{L}_{\text{ft}}(h)=\mathbb{E}_{x\sim P_{S},y\sim p^{*}(%
+\cdot\mid x),{x^{\prime}}\sim\mathcal{A}_{\text{ft}}(\cdot|x)}[\text{loss}_{%
+\text{ft}}(h(\widehat{\phi}({x^{\prime}})),\;y;\;\theta)]$ $\displaystyle\mathcal{L}_{\text{pretrain}}(\phi)=\mathbb{E}_{(x,x^{+})\sim S_{%
++}}[d_{+}(\phi(x),\phi(x^{+}))]-\mathbb{E}_{x,{x^{\prime}}\sim P_{U}}[d_{-}(%
+\phi(x),\phi({x^{\prime}}))].$ $93.8\%\rightarrow 94.9\%$ $61.3\%\rightarrow 67.8\%$ $d(z)$ $0.32\rightarrow 0.28$ $\{F_{\text{err}}(t_{i},w_{j})\}_{i=1,j=1}^{T,W}$ $\mathcal{A}_{\text{pre}}(\cdot\mid x)$ ${\bm{t}}_{\text{new}}=\frac{1+z^{\prime}}{1+z_{\text{orig}}}{\bm{t}}$ $\hat{p_{T}}(z^{\prime}|z)$ $78.84\pm 0.97$ $40.5\pm 1.6$ $\mathcal{A}(\cdot|x)$ $\mathcal{A}_{\text{prop}}$ $78.9\%\rightarrow 68.8\%$ $\gamma>\beta$ $\hat{p_{T}}(z^{\prime}\mid z)$ $46.4\pm 0.5$ $0.286\pm 0.007$ $30.4\rightarrow 31.2$ $30.4\%\rightarrow 37.2\%$ $10^{\text{Uniform}[-3,-2]}$ $36.1\pm 0.7$ $89.3\%\rightarrow 92.3\%$ $30.4\pm 0.6$ $\displaystyle\begin{cases}{\rho}&y_{1}=y_{2},d_{1}=d_{2}~{}\text{~{}~{}(same %
+class, same domain)}\\
+{\alpha}&y_{1}=y_{2},d_{1}\neq d_{1}\text{~{}~{}(same class, different domain)%
+}\\
+{\beta}&y_{1}\neq y_{2},d_{1}=d_{2}\text{~{}~{}(different class, same domain)}%
+\\
+{\gamma}&y_{1}\neq y_{2},d_{1}\neq d_{2}\text{~{}~{}(different class and %
+domain)}\\
+\end{cases},$ $68.75\pm 0.95$ $\mathbf{94.9\pm 0.4}$ $10^{\text{Uniform}[-5,-2]}$ $\mathcal{S}=\{1,2\}$ $F,F_{\text{err}}:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ $h:\mathbb{R}^{k}\rightarrow\mathbb{R}^{n}$ $77.40$ $\mathcal{Y}=\{1,\dots,k\}$ $\mathbf{80.54\pm 1.20}$ $S_{+}(x,{x^{\prime}})$ $71.59\pm 1.10$ $z_{\text{orig}}$ $\displaystyle\mathbb{E}_{(x,x^{+})\sim S_{+}}\left[\phi(x)^{\top}\phi(x^{+})%
+\right]+\mathbb{E}_{x,x^{\prime}\sim P_{U}}\left[\left(\phi(x)^{\top}\phi(x^{%
+\prime})\right)^{2}\right].$ $L_{T}(f)=\mathbb{E}_{x\sim P_{T},y\sim p^{*}(\cdot\mid x)}[\ell(f(x),y)]$ $89.3\pm 0.9$ $\mathbf{51.4\pm 0.6}$ $\mathcal{A}_{\text{pre}}$ ${\bm{X}}^{\prime},{\bm{X}}^{\prime}_{\text{err}}$ $89.3\rightarrow 92.3$ $65.15\pm 0.67$ ${\bm{w}}_{\text{new}}=\frac{1+z^{\prime}}{1+z_{\text{orig}}}{\bm{w}}$ $62.3\pm 1.9$ $36.3\%\rightarrow 37.2\%$ $47.5\pm 1.0$ $y_{x}=-1$ ${\rho},{\alpha},{\beta},{\gamma}$ $46.4\rightarrow 46.4$ $\mathbf{0.256\pm 0.005}$ ${\bm{X}}_{\text{err}}\in\mathbb{R}^{T\times W}$ $\epsilon\in\mathbb{R}^{W}$ $\widehat{\phi}$ $\widehat{\phi}:\mathcal{X}\to\mathbb{R}^{k}$ $d_{+}$ ${\bm{X}}\in\mathbb{R}^{T\times W}$ ${\alpha}>{\gamma}$ $({\bm{t}}_{\text{new}},{\bm{w}}_{\text{new}})$ $31.2\pm 0.6$ $0.289\pm 0.003$ $0.310\pm 0.006$ $\displaystyle\mathcal{L}_{\text{MAE}}(\phi)=\mathbb{E}_{x\sim P_{U},{x^{\prime%
+}}\sim\mathcal{A}_{\text{pre}}(\cdot\mid x)}[(\phi({x^{\prime}})-x)^{2}]$ $\mathbf{0.246\pm 0.015}$ $\frac{{\alpha}}{{\gamma}}$ $\tilde{{\bm{X}}^{\prime}}=10^{0.4(d(z^{\prime})-d(z_{\text{orig}}))}\{F^{%
+\prime}(t_{\text{new},i},w_{\text{new},j})\}_{i=1,j=1}^{T,W},$ $\mathcal{S}=\{x\in\mathcal{X}:d_{x}=1\}$ $92.3\%\rightarrow 96.7\%$ $\displaystyle\mathcal{L}(B)=\mathbb{E}_{x\sim P_{S}}\left[\ell(B\widehat{\phi}%
+(x),y_{x})\right]+\eta\|B\|_{F}^{2},$ $x\sim P_{S}$ $({\bm{t}},{\bm{w}})$ $\text{Uniform}[0.5,0.9]$ $S_{+}(x,x^{+})=\mathbb{E}_{\bar{x}\sim P_{U}}[\mathcal{A}_{\text{pre}}(x\mid%
+\bar{x})\mathcal{A}_{\text{pre}}(x^{+}\mid\bar{x})]$ $\{F^{\prime}(t_{\text{new},i},w_{\text{new},j})\}_{i=1,j=1}^{T,W}$ $\widehat{h}=\operatorname*{arg\,min}_{h}\mathcal{L}_{\text{ft}}(h)$ $5Ã$ $0.8\AA$ $\left(\,\overline{\text{Ch.}},\text{Ch.}\,\right)$ $match(Ch.,\cdot{})$ $\left(\,\text{Ch.},\overline{\text{Ch.}}\,\right)$ $\\
+A$ $\min(match)$ $\left(\,\text{H.2},\overline{\text{H.2}}\,\right)$ $10\AA$ $\max(d_{\rho})$ $match(\cdot{},Ch.)$ $B(\mathbf{p},\rho_{\mathbf{p}})$ $\gamma:M\rightarrow M^{\prime}$ $d_{\rho}(C,C^{\prime}):=\sqrt{\sum_{\mathbf{p}\in M}|\,\rho_{\mathbf{p}}-\rho_%
+{\gamma(\mathbf{p})}\,|^{2}},$ $\overline{\text{H.1}}$ $\AA$ $\mathbf{p}^{\prime}\in C^{\prime}$ $d_{\rho}(C,C^{\prime})$ $d_{\rho}$ $\overline{\text{Ch.}}$ $1.4\AA$ $\mathbf{p}_{N}$ $match(C,C^{\prime})$ $\left(\,\overline{\text{H.1}},\text{H.1}\,\right)$ $\alpha-\pi$ $V(C,\rho):=\iiint_{\bigcup_{\mathbf{p}\in{}C}B(\mathbf{p},\rho_{\mathbf{p}})}1dxdydz.$ $\overline{\text{H.2}}$ $\left(\,\overline{\text{H.2}},\text{H.2}\,\right)$ $L(C):=\sum_{j=1}^{N}\|\mathbf{p}_{j}-\mathbf{p}_{j-1}\|_{2}.$ $s(C):=\dfrac{1}{tortuousness(C)},$ $\left(\,\text{H.1},\overline{\text{H.1}}\,\right)$ $-\mathbf{v}$ $\alpha_{t}x_{t}+\beta_{t}\epsilon_{t}$ $\max_{x^{adv}}\ \ J(x^{adv},y)\ \ \ \ s.t.\left\|x-x^{adv}\right\|_{\infty}<\epsilon.$ $\max_{x^{adv}}\ \ \left\|f^{m}(x,p)-f^{m}(x^{adv},p)\right\|_{2}\ \ \ \ s.t.%
+\left\|x-x^{adv}\right\|_{\infty}<\epsilon.$ $f^{m}(\cdot)$ $\epsilon^{\prime}=0.01$ $\left\|x-x^{adv}\right\|_{p}<\epsilon$ $\displaystyle\mathbb{E}[\sum_{t=0}^{m}\gamma^{t}r(y_{{\mathrm{LM}},t})],{\rm s%
+.t.,}y_{{\mathrm{LM}},t}\sim M_{\mathrm{LM}}(\cdot|\hat{s_{t}},x),$ $c_{i}=Z_{1}(f_{i})$ ${}^{\clubsuit,\heartsuit}$ $M_{\mathrm{LM}}$ $\rm answer$ $s_{0}=[-1]$ $P^{\prime\prime}=\{c_{i}\}_{i=0}^{n-1}$ $S_{\mathrm{semantic}}$ $V=\{v_{1},...,v_{m}\}$ $P^{\prime}=\{f_{i}\}_{i=0}^{n-1}$ $\rm question$ $k=a_{t}$ $(\rm question,\rm context,\rm answer)$ $q=Z_{2}(l)$ $\zeta(y,\hat{y})=\lambda\cdot S_{\mathrm{textual}}(y,\hat{y})+(1-\lambda)\cdot
+S%
+_{\mathrm{semantic}}(y,\hat{y}),$ $\displaystyle\underset{\theta}{\mathrm{max}}$ $r=\alpha\cdot\zeta(y,\hat{y})$ $l=\mathrm{concat}(g,h)$ $P=\{p_{i}\}_{i=0}^{n-1}$ $\rm context$ $\hat{s_{t}}\sim\prod_{i=0}^{t}\pi_{\theta}(a_{i}|s_{ $S_{\mathrm{textual}}$ $\{v_{i}\}_{i=0}^{m}$ $s_{t}=\mathrm{append}(s_{t-1},a_{t})$ $\pi_{\theta}(a_{t}|s_{ $v_{t+1}=p_{k}$ $y_{\mathrm{LM}}$ ${}^{*~{}\heartsuit}$ $M_{\mathrm{LM}}(\cdot|v_{0},v_{1},...,v_{m},x)$ $(v_{0},...,v_{t})\times a_{t}\rightarrow(v_{0},...,v_{t},v_{t+1})$ $\underset{V\subset P}{\mathrm{max}}R(y_{\mathrm{LM}}\sim M_{\mathrm{LM}}(\cdot%
+|v_{0},v_{1},...,v_{m},x)),$ $s_{0}=(v_{0},x)$ $\begin{cases}1&\text{ if }S\left({V_{\left(i,j\right)}}\right)>1,\\
+x_{(i,j)}&\text{ if }S\left({V_{\left(i,j\right)}}\right)=1,\\
+0&\text{otherwise}\end{cases}$ $x_{\left(i,j\right)}$ $Q:\{0,1\}$ $u{\left({x}\right)}^{t}$ $MFA=\left\langle{\mathbb{Z}^{2},Q,V,F,w}\right\rangle$ $Q:{Wb}_{i}=0$ $f(0,x_{i+1},x_{i+2})=f(1,x_{i+1},x_{i+2})$ $\left[0,51,204,255\right]$ $\left({x_{i-1},x_{i},x_{i+1},x_{i+2}}\right)$ $\Delta m_{\text{init}-t}=\sum_{i=0}^{n}{x_{i}^{t_{0}}}-\sum_{i=0}^{n}{x_{i}^{t}}$ $f(x_{i},x_{i+1},x_{i+2})$ $\displaystyle x^{t+1}_{i}=\begin{cases}x^{t}&\text{if }w^{t}=1\\
+f\left({u\left({x_{i}}\right)}^{t}\right)&\text{otherwise}\end{cases}$ $Q:{Wb}_{i}=1$ $f(x_{i-1},x_{i},x_{i+1})$ $\begin{split}&\delta_{t}=\frac{\Delta(X_{t-1},X_{t})}{n}\quad\text{with}\quad t%
+\geq 1,n\in\mathbb{N}\quad\text{and}\\
+&\Delta(X_{t-1},X_{t})=X_{t-1}\oplus X_{t}\end{split}$ $m_{t}=\sum_{i=0}^{n}{x_{i}^{t}}$ $V\left(i_{0},j_{0}\right)=\left\{{\left({i,j}\right):\mid{i-i_{0}}\mid+\mid{j-%
+j_{0}}\mid\leq 1}\right\}$ $m_{\text{inter}}=\sum_{i=0}^{m}{\sum_{j=0}^{m}{x_{t_{n}}^{F,w_{0}}(i,j)}}-\sum%
+_{i=0}^{m}{\sum_{j=0}^{m}{x_{t_{n}}^{F,w_{k}}(i,j)}}$ $w=\left\{{H1V1},{H2V2},{H4V4},\text{cut\&rel}\right\}$ $\begin{cases}1&\text{ if }S\left({V_{\left(i,j\right)}}\right)=2,\\
+0&\text{otherwise}\end{cases}$ $x^{t_{0}}$ $\left({HV}\right)^{20}$ ${Wb}_{i}$ $f(x_{i-1},x_{i},0)=f(x_{i-1},x_{i},1)$ $\left[1,10,30,60\right]$ $\delta_{\text{inter}}=\frac{{X_{t_{n}}^{F,w_{0}}}\oplus{X_{t_{n}}^{F,w_{k}}}}{%
+N^{2}}$ $w^{t+1}={g\left({u\left({x}\right)}^{t}\right)}$ $x^{t+1}_{i}=\begin{cases}&\text{if $w^{t}_{i}=1$ and $f\left(x^{t}_{i-1},x^{t}%
+_{i},1\right)\neq f\left(x^{t}_{i-1},x^{t}_{i},0\right)$}\\
+x^{t}_{i}&\qquad\qquad\qquad\text{ or}\\
+&\text{if $w^{t}_{i}=1$ and $f\left(1,x^{t}_{i},x^{t}_{i+1}\right)\neq f\left(%
+0,x^{t}_{i},x^{t}_{i+1}\right)$}\\
+\\
+f(x^{t}_{i-1},x^{t}_{i},x^{t}_{i+1})&\text{otherwise}\end{cases}$ $\left({HV}\right)^{*}$ ${Wb}_{\left(i,j\right)}$ $\left({HHVV}\right)^{*}$ $t_{0},t_{1},\ldots,t_{n}$ ${H,V}$ $\begin{split}&\delta_{\text{init state}}=\frac{\Delta(X_{t_{0}},X_{t})}{n}%
+\quad\text{with}\quad t\geq 1,n\in\mathbb{N}\quad\text{and}\\
+&\Delta(X_{t_{0}},X_{t})=X_{t_{0}}\oplus X_{t}\end{split}$ $\displaystyle x^{t+1}_{i}=\begin{cases}0&\text{if }w^{t}=1\\
+f\left({u\left({x_{i}}\right)}^{t}\right)&\text{otherwise}\end{cases}$ $x(i,j)$ $x_{t_{n}}^{F,w}(i,j)[0\mapsto-1]=\begin{cases}x_{t_{n}}^{F,w}(i,j)&\text{ if }%
+x_{t_{n}}^{F,w}(i,j)=1\\
+-1&\text{ otherwise. }\end{cases}$ $\left({HHHHVVVV}\right)^{*}$ $Q=\{0,1\}$ $\begin{cases}1&\text{ if }S\left({V_{\left(i,j\right)}}\right)\geq 2,\\
+0&\text{otherwise}\end{cases}$ $w_{0}=\emptyset$ $x^{t+1}_{i}=f(x^{t}_{i-1},x^{t}_{i},x^{t}_{i+1})$ $V_{\left(i,j\right)}=\left[x_{\left(i-1,j\right)},x_{\left(i+1,j\right)},x_{%
+\left(i,j-1\right)},x_{\left(i,j+1\right)}\right]$ $\begin{cases}1&\text{ if }S\left({V_{\left(i,j\right)}}\right)>2,\\
+x_{(i,j)}&\text{ if }S\left({V_{\left(i,j\right)}}\right)=2,\\
+0&\text{otherwise}\end{cases}$ $m_{\text{intra}}=\sum_{i=0}^{m}{\sum_{j=0}^{m}{x_{t_{0}}^{F,w}(i,j)}}-\sum_{i=%
+0}^{m}{\sum_{j=0}^{m}}{x_{t_{n}}^{F,w}(i,j)}$ $\delta_{\text{intra}}=\frac{X_{t_{0}}^{F,w}\oplus X_{t_{n}}^{F,w}}{N^{2}}$ $\left[30,32,90,110,150\right]$ $\displaystyle\begin{split}{\text{Cid}(x_{t})}=\frac{\sum_{i=0}^{m}{\sum_{j=0}^%
+{m}{x_{t_{n}}^{F,w}(i,j)[0\mapsto-1]}}}{N^{2}}\end{split}$ $\displaystyle f_{ij}=MLP(FC_{q}(F_{i})\oplus FC_{k}(F_{j})),$ $(x_{i2},y_{i2})$ $FC_{q}$ $\{s_{kj},k=1,2,...,n\}$ $b_{i}=(x_{i1},y_{i1},x_{i2},y_{i2})$ $T=\{T_{i}|i=1,2,3,\ldots,K\}$ $(x_{i1},y_{i1})$ $FC_{k}$ $C\in\mathbb{Z}^{N\times N}$ $\displaystyle s_{ij}=\frac{\exp(f_{ij})}{\sum_{i=1}^{N}\exp(f_{ij})},$ $R\in\mathbb{R}^{N\times N}$ $\{T_{i}|i=1,2,3,\ldots,K\}$ $\displaystyle=\min\limits_{1\leq i\leq R}{\Delta_{l2_{i}}},$ $\displaystyle=\{[t_{1},t_{2},...,t_{m}]|t_{i}\in T_{j}\},\forall j\in[1..R],$ $\displaystyle L_{b_{j}}$ $\displaystyle=\{l_{2ij}|i\in D_{b_{j}}\},L_{n_{j}}=\{l_{2ij}|i\in D_{n}\},$ $>0.17$ $\text{\text{EmbMarker}}+\text{{CSE}}$ $>0.98$ ${\bm{e}}_{p2}$ $\lambda(S)$ $>0.10$ $j\in[1..R]$ $\displaystyle=\{\cos_{ij}|i\in D_{b_{j}}\},C_{n_{j}}=\{\cos_{ij}|i\in D_{n}\},$ ${\bm{e}}_{p1}$ $\displaystyle C_{b_{j}}$ ${\bm{e}}_{o}$ $\Delta_{cos}(\%)\downarrow$ $>0.20$ ${\bm{u}}^{\langle k+1\rangle}$ ${\bm{e}}_{s}$ $\displaystyle\Delta_{\cos_{k}}$ $\displaystyle=\frac{1}{|L_{b_{k}}|}\sum_{i\in L_{b_{k}}}i-\frac{1}{|L_{n_{k}}|%
+}\sum_{j\in L_{n_{k}}}j.$ $\displaystyle=\frac{{\bm{e}}_{i}\cdot{\bm{w}}_{j}}{||{\bm{e}}_{i}||\cdot||{\bm%
+{w}}_{j}||},\quad l_{2ij}=\biggl{|}\biggl{|}\frac{{\bm{e}}_{i}}{||{\bm{e}}_{i}%
+||}-\frac{{\bm{w}}_{j}}{||{\bm{w}}_{j}||}\biggl{|}\biggl{|}^{2},$ $\Delta_{l2}(\%)\uparrow$ $>0.47$ $\lambda(S)=\lambda_{1}(S)+\lambda_{2}(S)+…+\lambda_{R}(S)$ $\displaystyle\Delta_{l2_{k}}$ $\displaystyle D_{p}=Rank(D_{v})-Rank(D_{s}),$ $>0.57$ $>0.02$ ${\bm{u}}^{\langle k+1\rangle}={\bm{e}}^{\langle k\rangle}-\text{Proj}({\bm{e}}%
+^{\langle k\rangle},{\bm{c}}^{\langle k\rangle}).$ ${\bm{W}}=\{{\bm{w}}_{1},{\bm{w}}_{2},...,{\bm{w}}_{R}\}$ ${\bm{e}}_{p}$ $\displaystyle D_{b_{j}}$ ${\bm{e}}^{\langle k+1\rangle}$ $>0.26$ $\displaystyle D_{n}$ $\displaystyle\min_{\boldsymbol{\alpha}}\biggl{\|}{\bm{w}}-\sum_{k=1}^{K}{%
+\alpha}_{k}\cdot{\bm{c}}^{\langle k\rangle}\biggl{\|}^{2}.$ $>0.04$ $\text{p-value}_{j}$ $\displaystyle=\frac{1}{|C_{b_{k}}|}\sum_{i\in C_{b_{k}}}i-\frac{1}{|C_{n_{k}}|%
+}\sum_{j\in C_{n_{k}}}j,$ $>0.22$ $Rank$ $\displaystyle=\min\limits_{1\leq i\leq R}{\text{p-value}_{i}}.$ $>0.56$ $\text{Norm}\left({\bm{u}}^{\langle k+1\rangle}\right)=\frac{{\bm{u}}^{\langle k%
++1\rangle}}{||{\bm{u}}^{\langle k+1\rangle}||}$ $\displaystyle\Delta_{l2}$ $\text{Proj}({\bm{e}}^{\langle k\rangle},{\bm{c}}^{\langle k\rangle})=\frac{{%
+\bm{c}}^{\langle k\rangle}\cdot{\bm{e}}^{\langle k\rangle}}{||{\bm{c}}^{%
+\langle k\rangle}||}\cdot{\bm{c}}^{\langle k\rangle}.$ $\displaystyle{\bm{e}}_{p}=\text{Norm}\left((1-\sum_{r=1}^{R}\lambda_{r}(S))%
+\cdot{\bm{e}}_{o}+\sum_{r=1}^{R}\lambda_{r}(S)\cdot{\bm{w}}_{r}\right).$ ${\bm{e}}_{s2}$ $>10^{-3}$ $\displaystyle=\{[t_{1},t_{2},...,t_{m}]|t_{i}\notin T\}.$ ${\bm{c}}^{\langle k\rangle}$ $\Delta_{l2}(\%)$ $>0.36$ ${\bm{e}}_{p}=f({\bm{e}}_{o},t)$ $>0.62$ $>0.55$ $\Delta_{cos}(\%)$ ${\bm{e}}^{\langle 0\rangle}$ ${\bm{e}}^{\langle k\rangle}$ $>0.08$ $>0.83$ ${\bm{e}}_{s1}$ $\displaystyle=\max\limits_{1\leq i\leq R}{\Delta_{\cos_{i}}},$ $>10^{-4}$ $m=4,n=20,\text{and frequency interval}=[0.5\%,1\%]$ $\displaystyle\Delta_{\cos}$ $\displaystyle\cos_{ij}$ $C_{b_{j}}$ $<10^{-3}$ $\Theta_{v}$ $1,801,350$ $C_{n_{j}}$ $>0.21$ $Pops(\pi_{a})=\\
+\{\{1,7\},\{5\},\{11,37\},\{13,19\},\{15\},\{22\},\{24\}\}$ $\delta_{sgo}$ $\delta_{flex}=\frac{4}{9}=0.4$ $Pops(\pi_{b})=\\
+\{\{1,7\},\{5,11\},\{13,19\},\{15\},\{22\},\{24,37\}\}$ $sg\neq X$ $SubGoals(\pi_{b})=``XXXCBXXXXA"$ $D(\pi_{a},\pi_{b})=1-\delta(\pi_{a},\pi_{b})$ $\delta_{\text{sgo}}(\pi_{a},\pi_{b})=1-\frac{HDist(SubGoals(\pi_{a}),SubGoals(%
+\pi_{b}))}{max(SubGoals(\pi_{a}),SubGoals(\pi_{b}))}$ $HDist(SubGoals(\pi_{a}),SubGoals(\pi_{b}))=5$ $x\in\{a,s,c\}$ $SubGoals(\pi_{b})$ $\delta(\pi_{a},\pi_{b})\rightarrow{[0,1]}$ $state\leftarrow PerformAction(state,a)$ $subgoalLetter\leftarrow GetEncodedSubgoals(PI)$ $CBA$ $\delta_{flex}$ $\pi_{a}=\{1,7,5,11,37,13,19,15,22,24\}$ $\pi_{b}=\{1,7,5,11,13,19,15,22,24,37\}$ $sg\leftarrow GetSubGoal(state,PI)$ $state\leftarrow GetInitialState(PI)$ $\delta_{x}(\pi_{a},\pi_{b})=|A(\pi_{a})\cap A(\pi_{b})|/|A(\pi_{b})\cup A(\pi_%
+{a})|$ $seq\leftarrow AppendTo(seq,subgoalLetter[sg])$ $SubGoals$ $HDist(s_{\pi_{a}},s_{\pi_{b}})\rightarrow\mathbb{R}$ $max(a,b)\rightarrow\mathbb{N}$ $SubGoals(\pi_{a})=``XXBXXXXAXC"$ $\delta_{u}(\pi_{a},\pi_{b})={\begin{cases}1,\ if\ \pi_{a}\setminus\pi_{b}=%
+\emptyset\\
+1,\ if\ \pi_{a}\subset\pi_{b}\\
+0,\ otherwise\end{cases}}$ $\delta_{\text{flex}}(\pi_{a},\pi_{b})=\frac{|Pop(\pi_{a})\cap Pop(\pi_{b})|}{|%
+Pop(\pi_{a})\cup Pop(\pi_{b})|}$ $seq\leftarrow AppendTo(seq,``X")$ $seq\leftarrow``"$ $Pop(\pi)$ $\delta_{a}(\pi_{a},\pi_{b})=\frac{|1,7,5,11,37,13,19,15,22,24|}{|1,7,5,11,13,1%
+9,15,22,24,37|}=1$ $HDist$ $SubGoals(\pi_{a})$ $BAC$ $A(\pi)$ $\xi=0.031$ $\mathcal{D^{\prime}}(\mathcal{D}(x)+\eta)$ $||\delta||_{\infty}\leq\xi$ $Z_{l}(\cdot)$ $u\leftarrow\max(|\delta_{init}|)$ $s\in\{-1,1\}^{d}$ $\mathrm{Cos}$ $y\in\mathbb{R}^{2}=\{0,1\}$ $s\sim\mathrm{\textbf{Bernoulli}}(p),\quad\delta_{s}\leftarrow\delta_{p}\odot s.$ $\delta_{0}\leftarrow 0$ $f(x)_{i}$ $x_{adv}\leftarrow x_{K}$ $\delta\leftarrow x^{p}-x$ $\epsilon=0.062$ $\mathcal{D^{\prime}}(\cdot)$ $x_{adv}\leftarrow\mathrm{Clip}_{x,\ \epsilon^{\prime}-\kappa}(x_{idct}),$ $[0,\max(\delta_{init})]$ $m\leftarrow(l+u)/2$ $c(x_{t})=0$ $c(x_{adv})=0$ $x_{0}\leftarrow x_{r}$ $x_{idct}^{p}\leftarrow\mathcal{D’}(x_{dct}^{p}+\eta)$ $f:\mathbb{R}^{d}\rightarrow\mathbb{R}^{k}$ $\delta\leftarrow\mathrm{Clip}_{0,\ u}(\delta_{init})$ $x_{dct}^{p}\leftarrow\mathcal{D}(x+\delta)$ $\mathop{\mathrm{argmin}}\limits_{\delta}\ \mathrm{Cos}(Z_{l}(x),\ Z_{l}(x_{r}+%
+\delta)),\quad s.t.\quad||\delta||_{\infty}\leq\xi,$ $p=0.999$ $x_{idct}\leftarrow\mathcal{D^{\prime}}(\mathcal{D}(x_{adv})+\eta),\quad\eta%
+\sim\{-\gamma,\gamma\}^{d},$ $\delta\leftarrow\mathrm{\textbf{BinarySearch}}(x,\delta_{init})$ $i\in[1,\ k]$ $\gamma=1.75$ $||\delta||_{\infty}\leq\epsilon^{\prime},\epsilon^{\prime}\in(0,\max|\delta_{%
+init}|]$ $c(x)=\mathrm{argmax}_{i\in\{0,1\}}\ f(x)_{i}$ $Z_{l}(x)$ $\mathrm{Clip}_{x_{r},\ \xi}(\cdot)$ $\eta\sim\{-\gamma,\gamma\}^{d}$ $c(x^{p})=0$ $\mathrm{\#\ Perturbation\ Random\ Flip}$ $i\in[1,\ K]$ $50.34$ $\delta_{init}\leftarrow\mathrm{\textbf{CrossPerturbInit}}(x,x_{r},Z,f)$ $c(x+\delta)=0$ $Z_{l}(x_{r}+\delta)$ $\delta^{p}\leftarrow\delta\odot s$ $\mathrm{\#\ Frequency\ Noise\ Projection}$ $x^{p}\leftarrow\mathrm{Clip}_{x,\ \epsilon^{\prime}-\kappa}(x_{dct}^{p})$ $\delta\leftarrow\delta^{p}$ $x_{adv}\leftarrow x+\delta$ $J(x_{i-1},x)\leftarrow\mathrm{-Cos}(Z_{l}(x_{i-1}),\ Z_{l}(x))$ $\mathcal{D^{\prime}}(\mathcal{D}(x))$ $l\leftarrow m$ $s\sim\mathrm{\textbf{Bernoulli}}(p)$ $\min_{\delta}||\delta||_{\infty}\quad\mathrm{s.t.}\quad c(x+\delta)=0.$ $\delta_{i}\leftarrow\delta_{i-1}+\frac{\xi}{K}\cdot\mathrm{sign}(\nabla_{x_{i-%
+1}}J(x_{i-1},x))$ $c(x+\delta^{p})==0$ $x_{r}+\delta$ $p\in(0,1)^{d}$ $x_{i}\leftarrow\mathrm{Clip}_{x_{r},\ \xi}(x_{i-1}+\delta_{i})$ $x_{t}\leftarrow\mathrm{Clip}_{x_{r},\ m}(x+\delta_{init})$ $\mathrm{\#\ Perturbation\ Initialization}$ $\delta_{init}$ $\epsilon^{\prime}\leftarrow||\delta||_{\infty}$ $i,j,k\in[1,n]$ $\varepsilon_{2}=0.9\varepsilon$ $\langle\gamma\rangle_{1}=\sum_{i=1}^{n}\langle\gamma_{i}\rangle_{1}$ $d_{max}^{\prime}\leftarrow\mathsf{max}(d_{1}^{\prime},...,d_{n}^{\prime})$ $f=\langle f\rangle_{1}+\langle f\rangle_{2}$ $\langle v_{4},v_{5}\rangle$ $Lap(\frac{\triangle}{\varepsilon_{2}})$ $\displaystyle\frac{e^{\frac{-\varepsilon_{2}.|\widetilde{T}-T(G)|}{\triangle}}%
+}{e^{\frac{-\varepsilon_{2}.|\widetilde{T}-T(G^{\prime})|}{\triangle}}}=e^{%
+\frac{\varepsilon_{2}.(|\widetilde{T}-T(G^{\prime})|-|\widetilde{T}-T(G)|)}{%
+\triangle}}$ $Pr[\mathcal{M}_{i}(A_{i})\in S]\leq e^{\epsilon}Pr[\mathcal{M}_{i}(A_{i}^{%
+\prime})\in S]$ $\hat{T}(G,d_{max}^{\prime})$ $T^{\prime}(G,\varepsilon_{2},d_{max}^{\prime})$ $\displaystyle\mathbb{E}[l_{2}^{2}(T^{\prime}(G,\varepsilon_{2},d_{max}^{\prime%
+}),\hat{T}(G,d_{max}^{\prime}))]=\mathbb{V}[T^{\prime}(G,\varepsilon_{2},d_{%
+max}^{\prime})]$ $ds=[0]*n,\hat{A_{i}}=\varnothing$ $e=a-x,f=b-y,g=c-z$ $\langle u\rangle_{i}$ $\langle T^{\prime}\rangle_{2}=\langle T\rangle_{2}+\langle\gamma\rangle_{2}$ $\mathsf{Local2Rounds_{\triangle}}$ $\langle x\rangle_{2},\langle y\rangle_{2},\langle z\rangle_{2},\langle w%
+\rangle_{2},\langle o\rangle_{2},\langle p\rangle_{2},\langle q\rangle_{2}%
+\rightarrow S_{2}$ $\varepsilon_{1}=0.1\varepsilon$ $\Gamma(\beta)=\int_{0}^{\infty}x^{\beta-1}e^{-x}dx$ $O\binom{n}{2}$ $\gamma_{i}=(Gam_{1}-Gam_{2})$ $\langle\gamma\rangle_{2}=\sum_{i=1}^{n}\langle\gamma_{i}\rangle_{2}$ $\mathsf{Count}$ $\displaystyle e^{\frac{\varepsilon_{2}|T(G)-T(G^{\prime})|}{\triangle}}=e^{%
+\varepsilon_{2}}$ $\hat{A_{i}}$ $d_{max}^{\prime}$ $\hat{A}=\{\hat{A_{1}},...,\hat{A_{n}}\}$ $\langle a_{ij}\rangle_{1}$ $l_{2}^{2}(d_{max}^{\prime},d_{max})<0.009d_{max}$ $\langle v_{2},v_{1}\rangle$ $\mathbb{E}[l_{2}^{2}(T(G),\hat{T}(G,d_{max}^{\prime}))]=(T(G)-\hat{T}(G,d_{max%
+}^{\prime}))^{2}$ $\langle v_{j},v_{i}\rangle$ $v_{i},i\in[1,n]$ $Gam_{2}=\mathsf{Gamma}(n,\frac{d_{max}^{\prime}}{\varepsilon_{2}})$ $\langle x\rangle_{2}=(x-r)$ $r\in\mathbb{Z}_{2^{l}}$ $T(G^{\prime})$ $D^{\prime},d_{max}^{\prime}$ $\displaystyle\frac{Pr[d^{\prime}=d_{i}+x]}{Pr[d^{\prime}=d_{i}^{\prime}+x^{%
+\prime}]}=\frac{Pr[x=d^{\prime}-d_{i}]}{Pr[x^{\prime}=d^{\prime}-d_{i}^{\prime%
+}]}$ $\langle T\rangle_{1}=\langle T\rangle_{2}=0$ $\langle a_{ij}\rangle_{2}$ $\displaystyle\mathbb{V}[Lap(\frac{d_{max}^{\prime}}{\varepsilon_{2}})]=O(\frac%
+{d_{max}^{\prime 2}}{\varepsilon_{2}^{2}})$ $\langle x\rangle_{1}=r$ $v_{k},k\in[n]$ $\langle g\rangle_{i}=\langle c\rangle_{i}-\langle z\rangle_{i}$ $\displaystyle\frac{Pr[\widetilde{T}=T+(r_{1}+...+r_{n})]}{Pr[\widetilde{T}=T^{%
+\prime}+(r_{1}^{\prime}+...+r_{n}^{\prime})]}$ $\hat{A_{i}}\leftarrow A_{i}$ $a_{ij}=1,j\in[n]$ $|d_{i}-d_{i}^{\prime}|=1$ $T^{\prime}=\langle T^{\prime}\rangle_{1}+\langle T^{\prime}\rangle_{2}$ $D^{\prime}\leftarrow D^{\prime}\cup\{d_{i}^{\prime}\}$ $O(\frac{d_{max}^{2}}{\varepsilon^{2}})$ $r=\{r_{1},...,r_{n}\}$ $O(\theta)$ $+\langle x\rangle_{2}fg+\langle y\rangle_{2}eg+\langle z\rangle_{2}ef+efg$ $e=\langle e\rangle_{1}+\langle e\rangle_{2}$ $O(n^{2}+nd_{max}^{2})$ $d_{i}>d_{max}^{\prime}$ $(d_{i}-d_{max}^{\prime})$ $(i\in\{1,2\})$ $\displaystyle\frac{Pr[\widetilde{T}=T(G)+x]}{Pr[\widetilde{T}=T(G^{\prime})+x^%
+{\prime}]}=\frac{Pr[x=\widetilde{T}-T(G)]}{Pr[x^{\prime}=\widetilde{T}-T(G^{%
+\prime})]}$ $\langle x\rangle_{2},\langle y\rangle_{2},\langle z\rangle_{2},\langle w%
+\rangle_{2},\langle o\rangle_{2},\langle p\rangle_{2},\langle q\rangle_{2}$ $O(d_{max}^{\prime})$ $\langle e\rangle_{1}=\langle a_{ij}\rangle_{1}-\langle x\rangle_{1}$ $\langle\gamma_{i}\rangle_{1}\rightarrow S_{1}$ $\hat{ds}\leftarrow ds[1:d_{max}^{\prime}]$ $\mathsf{view}_{S_{i}}^{\Pi}$ $x=r_{1}+...+r_{n}$ $d_{max}^{\prime}\approx d_{max}$ $\langle e\rangle_{i}=\langle a\rangle_{i}-\langle x\rangle_{i},$ $\displaystyle\mathbb{E}[l_{2}^{2}(T^{\prime}(G,\varepsilon_{2},d_{max}^{\prime%
+}),\hat{T}(G,d_{max}^{\prime}))]$ $\times 10^{10}$ $k=j+1$ $e,f,$ $\displaystyle\frac{Pr[\widetilde{T}=\langle T\rangle_{1}+\langle T\rangle_{2}+%
+(r_{1}+...+r_{n})]}{Pr[\widetilde{T}=\langle T^{\prime}\rangle_{1}+\langle T^{%
+\prime}\rangle_{2}+(r_{1}^{\prime}+...+r_{n}^{\prime})]}$ $\mathsf{Sim}_{S_{i}}$ $\langle\gamma_{i}\rangle_{1},\langle\gamma_{i}\rangle_{2}$ $\langle u\rangle_{1}$ $r^{\prime}=\{r_{1}^{\prime},...,r_{n}^{\prime}\}$ $Pr[\mathcal{M}(D)\in S]\leq e^{\epsilon}Pr[\mathcal{M}(D^{\prime})\in S]$ $O\binom{d_{max}^{\prime}}{2}$ $d=\langle d\rangle_{1}+\langle d\rangle_{2}=\langle w\rangle_{1}+\langle xy%
+\rangle_{1}g+\langle xz\rangle_{1}f+\langle yz\rangle_{1}e+\langle x\rangle_{1%
+}fg+\langle y\rangle_{1}eg+\langle z\rangle_{1}ef+\langle w\rangle_{2}+\langle
+xy%
+\rangle_{2}g+\langle xz\rangle_{2}f+\langle yz\rangle_{2}e+\langle x\rangle_{2%
+}fg+\langle y\rangle_{2}eg+\langle z\rangle_{2}ef+efg$ $\langle x\rangle_{2}+\langle y\rangle_{1}+\langle y\rangle_{2}=x+y$ $\langle e\rangle_{2}=\langle a_{ij}\rangle_{2}-\langle x\rangle_{2}$ $D=\{d_{1},...,d_{n}\}$ $\hat{ds}$ $\langle T\rangle=\{\langle T\rangle_{1},\langle T\rangle_{2}\}$ $T\neq$ $\mathsf{CentralLap_{\triangle}}$ $\langle g\rangle_{2}=\langle a_{jk}\rangle_{2}-\langle z\rangle_{2}$ $d_{i}^{\prime}\leftarrow d_{i}+\mathsf{Lap}(\frac{1}{\varepsilon_{1}})$ $Lap(\frac{1}{\varepsilon_{1}})$ $\langle d\rangle_{i}=\langle w\rangle_{i}+\langle xy\rangle_{i}g+\langle xz%
+\rangle_{i}f+\langle yz\rangle_{i}e+\langle x\rangle_{i}fg+\langle y\rangle_{i%
+}eg+\langle z\rangle_{i}ef+(i-1)efg$ $u_{1}=\langle w\rangle_{1}+\langle xy\rangle_{1}g+\langle xz\rangle_{1}f+%
+\langle yz\rangle_{1}e$ $re(T,T^{\prime})=\frac{|T-T^{\prime}|}{T}$ $\langle f\rangle_{i}=\langle b\rangle_{i}-\langle y\rangle_{i},$ $Gam_{1}(n,\frac{\triangle}{\varepsilon_{2}})-Gam_{2}(n,\frac{\triangle}{%
+\varepsilon_{2}})$ $D,D^{\prime}\in\mathcal{X}^{n}$ $A_{ij}==1$ $u=a_{ij}\times a_{ik}\times a_{jk}(i $A_{i}=\{a_{i1},...,a_{in}\}$ $\varepsilon,n,d_{max},d_{max}^{\prime}$ $y\in range(\mathcal{M})$ $\mathsf{Project}$ $\displaystyle(\mathbb{E}[T^{\prime}(G,\varepsilon_{2},d_{max}^{\prime})]-\hat{%
+T}(G,d_{max}^{\prime}))^{2}+\mathbb{V}[T^{\prime}(G,\varepsilon_{2},d_{max}^{%
+\prime})]$ $\mathsf{Gamma}$ $\theta=1000$ $\mathsf{Perturb}$ $Gam_{2}$ $\mathsf{Perturb}()$ $O(nd_{max})$ $\mathsf{GraphProjection}$ $\langle x\rangle_{1},\langle y\rangle_{1},\langle z\rangle_{1},\langle w%
+\rangle_{1},\langle o\rangle_{1},\langle p\rangle_{1},\langle q\rangle_{1}$ $S\subseteq Range(\mathcal{M}_{i})$ $\mathbb{E}[l_{2}^{2}(T(G),\hat{T}(G,d_{max}^{\prime}))]=0$ $\varepsilon=\varepsilon_{1}+\varepsilon_{2}$ $\langle T^{\prime}\rangle_{1}=\langle T\rangle_{1}+\langle\gamma\rangle_{1}$ $\langle v_{i},v_{j}\rangle\in E$ $\langle y\rangle_{i}$ $ds[j]$ $\langle f\rangle_{1}=\langle a_{ik}\rangle_{1}-\langle y\rangle_{1}$ $Gam_{1}(n,\lambda)$ $\mathsf{project}$ $\hat{A_{i}}\leftarrow\hat{A_{i}}\cup\{0\}$ $O(\frac{e^{\varepsilon}}{(e^{\varepsilon}-1)^{2}}(d_{max}^{3}n+\frac{e^{%
+\varepsilon}}{\varepsilon^{2}}d_{max}^{2}n))$ $u=\langle u\rangle_{1}+\langle u\rangle_{2}=\langle x\rangle_{1}$ $\hat{A_{i}}\leftarrow\hat{A_{i}}\cup\{1\}$ $\langle T\rangle\leftarrow\{\langle T\rangle_{1},\langle T\rangle_{2}\}$ $Laplace$ $Pr[\mathcal{M}(G)=y]\leq e^{\varepsilon}Pr[\mathcal{M}(G^{\prime})=y]$ $\langle x\rangle_{i}$ $\langle T\rangle_{1}$ $1\times\sim 2\times$ $S_{i\in\{1,2\}}$ $A=\{A_{1},...,A_{n}\}$ $\langle T\rangle_{2}\leftarrow\langle T\rangle_{2}+u_{2}$ $G,G^{\prime}\in\mathcal{G}$ $\langle\gamma_{i}\rangle_{2}\rightarrow S_{2}$ $\mathsf{Lap(\frac{1}{\varepsilon_{1}})}$ $O(\frac{d_{max}^{\prime 2}}{\varepsilon^{2}})$ $u_{2}=\langle w\rangle_{2}+\langle xy\rangle_{2}g+\langle xz\rangle_{2}f+%
+\langle yz\rangle_{2}e$ $\mathbb{E}[l_{2}^{2}(T^{\prime}(G,\varepsilon_{2},d_{max}^{\prime}),\hat{T}(G,%
+d_{max}^{\prime}))]=O(\frac{d_{max}^{\prime 2}}{\varepsilon_{2}^{2}})$ $\langle v_{i},v_{j}\rangle$ $Gam_{1}$ $l_{2}(T,T^{\prime})=(T-T^{\prime})^{2}$ $=w+xyg+xzf+yze+xfg+yeg+zef+efg$ $\langle u\rangle_{2}$ $D^{\prime}=\{d_{1}^{\prime},...,d_{n}^{\prime}\}$ $\mathsf{Laplace}$ $w=xyz,o=xy,p=xz,q=yz$ $ds[j]\leftarrow\mathsf{DS}(d_{i},d_{j}^{\prime})$ $d_{max}^{\prime}\geq 0$ $\langle f\rangle_{2}=\langle a_{ik}\rangle_{2}-\langle y\rangle_{2}$ $S\subseteq Range(\mathcal{M})$ $\{d_{1}^{\prime},...,d_{n}^{\prime}\}$ $Gam_{2}(n,\lambda)$ $\displaystyle\frac{e^{-\varepsilon_{1}.|d^{\prime}-d_{i}|}}{e^{-\varepsilon_{1%
+}.|d^{\prime}-d_{i}^{\prime}|}}=e^{\varepsilon_{1}.(|d^{\prime}-d_{i}^{\prime}%
+|-|d^{\prime}-d_{i}|)}\leq e^{\varepsilon_{1}|d_{i}-d_{i}^{\prime}|}=e^{%
+\varepsilon_{1}},$ $(r_{1},...,r_{n})$ $\mathsf{view}_{S_{i}}^{\Pi}\approx\mathsf{Sim}_{S_{i}}$ $T(G)=\langle T\rangle_{1}+\langle T\rangle_{2}$ $\mathsf{Project}()$ $Gam_{1}=\mathsf{Gamma}(n,\frac{d_{max}^{\prime}}{\varepsilon_{2}})$ $Gamma(x,n,\lambda)=\frac{(1/\lambda)^{1/n}}{\Gamma(1/n)}x^{\frac{1}{n}-1}e^{-%
+\frac{x}{\lambda}},$ $g=\langle g\rangle_{1}+\langle g\rangle_{2}$ $d=a\times b\times c$ $Lap(.)$ $ds[k]$ $a_{ij}\times a_{ik}\times a_{jk}=1$ $(D^{\prime},d_{max}^{\prime})\leftarrow\mathsf{Max}(D,\varepsilon_{1})$ $\langle x\rangle=\langle x\rangle_{1}+\langle x\rangle_{2}$ $Lap(\lambda)=\sum_{i=1}^{n}[Gam_{1}(n,\lambda)-Gam_{2}(n,\lambda)],$ $\mathsf{Count}()$ $A=\{A_{1},A_{2},...,A_{n}\}$ $\langle v_{2},v_{5}\rangle$ $DS(d_{1},d_{2})$ $\mathsf{Max}(.)$ $u=a_{ij}\times a_{ik}\times a_{jk}$ $\langle T\rangle\leftarrow\mathsf{Count}(\hat{A})$ $|T(G)-T(G^{\prime})|=\triangle$ $+\langle x\rangle_{1}fg+\langle y\rangle_{1}eg+\langle z\rangle_{1}ef$ $d_{max}^{\prime}\geq d_{max}$ $\gamma_{i}=\langle\gamma_{i}\rangle_{1}+\langle\gamma_{i}\rangle_{2}$ $w=x\times y\times z,o=x\times y,p=x\times z,q=y\times z$ $Pr[\mathcal{M}(G)\in S]\leq e^{\epsilon}Pr[\mathcal{M}(G^{\prime})\in S]$ $d_{i}\leq d_{max}^{\prime}$ $\hat{A}\leftarrow\mathsf{Project}(A,D,D^{\prime},d_{max}^{\prime})$ $\langle g\rangle_{1}=\langle a_{jk}\rangle_{1}-\langle z\rangle_{1}$ $\langle x\rangle_{1},\langle y\rangle_{1},\langle z\rangle_{1},\langle w%
+\rangle_{1},\langle o\rangle_{1},\langle p\rangle_{1},\langle q\rangle_{1}%
+\rightarrow S_{1}$ $\langle T\rangle_{1}\leftarrow\langle T\rangle_{1}+u_{1}$ $x,y,z,w,o,p,q$ $O(\frac{d_{max}^{\prime 2}}{\varepsilon_{2}^{2}})$ $T^{\prime}\leftarrow\mathsf{Perturb}(\langle T\rangle,d_{max}^{\prime},%
+\varepsilon_{2})$ $T(G^{\prime})=\langle T^{\prime}\rangle_{1}+\langle T^{\prime}\rangle_{2}$ $x^{\prime}=r_{1}^{\prime}+...+r_{n}^{\prime}$ $a_{ij},a_{ik},a_{jk}$ $\varepsilon_{2}\geq 0$ $f(x,\lambda)=\frac{1}{2\lambda}e^{\frac{|x|}{\lambda}}$ $DS(d_{1},d_{2})=\frac{|d_{1}-d_{2}|}{d_{1}}$ $\langle T\rangle_{2}$ ${\bigl{\{}Stat[u_{k}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{P2P}[u_{k}%
+]\bigr{\}}_{\mathcal{E}}}\cdot\pi_{P2P}+{\bigl{\{}inDev[u_{k}]\bigr{\}}_{%
+\mathcal{E}}}\cdot\pi_{RT}$ $\pi_{FiT}$ ${\bigl{\{}Dev_{C}^{Tot}\bigr{\}}_{\mathcal{E}}}$ ${\bigl{\{}V^{P2P}\bigr{\}}_{\mathcal{E}}}$ $N_{U},{\bigl{\{}V^{P2P}\bigr{\}}_{\mathcal{E}}},{\bigl{\{}V^{Real}\bigr{\}}_{%
+\mathcal{E}}},Dev_{C}^{Tot},Dev_{P}^{Tot},\pi_{P2P},\pi_{RT}$ ${\bigl{\{}Stat[u_{k}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{P2P}[u_{k}%
+]\bigr{\}}_{\mathcal{E}}}\cdot\pi_{P2P}+{\bigl{\{}inDev[u_{k}]\bigr{\}}_{%
+\mathcal{E}}}\cdot\pi_{P2P}$ $\pi_{P2P}$ $KGen_{pe}(k)$ $each~{}m_{l}\pcin~{}M(u_{k})$ ${\bigl{\{}inDev\bigr{\}}_{\mathcal{E}}},{\bigl{\{}inDev_{M}\bigr{\}}_{\mathcal%
+{E}}}$ $\{.\}_{\mathcal{E}}$ $Bal^{Tot}_{sup}$ ${\bigl{\{}V^{Real}\bigr{\}}_{\mathcal{E}}}$ $V^{Real}$ ${\bigl{\{}Stat[m_{l}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{P2P}[m_{l}%
+]\bigr{\}}_{\mathcal{E}}}\cdot{\pi_{P2P}}+{\bigl{\{}inDev[m_{l}]\bigr{\}}_{%
+\mathcal{E}}}\cdot{\pi_{RT}}$ $Dev_{P}^{Tot}>Dev_{C}^{Tot}c$ ${\bigl{\{}Stat[m_{l}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{P2P}[m_{l}%
+]\bigr{\}}_{\mathcal{E}}}\cdot{\pi_{P2P}}+{\bigl{\{}inDev[m_{l}]\bigr{\}}_{%
+\mathcal{E}}}/Dev_{P}^{Tot}\cdot TotRev_{P}$ $Stat_{M}[u_{k}]$ $KGen_{pe}$ ${\bigl{\{}V^{Real}[u_{k}]\bigr{\}}_{\mathcal{E}}}$ ${\bigl{\{}inDev\bigr{\}}_{\mathcal{E}}}$ ${\bigl{\{}Stat\bigr{\}}_{\mathcal{E}}},{\bigl{\{}Stat_{M}\bigr{\}}_{\mathcal{E%
+}}}$ ${\bigl{\{}inDev_{M}\bigr{\}}_{\mathcal{E}}}$ ${{\bigl{\{}inDev[u_{k}]\bigr{\}}_{\mathcal{E}}}}$ $inDev$ ${\bigl{\{}Stat[m_{l}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{P2P}[m_{l}%
+]\bigr{\}}_{\mathcal{E}}}\cdot\pi_{P2P}+{\bigl{\{}inDev[m_{l}]\bigr{\}}_{%
+\mathcal{E}}}\cdot\pi_{P2P}$ $V^{P2P}$ $V^{Real}[u_{k}]$ ${\bigl{\{}inDev[u_{k}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{Real}[u_{%
+k}]\bigr{\}}_{\mathcal{E}}}-{\bigl{\{}V^{P2P}[u_{k}]\bigr{\}}_{\mathcal{E}}};$ $M(u_{k})$ $H{({\bigl{\{}V^{P2P}[u_{k}]\bigr{\}}_{\mathcal{E}}})}$ $PK_{sup}$ $\xrightarrow{}PK_{sup},SK_{sup}$ ${\bigl{\{}inDev_{M}[M(m_{l})]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{%
+Real}[M(m_{l})]\bigr{\}}_{\mathcal{E}}}-{\bigl{\{}V^{P2P}[M(m_{l})]\bigr{\}}_{%
+\mathcal{E}}};$ $Dev_{P}^{Tot}=Dev_{C}^{Tot}$ $Dev_{P}^{Tot}>Dev_{C}^{Tot}$ $TotRev_{P}=(Dev_{C}^{Tot}\cdot\pi_{P2P}+(Dev_{P}^{Tot}-Dev_{C}^{Tot})\cdot\pi_%
+{FiT})$ $\footnotesize{\bigl{\{}Dev_{C}^{Tot}\bigr{\}}_{\mathcal{E}}}\leftarrow\sum_{i=%
+0}^{N_{C}-1}{\bigl{\{}inDev_{C}[c_{i}]\bigr{\}}_{\mathcal{E}}}$ ${inDev_{P}[u_{k}]}/Dev_{P}^{Tot}$ ${\bigl{\{}stat^{Tot}[u_{k}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}stat^{%
+Tot}[u_{k}]\bigr{\}}_{\mathcal{E}}}+{\bigl{\{}stat[u_{k}]\bigr{\}}_{\mathcal{E%
+}}}$ $Dev^{Tot}$ $\pi_{P2P},\pi_{FiT},\pi_{RT}$ $N_{C}=N_{P}$ ${\bigl{\{}inDev[c_{i}]\bigr{\}}_{\mathcal{E}}}$ $each~{}u_{k}$ ${\bigl{\{}Dev_{P}^{Tot}\bigr{\}}_{\mathcal{E}}}$ ${Dev_{C}^{Tot}}$ $Bal_{sup}\leftarrow 0$ $\pi_{RT}$ $Stat$ ${Dev_{P}^{Tot}}$ $N_{C} ${\bigl{\{}Stat[m_{l}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{P2P}[m_{l}%
+]\bigr{\}}_{\mathcal{E}}}\cdot{\pi_{P2P}}+{\bigl{\{}inDev[m_{l}]\bigr{\}}_{%
+\mathcal{E}}}\cdot{\pi_{P2P}}$ $TotRev_{P}$ $Bal_{sup}\leftarrow-(Dev_{P}^{Tot}-Dev_{C}^{Tot})\cdot\pi_{FiT}$ $Stat[u_{k}]$ $\footnotesize{\bigl{\{}Dev_{P}^{Tot}\bigr{\}}_{\mathcal{E}}}\leftarrow\sum_{j=%
+0}^{N_{C}-1}{\bigl{\{}inDev_{P}[p_{j}]\bigr{\}}_{\mathcal{E}}}$ $H{({\bigl{\{}V^{Real}[u_{k}]\bigr{\}}_{\mathcal{E}}})}$ ${\bigl{\{}stat_{M}^{Tot}[m_{l}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}%
+stat^{Tot}[m_{l}]\bigr{\}}_{\mathcal{E}}}+{\bigl{\{}stat[m_{l}]\bigr{\}}_{%
+\mathcal{E}}}$ $stat^{Tot}$ $N_{C}>N_{P}$ $Bal_{sup}$ $(Dev_{P}^{Tot}-Dev_{C}^{Tot})$ $Dev_{P}^{Tot} ${\bigl{\{}Stat[u_{k}]\bigr{\}}_{\mathcal{E}}}\leftarrow{\bigl{\{}V^{P2P}[u_{k}%
+]\bigr{\}}_{\mathcal{E}}}\cdot{\pi_{P2P}}+{\bigl{\{}inDev[u_{k}]\bigr{\}}_{%
+\mathcal{E}}}/Dev_{P}^{Tot}\cdot TotRev_{P}$ $SK_{sup}$ ${\bigl{\{}V^{P2P}[u_{k}]\bigr{\}}_{\mathcal{E}}}$ $Bal_{sup}\leftarrow(Dev_{C}^{Tot}-Dev_{P}^{Tot})\cdot\pi_{RT}.$ $k(\mu,\mu^{\prime})=\left(1+\left\|(\mu-\mu^{\prime})/\sigma\right\|_{2}^{2}%
+\right)^{-1}$ $\displaystyle=-\log(\mathcal{N}(h(\mathbf{x});\mathbf{0},\mathbf{I})))-\log%
+\left|\det\mathbf{J}_{h}(\mathbf{x})\right|$ $K=D-d$ $\mathbf{x}=h^{-1}(\mathbf{z})$ $\mathcal{L}_{\mathrm{MMD}}(p(\mathbf{x}_{D}),p(h^{-1}([\mathbf{y}_{d},\mathbf{%
+z}_{K}])))$ $D>d$ $(x_{\text{soma}},y_{\text{soma}})$ $\displaystyle\mathcal{L}_{\mathrm{NLL}}=-\log(p(\mathbf{x}))$ $I(x,y)=\exp\left(-\frac{(x-x_{\text{stim}})^{2}+(y-y_{\text{stim}})^{2}}{2\rho%
+^{2}}-\frac{(x-x_{\text{soma}})^{2}+(y-y_{\text{soma}})^{2}}{2\lambda^{2}}%
+\right),$ $\{\mu\},\{\mu^{\prime}\}$ $g_{\theta}:\tilde{\mathbf{x}}\mapsto\mathbf{x}_{s}$ $\mathbf{z}\neq\mathbf{0}$ $\mathbf{J}_{h}(\mathbf{x})=\partial h(\mathbf{x})/\partial\mathbf{x}^{T}$ $h(\mathbf{x}_{D})=[h_{\mathbf{y}_{d}}(\mathbf{x}_{D}),h_{\mathbf{z}_{K}}(%
+\mathbf{x}_{D})]$ $P_{M^{\prime}}$ $h:\mathbf{x}_{D}\mapsto[\mathbf{y}_{d},\mathbf{z}_{K}]$ $h_{\mathbf{y}_{d}}(\mathbf{x}_{D})\approx s(\mathbf{x}_{D})$ $\mathcal{L}_{\mathrm{MMD}}(p_{M},p_{M^{\prime}})=\left(\mathbb{E}_{i,j}[k(\mu_%
+{i},\mu_{j})-2k(\mu_{i},\mu_{j}^{\prime})+k(\mu_{i}^{\prime},\mu_{j}^{\prime})%
+]\right)^{\frac{1}{2}},$ $\rho=400\,\mu m$ $\lambda=1550\,\mu m$ $\mathbf{y}_{d}$ $\mathbf{z}\sim\pi(\mathbf{z})=\mathcal{N}(\mathbf{z};\mathbf{0},\mathbf{I}))$ $\mathcal{L}_{\mathrm{NLL}}\simeq\frac{1}{2}\|h(\mathbf{x};\mathbf{c})\|_{2}^{2%
+}-\log\left|\det\mathbf{J}_{h}(\mathbf{x})\right|$ $\min_{\theta}\mathcal{L}_{\mathrm{MSE}}\left(\tilde{\mathbf{x}},f_{\phi}\left(%
+g_{\theta}\left(\tilde{\mathbf{x}}\right)\right)\right),$ $\mathbf{z}_{K}\sim\pi(\mathbf{z}_{K})=\mathcal{N}(\mathbf{z}_{K};\mathbf{0},%
+\mathbf{I}_{K}))$ ${\mathbf{x}_{s}}$ $\mathcal{L}_{\mathrm{MSE}}=\mathbb{E}[(\mathbf{y}_{d}-h_{\mathbf{y}_{d}}(%
+\mathbf{x}_{D}))^{2}]$ $\mu^{\prime}\sim p_{M^{\prime}}$ $p(\mathbf{x})=\pi(\mathbf{z}=h(\mathbf{x}))\left|\det\frac{\partial h(\mathbf{%
+x})}{\partial\mathbf{x}^{T}}\right|,$ $\mathbf{x}=h^{-1}(\mathbf{z};\mathbf{c})$ $s:\mathbf{x}_{D}\in\mathbb{R}^{D}\mapsto\mathbf{y}_{d}\in\mathbb{R}^{d}$ $\displaystyle\simeq\frac{1}{2}\|h(\mathbf{x})\|_{2}^{2}-\log\left|\det\mathbf{%
+J}_{h}(\mathbf{x})\right|.$ $(x_{\text{stim}},y_{\text{stim}})$ $\mathbf{z}=h(\mathbf{x};\mathbf{c})$ $\mathbf{z}=h(\mathbf{x})$ $\mathbf{z}_{K}$ $\mathcal{L}_{\mathrm{MMD}}(p(h(\mathbf{x}_{D})),p(\mathbf{y}_{d})p(\mathbf{z}_%
+{K}))$ $\mathbf{x}_{D}$ $\mathbf{y}_{p}$ $\mu\sim p_{M}$ $\Psi\in\mathbb{R}^{z\times h}$ $\bm{0.982}$ $24.90$ $\bm{0.861}$ $\bm{0.044}$ $z=8,192$ $r:\Re\times\Re^{3}\rightarrow\Re^{3}$ $24.88$ $\bm{0.047}$ $24.13$ $\bm{0.018}$ $\bm{32.70}$ $\bm{W}_{\Psi}$ $M^{\prime}_{\Delta}=(\bm{W}^{\prime}_{\Delta},\bm{b}^{\prime}_{\Delta})$ $\bm{W}_{\gamma}$ $27.89$ $\bm{0.714}$ $\bm{0.070}$ $\bm{0.025}$ $\bm{0.911}$ $\bm{h}^{\prime}_{\Delta}$ $\bm{28.14}$ $32.53$ $\bm{b}=\{\bm{b}_{\Delta},\bm{b}_{c},\bm{b}_{\psi}$ $29.57$ $\bm{h}_{c}$ $M^{\prime}_{\Delta}$ $\bm{W}=\{\bm{W}_{\Delta},\bm{W}_{c},\bm{W}_{\psi}$ $\bm{0.991}$ $\bm{0.093}$ $\bm{W}_{\mu},\bm{W}_{\gamma},\bm{W}_{\Psi},\bm{W}^{\prime}_{\Delta},\bm{W}^{%
+\prime}_{c}\}$ $\bm{37.72}$ $\bm{b}_{\Psi}$ $\bm{0.988}$ $\bm{35.41}$ $26.11$ $\bm{f}_{\Delta}=\sigma\left(\bm{h}_{\Delta}\right),$ $\bm{0.029}$ $\bm{h}_{c}^{\prime}=[\bm{\Psi}\otimes\bm{f}_{c},\mathbf{d}].$ $\bm{0.956}$ $\bm{32.34}$ $\bm{31.13}$ $\bm{26.51}$ $26,29$ $27.31$ $\bm{20.80}$ $\bm{\mu}=\bm{f}_{\mu}\otimes\bm{\gamma}.$ $\bm{36.76}$ $\bm{W}_{\psi}$ $\bm{\gamma}=tanh\left({\bm{W}_{\gamma}[\bm{h}_{\Delta},\bm{h}_{c}]+\bm{b}_{%
+\gamma}}\right),$ $\bm{27.56}$ $\bm{b}_{\mu},\bm{b}_{\gamma},\bm{b}_{\Psi},\bm{b}^{\prime}_{\Delta},\bm{b}^{%
+\prime}_{c}\}$ $\bm{W}_{\mu}$ $23.49$ $\bm{0.007}$ $\bm{h}_{\Delta}$ $\bm{0.103}$ $\bm{f}_{c}=\sigma\left({\bm{h}_{c}}\right),$ $\bm{0.068}$ $MSE(\cdot)$ ${\cal L}=MSE(r(\Delta,\bm{c}),\bm{g}),$ $32.45$ $\bm{0.009}$ $\bm{f}_{\psi}=\sigma\left({\bm{W}_{\psi}\bm{[h}_{\Delta},\bm{h}_{c}]+\bm{b}_{%
+\psi}}\right),$ $\bm{0.026}$ $\mathbf{l}=(x,y,z)$ $\bm{f}_{\mu}=\sigma\left({\bm{W}_{\mu}[\bm{h}_{\Delta},\bm{h}_{c}]+\bm{b}_{\mu%
+}}\right),$ $\bm{0.882}$ $\bm{0.837}$ $[\bm{h}_{\Delta},\bm{h}_{c}]$ $\bm{f}_{\Delta}$ $\bm{26.06}$ $\bm{b_{\mu}}$ $29.56$ $\bm{27.01}$ $26.05$ $F_{\Theta}:(\mathbf{l},\mathbf{d})\rightarrow(\mathbf{c},\Delta)$ $\operatorname{Inst}(\emptyset)$ $\prod_{s=1}^{t}2^{p_{s}-1}$ ${\rm D}_{\rm NP}\leq_{0\hbox{-}T}{\rm D}_{\rm P}$ $\p^{A}=\np^{A}$ $\p^{L}=\p$ $\langle M^{\prime},\epsilon\rangle$ $\np^{\mathcal{O}}$ $\langle M,1^{n},1^{t}\rangle\in{\rm U}_{\rm NP}$ $\np^{L}=\np$ $x\notin{\rm HP}$ $\np^{\p}=\np$ $\p$ $A\leq_{f(n)\hbox{-}T}B$ ${\rm D}_{\rm NP}\not\in\p^{{\rm D}_{\rm P}}$ $x\in\mathcal{O}$ $L\in{\rm\Sigma_{1}^{0}}$ $n,t\in{\mathbb{N}^{+}}$ $t^{\prime}\in{\mathbb{N}}$ $\Theta(2^{n})$ ${\rm D}_{\rm NP}=\{\langle M,1^{n}\rangle\mid n\in{\mathbb{N}}\land(\exists x%
+\in\{0,1\}^{n})[M$ $\langle M,1^{n}\rangle\in{\rm D}_{\rm NP}$ ${\rm U}_{\rm P}=\{\langle M,x,1^{t}\rangle\mid t\in{\mathbb{N}^{+}}\land M$ $M^{\mathcal{O}}$ ${\rm HP}=\{\langle M,w\rangle\mid M\text{ halts on input }w\}.$ $(\forall x)[x\in A\iff f(x)\in B]$ $z\notin{\rm D}_{\rm NP}$ $(\forall L\in{\rm\Sigma_{1}^{0}})[L\leq_{m}A]$ $\p^{\p}=\p$ $A\leq_{m}B$ ${\mathbb{N}^{+}}=\{1,2,3,\ldots\}$ ${\rm D}_{\rm NP}\in{\rm\Sigma_{1}^{0}}$ $\Omega(2^{n})$ $\p^{X}\neq\np^{X}$ $\p^{A}\neq\np^{A}\iff\p\neq\np$ $\p\neq\np$ $]\}$ ${\rm U}_{\rm NP}$ $L\leq_{m}{\rm D}_{\rm NP}$ $x]\}$ $\p^{\mathcal{O}}$ $\min(c,2^{n})$ ${\rm D}_{\rm NP}\leq_{m}{\rm D}_{\rm P}$ $z\not\in{\rm D}_{\rm NP}$ $y=\epsilon$ $\p^{\p}$ ${\rm U}_{\rm P}$ $\{0,1\}^{n}\cap L(M)$ $A\leq_{T}B$ ${\rm D}_{\rm P}$ $U_{\text{\bf P}}$ ${\rm PSPACE}$ $x\in L(M)$ $\p^{B}\neq\np^{B}$ $A\in\p$ $\langle M,x\rangle\in{\rm D}_{\rm P}\iff(\exists t>0)[\langle M,x,1^{t}\rangle%
+\in{\rm U}_{\rm P}]$ ${\mathbb{N}}=\{0,1,2,\ldots\}$ $A=L(M^{B})$ $\mathcal{C}^{\mathcal{D}}=\bigcup_{A\in\mathcal{D}}\mathcal{C}^{A}$ $\np$ ${\rm U}_{\rm NP}\not\in\p^{{\rm U}_{\rm P}}$ $\np^{\p}$ $f:{\mathbb{N}}\rightarrow{\mathbb{N}}$ $\p=\np$ $f(x)\in{\rm D}_{\rm NP}$ ${\rm\Sigma_{0}^{0}}$ ${\rm\Sigma_{0}^{0}}^{\mathcal{O}}$ $A\in{\rm\Sigma_{1}^{0}}$ $x\not\in\mathcal{O}$ $\p^{A}=\p$ ${\rm U}_{\rm NP}=\{\langle M,1^{n},1^{t}\rangle\mid n\in{\mathbb{N}}\land t\in%
+{\mathbb{N}^{+}}\land(\exists x\in\{0,1\}^{n})[M$ $\langle M,1^{n}\rangle$ $x\in{\rm HP}$ $\langle M,1^{n}\rangle\in{\rm D}_{\rm NP}\iff(\exists t>0)[\langle M,1^{n},1^{%
+t}\rangle\in{\rm U}_{\rm NP}]$ $x\}$ $w\in\{0,1\}^{n}$ $\epsilon\notin L(M^{\prime})$ $A\leq_{1\hbox{-}T}B$ ${\rm HP}\leq_{m}{\rm D}_{\rm NP}$ $\np^{A}=\np$ $(\forall x)[x\notin{\rm HP}\implies f(x)\notin{\rm D}_{\rm NP}]$ ${\rm\Sigma_{1}^{0}}$ ${\rm HP}$ $\langle M,w\rangle$ $(\forall x)[x\in{\rm HP}\iff f(x)\in{\rm D}_{\rm NP}]$ $\mathcal{C},\mathcal{D}$ $\epsilon\in L(M^{\prime})$ ${\rm D}_{\rm NP}$ ${\rm D}_{\rm NP}\leq_{h\hbox{-}T}{\rm D}_{\rm P}$ $L\in\p$ ${\rm D}_{\rm P}=\{\langle M,x\rangle\mid M$ ${\rm D}_{\rm NP}\leq_{1\hbox{-}T}{\rm D}_{\rm P}$ $c\in{\mathbb{N}^{+}}$ $(S^{\prime}\setminus\{x,y\})\cup\{v^{xy}\}$ $G\setminus X:=G[V(G)\setminus X]$ $V(C_{2})$ $p_{j}\in B$ $H\setminus u$ $\chi(G)\leq k$ $u\in V(H)$ $\{\overline{K_{2}\cup C_{2k+1}}\mid k\in\mathbb{N}\}$ $\Theta_{\Delta}=(\bm{W}_{\Delta},\bm{b}_{\Delta})$ $\bm{0.954}$ $(\Delta,\bm{c})$ $\bm{0.873}$ $\bm{0.796}$ $M_{\Delta}$ $\bm{0.055}$ $\bm{33.99}$ $\bm{b}_{\gamma}$ $\bm{0.122}$ $\bm{0.914}$ $24.54$ $\bm{36.46}$ $M^{\prime}_{c}=(\bm{W}^{\prime}_{c},\bm{W}^{\prime}_{c})$ $\bm{h}^{\prime}_{c}$ $20-30$ $\bm{f}_{\psi}$ $\bm{36.13}$ $30.91$ $\bm{f}_{\mu}$ $\bm{28.60}$ $\bm{h}_{\Delta}^{\prime}=[\bm{\Psi}\otimes\bm{f}_{\Delta},\mathbf{l}],$ $\bm{22.23}$ $\bm{0.010}$ $33.91$ $33.09$ $31.75$ $26,73$ $\bm{\Psi}=tanh\left({\bm{W}_{\Psi}\left(\bm{\mu}+\left(\bm{f}_{\psi}\otimes%
+\Psi\right)\right)+\bm{b}_{\Psi}}\right),$ $34.56$ $\bm{35.83}$ $\bm{b_{\psi}}$ $\Theta_{c}=(\bm{W}_{c},\bm{b}_{c})$ $\lfloor p^{2}\cdot r\rfloor$ $E_{SP}$ $E_{SE}$ $\displaystyle E_{SP}(sp,2i)=\sin(\frac{sp}{\Omega^{\frac{2i}{d}}})\quad E_{SP}%
+(sp,2i+1)=\cos(\frac{sp}{\Omega^{\frac{2i}{d}}})$ $E_{SE}(s,2i)$ $\displaystyle E_{P}(pos,2i)=\sin(\frac{pos}{\Omega^{\frac{2i}{d}}}),\text{ }E_%
+{P}(pos,2i+1)=\cos(\frac{pos}{\Omega^{\frac{2i}{d}}})$ $j\in\{i,...,n\}$ $p_{i}c_{j}$ $(p,M)$ $\displaystyle\leq\mathbb{P}\left[(\mathcal{T}+1)\operatorname{Gap}(\tilde{x},%
+\tilde{y})\geq A\right]+\mathbb{P}\left[\frac{1}{(\mathcal{T}+1)}\geq\frac{1}{%
+B}\right]$ $\hat{x}^{1,k},\hat{y}^{1,k}$ $s_{x}^{t}$ $F_{1},...,F_{M}:\mathbb{R}^{d}\to\mathbb{R}$ $\nabla F_{\mathcal{D}}(w^{t})$ $\displaystyle=\langle\nabla F_{\mathcal{D}}(w^{t})-\nabla f(w^{t};B^{t}),x^{t}%
+-x\rangle$ $\displaystyle\leq 2\tau q\|\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}};%
+B^{t})\|_{\infty}^{2}+2\tau\sum_{k=t_{0}}^{t-1}\Big{(}\|\nabla f(\hat{w}^{t_{0%
+}};B^{k})-\nabla f(w^{t_{0}};B^{k})\|_{\infty}^{2}\Big{)}$ $L_{2}^{F}$ $L_{2}^{G}\leq L_{2}^{F}D^{3}$ $\mathbb{E}\big{[}\min_{w\in\Delta_{x}}F_{\mathcal{D}}(w,\bar{y})-\min_{w\in%
+\Delta_{x}}F_{\mathcal{D}}(w,\tilde{y})\big{]}\leq\frac{2L_{1}}{T}+\frac{2L_{1%
+}^{2}}{\lambda T}+\lambda\log(d_{x}).$ $\displaystyle\qquad+\frac{L_{0}\sqrt{(TK/q+qK)\log(1/\delta)}\log(d_{x})}{n%
+\varepsilon}+\frac{L_{2}}{K}+\frac{L_{1}q\log(T/q)}{T}.$ $F(w^{T})-F(x)\leq\frac{1}{\sum_{j\in[T]}\beta_{j}}\bigg{(}\operatorname{Regret%
+}_{T}(x)+\sum_{t\in[T]}\langle\beta_{t}\nabla F(w^{t})-g_{t},x_{t}-x\rangle%
+\bigg{)}.$ $2\Delta_{s}-$ $\Lambda^{T}$ $\beta_{t}\in\mathbb{R}^{+},g^{t}\in\mathbb{R}^{d_{x}}$ $\color[rgb]{1,0,0}\frac{\sqrt{\ell}}{\sqrt{n}}+\left(\frac{\ell^{3/2}}{n%
+\varepsilon}\right)^{1/2}$ $\ell_{t}(x)=\langle g^{t},x\rangle$ $\mathbb{R}^{d_{x}}$ $\mathcal{Z}=\{z_{1},...,z_{|\mathcal{Z}|}\}$ $\displaystyle\leq\mathbb{E}\left[\max_{y\in\Delta_{|{\cal Q}|},x\in\Delta_{|%
+\mathcal{Z}|}}(F_{\mathcal{D}}(\tilde{x},y)-F_{\mathcal{D}}(x,\bar{y})).\right]$ $(g_{x},g_{y})=\text{BiasReducedGradient}(x,y,N,B)$ $\displaystyle\leq 2\tau\sum_{k=t_{0}}^{t-1}\Big{(}\|\nabla f(w^{t_{0}};B^{t})-%
+\nabla f(\hat{w}^{t_{0}};B^{t})\|_{\infty}^{2}+\|\nabla f(\hat{w}^{t_{0}};B^{k%
+})-\nabla f(w^{t_{0}};B^{k})\|_{\infty}^{2}\Big{)}$ $a(x)\leq Cb(x)$ $(\tilde{x}^{t},\tilde{y}^{t})_{t\in[T]}$ $\max\{L_{0},B\}$ $\displaystyle=\max_{j\in[d]}\left|\nabla_{j}F\left(\bar{a}\right)-\nabla_{j}F%
+\left(\bar{x}\right)\right|^{2}=\max_{j\in[d]}\left|\langle\nabla F(\bar{a})-%
+\nabla F(\bar{x}),e_{j}\rangle\right|^{2}$ $\mathbb{E}[(\mathcal{T}+1)\operatorname{Gap}(\bar{x},\bar{y})]\lesssim\sqrt{[(%
+\log(d_{x})+\log(d_{y})][L_{0}^{2}+L_{2}^{2}+(\log(d_{x})+\log(d_{y}))L_{1}^{2%
+}]U}+L_{2}\sqrt{U}.$ $\max\{L_{1},L_{2}\}$ $\tilde{x}^{t+1},\tilde{y}^{t+1}$ $\displaystyle\leq 5\sqrt{\frac{\log(|{\cal Q}|)}{n}},$ $\mathbb{P}\left[F(\bar{a}^{T})-F(\bar{x}^{T})\geq\frac{L_{1}D^{2}}{2}\sum_{t=1%
+}^{T}\lambda_{t}^{2}+\beta\frac{L_{0}D}{\sqrt{2}}\sqrt{\sum_{t=1}^{T}\lambda_{%
+t}^{2}}\right]\leq\exp(-\beta^{2}).$ $(\tilde{x},\tilde{y})$ $\min_{x\in\Delta_{x}}\max_{y\in\Delta_{y}}F_{\mathcal{D}}(x,y),$ $g_{y}=C_{M}2^{N}(\nabla_{y}f(\bar{x}_{+},\bar{y}_{+};B)-\nabla_{y}f(\bar{x}_{-%
+},\bar{y}_{-};B))+\nabla_{y}f(x_{0},y_{0};B)$ $\mathbb{E}\left[(\mathcal{T}+1)\operatorname{Gap}(\bar{x},\bar{y})\right]$ $i\in\{K_{x}+1,...,K_{x}+K_{y}\}$ $\textstyle s^{t}_{y}(S,j)=-\tau\left(\sum_{i=1}^{t}g^{i}_{y,j}\right)$ $\alpha_{t}=\lambda_{t}\left\langle\nabla F\left(\sum_{k=t}^{T}\lambda_{k}x^{k}%
++\sum_{k=1}^{t-1}\lambda_{k}a^{k}\right),a^{t}-x^{t}\right\rangle$ $(\alpha_{t})_{t\in[T]}$ $\displaystyle\|\mathbb{E}[\Phi_{\mathcal{D}}(x,y)-(g_{x},-g_{y})]\|_{\infty}$ $f:\mathcal{X}\times\mathcal{Y}\times\mathcal{Z}\to\mathbb{R}$ $\Delta(s_{y}^{t})$ $\tilde{y}^{1}$ $\{\mathcal{T}\geq t-1\}=\Big{\{}\sum_{k\in[t-1]}2^{N_{k}}\leq U\Big{\}}$ $\displaystyle\lesssim\frac{\log(d_{x})+\log(d_{y})}{\tau}+\tau[L_{0}^{2}+L_{2}%
+^{2}+(\log(d_{x})+\log(d_{y}))ML_{1}^{2}]\frac{U}{M}+\frac{L_{2}U}{M2^{M}}$ $\displaystyle\leq\|\nabla f(w^{t};B^{t})-\nabla f(w^{t_{0}};B^{t})\|_{\infty}%
+\|x^{t}-x\|_{1}$ $\displaystyle\langle\nabla f(w^{t};B^{t})-\nabla f(w^{t_{0}};B^{t}),x^{t}-x\rangle$ $\displaystyle\geq\mathbb{E}\left[\sum_{t=1}^{\mathcal{T}+1}2^{N_{t}}\right]=%
+\mathbb{E}\left[\sum_{t=1}^{\mathcal{T}+1}\mathbb{E}[2^{N_{t}}\mathbbm{1}_{(%
+\mathcal{T}+1\geq t)}|N_{t-1},...,N_{1}]\right]$ $(\mathcal{A}_{n})_{n\geq 1}$ $\displaystyle=|F(\Lambda(\lambda_{1},\mu_{1})+(x_{1},y_{1}))-F(\Lambda(\lambda%
+_{2},\mu_{2})+(x_{1},y_{1}))|$ $\displaystyle=\mathbb{P}\left[(\mathcal{T}+1)\operatorname{Gap}(\tilde{x},%
+\tilde{y})\cdot\frac{1}{\mathcal{T}+1}\geq\frac{A}{B}\right]$ $\tau\eqsim\min\left\{\sqrt{\frac{\log(d_{x})}{(L_{0}^{2}+(L_{0}^{2}+L_{1}^{2})%
+q\sqrt{\log(d_{x})/K}+L_{1}^{2}q/[\sqrt{\log(d_{x})}K^{3/2}])T}},\frac{1}{L_{0%
+}q},\frac{n\varepsilon}{TL_{0}\sqrt{(TK/q+qK)\log(1/\delta)}}\right\}.$ $\hat{x}^{t+1,k}$ $\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})]\lesssim L_{1}\frac{\sqrt{\log(%
+d_{x})+\log(d_{y})}}{\sqrt{T}}+\frac{(\log(d_{x})+\log(d_{y}))}{\tau T}+\tau L%
+_{0}^{2}+\frac{L_{2}}{K}.$ $\displaystyle=\tau\Big{|}2^{N_{r}}\Big{(}\nabla_{x}f(\bar{x}_{+}^{r},\bar{y}_{%
++}^{r};B^{r})-\nabla_{x}f(\bar{x}_{-}^{r},\bar{y}_{-}^{r};B^{r})\Big{)}+\nabla%
+_{x}f(x_{0}^{r},y_{0}^{r};B^{r})$ $\mathbb{E}\Big{\|}\nabla_{x}f(\bar{x}_{+},\bar{y}_{+};B_{k})-\nabla_{x}f(x,y;B%
+_{k})\Big{\|}_{\infty}^{2}\leq\frac{20L_{2}^{2}}{2^{2(k+1)}}+\frac{(12+2\log(d%
+_{x}))L_{1}^{2}}{2^{k+1}},$ $\bar{x}=\frac{1}{T}\sum_{t=1}^{T}x^{t}$ $\sum_{t=1}^{\mathcal{T}+1}2^{N_{t}}\leq U$ $\displaystyle\leq\tau\Big{|}\frac{2^{N_{r}}\alpha}{2^{N_{r}}}\Big{[}\Big{(}%
+\nabla_{x}f(\bar{x}_{+}^{r},\bar{y}_{+}^{r};z^{*})-\nabla_{x}f(\bar{x}_{+}^{r}%
+,\bar{y}_{+}^{r};z^{\prime*})\Big{)}+\Big{(}\nabla_{x}f(\bar{x}_{-}^{r},\bar{y%
+}_{-}^{r};z^{*})-\nabla_{x}f(\bar{x}_{-}^{r},\bar{y}_{-}^{r};z^{\prime*})\Big{%
+)}\Big{]}$ $\displaystyle\lesssim\begin{cases}q\tau\big{[}\frac{L_{1}^{2}}{\sqrt{\log(d_{x%
+})}K^{3/2}}+\frac{(L_{0}^{2}+L_{1}^{2})\sqrt{\log(d_{x})}}{\sqrt{K}}\big{]}&%
+\text{without second order smoothnes}\\
+q\tau\big{[}\frac{L_{2}^{2}}{K^{2}}+\frac{L_{1}^{2}\log(d_{x})}{K}\big{]}&%
+\text{with second order smoothness}\end{cases},$ $(w^{t},v^{t})$ $M=d$ $\mbox{TG}(p,M)$ $\displaystyle\quad+\mathbb{E}\big{[}\langle\nabla f(w^{t_{0}};B^{t})-\nabla f(%
+\hat{w}^{t_{0}};B^{t}),x^{t}-x\rangle\big{]}$ $i\in[K_{x}]$ $\langle\nabla F(\bar{a})-\nabla F(\bar{x}),e_{j}\rangle\leq\frac{|F(\bar{a}+re%
+_{j})-F(\bar{x}+re_{j})|+|F(\bar{x})-F(\bar{a})|}{r}+L_{1}r.$ $q=\sqrt{T/\log(d_{x})},K=T/q=\sqrt{T\log(d_{x})}$ $J:=\{e_{1},...,e_{d}\}$ $s_{y}^{t}$ $L_{2}\lesssim(L_{0}+L_{1})\left\{\frac{\sqrt{\log(d_{x})+\log(d_{y})}}{\sqrt{n%
+}}+\left(\frac{(\log(d_{x})+\log(d_{y}))^{3/2}\sqrt{\log(1/\delta)}}{n%
+\varepsilon}\right)^{1/2}\right\}$ $y^{1}=(1/d_{y},...,1/d_{y})$ $\displaystyle\lesssim\frac{2(\log(d_{x})+\log(d_{y}))}{\tau T}+5\tau L_{0}^{2}%
++\frac{1}{T}\sum_{t=1}^{T}\mathbb{E}\|\mathbb{E}[\Phi_{\mathcal{D}}(x^{t},y^{t%
+})-g^{t}\mid\mathcal{F}_{t}]\|_{\infty}.$ $2^{N_{t}}\leq 2^{M}$ $\frac{\sqrt{\ell}}{\sqrt{n}}+\left(\frac{\ell^{3/2}}{n\varepsilon}\right)^{2/5}$ $\displaystyle\leq\frac{2\log(|\mathcal{Z}|)}{\tau_{x}T}+\frac{\log(|{\cal Q}|)%
+}{\tau_{y}T}+18\tau_{x}+2\tau_{y}+\frac{1}{T}\sum_{t=1}^{T}\langle\nabla_{x}F_%
+{\mathcal{D}}(x^{t},y^{t})-g^{t}_{x},x^{t}-w^{t}\rangle+2\|-\nabla_{y}F_{%
+\mathcal{D}}(x^{t},y^{t})-g^{t}_{y}\|_{\infty}.$ $\operatorname{Regret}_{T}(x)\leq\frac{\log(d)}{\tau}+\frac{\tau}{2}\sum_{t\in[%
+T]}\|g^{t}\|_{\infty}^{2},$ $a^{t}\sim P_{x^{t}}$ $w^{t}=\frac{(t-1)w^{t-1}+x^{t}}{t}$ $\mathbb{E}\left[(\mathcal{T}+1)(\operatorname{Gap}(\tilde{x},\tilde{y})-%
+\operatorname{Gap}(\bar{x},\bar{y}))\right]$ $F_{\mathcal{D}}(w^{T})-F_{\mathcal{D}}(x)\leq\frac{1}{T}\left[\operatorname{%
+Regret}_{T}(x)+\sum_{t\in[T]}\langle\nabla F_{\mathcal{D}}(w^{t})-g^{t},x^{t}-%
+x\rangle\right],$ $f:\mathbb{R}^{d}\mapsto\mathbb{R}$ $X\sim{\cal N}(0,1/2)$ $\displaystyle=\max_{k\in[K_{x}+K_{y}]}\big{|}\nabla_{k}\langle\Lambda^{T}_{j},%
+\nabla F(\Lambda(\lambda_{1},\mu_{1})+(x_{1},y_{1}))-\nabla F(\Lambda(\lambda_%
+{2},\mu_{2})+(x_{1},y_{1}))\rangle\big{|}$ $\displaystyle=3\left(\frac{|F(\bar{x})-F(\bar{a})|^{2}}{r^{2}}+L_{1}^{2}r^{2}+%
+\max_{j\in[d]}\frac{|F(\bar{a}+re_{j})-F(\bar{x}+re_{j})|^{2}}{r^{2}}\right)$ $\displaystyle\leq 4\tau\sum_{k=t_{0}}^{t-1}\left\|\nabla f(\hat{w}^{t_{0}};B^{%
+k})-\nabla f(w^{t_{0}};B^{k})\right\|_{\infty}.$ $\displaystyle\lesssim\left(L_{0}^{2}+L_{2}^{2}+\log(d_{x})ML_{1}^{2}\right)%
+\mathbb{E}\left[\sum_{t=1}^{U}\mathbbm{1}_{(\mathcal{T}+1\geq t)}\right]$ $\tau\leq\frac{B\varepsilon}{8L_{0}\sqrt{2(TK/q+qK)\log(1/\delta)}}$ $\displaystyle\lesssim\sqrt{\frac{(L_{0}^{2}+L_{1}^{2}q\log(d_{x})/K+L_{2}^{2}q%
+/K^{2})\log(d_{x})}{T}}+\frac{L_{0}q\log(d_{x})}{T}$ $\displaystyle+\langle\nabla f(w^{t};B^{t})-\nabla f(w^{t_{0}};B^{t}),x^{t}-x\rangle$ $f_{i}:\mathcal{X}\times\mathcal{Z}\to[-B,B]$ $\hat{x}\sim P_{x}$ $\mathcal{Y}=\Delta_{y}$ $\tilde{x}_{j}^{t+1}\propto\tilde{x}_{j}^{t}\exp\big{(}-\tau\nabla_{j}f(w^{t_{0%
+}};B^{k})\big{)}$ $N(S)=N((\varepsilon_{n}(S))_{n\geq 1},(\delta_{n}(S))_{n\geq 1})$ $\displaystyle\leq\tau\alpha L_{0}(4+1/2^{N_{r}})\leq 4.5\tau\alpha L_{0}.$ $A\eqsim\sqrt{[(\log(d_{x})+\log(d_{y})][L_{0}^{2}+L_{2}^{2}+(\log(d_{x})+\log(%
+d_{y}))L_{1}^{2}]U}\quad,\quad B\eqsim U/M$ $\displaystyle\leq\frac{\log(|\mathcal{Z}|)}{\tau_{x}}+\frac{\tau_{x}}{2}\sum_{%
+t\in[T]}\|g^{t}_{x}\|_{\infty}^{2}+\frac{\log(|{\cal Q}|)}{\tau_{y}}+\frac{%
+\tau_{y}}{2}\sum_{t\in[T]}\|g^{t}_{y}\|_{\infty}^{2}+\sum_{t=1}^{T}\langle\Phi%
+_{\mathcal{D}}(x^{t},y^{t})-g^{t},(x^{t},y^{t})-(w,v)\rangle.$ $\lambda_{i}=\frac{1}{T}$ $(\lambda_{1},\ldots,\lambda_{T})\in\Delta_{T}$ $\mathbb{P}\left[\sum_{t=1}^{T}\alpha_{t}\geq\beta\frac{L_{0}D}{\sqrt{2}}\sqrt{%
+\sum_{t=1}^{T}\lambda_{t}^{2}}\right]\leq\exp(-\beta^{2}),$ $F_{\mathcal{D}}(x_{\mathcal{D}},\bar{y})=0$ $\displaystyle T\operatorname{Gap}(\bar{x},\bar{y})$ $x^{1},...,x^{T}\in\Delta_{d}$ $\|\nabla F(x^{1})-\nabla F(x^{2})\|_{\infty}\leq L_{1}\|x^{1}-x^{2}\|_{1}$ $\frac{4\tau L_{0}}{B}\leq\frac{\varepsilon}{2\sqrt{2(TK/q+(q+1)K)\log(1/\delta%
+)}}.$ $T\eqsim\min\left\{n,\frac{n\varepsilon}{\sqrt{(\log(d_{x})+\log(d_{y}))\log(1/%
+\delta)}}\right\},$ $T\eqsim\min\left\{n,\frac{(n\varepsilon)^{2/3}}{\log(1/\delta)^{1/3}}\right\}$ $|(x^{t}-\tilde{x}^{t})_{j}|\leq\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)}{\sum%
+_{i\in[d_{x}]}x^{t_{0}}_{i}\exp\left(G_{i}\right)}\underbrace{\max\left\{2\max%
+_{j\in[d_{x}]}|\hat{G}_{j}-G_{j}|,\exp\left(2\max_{j\in[d_{x}]}|\hat{G}_{j}-G_%
+{j}|\right)-1\right\}}_{:=\psi},$ $\frac{4L_{0}\tau}{B}$ $\displaystyle\leq L_{0}^{F}\sum_{i=1}^{K_{x}}|\lambda_{1,i}-\lambda_{2,i}|\|x_%
+{i}-x_{1}\|_{1}+L_{0}\sum_{j=1}^{K_{y}}|\mu_{1,j}-\mu_{2,j}|\|y_{j}-y_{1}\|_{1}$ $U=\min\left\{\frac{n\varepsilon\sqrt{L_{0}^{2}+L_{2}^{2}+(\log(d_{x})+\log(d_{%
+y}))L_{1}^{2}}}{\sqrt{4\cdot 48\cdot 81(\log(d_{x})+\log(d_{y}))\log(1/\delta)%
+}L_{0}},\frac{n}{2}\right\}$ $T=\min\left\{n,\frac{n\varepsilon}{\log(d_{x})\sqrt{\log(1/\delta)}}\right\}$ $\mathbb{E}\left[\sum_{t=1}^{\mathcal{T}+1}\langle\Phi_{\mathcal{D}}(x^{t},y^{t%
+})-(g_{x}^{t},g_{y}^{t}),(x^{t},y^{t})-(w^{t},v^{t})\rangle\right]\lesssim%
+\frac{L_{2}\mathbb{E}[\mathcal{T}+1]}{2^{M}}.$ $\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}};B^{t})$ $\mathbb{E}\|\nabla F\left(\frac{1}{T}\sum_{t\in[T]}a^{t}\right)-\nabla F\left(%
+\frac{1}{T}\sum_{t\in[T]}x^{t}\right)\|^{2}_{\infty}\lesssim\frac{L_{2}^{2}}{T%
+^{2}}+\frac{L_{1}^{2}(1+\log(d))}{T}$ $x_{i}-x_{1}$ $2^{N_{t}}$ $\|x\|_{1}=\sum_{j\in[d]}|x_{j}|$ $0 $\mathbb{E}[\nabla_{j}F(\bar{x})-\nabla_{j}F(\bar{a})]=\mathbb{E}[\langle\nabla
+F%
+(\bar{x})-\nabla F(\bar{a}),e_{j}\rangle]\leq\frac{4L_{1}}{rT}+L_{1}r.$ $\{x_{1},\ldots,x_{K_{x}}\}$ $M=\log_{2}(\sqrt{U})$ $\displaystyle\leq 6\sum_{t=1}^{\mathcal{T}+1}2^{N_{t}}\leq 6\sum_{t=1}^{%
+\mathcal{T}}2^{N_{t}}+6\cdot 2^{M}\leq 6(U-2^{M})+6\cdot 2^{M}=6U,$ $\displaystyle\leq 2L_{1}\|w^{t}-w^{t_{0}}\|_{1}\leq 2L_{1}\sum_{k=t_{0}}^{t-1}%
+\|w^{k+1}-w^{k}\|_{1}$ $\frac{2\tau L_{0}}{B}$ $|s_{x}^{t}-s_{x}^{\prime t}|$ $\displaystyle\leq\tau\|\nabla\operatorname{LSE}(-\tau G(S))\|_{1}\|G_{j}(S)-G_%
+{j}(S^{\prime})\|_{\infty}$ $\displaystyle 4\sum_{t=1}^{\mathcal{T}+1}2^{N_{t}}+2(\mathcal{T}+1)$ $\displaystyle\leq 0+\frac{4L_{1}q}{q\lfloor t/q\rfloor+1}+2\|\mathbb{E}_{\hat{%
+w}^{t_{0}}}[\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}};B^{t})]\|_{\infty}$ $K=T/\log(d_{x})$ $\displaystyle\leq\frac{2L_{1}}{T}+\frac{2L_{1}^{2}}{\lambda T}+\lambda\log(d_{%
+y}).$ $\tilde{x}^{t+1}$ $x^{1}=(1/d_{x},...,1/d_{x})$ $N:\mathbb{R}^{\infty}_{\geq 0}\times\mathbb{R}^{\infty}_{\geq 0}\to\mathbb{N}$ $\{y_{1},\ldots,y_{K_{y}}\}$ $\mathbb{P}(j=e_{i})\propto\exp\left(\frac{\varepsilon s(S,i)}{2\Delta_{s}}\right)$ $N^{t+1}\sim\mbox{TG}(0.5,M)$ $\left\|\mathbb{E}\left[\nabla F\left(\frac{1}{T}\sum_{t=1}^{T}a^{t}\right)-%
+\nabla F\left(\frac{1}{T}\sum_{t=1}^{T}x^{t}\right)\right]\right\|_{\infty}%
+\leq\frac{4L_{1}}{\sqrt{T}}.$ $\displaystyle\mathbb{E}\left[\max_{j\in[M]}|F_{j}(\bar{x})-F_{j}(\bar{a}^{T})|%
+^{2}\right]$ $\displaystyle\mathbb{E}[\langle\nabla F_{\mathcal{D}}(w^{t})-\nabla f(\hat{w}^%
+{t_{0}};B^{t}),x^{t}-x\rangle]$ $\displaystyle\leq\sum_{t\in[T]}[F_{\mathcal{D}}(x^{t},v)-F_{\mathcal{D}}(w,y^{%
+t})]\leq\sum_{t\in[T]}\langle\Phi_{\mathcal{D}}(x^{t},y^{t}),(x^{t},y^{t})-(w,%
+v)\rangle$ $\bar{x}^{T}=\sum_{t=1}^{T}\lambda_{t}x^{t}$ $T,K,q,\tau$ $\|g^{t}\|_{\infty}\leq L_{0}$ $\displaystyle\sum_{t=1}^{T}\langle\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t},(x^{t}%
+,y^{t})-(w,v)\rangle=\sum_{t=1}^{T}\langle\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t%
+})-g^{t}_{x},x^{t}-w\rangle+\langle-\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{%
+t}_{y},y^{t}-v\rangle$ $\displaystyle\operatorname{Gap}(\bar{x},\bar{y})\leq\frac{2(\log(d_{x})+\log(d%
+_{y}))}{\tau T}+5\tau L_{0}^{2}+\frac{1}{T}\sum_{t=1}^{T}\langle\Phi_{\mathcal%
+{D}}(x^{t},y^{t})-g^{t},(x^{t},y^{t})-(w^{t},v^{t})\rangle.$ $\displaystyle\leq\frac{2L_{1}}{T}+\frac{2L_{1}^{2}}{\lambda T}+\lambda\log(d_{%
+y})+\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})]+\frac{2L_{1}}{T}+\frac{2L_%
+{1}^{2}}{\lambda T}+\lambda\log(d_{x})$ $\displaystyle\mathbb{E}[\langle\nabla F_{\mathcal{D}}(w^{t})-\nabla f(\hat{w}^%
+{t};B^{t}),x^{t}-x\rangle]$ $s^{t}_{x}$ $f(\cdot,\cdot;z)$ $G_{j}(S)=\left(\sum_{k\in[i-1]}g^{k}_{j}\right)_{i\in[t]}$ $\hat{G}_{j}=-\tau\sum_{k=t_{0}}^{t-1}\nabla_{j}f(\hat{w}^{t_{0}};B^{k}),G_{j}=%
+-\tau\sum_{k=t_{0}}^{t-1}\nabla_{j}f(w^{t_{0}};B^{k})$ $\xi^{1},\xi^{2},...$ $\|g^{t}\|_{\infty}^{2}+\|\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t}\|_{\infty}^{2}%
+\leq 5L_{0}^{2}$ $T\eqsim\min\left\{n,\frac{(n\varepsilon)^{2/3}}{\log(1/\delta)^{1/3}}\right\},$ $\mathcal{T}+1$ $x^{t+1}_{j}\propto x^{t}_{j}\exp\left(-\tau g^{t}_{j}\right),\quad\forall j\in%
+[d_{x}]$ $\mathbb{E}[A_{1}]=\mathbb{E}[\langle\tilde{\Delta}^{t_{0}},\tilde{x}^{t}-x%
+\rangle]\leq\begin{cases}\frac{8L_{1}}{\sqrt{K}}&\text{without second order %
+smoothnes}\\
+\frac{4L_{2}}{K}&\text{with second order smoothness}\end{cases}.$ $\mathbb{E}[\|x^{t}\|_{\infty}]<\infty$ $x^{1}=(1/|\mathcal{Z}|,...,1/|\mathcal{Z}|),y^{1}=(1/|\mathcal{Q}|,...,1/|%
+\mathcal{Q}|)$ $\langle-\nabla F(\bar{a}),e_{j}\rangle\leq\frac{F(\bar{a}+re_{j})-F(\bar{a})}{%
+r}+\frac{L_{1}r}{2}.$ $F_{1}(\cdot)=F(\cdot)$ $2^{N_{t}+1}$ $\mathbb{P}\left[(\mathcal{T}+1)\operatorname{Gap}(\tilde{x},\tilde{y})\geq A%
+\right]\lesssim\frac{\sqrt{[(\log(d_{x})+\log(d_{y})][L_{0}^{2}+L_{2}^{2}+(%
+\log(d_{x})+\log(d_{y}))L_{1}^{2}]U}}{A}.$ $\lambda=L_{1}\frac{2}{\sqrt{T(\log(d_{x})+\log(d_{y}))}}$ $\displaystyle\lesssim\frac{\sqrt{[(\log(d_{x})+\log(d_{y})][L_{0}^{2}+L_{2}^{2%
+}+(\log(d_{x})+\log(d_{y}))L_{1}^{2}]U}}{A}$ $(x_{i}-x_{1},\mathbf{0}_{d_{y}})$ $\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})]\lesssim(L_{0}+L_{1}+L_{2})%
+{\small\bigg{[}\sqrt{\frac{\log(d_{x})+\log(d_{y})}{n}}+\left(\frac{(\log(d_{x%
+})+\log(d_{y}))^{3/2}\sqrt{\log(1/\delta)}}{n\varepsilon}\right)^{2/5}\bigg{]}.}$ $\displaystyle=\frac{x^{t_{0}}_{j}\exp\left(\hat{G}_{j}\right)}{\sum_{i\in[d_{x%
+}]}x^{t_{0}}_{i}\exp\left(\hat{G}_{i}\right)}-\frac{x^{t_{0}}_{j}\exp\left(G_{%
+j}\right)}{\sum_{i\in[d_{x}]}x^{t_{0}}_{i}\exp\left(G_{i}\right)}$ $\lambda=\sqrt{8}/\sqrt{\log(|{\cal Q}|)T}$ $\mathcal{A}_{1:N(S)}$ $L_{0}D$ $\bar{a}^{T}=\sum_{t\in[T]}\lambda_{t}a^{t}$ $L/\mu=\Omega(d)$ $\ell_{x}=\log(d_{x})$ $x^{1},\ldots,x^{T}\in\mathbb{R}^{d}$ $\displaystyle\leq\frac{|F(\bar{a}+re_{j})-F(\bar{x}+re_{j})|+|F(\bar{x})-F(%
+\bar{a})|}{r}+L_{1}r$ $\displaystyle\leq\frac{2(\log(d_{x})+\log(d_{y}))}{\tau}+\tau\sum_{t=1}^{T}(\|%
+g^{t}\|_{\infty}^{2}+\|\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t}\|_{\infty}^{2})$ $\displaystyle=\mathbb{E}\left[\nabla_{x}F_{\mathcal{D}}(\bar{x}_{+,M},\bar{y}_%
+{+,M})-\nabla_{x}F_{\mathcal{D}}(x_{-,0},y_{-,0})+\nabla_{x}F_{\mathcal{D}}(x_%
+{0},y_{0})\right]$ $g^{t}_{x,j}$ $\operatorname{Gap}(x,y)=\max_{v\in\mathcal{X},w\in\mathcal{Y}}(F_{\mathcal{D}}%
+(x,w)-F_{\mathcal{D}}(v,y))$ $\|x\|_{\infty}=\max_{j\in[d]}|x_{j}|$ $B=n/T$ $\displaystyle\leq\max_{i\in[K_{x}+K_{y}]}\|\Lambda^{T}_{i}\|_{1}\|\nabla F(%
+\Lambda(\lambda_{1},\mu_{1})+(x_{1},y_{1}))-\nabla F(\Lambda(\lambda_{2},\mu_{%
+2})+(x_{1},y_{1}))\|_{\infty}$ $\left\|\mathbb{E}\left[\nabla F\left(\frac{1}{T}\sum_{t=1}^{T}a^{t}\right)-%
+\nabla F\left(\frac{1}{T}\sum_{t=1}^{T}x^{t}\right)\right]\right\|_{\infty}%
+\leq\frac{2L_{2}}{T}.$ $w^{t}_{j}\propto\exp(\operatorname{LSE}(\tau G_{j}(S))).$ $\displaystyle\left(\sum_{i=1}^{K_{x}}\lambda_{i}(x_{i}-x_{1}),\sum_{j=1}^{K_{y%
+}}\mu_{j}(y_{j}-y_{1})\right).$ $\displaystyle\lesssim\sqrt{[(\log(d_{x})+\log(d_{y})]L_{1}^{2}\frac{U}{M}}$ $\tilde{x}^{t+1},\hat{x}^{t+1,k}$ $\displaystyle\leq A^{2}+C^{2}\log(M)+2C^{2}+2AC$ $\displaystyle\langle\nabla F_{\mathcal{D}}(w^{t})-\nabla f(\hat{w}^{t_{0}};B^{%
+t}),x^{t}-x\rangle$ $\mathbb{P}[F(\bar{a}^{T})-F\left(\bar{x}^{T}\right)\geq\alpha]\leq\mathbb{P}%
+\left[\sum_{t=1}^{T}\alpha_{t}+\frac{L_{1}D^{2}}{2}\sum_{t=1}^{T}\lambda_{t}^{%
+2}\geq\alpha\right]=\mathbb{P}\left[\sum_{t\in[T]}\alpha_{t}\geq\alpha^{\prime%
+}\right]$ $s_{x}^{t}=s_{x}^{\prime t}$ $s^{t}_{x}=-\tau\sum_{r=1}^{t}\Big{[}2^{N_{r}}\Big{(}\nabla_{x}f(\bar{x}_{+}^{r%
+},\bar{y}_{+}^{r};B^{r})-\nabla_{x}f(\bar{x}_{-}^{r},\bar{y}_{-}^{r};B^{r})%
+\Big{)}+\nabla_{x}f(x_{0}^{r},y_{0}^{r};B^{r})\Big{]}.$ $\mathbb{E}\left[\max_{j\in[M]}|F_{j}(\bar{x})-F_{j}(\bar{a}^{T})|^{2}\right]%
+\leq\frac{5L_{1}^{2}D^{4}}{4}\left(\sum_{t=1}^{T}\lambda_{t}^{2}\right)^{2}+%
+\frac{L_{0}^{2}D^{2}(6+\log(M))}{2}\sum_{t=1}^{T}\lambda_{t}^{2}.$ $\mathbb{E}\big{[}\sum_{t\in[T]}\alpha_{t}\big{]}=0$ $g^{t}\in\partial f_{t}(x)$ $\left\|\mathbb{E}\left[\nabla F\left(\frac{1}{T}\sum_{t=1}^{T}a^{t}\right)-%
+\nabla F\left(\frac{1}{T}\sum_{t=1}^{T}x^{t}\right)\right]\right\|_{\infty}%
+\lesssim\frac{L_{2}}{T}$ $\displaystyle\leq 2L_{0}^{2}+4C_{M}\sum_{k=0}^{M}2^{k}\mathbb{E}\Big{\|}\nabla%
+_{x}f(\bar{x}_{+},\bar{y}_{+};B_{k})-\nabla_{x}f(x,y;B_{k})\Big{\|}_{\infty}^{2}$ $\sum_{t\in[\mathcal{T}+1]}\max\{1,2^{N_{t}}/\alpha\}\leq n$ $\displaystyle\lesssim\frac{\log(d_{x})}{\tau T}+\tau\Big{[}L_{0}^{2}+\frac{L_{%
+1}^{2}q\log(d_{x})}{K}+\frac{L_{2}^{2}q}{K^{2}}\big{]}+\frac{L_{2}}{K}+\frac{L%
+_{1}q\log(T/q)}{T},$ $\operatorname{Regret}_{T}(x)\leq\frac{\log(d_{x})}{\tau}+\frac{\tau}{2}\sum_{t%
+\in[T]}\|g^{t}\|_{\infty}^{2}$ $\displaystyle\mathbb{E}\left[\max_{q\in{\cal Q}}|\mathbb{E}_{z}[q(z)]-q(\tilde%
+{S})|\right]$ $M=\log(\sqrt{U})$ $\mathbb{R}^{d_{x}},\mathbb{R}^{d_{y}}$ $\mathbb{P}\left[\frac{1}{\mathcal{T}+1}\geq\frac{1}{B}\right]=\mathbb{P}\left[%
+\mathcal{T}\leq B-1\right]\leq\mathbb{P}\left[\sum_{t\in[B]}2^{N_{t}}>U-2^{M}%
+\right]\leq\frac{B\mathbb{E}[2^{N_{1}}]}{U-2^{M}},$ $F_{\mathcal{D}}(\cdot)$ $\displaystyle\leq L_{0}^{F}D(\|\lambda_{1}-\lambda_{2}\|_{1}+\|\mu_{1}-\mu_{2}%
+\|_{1})$ $\displaystyle\lesssim(L_{0}+L_{1})\sqrt{\frac{\log(d_{x})}{T}}+\frac{L_{2}}{%
+\sqrt{T}\log(d_{x})^{1/4}}$ $(\hat{x}^{t,i})_{i\in[K]}\overset{\text{iid}}{\sim}P_{x^{t}}$ $\tau\eqsim\min\left\{\frac{1}{L_{0}}\sqrt{\frac{\log(d_{x})+\log(d_{y})}{T}},%
+\frac{n\varepsilon}{L_{0}T\sqrt{TK\log(1/\delta)}}\right\}$ $g^{t}=\nabla f(\hat{w}^{t};B^{t})$ $L_{1}^{G}\leq L_{1}^{F}D^{2}$ $g^{t}_{x}=\nabla_{x}f(x^{t},\hat{y}^{t};S)$ $\displaystyle=\mathbb{E}[\max_{v\in\Delta_{y}}F_{\mathcal{D}}(\tilde{x},v)-%
+\max_{v\in\Delta_{y}}F_{\mathcal{D}}(\bar{x},v)]+\mathbb{E}[\max_{v\in\Delta_{%
+y}}F_{\mathcal{D}}(\bar{x},v)-\min_{w\in\Delta_{x}}F_{\mathcal{D}}(w,\bar{y})]$ $\frac{\varepsilon}{2\sqrt{2T(K+1)\log(1/\delta)}}$ $\frac{4TL_{0}\tau_{y}}{n}\leq\frac{\varepsilon}{2\sqrt{2T\log(1/\delta)}}.$ $\displaystyle=\frac{(A+B)^{2}}{2}+2MC\left[C\exp\left(-(B/C)^{2}\right)+A%
+\mathbb{P}[X\geq B/C]\right]$ $\displaystyle+\langle\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}};B^{t})%
+,x^{t}-x\rangle.$ $\mathbb{E}[x_{t+1}|x_{1:t}]=0$ $M(\log(d_{x})+\log(d_{y}))$ $\mathbb{P}[X\geq B/C]\leq\exp\left(-(B/C)^{2})\right)$ $T=\frac{6n\varepsilon}{16\sqrt{2\log(1/\delta)}\log(|{\cal Q}|)},\tau_{x}=%
+\sqrt{\frac{\log(|\mathcal{Z}|)}{9T}},\tau_{y}=\frac{\log(|{\cal Q}|)}{6\sqrt{%
+\log(|\mathcal{Z}|)T}}$ $(\varepsilon^{\prime},T\delta+\delta^{\prime})$ $\tilde{x}^{t}\sim P_{x^{t}}$ $\sum_{t\in[T]}\langle\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{x},x^{t}-w%
+\rangle\\
+\leq\sum_{t\in[T]}\langle\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{x},x^{t%
+}-w^{t}\rangle+\frac{\log(d_{x})}{\tau}+\frac{\tau}{2}\sum_{t\in[T]}\|\nabla_{%
+x}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{x}\|_{\infty}^{2}.$ $\displaystyle\leq\mathbb{E}\left[(\mathcal{T}+1)\left(\frac{4L_{1}}{\mathcal{T%
+}+1}+\frac{2L_{1}\sqrt{\log(d_{x})+\log(d_{y})}}{\sqrt{\mathcal{T}+1}}\right)\right]$ $\displaystyle\leq\max_{k\in[K_{x}+K_{y}]}\sum_{i\in[K_{x}+K_{y}]}|\Lambda_{j,i%
+}|\|\Lambda^{T}_{k}\|_{1}\|\nabla\nabla_{i}F(\Lambda(\lambda_{1},\mu_{1})+(x_{%
+1},y_{1}))-\nabla\nabla_{i}F(\Lambda(\lambda_{2},\mu_{2})+(x_{1},y_{1}))\|_{\infty}$ $(f_{i})_{i\in[d_{y}]}$ $\sqrt{\log(d)/n}+(\log(d)^{3/2}/[n\varepsilon])^{2/5}$ $f:\mathcal{X}\times\mathcal{Z}\to\mathbb{R}$ $(g^{t}_{x},g^{t}_{y})=\operatorname{BiasReducedGradient}(x^{t},y^{t},N^{t},B^{%
+t})$ $w\in\Delta_{|\mathcal{Z}|},v\in\Delta_{|{\cal Q}|}$ $2^{N_{1}},...,2^{N_{t-1}}$ $\frac{4\tau L_{0}}{B}$ $\sum_{t\in[B]}2^{N_{i}}>U$ $\min_{x\in\mathcal{X}}\max_{y\in\mathcal{Y}}\{F_{\mathcal{D}}(x,y)=\mathbb{E}_%
+{z\sim\mathcal{D}}[f(x,y;z)]$ $\|\mathbb{E}[\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t}\mid\mathcal{F}_{t}]\|_{%
+\infty}\leq 2L_{2}/K$ $\mathbb{P}\left[\operatorname{Gap}(\tilde{x},\tilde{y})\lesssim\sqrt{\frac{[(%
+\log(d_{x})+\log(d_{y})][L_{0}^{2}+L_{2}^{2}+(\log(d_{x})+\log(d_{y}))L_{1}^{2%
+}]\log(U)^{2}}{U}}\right]\geq 0.99.$ $\phi(y)=\sum_{j\in[d_{y}]}y_{j}\log(1/y_{j})$ $\bar{x}^{T}=\sum_{t\in[T]}\lambda_{t}x^{t}$ $F(\cdot+re_{j})$ $\displaystyle\|\nabla\nabla_{j}G(\lambda_{1},\mu_{1})-\nabla\nabla_{j}G(%
+\lambda_{2},\mu_{2})\|_{\infty}=\max_{k\in[K_{x}+K_{y}]}\big{|}\nabla_{k,j}G(%
+\lambda_{1},\mu_{1})-\nabla_{k,j}G(\lambda_{2},\mu_{2})\big{|}$ $x^{t+1}_{j}\propto\exp\left\{-\tau\sum_{r=1}^{t}\Big{[}2^{N_{r}}\Big{(}\nabla_%
+{x}f(\bar{x}_{+}^{r},\bar{y}_{+}^{r};B^{r})-\nabla_{x}f(\bar{x}_{-}^{r},\bar{y%
+}_{-}^{r};B^{r})\Big{)}+\nabla_{x}f(x_{0}^{r},y_{0}^{r};B^{r})\Big{]}\right\},$ $w\in\mathcal{X}$ $\displaystyle+\mathbb{E}[\min_{w\in\Delta_{x}}F_{\mathcal{D}}(w,\bar{y})-\min_%
+{w\in\Delta_{x}}F_{\mathcal{D}}(w,\tilde{y})]$ $\displaystyle=\frac{5L_{1}^{2}D^{4}}{4}\left(\sum_{t=1}^{T}\lambda_{t}^{2}%
+\right)^{2}+\frac{L_{0}^{2}D^{2}(6+\log(M))}{2}\sum_{t=1}^{T}\lambda_{t}^{2}.$ $\displaystyle|G(\lambda_{1},\mu_{1})-G(\lambda_{2},\mu_{2})|$ $B^{r}=B^{\prime r}$ $\displaystyle=\mathbb{E}\left[\sum_{k=0}^{M}\big{(}\nabla_{x}F_{\mathcal{D}}(%
+\bar{x}_{+,k},\bar{y}_{+,k})-\nabla_{x}F_{\mathcal{D}}(\bar{x}_{-,k},\bar{y}_{%
+-,k})\big{)}+\nabla_{x}F_{\mathcal{D}}(x_{0},y_{0})\right]$ $\mathbb{E}[\langle\nabla F_{\mathcal{D}}(w^{t})-\nabla f(w^{t};B^{t}),x^{t}-x%
+\rangle]=0$ $\mathbb{E}[\operatorname{Gap}(\tilde{x},\bar{y})-\operatorname{Gap}(\bar{x},%
+\bar{y})]\leq\frac{4}{n}+6\sqrt{\frac{\log(|{\cal Q}|)}{n}}$ $g_{y}^{1}$ $\hat{G}_{j}$ $\displaystyle\leq L_{2}^{F}D^{3}\|(\lambda_{1},\mu_{1})-(\lambda_{2},\mu_{2})%
+\|_{1}.$ $\displaystyle\lesssim\frac{\log(d_{x})}{\tau T}+\tau L_{0}^{2}+\frac{1}{T}\sum%
+_{t\in[T]}\left(\frac{L_{2}}{K}+\mathbbm{1}_{(t>q)}\left[\frac{4L_{1}q}{q%
+\lfloor t/q\rfloor+1}+q\frac{L_{2}^{2}\tau}{K^{2}}+q\frac{L_{1}^{2}\log(d_{x})%
+\tau}{K}\right]\right)$ $\tau\eqsim\min\left\{\sqrt{\frac{\log(d_{x})}{(L_{0}^{2}+(L_{0}^{2}+L_{1}^{2})%
+q\sqrt{\log(d_{x})/K}+L_{1}^{2}q/[\sqrt{\log(d_{x})}K^{3/2}])T}},\frac{1}{L_{0%
+}q},\frac{n\varepsilon}{TL_{0}\sqrt{(TK/q+qK)\log(1/\delta)}}\right\}$ $\mathbb{E}\Big{\|}\nabla_{x}f(x,y;B_{k})-\nabla_{x}f(\bar{x}_{-},\bar{y}_{-};B%
+_{k})\Big{\|}_{\infty}^{2}\leq\frac{20L_{2}^{2}}{2^{2k}}+\frac{(12+2\log(d_{x}%
+))L_{1}^{2}}{2^{k}}.$ $\displaystyle\leq\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)}{\sum_{i\in[d_{x}]}%
+x^{t_{0}}_{i}\exp\left(G_{i}\right)}\left[2\max_{j\in[d_{x}]}|\hat{G}_{j}-G_{j%
+}|\right].$ $(\hat{w}^{t,i})_{i\in[K]}\overset{\text{iid}}{\sim}P_{w^{t}}$ $\min_{x\in\mathcal{X}}\{\max_{i\in[d_{y}]}F_{i}(x)\}=\min_{x\in\mathcal{X}}%
+\max_{y\in\Delta_{y}}\mathbb{E}_{z\sim\mathcal{D}}\left[\sum_{i\in[d_{y}]}y_{i%
+}f_{i}(x;z)\right].$ $(\bar{x}_{-,k},\bar{y}_{-,k})$ $\displaystyle\lesssim\frac{\log(d_{x})}{\tau T}+\tau\Big{[}L_{0}^{2}+\frac{L_{%
+1}^{2}q}{\sqrt{\log(d_{x})}K^{3/2}}+\frac{(L_{0}^{2}+L_{1}^{2})\sqrt{\log(d_{x%
+})}q}{\sqrt{K}}\Big{]}+\frac{L_{1}}{\sqrt{K}}+\frac{1}{T}\sum_{t=q+1}^{T}\left%
+[\frac{L_{1}q}{q\lfloor t/q\rfloor+1}\right]$ $\max_{j\in[d]}\left|s^{t}_{x}(S,j)-s^{t}_{x}(S^{\prime},j)\right|\leq\frac{2L_%
+{0}\tau}{B}$ $Q(S)=\frac{1}{n}\sum_{j\in[n]}Q(z_{i_{j}})$ $x^{1},x^{2}\in\mathcal{X}$ $a^{1},...,a^{T}$ $y^{t+1}_{i}\propto y^{t}_{i}\exp\left(\tau g^{t}_{y,i}\right),\quad\forall i%
+\in[d_{y}]$ $r=2/\sqrt{T}$ $TK/q+qK+K$ $s_{x}^{\prime t}$ $\sqrt{\log(d)/n}+\log(d)/\sqrt{n\varepsilon}$ $\displaystyle=\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)}{\sum_{i\in[d_{x}]}x^{%
+t_{0}}_{i}\exp\left(G_{i}\right)}-\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)%
+\exp\left(\hat{G}_{j}-G_{j}\right)}{\sum_{i\in[d_{x}]}x^{t_{0}}_{i}\exp\left(G%
+_{i}\right)\exp\left(\hat{G}_{i}-G_{i}\right)}$ $s^{t}_{x}(S,j)=-\tau\left(\sum_{i=1}^{t}g^{i}_{x,j}\right).$ $\varepsilon=\frac{\varepsilon^{\prime}}{2\sqrt{2T\log(1/\delta^{\prime})}}$ $\langle\nabla F(\bar{x}),e_{j}\rangle\leq\frac{F(\bar{x})-F(\bar{x}+re_{j})}{r%
+}+\frac{L_{1}r}{2},$ $\hat{w}^{t}=\hat{w}^{t-1}$ $\displaystyle\leq\sqrt{\mathbb{E}[\|Q(S)-\mathbb{E}_{z}[Q(z)]\|_{\infty}^{2}]}$ $\tau=\sqrt{\frac{(\log(d_{x})+\log(d_{y}))}{(L_{0}^{2}+L_{2}^{2}+(\log(d_{x})+%
+\log(d_{y}))L_{1}^{2})U}}$ $q(\mathcal{D})=\mathbb{E}_{z\sim\mathcal{D}}[q(z)]$ $g^{t}_{x}$ $\mathbb{E}[F(\bar{a}^{T})-F(\bar{x}^{T})]\leq\frac{L_{1}}{2}\sum_{t=1}^{T}%
+\lambda_{t}^{2}\mathbb{E}\left[\left\|a^{t}-x^{t}\right\|^{2}\right]$ $\displaystyle=\mathbb{E}\left[\max_{y\in\Delta_{|{\cal Q}|}}\sum_{j\in[|{\cal Q%
+}|]}y_{j}(\mathbb{E}_{z}[q(z)]-q(\tilde{S}))\right]$ $a,b:\mathbb{R}^{p}\mapsto\mathbb{R}$ $(w{t+1},v^{t+1})$ $\|x^{t}-a^{t}\|\leq D$ $(x_{0},y_{0})=(\hat{x}^{1},\hat{y}^{1})$ $U\geq 4$ $\displaystyle\leq 5A^{2}+(6+\log(M))C^{2}$ $\displaystyle\mathbb{P}\left[\operatorname{Gap}(\tilde{x},\tilde{y})\geq\frac{%
+A}{B}\right]$ $\displaystyle\leq 4L_{1}+2L_{1}\sqrt{[\log(d_{x})+\log(d_{y})]\mathbb{E}[(%
+\mathcal{T}+1)]}$ $\max\{L_{0},L_{1}\}$ $\Delta(s_{x}^{t})$ $\displaystyle\left\|\nabla F\left(\bar{a}\right)-\nabla F\left(\bar{x}\right)%
+\right\|^{2}_{\infty}$ $2\Delta(s^{t}_{y})$ $L_{2}^{G}$ $(\mathbb{R}^{d},\|\cdot\|_{1})$ $\frac{\sqrt{\ell}}{\sqrt{n}}+\left(\frac{\ell^{3/2}}{n\varepsilon}\right)^{1/3}$ $q=\sqrt{T}/\log(d_{x})$ $f(\cdot;z)$ $\sum_{t\in[\mathcal{T}+1]}2^{N_{t}}\leq n\alpha/2$ $\displaystyle=\mathbb{E}\left[\sum_{t=1}^{U}\mathbb{E}[\|g^{t}_{x}\|_{\infty}^%
+{2}\mathbbm{1}_{(\mathcal{T}\geq t-1)}\mid\mathcal{F}_{t}]\right]$ $\displaystyle\mathbb{E}\left[(\mathcal{T}+1)(\operatorname{Gap}(\tilde{x},%
+\tilde{y})-\operatorname{Gap}(\bar{x},\bar{y}))\right]$ $0<\varepsilon<8\log(1/\delta)$ $\operatorname{Regret}_{T}^{x}(w)\leq\frac{\log(d_{x})}{\tau}+\frac{\tau}{2}%
+\sum_{t\in[T]}\|g^{t}_{x}\|_{\infty}^{2}\text{ and }\operatorname{Regret}_{T}^%
+{y}(v)\leq\frac{\log(d_{y})}{\tau}+\frac{\tau}{2}\sum_{t\in[T]}\|g^{t}_{y}\|_{%
+\infty}^{2}$ $F_{\mathcal{D}}(x,y)$ $\mathbb{E}[a^{t}|a^{t-1},..,a^{1}]=x^{t}$ $\mathbb{E}[F_{\lambda}(\tilde{x})-F_{\lambda}(\bar{x})]\leq\frac{2L_{1}}{T}+%
+\frac{2L_{1}^{2}}{\lambda T}.$ $\displaystyle=\mathbb{E}\left[(\mathcal{T}+1)\mathbb{E}_{\tilde{x},\tilde{y}}%
+\left[\operatorname{Gap}(\tilde{x},\tilde{y})-\operatorname{Gap}(\bar{x},\bar{%
+y})\mid\mathcal{T}\right]\right]$ $f:{\cal X}\times{\cal Y}\mapsto\mathbb{R}$ $(\bar{x}_{+},\bar{y}_{+})=\frac{1}{2^{N+1}}\sum_{i\in[2^{N+1}]}(\hat{x}^{i},%
+\hat{y}^{i})$ $\log(d_{x})\simeq\log(d_{y})$ $\displaystyle\lesssim\left(L_{0}^{2}+L_{2}^{2}+\log(d_{x})ML_{1}^{2}\right)%
+\mathbb{E}[\mathcal{T}+1],$ $\Delta_{d}=\{x\in\mathbb{R}^{d}:\|x\|_{1}=1,x_{j}\geq 0\text{ for all }j\in[d]\}$ $F(\bar{x}+re_{j})\leq F(\bar{x})+\langle\nabla F(\bar{x}),re_{j}\rangle+\frac{%
+L_{1}r^{2}}{2}\|e_{j}\|_{1}^{2},$ $\displaystyle\lesssim\frac{\log(d_{x})+\log(d_{y})}{\tau}+\tau[L_{0}^{2}+L_{2}%
+^{2}+(\log(d_{x})+\log(d_{y}))ML_{1}^{2}]\mathbb{E}[\mathcal{T}+1]+\frac{L_{2}%
+\mathbb{E}[\mathcal{T}+1]}{2^{M}}.$ $\displaystyle=\frac{4L_{1}}{T}+\frac{4L_{1}^{2}}{\lambda T}+\lambda(\log(d_{x}%
+)+\log(d_{y}))+\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})].$ $\displaystyle\quad\mathbb{E}[g_{x}]=\mathbb{E}\left[\sum_{k=0}^{M}(\nabla_{x}f%
+(\bar{x}_{+,k},\bar{y}_{+,k};B_{k})-\nabla_{x}f(\bar{x}_{-,k},\bar{y}_{-,k};B_%
+{k}))+\frac{2^{-k}}{C_{M}}\nabla_{x}f(x_{0},y_{0};B_{k})\right]$ $U-\sqrt{U}\geq U/2$ $g^{t}=\nabla f(\hat{w}^{t_{0}};B^{t})$ $\displaystyle F(\bar{a}^{T})$ $\displaystyle\mathbb{E}\big{[}\max_{v\in\Delta_{y}}F_{\mathcal{D}}(\tilde{x},v%
+)-\max_{v\in\Delta_{y}}F_{\mathcal{D}}(\bar{x},v)\big{]}$ $\displaystyle\|\mathbb{E}[(g_{x},g_{y})]-\nabla F_{\mathcal{D}}(x,y)\|_{\infty}$ $\displaystyle=\mathbb{E}_{x^{t}}[\langle\mathbb{E}_{\hat{w}^{t}}[\nabla f(w^{t%
+};B^{t})-\nabla f(\hat{w}^{t};B^{t})\mid x^{t}],x^{t}-x\rangle]$ $\displaystyle\lesssim\begin{cases}\frac{L_{1}}{\sqrt{K}}+q\tau\big{[}\frac{L_{%
+1}^{2}}{\sqrt{\log(d_{x})}K^{3/2}}+\frac{(L_{0}^{2}+L_{1}^{2})\sqrt{\log(d_{x}%
+)}}{\sqrt{K}}\big{]}&\text{without second order smoothnes}\\
+\frac{L_{2}}{K}+q\tau\big{[}\frac{L_{2}^{2}}{K^{2}}+\frac{L_{1}^{2}\log(d_{x})%
+}{K}\big{]}&\text{with second order smoothness}\end{cases}$ $\displaystyle\mathbb{E}\|g_{x}\|_{\infty}^{2}$ $\tau\leq 1/(8qL_{0})$ $\mathbb{E}[F_{\cal D}(\hat{w}^{T})-F_{\cal D}(x)]\lesssim(L_{0}+L_{1})\left[%
+\sqrt{\frac{\log(d_{x})}{n}}+L_{0}\frac{\log(d_{x})^{7/10}\log(1/\delta)^{1/5}%
+}{(n\varepsilon)^{2/5}}\right]\\
++L_{1}\log(\sqrt{n}\log(d_{x}))\left[\frac{1}{\log(d_{x})\sqrt{n}}+\frac{\log(%
+1/\delta)^{1/5}}{\log(d_{x})^{4/5}(n\varepsilon)^{2/5}}\right].$ $\frac{L_{1}}{\sqrt{T}}$ $\displaystyle\qquad+\sum_{t=1}^{T}\langle\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t}%
+,(x^{t},y^{t})-(w^{t},v^{t})\rangle.$ $\alpha_{t}=\Big{\langle}\nabla F\Big{(}\sum_{k=t}^{T}\lambda_{k}x^{k}+\sum_{k=%
+1}^{t-1}\lambda_{k}a^{k}\Big{)},\lambda_{t}(a^{t}-x^{t})\Big{\rangle}$ $(\mathbf{0}_{d_{x}},y_{i}-y_{1})$ $\tilde{x}^{1},\tilde{y}^{1}$ $x^{t+1}_{j}\propto x^{t}_{j}\exp\left(-\tau g^{t}_{x,j}\right),\quad\forall j%
+\in[d_{x}]$ $U\geq 1$ $U\leq\min\left\{\frac{\varepsilon^{2}}{48\log(1/\delta)(9\tau\alpha L_{0})^{2}%
+},n\alpha/2,n/2\right\}$ $a^{1},...,a^{T}\in\Delta_{d}$ $\displaystyle=\langle\tilde{\Delta}^{t_{0}},x^{t}-\tilde{x}^{t}\rangle$ $\mathbb{E}[\left\|\nabla F\left(\bar{a}\right)-\nabla F\left(\bar{x}\right)%
+\right\|^{2}_{\infty}]\leq\frac{43L_{1}^{2}}{\sqrt{12+\log(d)}T^{3/2}}+\frac{1%
+7(L_{0}^{2}+L_{1}^{2})\sqrt{(12+\log(d))}}{\sqrt{T}}.$ $\displaystyle\leq L_{0}^{F}\|\Lambda(\lambda_{1},\mu_{1})-\Lambda(\lambda_{2},%
+\mu_{2})\|_{1}$ $x\in\Delta_{x},y\in\Delta_{y}$ $w^{t}=\frac{1}{t}\sum_{i\in[t]}x^{i}$ $T=\min\left\{n,\frac{(n\varepsilon)^{4/5}}{(\log(d_{x})\log(1/\delta))^{2/5}}\right\}$ $(v^{t})_{t\in[T]}$ $\tilde{y}^{t}\sim P_{y^{t}}$ $\displaystyle\lesssim\frac{\log(d_{x})}{\tau T}+\tau L_{0}^{2}+\frac{1}{T}\sum%
+_{t\in[T]}\left(\frac{L_{1}}{\sqrt{K}}+\mathbbm{1}_{(t>q)}\left[\frac{4L_{1}q}%
+{q\lfloor t/q\rfloor+1}+\frac{L_{1}^{2}q\tau}{\sqrt{\log(d_{x})}K^{3/2}}+\frac%
+{(L_{0}^{2}+L_{1}^{2})\sqrt{\log(d_{x})}q\tau}{\sqrt{K}}\right]\right)$ $(x_{-,0},y_{-,0})$ $r^{2}=\sqrt{\frac{8(12+\log(d))}{T}}$ $(\bar{x}_{+,k},\bar{y}_{+,k})$ $\mathbb{E}\left[\sum_{t=1}^{\mathcal{T}+1}\|g^{t}_{y}\|_{\infty}^{2}\right]%
+\lesssim L_{0}^{2}+L_{2}^{2}+\log(d_{y})ML_{1}^{2}\mathbb{E}[\mathcal{T}+1]$ $(\mathbf{E},\|\cdot\|)$ $\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})]\leq\frac{2\log(|\mathcal{Z}|)}%
+{\tau_{x}T}+\frac{\log(|{\cal Q}|)}{\tau_{y}T}+18\tau_{x}+2\tau_{y}+\frac{2}{T%
+}\sum_{t=1}^{T}\mathbb{E}[\|-\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y}%
+\|_{\infty}].$ $\|a^{t}-x^{t}\|_{1}\leq D$ $\mathbb{E}[\langle\nabla F_{\mathcal{D}}(w^{t})-\nabla f(\hat{w}^{t};B^{t}),x^%
+{t}-x\rangle]\leq\begin{cases}\frac{8L_{1}}{\sqrt{K}}&\text{without second %
+order smoothnes}\\
+\frac{4L_{2}}{K}&\text{with second order smoothness}\end{cases}.$ $\displaystyle\mathbb{E}[F_{\mathcal{D}}(w^{T})-F_{\mathcal{D}}(x)]$ $|\mathbb{E}[-\nabla_{y,i}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y,i}\mid\mathcal{%
+F}_{t}]|\leq 2L_{2}/K$ $\displaystyle+\frac{L_{1}\log(\sqrt{n\log(d_{x})})}{\sqrt{T\log(d_{x})}}+\frac%
+{L_{0}\sqrt{T\log(1/\delta)}\log^{3/2}(d_{x})}{n\varepsilon}.$ $T,\tau$ $w\in\Delta_{x},v\in\Delta_{y}$ $T,\tau,L$ $\displaystyle\lesssim\frac{\log(d_{x})+\log(d_{y})}{\tau}+\frac{\tau}{2}\sum_{%
+t=1}^{\mathcal{T}+1}(L_{0}^{2}+\|g^{t}_{x}\|_{\infty}^{2}+\|g^{t}_{y}\|_{%
+\infty}^{2})$ $F_{\mathcal{D}}(x)=\mathbb{E}_{z\sim\mathcal{D}}[f(x;z)]$ $\langle\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}};B^{t}),x^{t}-x\rangle$ $t_{0}=q\lfloor t/q\rfloor$ $C=\frac{L_{0}D}{\sqrt{2}}\sqrt{\sum_{t=1}^{T}\lambda_{t}^{2}}$ $s^{t}_{y}=-\tau\sum_{r=1}^{t}\Big{[}2^{N_{r}}\Big{(}\nabla_{y}f(\bar{x}_{+}^{r%
+},\bar{y}_{+}^{r};B^{r})-\nabla_{y}f(\bar{x}_{-}^{r},\bar{y}_{-}^{r};B^{r})%
+\Big{)}+\nabla_{y}f(x_{0}^{r},y_{0}^{r};B^{r})\Big{]}.$ $\Delta(s_{y}^{t})\leq 4.5\tau\alpha L_{0}$ $\hat{w}^{t_{0}}$ $\displaystyle\longmapsto$ $(\hat{x}^{i})_{i\in[2^{N+1}]}\overset{\text{iid}}{\sim}P_{x}$ $(\hat{y}^{i})_{i\in[2^{N+1}]}\overset{\text{iid}}{\sim}P_{y}$ $\{a^{i}\}_{i\in[T]}$ $\displaystyle=\frac{(A+B)^{2}}{2}+\sum_{j\in[M]}\int_{B/C}^{\infty}\mathbb{P}%
+\left[|F_{j}(\bar{x})-F_{j}(\bar{a}^{T})|\geq A+C\beta\right](A+C\beta)Cd\beta$ $\frac{2TL_{0}\tau}{n}$ $L_{1}=2$ $\delta^{\prime},\delta^{\prime\prime}>0$ $N((\varepsilon_{n})_{n\geq 1},(\delta_{n})_{n\geq 1})=\inf\left\{n:\varepsilon%
+<\sqrt{2\log\left(\frac{1}{\delta^{\prime}}\right)\sum_{m\leq n+1}\varepsilon_%
+{m}^{2}}+\frac{1}{2}\sum_{\sum_{m\leq n+1}}\varepsilon_{m}^{2}\text{ or }%
+\delta^{\prime\prime}<\sum_{m\leq n+1}\delta_{m}\right\}.$ $\hat{G}_{j},G_{j}$ $(x^{t},y^{t})_{t\in[T]}$ $\mathbb{P}\left[F\left(\sum_{t=1}^{T}\lambda_{t}a^{t}\right)-F\left(\sum_{t=1}%
+^{T}\lambda_{t}x^{t}\right)\geq\frac{L_{1}D^{2}}{2}\sum_{t=1}^{T}\lambda_{t}^{%
+2}+\beta\frac{L_{0}D}{\sqrt{2}}\sqrt{\sum_{t=1}^{T}\lambda_{t}^{2}}\,\right]%
+\leq\exp(-\beta^{2}).$ $\tau\eqsim\min\left\{\frac{1}{L_{0}}\sqrt{\frac{\log(d_{x})+\log(d_{y})}{T}},%
+\frac{n\varepsilon}{L_{0}T\sqrt{TK\log(1/\delta)}}\right\},K\eqsim\frac{T}{%
+\log(d_{x})+\log(d_{y})}.$ $\displaystyle=\int_{0}^{\infty}\mathbb{P}\left[\max_{j\in[M]}|F_{j}(\bar{x})-F%
+_{j}(\bar{a}^{T})|\geq\beta\right]\beta d\beta$ $\mathbb{E}\left[F(\bar{a}+re_{j})-F(\bar{x}+re_{j})+F(\bar{x})-F(\bar{a})%
+\right]\leq 4L_{1}/T$ $g^{t}_{x}=\nabla_{x}F_{\mathcal{D}}(\hat{x}^{t},\hat{y}^{t};B^{t})$ $(\tilde{x},\tilde{y})=\frac{1}{T}\sum_{t=1}^{T}(\tilde{x}^{t},\tilde{y}^{t})$ $\hat{y}^{2}$ $\displaystyle=\sqrt{\mathbb{E}\left[\left\|\sum_{j\in[n]}\frac{Q(z_{i_{j}})-%
+\mathbb{E}_{z}[Q(z)]}{n}\right\|_{\infty}^{2}\right]}$ $L_{2}=0$ $\displaystyle\leq\sqrt{2e\log(|{\cal Q}|)\sum_{j\in[n]}\left\|\frac{Q(z_{i_{j}%
+})-\mathbb{E}_{z}[Q(z)]}{n}\right\|_{\infty}^{2}}$ $\displaystyle\text{and}\quad\mathbb{E}[\|(g_{x},-g_{y})\|_{\infty}^{2}]$ $\displaystyle=\mathbb{E}[\|Q(S)-\mathbb{E}_{z}[Q(z)]\|_{\infty}]$ $N\sim\mbox{TG}(0.5,M)$ ${}^{(\ast)}$ $\hat{x}^{t+1,k},\hat{y}^{t+1,k}$ $\|\nabla F(x)\|_{\infty}\leq L_{0}$ $\operatorname{LSE}$ $\alpha=\left(\frac{2\varepsilon^{2}}{48\cdot 81\log(1/\delta)(\tau L_{0})^{2}n%
+}\right)^{1/3}$ $(x^{t}-\tilde{x}^{t})_{j}=\frac{x^{t_{0}}_{j}\exp\left(-\tau\sum_{k=t_{0}}^{t-%
+1}\nabla_{j}f(\hat{w}^{t_{0}};B^{k})\right)}{\sum_{i\in[d_{x}]}x^{t_{0}}_{i}%
+\exp\left(-\tau\sum_{k=t_{0}}^{t-1}\nabla_{i}f(\hat{w}^{t_{0}};B^{k})\right)}-%
+\frac{x^{t_{0}}_{j}\exp\left(-\tau\sum_{k=t_{0}}^{t-1}\nabla_{j}f(w^{t_{0}};B^%
+{k})\right)}{\sum_{i\in[d_{x}]}x^{t_{0}}_{i}\exp\left(-\tau\sum_{k=t_{0}}^{t-1%
+}\nabla_{i}f(w^{t_{0}};B^{k})\right)}.$ $\hat{x}^{t}=\frac{1}{K}\sum_{k\in[K]}\hat{x}^{t,k}$ $(x^{t},y^{t})$ $g^{t}_{y}$ $K\eqsim\frac{T}{\log(d_{x})+\log(d_{y})}$ $A=\frac{L_{1}D^{2}}{2}\sum_{t=1}^{T}\lambda_{t}^{2}$ $w^{1}\in\mathbb{R}^{d}$ $\mathbb{E}\left[\max_{q\in{\cal Q}}|\mathbb{E}_{z}[q(z)]-q(\tilde{S})|\right]%
+\leq\mathbb{E}[\operatorname{Gap}(\tilde{x},\bar{y})]\leq 20\sqrt{\frac{\log(|%
+{\cal Q}|)}{n}}+2(12+C/3)\frac{(\log(1/\delta)\log(|\mathcal{Z}|))^{1/4}\sqrt{%
+\log(|{\cal Q}|)}}{\sqrt{n\varepsilon}}.$ ${\cal X}\subseteq\mathbb{R}^{d_{x}}$ $\operatorname{Gap}(\tilde{x},\tilde{y})\leq A/B$ $\sqrt{\log(d)/n}+(\log(d)^{3/2}/[n\varepsilon])^{1/3}$ $\mathcal{A}:\mathcal{Z}^{n}\mapsto\mathcal{X}$ $\mathbb{E}[\operatorname{Gap}(\tilde{x},\bar{y})-\operatorname{Gap}(\bar{x},%
+\bar{y})]\leq\frac{4}{n}+\frac{8}{\lambda n}+\lambda\log(|{\cal Q}|).$ $\sum_{t=q+1}^{T}\left[\frac{1}{q\lfloor t/q\rfloor+1}\right]\leq q\left(\frac{%
+1}{q}+\frac{1}{2q}+...+\frac{1}{(T/q)q}\right)\lesssim\log\Big{(}\frac{T}{q}%
+\Big{)}$ $\mathbb{E}\|g_{x}\|_{\infty}^{2}\lesssim L_{0}^{2}+L_{2}^{2}+M\log(d_{x})L_{1}%
+^{2}$ $\displaystyle\lesssim\frac{\log(d_{x})}{\tau T}+\tau\Big{[}L_{0}^{2}+\frac{L_{%
+1}^{2}q\log(d_{x})}{K}+\frac{L_{2}^{2}q}{K^{2}}\big{]}+\frac{L_{2}}{K}+\frac{1%
+}{T}\sum_{t=q+1}^{T}\left[\frac{L_{1}q}{q\lfloor t/q\rfloor+1}\right]$ $(\hat{y}^{t,i})_{i\in[K]}\overset{\text{iid}}{\sim}P_{y^{t}}$ $\displaystyle=\max\{\|\mathbb{E}[g_{x}]-\nabla_{x}F_{\mathcal{D}}(x,y)\|_{%
+\infty},\|\mathbb{E}[g_{y}]-\nabla_{y}F_{\mathcal{D}}(x,y)\|_{\infty}\}.$ $\mathcal{T}+1\leq n/2$ $\displaystyle=\big{\|}\Lambda^{T}\big{[}\nabla F(\Lambda(\lambda_{1},\mu_{1})+%
+(x_{1},y_{1}))-\nabla F(\Lambda(\lambda_{2},\mu_{2})+(x_{1},y_{1}))\big{]}\big%
+{\|}_{\infty}$ $\mathbb{E}\left[F\left(\sum_{t=1}^{T}\lambda_{t}a^{t}\right)-F\left(\sum_{t=1}%
+^{T}\lambda_{t}x^{t}\right)\right]\leq\frac{L_{1}}{2}\sum_{t=1}^{T}\lambda_{t}%
+^{2}\mathbb{E}\left[\left\|a^{t}-x^{t}\right\|^{2}\right].$ $|{\cal Z}|$ $\displaystyle\sqrt{2\log(1/\delta)\sum_{t=1}^{\mathcal{T}+1}(4\cdot 2^{N_{t}}+%
+2)(9\tau\alpha L_{0})^{2}}+\frac{1}{2}\sum_{t=1}^{\mathcal{T}+1}(4\cdot 2^{N_{%
+t}}+2)(9\tau\alpha L_{0})^{2}\leq\varepsilon.$ $0<\varepsilon^{\prime}<1$ $t>q$ $\displaystyle\leq\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)}{\sum_{i\in[d_{x}]}%
+x^{t_{0}}_{i}\exp\left(G_{i}\right)}\left[\exp\left(2\max_{j\in[d_{x}]}|\hat{G%
+}_{j}-G_{j}|\right)-1\right],$ $\min_{x\in\Delta_{x}}F_{\mathcal{D}}(x)$ $\displaystyle\leq\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)}{\sum_{i\in[d_{x}]}%
+x^{t_{0}}_{i}\exp\left(G_{i}\right)}\left[1-\exp\left(-2\max_{j\in[d_{x}]}|%
+\hat{G}_{j}-G_{j}|\right)\right]$ $\displaystyle(x^{t}-\tilde{x}^{t})_{j}$ $\displaystyle\leq\begin{cases}\frac{8L_{1}}{\sqrt{K}}&\text{without second %
+order smoothnes}\\
+\frac{4L_{2}}{K}&\text{with second order smoothness}\end{cases}.$ $S=\{z_{i_{1}},...,z_{i_{n}}\}\overset{\text{iid}}{\sim}\mathcal{D}$ $\max\{1,2^{N^{t}}/\alpha\}$ $(\hat{w}^{T,i})_{i\in[K]}\overset{\text{iid}}{\sim}P_{w^{T}}$ $\tau\eqsim\min\left\{\left(\sqrt{\frac{\log(d_{x})}{(L_{0}^{2}+L_{1}^{2}q\log(%
+d_{x})/K+L_{2}^{2}q/K^{2})T}}\right),\frac{1}{L_{0}q},\frac{n\varepsilon}{TL_{%
+0}\sqrt{(TK/q+qK)\log(1/\delta)}}\right\},$ $\displaystyle=\mathbb{E}\left[\max_{y\in\Delta_{|{\cal Q}|}}F_{\mathcal{D}}(%
+\tilde{x},y)\right]$ $\ell=\log(d_{x})+\log(d_{y})$ $\displaystyle=\mathbb{E}[2^{N_{1}}]\mathbb{E}\left[\mathcal{T}+1\right]\eqsim M%
+\mathbb{E}\left[\mathcal{T}+1\right].$ $\ell_{1}/\ell_{q}$ $\displaystyle\qquad+\frac{\alpha}{2^{N_{r}}}\Big{(}\nabla_{x}f(x_{0}^{r},y_{0}%
+^{r};z^{*})-\nabla_{x}f(x_{0}^{r},y_{0}^{r};z^{\prime*})\Big{)}\Big{|}$ $\|\nabla\operatorname{LSE}(x)\|_{1}=1$ $\displaystyle=\mathbb{E}[\max_{v\in\Delta_{y}}F_{\mathcal{D}}(\tilde{x},v)-%
+\min_{w\in\Delta_{x}}F_{\mathcal{D}}(w,\tilde{y})]$ $a^{1},...,a^{t-1}$ $\displaystyle\leq 2\|\mathbb{E}_{\hat{w}^{t}}[\nabla f(w^{t};B^{t})-\nabla f(%
+\hat{w}^{t};B^{t})]\|_{\infty}$ $\mathcal{T}\leq B-1$ $(\bar{x},\bar{y})=\frac{1}{T}\sum_{t=1}^{T}(x^{t},y^{t})$ $\displaystyle\quad+4C_{M}\sum_{k=0}^{M}\mathbb{E}2^{k}\Big{\|}\nabla_{x}f(x,y;%
+B_{k})-\nabla_{x}f(\bar{x}_{-},\bar{y}_{-};B_{k})\Big{\|}_{\infty}^{2}.$ $y^{1}=(1/d_{y},...,1/d_{y}),t=1,N^{1}\sim\mbox{TG}(0.5,M)$ $\tau\leq 1/(4L_{0}q)$ $\log(n)\left(L_{0}+L_{2}+L_{1}\sqrt{(\log(d_{x})+\log(d_{y}))}\right)\sqrt{%
+\frac{\log(d_{x})+\log(d_{y})}{n}}\\
++\log(n)\sqrt{L_{0}\sqrt{L_{0}^{2}+L_{2}^{2}+L_{1}^{2}\log(n)(\log(d_{x})+\log%
+(d_{y}))}}\left(\frac{(\log(d_{x})+\log(d_{y}))^{3/2}\sqrt{\log(1/\delta)}}{n%
+\varepsilon}\right)^{1/2}.$ $\left|\mathbb{E}\left[F\left(\frac{1}{T}\sum_{t=1}^{T}a^{t}\right)-F\left(%
+\frac{1}{T}\sum_{t=1}^{T}x^{t}\right)\right]\right|\leq\frac{2L_{1}}{T}.$ $\displaystyle=\tau\|G_{j}(S)-G_{j}(S^{\prime})\|_{\infty}.$ $\color[rgb]{1,0,0}\frac{\sqrt{\ell_{x}}}{\sqrt{n}}+\frac{\ell_{x}^{2/3}}{(n%
+\varepsilon)^{2/3}}$ $\displaystyle+\frac{BM}{U-\sqrt{U}}.$ $\min_{x\in\mathcal{X}}\max_{y\in\mathcal{Y}}F(x,y)$ $\bar{x}=\frac{1}{T}\sum_{t\in[T]}x^{t},\bar{a}=\frac{1}{T}\sum_{t\in[T]}a^{t}$ $L_{1}^{F}$ $\mathcal{Q}=\{q:\mathcal{Z}\to[-1,1]\}$ $\displaystyle=\mathbb{E}[\langle\nabla f(w^{t};B^{t})-\nabla f(\hat{w}^{t};B^{%
+t}),x^{t}-x\rangle]$ $L_{1}^{G}$ $\max_{q\in{\cal Q}}|q(\tilde{S})-q(\mathcal{D})|$ $g^{t}=(g^{t}_{x},g^{t}_{y})$ $0\leq\phi(y)\leq\log(d_{y})$ $y_{i}-y_{1}$ $f(x,y;z)=\sum_{j\in[|\mathcal{Q}|]}y_{j}(q_{j}(z)-\langle q_{j},x\rangle)$ $S=\{z^{1},...,z^{n}\}$ $\displaystyle=\mathbb{E}\left[\sum_{t=1}^{U}\|g^{t}_{x}\|_{\infty}^{2}\mathbbm%
+{1}_{(\mathcal{T}+1\geq t)}\right]$ $\displaystyle\leq 4\tau\sum_{k=t_{0}}^{t-1}\|\nabla f(w^{t_{0}};B^{t})-\nabla f%
+(\hat{w}^{t_{0}};B^{t})\|_{\infty}\|\nabla f(\hat{w}^{t_{0}};B^{k})-\nabla f(w%
+^{t_{0}};B^{k})\|_{\infty}$ $\displaystyle\operatorname{Gap}(\bar{x},\bar{y})$ $\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})]\leq 10\sqrt{\frac{\log(|{\cal Q%
+}|)}{n}}+2(12+C/3)\frac{(\log(1/\delta)\log(|\mathcal{Z}|))^{1/4}\sqrt{\log(|{%
+\cal Q}|)}}{\sqrt{n\varepsilon}}.$ $\displaystyle\qquad-2^{N_{r}}\Big{(}\nabla_{x}f(\bar{x}_{+}^{r},\bar{y}_{+}^{r%
+};B^{\prime r})-\nabla_{x}f(\bar{x}_{-}^{r},\bar{y}_{-}^{r};B^{\prime r})\Big{%
+)}-\nabla_{x}f(x_{0}^{r},y_{0}^{r};B^{\prime r})\Big{|}$ $\displaystyle\leq\|\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}};B^{t})\|%
+_{\infty}\|x^{t}-\tilde{x}^{t}\|_{1}$ $\mathbb{E}_{\mathcal{A},S}\Big{[}\max_{q\in\mathcal{Q}}|\mathbb{E}_{z\sim%
+\mathcal{D}}[q(z)]-q(\tilde{S})|\Big{]}\leq 20\sqrt{\frac{\log(|{\cal Q}|)}{n}%
+}+2(12+C/3)\frac{(\log(1/\delta)\log(|\mathcal{Z}|))^{1/4}\sqrt{\log(|{\cal Q}%
+|)}}{\sqrt{n\varepsilon}}.$ $y^{t+1}_{j}\propto y^{t}_{j}\exp\left(\tau_{y}g^{t}_{y,j}\right),\quad\forall j%
+\in[|\mathcal{Q}|]$ $\displaystyle\Lambda:\Delta_{K_{x}}\times\Delta_{K_{y}}$ $\mathbb{E}[F_{\mathcal{D}}(\hat{w}^{T})-F_{\mathcal{D}}(w^{T})]\lesssim\frac{L%
+_{1}}{K}$ $\displaystyle\|x^{t}-\tilde{x}^{t}\|_{1}\leq\psi$ $\displaystyle\leq\frac{\log(d_{x})}{\tau T}+\frac{\tau L_{0}^{2}}{2}+\frac{1}{%
+T}\sum_{t\in[T]}\mathbb{E}\left[\langle\nabla F_{\mathcal{D}}(w^{t})-g^{t},x^{%
+t}-x\rangle\right]$ $\mathbb{E}\left[\max_{j\in[d]}\frac{|F(\bar{a}+re_{j})-F(\bar{x}+re_{j})|^{2}}%
+{r^{2}}\right]\leq\frac{20L_{1}^{2}}{T^{2}r^{2}}+\frac{8L_{0}^{2}(6+\log(d))}{%
+Tr^{2}}$ $\tilde{\Delta}^{t_{0}}$ $\displaystyle=\mathbb{E}\big{[}\max_{v\in\Delta_{y}}F_{\mathcal{D}}(\tilde{x},%
+v)-F_{\lambda}(\tilde{x})\big{]}+\mathbb{E}[F_{\lambda}(\tilde{x})-F_{\lambda}%
+(\bar{x})]+\mathbb{E}\big{[}F_{\lambda}(\bar{x})-\max_{v\in\Delta_{y}}F_{%
+\mathcal{D}}(\bar{x},v)\big{]}$ $\psi\leq 8\tau qL_{0}$ $\displaystyle\lesssim\frac{\log(d_{x})}{\tau T}+\tau\Big{[}L_{0}^{2}+\frac{L_{%
+1}^{2}q}{\sqrt{\log(d_{x})}K^{3/2}}+\frac{(L_{0}^{2}+L_{1}^{2})\sqrt{\log(d_{x%
+})}q}{\sqrt{K}}\Big{]}+\frac{L_{1}}{\sqrt{K}}+\frac{L_{1}q\log(T/q)}{T}.$ $\mathbb{E}\left[\operatorname{Gap}(\bar{x},\bar{y})\right]$ $\displaystyle\leq\sum_{t=1}^{T}\langle\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t})-g%
+^{t}_{x},x^{t}-w\rangle+2\|-\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y}\|%
+_{\infty}$ $\displaystyle\|x^{t}-\tilde{x}^{t}\|_{1}=\sum_{j\in[d_{x}]}|(x^{t}-\tilde{x}^{%
+t})_{j}|\leq\sum_{j\in[d_{x}]}\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)}{\sum_%
+{i\in[d_{x}]}x^{t_{0}}_{i}\exp\left(G_{i}\right)}\psi=\psi.$ $F(\bar{a}^{T})=F(\bar{x}^{T}+\bar{a}^{T}-\bar{x}^{T})$ $\mathcal{X}=\Delta_{x}$ $\displaystyle=\mathbb{E}\left[\nabla_{x}F_{\mathcal{D}}(\bar{x}_{+,M},\bar{y}_%
+{+,M})\right],$ $P_{w^{t-1}}$ $\mathbb{E}\left[\frac{|F(\bar{x})-F(\bar{a})|^{2}}{r^{2}}\right]\leq\frac{20L_%
+{1}^{2}}{T^{2}r^{2}}+\frac{48L_{0}^{2}}{Tr^{2}}.$ $T=\min\Big{\{}n,\frac{n\varepsilon}{\log(d_{x})\sqrt{\log(1/\delta)}}\Big{\}},%
+K=\sqrt{T\log(d_{x})},q=\sqrt{T/\log(d_{x})},$ $\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})-\operatorname{Gap}(\bar{x},%
+\bar{y})]\leq\frac{4L_{1}}{T}+\frac{2L_{1}\sqrt{\log(d_{x})+\log(d_{y})}}{%
+\sqrt{T}}.$ $f(x,y;z)=\sum_{i\in[d_{y}]}y_{i}f_{i}(x;z)$ $\displaystyle=\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)\exp\left(\hat{G}_{j}-G%
+_{j}\right)}{\sum_{i\in[d_{x}]}x^{t_{0}}_{i}\exp\left(G_{i}\right)\exp\left(%
+\hat{G}_{i}-G_{i}\right)}-\frac{x^{t_{0}}_{j}\exp\left(G_{j}\right)}{\sum_{i%
+\in[d_{x}]}x^{t_{0}}_{i}\exp\left(G_{i}\right)}$ $\displaystyle\leq\frac{F(\bar{a}+re_{j})-F(\bar{x}+re_{j})+F(\bar{x})-F(\bar{a%
+})}{r}+L_{1}r$ $\max_{v\in\Delta_{y}}F_{\mathcal{D}}(x,v)\leq F_{\lambda}(x)\leq\lambda\log(d_%
+{y})+\max_{v\in\Delta_{y}}F_{\mathcal{D}}(x,v)$ $2\max_{j\in[d_{x}]}|\hat{G}_{j}-G_{j}|\leq 1$ $i\in[d_{y}]$ $w^{t+1}=\operatorname{argmin}_{x\in\mathcal{X}}D_{\psi}(w^{t},x)+\langle\xi^{t%
+},x\rangle$ $\displaystyle\quad+2\tau q\|\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}}%
+;B^{t})\|_{\infty}^{2}+2\tau\sum_{k=t_{0}}^{t-1}\mathbb{E}\left[\|\nabla f(%
+\hat{w}^{t_{0}};B^{k})-\nabla f(w^{t_{0}};B^{k})\|_{\infty}^{2}\right]$ $\mathbb{E}[\left\|\nabla F\left(\bar{a}\right)-\nabla F\left(\bar{x}\right)%
+\right\|^{2}_{\infty}]\leq\frac{120L_{1}^{2}}{T^{2}r^{2}}+\frac{24L_{0}^{2}(12%
++\log(d))}{Tr^{2}}+3L_{1}^{2}r^{2}.$ $\mathbb{E}\|g_{y}\|_{\infty}^{2}\lesssim L_{0}^{2}+L_{2}^{2}+M\log(d_{y})L_{1}%
+^{2}$ $\operatorname{LSE}((x_{1},...,x_{k}))=\log(\exp(x_{1})+...+\exp(x_{k}))$ $\mathbb{E}\left[\max_{j\in[M]}|F_{j}(\bar{x})-F_{j}(\bar{a}^{T})|^{2}\right]%
+\leq\frac{(A+B)^{2}}{2}+2MC(C+A)\exp\left(-(B/C)^{2}\right).$ $|\alpha_{i}|\leq L_{0}D\lambda_{i}$ $\displaystyle\lesssim(L_{0}+L_{1})\sqrt{\frac{\log(d_{x})}{T}}+\frac{L_{2}}{T^%
+{3/4}\log(d_{x})^{1/4}}+\frac{L_{1}+L_{2}}{\sqrt{T\log(d_{x})}}$ $z^{\prime*}$ $x,y,N,B$ $p\leq C_{M}\leq 1$ $v^{1}=(1/d_{y},...,1/d_{y})\in\mathbb{R}^{d_{y}},v^{t+1}:=\operatorname{argmin%
+}_{y\in\Delta_{y}}\left(\tau\langle-\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g_{%
+y}^{t},y\rangle+\sum_{i\in[d_{y}]}v^{t}_{i}\log(v^{t}_{i}/y_{i})\right).$ $\mathbb{E}\left[\operatorname{Gap}(\tilde{x},\bar{y})\right]$ $\displaystyle\leq F(\bar{x}^{T}+\bar{a}^{T-1}-\bar{x}^{T-1})+\langle\nabla F(%
+\bar{x}^{T}+\bar{a}^{T-1}-\bar{x}^{T-1}),\lambda_{T}(a^{T}-x^{T})\rangle+\frac%
+{L_{1}\lambda_{T}^{2}\left\|a^{T}-x^{T}\right\|^{2}}{2}$ $F_{\lambda}(x):=\max_{y\in\Delta_{y}}[F_{\mathcal{D}}(x,y)+\lambda\phi(y)]$ $\hat{y}^{t}\sim P_{y^{t}}$ $\displaystyle=\mathbb{E}\left[\sum_{t=1}^{U}\mathbb{E}[\|g^{t}_{x}\|_{\infty}^%
+{2}]\mathbbm{1}_{(\mathcal{T}\geq t-1)}\right]$ $Q(z)=(q_{1}(z),...,q_{|{\cal Q}|}(z))$ $\mathbb{P}[\operatorname{Gap}(\tilde{x},\tilde{y})\geq A/B]\leq 0.01$ $\sum_{t\in[T]}\langle-\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y},y^{t}-v%
+\rangle\\
+\leq\sum_{t\in[T]}\langle-\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y},y^{%
+t}-v^{t}\rangle+\frac{\log(d_{y})}{\tau}+\frac{\tau}{2}\sum_{t\in[T]}\|-\nabla%
+_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y}\|_{\infty}^{2}.$ $\sum_{t\in[\mathcal{T}+1]}2^{N_{t}}\leq U$ $x^{t+1}_{j}\propto x^{t}_{j}\exp\left(-\tau_{x}g^{t}_{x,j}\right),\quad\forall
+j%
+\in[|\mathcal{Z}|]$ $\tilde{y}^{i}\sim P_{y^{i}}$ $\displaystyle\leq 4\tau\max_{j\in[d_{x}]}\sum_{k=t_{0}}^{t-1}\left|\nabla_{j}f%
+(\hat{w}^{t_{0}};B^{k})-\nabla_{j}f(w^{t_{0}};B^{k})\right|$ $\displaystyle\leq\langle\nabla\operatorname{LSE}(-\tau G_{j}(S)),\tau G_{j}(S^%
+{\prime})-\tau G_{j}(S)\rangle$ $\hat{x}^{1},\tilde{x}^{1},\hat{y}^{1},\tilde{y}^{1},...,\hat{x}^{t},\tilde{x}^%
+{t},\hat{y}^{t},\tilde{y}^{t}$ $\displaystyle\leq 4L_{0}^{2}+64L_{2}^{2}+3ML_{1}^{2}[\log(2d_{x})+\log(2d_{y})%
+]+8ML_{2},$ $\displaystyle\qquad+\sum_{t=1}^{\mathcal{T}+1}\langle\Phi_{\mathcal{D}}(x^{t},%
+y^{t})-(g_{x}^{t},g_{y}^{t}),(x^{t},y^{t})-(w^{t},v^{t})\rangle$ $\tau\eqsim\min\left\{\frac{1}{L_{0}}\sqrt{\frac{\log(d_{x})+\log(d_{y})}{T}},%
+\frac{n\varepsilon}{L_{0}T\sqrt{TK\log(1/\delta)}}\right\},K\eqsim\sqrt{\frac{%
+T}{\log(d_{x})+\log(d_{y})}}.$ $\max_{j\in[d_{x}]}\tau\|G_{j}(S)-G_{j}(S^{\prime})\|_{\infty}=\tau\max_{j\in[d%
+_{x}],i\in[t]}\left|\sum_{k\in[i-1]}(g^{k}_{j}-g^{\prime k}_{j})\right|\leq%
+\frac{2L_{0}}{B}.$ $\displaystyle\leq 3\max_{j\in[d]}\frac{|F(\bar{a}+re_{j})-F(\bar{x}+re_{j})|^{%
+2}+|F(\bar{x})-F(\bar{a})|^{2}}{r^{2}}+L_{1}^{2}r^{2}$ $w^{t}_{j}=\frac{1}{t}\sum_{i\in[t]}x^{i}_{j}\propto\sum_{i\in[t]}\exp\left(-%
+\tau\sum_{k\in[i-1]}g^{k}_{j}\right)=\exp\left(\log\left(\sum_{i\in[t]}\exp%
+\left(-\tau\sum_{k\in[i]}g^{k}_{j}\right)\right)\right).$ $\displaystyle\leq\operatorname{Regret}_{T}^{x}(w)+\operatorname{Regret}_{T}^{y%
+}(v)+\sum_{t=1}^{T}\langle\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t},(x^{t},y^{t})-%
+(w,v)\rangle.$ $t\in[T],k\in[K]$ $\displaystyle=4L_{1}+2L_{1}\sqrt{\log(d_{x})+\log(d_{y})}\mathbb{E}\left[\sqrt%
+{\mathcal{T}+1}\right]$ $\displaystyle\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})]$ $\hat{y}^{t}=\frac{1}{K}\sum_{k\in[K]}\hat{y}^{t,k}$ $(\tilde{x},\tilde{y})=\frac{1}{t}\sum_{i=1}^{t}(\tilde{x}^{i},\tilde{y}^{i})$ $\displaystyle\leq F\left(\bar{x}^{T}\right)+\sum_{t\in[T]}\underbrace{\Big{%
+\langle}\nabla F\Big{(}\bar{x}^{T}+\bar{a}^{t-1}-\bar{x}^{t-1}\Big{)},\lambda_%
+{t}(a^{t}-x^{t})\Big{\rangle}}_{\alpha_{t}}+\frac{L_{1}}{2}\sum_{t\in[T]}%
+\lambda_{t}^{2}\|a^{t}-x^{t}\|^{2}$ $2\Delta(s^{t}_{x})$ $(x^{t}-\tilde{x}^{t})_{j}$ $T\eqsim\min\left\{n,\left[\frac{n\varepsilon}{(\log(d_{x})+\log(d_{y}))^{1/4}%
+\sqrt{\log(1/\delta)}}\right]^{4/5}\right\},$ $\mathcal{T}+1\leq\sum_{t\in[\mathcal{T}+1]}2^{N_{t}}\leq U$ $S=\{z^{1},...,z^{n}\}\overset{\text{iid}}{\sim}\mathcal{D}$ $\|w^{t}-w^{t-1}\|_{1}=\|x^{t}-w^{t-1}\|_{1}/t\leq 2/t$ $\bar{x}=\frac{1}{T}\sum_{t\in[T]}x^{t}$ $\{\alpha_{i}\}_{i\in[T]}$ $\displaystyle=L_{0}^{F}\left\|\left(\sum_{i=1}^{K_{x}}(\lambda_{1,i}-\lambda_{%
+2,i})(x_{i}-x_{1}),\sum_{j=1}^{K_{y}}(\mu_{1,j}-\mu_{2,j})(y_{j}-y_{1})\right)%
+\right\|_{1}$ $\alpha=\frac{L_{1}D^{2}}{2}\sum_{t=1}^{T}\lambda_{t}^{2}+\alpha^{\prime}$ $\frac{4TL_{0}\tau}{n}$ $\mathbb{E}[a^{t}|a^{t-1},\ldots,a^{1}]=x^{t}$ $\displaystyle=\mathbb{E}[2^{N_{1}}]\mathbb{E}\left[\sum_{t=1}^{\mathcal{T}+1}%
+\mathbbm{1}_{(\mathcal{T}+1\geq t)}\right]$ $\hat{w}^{t}=\frac{1}{K}\sum_{k\in[K]}\hat{w}^{t,k}$ $\|\mathbb{E}[g_{x}]-\nabla_{x}F_{\mathcal{D}}(x,y)\|_{\infty}=\max_{j\in[d]}|%
+\mathbb{E}[g_{x,j}]-\nabla_{x,j}F_{\mathcal{D}}(x,y)|\leq\frac{2L_{2}}{2^{M}}$ $\mathcal{A}_{1:n-1}$ $\displaystyle\mathbb{E}[\|-\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y}\|_%
+{\infty}]$ $\tau\eqsim\min\left\{\left(\sqrt{\frac{\log(d_{x})}{(L_{0}^{2}+L_{1}^{2}q\log(%
+d_{x})/K+L_{2}^{2}q/K^{2})T}}\right),\frac{1}{L_{0}q},\frac{n\varepsilon}{TL_{%
+0}\sqrt{(TK/q+qK)\log(1/\delta)}}\right\}.$ $\mathcal{X}\subseteq\mathbb{R}^{d_{x}}$ $g_{x},g_{y}$ $\displaystyle\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})]$ $T=\min\left\{n,\frac{(n\varepsilon)^{4/5}}{(\log(d_{x})\log(1/\delta))^{2/5}}%
+\right\},q=\sqrt{T}/\log(d_{x}),K=T/\log(d_{x})$ $\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})]\lesssim(L_{0}+L_{1})\left[%
+\frac{\sqrt{\log(d_{x})+\log(d_{y})}}{\sqrt{n}}+\bigg{(}\frac{(\log(d_{x})+%
+\log(d_{y}))^{3/2}\sqrt{\log(1/\delta)}}{n\varepsilon}\right)^{1/3}\bigg{]}.$ $\displaystyle|s_{x}^{t}-s_{x}^{\prime t}|$ $\displaystyle=\langle\mathbb{E}_{\hat{w}^{t_{0}}}[\nabla f(w^{t_{0}};B^{t})-%
+\nabla f(\hat{w}^{t_{0}};B^{t})],\tilde{x}^{t}-x\rangle$ $\mathbb{E}[\hat{x}]=x$ $x^{1},...,x^{T}$ $\|\Lambda(\lambda_{1},\mu_{1})-\Lambda(\lambda_{2},\mu_{2})\|_{1}\leq D\|(%
+\lambda_{1},\mu_{1})-(\lambda_{2},\mu_{2})\|_{1}$ $\displaystyle\mathbb{E}[F_{\mathcal{D}}(\hat{w}^{T})-F_{\mathcal{D}}(x)]$ $\log(|{\cal Q}|)\leq C\log(|\mathcal{Z}|)$ $w^{t}=\sum_{i\in[t]}\frac{\beta_{i}x^{i}}{\sum_{j\in[t]}\beta_{j}}$ $\textstyle x^{t+1}_{j}\propto\exp\left\{-\tau\left(\sum_{i=1}^{t}g^{i}_{x,j}%
+\right)\right\},$ $|\mathcal{Q}|\leq|\mathcal{Z}|^{C}$ $8\tau qL_{0}\leq 2$ $\frac{\sqrt{\ell_{x}}}{\sqrt{n}}+\frac{\ell_{x}^{7/10}}{(n\varepsilon)^{2/5}}$ $q(S)=\frac{1}{|S|}\sum_{z\in S}q(z)$ $\Phi(x,y)=\big{(}\nabla_{x}f(x,y),-\nabla_{y}f(x,y)\big{)}$ $\varepsilon\leq 8\log(1/\delta)$ $g_{x}=C_{M}2^{N}(\nabla_{x}f(\bar{x}_{+},\bar{y}_{+};B)-\nabla_{x}f(\bar{x}_{-%
+},\bar{y}_{-};B))+\nabla_{x}f(x_{0},y_{0};B)$ $\displaystyle\operatorname{LSE}(-\tau G_{j}(S))-\operatorname{LSE}(-\tau G_{j}%
+(S^{\prime}))$ $\mathbb{E}\left[\langle\nabla F_{\mathcal{D}}(w^{t})-g^{t},x^{t}-x\rangle\right]$ $\displaystyle\leq L_{1}^{F}D^{2}\|(\lambda_{1},\mu_{1})-(\lambda_{2},\mu_{2})%
+\|_{1}.$ $\frac{4L_{0}\tau}{B}\leq\frac{\varepsilon}{2\sqrt{2T(K+1)\log(1/\delta)}}$ $B^{\prime r}$ $\frac{L_{1}^{2}}{T^{3/2}}+\frac{(L_{0}^{2}+L_{1}^{2})\sqrt{\log(d)}}{\sqrt{T}}$ $n,d_{x},d_{y}$ $F_{j}=\nabla_{j}F$ $\tau_{x},\tau_{y}$ $L_{0}^{G}\leq L_{0}^{F}D$ $x^{t}\in\Delta_{d}$ $\|\mathbb{E}[\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t}\mid\mathcal{F}_{t}]\|_{%
+\infty}\leq\frac{4L_{1}}{\sqrt{K}}$ $\mathbb{P}[\mathcal{A}(S)\in\mathcal{E}]\leq e^{\varepsilon}\mathbb{P}[%
+\mathcal{A}(S^{\prime})\in\mathcal{E}]+\delta$ $(\varepsilon_{n},\delta_{n})$ $\sum_{t=1}^{\mathcal{T}}2^{N_{t}}\leq U-2^{M}$ $\sum_{t\in[\mathcal{T}+1]}2^{N_{t}}\leq\min\{n,n\alpha/2\}$ $\color[rgb]{1,0,0}\frac{\ell}{\sqrt{n}}+\frac{\ell}{\sqrt{n\varepsilon}}{}^{(%
+\ast)}$ $\mathbb{P}\left[\operatorname{Gap}(\tilde{x},\tilde{y})\lesssim\sqrt{\frac{[(%
+\log(d_{x})+\log(d_{y})][L_{0}^{2}+L_{2}^{2}+(\log(d_{x})+\log(d_{y}))L_{1}^{2%
+}]M^{2}}{U}}\,\right]\geq 0.99.$ $x\in\mathcal{X},$ $\displaystyle\leq 2\|\mathbb{E}_{\hat{w}^{t_{0}}}[\nabla f(w^{t_{0}};B^{t})-%
+\nabla f(\hat{w}^{t_{0}};B^{t})]\|_{\infty}$ $\displaystyle(\tilde{x}^{t}-x^{t})_{j}$ $\lambda\phi(y)$ $\nabla f(w^{t};B^{t})$ $\displaystyle\langle\nabla F(\bar{x})-\nabla F(\bar{a}),e_{j}\rangle$ $\displaystyle\lesssim\frac{\log(d_{x})+\log(d_{y})}{\tau}+\tau[L_{0}^{2}+L_{2}%
+^{2}+(\log(d_{x})+\log(d_{y}))L_{1}^{2}]U+\frac{L_{2}U}{2^{M}}.$ $j\in[d_{x}]$ $\log(d)/\sqrt{n}+\log(d)/[n\varepsilon]^{1/2}$ $\displaystyle\mathbb{E}\left[\sum_{t=1}^{\mathcal{T}+1}\|g^{t}_{x}\|_{\infty}^%
+{2}\right]$ $\displaystyle\leq\sum_{t=1}^{T}\langle(\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t})-%
+g^{t}_{x},x^{t}-w^{t}\rangle)+\frac{\log(|\mathcal{Z}|)}{\tau_{x}}+\sum_{t=1}^%
+{T}\tau_{x}\|\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{x}\|_{\infty}^{2}+2%
+\|\nabla_{y}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{y}\|_{\infty},$ $B=C\sqrt{\log(M)}$ $\displaystyle\leq 2\sum_{k=0}^{M}C_{M}2^{k}\mathbb{E}\Big{\|}\nabla_{x}f(\bar{%
+x}_{+},\bar{y}_{+};B_{k})-\nabla_{x}f(\bar{x}_{-},\bar{y}_{-};B_{k})\Big{\|}_{%
+\infty}^{2}+\frac{2^{-k}}{C_{M}}\mathbb{E}\|\nabla_{x}f(x_{0},y_{0};B_{k})\|_{%
+\infty}^{2}$ $\displaystyle=\max_{i\in[K_{x}+K_{y}]}|\langle\Lambda^{T}_{i},\nabla F(\Lambda%
+(\lambda_{1},\mu_{1})+(x_{1},y_{1}))-\nabla F(\Lambda(\lambda_{2},\mu_{2})+(x_%
+{1},y_{1}))\rangle|$ $\displaystyle\sqrt{2\log(1/\delta)\sum_{t=1}^{\mathcal{T}+1}(2^{N_{t}+1}+1)(4%
+\Delta(s_{x}^{t})^{2}+4\Delta(s_{y}^{t})^{2})}+\frac{1}{2}\sum_{t=1}^{\mathcal%
+{T}+1}(2^{N_{t}+1}+1)(4\Delta(s_{x}^{t})^{2}+4\Delta(s_{y}^{t})^{2})\leq\varepsilon.$ $\min_{x\in\Delta_{|\mathcal{Z}|}}F_{\mathcal{D}}(x,\bar{y})\leq 0$ $U\leq\min\{n,n\alpha/2\}$ $n,1/\delta$ $g^{t}_{y}=\nabla_{y}f(x^{t},\hat{y}^{t};S)$ $\hat{w}^{T}=\frac{1}{K}\sum_{k\in[K]}\hat{w}^{T,k}$ $\tilde{y}^{t+1}$ $\min_{\lambda\in\Delta_{K_{x}}}\max_{\mu\in\Delta_{K_{y}}}[G(\lambda,\mu):=F(%
+\Lambda(\lambda,\mu)+(x_{1},y_{1}))].$ $\displaystyle(\mathcal{T}+1)\operatorname{Gap}(\bar{x},\bar{y})$ $\Delta(s^{t}_{x})$ $\displaystyle\leq\max_{k\in[K_{x}+K_{y}]}\|\Lambda^{T}_{k}\|_{1}\|\Lambda^{T}_%
+{j}\|_{1}L_{2}^{F}\|\Lambda(\lambda_{1},\mu_{1})-\Lambda(\lambda_{2},\mu_{2})%
+\|_{1}$ $w^{t},v^{t}$ $\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})]\lesssim(L_{0}+L_{1})\left[%
+\sqrt{\frac{\log(d_{x})+\log(d_{y})}{n}}+\left(\frac{\sqrt{\log(1/\delta)}(%
+\log(d_{x})+\log(d_{y}))^{3/2}}{n\varepsilon}\right)^{1/3}\right].$ $\mathcal{E}\subseteq\mathcal{X}$ $\tilde{x}=\frac{1}{n}\sum_{k=1}^{n}\tilde{x}^{k}$ $\tau\leq B\varepsilon/[8L_{0}\sqrt{2T(K+1)\log(1/\delta)}].$ $\displaystyle=\max_{k\in[K_{x}+K_{y}]}\left|\sum_{i\in[K_{x}+K_{y}]}\Lambda^{T%
+}_{j,i}\langle\Lambda^{T}_{k},\nabla\nabla_{i}F(\Lambda(\lambda_{1},\mu_{1})+(%
+x_{1},y_{1}))-\nabla\nabla_{i}F(\Lambda(\lambda_{2},\mu_{2})+(x_{1},y_{1}))\right|$ $\displaystyle\|\nabla G(\lambda_{1},\mu_{1})-\nabla G(\lambda_{2},\mu_{2})\|_{\infty}$ $L_{2}\lesssim(L_{0}+L_{1})\left\{\frac{\sqrt{\log(d_{x})+\log(d_{y})}}{\sqrt{n%
+}}+\left(\frac{(\log(d_{x})+\log(d_{y}))^{3/2}\sqrt{\log(1/\delta)}}{n%
+\varepsilon}\right)^{1/2}\right\},$ $\mathbb{E}[F_{\mathcal{D}}(\hat{w}^{T})-F_{\mathcal{D}}(x)]\lesssim\frac{\log(%
+d_{x})}{\tau T}+\tau\Big{[}L_{0}^{2}+\frac{L_{1}^{2}q}{\sqrt{\log(d_{x})}K^{3/%
+2}}+\frac{(L_{0}^{2}+L_{1}^{2})\sqrt{\log(d_{x})}q}{\sqrt{K}}\Big{]}+\frac{L_{%
+1}}{\sqrt{K}}+\frac{L_{1}q\log(T/q)}{T}.$ $\big{(}L_{1}+\frac{L_{1}^{2}}{\lambda}\big{)}$ $U=\min\left\{\frac{(n\varepsilon)^{2/3}}{(4\cdot 48\cdot 81\log(1/\delta)^{1/3%
+}(\tau L_{0})^{2/3}},\frac{n}{2}\right\}$ $F_{j}(\cdot)=F(\cdot+re_{j})$ $\mathbb{P}(j=e_{i})\propto\exp\left(s(S,i)\right)$ $\tilde{x}^{i}\sim P_{x^{i}}$ $P_{x^{t}}$ $\min_{x\in\Delta_{|\mathcal{Z}|}}\max_{y\in\Delta_{|\mathcal{Q}|}}\mathbb{E}_{%
+z\sim\mathcal{D}}\Big{[}\sum_{j\in[|\mathcal{Q}|]}y_{j}(q_{j}(z)-\langle q_{j}%
+,x\rangle)\Big{]}.$ $\displaystyle T[F_{\mathcal{D}}(\bar{x},v)-F_{\mathcal{D}}(w,\bar{y})]$ $s(S,j)=\operatorname{LSE}(\tau G_{j}(S))$ $\mathbb{E}\left\|\nabla F\left(\frac{1}{T}\sum_{t\in[T]}a^{t}\right)-\nabla F%
+\left(\frac{1}{T}\sum_{t\in[T]}x^{t}\right)\right\|^{2}_{\infty}\leq\frac{20L_%
+{2}^{2}}{T^{2}}+\frac{8L_{1}^{2}(6+\log(d))}{T}.$ $\displaystyle\leq\int_{0}^{A+B}\beta d\beta+\sum_{j\in[M]}\int_{A+B}^{\infty}%
+\mathbb{P}\left[|F_{j}(\bar{x})-F_{j}(\bar{a}^{T})|\geq\beta\right]\beta d\beta$ $L_{0}^{G}$ $\mathcal{T}+1\geq t$ $\displaystyle\mathbb{E}[(\mathcal{T}+1)\operatorname{Gap}(\bar{x},\bar{y})]$ $s^{t}(S,j)=-\tau\left(\sum_{i=1}^{t}g^{i}_{y,j}\right)$ $\displaystyle T[F_{\mathcal{D}}(\bar{x},v)-F_{\mathcal{D}}(w,\bar{y})]\leq%
+\operatorname{Regret}_{T}^{x}(w)+\operatorname{Regret}_{T}^{y}(v)+\sum_{t=1}^{%
+T}\langle\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t},(x^{t},y^{t})-(w,v)\rangle$ $\displaystyle\leq\max_{i\in[K_{x}+K_{y}]}\|\Lambda^{T}_{i}\|_{1}L_{1}^{F}\|%
+\Lambda(\lambda_{1},\mu_{1})-\Lambda(\lambda_{2},\mu_{2})\|_{1}$ $\langle\tilde{\Delta}^{t_{0}},x^{t}-x\rangle=\underbrace{\langle\tilde{\Delta}%
+^{t_{0}},\tilde{x}^{t}-x\rangle}_{=:A_{1}}+\underbrace{\langle\tilde{\Delta}^{%
+t_{0}},x^{t}-\tilde{x}^{t}\rangle}_{=:A_{2}},$ $\tilde{x}^{1},...,\tilde{x}^{n}\overset{\text{iid}}{\sim}P_{\bar{x}}$ $\mathbb{E}\left[\mathcal{T}+1\right]\lesssim U/\mathbb{E}[2^{N_{1}}]\eqsim U/M$ $\langle\tilde{\Delta}^{t_{0}},x^{t}-\tilde{x}^{t}\rangle\leq\|\tilde{\Delta}^{%
+t_{0}}\|_{\infty}\|x^{t}-\tilde{x}^{t}\|_{1}$ $\displaystyle\mathbb{E}_{\hat{w}^{t_{0}}}[\langle\tilde{\Delta}^{t_{0}},\tilde%
+{x}^{t}-x\rangle]$ $\max_{i\in[d_{y}]}\{F_{i}(x):=\mathbb{E}_{z\sim\mathcal{D}}[f_{i}(x;z)]\}$ $\sum_{i\in[t-1]}2^{N^{i}}\leq U-2^{M}$ $\mathbb{P}\left[(\mathcal{T}+1)\operatorname{Gap}(\tilde{x},\tilde{y})\geq A%
+\right]\leq\frac{\mathbb{E}\left[(\mathcal{T}+1)(\operatorname{Gap}(\tilde{x},%
+\tilde{y})-\operatorname{Gap}(\bar{x},\bar{y}))\right]+\mathbb{E}\left[(%
+\mathcal{T}+1)\operatorname{Gap}(\bar{x},\bar{y})\right]}{A}.$ ${\cal Y}\subseteq\mathbb{R}^{d_{y}}$ $a\eqsim b$ $\mathbb{P}[N=k]=p^{k}\mathbbm{1}_{(k\in\{0,1,...,M\})}/C_{M}$ $(w^{t})_{t\in[T]}$ $\displaystyle\leq\frac{(A+B)^{2}}{2}+2MC\left[C\int_{B/C}^{\infty}\exp\left(-%
+\beta^{2}\right)\beta d\beta+A\int_{B/C}^{\infty}\exp\left(-\beta^{2}\right)d%
+\beta\right]$ $\bar{a}^{T}=\sum_{t=1}^{T}\lambda_{t}a^{t}$ $g^{t}_{y}=-\nabla_{y}F_{\mathcal{D}}(\hat{x}^{t},\hat{y}^{t};B^{t})$ $U\leq\frac{\varepsilon^{2}}{48\log(1/\delta)(9\tau\alpha L_{0})^{2}}$ $4\sum_{t=1}^{\mathcal{T}+1}2^{N_{t}}+2(\mathcal{T}+1)\leq 6\sum_{t=1}^{%
+\mathcal{T}+1}2^{N_{t}}\leq\frac{\varepsilon^{2}}{8\log(1/\delta)(9\tau\alpha L%
+_{0})^{2}}.$ $\displaystyle=L_{0}^{F}D\|(\lambda_{1},\mu_{1})-(\lambda_{2},\mu_{2})\|_{1}.$ $\psi=4\max_{j\in[d_{x}]}|\hat{G}_{j}-G_{j}|$ $\tau\eqsim\min\left\{\frac{1}{L_{0}}\sqrt{\frac{\log(d_{x})+\log(d_{y})}{T}},%
+\frac{n\varepsilon}{L_{0}T\sqrt{TK\log(1/\delta)}}\right\},K=1.$ $\displaystyle\leq 2L_{1}\sum_{k=t_{0}}^{t-1}\frac{2}{k+1}\leq\frac{4L_{1}q}{q%
+\lfloor t/q\rfloor+1},$ $\displaystyle=\mathbb{E}\big{[}\langle\nabla F_{\mathcal{D}}(w^{t})-\nabla f(w%
+^{t};B^{t}),x^{t}-x\rangle\big{]}+\mathbb{E}\big{[}\langle\nabla f(w^{t};B^{t}%
+)-\nabla f(w^{t_{0}};B^{t}),x^{t}-x\rangle\big{]}$ $\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})]\lesssim(L_{0}+L_{1})\bigg{%
+[}\frac{\sqrt{\log(d_{x})+\log(d_{y})}}{\sqrt{n}}+\left(\frac{(\log(d_{x})+%
+\log(d_{y}))^{3/2}\sqrt{\log(1/\delta)}}{n\varepsilon}\right)^{1/2}\bigg{]}.$ $\textstyle\mathbb{E}\Big{[}\big{\|}\sum_{t=1}^{T}x^{t}\Big{\|}_{\infty}^{2}%
+\Big{]}\leq c\log(d)\sum_{t=1}^{T}\mathbb{E}[\|x^{t}\|_{\infty}^{2}].$ $b\lesssim a$ $\hat{y}^{t+1,k}$ $P_{w^{t}}$ $F(\bar{a}^{T})-F\left(\bar{x}^{T}\right)\leq\sum_{t=1}^{T}\alpha_{t}+\frac{L_{%
+1}D^{2}}{2}\sum_{t=1}^{T}\lambda_{t}^{2},$ $\displaystyle\mathcal{X}\times\mathcal{Y}-(x_{1},y_{1})$ $\mathbb{E}\left\|\nabla F\left(\frac{1}{T}\sum_{t\in[T]}a^{t}\right)-\nabla F%
+\left(\frac{1}{T}\sum_{t\in[T]}x^{t}\right)\right\|^{2}_{\infty}\leq\frac{43L_%
+{1}^{2}}{\sqrt{12+\log(d)}T^{3/2}}+\frac{17(L_{0}^{2}+L_{1}^{2})\sqrt{(12+\log%
+(d))}}{\sqrt{T}}.$ $\displaystyle=\mathbb{E}\left[\sum_{t=1}^{\mathcal{T}+1}\mathbb{E}[2^{N_{t}}]%
+\mathbbm{1}_{(\mathcal{T}+1\geq t)}\right]$ $9\tau\alpha L_{0}$ $\mathbb{E}[F_{\mathcal{D}}(w^{T})-F_{\mathcal{D}}(x)]\leq\frac{\log(d_{x})}{%
+\tau T}+\frac{\tau L_{0}^{2}}{2}+\frac{1}{T}\sum_{t\in[T]}\mathbb{E}\left[%
+\langle\nabla F_{\mathcal{D}}(w^{t})-g^{t},x^{t}-x\rangle\right].$ $\Delta(s^{t}_{y})$ $\mathbb{E}[F_{\cal D}(\hat{w}^{T})-F_{\cal D}(x)]\lesssim(L_{0}+L_{1})\left[%
+\sqrt{\frac{\log(d_{x})}{n}}+\frac{\log(d_{x})\log(1/\delta)^{1/4}}{\sqrt{n%
+\varepsilon}}\right]+\frac{L_{2}}{\sqrt{n}\log(d_{x})^{1/4}}\\
++\frac{L_{2}\log(d_{x})^{1/4}\log(1/\delta)^{1/4}}{\sqrt{n\varepsilon}}+L_{1}%
+\log(\sqrt{n\log(d_{x})})\Big{[}\frac{1}{\sqrt{n}}+\frac{\log(1/\delta)^{1/4}}%
+{\sqrt{n\varepsilon}}\Big{]}.$ $\frac{\sqrt{\ell_{x}}}{\sqrt{n}}+\frac{\ell_{x}}{\sqrt{n\varepsilon}}$ $\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})-\operatorname{Gap}(\bar{x},%
+\bar{y})]\leq 8L_{1}\sqrt{\log(d_{x})+\log(d_{y})}/\sqrt{T}$ $|\mathbb{E}[\nabla_{x,j}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{x,j}\mid\mathcal{F%
+}_{t}]|\leq\frac{2L_{2}}{K}$ $\Delta_{s}:=\max_{j\in J}\max_{S\sim S^{\prime}}|s(S,j)-s(S^{\prime},j)|$ $\displaystyle=\langle\nabla f(w^{t_{0}};B^{t})-\nabla f(\hat{w}^{t_{0}};B^{t})%
+,x^{t}-\tilde{x}^{t}\rangle$ $L_{0}^{F}$ $\displaystyle=\max_{k\in[K_{x}+K_{y}]}\left|\sum_{i\in[K_{x}+K_{y}]}\Lambda^{T%
+}_{j,i}\Big{[}\nabla_{k,i}F(\Lambda(\lambda_{1},\mu_{1})+(x_{1},y_{1}))-\nabla%
+_{k,i}F(\Lambda(\lambda_{2},\mu_{2})+(x_{1},y_{1}))\Big{]}\right|$ $\displaystyle\leq\frac{2L_{2}}{2^{M}},$ $x\in\Delta_{x}$ $(\bar{x}_{-},\bar{y}_{-})=\frac{1}{2^{N}}\sum_{i\in[2^{N}]}(\hat{x}^{i},\hat{y%
+}^{i})$ $s:{\cal Z}^{n}\times J\mapsto\mathbb{R}$ $\mathbb{E}[\langle\nabla F(\bar{x})-\nabla F(\bar{a}),e_{j}\rangle]\leq\frac{%
+\mathbb{E}\left[F(\bar{a}+re_{j})-F(\bar{x}+re_{j})+F(\bar{x})-F(\bar{a})%
+\right]}{r}+L_{1}r.$ $\hat{w}^{T}$ $\displaystyle(\lambda,\mu)$ $\operatorname{LSE}(x)$ $\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})]\leq\frac{2(\log(d_{x})+\log(d_%
+{y}))}{\tau T}+5\tau L_{0}^{2}+\frac{1}{T}\sum_{t=1}^{T}\mathbb{E}\|\mathbb{E}%
+[\Phi_{\mathcal{D}}(x^{t},y^{t})-g^{t}\mid\mathcal{F}_{t}]\|_{\infty}.$ $x_{\mathcal{D}}$ $\sum_{t\in[T]}\langle\tau(\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t})-g^{t}_{x}),w^%
+{t}-w\rangle\leq\log(d_{x})+\frac{1}{2}\sum_{t\in[T]}\tau^{2}\|\nabla_{x}F_{%
+\mathcal{D}}(x^{t},y^{t})-g^{t}_{x}\|_{\infty}^{2}.$ $\mathbb{E}[\mathcal{T}+1]$ $\sqrt{\log(d)/n}+\log(d)^{7/10}/[n\varepsilon]^{2/5}$ $\displaystyle=4\max_{j\in[d_{x}]}\tau\left|\sum_{k=t_{0}}^{t-1}[\nabla_{j}f(%
+\hat{w}^{t_{0}};B^{k})-\nabla_{j}f(w^{t_{0}};B^{k})]\right|$ $\mathcal{T}=\sup\left\{T\in\mathbb{N}:\sum_{t=1}^{T}2^{N_{t}}\leq U-2^{M}%
+\right\}.$ $\bar{y}=\sum_{t\in[T]}y^{t}$ $\displaystyle=\sum_{k=0}^{M}\mathbb{E}\Big{\|}C_{M}2^{k}[\nabla_{x}f(\bar{x}_{%
++},\bar{y}_{+};B_{k})-\nabla_{x}f(\bar{x}_{-},\bar{y}_{-};B_{k})]+\nabla_{x}f(%
+x_{0},y_{0};B_{k})\Big{\|}_{\infty}^{2}\cdot\frac{2^{-k}}{C_{M}}$ $\left|\mathbb{E}\left[\nabla_{j}F\left(\frac{1}{T}\sum_{t=1}^{T}a^{t}\right)-%
+\nabla_{j}F\left(\frac{1}{T}\sum_{t=1}^{T}x^{t}\right)\right]\right|\leq\frac{%
+2L_{2}}{T}.$ $w^{1}=(1/d_{x},...,1/d_{x})\in\mathbb{R}^{d_{x}},w^{t+1}:=\operatorname{argmin%
+}_{x\in\Delta_{x}}\left(\tau\langle\nabla_{x}F_{\mathcal{D}}(x^{t},y^{t})-g_{x%
+}^{t},x\rangle+\sum_{i\in[d_{x}]}w^{t}_{i}\log(w^{t}_{i}/x_{i})\right)$ $\mathbb{E}[\operatorname{Gap}(\tilde{x},\tilde{y})]\leq\frac{8L_{1}\sqrt{\log(%
+d_{x})+\log(d_{y})}}{\sqrt{T}}+\frac{2(\log(d_{x})+\log(d_{y}))}{\tau T}+5\tau
+L%
+_{0}^{2}+\frac{2\sqrt{2}L_{1}}{\sqrt{K}}.$ $\mathbb{E}[\operatorname{Gap}(\bar{x},\bar{y})]\leq\frac{2\log(|\mathcal{Z}|)}%
+{\tau_{x}T}+\frac{\log(|{\cal Q}|)}{\tau_{y}T}+18\tau_{x}+2\tau_{y}+10\sqrt{%
+\frac{\log(|{\cal Q}|)}{n}}.$ $\sum_{t\in[T]}\langle\xi^{t},w^{t}-w\rangle\leq D_{\psi}(w^{1},w)+\frac{1}{2}%
+\sum_{t\in[T]}\|\xi^{t}\|^{2}_{*}$ $20.28\pm 1.09$ $\textbf{20.60}\pm 0.89$ $20.84\pm 0.83$ $19.82\pm 1.05$ $20.46\pm 1.16$ $20.05\pm 1.10$ $20.30\pm 0.74$ $20.96\pm 0.86$ $19.74\pm 1.00$ $\textbf{20.41}\pm 1.19$ $20.80\pm 0.88$ $20.25\pm 1.10$ $20.94\pm 0.85$ $\textbf{21.44}\pm 0.80$ $20.19\pm 1.04$ $\{(o_{t},a_{t}),\dots,(o_{t+n},a_{t+n})\}$ $20.73\pm 0.84$ $20.09\pm 1.02$ $21.22\pm 0.87$ $20.95\pm 0.87$ $\textbf{20.54}\pm 1.21$ $20.24\pm 0.98$ $20.83\pm 0.94$ $97.89\%$ $\textbf{20.95}\pm 0.92$ $\textbf{21.58}\pm 1.10$ $\textbf{20.47}\pm 1.11$ $\textbf{20.41}\pm 0.90$ $20.12\pm 1.12$ $20.26\pm 1.05$ $21.15\pm 0.91$ $20.51\pm 1.02$ $20.20\pm 0.24$ $20.81\pm 0.90$ $20.31\pm 1.07$ $\textbf{20.97}\pm 0.92$ $21.01\pm 0.85$ $20.91\pm 0.79$ $20.96\pm 0.84$ $20.83\pm 0.82$ $256\times 160$ $20.33\pm 1.20$ $20.31\pm 1.12$ $w_{p_{xy}}=\frac{1}{2\pi\sigma^{2}}e^{-\frac{(x-m)^{2}+(y-n)^{2}}{2\sigma^{2}}},$ $p(I_{pred})$ $\displaystyle L_{VQVAE}=||x-D(e)||^{2}_{2}+||sg[E(x)]-e||^{2}_{2}+$ $T_{style}$ $I_{x_{ij}}=\mathop{\arg\min}_{\theta\in\Theta}Dist(Z_{q_{ij}},Z_{\theta}),$ $L_{VQVAE}$ $\displaystyle\beta||sg[e]-E(x)||^{2}_{2},$ $I_{GT}$ $i=1,{\ldots}\,,T$ $T_{text}$ $p_{xy}$ $S_{text}$ $P(I_{0},I_{1},...,I_{F}|I_{0})=\prod_{i=1}^{F}P(I_{i}|I_{0}),$ $w_{p_{xy}}$ $L_{Transformer}=-\sum p(I_{pred})\log q(I_{GT}),$ $Dist(Z_{q_{ij}},Z_{\theta})=\left\|Z_{q_{ij}}-Z_{\theta}\right\|_{2},$ $S_{style}$ $q(I_{GT})$ $P_{i_{xy}}=\frac{\sum_{p_{xy}>0}w_{p_{xy}}p_{xy}}{\sum_{p_{xy}>0}w_{p_{xy}}},$ $\displaystyle\text{Queries}^{\mathcal{P}_{n}}_{A_{n}}(G)\leq c\cdot\mathrm{RAC%
+}(\mathcal{P}_{n},G)$ $\displaystyle\text{Queries}^{\mathcal{P}_{n}}_{A_{n}}(f)\leq c\cdot\mathrm{RAC%
+}(\mathcal{P}_{n},f)$ $\mathrm{RAC}(\mathcal{P}_{H},\Delta)=O(n^{1/2}\log n)$ $H=S_{k}$ $v_{\sqrt{n}}$ $H=([h],E)$ $O(1/n^{c})$ $n/{2.2^{i}}$ $\Pr(E_{j}|\neg E_{1}\wedge\neg E_{2}\wedge\ldots\wedge\neg E_{j-1})=\Theta%
+\left(\frac{2^{t}}{1.1^{t}}\cdot\frac{1}{n^{1/10}}\right)$ $\mathrm{RAC}(\mathcal{P}_{H},\Delta)=O(n^{1/2})$ $\displaystyle\mathrm{RAC}(\mathcal{P},G)=\min_{A\in\mathcal{A}_{\mathcal{P}}}%
+\max_{\pi\in\Gamma}\text{Queries}^{\mathcal{P}}_{A}(\pi(G)),$ $|H|/(|B|-i)=\Theta(1/n)$ $x\neq y\in[n]$ $n^{1-c}/{1.1^{t}}$ $|C|=\Theta(n/\log n)$ $1\leq i\leq\sqrt{n}-1$ $I\in\mathcal{I}_{i}\setminus\mathcal{L}_{i}$ $\mathop{\mathbb{E}}_{G\leftarrow\Delta}\text{Queries}_{A}(G)=\Omega(n^{1/10}%
+\log(n))$ $n^{0.9}$ $\Pr(E)=1-1/\Theta(n^{1/4})$ $C\sqrt{n}$ $(1+o(1))p$ $\mathop{\mathbb{E}}_{G\leftarrow\Delta}\text{Queries}_{A}(G)=\Omega(n)$ $|V_{t}|=n-O(t)\geq n/2$ $\alpha n^{1/4}$ $I\in\mathcal{L}_{i}$ $O(n^{0.1})$ $\mathop{\mathbb{E}}[I_{i}]\leq\alpha q_{i}/n^{0.1}$ $x_{1},\ldots x_{h}\in[n]$ $\{x_{1},\ldots,x_{h}\}$ $\Pr(E)=1-O(1/\sqrt{n})$ $b_{t}=n^{9/10}/1.1^{t}$ $\mathop{\mathbb{E}}_{G\leftarrow\Delta}\mathrm{RAC}(\mathcal{P},G)=O(n^{1/10})$ $|\mathcal{L}_{i}|\leq\frac{\kappa n^{0.1}}{2^{i-2}}.$ $I\in\mathcal{R}_{i}$ $O(1/p)=O(1/n^{1/4})$ $H_{a,b,c}$ $y_{i},y_{j},y_{l}$ $2^{t}\cdot n^{9/10}/1.1^{t}$ $\text{deg}(v)=1$ $H=P_{k}$ $\mathcal{A}=\{A_{n}\}_{n\in\mathbb{N}}$ $f\circ\pi$ $V_{i}=\bigcup_{I\in\mathcal{I}_{i}}V(I)$ $Cn^{0.1}\log n$ $Q\subset I$ $\mathcal{A}_{\mathcal{P}}$ $\{G_{n}\}_{n\in\mathbb{N}}$ $p\in\{p_{i_{1}},\ldots,p_{i_{k}}\}$ $f(x_{u})=x_{v}$ $2^{i}<2^{t}$ $B(t)/|V_{t}|\leq 1/n^{0.1+\Omega(1)}$ $\mathcal{P}_{S_{k}}$ $\alpha\kappa$ $n^{1-c}/1.1^{i}$ $B(n)\leq n^{9/10}/1.1^{\log(n)/1000}=n^{9/10-\Omega(1)}$ $G^{\pi}=(V,E^{\pi})$ $2^{i-2}$ $f(x_{1})=\ldots=f(x_{k})$ $2/2^{i-1}=4/2^{i}$ $O(n^{3/4}\cdot\log^{4}n/n^{1/4})=o(n^{3/4})$ $B=[n]\setminus C$ $\displaystyle\text{Queries}^{\mathcal{P}_{n}}_{A_{n}}(f_{n})\geq\omega(n)\cdot%
+\mathrm{RAC}(\mathcal{P}_{n},f_{n}).$ $b_{i}=n^{9/10}/1.1^{i}$ $f\in\Delta$ $\frac{2^{t}}{1.1^{t}}\cdot\frac{1.1^{t}\cdot n}{n^{1-c}\cdot 2^{t}}=n^{c}.$ $\mathop{\mathbb{E}}_{f\leftarrow\Delta}RAC(\mathcal{P}_{H},f)=O(n^{3/4})$ $\frac{n^{1-c}\cdot 2^{t}}{n\cdot 1.1^{t}}$ $\Omega(n^{0.1}\log n)$ $2^{t}+4$ $\mathop{\mathbb{E}}\left[X_{j}|\neg E_{1}\wedge\neg E_{2}\wedge\ldots\wedge%
+\neg E_{j-1}\right]=O\left(\frac{2^{t}}{1.1^{t}}\right)$ $c=1/10$ $\Pr({E_{\text{long}}})=O(\kappa)$ $\frac{a_{i}\cdot 2^{i}}{n}=\frac{1}{1.1^{i}}$ $\omega\colon\mathbb{N}\to\mathbb{N}$ $C_{i_{1}},\ldots,C_{i_{T}}$ $C=\bigcup_{i=1}^{N}C_{i}$ $\mathcal{P}_{H}$ $\mathop{\mathbb{E}}[\text{\# red vertices encountered}]\leq\sum_{\frac{1}{1000%
+}\log n\leq i\leq\frac{1}{100}\log n}\alpha\frac{q_{i}}{n^{0.1}}\leq\frac{%
+\alpha}{n^{0.1}}cn^{0.1}\log n\leq\alpha c\log n,$ $N=\alpha n^{1/4}/\log(n)$ $\pi\colon[n]\to[n]$ $\bigcup_{i=1}^{N}C_{i}$ $\Omega(n^{c}\log n)$ $\Pr({E_{\text{long}}})$ $\Pr({E_{\text{long}}})\leq\sum_{I\in\mathcal{L}_{i}}\Pr(I\in\mathcal{R}_{i})%
+\leq|\mathcal{L}_{i}|\cdot(1+o(1))p\leq(1+o(1))\cdot\frac{\kappa n^{0.1}}{2^{i%
+-2}}\cdot\frac{2^{i}}{n^{0.1}}=O(\kappa).$ $\pi\colon V\to V$ $G=(V,E)\in\mathcal{P}$ $i=1,...,\Theta(\sqrt{n}\log n)$ ${E_{\text{short}}}$ $n^{9/10}/1.1^{t}$ $\text{Queries}_{A}(\cdot)$ $\mathcal{B}_{i}=\mathcal{I}_{i}\setminus\mathcal{R}_{i}$ $\mathrm{RAC}(\mathcal{P}_{S_{k}},\Delta)=O(n^{1/2})$ $\mathcal{P}_{S_{3}}$ $\mathcal{RAC}(\mathcal{P}_{H},f)=O(n^{3/4})$ $p=\frac{n^{9/10}/1.1^{i}}{n/2.2^{i}}=\frac{2^{i}}{n^{0.1}}$ $\tilde{\theta}(n^{1/4})$ $O(2^{t}/1.1^{t})$ $\frac{1}{1000}\log n\leq t\leq\frac{1}{100}\log n$ $\omega(n)=n^{\Omega(1)}$ $\mathcal{R}_{i}\subseteq\mathcal{I}_{i}$ $\alpha\frac{n}{\log n}$ $Cn^{1/10}\log n$ $p_{i_{1}},\ldots,p_{i_{T}}$ $1/|I|$ $\text{Queries}^{\mathcal{P}}_{A}(f,r)$ $\mathrm{RAC}(\mathcal{P},G)\in O(n^{1/10})$ $H=H_{a,b,c}$ $\displaystyle\text{Queries}_{A}(f)\leq\alpha\cdot\max_{\pi}\text{Queries}_{A^{%
+\prime}}(f\circ\pi)$ $(\pi(u),\pi(v))\in E^{\pi}$ $([3],\{1\to 3,2\to 3\})$ $a_{i}=n/2.2^{i}$ $\mathop{\mathbb{E}}_{f\leftarrow\Delta}\text{Queries}_{A}(f)=\Omega(n/\log n)$ $\frac{1}{1000}\log n\leq i\leq\frac{1}{100}\log n$ $I\notin\mathcal{R}_{i}$ $n^{1/10}$ $n^{1/10+\Omega(1)}$ ${E_{\text{short}}}=E\setminus{E_{\text{long}}}$ $\Pr(E)=1-o(1)$ $\text{Queries}_{A}(f)$ $\Omega(n^{1/10}\log n)$ $f\circ\pi\in\mathcal{P}$ $\Pr(E)=O(\kappa)$ $Cn^{1/2}$ $\sum_{i=0}^{t}2^{i}\cdot\frac{1}{1.1^{i}}+\sum_{i=t}^{\log n}2^{t}\cdot\frac{1%
+}{1.1^{i}}=O\left(\frac{2^{t}}{1.1^{t}}\right)$ $n^{\frac{3}{4}}$ $P\in\mathcal{P}_{t}$ $\displaystyle\text{Queries}_{A}(f)\geq\omega(n)\cdot\max_{\pi}\text{Queries}_{%
+A^{\prime}}(f\circ\pi).$ $t=o(n)$ $\Pr({E_{\text{short}}})=O(\kappa)$ $[\frac{\sqrt{n}}{4},\frac{3\sqrt{n}}{2}]$ $v_{0},v_{1},\ldots,v_{\sqrt{n}}$ $n^{\frac{1}{10}+\varepsilon}$ $n\geq\mathbb{N}$ $O(n^{1/10})$ $P_{1},...,P_{\sqrt{n}}$ $\{\mathcal{A}_{n}\}_{n\in\mathbb{N}}$ $O(\log^{4}n/\sqrt{n})$ $\frac{4}{2^{i}}\cdot(1+o(1))\cdot\frac{2^{i}}{n^{0.1}}=O\left(\frac{1}{n^{0.1}%
+}\right).$ $2^{i}/n^{c}$ $P\setminus\{u,v\}$ $f\colon[n]\to[n]$ $\{\mathcal{P}_{n}\}$ $\Theta(\sqrt{n}\log n)$ $\mathcal{P}=\{\mathcal{P}_{n}\}_{n\in\mathbb{N}}$ $\{A_{n}\}_{n\geq N}$ $Cn^{1/4}$ $\displaystyle\text{Queries}^{\mathcal{P}_{n}}_{A_{n}}(G_{n})\geq\omega(n)\cdot%
+\mathrm{RAC}(\mathcal{P}_{n},G_{n}).$ $(n^{1/4}/4,{n^{1/4}}/2)$ $f(y)=x$ $\text{Queries}^{\mathcal{P}}_{A}(f)=\mathop{\mathbb{E}}_{r}\text{Queries}^{%
+\mathcal{P}}_{A}(f,r)$ $\min(2^{i},2^{t})$ $2\leq i\leq\sqrt{n}-1$ $\frac{4}{2^{i}}$ $\Pr({E_{\text{short}}})$ $n^{1/4}/\log n$ $p_{i_{k}}$ $v_{0},\ldots,v_{\sqrt{n}}$ $W_{1},\ldots,W_{m}$ $I=\{\frac{1}{1000}\log n\leq i\leq\frac{1}{100}\log n:\text{there exists a %
+path of length $2^{i}$ with a red end}\}$ $i_{1},...,i_{T}$ $f_{n}\colon[n]\to[n]$ $O(\log^{1.1}n)$ $G\in\Delta$ $|Q|<2^{i-2}$ ${E_{\text{long}}}$ $E={E_{\text{long}}}\cup{E_{\text{short}}}$ $G^{\pi}\in\mathcal{P}$ $\displaystyle\mathrm{RAC}(\mathcal{P},f)=\min_{A\in\mathcal{A}_{\mathcal{P}}}%
+\max_{\pi}\text{Queries}_{A}(f\circ\pi),$ $i_{1},\ldots,i_{T}$ $I\setminus Q$ $n/2.2^{i}$ $cn^{0.1}\log n$ $\frac{1}{1000}\log n$ $\kappa n^{0.1}$ $\sqrt{n}-2$ $Path\mbox{ }of\mbox{ }length\mbox{ }\sqrt{n}$ $s\in B\Rightarrow s_{i}\in M_{B}\;\forall_{s_{i}\in s}\land((s_{0}\in E_{B}^{0%
+}\land s_{N}\in E_{B}^{1})\lor(s_{N}\in E_{B}^{0}\land s_{0}\in E_{B}^{1})).$ $3\times N$ $\mathrm{LM}_{k}(s):=\left\{||l_{k}-s_{i}||\forall s_{i}\in s\right\},$ $b\in\{1000,3000\}$ $5\,mm$ $s\equiv(s_{0},...,s_{N})$ $1\times N$ $56\times N$ $\mathrm{LM}[s]$ $W_{ij}=\left\{\begin{array}[]{ll}1&s_{j}\in R_{i}\\
+0&otherwise\end{array}\right.,$ $M_{B},E_{B}^{0},E_{B}^{1}$ $3\cdot\mathrm{10}^{-5}$ $J_{mm}(A,B)$ $x=a$ $C_{1},\ldots,C_{k}\cup(C\setminus\{v\})$ $z^{v}$ $\{2,\dots,s-1\}$ $S\cap(A\cup C)$ $H:=G[A\cup B\cup S]$ $z\in V(C)$ $N_{G}(p_{s})\cap V(H)=\{b_{2}\}$ $H\setminus C$ $k\geq\chi(\overline{G})\geq 2$ $H_{1},H_{2}\in\mathcal{M}_{\mathcal{C}}$ $\{p_{0},\dots,p_{s-1}\}$ $(x,c)$ ${\cal G}_{1}$ $q_{t}=y$ $q_{2},\dots,q_{t}\in B)$ $G\in\mathcal{G}_{k}$ $S=(S^{\prime}\setminus\{x,y\})\cup\{v^{xy}\}$ $X_{v}$ $\mathcal{M}_{\mathcal{G}_{2}}=\{\overline{K_{2}\cup C_{2k+1}}\mid k\in\mathbb{%
+N}\}$ $\{a_{i},b_{j}\}$ $\mathcal{O}(n^{\omega}\log n)$ $|V(H)|\geq 2$ $V(C)$ $q_{i},\ldots,q_{t},b_{1},b_{2},a_{2},q_{i}$ $N_{G}(v)\cap V(H)\subseteq A$ $G[N_{G}(v)]$ $G_{A}\setminus S^{\prime}$ $\mathcal{O}(n^{2k})$ $\mathbb{Z}_{k}$ $\{X_{v}\}_{v\in V(H)}$ $2K_{1}\vee K_{2}$ $G[\{a_{1},a_{2},b_{1},b_{2}\}\cup V(Q)]$ $v^{xy}\in S$ $S\cap A$ $(S^{\prime}\setminus\{x,y\})\cup\{v^{xy}\}\subseteq S$ $X_{w}\subseteq V(D)$ $\mathcal{G}_{k}=\mathcal{G}_{\mathcal{C}_{k}}$ $N_{G}[u]=V(G)$ $\mathcal{O}(n^{2.373}(n+m))$ $F\in{\cal F}$ $r_{i}\in C$ $G\setminus x$ $G\setminus N_{G}[v]$ $S^{*}:=(S\setminus\{v^{xy}\})\cup\{x,y\}$ $U:=\{v\in V(H)\mid X_{v}\cap V(C)\neq\emptyset\}$ $N_{A}\cap N_{B}=\emptyset$ $G\setminus(A\cup B)$ $N_{G}(q_{t})\cap V(H)=\{a_{1},b_{1}\}$ $S\subseteq S^{*}$ $K_{\ell+1}\in\mathcal{G}_{\mathcal{C}}$ ${\mathcal{C}_{k}}$ $V(H)=A\cup B$ $K_{\ell}$ $N_{G}(v)\cap V(H)=B$ $X_{w}$ $H\setminus a$ $H[A]$ $G^{\prime}[S]$ $G_{B}:=G[B\cup C]$ $S^{*}\subseteq S\cup(V(G)\setminus V(H))$ $G^{\prime}:=G/xy$ $p_{i},\dots,p_{s},b_{1},b_{3},a_{3},a_{2},q_{t},\dots,q_{0},p_{i}$ $q_{2},\dots,q_{t}\in Y$ $a_{1},q_{i}$ $\{2K_{1}\vee H\mid H\in\mathcal{M}_{\mathcal{C}}\}$ $p_{1},\dots,p_{i-1}\in A$ $v^{xy}$ $\mathcal{C}\subseteq\mathcal{G}_{\mathcal{C}}$ $S^{\prime}\setminus\{x,y\}$ $p_{1}=x$ $N_{G}(q_{t})\cap V(H)=\{a_{2}\}$ $G\setminus N_{G}[x]$ $\overline{\overline{G}}=G$ $H[S]\in\mathcal{C}$ $S^{\prime}=S$ $\bigcup_{v\in V(H)}X_{v}\not\subseteq V(H_{2})$ $c\in C\setminus S$ $N_{G}(p_{s})\cap V(H)=\{b_{1}\}$ $\{2K_{1}\vee H\mid H\in\mathcal{M}_{\mathcal{C}_{0}}\}=\{2K_{1}\vee K_{1}\}=\{%
+P_{3}\}$ $K_{\ell+1}$ $H\in\mathcal{H}$ $A=C\cup\{v\}$ $U=\{v\in V(H)\mid X_{v}\subseteq V(C)\}$ $S_{A}:=S\setminus B$ $2K_{1}\vee 3K_{1}$ $q_{j}\in C$ $p_{i-1},p_{j+1}\in C$ $\{a_{1},b_{1}\}$ $Y\subseteq V(G)\setminus\{x\}$ $\{a,b\}=\{x,y\}$ $\{X_{v}\}_{v\in\{a,b\}\cup V(H)}$ $|V(H)|=2k$ $(x,b)$ $C\cap S$ $\mathcal{O}(|V(G)|^{2})$ $G[a_{j},a_{k},b_{j},b_{k},q_{i},q_{i+1},\dots,q_{t}]$ $2K_{1}$ $a_{i},a_{j}$ $p_{i}\in B$ ${}_{mm\_morph}$ $c=\frac{max(|A|,|B|)}{min(|A|,|B|)}$ $J\_{mm\_syn}$ $C_{WALS}$ $m:\mathbb{L}\mapsto\mathbb{R}$ $\rho=0.69$ $J_{mm}$ $1\dots|Z|$ $J_{mm\_syn}$ $J_{mm}(\mathbf{a},\mathbf{b})=\frac{\sum_{j=1}^{|Z|}min(a_{j},b_{j})}{\sum_{j=%
+1}^{|Z|}max(a_{j},b_{j})}$ $\{Z=t(y):y\in Y\}=\{(y_{i},z_{j})\}$ $\{Y=m(x):x\in X\}=\{(x_{i},y_{i})\}$ $J_{mm\_morph}$ ${}_{mm}$ $1\dots|X|$ $J\_{mm\_morph}$ $ch\_ttr_{500}$ $s_{l}=r\cdot ch\_ttr_{l\_500}$ ${}_{morph}$ ${}_{mm\_syn}$ ${}_{syn}$ $S=\textbf{C}\circ\textbf{U}(X)$ $val_{1}$ $\sigma(G;D)$ $\log P(G|D,\lambda)\propto\log P(D|G)+\log P(G|\lambda)$ $\sigma(G;D)=\sum_{i=1}^{n}\sigma\left(v_{i},\operatorname{pa}\left(v_{i}\right%
+);D\right)$ $X=\{x_{1},x_{2},\cdots,x_{n}\}$ $sym_{1}$ $P\left(v_{i}\mid\text{pa}\left(v_{i}\right)\right)$ $val_{n}$ $x_{i_{1}}\rightarrow x_{j_{1}}$ $S{\prime}$ $\mathcal{A}=\{x_{i}\leadsto x_{j}\mid(x_{i},x_{j})\in S^{\prime}\}$ $P(G\mid D,\lambda)=\frac{P(D\mid G,\lambda)P(G\mid\lambda)}{P(D\mid\lambda)}$ $Domain$ $sym_{i}$ $val_{i}$ $\text{pa}(v_{i})$ $x\leadsto y$ $P(D|\lambda)$ $S=\{(x_{i},x_{j})\}$ $\sigma(G;D,\lambda)=\sigma(G;D)+\sigma(G;\lambda)$ $PromtU$ $sym_{n}$ $x_{i}\leadsto x_{j}\Rightarrow x_{j}\not\in\text{pa}(x_{i}),x_{i} $x\not\rightarrow y$ $S^{\prime}={(x_{i},x_{j})}$ $S=\{(x_{i_{1}},x_{j_{1}}),\cdots,(x_{i_{m}},x_{j_{m}})\mid x_{i_{k}},x_{j_{k}}%
+\in X\}$ $\frac{2\cdot\text{precision}\cdot\text{recall}}{\text{precision}+\text{recall}}$ $x_{i_{m}}\rightarrow x_{j_{m}}$ $S^{\prime}=\textbf{R}\circ\textbf{C}\circ\textbf{U}(X)$ $P(D\mid G,\lambda)=P(D\mid G)$ $T=\textbf{U}(X)$ $\sigma(G;\lambda)$ $x_{i}\leadsto x_{j}$ $T=\{t_{1},t_{2},\cdots,t_{n}\}$ $g_{i}(\textbf{x})$ $h_{j}(\textbf{x})$ $\displaystyle\mathrm{s.t.}:$ $\displaystyle f({\bf{x}}),x\in D$ $\displaystyle\mathrm{Min}:$ $\displaystyle g_{i}({\bf{x}})\leq 0,{i}=1,\dots,{p}$ $\displaystyle h_{j}({\bf{x}})\leq 0,{j}=1,\dots,{q}$ $S\neq\phi$ $W=\phi$ $W=W\cup(S\cap T)$ $E^{\prime}[i]=E^{\prime}[j]$ $\lvert E^{\prime}\rvert$ $\rm{LayoutLMv3_{LARGE}}$ $V[j]=V[j]+1$ $T=P[j*2-1]$ $batch\_size=8$ $S=S[j+1:\lvert S\rvert]$ $L=L[1:\lvert L\rvert]$ $E=E\cup ParseEntityValue(D,J^{\prime})$ $ $T.subtypes=\phi$ $G=\phi$ $\lvert E\rvert$ $M=\{``s.x|s.y"\mapsto s|s\in D.segments\}$ $G^{\prime}.value=\bigcup_{w\in W}w.text\_value$ $P[j*2]\notin M$ $\bigcup_{T^{\prime}\in T}MajorityVoting(\bigcup_{S^{\prime}\in S}DecodeForType%
+(ParseJson(S^{\prime}),T^{\prime},D))$ $800train/100dev/100test$ $E^{\prime}.subtypes=\bigcup_{T^{\prime}\in T.subtypes}DecodeForType(J^{\prime}%
+,T^{\prime},D)$ $R.split(E[i])$ $(segment~{}text,segment~{}identifier)$ $S=D.pages[i].segments$ $G^{\prime}.bounding\_box=\{\min(b.x),\min(b.y),\max(b.x),\max(b.y)\}_{w\in W,b%
+=w.bounding\_box}$ $E=\phi$ $F(S[1:j])\leq L$ $V=[0,0,...,0]\in\mathbb{R}\textsuperscript{$\lvert E\rvert$}$ $L=\{T\}$ $\lvert P\rvert/2$ $G=G\cup\{G^{\prime}\}$ $learning\_rate=2\cdot 10^{-5}$ $E=E\cup\{E^{\prime}\}$ $\mathbf{LayoutLMv3_{LARGE}}$ $T^{\prime}=L[0]$ $C=C\cup\{S[1:j]\}$ $\lvert D.pages\rvert$ $[x\textsubscript{center},y\textsubscript{center}]$ $E^{\prime}=\phi$ $E^{\prime}[j]$ $J^{\prime}=J[j][T.type]$ $j=\lvert S\rvert$ $C=\phi$ $S=M[P[j*2]]$ $E^{\prime}=E^{\prime}\cup\{e|e\in E[j],e.type=T^{\prime}\}$ $R=Regex(``(\char 92d\char 92d\char 92|\char 92d\char 92d)")$ $\mathbf{Donut}$ $L\neq\phi$ $L=L\cup T^{\prime}.subtypes$ $Donut$ $E[argmax(V)]$ $[x\textsubscript{min},y\textsubscript{min},x\textsubscript{max},y%
+\textsubscript{max}]$ $D=10/50/100/200$ $V[i]=V[i]+1$ $ $\displaystyle\bm{q}(t):=\begin{bmatrix}q_{0}^{(1)}(t)&q_{0}^{(2)}(t)&q_{1}^{(1%
+)}(t)&q_{1}^{(2)}(t)&\dots&q_{N-2}^{(1)}(t)&q_{N-2}^{(2)}(t)\end{bmatrix}^{T},$ $\displaystyle q_{3}:$ $40\text{\,}\mathrm{ms}$ $\displaystyle q_{0}:\quad$ $\displaystyle\;a_{i+1,i+1}=-\frac{1}{\psi_{\frac{i-1}{2}}}\left(\frac{\zeta_{%
+\frac{i-3}{2}}+\zeta_{\frac{i-1}{2}}}{2\zeta_{\frac{i-3}{2}}\zeta_{\frac{i-1}{%
+2}}}\right),$ $\displaystyle\lim_{t\to 0}\bm{q}(0,t)$ $M_{\textrm{left}}$ $(\xi_{n})_{n=0,\ldots,N-1}$ $n=1,2,...,N-1$ $\displaystyle\dot{\bm{q}}(t)$ $\bm{y}(t_{0})={(0.02,0.01)^{T}}$ $\displaystyle D_{x}\bm{q}(t)$ $\displaystyle=\bm{y}_{0}.$ $\dot{\bm{y}}(t)$ $\bm{v}(\bm{q}_{0}(t),\bm{y}(t),t)\in\mathbb{R}^{N}$ $\displaystyle\frac{243d^{5}}{120}(\partial^{5}_{\xi}q)_{0}.$ $\nicefrac{{\partial q(\xi(x),t)}}{{\partial\xi}}\rvert_{x_{n}}$ $\displaystyle\frac{d^{3}}{24}b_{1}$ $\displaystyle t\in[0,T],$ $\displaystyle c\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_{x_{0}}+%
+\frac{Nc}{N-1}\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_{x_{1}}\approx$ $\displaystyle\bm{v}(\bm{q}_{0}(t),\bm{y}(t),t)$ $\displaystyle\frac{81d^{3}}{24}b_{3}$ $\displaystyle\frac{d^{4}}{24}(\partial^{4}_{\xi}q)_{0}$ $\displaystyle\bm{v}(\bm{q}_{0}(t),\bm{y}(t),t):=\begin{bmatrix}\frac{2}{2+%
+\gamma\psi_{0}}\bm{f}^{(1)}(\bm{q}_{0}(t),\bm{y}(t),t)\\
+\frac{2}{2+\gamma\psi_{0}}\bm{f}^{(2)}(\bm{q}_{0}(t),\bm{y}(t),t)\\
+0\\
+\vdots\\
+0\end{bmatrix}.$ $\displaystyle M_{2}\;D_{x}^{2}\bm{q}(t)$ $\displaystyle\approx B_{2}\;D_{x}\bm{q}(t).$ $R=\nicefrac{{7}}{{3}}$ $-\nicefrac{{1}}{{2}}$ $\displaystyle\left(\frac{4Nc}{N-n}\right)\frac{\partial q(\xi(x),t)}{\partial x%
+}\biggr{\rvert}_{n}+\left(\frac{Nc}{N-n-1}\right)\frac{\partial q(\xi(x),t)}{%
+\partial x}\biggr{\rvert}_{n+1}$ $u^{(1)}=0$ $\displaystyle B_{1}:=\frac{1}{d}\begin{bmatrix}\frac{12\alpha dc-17\gamma}{18%
+\gamma}&0&\nicefrac{{1}}{{2}}&0&\nicefrac{{1}}{{2}}&0&-\nicefrac{{1}}{{18}}&0&%
+\dots&0\\
+0&\frac{12\alpha dc-17\gamma}{18\gamma}&0&\nicefrac{{1}}{{2}}&0&\nicefrac{{1}}%
+{{2}}&0&-\nicefrac{{1}}{{18}}&\ddots&0\\
+-3&0&0&0&3&0&0&0&\dots&0\\
+0&-3&0&0&0&3&0&0&\ddots&0\\
+\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\
+\vdots&\ddots&\ddots&\ddots&\ddots&0&-3&0&0&0\\
+0&\dots&\dots&\dots&\dots&\dots&0&-3&0&0\end{bmatrix},$ $\displaystyle+\frac{2}{\pi}\frac{(1-R)\lambda}{R}\int_{0}^{\infty}\frac{k^{4}%
+\gamma\sin(\lambda t)}{(k^{2}\gamma^{2}+(k^{2}-\alpha)^{2})(k^{4}+\lambda^{2})%
+\lambda}dk+$ $\displaystyle\;y_{0}^{(2)}+\frac{1-\cos(\lambda t)}{\lambda}+\frac{2}{\pi}\int%
+_{0}^{\infty}\;v_{0}^{(2)}\frac{\gamma(1-e^{-k^{2}t})}{k^{2}\gamma^{2}+(k^{2}-%
+\alpha)^{2}}\;dk+$ $\displaystyle d\;(\partial^{2}_{\xi}q)_{0}$ $\displaystyle\underbrace{\begin{bmatrix}\dot{\bm{q}}(t)\\
+\dot{\bm{y}}(t)\end{bmatrix}}_{=:\dot{\bm{\eta}}(t)}=\underbrace{\left[\begin{%
+array}[]{c | c}A_{s}&\begin{array}[]{c c}0&0\\
+\vdots&\vdots\\
+0&0\\
+\end{array}\\
+\hline\cr\\
+\begin{array}[]{c c c c c}1&0&0&\dots&0\\
+0&1&0&\dots&0\end{array}&\begin{array}[]{c c}0&0\\
+0&0\end{array}\end{array}\right]}_{=:A}\underbrace{\begin{bmatrix}\bm{q}(t)\\
+\bm{y}(t)\end{bmatrix}}_{=:\bm{\eta}(t)}+\underbrace{\begin{bmatrix}\bm{v}(\bm%
+{q}_{0}(t),\bm{y}(t),t)\\
+\bm{u}(\bm{y}(t),t)\end{bmatrix}}_{=:\bm{\omega}(\bm{q}_{0}(t),\bm{y}(t),t)}$ $R=\nicefrac{{1}}{{3}}$ $\displaystyle\frac{d^{2}}{2}a_{1}$ $\displaystyle\underbrace{\left(\mathbb{I}-\frac{\Delta t}{2}A\right)}_{=:M_{%
+\textrm{left}}}\bm{\eta}^{k+1}=\left(\mathbb{I}+\frac{\Delta t}{2}A\right)\bm{%
+\eta}^{k}+\frac{\Delta t}{2}\left(\bm{\omega}^{k}+\bm{\omega}^{k+1}\right),$ $\displaystyle\frac{\partial^{2}q(\xi(x),t)}{\partial x^{2}}\biggr{\rvert}_{x_{%
+n}}\approx\;\frac{1}{\psi_{n}}\left[\frac{\partial q(\xi(x),t)}{\partial x}%
+\biggr{\rvert}_{x_{n+\nicefrac{{1}}{{2}}}}-\frac{\partial q(\xi(x),t)}{%
+\partial x}\biggr{\rvert}_{n-\nicefrac{{1}}{{2}}}\right]$ $\displaystyle:=M_{2}-\frac{2c}{3\gamma}B_{2}M_{1}^{-1}\mathbb{P}.$ $\displaystyle-\frac{2c}{3\gamma}\left(\alpha q_{0}-\gamma(\partial_{x}q)_{0}\right)$ $\bm{v}(t_{0})$ $\displaystyle\frac{16d^{4}}{24}(\partial^{4}_{\xi}q)_{0}$ $R=\nicefrac{{7}}{{9}}$ $\bm{q}(0,t_{0})={(0,0.1)}^{T}$ $\left(\nicefrac{{d\xi(x)}}{{dx}}\right)_{x_{n}}$ $\displaystyle\approx M_{1}^{-1}\;B_{1}\bm{q}(t)+M_{1}^{-1}\;K_{1}(\bm{q}_{0}(t%
+),\bm{y}(t),t)+M_{1}^{-1}\;V_{1}(\dot{\bm{q}}_{0}(t)),$ $\displaystyle\approx B_{2}\;D_{x}\bm{q}(t),$ $\displaystyle\beta=\frac{\rho_{p}}{\rho_{f}},\quad R=\frac{1+2\beta}{3},\quad S%
+=\frac{a^{2}}{3T\nu},$ $a_{0},a_{1},b_{0},\hat{b}_{0},b_{1},\hat{b}_{1},b_{2},b_{3}\in\mathbb{R}$ $\displaystyle(\partial_{\xi}q)_{0}:\quad$ $\displaystyle M_{2}:=\begin{bmatrix}c&0&3\frac{Nc}{N-1}&0&0&0&0&0&\dots&0\\
+0&c&0&3\frac{Nc}{N-1}&0&\ddots&\ddots&\ddots&\ddots&0\\
+c&0&\frac{4Nc}{N-1}&0&\frac{Nc}{N-2}&0&\ddots&\ddots&\ddots&0\\
+0&c&0&\frac{4Nc}{N-1}&0&\frac{Nc}{N-2}&0&\ddots&\ddots&0\\
+0&0&\frac{Nc}{N-1}&0&\frac{4Nc}{N-2}&0&\frac{Nc}{N-3}&0&\ddots&0\\
+\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\
+\vdots&\ddots&\ddots&\ddots&\frac{Nc}{4}&0&\frac{4Nc}{3}&0&\frac{Nc}{2}&0\\
+\vdots&\ddots&\ddots&\ddots&0&\frac{Nc}{4}&0&\frac{4Nc}{3}&0&\frac{Nc}{2}\\
+0&\ddots&\ddots&\ddots&\ddots&0&\frac{Nc}{3}&0&\frac{4Nc}{2}&0\\
+0&0&\dots&\dots&\dots&\dots&0&\frac{Nc}{3}&0&\frac{4Nc}{2}\end{bmatrix},$ $\displaystyle+\frac{1}{2d}q_{2}(t)-\frac{1}{18d}q_{3}(t)+$ $\bm{q}(0,t_{0})={(0.5414,0)}^{T}$ $t\in[0,5]$ $\displaystyle a_{11}=$ $\displaystyle\frac{d^{3}}{6}a_{1}$ $m\in\{1,2,\dots,N-2\}$ $\displaystyle(\partial^{4}_{\xi}q)_{0}:\quad$ $\bm{u}(\bm{y}(t),t)$ $\displaystyle\left(\frac{Nc}{N-n+1}\right)\frac{\partial q(\xi(x),t)}{\partial
+x%
+}\biggr{\rvert}_{n-1}+$ $\displaystyle x>0,$ $n\in\{1,2,\dots,N-1\}$ $\displaystyle\frac{d^{2}}{2}(\partial^{3}_{\xi}q)_{0}$ $f^{(1)}\equiv 0$ $S=0.3$ $\displaystyle 3b_{3}$ $\bm{f}(\bm{q}_{0}(t),\bm{y}(t),t)$ $\frac{\partial}{\partial x}\bm{q}(\xi(x),t)$ $\Psi(x,y,t)=-U_{0}L\text{tanh}(y/L)+\sum_{i=1}^{3}A_{i}U_{0}L\text{sech}^{2}(y%
+/L)\cos(k_{i}x-\sigma_{i}t)$ $\displaystyle\frac{1}{d}b_{3}$ $\displaystyle\frac{d^{4}}{120}b_{1}$ $\displaystyle=\frac{D\bm{u}}{Dt}-\frac{1}{S}(\bm{v}-\bm{u})-\sqrt{\frac{3}{\pi
+S%
+}}\left\{\frac{1}{\sqrt{t}}\left(\bm{v}(0)-\bm{u}(0)\right)+\int_{0}^{t}\frac{%
+(\dot{\bm{v}}(s)-\dot{\bm{u}}(s))}{\sqrt{t-s}}ds\right\},$ $\displaystyle\alpha:=\frac{1}{RS},\quad\gamma:=\frac{1}{R}\sqrt{\frac{3}{S}}$ $d:=\xi_{n+1}-\xi_{n}=\frac{1}{N}$ $\bm{q}(0,t_{0})={(0.1,0)}^{T}$ $\displaystyle\frac{32d^{4}}{120}b_{2}$ $\displaystyle\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_{x_{0}}%
+\approx\frac{1}{2}\left[\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_%
+{x_{\nicefrac{{1}}{{2}}}}+\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert%
+}_{x_{-\nicefrac{{1}}{{2}}}}\right].$ $\displaystyle\frac{16d^{3}}{24}b_{2}$ $\displaystyle\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_{x_{n}}=%
+\frac{\partial q(\xi(x),t)}{\partial\xi}\biggr{\rvert}_{x_{n}}\cdot\frac{1}{c}%
+\left(1-\frac{n}{N}\right),$ $\displaystyle M_{1}\;D_{x}\bm{q}(t)$ $\displaystyle-\frac{2}{\pi}\frac{(1-R)\lambda}{R}\int_{0}^{\infty}\frac{k^{2}%
+\gamma\cos(\lambda t)}{(k^{2}\gamma^{2}+(k^{2}-\alpha)^{2})(k^{4}+\lambda^{2})%
+}dk.$ $\displaystyle q_{0}:$ $\bm{q}(x,t)$ $\displaystyle\frac{4d^{2}}{2}(\partial^{2}_{\xi}q)_{0}$ $\displaystyle\dot{\bm{y}}(t)$ $\displaystyle\frac{243d^{4}}{120}b_{3}.$ $M_{1}^{-1}$ $A_{s}\in\mathbb{R}^{N\times N}$ $(x_{n})_{n=0,\ldots,N-1}$ $\displaystyle\frac{d}{2}b_{1}$ $\bm{y}={(0.02,0.01)}^{T}$ $\displaystyle V_{1}(\dot{\bm{q}}_{0}(t))=\frac{2c}{3\gamma}\begin{bmatrix}\dot%
+{q}^{(1)}_{0}(t)\\
+\dot{q}^{(2)}_{0}(t)\\
+0\\
+0\\
+\vdots\\
+0\end{bmatrix}=\frac{2c}{3\gamma}\underbrace{\begin{bmatrix}1&0&0&\dots\\
+0&1&0&\ddots\\
+0&0&0&\ddots\\
+\vdots&\ddots&\ddots&\ddots\end{bmatrix}}_{=:\mathbb{P}}\dot{\bm{q}}(t)=\frac{%
+2c}{3\gamma}\;\mathbb{P}\;\dot{\bm{q}}(t).$ $\bm{y}(t_{0})={(0.02,0.01)}^{T}$ $\displaystyle\;a_{24}=\frac{\gamma}{\zeta_{0}(2+\gamma\psi_{0})}$ $q(\xi(x),t)$ $\displaystyle a_{i,i-2}=$ $\bm{q}(0,t_{0})={(0,0)}^{T}$ $\displaystyle(\partial_{\xi}q)_{0}+(\partial_{\xi}q)_{1}=-\frac{2c}{3\gamma}%
+\left(\alpha q_{0}-\frac{\gamma}{c}(\partial_{\xi}q)_{0})\right)+\frac{1}{d}%
+\left(\frac{12\alpha dc-17\gamma}{18\gamma}\right)q_{0}+\frac{1}{2d}q_{1}+%
+\frac{1}{2d}q_{2}-\frac{1}{18d}q_{3},$ $\dot{\bm{q}}(t)=D_{x}^{2}\bm{q}(t)$ $\displaystyle\bm{q}_{0}(t):=\begin{bmatrix}q_{0}^{(1)}(t)&q_{0}^{(2)}(t)\end{%
+bmatrix}^{T}.$ $\displaystyle:=\Psi^{-1}B_{2}M_{1}^{-1}K_{1}(\bm{q}_{0}(t),\bm{y}(t),t),$ $q^{(2)}(0,t_{0})>0$ $\xi_{n}:=\frac{n}{N}$ $\bm{y}(t_{0})={(1,0)}^{T}$ $\displaystyle a_{0}(\partial_{\xi}q)_{0}+a_{1}(\partial_{\xi}q)_{1}=\hat{b}_{0%
+}\left(\alpha q_{0}-\gamma(\partial_{x}q)_{0}\right)+\frac{1}{d}\left\{b_{0}q_%
+{0}+b_{1}q_{1}+b_{2}q_{2}+b_{3}q_{3}\right\},$ $\displaystyle\frac{8d^{2}}{6}b_{2}$ $\displaystyle+\frac{2}{\pi}\frac{(1-R)\lambda}{R}\int_{0}^{\infty}\frac{k^{2}%
+\gamma e^{-k^{2}t}}{(k^{2}\gamma^{2}+(k^{2}-\alpha)^{2})(k^{4}+\lambda^{2})}dk+$ $q^{(3)}_{0}(t)$ $\displaystyle\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_{x_{n+%
+\nicefrac{{1}}{{2}}}}\approx\;\frac{q_{n+1}(t)-q_{n}(t)}{2\zeta_{n}},$ $\bm{\omega}^{k+1}$ $\displaystyle\bm{q}_{t}(0,t)+\alpha\bm{q}(0,t)-\gamma\bm{q}_{x}(0,t)$ $\displaystyle\bm{\tilde{\eta}}^{k+1}=\left(\mathbb{I}+\Delta t\;A\right)\bm{%
+\eta}^{k}+\Delta t\;\bm{\omega}^{k}$ $\displaystyle q_{1}:$ $q^{(3)}_{1}(t)$ $\bm{y}(t_{0})={(0,0)}^{T}$ $\displaystyle 2b_{2}$ $R=7/9$ $\bm{q}(t_{0})={(0,0)}^{T}$ $\displaystyle\approx B_{2}M_{1}^{-1}B_{1}\bm{q}(t)+B_{2}M_{1}^{-1}\;K_{1}(\bm{%
+q}_{0}(t),\bm{y}(t),t).$ $\displaystyle\frac{d^{3}}{6}(\partial^{3}_{\xi}q)_{0}$ $\displaystyle x_{n}:=x(\xi_{n})=-c\ln(1-\xi_{n}),$ $\rho_{p}\to 0$ $\displaystyle\;a_{i+1,i-1}=\frac{1}{2\psi_{\frac{i-1}{2}}\zeta_{\frac{i-3}{2}}},$ $R=4/3$ $\displaystyle\frac{4d}{2}b_{2}$ $\displaystyle q_{0}$ $V_{1}(\dot{\bm{q}}_{0}(t))$ $\displaystyle\;y^{(1)}_{0}+u^{(1)}t+2\frac{v_{0}^{(1)}}{\pi}\int_{0}^{\infty}%
+\frac{\gamma(1-e^{-k^{2}t})}{(\alpha-k^{2})^{2}+(k\gamma)^{2}}dk,$ $\displaystyle\frac{d^{2}}{2}(\partial^{2}_{\xi}q)_{0}$ $\bm{q}(0,t)$ $\displaystyle\frac{d^{5}}{120}(\partial^{5}_{\xi}q)_{0},$ $\displaystyle=\bm{q}(0,t)+\bm{u}(\bm{y}(t),t),$ $S=0.01$ $\displaystyle\iff\xi(x)=1-e^{-\nicefrac{{x}}{{c}}}$ $S\in\{0.01,0.1,0.5,1,2,4\}$ $\displaystyle 3d(\partial_{\xi}q)_{0}$ $\displaystyle\approx\frac{3}{d}\left(q_{n+1}(t)-q_{n-1}(t)\right)$ $\displaystyle(\partial_{\xi}q)_{0}$ $\displaystyle(\partial^{2}_{\xi}q)_{0}:\quad$ $i=2m+1$ $\displaystyle M_{1}:=\begin{bmatrix}c&0&\frac{Nc}{N-1}&0&0&0&0&\dots&0\\
+0&c&0&\frac{Nc}{N-1}&0&\ddots&\ddots&\ddots&0\\
+c&0&\frac{4Nc}{N-1}&0&\frac{Nc}{N-2}&0&\ddots&\ddots&0\\
+0&c&0&\frac{4Nc}{N-1}&0&\frac{Nc}{N-2}&0&\ddots&0\\
+\vdots&0&\frac{Nc}{N-1}&0&\frac{4Nc}{N-2}&0&\frac{Nc}{N-3}&\ddots&0\\
+\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\
+\vdots&\ddots&\ddots&\frac{Nc}{4}&0&\frac{4Nc}{3}&0&\frac{Nc}{2}&0\\
+\vdots&\ddots&\ddots&\ddots&\frac{Nc}{4}&0&\frac{4Nc}{3}&0&\frac{Nc}{2}\\
+0&\ddots&\ddots&\ddots&\ddots&\frac{Nc}{3}&0&\frac{4Nc}{2}&0\\
+0&0&\dots&\dots&\dots&0&\frac{Nc}{3}&0&\frac{4Nc}{2}\end{bmatrix},$ $u^{(1)}=0.05$ $\bm{\omega}^{k}:=\bm{\omega}(\bm{q}_{0}(t^{k}),\bm{y}(t^{k}),t^{k})$ $115\times 86$ $\displaystyle\bm{q}(x,0)$ $\displaystyle=A_{s}\bm{q}(t)+\bm{v}(\bm{q}_{0}(t),\bm{y}(t),t)$ $\displaystyle 2d(\partial_{\xi}q)_{0}$ $\displaystyle-\frac{2c}{3\gamma}\left(f(\bm{q}_{0}(t),\bm{y}(t),t)-\frac{%
+\partial q(\xi(x),t)}{\partial t}\biggr{\rvert}_{x_{0}}\right).$ $\displaystyle\alpha\hat{b}_{0}$ $\displaystyle\bm{u}=\left\lVert\bm{y}(t)\right\lVert\omega\;\bm{e}_{\theta}=%
+\omega\begin{bmatrix}-y^{(2)}\\
+y^{(1)}\end{bmatrix},$ $\displaystyle=\bm{v}_{0}-\bm{u}_{0},$ $\displaystyle B_{2}:=\frac{1}{d}\begin{bmatrix}-\nicefrac{{17}}{{6}}&0&%
+\nicefrac{{3}}{{2}}&0&\nicefrac{{3}}{{2}}&0&-\nicefrac{{1}}{{6}}&0&\dots&0\\
+0&-\nicefrac{{17}}{{6}}&0&\nicefrac{{3}}{{2}}&0&\nicefrac{{3}}{{2}}&0&-%
+\nicefrac{{1}}{{6}}&\dots&0\\
+-3&0&0&0&3&0&0&0&\dots&0\\
+0&-3&0&0&0&3&0&\ddots&\ddots&0\\
+0&0&-3&0&0&0&3&0&\ddots&\vdots\\
+\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\
+\vdots&\ddots&\ddots&\ddots&\ddots&0&-3&0&0&0\\
+0&\dots&\dots&\dots&\dots&\dots&0&-3&0&0\end{bmatrix},$ $\bm{y}(t_{0})=(1,0)^{T}$ $1056\text{\,}\mathrm{s}$ $\displaystyle M_{2}D_{x}^{2}\bm{q}(t)\approx B_{2}\left(M_{1}^{-1}\;B_{1}\bm{q%
+}(t)+M_{1}^{-1}\;K_{1}(\bm{q}_{0}(t),\bm{y}(t),t)+M_{1}^{-1}\;V_{1}(\dot{\bm{q%
+}}_{0}(t))\right).$ $R=7/3$ $\displaystyle\frac{9d^{2}}{2}(\partial^{2}_{\xi}q)_{0}$ $\bm{f}\left(\bm{q}(0,t),\bm{y}(t),t\right):=\left(\frac{1}{R}-1\right)\frac{D%
+\bm{u}}{Dt}-\bm{q}(0,t)\cdot\nabla_{y}\bm{u}(\bm{y}(t),t)$ $R<1$ $\displaystyle\bm{u}(t)=\begin{bmatrix}u^{(1)}\\
+\sin(\lambda t)\end{bmatrix},$ $\displaystyle\frac{d^{4}}{24}a_{1}$ $\displaystyle\bm{y}(0)$ $\displaystyle\frac{32d^{5}}{120}(\partial^{5}_{\xi}q)_{0},$ $S=0.5$ $\displaystyle\frac{1}{d}b_{0}$ $\displaystyle\frac{\partial\xi(x)}{\partial x}\biggr{\rvert}_{x_{n}}$ $\displaystyle(\partial^{3}_{\xi}q)_{0}:\quad$ $\displaystyle d(\partial_{\xi}q)_{0}$ $\frac{d^{4}}{60}(\partial^{5}_{\xi}q)_{0}$ $\displaystyle=\bm{f}(\bm{q}(0,t),\bm{y}(t),t),$ $\displaystyle R\dot{\bm{v}}(t)$ $f^{(2)}(s)=\left(\frac{1}{R}-1\right)\left(\lambda\cos(\lambda s)\right)$ $\displaystyle:=\Psi^{-1}B_{2}M_{1}^{-1}B_{1},$ $\displaystyle\frac{27d^{2}}{6}b_{3}$ $\displaystyle\bm{q}_{t}(x,t)$ $\displaystyle a_{12}=$ $\displaystyle\frac{27d^{3}}{6}(\partial^{3}_{\xi}q)_{0}$ $\displaystyle y^{(2)}(t)=$ $\bm{q}(0,t_{0})={(0.00052558,-0.00064947)}^{T}$ $\displaystyle x(\xi)=-c\ln(1-\xi)$ $10\text{\,}\mathrm{ms}$ $\displaystyle a_{0}(\partial_{\xi}q)_{0}+a_{1}(\partial_{\xi}q)_{1}=\hat{b}_{0%
+}\left(\alpha q_{0}-\frac{\gamma}{c}(\partial_{\xi}q)_{0}\right)+\frac{1}{d}%
+\left\{b_{0}q_{0}+b_{1}q_{1}+b_{2}q_{2}+b_{3}q_{3}\right\}.$ $\displaystyle\frac{81d^{4}}{24}(\partial^{4}_{\xi}q)_{0}$ $\displaystyle(\partial^{5}_{\xi}q)_{0}:\quad$ $\displaystyle da_{1}$ $\displaystyle q_{2}:$ $\displaystyle\psi_{n}:=x_{n+\nicefrac{{1}}{{2}}}-x_{n-\nicefrac{{1}}{{2}}},%
+\quad\zeta_{n}:=x_{n+\nicefrac{{3}}{{4}}}-x_{n+\nicefrac{{1}}{{4}}}$ $R=1/3$ $\beta=2/3$ $\displaystyle(\partial_{\xi}q)_{0}:$ $\displaystyle\frac{9d}{2}b_{3}$ $\displaystyle\frac{1}{d}b_{2}$ $\bm{v}(\bm{q}_{0}(t),\bm{y}(t),t)$ $\displaystyle a_{i,i+2}=$ $\displaystyle c\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_{x_{0}}+%
+\left(\frac{3Nc}{N-1}\right)\frac{\partial q(\xi(x),t)}{\partial x}\biggr{%
+\rvert}_{x_{1}}\approx\frac{1}{d}\left(-\frac{17}{6}q_{0}(t)+\frac{3}{2}q_{1}(%
+t)+\frac{3}{2}q_{2}(t)-\frac{1}{6}q_{3}(t)\right).$ $\displaystyle\approx B_{1}\bm{q}(t)+K_{1}(\bm{q}_{0}(t),\bm{y}(t),t)+V_{1}(%
+\dot{\bm{q}}_{0}(t)),$ $R\neq 1$ $D_{x}\bm{q}(t)$ $\displaystyle\frac{\gamma}{c}\hat{b}_{0}$ $\displaystyle\frac{d^{4}}{24}(\partial^{5}_{\xi}q)_{0},$ $\displaystyle x>0,t\in(0,T],$ $\displaystyle\;a_{i+1,i+3}=\frac{1}{2\psi_{\frac{i-1}{2}}\zeta_{\frac{i-1}{2}}},$ $\displaystyle=\bm{q}_{xx}(x,t),$ $\bm{q}(0,t_{0})=(0,0.1)$ $n\in\{0,1,\dots,N-1\}$ $\bm{q}_{x}(0,t)$ $\displaystyle\;a_{22}=-\frac{\gamma+2\alpha\zeta_{0}}{\zeta_{0}(2+\gamma\psi_{%
+0})},$ $\displaystyle=\frac{1}{c}e^{-\nicefrac{{x_{n}}}{{c}}}=\frac{1}{c}\left(1-\xi_{%
+n}\right)=\frac{1}{c}\left(1-\frac{n}{N}\right).$ $\bm{y}={(0,0)}^{T}$ $\displaystyle(\partial_{\xi}q)_{1}:$ $\displaystyle+\frac{1}{d}\left(\frac{12\alpha dc-17\gamma}{18\gamma}\right)q_{%
+0}+\frac{1}{2d}q_{1}+\frac{1}{2d}q_{2}-\frac{1}{18d}q_{3}.$ $\displaystyle\frac{\partial q(\xi(x),t)}{\partial\xi}\biggr{\rvert}_{x_{n-1}}+%
+\frac{4\partial q(\xi(x),t)}{\partial\xi}\biggr{\rvert}_{x_{n}}+\frac{\partial
+q%
+(\xi(x),t)}{\partial\xi}\biggr{\rvert}_{x_{n+1}}\approx\frac{3}{d}\left(q_{n+1%
+}(t)-q_{n-1}(t)\right)$ $\displaystyle\left(\frac{12\alpha dc-17\gamma}{18\gamma d}\right)q_{0}(t)+%
+\frac{1}{2d}q_{1}(t)+$ $\displaystyle K_{1}:=\begin{bmatrix}-\frac{2c}{3\gamma}f^{(1)}(\bm{q}_{0}(t),%
+\bm{y}_{0}(t),t)\\
+-\frac{2c}{3\gamma}f^{(2)}(\bm{q}_{0}(t),\bm{y}_{0}(t),t)\\
+0\\
+0\\
+\vdots\\
+0\end{bmatrix}\quad\text{and}\quad V_{1}:=\frac{2c}{3\gamma}\begin{bmatrix}%
+\frac{\partial q^{(1)}(\xi(x),t)}{\partial t}\biggr{\rvert}_{0}\\
+\frac{\partial q^{(2)}(\xi(x),t)}{\partial t}\biggr{\rvert}_{0}\\
+0\\
+0\\
+\vdots\\
+0\end{bmatrix}.$ $q^{(2)}(x,t_{0})=0$ $R=\nicefrac{{4}}{{3}}$ $50\text{\,}\mathrm{Hz}$ $\displaystyle\frac{\partial q(\xi(x),t)}{\partial x}\biggr{\rvert}_{x_{n}}=%
+\frac{\partial q(\xi(x),t)}{\partial\xi}\biggr{\rvert}_{x_{n}}\cdot\frac{d\xi(%
+x)}{dx}\biggr{\rvert}_{x_{n}}.$ $\displaystyle\hat{b}_{0}=-\frac{2c}{3\gamma}\quad b_{0}=\frac{12\alpha dc-17%
+\gamma}{18\gamma}\quad b_{1}=\frac{1}{2}\quad b_{2}=\frac{1}{2}\quad b_{3}=-%
+\frac{1}{18},$ $\displaystyle a_{i,i}=$ $\displaystyle\frac{d^{2}}{6}b_{1}$ $(a_{i,j})_{1\leq i\leq 2N,\;1\leq j\leq 2N}$ $\displaystyle M_{2}\;D_{x}^{2}\bm{q}(t)-B_{2}M_{1}^{-1}V_{1}(\dot{\bm{q}}_{0}(%
+t))$ $\displaystyle\frac{8d^{3}}{6}(\partial^{3}_{\xi}q)_{0}$ $\displaystyle=\bm{v}(t),$ $\bm{\eta}^{k}:=\bm{\eta}(t^{k})$ $\displaystyle(\partial_{\xi}q)_{0},$ $2\text{\,}\mathrm{mm}$ $t\in[0,10]$ $\displaystyle y^{(1)}(t)=$ $\displaystyle\frac{\partial q(\xi(x),t)}{\partial\xi}\biggr{\rvert}_{x_{0}}+3%
+\frac{\partial q(\xi(x),t)}{\partial\xi}\biggr{\rvert}_{x_{1}}\approx\frac{1}{%
+d}\left(-\frac{17}{6}q_{0}(t)+\frac{3}{2}q_{1}(t)+\frac{3}{2}q_{2}(t)-\frac{1}%
+{6}q_{3}(t)\right).$ $\displaystyle q_{0},$ $\displaystyle\frac{1}{d}b_{1}$ $\displaystyle\frac{d^{3}}{6}(\partial^{4}_{\xi}q)_{0}$ $\displaystyle-\frac{2c}{3\gamma}\left(f(q_{0},x_{0},t)-(\partial_{t}q)_{0}\right)$ $\displaystyle c(\partial_{x}q)_{0}+\frac{Nc}{N-1}(\partial_{x}q)_{1}=$ $(\bar{1}+\bar{3})(\bar{2}+\bar{3})(1+2+3)$ ${{}^{2,1}}$ ${{}^{5,1}}$ ${{}^{3,1}}$ $(i,~{}j)$ $CS_{1}\diamond CS_{2}=\{u\cup v|u\in CS_{1},v\in CS_{2}\}$ $(N,~{}\preccurlyeq)$ ${}^{1,\text{\Letter}}$ ${}^{\text{\Letter}}$ $checking\_logic\_block$ $\Phi(n)$ $(\bar{1}+\bar{4})(\bar{2}+\bar{4})(1+2+4)$ $\Phi(n)=\left\{\begin{aligned} \{\{n\}\}&:n\in PIs\\
+\{\{n\}\}\cup\Phi(n_{1})\diamond\Phi(n_{2})&:otherwise\end{aligned}\right\}.$ $CS_{1}\diamond CS_{2}$ $ge\in N$ $\{|u\cup v|\leq k\}$ $(\bar{1}+4+5)(\bar{1}+\bar{3}+5)(\bar{1}+\bar{2}+5)(2+3+\bar{4}+\bar{5})(1+%
+\bar{5})$ $\mathbf{F_{q}}$ $\mathbf{F_{q}}\in\mathbb{R}^{H/8\times W/8\times C}$ $\mathbf{F_{s,s}}\in\mathbb{R}^{H/8\times W/8\times C}$ $\mathbf{P^{\prime}_{r}}=(1+\alpha\mathbf{W}\otimes\mathbf{P_{r}})$ $\mathbf{P}\in\mathbb{R}^{1\times 1\times C}$ $\mathbf{P^{\prime\prime}}$ $\mathbf{C_{train}}$ $\mathbf{M_{q}}$ $\mathbf{P_{r}}\in\mathbb{R}^{1\times 1\times C}$ $\mathbf{F_{r}}$ $\mathbf{M_{s}^{k}}$ $\mathbf{I_{s}^{k}}$ $\mathbf{M_{s}}$ $\mathbf{Q_{i}}=\{(\mathbf{I_{q}},\mathbf{M_{q}})\}_{i}$ $\mathbf{S_{i}}=\{(\mathbf{I_{s}^{k}},\mathbf{M_{s}^{k}}),k\in\{1,\dots,K\}\}_{i}$ $\mathbf{\hat{y}}=\mathrm{softmax}(\mathrm{cosine}(\mathbf{P^{\prime\prime}},%
+\mathbf{F^{\prime}_{q}}))$ $L=L_{seg}+\lambda_{1}L_{s}+\lambda_{2}L_{q}$ $\mathbf{P^{\prime}}=(1+\alpha\mathbf{W}\otimes\mathbf{P})$ $\mathbf{P^{\prime\prime}}\in\mathbb{R}^{1\times 1\times C}$ $\mathbf{P_{r}}$ $\mathbf{F^{\prime}_{r}}$ $L_{s}=\mathrm{BCE}(\mathrm{cosine}(\mathbf{P^{\prime}},\mathbf{F_{s,s}}),%
+\mathbf{M_{s}})\\
++\mathrm{BCE}(\mathrm{cosine}(\mathbf{P^{\prime}_{r}},\mathbf{F_{s,r}}),%
+\mathbf{M_{s}})$ $\mathbf{I_{q}}$ $\mathrm{MAP}$ $L_{q}=\mathrm{BCE}(\mathrm{cosine}(\mathrm{MAP}(\mathbf{F^{\prime}_{q}}),%
+\mathbf{F^{\prime}_{q}}),\mathbf{M_{q}})$ $\mathbf{F^{\prime}_{q}}$ $\mathbf{F_{s,r}}\in\mathbb{R}^{H/8\times W/8\times C}$ $\mathbf{C_{test}}$ $L_{seg}=\mathrm{BCE}(\mathbf{\hat{y}},\mathbf{M_{q}})$ $\mathbf{X}\in\mathbb{R}^{1\times 1\times 2C}$ $\mathbf{P^{\prime}}\in\mathbb{R}^{1\times 1\times C}$ $\mathbf{W}=\mathrm{sigmoid}(f_{2}(\sigma(f_{1}(\mathbf{X}))))$ $\mathbf{F_{r}}\in\mathbb{R}^{H/8\times W/8\times C}$ $\mathbf{F^{\prime}_{q}}\in\mathbb{R}^{H/4\times W/4\times C}$ $\mathbf{\hat{y}}\in\mathbb{R}^{H\times W\times 1}$ $e_{i}=\{\mathbf{S_{i}},\mathbf{Q_{i}}\}$ ${\sf msg^{*}}$ ${\sf sk_{sanit}}$ ${\sf AD}_{i}({\sf MODIFY})=1\}$ $({\sf msg},\sigma)$ ${\sf Fixed_{AD}(msg^{*})}={\sf Fixed_{AD}}({\sf msg}_{i})$ $\{\mathcal{S}_{1},\mathcal{F},\mathcal{S}_{2}\}$ ${\sf msg^{*},\sigma^{*},pk^{*}_{sig},pk_{sanit}}$ $({\sf sk_{sign},pk_{sign}})$ $\sigma_{2}^{{}^{\prime}}$ $({\sf msg_{1}},{\sf MODIFY}_{1})$ ${\sf sec_{k}}$ $({\sf pk*_{sanit},{\sf msg^{*}},\sigma^{*}})\leftarrow\mathcal{G}^{{\sf
+Signature%
+(\cdot,sk_{sign},\cdot,\cdot)}}({\sf pk_{sign}})$ ${\sf msg_{0}=\mathcal{H}(0||{\sf msg_{fix}}||{AD}||{\sf pk_{sanit}})}$ $\displaystyle\mbox{or }{\sf msg^{*}\notin\{MODIFY(msg_{i})\;|\;MODIFY\mbox{ %
+with }}$ $r_{(1)}\left(\bar{\delta}_{1},\dots,\bar{\delta}_{n}\right)=\dots=r_{(k)}\left%
+(\bar{\delta}_{1},\dots,\bar{\delta}_{n}\right)=0.$ $({\sf 0,{msg}^{*}_{fixed},AD^{*},pk^{*}_{sanit}})=(0,{\sf msg}^{*}_{{\sf fixed%
+},i},{\sf AD}^{*}_{i},{\sf pk}^{*}_{{\sf sanit},i})$ $({\sf 0,{msg}^{*}_{fixed},AD^{*},pk^{*}_{sanit}})$ $\{\mathcal{Q},\mathcal{X},\mathcal{Y}\}$ $\sigma^{*}=(\sigma_{1}^{*},\sigma_{2}^{*},{\sf AD}^{*})$ $({\sf msg^{*}_{fixed}},{\sf AD^{*},pk_{sign},pk^{*}_{sanit},\sigma_{1}^{*}})$ $({\sf pk*_{sanit},{\sf msg^{*}},\sigma^{*}})\leftarrow\mathcal{G}_{\sf sanit}^%
+{{\sf Signature(\cdot,sk_{sign},\cdot,\cdot)}}({\sf pk_{sign}})$ ${\sf pk_{sanit}}$ ${\sf x_{0}}=\mathcal{S}_{1}^{-1}({\sf y})\in\mathbb{F}_{q}^{m},{\sf x_{1}}=%
+\mathcal{F}_{1}^{-1}({\sf x_{0}})\in\mathbb{F}_{q}^{n}$ $\mathcal{H}:\{0,1\}^{*}\rightarrow\mathbb{F}^{m}$ $\alpha_{1}=\mathcal{S}^{-1}({\sf msg_{1}}),\beta_{1}=\mathcal{F}^{-1}(\alpha_{%
+1})$ $\displaystyle\mbox{and }{\sf Judge({\sf msg^{*},\sigma^{*},pk_{sign},pk*_{%
+sanit}})=Sig}$ $\forall i=1,2,\ldots,\Delta$ $({\sf pk^{*}_{sign},{\sf msg^{*}},\sigma^{*}})$ ${\sf msg}\in\{0,1\}^{*}$ ${\sf msg_{1}=\mathcal{H}(1||msg||pk_{sanit}||pk_{sign})}$ $(n=160,m=64)$ $(b=0)$ ${\sf msg^{*}_{fixed}}={\sf FIXED_{AD^{*}}(msg^{*})}$ ${\sf Fixed_{AD}}$ $\mathcal{X}:\mathbb{F}_{q}^{n}\rightarrow\mathbb{F}_{q}^{m}$ $\mathcal{S}_{2}:\mathbb{F}_{q}^{n}\rightarrow\mathbb{F}_{q}^{n}$ ${\sf msg^{\prime}}$ ${\sf msg_{2}=\mathcal{H}(1||msg^{\prime}||pk_{sanit}||pk_{sign})}$ $\delta(\kappa)={\sf Prob}[{\sf EXP\mbox{-}Immutability_{\mathcal{G}}^{SSS}}=1]$ $({\sf msg}_{i},{\sf AD}_{i},{\sf pk_{sign}},{\sf pk_{sanit}}_{i})$ $j=q+1,\dots,r$ $\delta(\kappa)={\sf Prob}[{\sf EXP\mbox{-}Sanitizer\mbox{-}Acc{\mathcal{G}}^{%
+SSS}}=1]$ ${\sf y}=\mathcal{P}({\sf x})$ ${\sf(msg_{0},{\sf MODIFY}_{0}),(msg_{1},{\sf MODIFY}_{1})}$ ${\sf msg^{\prime}}\in\{{\sf MODIFY(msg)}\;|\;{\sf MODIFY}\mbox{ with }{\sf AD(%
+MODIFY)}=1\}$ $\displaystyle{\sf msg^{\prime}}\leftarrow{\sf MODIFY(msg)}$ ${\sf(pk_{sanit},sk_{sanit})}\leftarrow{\sf KGen\mbox{-}Sanit(1^{\kappa})}$ $a\leftarrow\mathcal{G}^{{\sf Signature(\cdot,sk_{sign},\cdot,\cdot)},{\sf
+Sanitization%
+(\cdot,\cdot,sk_{sanit},\cdot)},{\sf LoRSanit(\cdot,\cdot,\cdot,sk_{sign},sk_{%
+sanit},b)}}({\sf pk_{sign},pk_{sanit}})$ ${\sf(msg^{\prime}_{j},\sigma^{\prime}_{j})\leftarrow Sanitization(msg_{j,b},%
+MODIFY_{j,b},\sigma_{j,b},pk_{sign},sk_{sanit})}$ $({\sf msg_{j,0}},{\sf Modify_{j,0}}),({\sf msg_{j,1}},{\sf Modify_{j,0}},)$ ${\sf LoRSanit(\cdot,\cdot,\cdot,sk_{sign},sk_{sanit},b)}$ ${\sf msg_{fixed}}\leftarrow{\sf FIXED}_{{\sf AD}_{i}}({\sf msg}_{i})$ $(\cdot,{\sf sk_{sign}},\cdot,\cdot)$ $\mathcal{R}=\left(r_{(1)}(\delta,\dots,\delta_{n}),\dots,r_{(k)}(\delta_{1},%
+\dots,\delta_{n})\right)$ ${\sf msg,{\sf MODIFY},\sigma,pk_{sign},}\\
+{\sf sk_{sanit}}$ $\mathcal{G}_{\sf sanit}$ $\sigma_{j,b}\leftarrow{\sf Signature({\sf msg_{j,b},sk_{sign},pk_{sanit},AD_{j%
+}})}$ ${\sf EXP\mbox{-}Unforgeability_{\mathcal{G}}^{SSS}}$ $\mathcal{R}(\sigma_{2})\stackrel{{\scriptstyle?}}{{=}}{\sf msg_{2}}$ $(\bar{\delta}_{1},\dots,\bar{\delta}_{n})\in\mathbb{F}_{q}^{n}$ $\mathcal{S}_{1}\circ\mathcal{F}\circ\mathcal{S}_{2}$ $({\sf msg^{\prime}}_{i},{\sf\sigma^{\prime}}_{i},{\sf pk}_{{\sf sanit},i})$ ${\sf msg_{FIX}=FIXED_{AD}(msg)}$ ${\sf(pk_{sign},sk_{sign})}\leftarrow{\sf KGen\mbox{-}Sign(1^{\kappa})}$ ${\sf pk^{*}_{sign}},{\sf msg^{*}})\neq({\sf pk}_{{\sf sign},i},{\sf msg^{%
+\prime}}_{i})$ ${\sf\sigma_{1}}$ ${\sf Mul\mbox{-}SAN}$ $GF(16)$ $1,{\sf msg^{*},\sigma^{*},pk^{*}_{sanit},pk_{sign}}$ ${\sf msg_{fix}}$ ${\sf MODIFY}$ $(b=1)$ $\mathcal{Y}:\mathbb{F}_{q}^{n}\rightarrow\mathbb{F}_{q}^{n}$ ${\sf msg,sk_{sign},pk_{sanit},AD}$ $\delta(\kappa)$ ${\sf msg}_{i}$ $\mathcal{O}_{\sf LoR}$ ${\sf EXP\mbox{-}Immutability_{\mathcal{G}}^{SSS}}$ ${\sf pub_{k}}$ ${\sf msg,\sigma,pk_{sign},pk_{sanit}}$ ${\sf AD}_{j}$ $\mathcal{R}=\mathcal{Q}\circ\mathcal{X}\circ\mathcal{Y}:\mathbb{F}_{q}^{n}%
+\rightarrow\mathbb{F}_{q}^{m}$ ${\sf Sig}$ $({\sf pk^{*}_{sanit},{\sf msg^{*}},\sigma^{*}})\leftarrow\mathcal{G}_{\sf sanit%
+}({\sf pk_{sign}})$ $i=1,\dots,\Delta$ ${\sf AD(MODIFY)}\in\{0,1\}$ ${\sf(pk_{sanit},sk_{sanit})}\leftarrow$ ${\sf msg^{*}\notin\{MODIFY(msg)\;|\;MODIFY\mbox{ with }}$ $\displaystyle{\sf Verification({\sf msg^{*},\sigma^{*},pk_{sign},pk^{*}_{sanit%
+}})=true}$ ${\sf Signature(\cdot,sk_{sign},\cdot,\cdot)}$ ${\sf msg_{fix}=Fixed_{AD}({\sf msg})}$ $\frac{m(n+2)(n+1)}{2}$ $\displaystyle{\sf AD}_{i}({\sf MODIFY})=1\}$ ${\sf pub_{k},sec_{k}\leftarrow Kg(\kappa}$ ${\sf AD(MODIFY)}=1$ ${\sf msg^{*},\sigma^{*},pk^{*}_{sign},pk_{sanit}}$ $({\sf 1,{msg}^{*},AD^{*},pk^{*}_{sanit},pk_{sign}})$ $(\mathbb{F}_{q})$ $r_{(i)}\in\mathbb{F}_{q}[\delta_{1},\dots,\delta_{n}]$ $\sigma^{\prime}=(\sigma_{1},\sigma^{\prime}_{2},{\sf AD})$ $({\sf msg^{*}},\sigma^{*})\leftarrow\mathcal{G}^{{\sf Signature(\cdot,sk_{sign%
+},\cdot,\cdot)},{\sf Sanitization(\cdot,\cdot,sk_{sanit},\cdot)}}({\sf pk_{%
+sign},pk_{sanit}})$ $\in\{0,1\}^{*}$ ${\sf EXP\mbox{-}Signer\mbox{-}Acc_{\mathcal{G}_{sign}}^{SSS}}$ $\mathcal{P}(\sigma_{2})\stackrel{{\scriptstyle?}}{{=}}{\sf msg}_{1}$ ${\sf msg^{\prime}\leftarrow MODIFY(msg)}$ $({\sf msg}_{j},{\sf MODIFY}_{j},\sigma_{j},{\sf pk}_{{\sf sign},i})$ $\mathcal{S}_{1}:\mathbb{F}_{q}^{m}\rightarrow\mathbb{F}_{q}^{m}$ $\displaystyle({\sf pk_{sanit}^{*}},{\sf msg^{*}})\neq({\sf pk}_{{\sf sanit},i}%
+,{\sf msg^{\prime}}_{j})\forall i=1,2,\dots,q$ ${\sf pk*_{sanit}\neq pk_{{sanit},i}}$ $\delta(\kappa)={\sf Prob}[{\sf EXP\mbox{-}San\mbox{-}Acc_{\mathcal{G}_{sanit}}%
+^{SSS}}]$ $\sigma^{*}_{2}$ ${\sf EXP\mbox{-}San\mbox{-}Acc_{\mathcal{G}_{sanit}}^{SSS}}$ $i=1,2,\ldots,q$ $\mathcal{Q}:\mathbb{F}_{q}^{m}\rightarrow\mathbb{F}_{q}^{m}$ $\displaystyle{\sf Verification({\sf msg^{*},\sigma^{*},pk_{sign},pk*_{sanit}})%
+=true}$ ${\sf(msg_{0},}{\sf MODIFY}_{0})$ ${\sf 0/1\leftarrow Ver(x,pub_{k})}$ $y\in\mathbb{F}_{q}^{m}$ ${\sf sk_{sign}}$ $({\sf msg}_{i},{\sf pk}_{{\sf sanit},i},{\sf pk}_{{\sf sign},i})$ $\displaystyle\mbox{and }\forall i=1,2,\dots q,\;\;{\sf pk^{*}_{sanit}\neq pk_{%
+{sanit},i}}$ ${\sigma}_{1}$ $i=1,2,\dots,q$ ${\sf FIXED_{AD}}$ ${\sf msg_{0}}$ ${\sigma_{i,2}}$ $({\sf pk*_{sign},{\sf msg^{*}},\sigma^{*}})\leftarrow\mathcal{G}_{\sf sign}^{{%
+\sf Sanitization(\cdot,\cdot,\cdot,sk_{sign})}}({\sf pk_{sanit}})$ ${\sf msg^{*},\sigma^{*},pk_{sign},pk*_{sanit}}$ ${\sf y}$ ${\sf msg^{\prime}},\sigma^{\prime})\leftarrow$ $\sigma^{\prime}_{2}=\mathcal{Y}^{-1}(\beta_{2})$ $\sigma_{1}=\mathcal{T}^{-1}(\beta_{0})$ $({\sf msg}_{j,0},{\sf MODIFY}_{j,0},{\sf AD}_{j})\equiv({\sf msg}_{j,1},{\sf
+MODIFY%
+}_{j,1},{\sf AD}_{j})$ ${\sf pk_{sign}},{\sf msg^{*}})\neq({\sf pk}_{{\sf sign},j},{\sf msg^{\prime}}_%
+{j})$ ${\sf AD(MODIFY)}=\begin{cases}1&\mbox{ if modifications are valid \newline
+with respect }\\
+&\mbox{ to }{\sf AD};\\
+0&\text{ otherwise.}\end{cases}$ $\sigma=(\sigma_{1},\sigma_{2},{\sf AD})$ $(0,{\sf msg_{fix},AD,pk_{sanit}})$ $\alpha_{0}=\mathcal{S}^{-1}({\sf msg_{0}}),\beta_{0}=\mathcal{F}^{-1}(\alpha_{%
+0})$ $\mathcal{B}^{\prime}s$ ${\sf EXP\mbox{-}Privacy_{\mathcal{G}}^{SSS}}$ ${\sf pk_{sanit}^{*}},{\sf msg^{*}})\neq({\sf pk}_{{\sf sanit},i},{\sf msg^{%
+\prime}}_{j})$ ${\sf AD}_{i}={\sf AD^{*}}$ ${\sf(pk_{sign},sk_{sign})}\leftarrow$ ${\sf x\leftarrow Sig(y,sec_{k})}$ $({\sf msg}_{i},{\sf AD}_{i},{\sf pk}_{{\sf sanit},i})$ $\mathcal{P}{\sf(\sigma_{1})\stackrel{{\scriptstyle?}}{{=}}msg_{0}}$ $\mathcal{G}_{\sf signer}$ ${\sf AD}$ ${\sf pk_{sign}}$ $\{\mathcal{S},\mathcal{F},\mathcal{T}\}$ ${\sf FIXED_{AD}(msg^{\prime})}\neq{\sf FIXED_{AD}(msg)}$ $\sigma_{2}=\mathcal{T}^{-1}(\beta_{1})$ $n^{2}+m^{2}+C$ ${\sf msg_{1}}$ $({\sf msg^{*},\sigma^{*},pk^{*}_{sanit}})$ $\alpha_{2}=\mathcal{Q}^{-1}({\sf msg_{2}}),\beta_{2}=\mathcal{X}^{-1}(\alpha_{%
+2})$ $({\sf msg^{\prime}_{j}},\sigma^{\prime}_{j})$ ${\sf pk*_{sanit}},{\sf msg^{*}})\neq({\sf pk}_{{\sf sanit},i},{\sf msg}_{i})$ $i\in\{1,\ldots,\Delta\}$ $({\sf pk^{*}_{sanit},{\sf msg^{*}},\sigma^{*}})\leftarrow\mathcal{G}({\sf pk_{%
+sign}})$ ${\sf MODIFY}({\sf msg}_{i})$ ${\sf San}$ $\mathcal{F}:\mathbb{F}_{q}^{n}\rightarrow\mathbb{F}_{q}^{m}$ $({\sf sk_{sanit},pk_{sanit}})$ $(\sf x,y)$ ${\sf x}$ ${\sf x}=\mathcal{S}_{2}^{-1}({\sf x_{1}})\in\mathbb{F}_{q}^{n}$ $\mathcal{P}(\sigma_{2})\stackrel{{\scriptstyle?}}{{=}}{\sf msg_{1}}$ ${\sf msg,{\sf MODIFY},\sigma,pk_{sign},sk_{sanit}}$ $j\in\{\Delta+1,\ldots,r\}$ $\sigma=(\sigma_{i,1},\sigma_{i,2},{\sf AD}_{i})$ $\mathcal{P}=\mathcal{S}\circ\mathcal{F}\circ\mathcal{T}:\mathbb{F}_{q}^{n}%
+\rightarrow\mathbb{F}_{q}^{m}$ $\hat{p}_{i}\in\mathbb{R}^{T}$ $\mathcal{L}=\sum_{i}\ell_{\textrm{FL}}\left(\hat{p}_{i},g_{i}\right)$ $\mathbf{A}\in\mathbb{B}^{L\times T}$ $g_{i}\in\mathbb{B}^{T}$ $\mathcal{C}\setminus c$ $\{t^{\prime}_{j}\}$ $\hat{\textbf{p}}$ $\textbf{p}_{\textbf{cf}}$ $F_{avg}=70.09.$ $`\backslash w+^{\prime}$ $F_{avg}=50.1$ $PP({\tilde{p}},q)=b^{H({\tilde{p}},q)}=b^{\mathbb{E}_{\tilde{p}}[log_{b}q]}$ $\begin{split}\arg\max_{i}\frac{P(y_{i}|x)}{P(y_{i}|x_{cf})}\approx\arg\max_{i}%
+\frac{P(y_{i}|x)}{P(y_{i}|x_{cf})}.\end{split}$ $P(y_{i})\text{ or }p_{cf}$ $(W\hat{p})_{i}=\frac{\hat{p}_{i}}{(p_{cf})_{i}}=\frac{P(y_{i}|x)}{P(y_{i}|x_{%
+cf})}$ $F_{avg}$ $F_{avg}=\frac{F_{\text{favor}}+F_{\text{against}}}{2}$ $\text{argmax}(\textbf{q})$ $\arg\max_{i}\prod_{j=1}^{i}\log P(y_{j}\mid x,y_{1,\cdots,j-1})$ $P(y_{i}|x_{cf})$ $\frac{P(y_{i}|x)}{P(y_{i})}$ $H({\tilde{p}},q)=-\sum_{i=1}^{n}\tilde{p}(x_{i})\log_{b}q(x_{i})$ $F_{avg}=72.6$ $F_{avg}=82.29$ $F_{avg}=84.43$ $\langle tweet\rangle$ $F_{avg}=71.0$ $F_{avg}=84.28$ $F=\frac{2\cdot Precision\cdot Recall}{Precision+Recall}$ $\text{PMI}(x,y_{i})$ $\textbf{W}=\text{diag}(\textbf{p}_{\textbf{cf}})^{-1}$ $77.11$ $78.43$ $\textbf{q}=softmax(\textbf{W}\hat{\textbf{p}}+\textbf{b})$ $t+5\Delta$ $\mathcal{WJ}$ $B^{\prime}_{k}$ $\mathcal{C}_{v^{\prime\prime}}(B_{k^{\prime\prime}})$ $\mathsf{status}$ $v^{\prime}-1\geq v+1$ $=\delta$ $\max(t_{g},t)+2\Delta$ $n-2f^{\prime}$ $\mathcal{C}_{v-1}(B_{k-1})$ $\mathcal{T}_{v-1}$ $\langle\mathsf{fb\text{-}vote},H(B_{h}),v\rangle_{i}$ $v^{*}>v$ $\mathcal{C}_{v}^{\prime}(B_{k^{\prime}})$ $v^{\prime}>v+1$ $55.7\%-69.0\%$ $\mathcal{C}_{v}(B_{k})$ $f^{\prime}=33$ $\mathcal{C}_{v^{\prime}-1}(B_{k^{\prime}-1})$ $\langle\mathsf{status},v,{\sf lock}_{i}\rangle$ $\langle{\sf opt\text{-}propose},B_{k+1},v+1\rangle$ $\mathcal{C}_{v}=\mathcal{C}_{v}(B_{k})$ $B_{k^{\prime}}=B_{k+1}$ $t\geq t_{g}$ $\mathcal{T}_{v^{\prime\prime}-1}$ $\langle\mathsf{propose},B_{k^{\prime}},\mathcal{C}_{v^{\prime\prime}}(B_{h}),v%
+^{\prime}\rangle$ $675\%$ $\mathcal{C}^{f}_{v}(B_{l})$ $\mathcal{TC}_{v}$ $\mathcal{C}_{v^{\prime}-1}$ $\Delta=500ms$ $f^{\prime}=0$ $\mathcal{C}_{v^{\prime\prime}-1}$ ${\sf view\text{-}timer}$ $v^{\prime}=v+1$ $=3\Delta$ $\langle\mathsf{commit},H(B_{k}),v\rangle_{i}$ $\langle\mathsf{timeout},v-1,{\sf lock}_{i}\rangle_{i}$ $\mathcal{C}^{n}_{v}(B_{l})$ $\mathcal{C}^{f}_{v}(B_{h})$ $5f-1$ $448\%$ $v^{*}\geq v+1$ $v\geq v^{\prime}$ $\langle{\sf fb\text{-}propose},B_{k},\mathcal{C}_{v^{\prime}}(B_{h}),\mathcal{%
+TC}_{v-1},v\rangle$ $t $\mathcal{C}_{v^{\prime\prime}}(B_{h})\geq{\sf lock}_{i}$ $n-3f^{\prime}$ $2f^{\prime}$ ${\sf timeout\_view}_{i} $\geq 2\delta$ $\langle\mathsf{propose},B_{k^{\prime}},\mathcal{C}_{v^{\prime\prime}}(B_{k^{%
+\prime\prime}}),v^{\prime}\rangle$ $174\%$ $\mathsf{fb\text{-}vote}$ $\langle\mathsf{propose},B_{h+1},\mathcal{C}_{v-1}(B_{h}),v\rangle$ $=(f+1)\delta$ $\mathcal{C}_{v^{\prime\prime}}(B_{k^{\prime\prime}})\geq{\sf lock}_{i}$ $=\Lambda+2\rho$ ${\sf lock}_{i}>\mathcal{C}_{v-1}(B_{h})$ $\mathcal{C}_{v+1}(B_{k^{\prime}})$ $\mathcal{WM}$ $\langle\mathsf{vote},H(B_{k}),v\rangle_{i}$ $\langle\mathsf{opt\text{-}vote},H(B_{k}),v\rangle_{i}$ $B^{\prime}_{k^{\prime}}$ $P_{j}\neq P_{i}$ ${\sf view\text{-}timer}_{i}$ $v^{\prime}+1>v$ $B_{l}=B_{k}$ $t+\lambda$ $,P_{i}$ $C_{v}(B_{k})$ ${\sf timeout\_view}_{i}\geq v$ $\mathcal{C}_{v}(B_{h})$ $\mathcal{C}_{v}(B_{l})$ $=2\Lambda+\rho$ $L_{v+1}$ $\langle{\sf opt\text{-}propose},B_{k},v\rangle$ $B_{k^{\prime\prime}}$ $\mathcal{TC}_{v+1}$ $43\%\text{--}54\%$ $214\%-230\%$ $(f+1)\delta$ ${\sf lock}_{i}<\mathcal{C}_{v}(B_{k})$ $\mathsf{opt\text{-}vote}$ $\mathcal{C}_{v+1}$ $\mathcal{C}_{v^{\prime}}(B_{l})$ $\lx@sectionsign\ref{sec:commit-moonshot}$ $t_{g}+2\Delta<\max(t_{g},t)+3\Delta$ $\mathcal{T}_{v^{*}}$ $B_{k^{\prime}}$ $7\delta^{*}$ $702\%$ $\langle\mathsf{propose},B_{h},\mathcal{C}_{v^{\prime}}(B_{h-1}),v\rangle$ ${\sf lock}_{i}$ $\mathsf{timeout}_{v}$ $\mathcal{TC}_{v-1}$ $\mathcal{T}_{v^{\prime}}$ $\mathcal{C}_{v}\leq\mathcal{C}_{v^{\prime}}$ $B_{k}\neq B^{\prime}_{k}$ $\max(t_{g},t)+3\Delta$ $\langle\mathsf{propose},B_{l},\mathcal{C}_{v^{\prime}}(B_{h}),v+1\rangle$ $\mathcal{C}_{v+1}(B_{k+1})$ $\mathcal{C}_{v^{\prime}}(B_{h})=\mathcal{C}_{v}(B_{k})$ $\mathsf{vote}$ $t+4\Delta$ $\lceil\frac{n+f+1}{2}\rceil$ $t_{g}+3\Delta=\max(t_{g},t)+3\Delta$ $\langle\mathsf{commit},H(B_{k}),v\rangle_{*}$ $\mathcal{C}_{v-1}(B_{h})$ $\mathcal{T}_{v^{\prime}-1}$ $\delta\leq\Delta$ $\rho<\Lambda$ $v^{\prime\prime}\geq v$ $\mathcal{C}^{n}_{v+1}(B_{k^{\prime}})$ $\mathcal{C}^{o}_{v}(B_{h})$ $B_{k^{\prime}}=B_{k}$ $\mathcal{C}_{v^{*}}(B_{k^{*}})$ $p\leq 1.8$ $\mathcal{T}_{v^{\prime\prime}}$ $\langle{\sf fb\text{-}propose},B_{h},\mathcal{C}_{v^{\prime}}(B_{h-1}),%
+\mathcal{TC}_{v-1},v\rangle$ $\mathcal{C}_{v^{\prime\prime}}$ $\langle\mathsf{timeout},v,{\sf lock}_{i}\rangle_{i}$ $3f^{\prime}$ $v\leq v^{\prime\prime} ${\sf timeout\_view} $v^{\prime\prime} $\langle\mathsf{timeout},v\rangle_{i}$ $\lx@sectionsign\ref{sec:moonshot_v2}$ ${\color[rgb]{0.0078125,0.38671875,0.46875}\definecolor[named]{pgfstrokecolor}{%
+rgb}{0.0078125,0.38671875,0.46875}\pgfsys@color@rgb@stroke{0.0078125}{0.386718%
+75}{0.46875}\pgfsys@color@rgb@fill{0.0078125}{0.38671875}{0.46875}\checkmark}$ $\langle{\sf fb\text{-}propose},B_{k^{\prime}},\mathcal{C}_{v^{\prime\prime}}(B%
+_{k^{\prime\prime}}),\mathcal{TC}_{v^{\prime}-1},v^{\prime}\rangle$ $\mathcal{C}_{v-1}(B_{h-1})$ $\mathcal{C}_{v^{\prime}}(B_{h})$ $C_{v-1}$ $\lx@sectionsign\ref{sec:moonshot_v1}$ $2f+1th$ $\max(t_{g},t)+\Delta$ $70.3\%-74.4\%$ $\langle\mathsf{propose},B_{k},\mathcal{C}_{v-1}(B_{k-1}),v\rangle$ $\mathcal{TC}_{v^{\prime\prime}-1}$ $v\leq v^{*} $\mathcal{C}_{v-1}$ ${\sf lock}_{i}\leq\mathcal{C}_{v^{\prime}}(B_{h})$ $\mathcal{C}^{o}_{v}(B_{k})$ $\langle\mathsf{propose},B_{l},\mathcal{C}_{v^{\prime}}(B_{h}),v\rangle$ $t_{g}+\Delta$ $p\leq 9$ $7\Delta$ $=3\delta$ $\geq 4\delta$ $\mathcal{C}^{n}_{v}(B_{h})$ $H(B_{k-1})$ $v^{\prime\prime}>v$ $\mathcal{C}_{v^{\prime}}(B_{h-1})$ $\langle\mathsf{propose},B_{h+1},\mathcal{C}_{v^{\prime}}(B_{h}),v\rangle$ $70.5\%-71.8\%$ $B_{k^{\prime}-1}$ $v^{\prime}+1$ $51\%\text{--}53\%$ $\mathsf{timeout}_{v^{*}}$ $B_{k}=B_{h+1}$ $B_{k}=B_{l}$ $\mathcal{C}^{o}_{v+1}(B_{k^{\prime}})$ $P_{i}\in\mathcal{V}$ $B_{k}\neq B_{l}$ $f^{\prime}\leq f=\lfloor\frac{n-1}{3}\rfloor$ $v^{\prime\prime}+1$ $\mathcal{V}=($ $B_{k}:=(b_{v},H(B_{k-1}))$ $=4\delta$ $v^{\prime}-1$ $5\Delta$ $\mathcal{C}_{v+1}(B_{l})$ $t+3\Delta$ $\mathcal{TC}_{v^{\prime}-1}$ $B_{k}=B^{\prime}_{k}$ $\mathcal{C}_{v^{\prime\prime}}(B_{h})$ $\mathcal{TC}_{v^{\prime}}$ $\mathcal{C}_{v}(B_{h+1})$ $\mathcal{C}^{n}_{v}(B_{k})$ $\mathcal{C}_{v^{\prime\prime}}(B_{k^{\prime\prime}})\geq\mathcal{C}_{v}(B_{k})$ $B_{h+1}$ $\mathcal{C}_{v^{\prime}}(B_{k^{\prime}})$ $=2\delta$ $MaxStepsBacktracking=NumberOfEmptyCells$ $\textit{Success Rate}\boldsymbol{\propto}\textit{max steps}*\textit{$\frac{1}{%
+emptycells}$}$ $s+r\approx o$ $\mathbf{I}_{t}^{s,r}\in\mathbb{Z}^{|\mathcal{E}|\times|\mathcal{R}|\times|%
+\mathcal{E}|}$ $\textbf{W}^{(l)}_{6}$ $\textbf{t}_{i}^{\prime\prime}=\sigma(\sum_{r\in\mathcal{R}_{TG}}\frac{1}{%
+\mathcal{N}_{i}}\textbf{W}_{r}\textbf{t}_{j}^{\prime}+\textbf{W}_{11}\textbf{t%
+}_{i}^{\prime})\textnormal{,}$ $y^{e}_{t}$ $\textbf{W}_{r}\in\mathbb{R}^{d\times d}$ $\textbf{W}_{9}\in\mathbb{R}^{d\times 32}$ $\alpha_{o,s}$ $\textbf{o}^{(l+1)}_{t}=\sigma(\sum_{(s,r,o)\in\mathcal{F}_{t}}\textbf{W}^{(l)}%
+_{1}(\psi(\textbf{s}^{(l)}_{t}||\textbf{r}_{t}))+\textbf{W}^{(l)}_{2}\textbf{o%
+}^{(l)}_{t})\textnormal{,}$ $\textbf{UE}_{t-k:t-1}$ $\textbf{W}_{11}\in\mathbb{R}^{d\times d}$ $f_{q}\in\{G_{t-k:t-1}\}$ $\textbf{R}_{t}=\mathrm{GRU}(\textbf{R}_{t-1},[pooling(\textbf{E}^{\mathcal{R}}%
+_{t-1})||\textbf{R}])\textnormal{,}$ $(s,r,o,t)$ $\mathcal{R}_{TG}$ $\textbf{E}^{\mathcal{R}}$ $t>t_{T}$ $\alpha^{(l)}_{o,s}=\frac{exp(\textbf{W}^{(l)}_{3}\sigma(\textbf{W}^{(l)}_{4}[%
+\textbf{s}^{(l)}||\textbf{r}||\textbf{o}^{(l)}||\textbf{t}^{\prime\prime}]))}{%
+\sum_{s^{\prime}\in\mathcal{N}_{(o)}}exp(\textbf{W}^{(l)}_{3}\sigma(\textbf{W}%
+^{(l)}_{4}[\textbf{s'}^{(l)}||\textbf{r}||\textbf{o}^{(l)}||\textbf{t}^{\prime%
+\prime}]))}\textnormal{,}$ $q=(s,r,?,t)$ $p_{R}(o|s,r,t,G_{t-1})=softmax(\mathbf{ConvTransE}(\textbf{s},\textbf{r},%
+\textbf{t}^{\prime\prime})\textbf{GE}^{\top}_{t})\textnormal{.}$ $\textbf{R}_{t}$ $\textbf{t}^{\prime}=\textbf{W}_{8}(\textbf{W}_{9}\textbf{t}||\sigma(\textbf{W}%
+_{10}\textbf{t}))\textnormal{,}$ $\mathbf{W}^{(l)}_{1}$ $\displaystyle=\beta\sum_{(s,r,t)\in\mathcal{Q}^{e}_{t}}y^{e}_{t}\log p(o|s,r,t%
+,G_{t-1})$ $\textbf{W}_{10}\in\mathbb{R}^{d\times 32}$ $\textbf{s},\textbf{o}\in\textbf{UE}_{t-k:t-1}$ $\textbf{W}^{(l)}_{3}\in\mathbb{R}^{4d}$ $p_{H}(o|s,r,t,G_{t-1})=softmax(\mathbf{ConvTransE}(\textbf{s},\textbf{r},%
+\textbf{t}^{\prime\prime})\textbf{GE}^{\top}_{t}\odot\mathbf{I}_{t}^{s,r})%
+\textnormal{,}$ $\mathcal{Q}^{r}_{t}$ $\textbf{UE}_{t}$ $\Theta\in\mathbb{R}^{|\mathcal{E}|\times d}$ $\{\textbf{E}_{t-k+1},\textbf{E}_{t-k+2},...,\textbf{E}_{t}\}$ $\mathcal{N}_{(o)}$ $y^{r}_{t}$ $G=\{\mathcal{E},\mathcal{R},\mathcal{F},\mathcal{T}\}$ $TG=\{\mathcal{E}_{TG},\mathcal{R}_{TG}\}$ $\textbf{W}_{8}\in\mathbb{R}^{d\times 2d}$ $\mathcal{Q}^{e}_{t}$ $\textbf{W}^{(l)}_{4}\in\mathbb{R}^{4d\times 4d}$ $\mathsf{LMS}$ $\textbf{W}_{7}\in\mathbb{R}^{1\times d}$ $\textbf{UE}_{t-k:t-1}=\mathrm{MEAN}(\sum_{i=t-k+1}^{t}\textbf{E}_{i})%
+\textnormal{.}$ $\displaystyle+(1-\beta)\sum_{(s,o,t)\in\mathcal{Q}^{r}_{t}}y^{r}_{t}\log p(r|s%
+,o,t,G_{t-1})\textnormal{,}$ $\textbf{E}_{t}=\mathrm{GRU}(\textbf{E}_{t-1},\textbf{G}_{t-1})\textnormal{.}$ $\textbf{W}^{(l)}_{5}$ $\mathbf{I}_{t}^{s,r}$ $p(o|s,r,t,G_{t-1})=\alpha p_{H}(o|s,r,t,G_{t-1})+(1-\alpha)p_{R}(o|s,r,t,G_{t-%
+1})\textnormal{.}$ $\mathbf{W}^{(l)}_{2}$ $\textbf{o}^{(l+1)}=\sigma(\sum_{(s,r,o)\in UG}\alpha^{(l)}_{o,s}\textbf{W}^{(l%
+)}_{5}\psi(\textbf{s}^{(l)}||\textbf{r})+\textbf{W}^{(l)}_{6}\textbf{o}^{(l)})%
+\textnormal{,}$ $\textbf{GE}_{t}=\sigma(\textbf{W}_{7}\Theta_{e})\textbf{E}_{t}+(1-\sigma(%
+\textbf{W}_{7}\Theta_{e}))\textbf{UE}_{t}\textnormal{,}$ $\mathcal{E}_{TG}$ $G=\{G_{1},G_{2},...,G_{\mathcal{T}}\}$ $p(o|s,r,t,G_{t-1})$ $1-A_{h}$ ${\mathcal{X}_{1}}\in\mathbb{R}^{{1024}\times{{3}}}$ $P=\{{\mathcal{P}_{1}}\in\mathbb{R}^{{2}\times{{N}\times{C}}},{\mathcal{P}_{2}}%
+\in\mathbb{R}^{{2}\times{{N_{1}}}\times{C_{1}}}$ ${\mathcal{X}_{3}}\in\mathbb{R}^{{128}\times{{3}}}$ $\displaystyle+||(\Pi(\mathcal{J}(\mathcal{M}^{h}_{MANO}))-\Pi(\mathcal{J}(\hat%
+{\mathcal{M}}^{h}_{MANO})))||_{2}.$ $X_{h}=\{X_{l}\in\mathbb{R}^{{N}\times{C}},X_{r}\in\mathbb{R}^{{N}\times{C}}\}$ $i $\mathcal{\hat{P}}=\mathcal{P}\odot\bm{\alpha}+\bm{\beta},(\bm{\alpha},\bm{%
+\beta})=\psi(\mathcal{F}).$ $\mathcal{L}_{V}=\sum_{h\in\{L,R\}}||\mathcal{M}^{h}_{GCN}-\hat{\mathcal{M}}^{h%
+}_{GCN}||_{1}+||\mathcal{M}^{h}_{MANO}-\hat{\mathcal{M}}^{h}_{MANO}||_{1}.$ ${\mathcal{F}_{3}}\in\mathbb{R}^{{\frac{H}{4}}\times{\frac{W}{4}}\times{256}}\}$ ${G_{out}}=\sum_{k=0}^{K-1}C_{k}(\hat{L})G_{in}W_{k}.$ $M=\{M_{l}\in\mathbb{R}^{{H}\times{{W}}},M_{r}\in\mathbb{R}^{{H}\times{{W}}}\}$ $\hat{P_{i}},NumPoints\_{i},BallRadius\_{i}$ $\mathcal{I}_{c}\in\mathbb{R}^{{H}\times{W}\times{3}}$ $num\_layers$ $\psi_{i}(\hat{F_{i}},P_{i})$ $\displaystyle\mathcal{L}_{rep}$ $cat(P_{i},PointNet(group(S_{i})))$ $BallRadius$ $(G\odot(\alpha+1)+\beta)$ $NumPoints$ $(P_{i}\odot(\alpha+1)+\beta)$ $\hat{P_{i}}$ $A_{h}\in[0,1]$ $F=\{{\mathcal{F}_{1}}\in\mathbb{R}^{{H}\times{{W}\times{3}}},{\mathcal{F}_{2}}%
+\in\mathbb{R}^{{\frac{H}{2}}\times{\frac{W}{2}}\times{64}}$ $P_{ct}=\{P_{l}\in\mathbb{R}^{2},P_{r}\in\mathbb{R}^{2}\}$ $\psi_{i+1}(C,G)$ $\mathcal{G}\in\mathbb{R}^{{2}\times{1024}\times{1}}$ $\mathcal{I}_{d}\in\mathbb{R}^{{H}\times{W}\times{1}}$ $Fetch(F_{i}|u,v)$ $\mathcal{L}_{J}=\sum_{h\in\{L,R\}}||\mathcal{J}(\mathcal{M}^{h}_{MANO})-%
+\mathcal{J}(\hat{\mathcal{M}}^{h}_{MANO})||_{1}.$ $K^{-1}X_{i}$ $\mathcal{L}_{m}=||M-\hat{M}||_{1},$ $[1,num\_layers]$ $W_{k}\in\mathbb{R}^{{C_{in}}\times{C_{out}}}$ $G_{in}\in\mathbb{R}^{{N}\times{C_{in}}}$ $PointNet(\hat{P_{i}})$ $\hat{L}\in\mathbb{R}^{{N}\times{N}}$ $\mathcal{L}_{root}=\sum_{h\in\{L,R\}}||Root^{h}-\hat{Root^{h}}||_{1}.$ $\mathcal{L}_{smooth}=\sum_{i=1}^{3}||e_{i}\cdot\hat{n}||_{1}+||e-\hat{e}||_{1},$ ${\mathcal{P}_{3}}\in\mathbb{R}^{{2}\times{{N_{2}}}\times{C_{2}}}\}$ $\mathcal{G_{V}}\in\mathbb{R}^{{N}\times{C}},(N=63,126,252),(C=512,256,128)$ $\mathcal{L}_{c}=\sum_{h\in\{L,R\}}(1-A_{h})^{\gamma}\log(A_{h}),$ ${\mathcal{X}_{2}}\in\mathbb{R}^{{512}\times{{3}}}$ $G_{out}\in\mathbb{R}^{{N}\times{C_{out}}}$ $\displaystyle=\sum_{h\in\{L,R\}}||(\Pi(\mathcal{M}^{h}_{MANO})-\Pi(\hat{%
+\mathcal{M}}^{h}_{MANO}))||_{2}$ $F(t)=\frac{(1-t)(1-2t)-\sqrt{(1-t)(1-5t)}}{2t(2-t)}.$ $\delta(R)$ $F(t)=F(t,1,1,1,1)=t^{5}C(t)^{4}$ $v_{n}\sim 4^{n+2}/\sqrt{\pi}n^{-3/2}$ $W(t)=\frac{1}{1-P(t)/t}$ $-(5t^{6}-16t^{5}+15t^{4}-28t^{3}+23t^{2}-8t+1)(t-1)^{2}F-(2t^{5}-5t^{4}+4t^{3}%
+-10t^{2}+6t-1)(t-1)^{2}=0$ $(2143,3\underline{41}2)$ $F(t,x_{1},x_{2},x_{3},x_{4})$ $F(t)=\frac{t(1-t)(1-7t+16t^{2}-11t^{3}+2t^{4})}{(1-4t+2t^{2})(1-3t+t^{2})^{2}}.$ $\mathrm{mod}\ 4$ $2\underline{14}3,3\underline{41}2$ $P_{1}=$ $(3,2,1,2)$ $\left.{D=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}\frac{1}{1-\frac{tA}{1-t}}%
+-1\right)A}\right\}$ $e_{m}:=[t^{m}]\prod_{i=1}^{4}(1+tx_{i})$ $F(t,x_{1},x_{2},x_{3},x_{4})=t^{5}\frac{1}{2\alpha}\left(\beta-\sqrt{\beta^{2}%
+-4\alpha e_{4}^{2}}\right)$ $(n+4)v_{n}-6(n+2)v_{n-1}+4(2n-1)v_{n-2}=0$ $\left.\left(\frac{1}{(1-t)^{2}}-1\right)D,\ \ {D=\frac{t^{2}}{1-t}+\left(\frac%
+{1}{(1-t)^{2}}-1\right)A+\frac{S_{A}^{2}}{(1-t)^{2}}}\right\}$ $\alpha F^{2}-\beta F+e_{2}^{4}$ $\left.\frac{t^{2}}{1-t}+\frac{tS_{A}}{1-t}\right\}$ $\left.{S_{D}=\frac{t^{2}}{1-t}+\frac{tS_{A}}{1-t}}\right\}$ $V(t)=tC^{2}(t)(1+t^{2}C^{4}(t))$ $(1,2,3,1)$ ${A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}-1\right)A}$ $(45312,213)$ $F(t)=\frac{1-3t-t^{2}+2t^{3}-\sqrt{1-6t+7t^{2}+2t^{3}+t^{4}}}{2t^{2}(2-t)}.$ $\left\{S_{A}=\frac{t^{2}}{1-t}+\frac{tS_{D}}{1-t},\ \ S_{D}=\frac{(t+S_{A})^{2%
+}}{1-(t+S_{A})}\right\}$ $O(\sqrt{n}\ln n)$ $F(t)=\frac{1-3t+t^{2}-\sqrt{1-6t+7t^{2}-2t^{3}+t^{4}}}{2t}.$ $\left\{{A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}-1\right)D},\right.$ $F(t)=\frac{1-t-\sqrt{1-6t+t^{2}}}{2}$ $\alpha:=\prod_{i=1}^{4}(1-x_{i}+tx_{i}^{2})$ $P_{4}=\leavevmode\hbox to15.51pt{\vbox to15.51pt{\pgfpicture\makeatletter%
+\raise-3.91434pt\hbox{\hskip 0.5pt\lower-0.5pt\hbox to 0.0pt{%
+\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
+\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
+{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
+to%
+ 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
+{}{{}}{}
+{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.0pt}%
+\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{7.25558pt}\pgfsys@lineto{14.51118pt}{%
+7.25558pt}\pgfsys@stroke\pgfsys@invoke{ }
+\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
+{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.0pt}%
+\pgfsys@invoke{ }{}\pgfsys@moveto{10.15784pt}{7.25558pt}\pgfsys@lineto{10.1578%
+4pt}{14.51118pt}\pgfsys@stroke\pgfsys@invoke{ }
+\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
+{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.0pt}%
+\pgfsys@invoke{ }{}\pgfsys@moveto{4.35333pt}{0.0pt}\pgfsys@lineto{4.35333pt}{7%
+.25558pt}\pgfsys@stroke\pgfsys@invoke{ }
+\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
+\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
+\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}%
+\lxSVG@closescope\endpgfpicture}}$ $P_{4}=$ $P_{3}=$ $2143,3\underline{41}2$ $\left\{{S_{A}=\frac{t^{2}}{1-t}+\left(\frac{1}{1-\frac{tS_{D}}{1-t}}\frac{1}{(%
+1-t)^{2}}-1\right)S_{D}},\right.$ $t^{8}(t-2)^{2}F^{4}-t^{3}(t^{2}-3t+2)(t^{5}-7t^{4}+4t^{3}-6t^{2}+5t-1)F^{3}-t(%
+t-1)(4t^{7}-22t^{6}+37t^{5}-42t^{4}+53t^{3}-35t^{2}+10t-1)F^{2}$ $\left\{{A=\frac{(t+D)^{2}}{1-(t+D)}},\ \ {D=\frac{(t+A)^{2}}{1-(t+A)}}\right\}$ $532642x_{1}=2x_{1}\times 11^{2}\times 31\times 71$ $(2143,3412)$ $\beta:=(2e_{4}t^{2}-4t(e_{4}-3e_{3}+2e_{2})+e_{4}-e_{3}+e_{2}-e_{1}+2)e_{4}$ $\left\{{S_{A}=\frac{t^{2}}{1-t}+\frac{S_{D}}{1-t}},\ \ S_{D}=\frac{t^{2}}{1-t}%
++\right.$ $tF^{3}+2tF^{2}+(2t-1)F+t=0.$ $\!,$ $\left\{A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}\frac{1}{1-\frac{tD}{1-t}}%
+-1\right)D,\ \ D=\frac{(t+A)^{2}}{1-(t+A)}\right\}$ $C(t)=\frac{1-\sqrt{1-4t}}{2t}$ $2x_{1}x_{2}^{2}x_{3}x_{4}$ $2413,3142)$ $Z(t)=\frac{t(1-2t)}{1-4t+2t^{2}}$ $2\underline{14}3,45312$ $\left\{{A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}-1\right)D},\ \ {D=\frac{%
+(t+A)^{2}}{1-(t+A)}}\right\}$ $(2413,3142)$ $(2\underline{14}3,231)$ $P_{3}=\leavevmode\hbox to15.51pt{\vbox to15.51pt{\pgfpicture\makeatletter%
+\raise-3.91434pt\hbox{\hskip 0.5pt\lower-0.5pt\hbox to 0.0pt{%
+\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
+\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
+{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
+to%
+ 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
+{}{{}}{}
+{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.0pt}%
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+4.51118pt}\pgfsys@stroke\pgfsys@invoke{ }
+\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
+{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.0pt}%
+\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{4.35333pt}\pgfsys@lineto{7.25558pt}{4%
+.35333pt}\pgfsys@stroke\pgfsys@invoke{ }
+\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
+{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.0pt}%
+\pgfsys@invoke{ }{}\pgfsys@moveto{7.25558pt}{10.15784pt}\pgfsys@lineto{14.5111%
+8pt}{10.15784pt}\pgfsys@stroke\pgfsys@invoke{ }
+\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
+\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
+\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}%
+\lxSVG@closescope\endpgfpicture}}$ $\delta(\mathcal{R})$ $(2143,45312)$ $d(f(t),g(t))=2^{-\operatorname{val}(f(t)-g(t))}$ $P(t)/t$ ${A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}-1\right)A+\frac{S_{D}^{2}}{(1-t%
+)^{2}}}$ $F(t)=\frac{t(1-2t)}{1-4t+2t^{2}}.$ $(2143,231)$ $\!\}.$ $[1..b]$ $2143,3412$ $F(t)=t+A(t)+D(t)$ $21354,45312$ $2143$ $\operatorname{val}(f(t))=+\infty$ $V(t)=W(t)Z(t)=\frac{(1-2t)\big{(}\,1-4t+2t^{2}+(1-2t)\sqrt{1-4t}\,\big{)}}{2t^%
+{3}}=tC^{2}(t)(1+t^{2}C^{4}(t)),$ $\left\{{S_{A}=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}-1\right)S_{D}},\ \ S%
+_{D}=\right.$ $A(t)=D(t)$ $\left\{{A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}-1\right)D+\frac{S_{D}^{2%
+}}{(1-t)^{2}}},\ \ {D=\frac{(t+A)^{2}}{1-(t+A)}}\right\}$ $\{\!$ $f(t)=\sum_{n\geq 0}f_{n}t^{n}$ $(2\underline{14}3,3\underline{41}2)$ $a_{i+1}>1$ $P_{2}=$ $(2\underline{14}3,45312)$ $P(t)=t^{5}C^{4}(t)\left(\frac{2}{1-\left(\frac{1}{(1-t)^{2}}-1\right)}-1\right).$ $2143,45312$ $t^{4}(t-2)^{2}F^{4}+t(t-2)(4t^{3}-7t^{2}+6t-1)F^{3}+(2t^{4}-t^{3}-2t^{2}+5t-1)%
+F^{2}-(4t^{3}-7t^{2}+6t-1)F+t^{2}=0$ $2\underline{14}3$ $F(t)=\frac{1-t-\sqrt{1-6t+t^{2}}}{2}.$ $\left.\left(\frac{1}{(1-t)^{2}}-1\right)S_{A}+\left(\frac{1}{1-t}\frac{t^{2}}{%
+1-2t}\right)^{2}\right\}$ $\left\{{A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}\frac{1}{1-\frac{tD}{1-t}%
+}-1\right)D},\ \ {D=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}-1\right)A+%
+\frac{S_{A}^{2}}{(1-t)^{2}}}\right\}$ $1,2,\ldots,b$ $F(t,x_{1},11,31,71)$ $(21354,45312)$ ${A=\frac{t^{2}}{1-t}+\left(\frac{1}{(1-t)^{2}}\frac{1}{1-\frac{tA}{1-t}}-1%
+\right)A}$ $f_{n}\neq 0$ $\left\{A=\frac{t^{2}}{1-t}+\right.$ $(1,b,1,1)$ $A=\frac{(t+A)^{2}}{1-(t+A)}$ $F(t)=\frac{t(1-16t+11t^{2}-434t^{3}+1045t^{4}-1590t^{5}+1508t^{6}-846t^{7}+252%
+t^{8}-30t^{9})}{(1-2t)^{4}(1-3t+t^{2})^{2}(1-4t+2t^{2})}.$ $t^{5}C^{4}(t)$ $\begin{array}[]{lcccl}\text{1.}\ (a,\ b,\ 1,\ 1)\ \longrightarrow\ (1,\ b+1,\ %
+1,\ 1),&&&&\text{3.}\ (1,\ b,\ c,\ d)\ \longrightarrow\ (1,\ b,\ [1..c],\ d+1)%
+,\\
+\text{2.}\ (1,\ b,\ c,\ 1)\ \longrightarrow\ (1,\ [1..b],\ c+1,\ 1),&&&&\text{%
+4.}\ (a,\ b,\ c,\ d)\ \longrightarrow\ (a+1,\ b,\ c,\ [1..d]).\end{array}$ $\displaystyle\begin{split}&F(t,x_{1},x_{2},x_{3},x_{4})=t^{5}x_{1}x_{2}x_{3}x_%
+{4}+tx_{1}x_{2}x_{3}x_{4}[x_{3}x_{4}]F(t,1,x_{2},x_{3},x_{4})\\
+&+tx_{1}x_{3}x_{4}\frac{[x_{1}x_{4}]F(t,x_{1},x_{2},x_{3},x_{4})-x_{2}[x_{1}x_%
+{4}]F(t,x_{1},1,x_{3},x_{4})}{x_{2}-1}\\
+&+tx_{1}x_{4}\frac{[x_{1}]F(t,x_{1},x_{2},x_{3},x_{4})-x_{3}[x_{1}]F(t,x_{1},x%
+_{2},1,x_{4})}{x_{3}-1}\\
+&+tx_{1}\frac{F(t,x_{1},x_{2},x_{3},x_{4})-x_{4}F(t,x_{1},x_{2},x_{3},1)}{x_{4%
+}-1}.\end{split}$ $21354$ $(\mathbf{c},\sigma)=F_{\theta}^{c}(\mathbf{\gamma}(\mathbf{x}),\mathbf{\gamma}%
+(\mathbf{d}))$ $\mathbf{f}_{z}\in\mathbb{R}^{4}$ $G\setminus\{a_{1},b_{1}\}$ $y\in V(F)$ $q_{1}=c$ $v\in V(G)\setminus V(H)$ $G[q_{j},\dots,,q_{t},b_{1},a_{1},a_{2}]$ $N_{G}(q_{t})\cap V(H)=B$ $G[S]=H[S]$ $\omega<2.3728596$ $x\in A$ $G_{i}-v_{i}$ $H\in\mathcal{C}$ $\{2K_{1}\vee H\mid H\in\mathcal{M}_{\mathcal{C}_{1}}\}=\{2K_{1}\vee 2K_{1}\}=%
+\{C_{4}\}$ $N_{G}(v)\cap V(H)\subseteq B$ $a_{2},a_{3},b_{1}$ $Q=q_{0},\dots,q_{t}$ $H\cong 2K_{1}$ $\mathcal{G}_{k}$ $V(G)\setminus V(H)$ $E(H_{1})\cup E(H_{2})\cup\{v_{1}v_{2}\mid v_{1}\in V(H_{1}),v_{2}\in V(H_{2})\}$ $G[S^{\prime}]\in\mathcal{C}$ $G_{A}:=G[A\cup C]$ $\bigcup_{v\in V(H)}X_{v}\subseteq V(2K_{1})$ $\mathcal{O}(n^{2}(n+m))$ $K_{2,3}$ $p_{1},\dots,p_{i},q_{j},\dots,q_{t}$ $K_{p,q}$ $\mathcal{F}_{\mathcal{C}}$ $S^{\prime}\subseteq S^{*}$ $G\in\mathcal{C}_{k}$ $\mathcal{O}(n^{4})$ $r_{1},\dots,r_{i},c$ $S=V(G)$ $\mathcal{O}(n^{2k+2})$ $S\subseteq A\cup C$ $S^{\prime}\subseteq S\cap(A\cup C)$ $S\subseteq V(G^{\prime})\setminus\{a,b,v^{xy}\}=V(G)\setminus\{a,b,x,y\}$ $A=\{a_{1},a_{2},a_{3}\}$ $G/e$ $p_{s}=c$ $S\subseteq V(G^{\prime})\setminus\{v^{xy}\}=V(G)\setminus\{x,y\}$ $a_{i},a_{j},a_{k}\in A$ $G[a_{1},a_{2},b_{1},b_{2},p_{0},\dots,p_{s}]$ $x,y\in V(G^{\prime})$ $S\subseteq V(G)\setminus\{a,b\}$ $\sum_{x\in X}w(x)$ $S_{v}=\{v,z^{v}\}$ $\mathcal{O}(n^{2+\epsilon})$ $p_{1},\ldots,p_{k}$ $r_{1}=x$ $K_{2,k+1}\notin\mathcal{G}_{k}$ $p_{1},\dots,p_{i-1},p_{j+1},\dots,p_{s}$ $P^{3}$ $p_{1},\dots,p_{i},q_{j+1},\dots,q_{t}$ $\chi(\overline{G})\leq k$ $r_{i},c\in C$ $G[S^{*}]\in\mathcal{C}$ $F\in{\mathcal{C}}$ $\overline{C_{6}}$ $|V(H)|-1$ $G\setminus S^{\prime}$ $S^{*}\cap V(H)\subseteq S$ $G^{\prime}[S]=G[S]$ $\overline{K_{2}\cup C_{2k+1}}\cong 2K_{1}\vee\overline{C_{2k+1}}$ $S\cap B=\emptyset$ $h_{1},\dots,h_{i}$ $y=b$ $a_{2},q_{j},b_{1}$ $K_{2,k}$ $a,b\in V(G)\setminus S$ $y\in B$ $N_{G}(q_{t})\cap V(H)=\{b_{1}\}$ $\mathcal{M}_{\mathcal{G}_{0}}=\{P_{3}\}$ $\mathcal{O}(n^{3+\epsilon})$ $K_{k-1}$ $\mathcal{O}(n^{2+\epsilon}).$ $p_{i}\in C$ $r_{k}=y$ $K_{2,k+1}$ $V(C_{1})$ $N_{G}[y]\setminus N_{G}[x]$ $p_{s}\in N_{B}$ ${\mathcal{C}}$ $N_{G}[x]:=\{x\}\cup N_{G}(x)$ $N_{B}=\emptyset$ $\alpha(G^{*})=\alpha(G)+|E(G)|$ $G[v,a_{i},a_{j},a_{k}]$ $p_{1}\in A$ $G[x_{1},\dots,x_{t}]$ $G^{\prime}\in\mathcal{G}_{\mathcal{C}}$ $K_{\ell}\notin\mathcal{C}$ $V(H)\setminus U$ $G\in\mathcal{C}$ $uw\in E(H)$ $a_{2}\in N_{G}(p_{s})\cap V(H)\subseteq\{a_{2},b_{2}\}$ $G[A\cup B\cup\{q_{0},q_{1}\}]$ $\{a_{i},b_{i}\}$ $|N_{G}(v)\cap V(H)|\geq 2$ $a,b\in V(G^{\prime})\setminus S$ $X=\{x\}$ $E(H_{1})\cup E(H_{2})$ $\tilde{z}^{i}_{t},{z}^{i}_{t}$ $\mathcal{L}_{rec}=\sum_{\mathbf{r}\in\mathcal{R}}\left\|Z^{i}(\mathbf{r})-\hat%
+{Z}^{i}(\mathbf{r})\right\|^{2}$ $\hat{Z^{i}}(\mathbf{r})$ $\mathcal{L}_{Mrec}$ $\mathcal{L}_{ref}=\sum_{\mathbf{r}\in\mathcal{R}}\left\|Z^{i}(\mathbf{r})-%
+\tilde{Z}^{i}(\mathbf{r})\right\|^{2}\hskip 2.84544pt,\textrm{where}\hskip 2.8%
+4544pt\tilde{z}^{i}={F}_{\phi}(\hat{z}^{i})$ $\hat{Z}^{i}(\mathbf{r})=\int_{t_{n}}^{t_{f}}T(t)\mathbf{\sigma}(\gamma(t))%
+\mathbf{f}_{z}(\mathbf{r}(t),d)dt,\hskip 2.84544pt\textrm{where}\hskip 2.84544%
+ptT(t)=\textrm{exp}\left(\int_{t_{n}}^{t}\sigma(\mathbf{r}(s))ds\right).$ $t\mathbf{d}$ $\nabla\mathcal{L}_{MDDS}$ $\nabla_{\theta}\mathcal{L}_{\mathrm{DDS}}=\nabla_{\theta}\mathcal{L}_{\mathrm{%
+SDS}}(\mathbf{z},y_{src})-\nabla_{\theta}\mathcal{L}_{\mathrm{SDS}}(\hat{%
+\mathbf{z}},y_{trg}),$ $\lambda_{im}$ $\mathcal{L}_{rtot}=\lambda_{rec}\mathcal{L}_{rec}+\lambda_{ref}\mathcal{L}_{ref}$ $\lambda_{ref}$ $y_{src}$ $\mathcal{L}_{MDDS}$ $\tilde{{Z}}^{i}$ ${F_{\phi}(\cdot)}$ $\nabla_{\theta,\phi}\mathcal{L}_{\mathrm{DDS}}=\nabla_{\theta,\phi}\mathcal{L}%
+_{\mathrm{SDS}}(\mathbf{z}^{i},y_{src})-\nabla_{\theta,\phi}\mathcal{L}_{%
+\mathrm{SDS}}(\tilde{\mathbf{z}}^{i},y_{trg}).$ $(\mathbf{f}_{z},\sigma)=F_{\theta}(\mathbf{\gamma}(\mathbf{x}),\mathbf{\gamma}%
+(\mathbf{d}))$ $\hat{C}(r)=\int_{t_{n}}^{t_{f}}T(t)\mathbf{\sigma}(\mathbf{r}(t))\mathbf{c}(%
+\mathbf{r}(t),d)dt,\hskip 2.84544pt\textrm{where}\hskip 2.84544ptT(t)=\textrm{%
+exp}\left(-\int_{t_{n}}^{t}\sigma(\mathbf{r}(s))ds\right).$ $\nabla_{\theta}\mathcal{L}_{\mathrm{SDS}}(\mathbf{z},y_{trg},\epsilon,t)=%
+\omega(t)(\epsilon_{\psi}\left(\mathbf{z}_{\mathbf{t}},y_{trg},t\right)-%
+\epsilon)\frac{\partial\mathbf{z}_{\mathbf{t}}}{\partial\theta}$ $504\times 378$ $\lambda_{rec}$ $\mathcal{L}_{\mathrm{Mrec}}=\lambda_{im}\cdot\mathcal{M}\cdot\mathcal{L}_{%
+\mathrm{rtot}}+\lambda_{om}\cdot(1-\mathcal{M})\cdot\mathcal{L}_{\mathrm{rtot}}.$ $I=\{I^{i}\}_{i=1}^{N}$ ${z}:=\{{z}^{i}\}_{i=1}^{N}$ $\tilde{z}^{i}$ ${z^{i}}=\mathcal{E}({{I^{i}}})\in\mathbb{R}^{64\times 64\times 4}$ $y_{trg}$ $\nabla_{\theta,\phi}\mathcal{L}_{\mathrm{MDDS}}=\mathcal{M}\cdot(\nabla_{%
+\theta,\phi}\mathcal{L}_{\mathrm{DDS}}),$ $[t_{near},t_{far}]$ $\mathcal{L}_{\mathrm{tot}}=\mathcal{L}_{\mathrm{MDDS}}+\mathcal{L}_{\mathrm{%
+Mrec}}$ $\lambda_{om}$ $\tilde{z}^{i}={F}_{\phi}(\hat{z}^{i}).$ $\mathcal{L}_{rtot}$ $\mathbf{r}(t)=\mathbf{o}$ $[T\cdot G,Q]$ $\displaystyle+R_{\text{upright}}+R_{\text{heading}}$ $205\pm 8$ $10P$ $\displaystyle+R_{\text{effort}}+R_{\text{act}}+R_{\text{dof}}$ $P=100,T=50,W=10$ $12+3\mathbb{A}+2\mathbb{F}$ $\mathbb{A},\mathbb{F}$ $\displaystyle+R_{\text{death}}\times\mathbf{1}_{\{\text{head\_height}\leq\text%
+{termination\_height}\}}$ $\displaystyle\ R_{\text{progress}}+R_{\text{alive}}\times\mathbf{1}_{\{\text{%
+head\_height}\geq\text{termination\_height}\}}$ $G=\lfloor(Q-P)/T\rfloor$ $(\sum_{i=0}^{\left\lfloor\frac{m-1}{6}\right\rfloor}\frac{1}{2}\Phi(p^{m-6i}))+1$ $v\in\mathbb{Z}_{p^{m}}$ $\tau^{\prime}\circ\tau_{i}(E)=\tau^{\prime}(E)=E^{\prime},\ \forall\ i\in\{1,2%
+,3,...,t\}$ $j(E_{1})\neq 0,1728$ $\begin{split}\quad y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}\end{split}$ $a^{\frac{p-1}{2}}\equiv 1\mod p)$ $N_{R}(\mathbb{Z}_{p})=\Phi(p^{2})$ $N_{R}(\mathcal{R})$ $\ \ \frac{\Phi(n^{4})}{|Aut(E)|}$ $32505856$ $E_{2}/\mathbb{Z}_{p^{m}}:y^{2}=x^{3}+\bar{a}x+\bar{b}$ $\displaystyle b_{6}$ $\Delta^{v}(p^{m})$ $2q-4+6+4=2q+6$ $E_{1}/\mathbb{F}_{q}:y^{2}=x^{3}+ax+b$ $|Aut(E)|=4$ $\{u^{\prime},\alpha u^{\prime},\alpha^{2}u^{\prime},-u^{\prime},-\alpha u^{%
+\prime},-\alpha^{2}u^{\prime}\}$ $\{6,19\}$ $\begin{split}C_{R}(\mathbb{Z}_{n})=\sum_{\mathbb{E}_{k}}C^{(k)}_{R}(\mathbb{Z}%
+_{n})\end{split}$ $34091302912$ $15400$ $(3^{-1})^{3}a^{3}\equiv-(2^{-1})^{2}b^{2}\ \mod p$ $\mathbb{Z}_{2^{m}}$ $k_{2}=4$ $6\nmid char(\mathbb{Z}_{n})$ $\displaystyle(E)=c_{4}^{3}/\Delta$ $\displaystyle b_{8}$ $3.5115653\times 10^{13}$ $\frac{q-1}{|Aut(E)|}$ $\{11,14\}$ $2p^{m}-4$ $28672$ $P(1,1)$ $N_{q}=2q+3+\left(\frac{-4}{q}\right)+2\left(\frac{-3}{q}\right)$ $N_{G}(\mathbb{Z}_{n})=\Phi(n^{5})$ $x=X/Z,y=Y/Z$ $Aut(E^{\prime})$ $y^{2}=x^{3}+3x$ $a=0\;\text{and}\;b\neq 0$ $|Aut(E)|=6$ $\tau^{\prime}\neq\tau_{i}$ $2p^{i-1}$ $|Aut(E_{1})|=6$ $q\equiv 1\mod k$ $E/\mathbb{F}_{q}$ $\{2,23\}$ $|Aut(\mathbb{E}_{k})|\;\text{over}\;\mathbb{Z}_{n}=\prod\limits_{i=1}^{l}|Aut(%
+\mathbb{E}_{k_{i}})|\;\text{over}\;\mathbb{Z}_{p_{i}}$ $\mathbb{F}_{2^{m}},1\leq m\leq 10$ $F(X,Y,Z)=Y^{2}Z+a_{1}XYZ+a_{3}YZ^{2}-X^{3}-a_{2}X^{2}Z-a_{4}XZ^{2}-a_{6}Z^{3}=0$ $N^{\prime\prime}_{R}(\mathbb{Z}_{n})$ $y^{2}=x^{3}+4x+2$ $\mathbb{Z}_{p^{m}}^{*}$ $Aut(E)=\{\tau_{1},\tau_{2},...,\tau_{t}\}$ $\mathbb{Z}_{7^{m}}$ $y^{2}=x^{3}+x$ $p^{2(i-1)}$ $2q-4$ $\tau:(x,y)\rightarrow(u^{2}x,u^{3}y),\;u\in\;\mathbb{Z}_{n}^{*}$ $2058$ $y^{2}=x^{3}-x+9$ $N_{R}(\mathbb{Z}_{n})$ $|Aut(E_{k})|=2,4$ $a=-3c^{2}$ $\begin{split}s_{1}^{3}+s_{2}^{2}&\equiv 0\mod p\\
+\implies s_{1}^{3}&\equiv-s_{2}^{2}\mod p\end{split}$ $\displaystyle=a_{1}^{2}a_{6}+4a_{2}a_{6}-a_{1}a_{3}a_{4}+a_{2}a_{3}^{2}-a_{4}^%
+{2}$ $\displaystyle=a_{1}^{2}+4a_{2}$ $\displaystyle\implies k_{1}$ $\tau:(x,y)\rightarrow(u^{2}x+r,u^{3}y+u^{2}sx+t),\;u\in\mathbb{K}^{*}$ $\Delta=-16(4a^{3}+27b^{2})$ $\tau:(x,y)\rightarrow(u^{2}x+r,u^{3}y+u^{2}sx+t),u\in\;\mathbb{Z}_{n}^{*},r,s,%
+t\in\mathbb{Z}_{n}$ $E(\mathbb{K})$ $\displaystyle=2a_{4}+a_{1}a_{3}$ $p^{m}=2j+1,\ i,j\in\mathbb{N}$ $\gcd(p,6)=1$ $\begin{split}E:Y^{2}Z+a_{1}XYZ+a_{3}YZ^{2}&=X^{3}+a_{2}X^{2}Z+a_{4}XZ^{2}\\
+&\ \ \ +a_{6}Z^{3}\end{split}$ $|A^{3}_{0}|=p^{\lfloor\frac{2m}{3}\rfloor}$ $3\mid\Phi(p^{m})$ $\begin{split}E:y^{2}+a_{1}xy+a_{3}y=&x^{3}+a_{2}x^{2}+a_{4}x+a_{6}\\
+&\text{where}\ a_{i}\in\;\mathbb{Z}_{n}\end{split}$ $C_{G}(\mathbb{Z}_{n})$ $\left(.\right)$ $|A^{2}_{0}|=p^{\lfloor\frac{m}{2}\rfloor}$ $E_{2}/\mathbb{F}_{q}:y^{2}=x^{3}+\bar{a}x+\bar{b}$ $u^{4}\bar{a}=a$ $q^{5}-q^{4}$ $(\sum_{i=0}^{\left\lfloor\frac{m-1}{2}\right\rfloor}\frac{1}{2}\Phi(p^{m-2i}))%
++1.$ $Aut(E)$ $1\leq i\leq\left\lfloor\frac{m-1}{2}\right\rfloor+1$ $\gcd(p^{m},6)=1$ $\bar{a}=0$ $\displaystyle=c_{4}^{3}/\Delta$ $\frac{\sum\limits_{i=0}^{\left\lfloor\frac{m-1}{3}\right\rfloor}\Phi(p^{m-3i})%
+}{3}+1.$ $N_{R}(\mathbb{Z}_{p^{m}})=\Phi(p^{2m})$ $\tau^{\prime}\circ\tau_{i}=\tau^{\prime}\circ\tau_{j}$ $A^{k}_{u}(p^{m})$ $|\#E(\mathbb{F}_{q})-q-1|\leq 2\sqrt{q}$ $\mathbb{Z}_{p}^{m}$ $y^{2}=x^{3}+1$ $16807$ $p^{m}\equiv 1\mod 4$ $\frac{\partial F}{\partial X},\frac{\partial F}{\partial Y},\frac{\partial F}{%
+\partial Z}$ $n^{3}\Phi(n)=\Phi(n^{4})$ $2\leq l\leq 4$ $\begin{split}C^{(k)}_{R}(\mathbb{Z}_{n})&=\frac{\prod\limits_{i=1}^{l}N^{(k_{i%
+})}_{R}(\mathbb{Z}_{p_{i}}).\prod\limits_{i=1}^{l}|Aut(\mathbb{E}_{k_{i}})|}{%
+\prod\limits_{i=1}^{l}\Phi(p_{i})}\\
+&=\prod\limits_{i=1}^{l}\frac{N^{(k_{i})}_{R}(\mathbb{Z}_{p_{i}}).|Aut(\mathbb%
+{E}_{(k_{i})})|}{\Phi(p_{i})}\\
+&=\prod\limits_{i=1}^{l}C^{(k_{i})}_{R}(\mathbb{Z}_{p_{i}})\end{split}$ $\displaystyle=-1728\frac{4a^{3}}{\Delta}$ $\frac{p-1}{6}$ $C^{(k)}_{R}(\mathbb{Z}_{n})$ $\mathbb{Z}_{p_{i}}(1\leq i\leq l)$ $\sum_{E_{k}}\frac{\Phi(n)}{|Aut(E_{k})|}=\Phi(n^{2})$ $y^{2}=x^{3}+3x+2$ $\Phi(n^{5})$ $\tau:(x,y)\rightarrow(u^{2}x+r,u^{3}y+u^{2}sx+t),\ u\in\mathbb{Z}_{n}^{*}$ $u^{\prime}\in\mathbb{F}_{q}^{*}$ $A^{2}_{u}(25)$ $\begin{split}s_{1}^{3}+s_{2}^{2}&\equiv 0\mod(p^{m})\\
+\implies s_{1}^{3}&\equiv-s_{2}^{2}\mod(p^{m})\end{split}$ $p\equiv 3\mod 4$ $N_{G}(\mathcal{R})$ $|Aut(E_{1})|=4$ $n=p_{1}p_{2}\ldots p_{k}$ $\mathbb{F}_{(2g+1)^{n}}$ $y^{2}=x^{3}+x+2$ $\displaystyle=-16(4a^{3}+27b^{2})\in\mathbb{Z}_{n}^{*}$ $Aut(E^{\prime})=t$ $char(\mathbb{Z}_{n})\nmid 2,3$ $\bar{b}=0$ $\gcd(q,6)=1$ $\displaystyle(\frac{\Phi(p^{m})}{6})k_{1}$ $E(\mathbb{Z}_{n})\cong E(\mathbb{Z}_{p_{1}})\oplus E(\mathbb{Z}_{p_{2}})\oplus%
+\ldots\oplus E(\mathbb{Z}_{p_{l}})$ $q\equiv 1\mod 12$ $p=3j+1$ $\displaystyle=b_{2}^{2}-24b_{4}$ $\Phi(n),n,n,n$ $p\equiv 2\mod 3$ $N_{R}(\mathbb{Z}_{p^{m}})$ $A^{k}_{0}(p^{m})=\{x\ |\ x^{k}\equiv 0\mod p^{m}\}$ $|Aut(E)|=2$ $q\equiv 1,5,7,11\mod 12$ $E_{1}/\mathbb{Z}_{p^{m}}:y^{2}=x^{3}+ax+b$ $p^{m}\equiv 1\mod 2$ $\begin{split}C_{R}(\mathbb{Z}_{n})&=\sum_{\mathbb{E}_{k}}C^{(k)}_{R}(\mathbb{Z%
+}_{n})\\
+&=\sum_{\mathbb{E}_{k}}\prod\limits_{i=1}^{l}C^{(k_{i})}_{R}(\mathbb{Z}_{p_{i}%
+})\end{split}$ $(3^{-1}a,2^{-1}b)=(s_{1},s_{2})$ $r,s,t\in\mathbb{K}$ $\mathbb{Z}_{3125}$ $u^{6}\bar{b}=b$ $\tau^{\prime}(E)=E^{\prime}$ $j(E_{1})=0$ $4\mid\Phi(p^{m})$ $\mathbb{Z}_{p^{m}}$ $p^{m}=3j+1,\ i,j\in\mathbb{N}$ $C^{(k)}_{R}(\mathbb{Z}_{n})=\frac{N^{(k)}_{R}(\mathbb{Z}_{n}).|Aut(\mathbb{E}_%
+{k})|}{\Phi(n)}$ $E:y^{2}=x^{3}+ax+b$ $\frac{q^{4}-q^{3}}{|Aut(E)|}$ $2\mid\Phi(p^{m})$ $\{3,22\}$ $\frac{p-1}{2}\mid(p-1)$ $983040$ $\mod p$ $C_{R}(\mathbb{Z}_{n})=\prod\limits_{i=1}^{l}C_{R}(\mathbb{Z}_{p_{i}^{e_{i}}})$ $k_{1}+k_{2}+k_{3}=6+4+2p^{m}-4=2p^{m}+6$ $\forall e_{i}\geq 1$ $\#E(\mathbb{F}_{q})$ $p^{m}=4j+1,\ i,j\in\mathbb{N}$ $\mathbb{Z}_{5^{m}}$ $y^{2}=x^{3}+x+1$ $\displaystyle b_{4}$ $n=p_{1}p_{2}\ldots p_{l}$ $|A^{2}_{u}(25)|$ $\Delta\equiv 0\mod(p^{m})$ $P^{2}(\overline{\mathbb{K}})$ $d=\gcd(6,3j+1)=2$ $c=-3b/2a$ $A^{k}_{u}(p^{m})=\{x\ |\ x^{k}\equiv u\mod p^{m}\}$ $p=3j+2$ $p\equiv 1\ \mod 12$ $N_{G}(\mathbb{Z}_{n})$ $2\Phi(p^{m})$ $N^{\prime\prime}_{R}(\mathbb{Z}_{p^{m}})=(p^{2m}-\Delta^{0}(p^{m}))\geq N_{R}(%
+\mathbb{Z}_{p^{m}})$ $n=p^{m}$ $1901$ $\{9,16\}$ $1056964608$ $\mathbb{Z}_{p_{i}}$ $3p^{2(i-1)}$ $|Aut(E_{k})|=6$ $\Phi(p^{m})$ $\{u^{\prime},\beta u^{\prime},\beta^{2}u^{\prime},\beta^{3}u^{\prime}\}$ $E:y^{2}=x^{3}+ax+b,\ char(\mathbb{K})\neq 2,3.$ $E(\mathbb{Z}_{{p_{i}}^{e_{i}}})$ $\Phi(p^{2m})-2\Phi(p^{m})=\Phi(p^{m})(p^{m}-2)$ $u\in{\mathbb{Z}_{p^{m}}^{*}}$ $m=3.20^{-1}$ $\Delta^{i}(n)$ $|Aut(E_{1})|$ $10^{e}$ $\begin{split}N_{R}(\mathbb{Z}_{n})&=\sum_{\mathbb{E}_{k}}N^{(k)}_{R}(\mathbb{Z%
+}_{n})\\
+&=\sum_{\mathbb{E}_{k}}\prod_{i=1}^{l}N^{(k_{i})}_{R}(\mathbb{Z}_{p_{i}})\end{split}$ $12500$ $\gcd(x^{2}+y^{2}+z^{2}+w^{2}+t^{2},n)=1$ $(\frac{p-1}{2}\times 1\times 2)+1=p$ $y^{2}=x^{3}+2x+1$ $p=2i+1$ $a\neq 0\ \text{and}\ b\neq 0\ (j(E)\neq 0,1728$ $\tau_{i}(E)=E$ $n=p_{1}^{e_{1}}p_{2}^{e_{2}}\ldots p_{l}^{e_{l}}$ $|Aut(E_{1})|=2$ $p^{m}\equiv 1\mod 3$ $x^{k}\equiv a\mod p$ $y^{2}=x^{3}+24x+1$ $char(\mathbb{F}_{q})\neq 2,3$ $E:y^{2}=x^{3}+ax+b,\ \ a,b\in\mathbb{Z}_{n}\ \text{and}\ \gcd(6,n)=1$ $6\nmid n$ $117012$ $\{12,13\}$ $\{1,24\}$ $1\leq i\leq\left\lfloor\frac{m-1}{3}\right\rfloor+1$ $\displaystyle=-16(4a^{3}+27b^{2})$ $|Aut(E)|$ $\frac{p-1}{6}\mid(p-1)$ $E(\mathbb{Z}_{n})$ $(\frac{p-1}{6}\times 3\times 2)+1=p$ $\frac{p-1}{2}$ $\displaystyle=\Phi(p^{m})$ $a^{\frac{p-1}{d}}\equiv 1\mod p$ $a\neq 0\ \text{and}\ b=0\ (j(E)=1728$ $d=\gcd(6,3j)=6$ $N^{(k)}_{R}(\mathbb{Z}_{n})$ $y(a-y)=x^{3}-x$ $\displaystyle=-b_{2}^{2}b_{8}-8b_{4}^{3}-27b_{6}^{2}+9b_{2}b_{4}b_{6}$ $\mathbb{Z}_{25}$ $\tau:(x,y)\rightarrow(u^{2}x+r,u^{3}y+u^{2}sx+t),u\in\;\mathbb{F}_{q}^{*}$ $1924$ $\Delta\in\mathbb{Z}_{n}^{*}$ $N_{R}(\mathbb{Z}_{n})=\Phi(n^{2})$ $\Delta\equiv 0\mod p$ $\mathbb{F}_{q}\ (q=p^{r})$ $\Delta=-16(27b^{2})$ $\bar{b}=u^{-6}b$ $N^{\prime}_{R}(\mathbb{Z}_{n})$ $p\equiv 1\mod 12$ $u\in\mathbb{F}_{q}^{*}$ $7^{1}$ $E_{1}/\mathbb{K}$ $(\sum_{i=0}^{\left\lfloor\frac{m-1}{3}\right\rfloor}\Phi(p^{m-3i}))+1$ $100842$ $\tau:(x,y)\rightarrow(u^{2}x,u^{3}y)$ $\mathbb{Z}_{3125},(5^{4}=3125)$ $\sum_{E_{k}}\frac{q-1}{|Aut(E_{k})|}=q^{2}-q$ $\sigma=\left\{\,\begin{array}[]{lll}2q+6&when&q\equiv 1\mod 12\\
+2q+2&when&q\equiv 5\mod 12\\
+2q+4&when&q\equiv 7\mod 12\\
+2q&when&q\equiv 11\mod 12\\
+\end{array}\right.\\$ $Q(21,4)$ $N^{(k)}_{R}(\mathbb{Z}_{n})=\prod\limits_{i=1}^{l}N^{(k_{i})}_{R}(\mathbb{Z}_{%
+p_{i}})$ $u=-u^{\prime}$ $1.0952167\times 10^{12}$ $(x,y)\rightarrow(\frac{x-3b_{2}}{36},\frac{y}{216})$ $N^{{}^{\prime\prime}}_{R}(\mathbb{Z}_{p^{m}})$ $a\neq 0\;\text{and}\;b=0$ $y^{2}=x^{3}+4x$ $p^{2m-1}$ $p\equiv 1\ \mod l$ $4a^{3}+27b^{2}=0$ $d=\gcd(k,p-1)$ $x^{m}\equiv 1\mod p$ $u\in\mathbb{Z}_{p^{m}}$ $C_{R}(\mathbb{Z}_{p_{i}})=\left\{\,\begin{array}[]{lll}2p+6&when&p\equiv 1\mod
+1%
+2\\
+2p+2&when&p\equiv 5\mod 12\\
+2p+4&when&p\equiv 7\mod 12\\
+2p&when&p\equiv 11\mod 12\\
+\end{array}\right.\\$ $N^{\prime\prime}_{R}(\mathbb{Z}_{p^{m}})$ $P(X,Y,Z)\in P^{2}(\overline{\mathbb{K}})$ $|Aut(E)|=t$ $(\tau^{\prime})^{-1}\circ\tau^{\prime}\circ\tau_{i}=(\tau^{\prime})^{-1}\circ%
+\tau^{\prime}\circ\tau_{j}$ $\displaystyle=a_{3}^{2}+4a_{6}$ $y^{2}=x^{3}+2x$ $\mathbb{Z}_{3^{m}}$ $\begin{split}\Delta^{0}(p^{m})&=(\sum_{i=0}^{\left\lfloor\frac{m-1}{6}\right%
+\rfloor}\frac{1}{6}\Phi(p^{m-6i})\cdot 3p^{2i}\cdot 2p^{i})\\
+&+p^{\left\lfloor\frac{m}{2}\right\rfloor+\left\lfloor\frac{2m}{3}\right%
+\rfloor}\\
+&=(\sum_{i=0}^{\left\lfloor\frac{m-1}{6}\right\rfloor}p^{3i}\Phi(p^{m-6i}))+p^%
+{\left\lfloor\frac{m}{2}\right\rfloor+\left\lfloor\frac{2m}{3}\right\rfloor}\\
+&=(\sum_{i=0}^{\left\lfloor\frac{m-1}{6}\right\rfloor}\Phi(p^{m-3i}))+p^{\left%
+\lfloor\frac{m}{2}\right\rfloor+\left\lfloor\frac{2m}{3}\right\rfloor}\\
+\end{split}$ $p^{2}-p=\Phi(p^{2})$ $p\equiv 1\mod 3$ $C_{G}(\mathcal{R})$ $E/\mathbb{Z}_{n}$ $\{8,17\}$ $|Aut(E_{k})|=4$ $p\equiv 1,5,7,11\ \mod 12$ $r,s,t\in\mathbb{Z}_{n}$ $a^{\frac{p-1}{6}}\equiv 1\mod p$ $(x,y)\xrightarrow{}(x,y-\frac{a_{1}}{2}x-\frac{a_{3}}{2})$ $A^{2}(\overline{\mathbb{K}})=\overline{\mathbb{K}}\times\overline{\mathbb{K}}$ $\tau_{i}=\tau_{j}$ $\frac{p-1}{3}$ $j(E_{1})=j(E_{2})$ $m\mid(p-1)$ $\Delta\equiv i\mod n$ $b=2c^{3}$ $C_{R}(\mathbb{Z}_{n})$ $1.1248004\times 10^{15}$ $(\sum\limits_{i=0}^{\left\lfloor\frac{m-1}{6}\right\rfloor}\frac{1}{6}\Phi(p^{%
+m-6i}))+1.$ $\tau:(x,y)\rightarrow(u^{2}x,u^{3}y),u\in\;\mathbb{Z}_{n}^{*}$ $u^{2}=\frac{\bar{a}b}{a\bar{b}}$ $p=4i+1$ $i\in\{1,2,3,...,t\}$ $a=0\ \text{and}\ b\neq 0\ (j(E)=0$ $E/\mathbb{K}$ $C_{R}(\mathbb{Z}_{p^{m}})=\left\{\,\begin{array}[]{lll}2p^{m}+6&when&p\equiv 1%
+\mod 12\\
+2p^{m}+2&when&p\equiv 5\mod 12\\
+2p^{m}+4&when&p\equiv 7\mod 12\\
+2p^{m}&when&p\equiv 11\mod 12\\
+\end{array}\right.\\$ $\begin{split}C_{R}(\mathbb{Z}_{n})&=\prod\limits_{i=1}^{l}\sum_{\mathbb{E}_{k_%
+{i}}}C^{(k_{i})}_{R}(\mathbb{Z}_{p_{i}})\\
+&=\prod\limits_{i=1}^{l}C_{R}(\mathbb{Z}_{p_{i}}).\end{split}$ $\begin{split}N_{R}(\mathbb{Z}_{n})&=\prod_{i=1}^{l}\sum_{\mathbb{E}_{k_{i}}}N^%
+{(k_{i})}_{R}(\mathbb{Z}_{p_{i}})\\
+N_{R}(\mathbb{Z}_{n})&=\prod_{i=1}^{l}N_{R}(\mathbb{Z}_{p_{i}})\end{split}$ $n=3^{m}$ $a\neq 0\;\text{and}\;b\neq 0$ $gcd(n,6)=1$ $n=p_{1}^{e_{1}}p_{2}^{e_{2}}\ldots p_{k}^{e_{k}}$ $\Phi(p^{m})=p^{m}-p^{m-1}$ $\{4,21\}$ $NQR$ $\frac{\Phi(n)}{|Aut(E)|}$ $\{7,18\}$ $N_{R}(\mathbb{Z}_{n})=\prod\limits_{i=1}^{l}N_{R}(\mathbb{Z}_{p_{i}})$ $j(E_{1})=1728$ $2310$ $C_{R}(\mathbb{Z}_{n})=\prod\limits_{i=1}^{l}C_{R}(\mathbb{Z}_{p_{i}})$ $C_{R}(\mathcal{R})$ $u,r,s$ $p\equiv 1\mod 2$ $N^{{}^{\prime}}_{R}(\mathbb{Z}_{p^{m}})$ $\bar{a}=u^{-4}a$ $p=3i+1$ $|Aut(E_{k})|=2$ $\Phi(p^{2m})$ $char(\mathbb{K})\neq 2,3$ $y^{2}=x^{3}+2$ $\displaystyle j(E)$ $y^{2}=x^{3}+4x+1$ $E_{2}/\mathbb{K}$ $\mathbb{Z}_{p^{m}},\ \gcd(p,6)=1$ $\{0,5,10,15,20\}$ $p\equiv 1\mod 4$ $j\in 2\mathbb{Z}$ $q^{2}-q-2(q-1)=(q-1)(q-2)$ $N^{\prime}_{R}(\mathbb{Z}_{p^{m}})$ $s(w,A,B)=\frac{1}{|A|}\sum_{a\in A}cos(\vec{w},\vec{a})-\frac{1}{|B|}\sum_{b%
+\in B}cos(\vec{w},\vec{b})$ $T(V,W)=\frac{T^{V}_{obs}}{\sqrt{var(T^{V}_{random})}}-\frac{T^{W}_{obs}}{\sqrt%
+{var(T^{W}_{random})}}$ $T^{V}_{obs}$ $cos(\vec{a},\vec{b})=\frac{\vec{a}\cdot\vec{b}}{\|\vec{a}\|\|\vec{b}\|}$ $\begin{split}T_{South}(1980-2009,1860-1889)=2.4774\\
+T_{Northeast}(1980-2009,1860-1889)=0.8116\end{split}$ $s(w,A,B)$ $\alpha\leq 0.05$ $s(X,Y,A,B)=\sum_{x\in X}s(x,A,B)-\sum_{y\in Y}s(y,A,B)$ $T^{W}_{obs}$ $s(w,\ A,\ B)$ $\text{Gini}(p)=1-\sum_{i=1}^{K}p_{i}^{2},$ $F^{s}_{in}$ $F^{b}_{in}$ $F_{out}^{s}$ $F_{i}^{s^{\prime}}=\mathrm{SAR}\left(F_{i}^{s},F_{i}^{b}\right).$ $F^{s^{\prime}}$ ${A}={\left[\begin{array}[]{c}{a^{s}};{a^{b}}\end{array}\right]}={\sigma}(%
+Aggregate([{F^{s}_{in}};{F^{b}_{in}}])),$ $Aggregate$ $F_{i+1}^{b}$ $a^{s}$ $L_{s_{out}}$ $F_{i}^{b}$ $P_{s}\in\mathbb{R}^{n\times h\times w}$ $A\in{[0,1]}^{2\times h\times w}$ $F_{in}^{s}$ $F^{u}=conv(F_{in}^{u}\bigodot a^{u})+F_{in}^{u},u\in\left\{s,b\right\},$ $L=L_{s_{out}}+\lambda_{s}L_{s_{i}}+\lambda_{b}L_{b_{i}},i\in\{1,2,3,4\},$ $L_{s_{i}}$ $F_{in}^{b}$ $P_{b}\in\mathbb{R}^{1\times h\times w}$ $F_{i}^{s^{\prime}}$ $Attention(q,k,v)=softmax(\frac{q*k^{t}}{\sqrt{d_{k}}})\cdot v,$ ${F_{i}^{s}},{F_{i}^{b}}=\left\{\begin{matrix}\mathrm{SME}_{i}\left(F_{i+1},%
+\left[F_{i+1};F_{1}\right]\right),&\mathrm{if}\;\;i=4,\\
+\mathrm{SME}_{i}\left(\left[{F_{i+1}^{s}};{F_{i+1}}\right],\left[{F_{i+1}^{b}}%
+;{F_{1}}\right]\right),&\mathrm{else}.\end{matrix}\right.$ $L_{b_{i}}$ $F^{b}$ $\mathcal{V}_{rc}$ $\mathbf{f}_{u}$ $\displaystyle=\boldsymbol{W}_{b}\left(\sigma\left(\boldsymbol{W}_{a}\mathrm{c}%
+_{u}^{t}+\boldsymbol{B}_{a}\right)\right)-\mathbb{I}\left[u\notin\mathcal{A}^{%
+t}\right]\infty,$ $\displaystyle=\operatorname{MLP}\left(\|_{k=1}^{K}\sigma\left(\sum_{j\in%
+\mathcal{N}_{i}}\alpha_{ij}^{k}\boldsymbol{h}_{j}^{(t)}\mathbf{W}^{k}\right)\right)$ $\mathcal{A}^{t>0}=\{v|v\in\operatorname{ONE-HOP}(\hat{\mathcal{V}}_{lg}^{t}\}%
+\cup\{\text{STOP}\}$ $\mathrm{EGAT}(\cdot)$ $\displaystyle s_{u}^{t}$ $\displaystyle\boldsymbol{h}_{i}^{(t+1)}$ $\hat{\mathcal{V}}_{lg}^{t}$ $\mathbb{I}[u\notin\mathcal{A}^{t}]\,(\mathbb{I}[\text{STOP}\notin\mathcal{A}^{%
+t}])\mapsto\{0,1\}$ $\mathcal{V}_{hg}$ $\boldsymbol{h}_{i}^{(t)}\in R^{1\times d}$ $\mathbf{x}_{explored}$ $\boldsymbol{h}_{i}^{(0)}$ $\mathcal{A}^{t>0}=\{v|v\in\operatorname{ONE-HOP}(\hat{\mathcal{G}}_{rc}^{t})\}%
+\cup\{\text{STOP}\}$ $\mathcal{G}_{hg}=(\mathcal{V}_{hg},\mathcal{E}_{hg})$ $\boldsymbol{W}_{b}$ $\boldsymbol{W}_{a}$ $\hat{\mathcal{V}}_{lg}^{T}$ $\mathbf{H}^{(t)}$ $\mathcal{E}_{rc}=\left\{(u,v)\mid u,v\in\mathcal{V}_{rc},(u,v)\in\mathcal{E}_{%
+p}\right\}$ $\displaystyle=\boldsymbol{W}_{b}\left(\sigma\left(\boldsymbol{W}_{a}\mathrm{c}%
+_{\mathcal{G}}^{t}+\boldsymbol{B}_{a}\right)\right)-\mathbb{I}\left[\mathrm{%
+STOP}\notin\mathcal{A}^{t}\right]\infty,$ $\hat{\mathcal{V}}_{rc}^{t=0}=\emptyset$ $A^{k}\in R^{3d\times d}$ $\mathbf{h}_{\mathcal{G}}^{t}$ $\mathbf{a}^{k}\in R^{1\times d}$ $\mathbf{f}_{ij}^{(t)}\in R^{1\times d}$ $\displaystyle e_{ij}$ $\mathbf{H}^{(t)}=[\boldsymbol{h}_{1}^{(t)};\boldsymbol{h}_{2}^{(t)};\cdots;%
+\boldsymbol{h}_{n}^{(t)}]\in R^{n\times d}$ $\displaystyle=\boldsymbol{W}_{b}\left(\sigma\left(\boldsymbol{W}_{a}\left(%
+\mathbf{h}_{\mathcal{G}}^{t}+\mathbf{c}_{\mathcal{G}}\right)+\boldsymbol{B}_{a%
+}\right)\right)-\mathbb{I}\left[\mathrm{STOP}\notin\mathcal{A}^{t}\right]\infty,$ $\left\{\mathbf{c}_{u}^{t}\right\}=\operatorname{EGAT}\left(\mathcal{G}_{p},%
+\left\{\mathbf{x}_{u}+\mathbb{I}[u\in\hat{\mathcal{V}}_{rc}^{t}]\mathbf{x}_{%
+explored}\right\},\left\{\mathbf{x}_{uv}\right\}_{v\in\mathcal{N}(u)}\right).$ $\mathcal{G}_{p}=(\mathcal{V}_{p},\mathcal{E}_{p})$ $\displaystyle=\boldsymbol{W}_{b}\left(\sigma\left(\boldsymbol{W}_{a}\left(%
+\mathbf{h}_{u}^{t}+\mathbf{c}_{\mathcal{G}}\right)+\boldsymbol{B}_{a}\right)%
+\right)-\mathbb{I}\left[u\notin\mathcal{A}^{t}\right]\infty,$ $|\mathcal{E}_{hg}|$ $\hat{\mathcal{G}}_{rc}^{t}$ $\operatorname{ELU}(x)=\alpha(\exp(x)-1)$ $|\mathcal{V}_{p}|$ $\hat{\mathcal{V}}_{rc}^{t}$ $\mathcal{A}^{0}=\{v|v\in\mathcal{V}_{p}\}$ $s_{t}=\{\mathcal{G}_{p},\hat{\mathcal{V}}_{rc}^{t}\}$ $\displaystyle\boldsymbol{f}_{ij}^{(t+1)}$ $\left\{\mathbf{h}_{u}^{t}\mid u\in\mathcal{V}_{hg}\right\}$ $\mathcal{V}_{rc}\subseteq\mathcal{V}_{p}$ $\mathcal{A}^{0}=\{v|v\in\mathcal{V}_{hg}\}$ $\mathbf{h}_{\mathcal{G}}^{t}=\sum_{u\in\mathcal{V}_{hg}}\mathbf{h}_{u}^{t}$ $\boldsymbol{B}_{a}$ $\mathcal{G}_{rc}=(\mathcal{V}_{rc},\mathcal{E}_{rc})$ $\left\{\mathbf{c}_{u}^{t}\mid u\in\mathcal{V}_{p}\right\}$ $e_{ij}\in R$ $\mathbf{W}\in R^{d\times d}$ $\mathbf{D}_{v}\in R^{n\times n},\mathbf{D}_{e}\in R^{e\times e},\mathbf{W}\in R%
+^{e\times e}$ $L^{CLIP}(\theta)=\hat{E}_{t}[\min(r_{t}(\theta)\hat{A}_{t},\operatorname{clip}%
+\left(r_{t}(\theta),1-\epsilon,1+\epsilon\right)\hat{A}_{t})].$ $|\mathcal{V}_{hg}|$ $\mathbf{a}\in R^{1\times d}$ $\mathbf{f}_{explored}$ $s_{u}^{t}\in\mathbb{R}^{1},s_{s}^{t}\in\mathbb{R}^{1}$ $\mathcal{E}_{hg}$ $\boldsymbol{E}_{ $\displaystyle=\mathbf{a}\cdot{\boldsymbol{f}_{ij}^{(t+1)}}^{T},\alpha_{ij}=%
+\frac{\exp\left(e_{ij}\right)}{\sum_{k\in\mathcal{N}_{i}}\exp\left(e_{ik}%
+\right)},$ $\mathrm{HGNN}(\cdot)$ $\hat{\mathcal{V}}_{rc}^{T}$ $\hat{\mathcal{V}}_{lg}^{t=0}=\emptyset$ $\boldsymbol{E}=\left(e_{0},\cdots,e_{t},\cdots,e_{T}\right)$ $\mathcal{G}_{rc}$ $\mathbb{I}[u\in\hat{\mathcal{V}}_{rc}^{t}]\mapsto\{0,1\}$ $\mathbf{\Theta}^{l}\in R^{d^{l}\times d^{l+1}}$ $\displaystyle\mathbf{X}^{l+1}=\sigma\left(\mathbf{D}_{v}^{-\frac{1}{2}}\mathbf%
+{HWD}_{e}^{-1}\mathbf{H}^{\top}\mathbf{D}_{v}^{-\frac{1}{2}}\mathbf{X}^{l}%
+\boldsymbol{\Theta}^{l}\right)$ $\mathrm{SOFTMAX}$ $\mathbf{c}_{\mathcal{G}}^{t}=\sum_{u\in\mathcal{V}_{p}}\mathbf{c}_{u}^{t}$ $\mathbf{c}_{\mathcal{G}}$ $\displaystyle s_{s}^{t}$ $\mathbb{I}[u\in\hat{\mathcal{V}}_{lg}^{t}]\mapsto\{0,1\}$ $p(\boldsymbol{E}\mid\boldsymbol{p})=\prod_{t=1}^{T}p\left(e_{t}\mid\boldsymbol%
+{p},\boldsymbol{E}_{ $\mathcal{V}_{lg}$ $\boldsymbol{f}_{ij}^{(0)}$ $\displaystyle=\text{ LeakyReLU }\left(\left[\boldsymbol{h}_{i}^{(t)}\mathbf{W}%
+\left\|\boldsymbol{f}_{ij}^{(t)}\right\|\boldsymbol{h}_{j}^{(t)}\mathbf{W}%
+\right]A\right),$ $A\in R^{3d\times d}$ $\mathrm{X}^{l}\in R^{n\times d^{l}}$ $s_{t}=\{\mathcal{G}_{p},\mathcal{G}_{hg},\hat{\mathcal{V}}_{lg}^{t}\}$ $\mathrm{H}\in R^{n\times e}$ $\mathbf{c}_{\mathcal{G}}^{t}$ $\mathbf{W}^{k}\in R^{d\times d}$ $\mathcal{G}_{hg}$ $\boldsymbol{E}=\left(e_{0}\right)$ $\alpha_{ij}\in R$ $\left\{\mathbf{h}_{u}^{t}\right\}=\operatorname{HGNN}\left(\mathcal{G}_{hg},%
+\left\{\mathbf{f}_{u}+\mathbb{I}[u\in\hat{\mathcal{V}}_{lg}^{t}]\mathbf{f}_{%
+explored}\right\}\right).$ $p(e_{0}\mid\boldsymbol{p})=1$ $\text{{TPR}}(f)=\frac{{\text{{TP}}(f)}}{{\text{{P}}}}$ $\epsilon=\frac{FP+FN}{TP+TN+FP+FN}\times 100\%$ $prob(\text{positive-data})=\sigma(\sum_{j}y_{j}^{2}-\theta))$ $\text{ROC AUC score}=\int_{-\infty}^{\infty}\text{TPR}(f)\cdot\text{FPR}(f)%
+\text{d}f$ $\text{{FPR}}(f)=\frac{{\text{{FP}}(f)}}{{\text{{N}}}}$ $insertBST(\sf T_{{\sf binNumber}},\sf gBinNumber[{\sf idNumber}(I_{t})],B-{\sf
+cost%
+}(I_{t}))$ ${\sf idNumber}(I_{t^{\prime}})$ $[\ell_{q},r_{q}-\epsilon]$ $\sf IntervalGenerator$ $\sf gBinNumber[{\sf idNumber}(I_{t})]$ $I\in List_{i}$ $\sf VariableSizeDS$ $\alpha=\frac{B_{max}}{B_{min}}$ ${\sf idNumber}=i(1\leq i\leq g)$ $L_{i}\leftarrow L_{i}+1$ ${\sf binNumber}(I_{t^{\prime}})\leftarrow\sf idTobinNumber[{\sf idNumber}(I_{t%
+^{\prime}})]$ $\sf insertHeap(\sf H^{min}_{F{\sf idNumber}},{\sf idNumber}(I_{t^{\prime\prime%
+}}))$ ${\sf cost}(q)=\min_{p\leq i $\sf H^{min}_{L}$ $\ell_{q}=T_{i^{\prime}}$ $l_{t^{\prime}}=t$ $\{(i,1),(i,2),\cdots,(i,L_{i})\}$ ${\sf Color}$ $t=l_{t^{\prime}}$ $root.max<{\sf cost}(I_{t})$ $\sf deleteBST(\sf T_{U{\sf idNumber}},{\sf idNumber}(I_{t^{\prime\prime}}))$ $\sf OPT_{off}^{B}$ $\sf searchMap()$ $\sf T_{U{\sf idNumber}}$ ${\sf binNumber}(I)\leftarrow L_{i}$ $\sf binToCapacity[{\sf idNumber}(I_{t^{\prime}})]\leftarrow\sf binToCapacity[{%
+\sf idNumber}(I_{t^{\prime}})]-{\sf cost}(I_{t^{\prime}})$ ${\sf binNumber}(I_{t^{\prime}})$ $({\sf idNumber}(I_{t^{\prime}}),{\sf binNumber}(I_{t^{\prime}}))$ $10-11$ $\sf extractMin()$ ${\sf cost}(i,j)$ $p=(x_{p},y_{p})$ $r_{q}=T_{j^{\prime}}$ $\sf L_{i}$ $OPT\geq g$ $\sf insertHeap(\sf H^{min}_{L},(l_{t},t))$ $decreaseKey(T,node,c)$ $node.bin=\sf gBinNumber[{\sf idNumber}(I_{t})]$ $rem-{\sf cost}(I)$ $node.bin=1$ $node.left$ ${\sf Color}[t]$ $\sf searchMap(\sf idTobinNumber,{\sf idNumber}(I_{t^{\prime}}))$ $\sf gBinNumber[{\sf idNumber}(I_{t})]\leftarrow\sf gBinNumber[{\sf idNumber}(I%
+_{t})]+1$ $13-14$ ${\sf Color}(I)=({\sf idNumber}(I),{\sf binNumber}(I))$ $I_{q}=[\ell_{q},r_{q}]$ $root.max ${\sf Color}(I_{t})$ $r_{q}=\ell_{p}$ $\mathcal{S}=\{S_{1},S_{2},\cdots,S_{k}\}$ $\ell_{q}=T_{i}$ $root.left.max<{\sf cost}(I_{t})$ ${\sf ALG}=\sum_{i=1}^{{\sf gIdNumber}}{L_{i}}\leq 2\cdot{\left(\sum_{i=1}^{{%
+\sf gIdNumber}}{\frac{{\sf cost}(L_{i})}{B}}\right)}+{\sf gIdNumber}\\$ $root.bin$ ${\sf binNumber}(I_{t})$ ${\sf cost}(i,1)+2\cdot\sum_{j=2}^{L_{i}-1}{\sf cost}(i,j)+{\sf cost}(i,L_{i})>%
+\sum_{j=1}^{L_{i}-1}B_{i}^{j}$ $B_{min}=\underset{1\leq i\leq g;1\leq j\leq L_{i}}{\min}B_{i}^{j}$ $r_{t^{\prime}}=t$ $S_{i}=(x_{i},y_{i})$ $root.left.max\geq{\sf cost}(I_{t})$ $1\leq j\leq L_{i}-1$ $\sf VariableSizeDS(\mathcal{I})$ $j\in M\quad and\quad i $l_{t^{\prime\prime}}\leftarrow\sf extractMin(\sf H^{min}_{L})$ $\sum_{i=1}^{g}\sum_{j=1}^{L_{i}-1}B_{i}^{j}<2\cdot\sum_{i=1}^{g}\sum_{j=1}^{L_%
+{i}}{\sf cost}(i,j)$ $(r_{t},t)$ $5-6$ ${\sf ALG}\leq 2\cdot{\left(\sum_{i=1}^{{\sf gIdNumber}}{\frac{{\sf cost}(L_{i}%
+)}{B}}\right)}+{\sf gIdNumber}\leq 2\cdot\sf OPT_{off}^{B}+\sf OPT_{off}\leq 3%
+\cdot\sf OPT_{off}^{B}\\$ $\{1,\cdots,{\sf gIdNumber}\}$ $I_{t}\neq I_{t^{\prime}}$ $root.rem\geq{\sf cost}(I_{t})$ $\sf insertBST(\sf T_{U{\sf idNumber}},{\sf idNumber}(I_{t^{\prime}}))$ $7-9$ $\sf binToCapacity$ $\sf insertMap(\mathcal{I}, ${\sf Color}(I)$ $decreaseKey(\sf T_{{\sf binNumber}},root,{\sf cost}(I_{t}))$ $\sf T_{{\sf binNumber}}$ $rem\leftarrow b$ $\sum_{i=1}^{{\sf gIdNumber}}{W(i)}\leq 1.7\cdot\sf OPT_{off}^{B}$ $root.left=NULL$ $({\sf idNumber}(l_{t^{\prime\prime}}),$ $root.left\neq NULL$ $({\sf idNumber}(I_{t}),{\sf binNumber}(I_{t}))$ ${\sf gIdNumber}\leq\sf OPT_{off}$ $List_{i}\leftarrow\emptyset$ $(I_{t^{\prime}})$ $\sf gBinNumber[{\sf idNumber}(I_{t})]=1$ $node.max$ $node.rem=B-{\sf cost}(I_{t})$ $\sf OPT_{off}$ $root.right$ $root.max\geq{\sf cost}(I_{t})$ $node.rem=c$ $(root,I_{t})$ ${\sf ALG_{f}}$ $\sf insertMap(\sf binToCapacity,<{\sf idNumber}(I_{t^{\prime}}),B-{\sf cost}(I%
+_{t^{\prime}})>)$ $B_{max}=\underset{1\leq i\leq g;1\leq j\leq L_{i}}{\max}B_{i}^{j}$ $root.rem$ $I_{t}\neq NULL$ ${\sf ALG_{f}}=\sum_{i=1}^{{\sf gIdNumber}}{L_{i}}\leq\sum_{i=1}^{{\sf gIdNumber%
+}}{\left(W(i)+1\right)}=\sum_{i=1}^{{\sf gIdNumber}}{W(i)}+{\sf gIdNumber}\\$ $\sum_{i=1}^{g}L_{i}=\sum_{i=1}^{g}(L_{i}-1)+g\leq\sum_{i=1}^{g}\sum_{j=1}^{L_{%
+i}-1}(\frac{B_{i}^{j}}{B_{min}})+g$ ${\sf ALG}$ ${\sf idNumber}=i$ $\sf insertHeap()$ $({\sf idNumber}(I_{t}),{\sf binNumber}(I_{t}))\neq({\sf idNumber}(I_{t^{\prime%
+}}),$ $I_{t^{\prime\prime}}$ ${\sf idNumber}(I_{t^{\prime}})\leftarrow\sf extractMin(\sf H^{min}_{F{\sf
+idNumber%
+}})$ $r_{t^{\prime\prime}}\leftarrow\sf extractMin(\sf H^{min}_{R})$ $r_{t^{\prime\prime}}=t$ $(l_{t},t)$ $L_{i}\leftarrow 0$ ${\sf cost}(I)\leq rem$ $rem\leftarrow$ ${\sf cost}(\cdot)$ $T_{1},\cdots,T_{k}$ $\sf idTobinNumber[{\sf idNumber}]$ $q=(x_{q},y_{q})$ $(S_{i},q,S_{j})$ ${\sf binNumber}(l_{t^{\prime\prime}}))>)$ ${\sf idNumber}(I_{t^{\prime}})\leftarrow{\sf gIdNumber}$ ${\sf gIdNumber}$ $\sf DSusingBPC(I_{t}=[l_{t},r_{t}])$ $node.right$ $\sf insertBST()$ $l_{t}\geq t$ ${\sf binNumber}$ $\sf insertHeap(\sf H^{min}_{R},(r_{t},t))$ $List_{i}$ $\frac{{\sf cost}(L_{i})}{B}\geq\frac{L_{i}-1}{2}$ $\sf deleteHeap()$ ${\sf idNumber}$ ${\sf cost}(L_{i})$ $r_{t^{\prime\prime}}==t$ $I_{t}=[l_{t},r_{t}]$ ${\sf binNumber}=L_{i}$ $\sf binToCapacity[{\sf idNumber}]$ $L_{i}\leq W(i)+1$ $\sf binToCapacity[{\sf idNumber}(I_{t^{\prime}})]\leftarrow B-{\sf cost}(I_{t^%
+{\prime}})$ $(i^{\prime},j^{\prime})=(i,j)$ ${\sf ALG_{f}}\leq{\sf gIdNumber}+\sum_{i=1}^{{\sf gIdNumber}}{W(i)}\leq 1.7\sf
+OPT%
+_{off}^{B}+\sf OPT_{off}\leq 2.7\sf OPT_{off}^{B}\\$ ${\sf binNumber}(I_{t^{\prime}}))$ $(root.left,I_{t})$ $\sf insertMap({\sf Color}, $\sf idTobinNumber[{\sf idNumber}(I_{t^{\prime}})]\leftarrow\sf idTobinNumber[{%
+\sf idNumber}(I_{t^{\prime}})]+1$ $L_{i}\leq 2\cdot\frac{{\sf cost}(L_{i})}{B}+1$ $\sf insertMap()$ $insertBST(T,i,c)$ $OPT\geq\frac{\sum_{i=1}^{g}\sum_{j=1}^{L_{i}}{\sf cost}(i,j)}{B_{max}}$ $\sf H^{min}_{F{\sf idNumber}}$ $\sf insertMap(\sf idTobinNumber,<{\sf idNumber}(I_{t^{\prime}}),1>)$ $I_{t}\cap I_{t^{\prime}}\neq\emptyset$ $root.rem<{\sf cost}(I_{t})$ $\sf deleteHeap(\sf H^{min}_{L})$ $\{1,2,\cdots,g\}$ $\sf(I_{t^{\prime\prime}})$ $\sf gBinNumber$ $\sf W(i)$ $\sf OPT_{off}^{B}\geq\sum_{i=1}^{{\sf gIdNumber}}{\frac{{\sf cost}(L_{i})}{B}}$ $node.rem=node.max=B-{\sf cost}(I_{t})$ $t^{\prime\prime} ${\sf cost}(q)>(S_{i},q,S_{j})$ $I_{t^{\prime}}=[l_{t^{\prime}},r_{t^{\prime}}]$ $[\ell_{p}+\epsilon,r_{p}]$ $node.rem$ $l_{t^{\prime\prime}}=t$ ${\sf cost}(I_{t})$ $l_{t^{\prime\prime}}==t$ ${\sf idNumber}(I_{t})$ $\sf binToCapacity[{\sf idNumber}(I_{t^{\prime}})]\geq{\sf cost}(I_{t^{\prime}})$ $node.bin$ $root.left$ ${\sf gIdNumber}\leftarrow{\sf gIdNumber}+1$ $List_{{\sf idNumber}(I)}$ $\sf deleteHeap(\sf H^{min}_{R})$ $\sf deleteHeap(\sf H^{min}_{F{\sf idNumber}})$ ${\sf cost}(q)=(S_{i},q,S_{j})$ $I_{p}=[\ell_{p},r_{p}]$ $\sf OPT_{off}\leq\sf OPT_{off}^{B}$ ${\sf cost}(i,j)+{\sf cost}(i,j+1)>B_{i}^{j}$ $\sf DSusingBPC$ ${\sf cost}(I_{q})={\sf cost}(q)$ $\sf H^{min}_{R}$ $r_{q}=T_{j}$ $\sf idTobinNumber$ $(root.right,I_{t})$ $\sf deleteBST()$ $O(k^{2}\cdot\mathcal{T})$ $\sum_{i=1}^{g}L_{i}\leq\frac{2\cdot\sum\limits_{i=1}\limits^{g}\sum\limits_{j=%
+1}\limits^{L_{i}}{\sf cost}(i,j)}{B_{min}}+g\leq\frac{2\cdot\alpha\cdot\sum%
+\limits_{i=1}\limits^{g}\sum\limits_{j=1}\limits^{L_{i}}{\sf cost}(i,j)}{B_{%
+max}}+g\leq(2\alpha+1)OPT.\\$ ${\bf q}_{orig}$ ${\bf 2:}(3,8,7)$ ${\bf 10:}(14,17,13)$ $\partial_{j}$ $\displaystyle C=\frac{1}{M}{\bf X}^{T}{\bf X}=\frac{1}{M}(V\Sigma U^{T})(U%
+\Sigma V^{T})=\frac{1}{M}V\Sigma^{2}V^{T}.$ $\mu^{(h)}$ ${\bf X}^{(h)}$ $N\cdot p_{h}$ $n_{\ell}\leq N$ ${\cal\bf\cal R}=0.9$ ${\cal O}(MN$ $EntrySize$ ${\bf 4:}(12,9.2)$ $R^{(h)}$ $\displaystyle C=\frac{1}{M}{\bf X}^{T}{\bf X}=V\Lambda V^{T},$ ${\cal\bf R}$ ${\bf 11:}(7,14,12)$ $\lambda_{i}=\sigma^{2}_{i}/M\mbox{ and conversely }\sigma_{i}=\sqrt{M\lambda_{%
+i}},1\leq i\leq N.$ $\displaystyle{\bf X}={\bf U}\Sigma V^{T},$ ${F_{\ell}}\approx S/EntrySize(n_{\ell})$ $m_{1},\ldots,m_{H}$ $p=0.30$ ${\cal O}(MN^{2})$ $H\cdot N$ $C({\bf q})=A({\bf q})\cap B({\bf q})$ $\partial_{j}^{th}$ $8\times(8.333/1000)\approx 0.067$ $D^{2}({\bf u},{\bf v})=||{\bf u}||^{2}+||{\bf v}||^{2}-2{\bf u}\times{\bf v}$ $\lambda_{i}^{(h)}m_{h},\forall{i},\forall{h}$ $\displaystyle=\frac{\sum_{h=1}^{H}m_{h}\sum_{j=n_{h}+1}^{N}\lambda_{j}^{(h)}}{%
+\sum_{h=1}^{H}m_{h}\sum_{j=1}^{N}\lambda_{j}^{(h)}}.$ ${\bf 3:}(9,10,8)$ $m_{h}\cdot p_{h}$ $\sigma_{i}\geq\sigma_{i+1},i=1,\ldots,N-1.$ ${\bf Y}^{(}h)$ $({\bf X}_{i,j}-\bar{x}_{i})/s_{i},1\leq i\leq N\mbox{ where }s_{i}\mbox{is the%
+ std deviation },\forall{j}.$ ${\bf 5:}(8,7,20)$ ${\bf 9:}(11,11,15)$ $\kappa_{j}$ $\Lambda=\lambda_{1},\ldots,lambda_{N}$ ${\bf 6:}(6.6.23)$ $A({\bf q})$ ${\cal\bf P}$ ${\bf 12:}(10,12,3)$ ${\cal\bf R}=0.8$ ${\bf Y}^{(h)},1\leq h\leq H$ ${\bf 1:}(1,2,5)$ ${\cal\bf R}\approx 1$ $\kappa_{j}^{th}$ ${\bf X}^{(h)},\;h=1,\ldots,H$ ${\bf 8:}(2,13.9)$ ${\cal R}=0.8$ $\displaystyle NMSE=\frac{\sum_{i=1}^{M}\sum_{j=n+1}^{N}y^{2}_{i,j}}{\sum_{i=1}%
+^{M}\sum_{j=1}^{N}y^{2}_{i,j}}=\frac{\sum_{j=n+1}^{N}{\lambda_{j}}}{\sum_{j=1}%
+^{N}\lambda_{j}}$ ${\cal O}(N^{3})$ $\displaystyle NMSE$ $D^{2}({\bf u},{\bf v})=\left[\sum_{i=1}^{n}|{u}_{i}-{v}_{i}|^{p}\right]^{1/p}.$ $\displaystyle{\bf Y}={\bf X}V.$ ${\cal\bf P}=E[|C({\bf q})|/|B({\bf q})|].$ ${\sum\nolimits_{k=1}^{n_{\ell}}{\lambda_{k}}}/{\sum\nolimits_{k=1}^{N}{\lambda%
+_{k}}}\geq min(\ell\times p,1).$ ${\cal\bf R}=E[|\frac{C({\bf q}|)}{|(B{\bf q})}|].$ $B({\bf q})$ $\max{\left\{\left[D({\bf q},\mu^{(i)}-R^{(i)}\right],0\right\}}.$ ${\bf 7:}(0,3,27)$ $K^{*}(k)=k{\bf\cal R}/{\bf\cal P}$ $V=N\cdot H+\sum_{h=1}^{H}{\left(N\cdot p_{h}+m_{h}\cdot p_{h})\right)},$ $\displaystyle=\frac{\sum_{h=1}^{H}\sum_{i=1}^{m_{h}}\sum_{j=n_{h}+1}^{N}(y^{(h%
+)}_{i,j})^{2}}{\sum_{h=1}^{H}\sum_{i=1}^{m_{h}}\sum_{j=1}^{N}(y^{(h)}_{i,j})^{%
+2}}$ $1302_{1296}$ $450_{190}$ ${}_{m}\textit{AoC}(2013)=9.54$ $7261_{4619}$ $475_{471}$ $2112_{885}$ $2974_{2357}$ $1530_{1522}$ $191_{190}$ $460_{452}$ $2355_{2344}$ $155_{153}$ $1306_{974}$ ${}_{m}AoC(t)=\frac{1}{N}\sum_{j=1}^{N}\overline{\textit{AoC}}(x_{j})$ $13,775_{13,667}$ $t=2014,\ldots,2022$ $778_{770}$ $4979_{4724}$ $y_{t}{=}{{}_{m}\textit{AoC}(t)}$ $1209_{717}$ $1245_{1239}$ $4914_{3012}$ $396_{355}$ $8841_{4280}$ $3476_{3461}$ $2671_{2646}$ $0_{0}$ ${}_{m}\textit{AoC}(2013)=8.56$ $1237_{1216}$ $\displaystyle\textit{$PoOC$}(x)=\frac{|\mathfrak{O}_{k}(x)|}{M}$ $x\mapsto 2\frac{x-\min{\mathbf{x}}}{\max{(\mathbf{x}})-\min{(\mathbf{x}})}-1$ $91_{91}$ $426_{417}$ $2598_{2589}$ $1193_{1167}$ $389_{388}$ $1441_{1436}$ $704_{694}$ $1383_{932}$ $2393_{2382}$ $1065_{1060}$ $999_{974}$ $\mathfrak{O}_{k}(x)=\{y\,|\,\textit{AoC}(x,y)\geq k\}$ $2013_{2007}$ $4623_{4259}$ $241_{240}$ $4516_{4287}$ $4751_{4432}$ $696_{405}$ $3202_{3194}$ $\approx 8k$ $219_{219}$ ${}_{m}\textit{AoC}=6.15$ $2631_{2604}$ $978_{975}$ $4936_{4569}$ $1916_{1126}$ $994_{987}$ $23,965_{23,845}$ $1313_{1306}$ $386_{384}$ $2211_{4193}$ $2114_{1289}$ $2340_{1122}$ $47,364_{43,331}$ $179_{176}$ $30,867_{23,629}$ $662_{511}$ $137_{137}$ $2250_{2246}$ $85_{85}$ $\frac{y_{t}}{y_{t-1}}-1$ $\approx 4k$ $139,769_{110,024}$ $13,087_{6819}$ $1937_{1910}$ $4736_{2227}$ $1187_{1184}$ $1313_{1297}$ $2242_{2231}$ $\approx 3k$ $12,695_{9849}$ $601_{153}$ $2418_{2414}$ $1122_{537}$ ${}_{m}\textit{AoC}(2022)=5.93$ $3882_{3798}$ $1238_{1231}$ $587_{355}$ $108,288_{102,532}$ $4574_{4445}$ $29964_{2951}$ $4571_{4447}$ $4797_{4708}$ $1004_{997}$ $14,625_{12476}$ $4598_{4454}$ $91_{90}$ $2130_{1273}$ $4006_{3990}$ $2160_{2134}$ $692_{176}$ $1859_{1166}$ ${}_{m}\textit{AoC}=10.9$ $765_{759}$ $331_{331}$ $2339_{2314}$ $1053_{1050}$ $2843_{2806}$ $739_{418}$ $784_{775}$ $\textit{AoC}(x,y_{i})=\textit{YoP}(x)-\textit{YoP}(y_{i})$ $2541_{2511}$ $402_{373}$ $1922_{1425}$ $7133_{5640}$ $112_{112}$ $540_{529}$ $13,529_{8316}$ $2820_{2811}$ $1442_{862}$ $4642_{4193}$ ${}_{m}\textit{AoC}(2022)=7.61$ $6578_{4876}$ $1318_{1295}$ $1337_{1323}$ $715_{704}$ $99_{99}$ $4170_{3198}$ $386_{382}$ $1662_{1652}$ ${}_{m}\textit{AoC}(2022)=5.10$ ${}_{m}\textit{AoC}(2022)=7.24$ $5582_{4109}$ $402_{399}$ $1523_{717}$ ${}_{m}\textit{AoC}(2013)=17.61$ $3084_{1166}$ $2835_{2827}$ ${}_{m}\textit{AoC}(2013)=10.91$ $1295_{1289}$ $13,754_{7949}$ $1055_{1051}$ $10,888_{10,786}$ $2789_{2589}$ $2279_{2243}$ $1963_{1950}$ $12,505_{12,435}$ $2407_{2403}$ $1319_{1314}$ $118_{117}$ $542_{534}$ $597_{529}$ $875_{554}$ $390_{386}$ $65,685_{48,391}$ $60,346_{59,822}$ $637_{629}$ $98_{98}$ $32,530_{32,229}$ $2467_{2467}$ $919_{916}$ $1356_{1351}$ $11,097_{5214}$ $2964_{2951}$ $11,000_{8546}$ $873_{867}$ $\displaystyle\textit{${}_{m}PoOC$}(t)=\frac{1}{N}\sum_{x=1}^{N}\textit{$PoOC$}%
+(x)$ $10,752_{10,660}$ $8489_{6324}$ $838_{830}$ $1210_{721}$ $14,741_{14,531}$ $1962_{1955}$ $3952_{3815}$ $5174_{3506}$ $587_{474}$ $\displaystyle\overline{\textit{AoC}}(x)=\frac{1}{M}\sum_{i=1}^{M}\textit{AoC}(%
+x,y_{i})$ $4_{4}$ $3670_{3667}$ $1096_{722}$ $766_{759}$ $57,935_{29,688}$ $663_{487}$ $m^{\rm rec}_{i}(t)\geq M$ $\Delta t=$ $2.3\times 10^{4}$ $R_{i}(t)\equiv m^{\rm rec}_{i}(t)/m^{\rm sent}_{i}(t)$ $\ell_{i}(t)$ $N_{model}$ $n^{\rm rec}_{i}(t)\geq T$ $Q_{i}(t)\equiv n^{\rm rec}_{i}(t)/n^{\rm sent}_{i}(t)$ $-97\%$ $\{M,\alpha,T,\beta,L,\tau\}$ $n_{i}^{\rm rec}(t)$ $R_{i}(t)\geq\alpha$ $N_{empirical}$ $m^{\rm sent}_{i}(t)$ $Q_{i}(t)\geq\beta$ $\ell_{i}(t)\geq L$ $m^{\rm rec}_{i}(t)$ $\phi_{i}(t)\leq\tau$ $n^{\rm sent}_{i}(t)$ $\phi_{i}(t)$ $2.3\times 10^{5}$ $\Delta t=30$ $u^{B}$ $u^{B}_{i}$ $\mathcal{L}=\mathcal{L}_{\text{nll}}+\mathcal{L}_{\text{complete}}+\mathcal{L}%
+_{\text{consist}},$ $p^{g}_{j}$ $\mathcal{L}_{\text{complete}}=-\sum_{t}\log p(\mathcal{P}^{\text{new}}_{t}\mid%
+\mathcal{P}^{\text{new}}_{ $\mathcal{P}^{g}=\{p^{g}_{1},p^{g}_{2},\cdots,p^{g}_{k}\}$ $S\in\mathbb{R}^{m\times k}$ $p_{B_{i}}$ $s_{i,j}=\text{sim}(E(p_{i}),E(p^{g}_{j})),$ $p^{g}_{I(i)}$ $\mathcal{L}_{\text{nll}}$ $\mathcal{P}^{\text{new}}$ $s_{i,I(i)}\geq\tau$ $\mathcal{P}^{g}_{A}$ $\mathcal{P}_{A}=\{p_{A_{1}},p_{A_{2}},\cdots,p_{A_{m}}\}$ $\mathcal{P}_{B}=\{p_{B_{1}},p_{B_{2}},\cdots,p_{B_{r}}\}$ $\mathcal{L}_{\text{gen}}=-\frac{1}{N}\sum^{N}_{t=1}\log p(u^{B}_{t}\mid u^{B}_%
+{ $\mathcal{U}=\{u^{A}_{1},u^{B}_{1},u^{A}_{2},u^{B}_{2}\cdots,u^{A}_{n}\}$ $\mathcal{P}^{\text{con}}=\{p^{g}_{I(i)}\mid s_{i,I(i)}\geq\tau\}$ $\mathcal{L}_{\text{consist}}=-\sum_{p^{g}_{i}\in\mathcal{P}^{\text{con}}}\log{%
+\frac{\exp(\text{sim}(h_{i},h_{\mathcal{P}}))}{\sum_{p^{g}_{j}\in\mathcal{P}^{%
+g}}\exp(\text{sim}(h_{j},h_{\mathcal{P}}))}},$ $p_{A_{i}}$ $\mathcal{L}_{\text{complete}}$ $\mathcal{L}_{\text{consist}}$ $\mathcal{P}^{\text{con}}$ $\mathcal{L}_{\text{nll}}=-\sum_{t}\log p(\mathcal{P}_{t}\mid\mathcal{P}_{ $u^{B}_{n}$ $\mathcal{P}^{\text{new}}=\mathcal{P}^{\text{con}}\cup\mathcal{P}^{\text{miss}}.$ $\mathcal{P}^{\text{miss}}=\{p_{i}\mid\forall j\,s_{i,j}<\tau\}$ $\mathcal{P}^{\text{miss}}$ $h_{\mathcal{P}}$ $\mathcal{P}^{g}_{A}=\text{PESS}(\mathcal{U}_{A})=\{p^{g}_{A_{1}},p^{g}_{A_{2}}%
+,\cdots,p^{g}_{A_{k}}\}$ $\mathcal{U}_{A}=\{u^{A}_{1},u^{A}_{2},\cdots,u^{A}_{n}\}$ $u^{A}_{n}$ $u^{A}_{i}$ $I(i)={\text{argmax}}_{j}s_{i,j}\,.$ $\mathcal{P}^{g}$ $i\in\{1,\dots,t-1\}$ $V(G)=V(G^{\prime})\setminus\{a,b\}$ $G\in\mathcal{G}_{\mathcal{C}}$ $\mathcal{M}_{\mathcal{G}_{\mathcal{C}}}=\{2K_{1}\vee H\mid H\in\mathcal{M}_{%
+\mathcal{C}}\}$ $N_{G}(p_{0})\cap V(H)=\{a_{1},b_{1}\}$ $G_{A}\in\mathcal{G}_{\mathcal{C}}$ $\Big{(}\bigcup_{v\in V(H)}X_{v}\Big{)}\cap V(C)\neq\emptyset$ $(A,B,S)$ $V(H)\setminus\{a_{1},b_{1}\}$ $\mathcal{O}(|V(H)|+|E(H)|)$ $v\in C_{1}$ $\{2,\dots,k-1\}$ $G_{A}\setminus S$ $S\cap B$ $G\setminus S_{A}$ $\mathcal{M}_{\mathcal{G}_{2}}$ $p_{i},\dots,p_{s},a_{2},a_{3},b_{3},b_{1},p_{i}$ $v^{xy}=a$ $\{2,\dots,t-1\}$ $\{\overline{C_{2k+1}}\mid k\in\mathbb{N}\}$ $\mathcal{C}_{k}$ $C\neq C_{1}$ $(a^{\prime},b^{\prime})$ $F\in\mathcal{C}$ $q_{0}\neq q_{t}$ $\mathcal{O}(|V(H)|^{2})$ $C_{1},\ldots,C_{k}$ $p_{1},\dots,p_{s},q_{2},\dots,q_{t}$ $G\setminus S_{B}$ $v\in V(H)$ $X_{w}\cap V(C)$ $N_{G}(v)\cap V(H)\subseteq\{a_{i},b_{i}\}$ $v^{xy}\notin\{a,b\}$ $a^{\prime},b^{\prime}$ $p_{i-1},p_{j+1}$ $2\leq i\leq j\leq s-1$ $\{0,\dots,s\}$ $A,B\subseteq V(G)$ $G/xy$ $N_{G}(v)\cap V(H)=\{a_{i},b_{i}\}$ $uv\in E(H)$ $B=\{b_{1},\dots,b_{n}\}$ $p_{0}\in N_{A}$ $H[C]\in\mathcal{C}$ $S\setminus\{v^{xy}\}=S^{\prime}\setminus\{x,y\}$ $\{x,y\}\cap\{a,b\}\neq\emptyset$ $\mathcal{O}(|V(H)|)$ $a,b\in V(G)$ $w\in V(H)\setminus U$ $\mathcal{G}_{0}\subseteq\mathcal{G}_{1}\subseteq\mathcal{G}_{2}\subseteq\dots$ $B=\{b_{1},b_{2},b_{3}\}$ $\bigcup_{v\in V(H)}X_{v}\subseteq V(C)$ $G\in{\cal G}$ $r_{1},\dots,r_{k}$ $G[N_{G}(x)]$ $G\setminus(V(H)\cup V(P))$ $G[N_{G[v_{1},\ldots,v_{i}]}(v_{i})]$ $S\setminus\{a,b\}$ $X_{v}\subseteq V(C)$ $2K_{1}\vee H_{1}$ $C\neq C_{2}$ $A\setminus N_{G}(v)$ $H[B]$ $p_{1},p_{s}\in A\cup C$ $G[v,a_{i},a_{j},b_{i},b_{j}]$ $\mathcal{C}=\mathcal{C}_{2}$ $S\cup(V(G)\setminus V(H))$ $H[N_{H}(v)]$ $x,y\in V(G)$ $S\subseteq V(G^{\prime})\setminus\{v^{xy}\}$ $S^{\prime}\setminus\{y\}$ $N_{G}(v)\cap V(H)=A$ $S\subseteq V(G^{\prime})$ $H[S]=G[S]$ $i>j$ $\{x,y\}\cap\{a,b\}=\emptyset$ $\mathcal{O}(n(n+m))$ $\mathcal{M}_{\mathcal{C}_{0}}=\{K_{1}\}$ $X_{u}\cap V(C_{1})$ $x\in S^{\prime}$ $\{X_{v}\}_{v\in V(2K_{1}\vee H_{1})}$ $a_{2},q_{t},b_{1}$ $\mathcal{G}_{\mathcal{C}}=\mathcal{G}_{2}$ $X_{u}\subseteq V(C)$ $\bigcup_{v\in V(H_{1})}X_{v}\subseteq V(H_{2})$ $|N_{G}(v)\cap A|\geq 2$ $G[v_{1},\dots,v_{i}]$ $C_{1}\cup\{a\},C_{2}\cup\{b\},C_{3},\dots,C_{k}$ $p_{s}=y$ $F\setminus u$ $q_{j+1},\dots,q_{t}\in B$ $q_{j},\ldots,q_{t},b_{1},a_{1},q_{j}$ $(2K_{1}\vee H_{1})\setminus V(H)\cong H_{1}$ $u,v\in V(H)$ $S=S^{\prime}$ $K_{2,3}\in\mathcal{G}_{\mathcal{C}}$ $S_{B}:=S\setminus A$ $y\in A\cup C$ $D\neq C$ $F\in\mathcal{F}_{\mathcal{C}}$ $\mathbf{V_{M}}$ $\lambda_{us}$ $D_{S}(y|x)$ $\{<\mathbf{v_{i}},\mathbf{p_{i}}>|i=1,...,N,\mathbf{v}\in\mathbb{R}^{1\times d%
+},\mathbf{p}\in\mathbb{R}^{L\times d}\}$ $err_{S}$ ${D_{T}}$ $\displaystyle d_{\mathcal{H}}(S,T)$ $\mathcal{L}_{epa}(X,D)=\lambda_{1}\min D(\mathbf{X}_{i<|\mathbf{p}|\times M})+%
+\lambda_{2}\max D(\mathbf{X}_{i\geq|\mathbf{p}|\times M})$ $\mathrm{L}1$ $\theta_{t}^{\prime}$ ${D_{S}}$ $C\xrightarrow{}F$ $\mathcal{D}_{S}(x,y)$ $\displaystyle+\frac{1}{q-p}\sum_{j=p+1}^{q-p}D[\mathbb{U}(y_{j}=1)]])$ $\lambda_{epa}$ $\lambda_{dpa}$ $2e-06$ $\displaystyle=2(1-\varepsilon_{D_{S}}^{S}-\varepsilon_{D_{T}}^{T}) $\mathbf{V_{M}}=\mathrm{argmax}\sum_{i=1}^{M}\psi(\mathbf{V},\gamma(x))$ $d_{\mathcal{H}}$ $\theta_{t}^{\prime}=\alpha\theta_{t-1}^{\prime}+(1-\alpha)\theta_{t}$ $C\xrightarrow{}B$ $\mathbb{U}=\{x_{i},y=0\}_{i=1}^{p}\cup\{x_{j},y=1\}_{j=p+1}^{q}$ $\displaystyle=2(1-\varepsilon_{D}^{S}-\varepsilon_{D}^{T})$ $\mathcal{L}_{epa}$ $err_{T}$ $\displaystyle d_{{\mathcal{H}_{S},\mathcal{H}_{T}}}(S,T)$ $\mathcal{L}_{dpa}$ $\mathcal{L}=\lambda_{s}\mathcal{L}_{sup}+\lambda_{us}\mathcal{L}_{unsup}+%
+\lambda_{epa}\mathcal{L}_{epa}+\lambda_{dpa}\mathcal{L}_{dpa}$ $\displaystyle=2(1-\min_{D\in\mathcal{H}}[\frac{1}{p}\sum_{i=1}^{p}D[\mathbb{U}%
+(y_{i}=0)]$ $2e-04$ $S\xrightarrow{}C$ $\mathcal{D}_{T}(x)$ $\mathbf{L}\times C\times 2$ $\mathit{fineTuningCorpus}$ $\mathit{chaseCorpus}\cup\{\mathit{chasePromptResp}$ $D,\Sigma,G,\mathit{model},\mathit{nlpTask}$ $\mathsf{postprocess}(\mathit{chaseCorpus})$ $\mathit{chasePromptResp}$ $\textit{Position}(\textit{EGTech},0.3,37.2,1)$ $\mathsf{generate}(\mathsf{preprocess}(\mathit{verbPlan},\mathit{nlpTask}))$ $\mathsf{composeBack}(\mathit{step},\mathit{chase})$ $6.9\$$ $\mathit{Open}(\mathit{EGTech},0.3,1),\neg\mathit{MarketClose}(1)\to\mathit{%
+Accepted}(\mathit{EGTech},0.3,1)$ $\displaystyle\textit{Accepted}(x,y,t_{1}),\textit{Price}(p_{1},t_{1}),k=y*p_{1}$ $\displaystyle\to\textit{Position}(x,y,k,t_{1})$ $\mathit{chase}$ $\mathit{verbChase}$ $\mathsf{verbalizeStep}(\mathit{step},\mathit{stepAggrContrib},G)$ $\mathsf{fineTune}(\mathit{model},\mathit{fineTuningCorpus})$ $\mathsf{map}(\mathit{tokenizedCorpus},\mathit{verbStep})$ $\mathit{tokenizedCorpus}$ $\mathit{chaseCorpus}\setminus\{\langle\mathit{prompt},\mathit{resp}\rangle$ $\displaystyle\textit{Close}(x,t_{2}),\textit{Price}(p_{2},t_{2}),\textit{%
+Position}(x,y,k,t_{1}),$ $\mathit{ftModel}$ $\mathit{chaseCorpus}$ $\mathit{qualityScore}$ $\mathit{stepAggrContrib}$ $\mathsf{hasAggregate}(\mathit{step}.\mathsf{getRule}())$ $147\$$ $\mathit{verbChase}\cup\{\mathit{verbStep}$ $\textsc{Vadalog}.\mathsf{reason}(D,\Sigma)$ $\textit{Accepted}(\textit{EGTech},0.3,1)$ $\displaystyle t_{2}>t_{1},pl=y*p_{2}-k$ $\mathit{qualityScore}\leq\mathit{threshold}$ $y*p_{2}-k$ $\displaystyle\to\textit{Return(x,pl)}$ $\mathsf{verbalizePlan}(\Sigma.\mathsf{getLogicPlan}())$ $p_{1}*y$ $\displaystyle\textit{Open}(x,y,t_{1}),\neg\,\textit{MarketClosed}(t_{1})$ $\mathsf{checkQuality}(\langle\mathit{prompt},\mathit{resp}\rangle,\mathit{%
+nlpTask},\mathit{verbChase})$ $\mathit{verbPlan}$ $\langle\mathit{prompt},\mathit{resp}\rangle$ $\displaystyle\to\textit{Accepted}(x,y,t_{1})$ $\mathit{verbStep}$ $\mathit{chaseCorpus}\cup\mathsf{paraphrase}(\langle\mathit{prompt},\mathit{%
+resp}\rangle)$ $\Sigma(D)=D$ $\bar{\boldsymbol{l}}\in\mathbb{R}^{L\times 64}$ $\tilde{\boldsymbol{\alpha}}\in\mathbb{R}^{512}$ $\displaystyle\boldsymbol{l}=\text{MLPs}(\boldsymbol{\Phi}_{D}^{c}(\bar{%
+\boldsymbol{l}}+\text{MLPs}(\hat{\boldsymbol{\alpha}}))).$ $(\boldsymbol{\theta}_{id}^{n},\boldsymbol{\theta}_{id})$ $\displaystyle\mathcal{L}_{con}=-\log\left[\frac{\exp\left(\mathcal{S}\left(%
+\boldsymbol{\theta}_{id}^{p},\boldsymbol{\theta}_{id}\right)\right)}{\exp\left%
+(\mathcal{S}\left(\boldsymbol{\theta}_{id}^{p},\boldsymbol{\theta}_{id}\right)%
+\right)+\exp\left(\mathcal{S}\left(\boldsymbol{\theta}_{id}^{n},\boldsymbol{%
+\theta}_{id}\right)\right)}\right].$ $\tilde{\boldsymbol{m}}=1-\boldsymbol{m}$ $\boldsymbol{m}_{i}=\boldsymbol{\mathcal{G}}_{MOD}(\hat{\boldsymbol{I}}_{i-1})$ $\hat{\boldsymbol{X}}$ $\boldsymbol{I}_{ad}^{i}$ $\boldsymbol{I}_{id}$ $\boldsymbol{I}_{id}^{i}=\hat{\boldsymbol{I}}_{1}$ $\hat{\boldsymbol{I}}_{i-1}$ $\mathcal{L}_{id}=\left\|\boldsymbol{\alpha}-\hat{\boldsymbol{\alpha}}\right\|_%
+{2}$ $\boldsymbol{\mathcal{F}}_{MB}$ $\boldsymbol{\theta}_{id}\in\mathbb{R}^{512}$ $\displaystyle\hat{\boldsymbol{\beta}}=\text{MLPs}(\boldsymbol{\Phi}_{D}^{e}(%
+\boldsymbol{\theta}_{e}+\bar{\boldsymbol{l}}+\text{MLPs}(\hat{\boldsymbol{%
+\alpha}}))).$ $\boldsymbol{I}_{ad}$ $\boldsymbol{F}_{sc}=\boldsymbol{\mathcal{F}}_{SC}(\boldsymbol{\mathcal{E}}(%
+\boldsymbol{I}_{rd})$ $(\boldsymbol{\theta}_{id}^{p},\boldsymbol{\theta}_{id})$ $\boldsymbol{\Phi}_{D}^{e}$ $\boldsymbol{\mathcal{F}}_{SC}(\boldsymbol{\mathcal{E}}(\boldsymbol{I}_{rd}))$ $\bar{\boldsymbol{l}}$ $p\textless 0.8$ $\displaystyle\boldsymbol{\theta}_{e}=\text{Avg}(\boldsymbol{\Phi}_{E}^{e}(%
+\text{MLPs}(\boldsymbol{A})+\text{PE})),$ $\hat{\boldsymbol{I}}_{1}=\boldsymbol{\mathcal{D}}(\text{CCF}(\boldsymbol{z}_{T%
+},\boldsymbol{\mathcal{E}}(\boldsymbol{I}_{rd}^{1}),\boldsymbol{Y})$ $\boldsymbol{\mathcal{G}}_{MOD}$ $\hat{\boldsymbol{I}}_{i}=\boldsymbol{\mathcal{D}}^{{}^{\prime}}(\text{CCF}(%
+\boldsymbol{z}_{T},\boldsymbol{\mathcal{E}}([\boldsymbol{I}_{rd}^{i},%
+\boldsymbol{I}_{id}^{i},\boldsymbol{I}_{ad}^{i}]),\boldsymbol{Y}),\boldsymbol{%
+\mathcal{E}}(\boldsymbol{I}_{bg}^{i}),\boldsymbol{m}_{i})$ $\boldsymbol{\mathcal{D}}^{{}^{\prime}}$ $\boldsymbol{\alpha}\in\mathbb{R}^{80}$ $\mathcal{L}_{reg}=\left\|\boldsymbol{l}-\boldsymbol{\beta}\right\|_{2}$ $\boldsymbol{\theta}_{id}$ $\boldsymbol{F}^{k}_{ve}$ $\boldsymbol{I}_{rd}$ $\displaystyle\bar{\boldsymbol{l}}=\boldsymbol{\Phi}_{E}^{c}(\text{MLPs}(%
+\boldsymbol{A})+\text{PE}),$ $i=2...H$ $\left\{\hat{\boldsymbol{Y}}_{1},\dots,\hat{\boldsymbol{Y}}_{N}\right\}$ $\boldsymbol{I}_{bg}^{i}=\boldsymbol{\mathcal{G}}_{IA}(\hat{\boldsymbol{I}}_{1})$ $\hat{\boldsymbol{V}}$ $\hat{\boldsymbol{\alpha}}\in\mathbb{R}^{80}$ $\boldsymbol{\mathcal{G}}_{IA}(\hat{\boldsymbol{I}}_{1})$ $\boldsymbol{\beta}\in\mathbb{R}^{L\times 64}$ $\boldsymbol{\Phi}_{D}^{c}$ $\boldsymbol{\mathcal{E}}(\boldsymbol{I}_{ad}))$ $\hat{\boldsymbol{I}_{1}}$ $\bar{\boldsymbol{\alpha}}\in\mathbb{R}^{512}$ $\boldsymbol{I}_{id}^{i}$ $\bar{\boldsymbol{\theta}}_{id}\in\mathbb{R}^{L\times 512}$ $\boldsymbol{\mathcal{E}}(\boldsymbol{I}_{rd}^{1})$ $\boldsymbol{\mathcal{E}}(\boldsymbol{I}_{id})$ $\boldsymbol{\theta}_{e}\in\mathbb{R}^{512}$ $\boldsymbol{I}_{bg}^{i}$ $\boldsymbol{\Phi}_{E}^{id}$ $\displaystyle\mathbb{E}_{\boldsymbol{z}_{0},\boldsymbol{\varepsilon}\sim N(0,%
+\boldsymbol{I}),t,\boldsymbol{F}_{sc}}\left\|\boldsymbol{\varepsilon}-%
+\boldsymbol{\varepsilon}_{\theta}\left(\boldsymbol{z}_{t},t,\boldsymbol{Y},%
+\boldsymbol{F}_{sc}\right)\right\|_{2}^{2},$ $\boldsymbol{I}_{bg}$ $\boldsymbol{F}^{k}_{vd}$ $\boldsymbol{\theta}_{e}$ $\boldsymbol{\mathcal{E}}$ $\boldsymbol{I}_{rd}^{i}$ $\displaystyle\hat{\boldsymbol{F}}_{vd}^{k}=\boldsymbol{F}_{vd}^{k}\otimes%
+\boldsymbol{m}+\text{Conv}(\boldsymbol{F}_{ve}^{k})\otimes\tilde{\boldsymbol{m%
+}},$ $\boldsymbol{\Phi}_{E}^{c}$ $\displaystyle\bar{\boldsymbol{\alpha}},\bar{\boldsymbol{\theta}}_{id}=%
+\boldsymbol{\Phi}_{E}^{id}([\tilde{\boldsymbol{\alpha}},\text{MLPs}(%
+\boldsymbol{A})]+\text{PE}),$ $\boldsymbol{\Phi}_{E}^{e}$ $\displaystyle\mathbb{E}_{\boldsymbol{z}_{0},\boldsymbol{\varepsilon}\sim N(0,%
+\boldsymbol{I}),t,\boldsymbol{Y}}\left\|\boldsymbol{\varepsilon}-\boldsymbol{%
+\varepsilon}_{\theta}\left(\boldsymbol{z}_{t},t,\boldsymbol{Y}\right)\right\|_%
+{2}^{2}.$ $\boldsymbol{l}\in\mathbb{R}^{L\times 64}$ $\boldsymbol{I}_{ad}^{i}=\boldsymbol{\mathcal{M}}(\hat{\boldsymbol{I}}_{i-1})$ $\boldsymbol{\mathcal{G}}_{IA}$ $\boldsymbol{\mathcal{F}}_{SC}$ $\boldsymbol{\mathcal{F}}_{TI}$ $\displaystyle\hat{\boldsymbol{\alpha}}=\text{MLPs}(\bar{\boldsymbol{\alpha}}),%
+\boldsymbol{\theta}_{id}=\text{Avg}(\bar{\boldsymbol{\theta}}_{id}),$ $\left\{\boldsymbol{Y}_{1},\dots,\boldsymbol{Y}_{M}\right\}$ $\mathcal{L}_{lip}=-\boldsymbol{X}\text{log}P(\hat{\boldsymbol{X}}|\boldsymbol{%
+V})$ $\boldsymbol{A}\in\mathbb{R}^{L\times 1280}$ $\left\{\boldsymbol{I}_{rd}^{i}\right\}_{i=1}^{H}$ $\hat{\boldsymbol{I}}_{1}$ $****$ $P(y|x,v)$ $\hat{y}_{test}$ $F_{S}(\boldsymbol{x}|\boldsymbol{P}_{j},\boldsymbol{B})$ $\begin{split}\mathcal{L}_{color}\left(\boldsymbol{x}\right)=\sum_{j}^{v}|\left%
+(F_{S}\left(\boldsymbol{x}\mid\boldsymbol{P}_{j},\boldsymbol{S},\boldsymbol{C}%
+_{\boldsymbol{T}}\right)-\boldsymbol{I}_{j}^{\textit{{r}}}\right)\odot\\
+F_{S}\left(\boldsymbol{x}\mid\boldsymbol{P}_{j},\boldsymbol{B}\right)|,\end{split}$ $\boldsymbol{x}=\left(\boldsymbol{E}+\lambda\boldsymbol{L}\right)^{-1}%
+\boldsymbol{\mu},$ $\boldsymbol{C}_{\boldsymbol{T}}$ $\mathcal{L}\left(\boldsymbol{x}\mid\Theta,\boldsymbol{I}^{\textit{{r}}}\right)%
+=\sum_{j}^{v}\left|F(\boldsymbol{x}\mid\Theta_{j})-\boldsymbol{I}_{j}^{\textit%
+{{r}}}\right|.$ ${\boldsymbol{T}}$ $\mathcal{L}_{depth}\left(\boldsymbol{x}\right)=\sum_{j}^{v}|\left(F_{D}\left(%
+\boldsymbol{x}\mid\boldsymbol{P}_{j}\right)-F_{D}\left(\boldsymbol{\tilde{x}}%
+\mid\boldsymbol{P}_{j}\right)\right)|,$ $\mathcal{M}=\left(\mathcal{X},\mathcal{E}\right)$ $\boldsymbol{L}\in\mathbb{R}^{n\times n}$ $\mathcal{L}_{normal}\left(\boldsymbol{x}\right)=\sum_{j}^{v}|\left(F_{N}\left(%
+\boldsymbol{x}\mid\boldsymbol{P}_{j}\right)-F_{N}\left(\boldsymbol{\tilde{x}}%
+\mid\boldsymbol{P}_{j}\right)\right)|,$ $\mathcal{L}=\mathcal{L}_{color}+\mathcal{L}_{depth}+\mathcal{L}_{normal}.$ $\boldsymbol{E}\in\mathbb{I}^{n\times n}$ $\boldsymbol{S}\in\mathcal{\mathbb{R}}^{9\times 3}$ $\boldsymbol{I}^{\textit{{r}}}\in\mathbb{R}^{w\times h\times c}$ $\frac{\partial\mathcal{L}}{\partial\boldsymbol{x}}$ $\boldsymbol{P}_{j}\in\mathbb{PL}\left(3\right)$ $\boldsymbol{x}\leftarrow\boldsymbol{x}-\eta\left(\boldsymbol{E}+\lambda%
+\boldsymbol{L}\right)^{-2}\frac{\partial\mathcal{L}}{\partial\boldsymbol{x}}.$ $\boldsymbol{L}_{ij}=\begin{cases}-w_{ij},&\textrm{if}\quad(i,j)\in\mathcal{E}%
+\\
+\sum_{(i,k)\in\mathcal{E}}w_{ik}&\mathrm{if}\quad i=j\\
+0&\mathrm{otherwise},\end{cases}$ $\boldsymbol{x}\in\mathbb{R}^{n\times 3}$ $\boldsymbol{p}(\textit{x},\textit{y})$ $\boldsymbol{I}(\boldsymbol{p}(x,y))=F(\boldsymbol{x};\Theta),$ $\boldsymbol{C}_{\boldsymbol{T}}\in\mathcal{\mathbb{R}}^{t\times t\times 3}$ $F_{S}:\mathbb{R}^{n\times 3}\rightarrow\mathbb{R}^{w\times h\times 3}$ $F_{D}:\mathbb{R}^{n\times 3}\rightarrow\mathbb{R}^{w\times h}$ $\boldsymbol{B}\in\left\{0,1\right\}^{t\times t\times 3}$ $F_{N}:\mathbb{R}^{n\times 3}\rightarrow\mathbb{R}^{w\times h\times 3}$ $\boldsymbol{\mu}\leftarrow\boldsymbol{\mu}-\eta\frac{\partial\boldsymbol{x}}{%
+\partial\boldsymbol{\mu}}\frac{\partial\mathcal{L}}{\partial\boldsymbol{x}},$ $\boldsymbol{\tilde{x}}\in\mathbb{R}^{m\times 3}$ $Loss=\sum_{i=1}^{N}BCE(v_{i},\hat{v}_{i})+\sum_{i=1}^{N}\sum_{j=1}^{N}\hat{v}_%
+{i}\hat{v}_{j}sim(i,j)$ $sim(i,j)=h_{i}^{L}*h_{j}^{L}$ $\bigtriangledown_{\boldsymbol{z}}f(v)$ $a_{i}^{*}=\frac{\|\boldsymbol{a}^{*}\|}{\mu}\Big{(}\tau-\gamma\eta^{*}\psi_{i}%
+(c_{i})-\log(1+e^{-y^{i}\cdot(\boldsymbol{w}^{*})^{T}\boldsymbol{x}^{i}})\Big{%
+)}^{+},$ $\psi_{i}(c_{i})\leq p+\frac{1}{m^{k}}.$ $\displaystyle+\int_{y_{i}=z_{i}}\epsilon_{i}(y,z_{-i})dy_{i}\Big{)}f_{i}(z_{i}%
+)dz_{i}f_{-i}(z_{-i})dz_{-i}.$ $t_{i}(0)\geq\int_{0}^{\infty}\epsilon_{i}(z)dz$ $c_{i}\cdot\epsilon_{i}(c)-t_{i}(c)$ $\Gamma(n,1)$ $\gamma\eta^{*}\psi_{i}(c_{i})-\log(1+e^{-y^{i}\cdot(\boldsymbol{w}^{*})^{T}%
+\boldsymbol{x}^{i}})$ $\boldsymbol{\epsilon}\in\mathbb{R}^{m}$ $p\gamma$ $\{\beta,\mu,\sigma,\gamma\}$ $t_{i}(D,\boldsymbol{c^{\prime}})=\Psi_{i}(c_{i})\epsilon_{i}(D,\boldsymbol{c^{%
+\prime}}),\quad\Psi_{i}(c)=c+\frac{F_{i}(c)}{f_{i}(c)}$ $\displaystyle=\mathbb{E}_{S}\big{[}\sup_{\|w\|\leq\beta}(\mathbb{E}[%
+\boldsymbol{w}]-\mathbb{\hat{E}}_{S}(\boldsymbol{w}))\big{]}$ $m\rightarrow\infty,$ $\lambda>\lambda_{conv}$ $\eta=\sum_{i=1}^{m}\epsilon_{i}=m\epsilon_{avg}$ $\displaystyle\qquad-\log(1+e^{-y^{i}\cdot\boldsymbol{w}^{T}\boldsymbol{x}^{i}}%
+)\big{)}\Big{]}\bigg{]}$ $V\subset\mathbb{R}^{n},$ $z_{i}=\log(\frac{1}{m})\ \forall i$ $\boldsymbol{w(D,\boldsymbol{c})}$ $\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}\cdot\boldsymbol{w}^{T}\boldsymbol{x}^{i}})%
++2\boldsymbol{b}^{T}\boldsymbol{w}/\eta$ $\log(1+e^{-\frac{\delta}{\sqrt{p\gamma}}})+\sqrt{\sigma p\gamma+2||b||\sqrt{p%
+\gamma}}<1.$ $\hat{\mathbb{L}}_{S}[\boldsymbol{w}]=\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}\cdot%
+\boldsymbol{w}^{T}\boldsymbol{x}^{i}})+\boldsymbol{b^{\prime}}^{T}\boldsymbol{%
+w},$ $\mathbb{L}(D,\boldsymbol{c};\boldsymbol{\epsilon},\boldsymbol{\theta})$ $\displaystyle=\Big{(}\tau-\gamma\eta^{*}\psi_{i}(c_{i})-\log(1+e^{-y^{i}\cdot(%
+\boldsymbol{w}^{*})^{T}\boldsymbol{x}^{i}})\Big{)}^{+}\Big{(}\frac{\|%
+\boldsymbol{a}^{*}\|}{\mu}\Big{)},$ $\psi_{i}(c_{i})\leq p+\frac{1}{m^{k}}$ $\epsilon_{i}(c)$ $\displaystyle\!\!-\!b(x_{1}^{i}y^{i})\begin{bmatrix}b(x_{1}^{i}y^{i})\!&\!\!-2%
+\lambda_{0}^{i}\frac{x_{2}^{i}y^{i}}{x_{1}^{i}y^{i}}&\!\!-2\lambda_{0}^{i}%
+\frac{x_{3}^{i}y^{i}}{x_{1}^{i}y^{i}}&\ldots&\!\!-2\lambda_{0}^{i}\frac{x_{n}^%
+{i}y^{i}}{x_{1}^{i}y^{i}}\\
+b(x_{2}^{i}y^{i})&2\lambda_{0}^{i}&0&\ldots&0\\
+.&.&.&\ldots&.\\
+.&.&.&\ldots&.\\
+.&.&.&\ldots&.\\
+b(x_{n}^{i}y^{i})&0&0&\ldots&2\lambda_{0}^{i}\\
+\end{bmatrix}\!.$ $\boldsymbol{\epsilon},\boldsymbol{c^{\prime}},\boldsymbol{t}$ $\displaystyle\leq f(v^{t})+\langle\bigtriangledown f(v^{t}),v^{t+1}-v^{t}%
+\rangle+\frac{K}{2}\|v^{t+1}-v^{t}\|^{2}$ $h(\boldsymbol{b^{\prime}}_{1})$ $\displaystyle=\mathbb{E}_{S}\big{[}\sup_{\|w\|\leq\beta}\mathbb{E}_{S^{\prime}%
+}[\hat{\mathbb{E}}_{S^{\prime}}(\boldsymbol{w})-\mathbb{\hat{E}}_{S}(%
+\boldsymbol{w})]\big{]}$ $\displaystyle\bigg{[}\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}\cdot\boldsymbol{w}^{T%
+}\boldsymbol{x}^{i}})+\frac{2\boldsymbol{b}^{T}\boldsymbol{w}}{\eta}$ $\displaystyle\frac{\mathbb{P}[\boldsymbol{w}(\boldsymbol{x}^{1},\ldots,%
+\boldsymbol{x}^{m}=\boldsymbol{d},y^{1},\ldots,y^{m}=y)\in V]}{\mathbb{P}[%
+\boldsymbol{w}(\boldsymbol{x}^{1},\ldots,\boldsymbol{x}^{m}=\boldsymbol{d^{%
+\prime}},y^{1},\ldots,y^{m}=y^{\prime})\in V]}=\frac{h(\boldsymbol{b^{\prime}}%
+_{1})}{h(\boldsymbol{b^{\prime}}_{2})}$ $z^{i}=(\boldsymbol{x}^{i},y^{i}),$ $\displaystyle\lim_{m\rightarrow\infty}\min_{\boldsymbol{w},\boldsymbol{%
+\epsilon},\|\boldsymbol{w}\|\leq\beta}\mathbb{E}[\mathbb{I}_{\{sign(%
+\boldsymbol{w}^{T}\boldsymbol{x})\neq y\}}]+\gamma\sum_{i=1}\epsilon_{i}\Psi_{%
+i}(c_{i})$ $\gamma K\leq 1$ $D=\{(\boldsymbol{x}^{1},y^{1}),\ldots,(\boldsymbol{x}^{m},y^{m})\}$ $\mathbb{L}(D,\boldsymbol{c};\boldsymbol{\epsilon},\boldsymbol{\boldsymbol{w}})%
+\leq\hat{\mathbb{L}}(D,\boldsymbol{a},\boldsymbol{w},\eta)+\mu\|a\|+\frac{%
+\sigma}{\eta}.$ $\sup_{\|\boldsymbol{w}\|\leq\beta}\Big{|}\mathbb{L}(D,\boldsymbol{c};%
+\boldsymbol{\epsilon},\boldsymbol{\boldsymbol{w}})-\hat{\mathbb{L}}(D,%
+\boldsymbol{a},\boldsymbol{w},\eta)\Big{|}\leq\mu\|a\|+\frac{\sigma}{\eta}.$ $0 $(\lambda-\lambda_{conv})$ $\{\boldsymbol{z}:\sum e^{z_{i}}=1,\eta e^{z_{i}}\leq\epsilon_{i}\}$ $\sum_{i=1}^{m}\Big{(}\tau-\gamma\eta^{*}\psi_{i}(c_{i})-\log(1+e^{-y^{i}\cdot(%
+\boldsymbol{w}^{*})^{T}\boldsymbol{x}^{i}})\Big{)}^{+}=\frac{\mu}{\|%
+\boldsymbol{a}^{*}\|}.$ $\boldsymbol{\epsilon(D,\boldsymbol{c^{\prime}})}$ $\Psi_{i}(\boldsymbol{c_{i}})$ $f^{*}_{\eta}=\inf_{\boldsymbol{w},\boldsymbol{z}}f(\boldsymbol{w},\boldsymbol{%
+z},\eta)$ $\boldsymbol{w}\leftarrow\boldsymbol{w}-\alpha\frac{d}{d\boldsymbol{w}}g(%
+\boldsymbol{w},\boldsymbol{z},\epsilon_{\text{avg}}),$ $\displaystyle P(|\phi(S)-\mathbb{E}_{S}[\phi(S)]|>t)$ $\|\boldsymbol{w}\|\leq\beta,\eta>0,\boldsymbol{a}>0$ $\sum_{i}a_{i}=1,a\geq 0,\eta\geq 0,a_{i}\eta\leq\epsilon_{i}$ $\mathbb{E}[\mathbb{I}_{sign(\boldsymbol{w}^{T}\boldsymbol{x})\neq y}]$ $\displaystyle a=e^{z_{i}}\log(1+e^{-\boldsymbol{w}^{T}\boldsymbol{x}^{i}\cdot y%
+^{i}}),$ $e^{-\epsilon_{i}}\leq\frac{\mathbb{P}[\mathbb{A}(S)\in V]}{\mathbb{P}[\mathbb{%
+A}(S^{\prime})\in V]}\leq e^{\epsilon_{i}}\quad\forall i\in\{1,2,\ldots,|S|\}.$ $v(t)=\sum_{i=0}^{n-1}\frac{(t/2)^{i}}{i!}e^{-\frac{t}{2}}$ $L_{1}>0$ $\eta=m\cdot\epsilon_{avg}$ $\epsilon_{i}(D,\boldsymbol{c})$ $\mathbb{L}(D,\boldsymbol{c};\boldsymbol{\epsilon},\boldsymbol{\boldsymbol{w}})%
+=\mathbb{E}[\mathbb{I}_{\{sign(\boldsymbol{w}^{T}\boldsymbol{x})\neq y\}}]$ $\max(0,f(x)).$ $\boldsymbol{b^{\prime}}=\frac{2\boldsymbol{b}}{\eta}$ $\boldsymbol{b^{\prime}}_{1}$ $\mathrm{min}_{\boldsymbol{w}}\hat{\mathbb{L}}(D,\boldsymbol{w},\boldsymbol{a},\eta)$ $(\boldsymbol{a},\eta)\in\mathbb{F}$ $(f(x))^{+}$ $\min_{\boldsymbol{w},\boldsymbol{\epsilon}(\cdot)}\mathbb{E}_{\boldsymbol{c}}%
+\bigg{[}\mathbb{L}(D,\boldsymbol{c};\boldsymbol{\epsilon},\boldsymbol{\theta})%
++\gamma\cdot\sum_{i=1}^{m}\Psi_{i}(c_{i})\epsilon_{i}(D,\boldsymbol{c})\bigg{]}.$ $c_{i}\epsilon_{i}.$ $\displaystyle\qquad+\frac{2\boldsymbol{b}^{T}\boldsymbol{w}}{\eta}+\mu\|%
+\boldsymbol{a}\|+\sigma\frac{1}{\eta}\bigg{]}+\gamma\eta\sum a_{i}\Psi_{i}(c_{%
+i})=0.$ $\displaystyle\qquad\leq\exp\Bigg{(}\frac{-2t^{2}}{\sum a_{i}^{2}\log^{2}(1+e^{%
+\beta})+(\frac{2\beta v^{-1}(\delta^{\prime})}{\eta})^{2}}\Bigg{)}.$ $\displaystyle\qquad\qquad\bigg{(}\frac{1}{\log(1+e^{-\boldsymbol{w}^{T}%
+\boldsymbol{x}^{i}y^{i}})}-\ln 2\cdot e^{\boldsymbol{w}^{T}\boldsymbol{x}^{i}y%
+^{i}}\bigg{)}.$ $\displaystyle 2\lambda>\bigg{(}\frac{1}{\ln 2}\bigg{)}^{2}\cdot\max_{i}\|%
+\boldsymbol{x}^{i}\|^{2}\bigg{(}\frac{e^{-\boldsymbol{w}^{T}\boldsymbol{x}^{i}%
+y^{i}}}{1+e^{-\boldsymbol{w}^{T}\boldsymbol{x}^{i}y^{i}}}\bigg{)}^{2}$ $\displaystyle+\mathbb{E}_{S^{\prime},\sigma_{i}}\bigg{[}\sup_{\|w\|\leq\beta}%
+\Big{[}\sum_{i=1}^{m}-a_{i}\sigma_{i}\big{(}\log(1+e^{-y^{\prime i}\cdot%
+\boldsymbol{w}^{T}\boldsymbol{x^{\prime}}^{i}})\Big{]}\bigg{]}$ $(\boldsymbol{w},\boldsymbol{z})$ $\epsilon_{avg}\in[0,L]$ $\displaystyle\qquad+\mu\|\boldsymbol{a}\|+\sigma\frac{1}{\eta}+\gamma\sum_{i=1%
+}^{m}\epsilon_{i}\Psi_{i}(c_{i})\bigg{]},$ $\hat{\boldsymbol{w_{2}}}$ $t_{i}(D,\boldsymbol{c^{\prime}})=\Psi_{i}(c_{i})\epsilon_{i}(D,\boldsymbol{c^{%
+\prime}}),$ $\|\boldsymbol{x}\|,\|\boldsymbol{w}\|$ $\eta=\frac{1}{m^{k^{\prime}}}$ $\displaystyle=f(v)-f(v^{*})+f(v^{*})-L$ $\mu(\delta,\beta)=\Big{(}\frac{3\ln\frac{1}{\delta}}{\sqrt{2}}\Big{)}\log(1+e^%
+{\beta})+\frac{\beta}{\ln 2}$ $\mathbb{B}_{\boldsymbol{c}_{-i}}[t_{i}(D,c,\boldsymbol{c}_{-i})]$ $\sigma(\delta,\delta^{\prime},\beta)=\Big{(}\frac{6\ln\frac{1}{\delta}}{\sqrt{%
+2}}+1\Big{)}\Big{(}2\beta v^{-1}(\delta^{\prime})\Big{)}$ $\displaystyle\!\!\begin{bmatrix}a&b(x_{1}^{i}y^{i})&0&\ldots&0\\
+b(x_{1}^{i}y^{i})&2\lambda_{0}^{i}+c(x_{1}^{i}y^{i})^{2}&-2\lambda_{0}^{i}%
+\frac{x_{2}^{i}y^{i}}{x_{1}^{i}y^{i}}&\ldots&-2\lambda_{0}^{i}\frac{x_{n}^{i}y%
+^{i}}{x_{1}^{i}y^{i}}\\
+b(x_{2}^{i}y^{i})&c(x_{1}^{i}y^{i})(x_{2}^{i}y^{i})&2\lambda_{0}^{i}&\ldots&0%
+\\
+.&.&.&\ldots&.\\
+.&.&.&\ldots&.\\
+.&.&.&\ldots&.\\
+b(x_{n}^{i}y^{i})&c(x_{1}^{i}y^{i})(x_{n}^{i}y^{i})&0&\ldots&2\lambda_{0}^{i}%
+\\
+\end{bmatrix}\!.$ $\displaystyle\qquad+\tau(1-\sum_{i=1}^{m}a_{i})-\sum_{i=1}^{m}\zeta_{i}a_{i}-%
+\kappa\eta.$ $\displaystyle\qquad\qquad+\mu\|\boldsymbol{a}\|+\sigma\frac{1}{\eta}\bigg{]}+%
+\gamma\eta\sum a_{i}\Psi_{i}(c_{i})$ $\displaystyle\min_{\boldsymbol{w}}\Bigg{[}\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}%
+\cdot\boldsymbol{w}^{T}\boldsymbol{x}^{i}})+\frac{\lambda}{2}\|\boldsymbol{w}%
+\|^{2}+\frac{2\boldsymbol{b}^{T}\boldsymbol{w}}{\eta}\Bigg{]},$ $\mathbb{F}=\{(\boldsymbol{a},\eta):\eta>0,\sum_{i}a_{i}=1,a_{i}>0,a_{i}\eta%
+\leq\epsilon_{i}\ \forall i\}$ $h_{i}(c)$ $\displaystyle\leq\frac{\|\boldsymbol{w}\|}{\ln 2}\mathbb{E}_{\sigma}[\sum_{i}a%
+_{i}\sigma_{i}\boldsymbol{x}^{i}]+\frac{\beta v^{-1}(\delta^{\prime})}{\eta}$ $\mathbb{I}_{\{\cdot\}}$ $\displaystyle\displaystyle\min_{\boldsymbol{a},\eta,\boldsymbol{w}}\bigg{[}%
+\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}\cdot\boldsymbol{w}^{T}\boldsymbol{x}^{i}})%
++\frac{2\boldsymbol{b}^{T}\boldsymbol{w}}{\eta}$ $\eta=\sum_{i}\epsilon_{i}$ $\sigma/\eta$ $a_{i}=\epsilon_{i}/\eta$ $\{(\boldsymbol{x}^{1},y^{1}),\ldots,(\boldsymbol{x}^{m-1},y^{m-1}),(%
+\boldsymbol{d^{\prime}},y^{\prime})\}$ $\hat{\boldsymbol{w_{1}}}$ $(2\lambda_{0}^{i})^{(n-1)}(2\lambda_{0}^{i}+c\|\boldsymbol{x}\|^{2})$ $\|\boldsymbol{w}\|^{2}$ $\displaystyle\lim_{m\rightarrow\infty}\|a\|=0$ $\mu,\sigma,\eta\in\mathbb{R}_{+}$ $\{(\boldsymbol{x}^{i},y^{i}),c_{i}\},$ $\lim_{p\rightarrow 0}\lim_{m\rightarrow\infty}\min_{\begin{subarray}{c}%
+\boldsymbol{w},\boldsymbol{\epsilon}\\
+\beta,\|\boldsymbol{w}\|\leq\beta\end{subarray}}\mathbb{E}[\mathbb{I}_{\{sign(%
+\boldsymbol{w}^{T}\boldsymbol{x})\neq y\}}]\!+\!\gamma\sum_{i=1}^{m}\epsilon_{%
+i}\Psi_{i}(c_{i})\!\leq\!1.$ $\displaystyle\mathbb{E}\big{[}\mathbb{\hat{L}}_{S}[\boldsymbol{w}]\big{]}$ $a_{i}=\frac{1}{N}$ $\sum_{i=0}^{n-1}\frac{(\frac{\eta r}{2\beta})^{i}}{i!}e^{-\frac{\eta r}{2\beta%
+}}=\delta^{\prime}.$ $\displaystyle\displaystyle\min_{\boldsymbol{a},\eta,\boldsymbol{w},\boldsymbol%
+{\epsilon}}$ $(\boldsymbol{a^{*}},\boldsymbol{w^{*}},\eta^{*})$ $\big{|}\|\boldsymbol{b^{\prime}}_{1}\|-\|\boldsymbol{b^{\prime}}_{2}\|\big{|}<%
+2a_{m}$ $\epsilon_{i}=a_{i}\eta.$ $\gamma\eta\sum_{i=1}^{m}e^{z_{i}}\psi_{i}(c_{i})$ $g(\boldsymbol{w},\boldsymbol{z},\eta)$ $\displaystyle\mbox{COST}(c_{i},\boldsymbol{c}_{-i},c_{i},\boldsymbol{c}_{-i},z%
+^{i};\epsilon_{i},t_{i})$ $\|\log(1+e^{-\boldsymbol{w}^{T}\boldsymbol{x}y})\|$ $\boldsymbol{b^{\prime}}_{2}$ $\displaystyle\!a\cdot\!\begin{bmatrix}2\lambda_{0}^{i}+c(x_{1}^{i}y^{i})^{2}&%
+\!\!-2\lambda_{0}^{i}\frac{x_{2}^{i}y^{i}}{x_{1}^{i}y^{i}}&\!\!-2\lambda_{0}^{%
+i}\frac{x_{3}^{i}y^{i}}{x_{1}^{i}y^{i}}&\ldots&\!\!-2\lambda_{0}^{i}\frac{x_{n%
+}^{i}y^{i}}{x_{1}^{i}y^{i}}\\
+c(x_{1}^{i}y^{i})(x_{2}^{i}y^{i})&2\lambda_{0}^{i}&0&\ldots&0\\
+.&.&.&\ldots&.\\
+.&.&.&\ldots&.\\
+.&.&.&\ldots&.\\
+c(x_{1}^{i}y^{i})(x_{n}^{i}y^{i})&0&0&\ldots&2*\lambda_{0}^{i}\\
+\end{bmatrix}$ $2\boldsymbol{c}-e^{-4}$ $f(\boldsymbol{w}^{t},\boldsymbol{z}^{t},\eta)-f^{*}_{\eta}\leq\alpha^{t}(f(%
+\boldsymbol{w}^{0},\boldsymbol{z}^{0},\eta)-f^{*}_{\eta}).$ $\displaystyle\leq\frac{\beta}{\ln 2}\sqrt{\sum_{i}a_{i}^{2}}+\frac{\beta v^{-1%
+}(\delta^{\prime})}{\eta},$ $\displaystyle\leq\lim_{m\rightarrow\infty}\log(1+e^{-\delta m^{k^{\prime\prime%
+}}})+\frac{2||b||m^{k^{\prime\prime}}}{m^{k^{\prime}}}+\sigma\frac{1}{m^{k^{%
+\prime}}}+\gamma\frac{m^{k^{\prime}}}{m^{k}}$ $\beta=\frac{1}{\sqrt{p\gamma}}$ $\displaystyle\qquad\qquad+\mu\|\boldsymbol{a}\|+\sigma\frac{1}{\eta}+\gamma%
+\eta\sum_{i=1}^{m}a_{i}\Psi_{i}(c_{i})\bigg{]}$ $\displaystyle\qquad\qquad\bigg{(}\frac{1}{\log(1+e^{-\boldsymbol{w}^{T}%
+\boldsymbol{x}^{i}y^{i}})}-\ln 2\cdot e^{\boldsymbol{w}^{T}\boldsymbol{x}^{i}y%
+^{i}}\bigg{)},$ $\displaystyle\mathbb{E}_{S}[\phi(S)]$ $\eta^{*}\rightarrow\infty$ $\mbox{COST}(c_{i},\boldsymbol{c}_{-i},c^{\prime}_{i},\boldsymbol{c}_{-i},z^{i}%
+;\epsilon_{i},t_{i})\leq 0\quad\forall i,c^{\prime}_{i},\boldsymbol{c}$ $\epsilon_{i}=a_{i}\eta$ $\sum_{i}t_{i}(D,\boldsymbol{c^{\prime}})$ $\displaystyle=e^{\eta(\|\boldsymbol{b^{\prime}}_{1}\|-\|\boldsymbol{b^{\prime}%
+}_{2}\|)/2}\leq e^{a_{m}\eta}\leq e^{\epsilon_{m}},$ $\displaystyle\lim_{m\to\infty}m\cdot\mathbb{P}\big{(}\psi_{i}(c_{i})\leq p+1/m%
+^{k}\big{)}\rightarrow\infty.$ $\|\boldsymbol{x}^{i}\|\leq 1\ \ \forall i$ $\alpha\in(0,1],\ g_{\min}=\infty$ $\displaystyle a_{i}^{*}$ $\quad\quad\quad\forall\boldsymbol{w}\ s.t.\ \|\boldsymbol{w}\|\leq\beta,\ (%
+\boldsymbol{a},\eta)\in\mathbb{F}$ $t_{i}(0)\geq\int_{0}^{c_{i}}\epsilon_{i}(z)dz.$ $\displaystyle\lim_{m\rightarrow\infty}\min_{\begin{subarray}{c}\boldsymbol{a},%
+\eta,\boldsymbol{w}\\
+\|w\|\leq\beta,\beta\end{subarray}}\bigg{[}\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}%
+\cdot\boldsymbol{w}^{T}\boldsymbol{x}^{i}})+\frac{2\boldsymbol{b}^{T}%
+\boldsymbol{w}}{\eta}$ $(f(x))^{+}=\max(0,f(x))$ $\displaystyle\leq\lim_{m\rightarrow\infty}\min_{\beta,\eta}\bigg{[}\log(1+e^{-%
+\delta\|x\|\cdot\|w\|})+\|\boldsymbol{b}\|\frac{2\beta}{\eta}$ $\epsilon_{i},$ $\mathbb{E}_{\boldsymbol{c}_{-i}}[\epsilon_{i}(D,c,\boldsymbol{c}_{-i})]$ $\|\boldsymbol{b}\|\sim\Gamma(n,1)$ $\displaystyle=\sum_{i}a_{i}\mathbb{E}[\log(1+e^{-y\cdot\boldsymbol{w}^{T}%
+\boldsymbol{x}})]+\mathbb{E}[\boldsymbol{b^{\prime}}^{T}\boldsymbol{w}]$ $(1-\delta)(1-\delta^{\prime})$ $\mathbb{U}[e^{-4},5e^{-4}]$ $\mu\|\boldsymbol{a}\|+\sigma/\eta$ $\displaystyle\Big{|}\mathbb{L}(D,\boldsymbol{c};\boldsymbol{\epsilon},%
+\boldsymbol{\boldsymbol{w}})-\hat{\mathbb{L}}(D,\boldsymbol{a},\boldsymbol{w},%
+\eta)\Big{|}$ $\frac{1}{1+e^{y^{\prime}\hat{\boldsymbol{w_{2}}}^{T}\boldsymbol{d^{\prime}}}}<1$ $\displaystyle\leq\frac{1}{\ln 2}E_{\sigma}[\sup_{\boldsymbol{w}}\sum_{i}a_{i}%
+\sigma_{i}(-y^{i}\boldsymbol{w}^{T}\boldsymbol{x}^{i})]+\frac{\beta v^{-1}(%
+\delta^{\prime})}{\eta}$ $\|\boldsymbol{b^{\prime}}_{1}-\boldsymbol{b^{\prime}}_{2}\|<2a_{m}$ $\displaystyle\leq\mu(\delta,\beta)\|a\|+\sigma(\delta,\delta^{\prime},\beta)%
+\Big{(}\frac{1}{\eta}\Big{)}.$ $\epsilon_{\text{avg}}$ $a_{i}=e^{z_{i}}$ $\boldsymbol{z}_{\text{opt}}\leftarrow\boldsymbol{z},$ $z^{i}=(\boldsymbol{x}^{i},y^{i}).$ $\displaystyle c=e^{z_{i}}\frac{e^{-\boldsymbol{w}^{T}\boldsymbol{x}^{i}y^{i}}}%
+{(1+e^{-\boldsymbol{w}^{T}\boldsymbol{x}^{i}y^{i}})^{2}}\frac{1}{\ln 2}.$ $\|\boldsymbol{x}^{i}\|\leq 1$ $(\boldsymbol{a},\boldsymbol{w})$ $\hat{\boldsymbol{w}}=\mathrm{argmin}_{\boldsymbol{w}}\hat{\mathbb{L}}(D,%
+\boldsymbol{w},\boldsymbol{a},\eta)$ $\boldsymbol{z}\leftarrow\boldsymbol{z}-\alpha\frac{d}{d\boldsymbol{z}}g(%
+\boldsymbol{w},\boldsymbol{z},\epsilon_{\text{avg}}),$ $\gamma\eta\sum_{i=1}^{m}a_{i}\Psi_{i}(c_{i})$ $R_{m}(\boldsymbol{w})$ $\sum_{i=1}^{m}\lambda_{0}^{i}=\lambda$ $\sum_{i}e^{z_{i}}=1,\eta\geq 0,\eta e^{z_{i}}\leq\epsilon_{i}$ $\lambda_{conv}$ $\boldsymbol{c}=[c_{1},c_{2},\ldots,c_{m}].$ $\big{|}\boldsymbol{b^{\prime}}^{T}\boldsymbol{w}\big{|} $N\rightarrow m\mathbb{P}\big{(}\psi_{i}(c_{i})\leq\frac{1}{m^{k}}\big{)}$ $0 $\displaystyle\qquad+\mu\|\boldsymbol{a}\|+\sigma\frac{1}{\eta}+\gamma\eta\sum a%
+_{i}\Psi_{i}(c_{i})$ $\eta r/\beta=t$ $\epsilon_{avg}$ $\mathbb{L}(D,\boldsymbol{c^{\prime}};\boldsymbol{\epsilon,\theta})$ $\boldsymbol{d^{\prime}}$ $\log(1+e^{-\frac{\delta}{\sqrt{p\gamma}}})+2\sqrt{\sigma p\gamma+2\|%
+\boldsymbol{b}\|\sqrt{p\gamma}}<1.$ $\boldsymbol{b^{\prime}}_{1}-\frac{a_{m}\cdot\boldsymbol{d}y}{1+e^{y\hat{%
+\boldsymbol{w_{1}}}^{T}\boldsymbol{d}}}=\boldsymbol{b^{\prime}}_{2}-\frac{a_{m%
+}\cdot\boldsymbol{d^{\prime}}y^{\prime}}{1+e^{y^{\prime}\hat{\boldsymbol{w_{2}%
+}}^{T}\boldsymbol{d^{\prime}}}}.$ $\displaystyle f(v)-L$ $\displaystyle\qquad\qquad+\mu\|\boldsymbol{a}\|+\sigma\frac{1}{\eta}+\gamma%
+\eta\sum_{i=1}^{m}a_{i}\Psi_{i}(c_{i})\bigg{]},$ $\displaystyle=\mathbb{E}_{S,S^{\prime}}\bigg{[}\sup_{\|w\|\leq\beta}\Big{[}%
+\sum_{i=1}^{m}a_{i}\sigma_{i}\big{(}\log(1+e^{-y^{\prime i}\cdot\boldsymbol{w}%
+^{T}\boldsymbol{x^{\prime}}^{i}})$ $\|\boldsymbol{w}\|$ $a_{i}=1/m$ $\displaystyle\lim_{m\rightarrow\infty}\min_{\boldsymbol{a},\eta,\boldsymbol{w}%
+,\|w\|\leq\beta,\beta}\bigg{[}\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}\cdot%
+\boldsymbol{w}^{T}\boldsymbol{x}^{i}})+\frac{2\boldsymbol{b}^{T}\boldsymbol{w}%
+}{\eta}$ $\sum_{i=1}^{m}a_{i}=1$ $\displaystyle=f(v^{t})-\frac{\gamma}{2}(2-K\gamma)\|\bigtriangledown f(v^{t})%
+\|^{2}$ $\psi_{i}(c_{i})$ $\displaystyle=\mathbb{E}_{c_{i}}[c_{i}\epsilon_{i}(c_{i})]+\mathbb{E}_{c_{i}}[%
+\int_{c_{i}}\epsilon_{i}(z)dz]$ $R_{m}(\boldsymbol{w})=\mathbb{E}_{\sigma,S}[\sup_{\|w\|\leq\beta}\sum_{i=1}^{m%
+}a_{i}\sigma_{i}\log(1+e^{-y^{i}\cdot\boldsymbol{w}^{T}\boldsymbol{x}^{i}})],$ $\displaystyle 2\lambda_{0}^{i}>\frac{b^{2}-ac}{a}\|\boldsymbol{x}^{i}\|^{2}$ $\epsilon_{\text{opt}}\leftarrow\epsilon_{\text{avg}},$ $\Psi_{i}(c_{i})=c_{i}+\frac{F_{i}(c_{i})}{f_{i}(c_{i})}\ \ \forall i\in\mathbb%
+{N},c_{i}\in\mathbb{R}$ $\mbox{COST}(c_{i},\boldsymbol{c}_{-i},c^{\prime}_{i},\boldsymbol{c}_{-i};%
+\epsilon_{i},t_{i})=c_{i}\cdot\epsilon_{i}-t_{i}.$ $c_{i}\epsilon_{i}(c_{i})+\int_{c_{i}}\epsilon_{i}(z)dz$ $a_{i} $\|\boldsymbol{w}\|\leq\beta$ $\displaystyle=\Big{(}\frac{\sigma+2\boldsymbol{b}^{T}\boldsymbol{w}}{\gamma%
+\sum_{i=1}^{m}\psi_{i}(c_{i})a_{i}^{*}}\Big{)}^{1/2}.$ $z^{i}=(\boldsymbol{x}^{i},y^{i})$ $\displaystyle\textrm{s.t.}\|\boldsymbol{w}\|\leq\beta,(\boldsymbol{a},\eta)\in%
+\mathbb{F}.$ $\sqrt{\sigma p\gamma+2\|\boldsymbol{b}\|\sqrt{p\gamma}}/\gamma$ $\|\boldsymbol{x}^{i}\|\leq 1\ \forall i$ $\hat{\mathbb{L}}(D,\boldsymbol{a},\boldsymbol{w},\eta)$ $\exists\ k>0,$ $t_{i}(\boldsymbol{c})=\Psi_{i}(c_{i})\epsilon_{i}(\boldsymbol{c}),$ $\displaystyle\!\leq\!\hat{R}_{S}(\boldsymbol{w})\!+\!\sqrt{\frac{\ln\frac{1}{%
+\delta}}{2}}\Bigg{(}\sqrt{\sum a_{i}^{2}\log^{2}(1+e^{\beta})}+\frac{2\beta v^%
+{-1}(\delta^{\prime})}{\eta}\Bigg{)}.$ $\displaystyle\mathbb{E}[\mathbb{I}_{\{sign(\boldsymbol{w}^{T}\boldsymbol{x})%
+\neq y\}}]\leq\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}\boldsymbol{w}^{T}\boldsymbol%
+{x}^{i})})+\boldsymbol{b^{\prime}}^{T}\boldsymbol{w}$ $\displaystyle=\mathbb{E}_{S,S^{\prime}}\big{[}\sup_{\|w\|\leq\beta}[\mathbb{%
+\hat{E}}_{S^{\prime}}(\boldsymbol{w})-\mathbb{\hat{E}}_{S}(\boldsymbol{w})]%
+\big{]}$ $\forall\ \delta>0$ $\|\bigtriangledown f(v^{t})\|^{2}$ $\displaystyle\mathbb{E}_{c_{i}}[t_{i}(c_{i})]$ $\epsilon_{i}(D,\boldsymbol{c^{\prime}})$ $\displaystyle\leq\log(1+e^{-\frac{\delta}{\sqrt{p\gamma}}})+\sqrt{\sigma p%
+\gamma+2\|\boldsymbol{b}\|\sqrt{p\gamma}}.$ $\displaystyle g(\boldsymbol{w},\boldsymbol{z},\eta)=f(\boldsymbol{w},%
+\boldsymbol{z},\eta)+\eta\gamma\sum_{i=1}^{m}e^{z_{i}}\Psi_{i}(c_{i}).$ $\epsilon_{i}(c_{i})\geq 0$ $\displaystyle\leq\delta+\frac{1}{2\mu}\|\bigtriangledown f(v)\|^{2}.$ $\mu||\boldsymbol{a}||$ $A+2\lambda_{0}^{i}I$ $\|w\|\leq\beta$ $\mathbb{N}(0,1)$ $\log(1+e^{-\frac{\delta}{\sqrt{p\gamma}}})$ $L\in\mathbb{R}_{+}.$ $\boldsymbol{\epsilon}=(\epsilon_{i})_{i=1}^{m}\in\mathbb{R}_{+}^{m}$ $\displaystyle\!\!\!f(\boldsymbol{w},\boldsymbol{z},\eta)$ $\mathbb{E}[\boldsymbol{w}]=0$ $c_{i}^{\prime}=c_{i}.$ $\displaystyle\sup_{\|\boldsymbol{w}\|\leq\beta}$ $\epsilon_{i}\leq k\epsilon_{avg}$ $\displaystyle\lim_{p\rightarrow 0}\lim_{m\rightarrow\infty}\min_{\boldsymbol{w%
+},\boldsymbol{\epsilon}}\mathbb{E}[\mathbb{I}_{\{sign(\boldsymbol{w}^{T}%
+\boldsymbol{x})\neq y\}}]+\gamma\sum_{i=1}^{m}\epsilon_{i}\Psi_{i}(c_{i})$ $\displaystyle\qquad-e^{z^{2}}(\log(1+e^{-\boldsymbol{w^{2}}^{T}\boldsymbol{x}y%
+})+\frac{e^{z^{2}}}{\|(e^{z^{2}})\|}\|^{2}.$ $a_{i}\eta=\epsilon_{i}\ \forall i$ $\displaystyle\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}\cdot\boldsymbol{w}^{T}%
+\boldsymbol{x}^{i}})+\frac{\lambda}{2}\|\boldsymbol{w}\|^{2}+\frac{2%
+\boldsymbol{b}^{T}\boldsymbol{w}}{\eta}$ $\big{\{}(\boldsymbol{x}^{1},y^{1}),\ldots,(\boldsymbol{x}^{m},y^{m})\big{\}}$ $(\boldsymbol{w}^{t},\boldsymbol{z}^{t})_{t\in\mathbb{N}}$ $\displaystyle\leq\delta+\langle\bigtriangledown f(v),v-v^{*}\rangle-\frac{\mu}%
+{2}\|v^{*}-v\|^{2}$ $\displaystyle\|\bigtriangledown f(v^{1})-\bigtriangledown f(v^{2})\|^{2}$ $\mathbb{E}_{c_{i}}[t_{i}(c_{i})]$ $(\boldsymbol{a},\eta)$ $\boldsymbol{w}_{\text{opt}}\leftarrow\boldsymbol{w},$ $\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}\cdot\boldsymbol{w}^{T}\boldsymbol{x}^{i}})$ $\displaystyle\min_{\boldsymbol{a},\eta,\boldsymbol{w}}$ $\|\boldsymbol{x}^{i}\|\leq 1,$ $c\|\boldsymbol{x}\|^{2}$ $\displaystyle\hat{R}_{S}(\boldsymbol{w})$ $\displaystyle b=e^{z_{i}}\frac{e^{-\boldsymbol{w}^{T}\boldsymbol{x}^{i}y^{i}}}%
+{1+e^{-\boldsymbol{w}^{T}\boldsymbol{x}^{i}y^{i}}}\frac{1}{\ln 2},$ $m\rightarrow\infty.$ $\Psi_{i}(c_{i})=c_{i}+\frac{F_{i}(c_{i})}{f_{i}(c_{i})}$ $\mathbb{E}_{c_{i}}[t_{i}(c_{i})]=\mathbb{E}_{\boldsymbol{c}}[\Psi_{i}(c_{i})%
+\epsilon_{i}(\boldsymbol{c})],$ $a_{i}=\frac{1}{m}$ $\displaystyle R_{m}(\boldsymbol{w})\leq\hat{R}_{S}(\boldsymbol{w})$ $\Psi_{i}(c_{i})$ $\frac{\lambda}{2}||\boldsymbol{w}||^{2}$ $\hat{R}_{S}(\boldsymbol{w})$ $f(v^{*})-L<\delta$ $\mathbb{L}(D,\boldsymbol{c};\boldsymbol{\epsilon},\boldsymbol{\boldsymbol{w}})$ $\displaystyle\qquad+\|e^{z^{1}}(\log(1+e^{-\boldsymbol{w^{1}}^{T}\boldsymbol{x%
+}y})+\frac{e^{z^{1}}}{\|(e^{z^{1}})\|}$ $\displaystyle\qquad+\Big{(}\frac{6\ln\frac{1}{\delta}}{\sqrt{2}}+1\Big{)}\Big{%
+(}\frac{2\beta v^{-1}(\delta^{\prime})}{\eta}\Big{)}.$ $\displaystyle=\mathbb{E}[\log(1+e^{-y\cdot\boldsymbol{w}^{T}\boldsymbol{x}})]=%
+\mathbb{L}[\boldsymbol{w}].$ $\|\boldsymbol{x}\|\leq 1$ $\bigtriangledown_{\boldsymbol{w}}f(v)$ $\displaystyle\leq\mathbb{E}_{S,\sigma_{i}}\bigg{[}\sup_{\|w\|\leq\beta}\Big{[}%
+\sum_{i=1}^{m}a_{i}\sigma_{i}\big{(}\log(1+e^{-y^{i}\cdot\boldsymbol{w}^{T}%
+\boldsymbol{x}^{i}})\Big{]}\bigg{]}$ $\mathbb{U}[p,q]$ $\displaystyle\phi(S)$ $\psi_{i}(c_{i})\in[p,q]$ $\displaystyle=\bigg{[}\sum_{i=1}^{m}e^{z_{i}}\log(1+e^{-y^{i}\cdot\boldsymbol{%
+w}^{T}\boldsymbol{x}^{i}})+\frac{\lambda}{2}\|\boldsymbol{w}\|^{2}$ $\mathbb{E}[\mathbb{I}_{\{sign(\boldsymbol{w}^{T}\boldsymbol{x})\neq y\}}]$ $\eta=\frac{\sqrt{(\sigma+2\beta||b||)}}{\sqrt{p\gamma}}$ $f(v^{t})-L\leq(1-\gamma\mu)^{t}(f(v^{0})-L)+\delta.$ $\sum_{i=1}^{m}\Big{(}\tau-\gamma\eta^{*}\psi_{i}(c_{i})-\log(1+e^{-y^{i}\cdot(%
+\boldsymbol{w}^{*})^{T}\boldsymbol{x}^{i}})\Big{)}^{+}=\frac{\mu}{\|%
+\boldsymbol{a}^{*}\|},$ $\displaystyle=\int_{z_{-i}}\int_{z_{i}}\Big{(}z_{i}\epsilon_{i}(z_{i},z_{-i})$ $\displaystyle+\sqrt{\frac{\ln\frac{1}{\delta}\Big{(}\sum a_{i}^{2}\log^{2}(1+e%
+^{\beta})+\big{(}\frac{2\beta v^{-1}(\delta^{\prime})}{\eta})^{2}\Big{)}}{2}}.$ $c=c_{i}$ $v^{*}\in\mathbb{R}^{m+n}$ $\displaystyle\qquad\qquad+\sqrt{\frac{\ln\frac{1}{\delta}\Big{(}\sum_{i}a_{i}^%
+{2}\log^{2}(1+e^{\beta})+\big{(}\frac{2\beta v^{-1}(\delta^{\prime})}{\eta})^{%
+2}\Big{)}}{2}}$ $\displaystyle\qquad\qquad+\frac{2\boldsymbol{b}^{T}\boldsymbol{w}}{\eta}+\mu\|%
+e^{\boldsymbol{z}}\|+\sigma\frac{1}{\eta}\bigg{]}.$ $\{\lambda,\mu,\sigma\}$ $h(\boldsymbol{b^{\prime}})\propto e^{-\frac{\eta}{2}\|\boldsymbol{b^{\prime}}\|}$ $\displaystyle\leq\mbox{COST}(c_{i},\boldsymbol{c}_{-i},c^{\prime}_{i},%
+\boldsymbol{c}_{-i},z^{i};\epsilon_{i},t_{i})\quad\forall i,c^{\prime}_{i},%
+\boldsymbol{c}.$ $c_{i}\cdot\epsilon_{i}(c_{i})-t_{i}(c_{i})\leq c_{i}\cdot\epsilon_{i}(c^{%
+\prime}_{i})-t_{i}(c^{\prime}_{i}).$ $\displaystyle\leq\min_{\beta}\log(1+e^{-\delta\beta})+\sqrt{\sigma+2\beta\|%
+\boldsymbol{b}\|}\sqrt{p\gamma}$ $\big{|}\boldsymbol{b^{\prime}}^{T}\boldsymbol{w}\big{|}$ $r=\frac{\beta v^{-1}(\delta^{\prime})}{\eta}.$ $y^{i}\in\{+1,-1\}$ $\{\lambda,\mu,\sigma,\gamma\}$ $t_{i}(c_{i})\geq c_{i}\epsilon_{i}(c_{i})+\int_{c_{i}}\epsilon_{i}(z)dz.$ $\eta^{*}=\Big{(}\frac{\sigma+2\boldsymbol{b}^{T}\boldsymbol{w}}{\gamma\sum_{i=%
+1}^{m}\psi_{i}(c_{i})a_{i}^{*}}\Big{)}^{1/2}.$ $\displaystyle\qquad+\Bigg{[}\Big{(}\frac{3\ln\frac{1}{\delta}}{\sqrt{2}}\Big{)%
+}\log(1+e^{\beta})+\frac{\beta}{\ln 2}\Bigg{]}\sqrt{\sum_{i}a_{i}^{2}}$ $\beta=m^{k^{\prime\prime}}$ $\mathbb{E}[\mathbb{I}_{\{sign(\boldsymbol{w}^{T}\boldsymbol{x})\neq y}\}]\leq%
+\mathbb{E}[\log(1+e^{-y\cdot\boldsymbol{w}^{T}\boldsymbol{x}})]=\mathbb{L}[%
+\boldsymbol{w}].$ $\hat{\mathbb{L}}(D,\boldsymbol{w},\boldsymbol{a},\eta)$ $\boldsymbol{c^{\prime}}$ $\displaystyle\leq\lim_{m\rightarrow\infty}\min_{\beta,\eta}\bigg{[}\log(1+e^{-%
+\delta\beta})+\|\boldsymbol{b}\|\frac{2\beta}{\eta}$ $\|\boldsymbol{b}\|$ $t_{i}(c_{i})=t_{i}(0)+c_{i}\epsilon_{i}(c_{i})-\int_{0}^{c_{i}}\epsilon_{i}(z)dz.$ $g_{\min}>g(\boldsymbol{w},\boldsymbol{z},\epsilon_{\text{avg}})$ $\displaystyle\bigg{[}\sum_{i=1}^{m}a_{i}\log(1+e^{-y^{i}\cdot\boldsymbol{w}^{T%
+}\boldsymbol{x}^{i}})+\frac{2\boldsymbol{b}^{T}\boldsymbol{w}}{\eta}+\mu\|%
+\boldsymbol{a}\|+\frac{\sigma}{\eta}\bigg{]}$ $\mathbb{E}_{\boldsymbol{c}}\bigg{[}\mathbb{L}(D,\boldsymbol{c^{\prime}};%
+\boldsymbol{\epsilon,\theta})+\gamma\sum_{i}t_{i}(D,\boldsymbol{c^{\prime}})%
+\bigg{]},$ $\displaystyle\begin{bmatrix}a&b(x_{1}^{i}y^{i})&\ldots&b(x_{n}^{i}y^{i})\\
+b(x_{1}^{i}y^{i})&2\lambda_{0}^{i}+c(x_{1}^{i}y^{i})^{2}&\ldots&c(x_{1}^{i}y^{%
+i})(x_{n}^{i}y^{i})\\
+b(x_{2}^{i}y^{i})&c(x_{1}^{i}y^{i})(x_{2}^{i}y^{i})&\ldots&c(x_{2}^{i}y^{i})(x%
+_{n}^{i}y^{i})\\
+.&.&\ldots&.\\
+.&.&\ldots&.\\
+.&.&\ldots&.\\
+b(x_{n}^{i}y^{i})&c(x_{1}^{i}y^{i})(x_{n}^{i}y^{i})&\ldots&c(x_{n}^{i}y^{i})^{%
+2}+2\lambda_{0}^{i}\\
+\end{bmatrix},$ $\displaystyle\eta^{*}$ $N\rightarrow m\mathbb{P}\big{(}\psi_{i}(c_{i})\leq p+\frac{1}{m^{k}}\big{)}$ $\epsilon_{avg}=\sum_{i}\epsilon_{i}/m$ $\frac{1}{1+e^{y\hat{\boldsymbol{w_{1}}}^{T}\boldsymbol{d}}}<1$ $\mathbb{E}_{\boldsymbol{c}_{-i}}[\mathbb{L}(D,\boldsymbol{c};\boldsymbol{%
+\epsilon},\boldsymbol{\theta})]$ $\hat{\mathbb{L}}(D,\boldsymbol{w},\boldsymbol{a},\eta)=\sum_{i=1}^{m}a_{i}\log%
+(1+e^{-y^{i}\cdot\boldsymbol{w}^{T}\boldsymbol{x}^{i}})+\boldsymbol{b^{\prime}%
+}^{T}\boldsymbol{w}$ $\sum_{i}a_{i}=1$ $(\boldsymbol{x}_{i},y_{i})$ $\boldsymbol{a^{*}},\boldsymbol{w^{*}},\eta^{*})$ $(2\lambda_{0}^{i})^{(n-1)}b^{2}\|\boldsymbol{x}\|^{2}$ $\displaystyle\leq f(v^{t})-\frac{\gamma}{2}\|\bigtriangledown f(v^{t})\|^{2}.$ $\displaystyle\leq\mathbb{E}_{S}[\phi(S)]$ $\|\boldsymbol{a}\|$ $t_{i}(D,\boldsymbol{c^{\prime}})$ $\Psi_{i}(c_{i})\epsilon_{i}(D,\boldsymbol{c^{\prime}})$ $h(\boldsymbol{b^{\prime}}_{2})$ $\boldsymbol{b^{\prime}}$ $\eta=m\min_{i}\epsilon_{i}$ $\phi(S)=\sup_{\boldsymbol{w}\in\mathbb{R}^{n}}(\mathbb{L}[\boldsymbol{w}]-%
+\mathbb{\hat{L}}_{S}[\boldsymbol{w}])$ $\|(1/\|(e^{\boldsymbol{z}})\|)\|<\sqrt{m}$ $\epsilon_{i}.$ $\psi_{i}(c_{i})\leq\frac{1}{m^{k}}$ $\displaystyle\min_{\boldsymbol{z},\eta,\boldsymbol{w}}f(\boldsymbol{w},%
+\boldsymbol{z},\eta),$ $a_{i}^{*}=\Big{(}\tau-\gamma\eta^{*}\psi_{i}(c_{i})-\log(1+e^{-y^{i}\cdot(%
+\boldsymbol{w}^{*})^{T}\boldsymbol{x}^{i}})\Big{)}^{+}\Big{(}\frac{\|%
+\boldsymbol{a}^{*}\|}{\mu}\Big{)},$ $\|\boldsymbol{b^{\prime}}\|\sim\Gamma(n,\frac{2}{\eta})$ $F(c_{i})$ $\eta=m\epsilon_{avg}$ $\displaystyle=2R_{m}(\boldsymbol{w}),$ $\{(\boldsymbol{x}^{1},y^{1}),\ldots,(\boldsymbol{x}^{m-1},y^{m-1}),(%
+\boldsymbol{d},y)\}$ $\inf_{v}f(v)=L$ $\|\boldsymbol{b^{\prime}}\| $\boldsymbol{w^{*}}^{T}\boldsymbol{x^{i}}y^{i}\geq\delta\ \forall i\$ $\displaystyle\leq\lim_{p\rightarrow 0}\lim_{m\rightarrow\infty}\min_{%
+\boldsymbol{a},\eta,\boldsymbol{w},\|w\|\leq\beta,\beta}\bigg{[}\sum_{i=1}^{m}%
+a_{i}\log(1+e^{-y^{i}\cdot\boldsymbol{w}^{T}\boldsymbol{x}^{i}})$ $\displaystyle=\|\bigtriangledown_{\boldsymbol{w}}f(v_{1})-\bigtriangledown_{%
+\boldsymbol{w}}f(v_{2})\|^{2}$ $\displaystyle f(v^{t+1})$ $g_{\min}\leftarrow g(\boldsymbol{w},\boldsymbol{z},\epsilon_{\text{avg}})$ $\sum_{i=1}^{m}\bigg{[}e^{z_{i}}\log(1+e^{-\boldsymbol{w}^{T}\boldsymbol{x}^{i}%
+\cdot y^{i}})+\frac{\lambda_{0}^{i}}{2}\cdot\|\boldsymbol{w}\|^{2}+\gamma\cdot
+m%
+\epsilon_{avg}e^{z_{i}}\Psi_{i}\bigg{]},$ $\theta(D,\boldsymbol{c^{\prime}})$ $\displaystyle 2\lambda_{0}^{i}>\|\boldsymbol{x}^{i}\|^{2}\bigg{(}\frac{1}{\ln 2%
+}\bigg{)}^{2}\cdot e^{z_{i}}\bigg{(}\frac{e^{-\boldsymbol{w}^{T}\boldsymbol{x}%
+^{i}y^{i}}}{1+e^{-\boldsymbol{w}^{T}\boldsymbol{x}^{i}y^{i}}}\bigg{)}^{2}$ $\displaystyle\qquad\leq\log(1+e^{-\frac{\delta}{\sqrt{p\gamma}}})+2\sqrt{%
+\sigma p\gamma+2\|\boldsymbol{b}\|\sqrt{p\gamma}},$ $\boldsymbol{w^{*}}$ $\displaystyle\qquad\qquad+\mu\|\boldsymbol{a}\|+\sigma\frac{1}{\eta}\bigg{]}+%
+\gamma\eta\sum_{i=1}^{m}a_{i}\Psi_{i}(c_{i})$ $\displaystyle=\mathbb{E}_{\sigma}[\sup_{\boldsymbol{w}\in R^{n}}\sum_{i}a_{i}%
+\sigma_{i}\log(1+e^{-y^{i}\boldsymbol{w}^{T}\boldsymbol{x}^{i}})+\boldsymbol{b%
+^{\prime}}^{T}\boldsymbol{w}]$ $\displaystyle\leq f(v^{t})-\gamma||\bigtriangledown f(v^{t})||^{2}+\frac{K%
+\gamma^{2}}{2}\|\bigtriangledown f(v^{t})\|^{2}$ $\eta=\sum\epsilon_{i}$ $\Gamma(n,\frac{2}{\eta})$ $\displaystyle=\|\bigtriangledown_{\boldsymbol{w}}f(v^{1})-\bigtriangledown_{%
+\boldsymbol{w}}f(v^{2})\|^{2}+||\bigtriangledown_{\boldsymbol{z}}f(v^{1})-%
+\bigtriangledown_{\boldsymbol{z}}f(v^{2})\|^{2}$ $\displaystyle 2\lambda>\sum_{i}e^{z_{i}}\|\boldsymbol{x}^{i}\|^{2}\bigg{(}%
+\frac{1}{\ln 2}\bigg{)}^{2}\cdot\bigg{(}\frac{e^{-\boldsymbol{w}^{T}%
+\boldsymbol{x}^{i}y^{i}}}{1+e^{-\boldsymbol{w}^{T}\boldsymbol{x}^{i}y^{i}}}%
+\bigg{)}^{2}$ $\boldsymbol{z}\leftarrow\mbox{\text{Proj}}_{S}(\boldsymbol{z})$ $S=\{\boldsymbol{z}:\sum_{i=1}^{m}e^{z_{i}}=1\}$ $w_{i\rightarrow j,n}$ $<\delta_{avg}^{x}$ $S_{BA}$ $p(\mathbf{b}|\mathbf{I},\mathbf{x}_{q})$ $\mathbf{x_{s}}$ $\mathbf{X}=[\mathbf{a};\mathbf{b}]$ $p(\mathbf{a}|\mathbf{I},\mathbf{x}_{q})\!=\!\frac{\Gamma(\frac{1+S}{2})}{%
+\Gamma(\frac{1}{2})\pi^{\frac{S}{2}}|\mathbf{\Sigma}_{a}|^{\frac{1}{2}}[1\!+\!%
+(\mathbf{\mathbf{a}}\!-\!\bm{\mu}_{a})^{T}\mathbf{\Sigma}_{a}^{-1}(\mathbf{%
+\mathbf{a}}\!-\!\bm{\mu}_{a})]^{\frac{1+S}{2}}}$ $\mathbf{\Phi}\leq Q(\gamma_{u})$ $\mathbf{V}=\mathcal{G}_{v}(\mathbf{F}^{K})$ $\displaystyle\text{LEAP}_{i\rightarrow j}(\mathbf{x}_{i,n})\|_{\rho},$ $\gamma_{v}=0.9,\gamma_{d}=0.9,\gamma_{u}=0.8,\gamma_{track}=3$ $\mathbf{F}=[\mathbf{f}_{1},\ldots,\mathbf{f}_{S}]$ $\mathbf{\Sigma}_{a}=K(\mathbf{F}_{a},\mathbf{F}_{a})+\sigma\mathbf{I}$ $\mathbf{Y_{s}}=\mathcal{F}(\mathbf{I}_{s})$ $(\mathbf{X},\mathbf{F})$ $S_{KF}=2$ $\mathbf{m}_{d}=\text{avgpool}(\mathcal{G}_{d}([\mathbf{X}^{K};\mathbf{X}^{K}_{%
+A}],[\mathbf{F}^{K};\mathbf{F}^{K}_{A}])).$ $N_{a}=64$ $\mathbf{f}_{q}$ $\gamma_{track}$ $\mathbf{I}_{s}\in\mathbf{I}$ $\phi(\mathbf{x}_{s})=\mathbf{\Sigma}_{a}[s,s]+\mathbf{\Sigma}_{b}[s,s]$ $\displaystyle\mathcal{L}_{vis}=(1-\mathbf{V}^{*})\log(1-\mathbf{V})+\mathbf{V}%
+^{*}\log\mathbf{V}.$ $\mathbf{V}^{*},\mathbf{m}_{d}^{*}$ $\mathbf{b}\in\mathbb{R}^{S}$ $\mathbf{I}=[\mathbf{I}_{1},...,\mathbf{I}_{S}],\mathbf{I}_{s}\in\mathbb{R}^{3%
+\times H\times W}$ $[\bm{\mu}_{a};\bm{\mu}_{b}]=\mathbf{X}$ $\Delta\mathbf{D}^{(k)}$ $\mathbf{x}_{q}\in\mathbb{R}^{2}$ $\mathbf{x}_{s_{q}}=\mathbf{x}_{q}$ $\mathbf{I}_{s_{q}}$ $\mathbf{F}_{b}^{k}=\mathcal{G}_{b}(\mathbf{F}^{k})$ $\displaystyle\mathcal{L}_{dyn}=(1-\mathbf{m}_{d}^{*})\log(1-\mathbf{m}_{d})+%
+\mathbf{m}_{d}^{*}\log\mathbf{m}_{d}.$ $\displaystyle\mathcal{L}_{main}=\sum_{k}^{K}\gamma^{K-k}\mathcal{L}_{NLL}(%
+\mathbf{X}^{k},\mathbf{X^{*}},\mathbf{\Sigma}_{a}^{k},\mathbf{\Sigma}_{b}^{k}),$ $k=8,N_{a}=64$ $K_{BA}$ $S_{LP}=12$ $p(\mathbf{X}|\mathbf{I},\mathbf{x}_{q})=p(\mathbf{a}|\mathbf{I},\mathbf{x}_{q}%
+)\cdot p(\mathbf{b}|\mathbf{I},\mathbf{x}_{q})$ $\mathbf{C}[\mathbf{X}^{k}]$ $\mathbf{V}=[v_{1},...,v_{S}]$ $(\mathbf{\Sigma}_{a},\mathbf{\Sigma}_{b})$ $\mathbf{X}_{i\rightarrow j}$ $\begin{split}(\Delta\mathbf{X},\Delta\mathbf{F})&=\text{Refiner}(\mathbf{F}^{k%
+},\text{pos}(\mathbf{X}^{k}-\mathbf{x_{q}}),\mathbf{C}^{k}[\mathbf{X}^{k}]),\\
+\mathbf{X}^{k+1}&\leftarrow\mathbf{X}^{k}+\Delta\mathbf{X},\quad\mathbf{F}^{k+%
+1}\leftarrow\mathbf{F}^{k}+\Delta\mathbf{F},\end{split}$ $\mathcal{A},|\mathcal{A}|=N_{a}$ $(\bm{\mu}_{a},\mathbf{\Sigma}_{a},\bm{\mu}_{b},\mathbf{\Sigma}_{b})$ $\mathbf{V},\mathbf{m}_{d}$ $\mathbf{X}=[\mathbf{x}_{1},...,\mathbf{x}_{S}],\mathbf{x}_{s}\in\mathbb{R}^{2}$ $\mathbf{Y}_{s}$ $S_{KF}$ $\mathbf{a}\in\mathbb{R}^{S}$ $\mathbf{m}_{d}$ $K(\mathbf{x},\mathbf{y})=\mathbf{x}^{T}\mathbf{y}$ $\text{LEAP}_{i\rightarrow j}$ $\Delta\xi^{(k)}\in\mathfrak{se}(3)~{}\textrm{(lie-algebra corresponding to}~{}%
+\mathbf{T})$ $\mathbf{F}_{a}^{k}=\mathcal{G}_{a}(\mathbf{F}^{k})$ $\mathbf{\Sigma}_{b}=K(\mathbf{F}_{b},\mathbf{F}_{b})+\sigma\mathbf{I}$ $(\mathbf{X},\mathbf{V})=\text{TAP}(\mathbf{I},\mathbf{x}_{q},s_{q}).$ $\frac{N_{a}}{k^{2}}$ $(\mathbf{G}_{x},\mathbf{G}_{y})$ $\mathbf{I}_{t-S_{LP}:t-1}$ $w_{1}=1.0,w_{2}=0.5,w_{3}=0.5$ $\mathbf{I}_{t-S_{LP}+1:t}$ $S_{LP}=16$ $i,j\in[t-S_{LP}+1,t-1]$ $Q:[0,1]\rightarrow\mathbb{R}$ $\displaystyle\sum_{i}\sum_{j\in|i-j|\leq S_{BA}}\sum_{n}w_{i\rightarrow j,n}\|%
+\mathcal{P}(\mathbf{T}_{i},\mathbf{T}_{j},\mathbf{K},d_{i,n})-$ $\displaystyle\mathcal{L}_{total}=w_{1}\mathcal{L}_{main}+w_{2}\mathcal{L}_{vis%
+}+w_{3}\mathcal{L}_{dyn}.$ ${\mathbf{x}_{q}}$ $v_{s}\in\{0,1\}$ $p(\mathbf{X}|\mathbf{I},\mathbf{x}_{q})$ $\mathbf{F}^{K}$ $\text{pos}(\cdot)$ $||{\mathrm{D}}||$ $({\cal H}_{Q^{\prime}},{\cal H}_{\mathcal{V}})$ $\langle a_{1},\ldots,a_{\rho}\rangle\in r^{\mathcal{A}}$ $(V,H)$ $\mathit{vars}(\mathit{atoms}(Q))$ $\mathit{color}(Q)$ $\{X,Y\}\subseteq(h\setminus\bar{W})$ ${\cal FH}(Q_{0},\{A,B,C,D\})$ $\mathit{Fr}(A,\Lambda)=\mathit{Fr}(B,\Lambda)=\mathit{Fr}(C,\Lambda)=\emptyset$ ${\cal H}\leq{\cal H}_{a}\leq{\cal H}^{\prime}$ ${\cal H}_{\mathcal{V}_{0}}$ $h\subseteq V$ $r^{\mathcal{A}}\subseteq A^{\rho}$ $\mathit{vars}(A)$ $\mathrm{count}(Q,{\mathrm{D}})$ $\exists D,...,I\ \Phi\wedge\Phi_{f}$ $J\!T$ $\pi_{W_{1}}(S_{1}\bowtie S_{2})$ $\{D,G\}$ $\bar{X}=X_{1},...,X_{n}$ $\Phi=r_{1}({\bf u_{1}})\wedge...\wedge r_{m}({\bf u_{m}})$ $r_{1},...,r_{m}$ $\mathit{atoms}(Q)$ $X=X_{0},\ldots,X_{\ell}=Y$ $\mathsf{param}\textup{-}\textsc{\#Clique}[\mathbb{N}]$ ${\cal H}_{Q_{0}}$ ${\cal H}_{Q^{\prime}}$ $\mathit{form}(Q)$ $\mathit{Frontiers}(\mathbf{C})=\{{\cal FH}(Q^{\prime},\mathit{free}(Q))\mid Q%
+\in\mathbf{C}$ $\mathit{Frontiers}(\mathbf{C})$ $\theta:\mathit{vars}(Q)\mapsto D$ $\theta^{\prime}(t)=t$ $\mathit{nodes}({\cal H})$ $\mathit{Fr}(Y,\bar{W},{\cal H})$ $\mathit{nodes}({\cal H})\setminus\bar{W}$ $\mathit{Fr}(A,\{D,E,G\},{\cal H}_{Q_{0}})=\{D,E\}$ $\Phi^{\prime}=mw(A,B,I)\wedge wt(B,D)\wedge wi(B,E)\wedge pt(C,D)\wedge st(D,F%
+)\wedge rr(F,H)\wedge rr(D,H).$ $(I,\pi)\subseteq\Sigma^{*}\times\mathbb{N}$ $\bar{W}\subseteq\mathit{vars}(Q)$ ${\cal FH}(Q_{0}^{\prime},\mathit{free}(Q_{0}^{\prime}))$ $st(D,G)\wedge rr(G,H)$ ${\rm FPT}=\rm W[1]$ $Q\in\mathbf{C}$ $\{D,F\}$ $\mathsf{param}\textup{-}\textsc{Clique}[\mathbb{N}]$ ${\cal FH}(Q_{0}^{\prime},\{A,B,C\})$ $\begin{array}[]{lll}\Phi_{f}&=&r_{A}(A)\wedge r_{B}(B)\wedge r_{C}(C).\\
+\end{array}$ $st(task,subtask)$ $A\cap\mathcal{U}$ $\deg_{{\mathrm{D}}}(F,v)$ $\exists\bar{X}\Phi$ $\mathit{bound}({\mathrm{D}},{\rm HD})$ $r(a_{1},...,a_{\rho})$ $\langle a_{1},...,a_{\rho}\rangle$ ${\cal H}_{Q^{\prime}_{0}}$ ${\cal H}\leq{\cal H}^{\prime}$ $\mathit{vars}(Q)$ ${\cal H}_{\mathcal{V}}=(N,H)$ $h\in\mathit{edges}({\cal H})$ $\pi_{W_{2}}(\{\theta\})\subseteq S_{2}$ $rr(task,resource)$ $H=\{\mathit{Fr}(Y,\bar{W},{\cal H}_{Q^{\prime}})\mid Y\in\mathit{vars}(Q^{%
+\prime})\}\cup\{e\in\mathit{edges}({\cal H}_{Q^{\prime}})\mid e\subseteq\bar{W}\}$ ${\tt cores}(Q)$ $\rm\#P$ $\mathrm{\#CQ}[\mathbf{C}]$ $w_{q}\in\mathit{views}(Q)$ ${\cal H}^{\prime}$ $\{G,H\}$ $r_{i}\in\tau_{Q}$ $\{\mathit{Fr}(Y,\Lambda,{\cal H}_{Q^{\prime}_{0}})\mid Y\in\{A,B,C,D,E,F,H,I\}%
+\}\cup\{e\in\mathit{edges}({\cal H}_{Q^{\prime}_{0}})\mid e\subseteq\Lambda\}$ ${\rm W[1]}$ $\mathit{edges}(C)$ ${\rm HD}$ $\{X_{1},...,X_{n}\}$ $i\in\{0,...,\ell\mbox{-}1\}$ $W\subseteq\mathit{vars}(Q)$ $\langle A,B,C\rangle$ $\mathit{Fr}(Y,\bar{W},{\cal H})=\bar{W}\cap\mathit{nodes}(\mathit{edges}(C))$ ${\cal H}_{Q_{0}^{\prime}}$ ${\mathrm{D}}_{Q}$ $\mathit{vars}(Q^{\prime})\subseteq\mathit{vars}(Q)$ $\langle h(a_{1}),\ldots,h(a_{\rho})\rangle\in r^{\mathcal{B}}$ $\sigma_{\theta}(S)$ $(I,\pi)\in J$ $Y\in\bar{W}$ $\mathit{Fr}(H,\{D,E,G\},{\cal H}_{Q_{0}})=\{D,G\}$ $\textsc{Clique}[\mathbb{N}]$ $\mathit{free}(Q)=$ ${\rm FPT}\subseteq{\rm W[1]}\subseteq{\rm W[2]}\subseteq\cdots$ $\begin{array}[]{ll}\Phi=&mw(A,B,I)\wedge wt(B,D)\wedge wi(B,E)\wedge pt(C,D)\ %
+\wedge\\
+&st(D,F)\wedge st(D,G)\wedge rr(G,H)\wedge rr(F,H)\wedge rr(D,H).\\
+\end{array}$ ${\cal H}_{1}\leq{\cal H}_{2}$ $D\subseteq\mathcal{U}$ $\pi_{\mathit{free}(Q)}(Q^{\mbox{\rm\tiny D}})$ $\mathit{nodes}({\cal H})\subseteq\mathit{vars}(Q)$ $\mathit{vars}(Q_{0}^{\prime})=\mathit{vars}(Q_{0})\setminus\{G\}$ $\mathit{free}(Q_{0}^{\prime})=\mathit{free}(Q_{0})=\{A,B,C\}$ $W_{1}\cup W_{2}$ $\rm FPT$ ${\cal H}_{2}$ $r_{i}^{\mathcal{Q}}$ $\mbox{\rm P}=\mbox{\rm NP}$ $\{h\in\mathit{edges}({\cal H})\mid h\cap C\neq\emptyset\}$ $\mathit{Fr}(Y,\bar{W},{\cal H})=\emptyset$ $w^{\mbox{\rm\tiny D}}$ $wt(worker\_id,task)$ $\tilde{O}(N^{\#subw})$ $Q_{0}^{\prime}=\Phi^{\prime}\wedge\Phi_{f}$ $({\cal H},{\cal H}^{\prime})$ $w^{\mbox{\rm\tiny D}}\supseteq\pi_{\mathit{vars}(w)}(Q^{\mbox{\rm\tiny D}})$ $\mathit{nodes}(\mathit{edges}(\{A,B,I\})=\{A,B,D,E,I\}$ $\mathit{color}(Q)\}$ $r_{v}=\ \pi_{\chi(v)}(\bowtie_{q\in\lambda(v)}q^{\mbox{\rm\tiny D}})$ ${Q_{0}}$ $h(\langle a_{1},\ldots,a_{\rho}\rangle)$ ${\cal FH}(Q_{0}^{\prime},\Lambda)$ ${\cal H}_{\mathit{color}(Q_{0})}$ $\Lambda=\{A,B,C\}$ $(\mathit{vars}(Q^{\prime})\cup\bar{W},H)$ $r_{X}(X)$ $\deg_{{\mathrm{D}}}(B,wt)$ ${\cal H}_{1}$ ${\cal H}_{1}\leq{\cal H}_{a}\leq{\cal H}_{2}$ $S_{1}\ltimes S_{2}$ $\{B,C\}$ $\mathit{edges}(\{A,B,I\})$ $t\in\mathit{vars}(Q)$ ${\cal FH}(Q^{\prime},\bar{W})$ $q\in\mathit{atoms}(Q)$ $({\cal H}_{1},{\cal H}_{2})$ $\mathit{free}(Q_{0})=\{A,B,C\}$ $\mathit{vars}(Q)\setminus$ $\deg_{{\mathrm{D}}}(C,pt)$ $\{D,F,G,H\}$ ${\bf u_{1}},...,{\bf u_{m}}$ $S_{1}\bowtie S_{2}$ $\{D,G,H\}$ $\mathit{Fr}(A,\{D,E,G\},{\cal H}_{Q_{0}})=\{A,B,D,E,I\}\cap\{D,E,G\}=\{D,E\}$ $h:A\mapsto B$ $Q^{\mbox{\rm\tiny D}}$ $\bigcup_{h\in H}h$ $\{A,B,I\}$ $\pi_{F}(r_{v})$ ${\mathit{bound}({\mathrm{D}},{\rm HD})}$ $Q_{0}^{\prime}$ $\mathit{edges}({\cal H})$ $\tau_{Q}$ $\mathit{nodes}(H)$ ${\cal FH}(Q^{\prime},\mathit{free}(Q))\leq{\cal H}_{a}$ $r^{\mbox{\rm\tiny D}}$ $pt(project,task)$ $f(\pi)\times|I|^{c}$ $st(D,F)\wedge rr(F,H)$ $r_{i}({\bf u})\in\mathit{atoms}(Q)$ $mw(machine,worker\_id,machine\_hours)$ $X\in\mathit{free}(Q)$ $wi(worker\_id,worker\_info)$ $\pi_{W}(S)$ $\mathit{Fr}(Y,\bar{W})$ $\{D,E,G\}$ $\deg_{{\mathrm{D}}}(X,r)$ $\pi_{W_{1}}(\{\theta\})\subseteq S_{1}$ $\{F,H\}$ $\theta^{\prime}(t)=\theta(t)$ $\mathsf{param}\textup{-}\mathrm{\#CQ}[\mathbf{C}]$ $U_{Q}$ $w_{q}^{\mbox{\rm\tiny D}}\subseteq q^{\mbox{\rm\tiny D}}$ ${\cal H}_{Q^{\prime}}\leq{\cal H}_{a}\leq{\cal H}_{\mathcal{V}}$ $J\subseteq\Sigma^{*}\times\mathbb{N}$ ${\cal FH}(Q^{\prime},\mathit{free}(Q))$ $\langle h(a_{1}),\ldots,h(a_{\rho})\rangle$ $\mathit{color}(Q_{0})$ $Q_{0}=\exists D,...,I\ \Phi$ $\mathit{views}(Q)$ ${\rm HD}=\langle T,\chi,\lambda\rangle$ $A\subseteq\mathcal{U}\cup\mathcal{X}$ $\{\theta^{\prime}\in S\mid\pi_{W}(\{\theta^{\prime}\})=\{\theta\})$ ${\cal H}_{a}$ $\theta^{\prime}(r_{\alpha_{i}}({\bf u_{i}}))\in{\mathrm{D}}$ ${\cal FH}(Q_{0},\{A,B,C\})$ $h(c)=c$ $\begin{array}[]{ll}\exists D,E,F,G,H,I&mw(A,B,I)\wedge wt(B,D)\wedge wi(B,E)%
+\wedge pt(C,D)\ \wedge\\
+&st(D,F)\wedge st(D,G)\wedge rr(G,H)\wedge rr(F,H)\wedge rr(D,H).\\
+\end{array}$ $\mathrm{\#CQ}$ ${\cal H}_{Q}$ $r^{\mathcal{A}}\subseteq r^{\mathcal{B}}$ $D\in(1,2)$ ${(D_{i},D_{i+1})\subseteq(D^{\prime}_{j},D^{\prime}_{j+1})}$ $\displaystyle=\widetilde{V}(D)-V(D)$ $\widetilde{\Gamma}_{D}\cap\mathcal{F}_{1}=\emptyset$ ${p_{2}=(e_{3},e_{7},e_{4})}$ ${D=D_{i}}$ $\mathcal{R}^{\operatorname{act}}_{D}$ $\bar{\beta}=\beta^{\top}f^{\delta}$ ${\lambda^{\operatorname{vec}}(T)=\lambda^{\operatorname{vec}}(D_{i})+\delta C^%
+{i}}$ ${\widetilde{\mathcal{R}}^{\operatorname{act}}_{D}=\mathcal{R}^{\operatorname{%
+act}}_{D}\cap(\mathcal{S}^{\texttt{rem}})^{c}}$ $A_{\mathcal{Q}}$ $C_{p}(f^{T})=\delta\lambda^{M}T+\bar{\beta}$ $\delta\widetilde{\lambda}^{\widetilde{M}}>\delta\lambda^{M}$ ${0 $\widehat{\Gamma}_{D}\cap\mathcal{F}_{1}\neq\emptyset$ $\displaystyle=C_{p}(f^{D})+\epsilon A_{p}f^{\delta}$ $\delta\lambda^{i}$ $C(f)=Af+\beta$ $\mathcal{Q},\mathcal{R},\widetilde{\mathcal{Q}},\widetilde{\mathcal{R}}%
+\subseteq\mathcal{P}$ $D_{2}=2$ $\displaystyle\mymathbf{1}^{\top}f$ $[D^{\prime\prime}_{k-1},D_{i+1})$ $\displaystyle=\widetilde{\lambda}^{\operatorname{WE}}(\widetilde{D}_{%
+\widetilde{M}})+(T-\widetilde{D}_{\widetilde{M}})\delta\widetilde{\lambda}^{%
+\widetilde{M}}.$ $A_{i}\in{}^{1\times n}$ ${D\in[D_{i},D_{i+1})}$ $f^{T_{i}}\in\mathcal{W}_{T_{i}}$ $[\frac{7}{2},\frac{35}{9}]$ $\lambda^{\operatorname{vec}}_{p}(D_{i})<\lambda^{\operatorname{vec}}_{r}(D_{i})$ $C_{p}(f):=\sum_{e_{k}\in p}C_{e_{k}}(f_{e_{k}}).$ $D^{\prime}\not=D$ $D^{+}\leq D$ $[D_{i},D_{i+1}]$ $\Gamma_{D}=\ker(A)\cap\mathcal{M}_{D}$ $\widecheck{(\cdot)}$ $\displaystyle\cdots,C_{n}(f)\big{)}^{\top},\text{ and }\,\mathcal{C}:=\{C_{e_{%
+k}}\}_{e_{k}\in\mathcal{E}},$ $\widecheck{\mathcal{P}}:=\mathcal{P}\setminus\mathcal{S}^{\texttt{rem}}_{i,D}$ $f^{\delta}\in\ker(A)\cap\mathcal{M}_{D}$ $\widetilde{\Gamma}_{D}\subseteq\Gamma_{D}$ $f^{\top}Af+f^{\top}\beta\geq D(\delta\lambda^{M}D+\bar{\beta}D).$ $\displaystyle\int_{0}^{D}\widetilde{\lambda}^{\operatorname{WE}}(z)-\lambda^{%
+\operatorname{WE}}(z)dz$ $\widetilde{f}^{\delta}\in\operatorname{SOL}(\widetilde{\mathcal{M}},A)$ $f^{\mu}\in\mathcal{W}_{T_{\mu}}$ $D_{M}\leq D $f^{*}_{p}=0$ $D^{-}\in(D_{i},D_{i+1})$ $f^{\delta}_{p}<0$ ${\Gamma_{D}=\operatorname{SOL}(\mathcal{M}_{D},A)}$ $V(T)\leq\widetilde{V}(T)$ ${[n]:=\{1,2,\cdots,n\}}$ $\mathcal{R}^{\operatorname{act}}_{D_{i+1}}$ $p^{\prime}\in\mathcal{J}^{\operatorname{act}}_{i}\subseteq\mathcal{J}^{%
+\operatorname{act}}_{i+1}$ $\displaystyle=(f^{\prime})^{\top}C(f),$ $(e_{i})_{j}=0$ $\displaystyle f^{\delta}\in\mathcal{F}_{1}.$ $f^{D}$ $\mu_{r}\in(0,1)$ $\mathcal{F}_{D}:=\Bigl{\{}f\in{\mathbb{R}}_{\geq 0}^{n}\;|\;\sum_{p\in\mathcal%
+{P}}f_{p}=D\Bigr{\}}.$ $\widetilde{\mathcal{P}}\subseteq\mathcal{P}$ $\delta\widetilde{\lambda}^{j}<\delta\widetilde{\lambda}^{j+1}$ $\displaystyle\delta\lambda^{+}(D)$ $D\in[\widetilde{D}_{\widetilde{M}},\infty)$ $f^{r}_{r}>0$ $\displaystyle=\delta\lambda^{M}T+\bar{\beta}+(D-T)\delta\lambda^{M},$ $\widetilde{\mathcal{M}}=\mathcal{M}_{D}$ $\displaystyle\leq(f^{\delta_{1}}-f^{\delta_{2}})^{\top}A(f^{\delta_{1}}-f^{%
+\delta_{2}})$ $\mathcal{M}_{D}\subseteq\mathcal{M}_{D_{i}}$ $\delta C^{i}=Af^{\delta}$ ${f^{T}=f^{D}+(T-D)f^{\delta}}$ $f\in\mathcal{M}_{D}$ $j\notin\{i-1,i,i+1\}$ $\Gamma_{D}\cap\operatorname{SOL}(\mathcal{F}_{1},A)$ $\lambda^{\prime\operatorname{WE}}(D_{i})=\lambda^{\operatorname{WE}}(D_{i})$ $f^{\mu}:=\operatorname{coco}_{\mu}(f^{D_{i}},f^{D_{i+1}})$ $\delta\widetilde{\lambda}^{i}$ ${\mathcal{R}^{\operatorname{act}}_{D^{-}}=\mathcal{R}^{\operatorname{act}}_{D^%
+{+}}}$ ${f^{\delta}\in\ker(A)\cap\mathcal{M}_{D}}$ $f^{r}\in\mathcal{W}_{D}$ $\Gamma^{i}\subset\mathcal{H}_{1}$ $\widetilde{\mathcal{M}}^{-}\subseteq\mathcal{M}^{-})$ $\mathcal{S}^{\texttt{nec}}\subseteq\mathcal{P}$ $\delta C^{i}=\delta C^{j}$ $D^{\prime}_{j},D^{\prime}_{j+1}\in\mathcal{D}^{\prime}$ $D_{i+1}>0$ $\epsilon\in[0,\bar{\epsilon}]$ ${\delta C^{M}_{p}=\min_{r\in\mathcal{P}}\delta C^{M}_{r}=\delta\lambda^{M}}$ $(f^{T_{i}}_{r}-f^{D}_{r})\geq-t$ $\Gamma_{D}=\Gamma^{M}$ $D_{j}>D_{j-1}$ $f^{D^{-}}_{p}>0$ ${\operatorname{SOL}(\mathcal{M}_{D},A)\subseteq\Gamma_{D}}$ $f^{T}=f^{D_{i}}+(T-D_{i})f^{\delta}.$ $\widetilde{\mathcal{P}}=\mathcal{P}\setminus\mathcal{S}^{\texttt{rem}}$ $\lambda^{\operatorname{vec}}_{r}(T)<\lambda^{\operatorname{vec}}_{p}(T)$ ${\Gamma_{D}^{-}=\operatorname{SOL}(\mathcal{M}_{D}^{-},A)}.$ $\delta C_{p}^{M}=\delta\lambda^{M}$ $f^{D}_{\mathcal{R}^{\operatorname{use}}_{D}}>0$ $\widetilde{f}^{D}\in\mathcal{W}_{D}$ $\delta\widetilde{\lambda}^{\widetilde{M}}<\delta\lambda^{M}$ $\widetilde{\lambda}^{\operatorname{WE}}(D)=\frac{D}{2}+1.$ ${f^{\delta}\in\Gamma_{D}}$ ${i\in[M]_{0}}$ $\lambda^{\operatorname{vec}}(T)=\lambda^{\operatorname{vec}}(D_{i})+(T-D_{i})%
+\delta C^{i},$ ${\lambda^{\operatorname{WE}}:{\mathbb{R}}_{\geq 0}\rightarrow{\mathbb{R}}_{%
+\geq 0}}$ $\displaystyle\geq\lambda^{\operatorname{WE}}(D_{M}).$ $\mathcal{R}^{\operatorname{use}}_{D_{i+1}}\subseteq\mathcal{J}^{\operatorname{%
+use}}_{i}\subseteq\mathcal{J}^{\operatorname{act}}_{i}$ $\mathcal{M}_{D}^{-}:=\{f^{\delta}\in\mathcal{H}_{-1}\;|\;f_{\mathcal{R}^{%
+\operatorname{act}}_{D}\setminus\mathcal{R}^{\operatorname{use}}_{D}}^{\delta}%
+\geq 0,\enskip f_{(\mathcal{R}^{\operatorname{act}}_{D})^{c}}^{\delta}=0\}.$ $\mathcal{S}^{\texttt{nec}}$ $\mathcal{M}^{-}_{D_{i}}$ $T>D_{M}$ $\widetilde{f}^{\delta}\in\mathcal{F}_{1}$ $\mathcal{M}=\mathcal{M}_{D}$ $(\mathcal{P}\setminus\mathcal{S}^{\prime},\mathcal{C})$ $\displaystyle\subseteq\mathcal{J}^{\operatorname{use}}_{i}\subseteq\mathcal{J}%
+^{\operatorname{act}}_{i}\subseteq\mathcal{R}^{\operatorname{act}}_{D_{i}},$ $(\mathcal{J}^{\operatorname{use}}_{i})^{c}\notin\mathcal{N}_{D}$ $\displaystyle=f^{D_{i}}+(T_{\mu}-D_{i})\frac{f^{D_{i+1}}-f^{D_{i}}}{{D_{i+1}}-%
+D_{i}}$ $A_{p}f^{\delta}=\delta\lambda^{M}$ $\displaystyle\leq(f^{\delta})^{\top}A\widetilde{f}^{\delta},$ $C_{p}(f^{T})\geq\delta\lambda^{M}T+\bar{\beta}$ $f^{D}\in\mathcal{W}_{D}$ $f^{\delta}_{p}>0$ ${\delta C^{i+1}\not=\delta C^{i}}$ ${C_{p}(f^{\prime})=\delta\lambda^{M}D+\bar{\beta}}$ $D_{i}=0$ $f^{T}=\operatorname{coco}_{\mu}(f^{D^{-}},f^{D^{+}})$ $\lambda^{\operatorname{WE}}(D)=\begin{cases}2D\quad&\text{if }0\leq D\leq 1,\\
+2\quad&\text{if }1\leq D\leq 2,\\
+\frac{D}{2}+1\quad&\text{if }2\leq D.\end{cases}$ $p\in\mathcal{J}^{\operatorname{act}}_{M}$ $\operatorname{SOL}(\mathcal{M}_{D},A)\subseteq\Gamma_{D}$ $W(D)\leq 0$ $f^{D}_{\mathcal{S}^{\texttt{nec}}}\neq 0\text{ for all }f^{D}\in\mathcal{W}_{D}.$ $p_{4}=(e_{3},e_{2})$ $\displaystyle\in\mathcal{H}_{1}\;|\;\exists f^{D}\in\mathcal{W}_{D},\hskip 2.0%
+pt\bar{\epsilon}>0\text{ such that }f^{D}+\epsilon f^{\delta}\in\mathcal{W}_{D%
++\epsilon}\hskip 2.0pt\forall\epsilon\in[0,\bar{\epsilon}]\}.\hbox{$\bullet$}$ $\lambda^{\operatorname{vec}}(\cdot)$ $\mathcal{S}^{\texttt{rem}}\notin\mathcal{N}_{D^{+}}$ $f^{D+\epsilon}_{p}>0$ $\displaystyle C_{e_{4}}(f_{e_{4}})=2f_{e_{4}},\hskip 8.0pt$ $D=D_{M}$ ${(f^{\prime})^{\top}Af^{\prime}+(f^{\prime})^{\top}\beta}$ $\mathcal{M}=\mathcal{M}^{-}_{D}$ $f_{e_{k}}\mapsto C_{e_{k}}(f_{e_{k}})$ $(D_{M},\infty)$ $\mathcal{I}_{1}\subseteq\mathcal{J}^{\operatorname{act}}_{M}$ ${\widetilde{f}^{D}_{\mathcal{S}^{\texttt{rem}}}=0}$ $\lambda^{\operatorname{vec}}(D_{M})\geq\lambda^{\operatorname{WE}}(D_{M})%
+\mymathbf{1}$ $r\in\mathcal{R}^{\operatorname{act}}_{D^{-}}$ $V(D)=\widetilde{V}(D)$ $\lambda^{\operatorname{WE}}(D)=\delta\lambda^{M}D+\bar{\beta}.$ $\mathcal{N}_{D}$ $\widetilde{\lambda}^{\operatorname{vec}}(D)=\lambda^{\operatorname{vec}}(D)$ $\widetilde{D}_{j},\widetilde{D}_{j+1}\in\widetilde{D}$ $C_{p}(f^{D})\leq C_{r}(f^{D})\quad\text{for all }r\in\mathcal{P}.$ $(D_{i-1},D_{i})$ $D^{-}>0$ ${f^{\delta}\in\Gamma^{M}}$ $i\in[\widetilde{M}]_{0}$ $f^{0}=(-1,-1,1,1)^{\top}$ $\displaystyle C_{e_{1}}(f_{e_{1}})=2f_{e_{1}},$ $r\in\mathcal{Q}$ $\lambda^{\operatorname{vec}}_{p}(T)<\lambda^{\operatorname{vec}}_{r}(T)$ $\displaystyle=\mathcal{R}^{\operatorname{use}}_{D_{i}}\Rightarrow\delta\lambda%
+^{i-1}>\delta\lambda^{i},\qquad\mathcal{J}^{\operatorname{use}}_{i}=\mathcal{R%
+}^{\operatorname{use}}_{D_{i}}\Rightarrow\delta\lambda^{i-1}<\delta\lambda^{i},$ $f^{\delta-}$ $f^{\delta}$ $\displaystyle\lambda^{\operatorname{vec}}(D)=\begin{cases}\left(\begin{array}[%
+]{cccc}1+D,&1+D,&2D,&2.1\end{array}\right)^{\top}&\text{for }D\in[0,1],\\
+\left(\begin{array}[]{cccc}2,&2,&2,&2.1\end{array}\right)^{\top}&\text{for }D%
+\in[1,2],\\
+\left(\begin{array}[]{cccc}1+\frac{D}{2},&1+\frac{D}{2},&D,&2.1\end{array}%
+\right)^{\top}&\text{for }D\in[2,2.2],\\
+\left(\begin{array}[]{cccc}2.1,&2.1,&2.2,&2.1\end{array}\right)^{\top}&\text{%
+for }D\in[2.2,\infty).\end{cases}$ $f^{D}_{p}>0$ $\displaystyle=C\big{(}f^{D_{i}}+(T_{\mu}-D_{i})f^{\delta_{0}}\big{)}$ $D^{\prime\prime}_{k-1}>0$ $T\in(D,D_{i+1}]$ ${C_{p}(f^{D+\epsilon})=\min_{r\in\mathcal{P}}C_{r}(f^{D+\epsilon})}$ $(D_{i+1},D_{i+2})$ $D\in(D_{i+1},D_{i+2})$ $D>D_{M}$ $\mathcal{J}^{\prime\operatorname{act}}_{j}=\mathcal{J}^{\prime\operatorname{%
+use}}_{j}=\mathcal{P}^{\prime}$ ${A_{p}f^{\delta}=\min_{r\in\mathcal{Q}}A_{r}f^{\delta}}$ $p\in\mathcal{R}^{\operatorname{use}}_{D_{i}}$ $\displaystyle f_{\mathcal{R}}$ $\mathcal{J}^{\prime\operatorname{use}}_{j}=\mathcal{P}^{\prime}$ $p\in\mathcal{J}^{\operatorname{use}}_{M}$ $\delta C^{1}=\delta C^{3}=\mymathbf{0}$ $(Af^{\delta_{0}})^{\top}(f-f^{\delta_{0}})\geq 0$ $0\leq D^{-} $\mathcal{P}=\mathcal{I}_{1}\cup\mathcal{I}_{2}\cup\mathcal{I}_{3}$ $C_{e_{6}}(f_{e_{6}}):=2.1.$ ${f^{\delta}=(T-D)^{-1}(f^{T}-f^{D})}$ $D\in\real$ $(Af^{\delta})^{\top}(f-f^{\delta})=0$ $C(f)=A_{\mathcal{Q}}f+b_{\mathcal{Q}}$ $C_{p}(f^{T})\leq C_{r}(f^{T})$ $f^{\top}Af+f^{\top}\beta$ $A\in{\mathbb{R}}_{\geq 0}^{n\times n}$ $\mathcal{R},\mathcal{Q}\subseteq\mathcal{P}$ ${f^{\delta}\in\operatorname{SOL}(\mathcal{F}_{1},A)}$ $f^{D^{\prime}}=\mu f^{D}+(1-\mu)f^{T}$ $\mathcal{J}^{\operatorname{use}}_{i}\subseteq\mathcal{J}^{\operatorname{act}}_%
+{i}$ $\mathcal{J}^{\operatorname{use}}_{M}=\mathcal{P}$ $D\leq{\widetilde{D}_{i+1}}$ $\mathcal{J}^{\operatorname{act}}_{i-1}=\mathcal{R}^{\operatorname{act}}_{D}%
+\neq\mathcal{R}^{\operatorname{act}}_{D_{i}}$ $\displaystyle\forall r\in\mathcal{I}_{1},$ $\{p_{3}\}$ $\lambda^{\operatorname{WE}}(D)\leq\widetilde{\lambda}^{\operatorname{WE}}(D)$ $f_{\mathcal{R}^{\operatorname{act}}_{D}\setminus\mathcal{R}^{\operatorname{use%
+}}_{D}}^{D}=0$ $T_{i}>D$ $\displaystyle\mathcal{M}$ $\displaystyle C_{e_{3}}(f_{e_{3}})=f_{e_{3}}+1,$ $\mathcal{R}^{\operatorname{act}}_{D^{-}}\neq\mathcal{R}^{\operatorname{act}}_{%
+D^{+}}$ $D^{\prime\prime}_{k}=0$ ${f^{\delta_{1}},f^{\delta_{2}}\in\operatorname{SOL}(\mathcal{M},A)}$ $Af^{\delta_{0}}=\delta C$ $r\in\mathcal{P}$ $p\in\widetilde{\mathcal{P}}$ ${p\in\mathcal{J}^{\operatorname{use}}_{M}\subseteq\mathcal{J}^{\operatorname{%
+act}}_{M}}$ $\beta^{\top}f^{\delta}$ $T_{\mu}=\operatorname{coco}_{\mu}(D_{i},T)$ $(f^{\prime})^{\top}Af^{\prime}+(f^{\prime})^{\top}\beta$ $\mathcal{R}^{\operatorname{use}}_{D}:=\{p\in\mathcal{P}\;|\;\exists f^{D}\in%
+\mathcal{W}_{D}\text{ such that }f^{D}_{p}>0\}.$ $p^{\prime}\in\mathcal{J}^{\operatorname{act}}_{i}$ $f^{D}=\begin{cases}\left(\begin{array}[]{ccc}0,&0,&D\end{array}\right)^{\top}&%
+\text{for }D\in[0,1],\\
+\left(\begin{array}[]{ccc}D-1,&D-1,&2-D\end{array}\right)^{\top}&\text{for }D%
+\in[1,2],\\
+\left(\begin{array}[]{ccc}\frac{D}{2},&\frac{D}{2},&0\end{array}\right)^{\top}%
+&\text{for }D\in[2,\infty).\end{cases}$ $T_{\mu}:=\operatorname{coco}_{\mu}(D,T)$ $\mathcal{M}_{\mathcal{C}_{1}}=\{2K_{1}\}$ $S^{*}\cap V(H)$ $x,y\notin S^{\prime}$ $p_{s}\in B$ $G^{\prime}[S]\in\mathcal{C}$ $p_{1},\dots,p_{s}$ $H\in\mathcal{M}_{\mathcal{C}}$ $N_{A}=\emptyset$ $V(H_{1})\cup V(H_{2})$ $\bigcup_{v\in V(2K_{1}\vee H_{1})\setminus V(H)}X_{v}\subseteq V(H_{2})$ $\bigcup_{v\in V(H_{1})}X_{v}\not\subseteq V(H_{2})$ $b_{1}b_{3}$ $B=V(G)\setminus A$ $j\in\{1,\dots,n\}$ $2K_{1}\vee H$ $q_{1},\dots,q_{t}$ $P^{1},P^{2},P^{3}$ $h_{0}h_{1},h_{0}h_{k-1},h_{i}h_{i+1},h_{i+1}h_{i+2}$ $A\cup C$ $S^{\prime}\subsetneqq S$ $(2K_{1}\vee H_{1})\setminus V(H)$ $K_{2,3}.$ $G,s$ $3K_{1}$ $h_{0}$ $\mathcal{G}_{\mathcal{C}}$ $\bigcup_{v\in V(H)}X_{v}=V(2K_{1})$ $H[N_{W}(v)]$ $\overline{H_{2}}$ $a_{2},b_{1},b_{3}$ $S\subseteq C$ $2K_{1}\vee H_{2}$ $e=xy$ $\mathcal{O}(n^{3+o(1)})$ $P=p_{0},\dots,p_{s}$ $N_{A}\cup N_{B}\neq\emptyset$ $H=h_{0},h_{1},\dots,h_{k-1},h_{0}$ $N_{G}(q_{0})\cap V(H)=A$ $G\setminus V(H)$ $\mathcal{M}_{\mathcal{G}_{1}}=\{C_{4}\}$ $\mathcal{O}(n^{\omega}\log n),$ $w:V\to\mathbb{Q}_{+}$ $\mathcal{M}_{\mathcal{C}_{2}}=\{\overline{C_{2k+1}}\mid k\in\mathbb{N}\}$ $N_{G}[y]\setminus N_{G}[x]=N_{\overline{G}}(x)\setminus N_{\overline{G}}(y)$ $G_{A}[S^{\prime}]=G[S^{\prime}]$ $w\in V(H)\setminus\{u\}$ $X_{w}\cap V(C)\neq\emptyset$ $X=\{x_{1},\dots,x_{t}\}$ $q_{1}=x$ $V(H)\subsetneqq V(G)$ $k\in\{1,\dots,n\}\setminus\{j\}$ $C_{1}\cup\{u\},C_{2},\ldots,C_{k}$ $H_{1}\vee H_{2}$ $v_{1},\dots,v_{n}$ $U=V(H)$ $\mathcal{O}(|V(G)|)$ $\mathcal{C}=\mathcal{C}_{k}$ $G^{\prime}\in\mathcal{G}_{k}$ $\overline{H_{1}}$ $r_{1},\dots,r_{i}$ $\{1,\dots,s\}$ $C_{1},\dots,C_{k}$ $N_{G}(p_{0})\cap V(H)=\{a_{1}\}$ $S^{*}\cap V(H)=S$ $G[S^{\prime}]$ $G[S]\in\mathcal{C}$ $C\setminus S$ $\mathcal{G}_{\mathcal{C}}=\mathcal{G}_{k}$ $.20{\scriptstyle\ \pm.01}$ $.29{\scriptstyle\ \pm.01}$ $\Big{(}\hat{p}_{i}-\underbrace{\frac{1}{|\mathcal{C}(i)|}\sum_{j\in\mathcal{C}%
+(i)}\mathbf{1}(\hat{a}_{j}\text{ is correct})}_{\text{Cluster accuracy (target%
+)}}\Big{)}^{2}.$ $.37{\scriptstyle\ \pm.01}$ $1.4\times 10^{-5}$ $\underline{\mathbf{.02}}{\scriptstyle\ \pm.00}$ $.26{\scriptstyle\ \pm.01}$ $.69{\scriptstyle\ \pm.02}$ $.79{\scriptstyle\ \pm.01}$ $.50{\scriptstyle\ \pm.01}$ $.39{\scriptstyle\ \pm.28}$ $.35{\scriptstyle\ \pm.01}$ $\underline{\mathbf{.73}}{\scriptstyle\ \pm.02}$ $.63{\scriptstyle\ \pm.02}$ $.25{\scriptstyle\ \pm.01}$ $\underline{\mathbf{.82}}{\scriptstyle\ \pm.02}$ $.49{\scriptstyle\ \pm.02}$ $5.12\times 10^{-5}$ $.54{\scriptstyle\ \pm.01}$ $.04{\scriptstyle\ \pm.01}$ $\underline{\mathbf{.18}}{\scriptstyle\ \pm.00}$ $.30{\scriptstyle\ \pm.01}$ $.00{\scriptstyle\ \pm.08}$ $.34{\scriptstyle\ \pm.01}$ $0.03327$ $.61{\scriptstyle\ \pm.02}$ $.20{\scriptstyle\ \pm.00}$ $.14{\scriptstyle\ \pm.00}$ $.69{\scriptstyle\ \pm.01}$ $.14{\scriptstyle\ \pm.01}$ $\underline{\mathbf{.82}}{\scriptstyle\ \pm.01}$ $.71{\scriptstyle\ \pm.03}$ $\displaystyle\delta\widetilde{\lambda}^{+}(D)$ $[\frac{1}{2},\frac{7}{2}]$ $f^{\prime D}\in\mathcal{W}_{D}$ $D\in[1,2)$ $v_{d}\in\mathcal{V}$ $\operatorname{SOL}(\mathcal{X},G)$ $\displaystyle=(f^{\delta})^{\top}\lambda^{\operatorname{vec}}(D_{M})$ $\displaystyle\widetilde{\mathcal{P}}$ $f^{\delta}\geq 0$ $\lambda^{\operatorname{WE}}(D)$ $\Gamma_{D}\subseteq\mathcal{M}_{D}$ $\delta C^{i}$ $\mathcal{S}^{\texttt{rem}}\notin\mathcal{N}_{T}$ ${f^{\delta-}\in\operatorname{SOL}(\mathcal{M}^{-},A)}$ ${f^{\delta_{0}}\in\Gamma_{D}\cap\operatorname{SOL}(\mathcal{M}_{D},A)}$ $Af^{\delta}=\delta C^{i}$ $\mathcal{R}^{\operatorname{use}}_{D_{i+1}}\neq\emptyset$ $\mathcal{R}^{\operatorname{act}}_{T}\neq\mathcal{R}^{\operatorname{act}}_{D^{-}}$ $p\notin\mathcal{R}^{\operatorname{use}}_{T}$ $f^{\mu}\geq 0$ $\displaystyle(f^{\delta})^{\top}\Big{(}A\big{(}f^{D_{M}}+(D-D_{M})f^{\delta}%
+\big{)}+\beta\Big{)}$ $\widetilde{\Gamma}^{i}\neq\widetilde{\Gamma}^{i+1}$ $D\mapsto f^{D}$ $\displaystyle\geq\delta\lambda^{-}(D).$ $T=D_{i+1}$ $f^{D_{M}}\in\mathcal{W}_{D_{M}}$ ${\delta C_{p}^{M}=\delta\lambda^{M}}$ $\mathcal{P}^{\prime\prime}\subset\mathcal{P}^{\prime}\subset\mathcal{P}$ $J(D)=0$ $C(f^{D})=C(\widehat{f}^{D})$ $\displaystyle\delta\lambda^{-}(D)$ $f_{(\mathcal{R}^{\operatorname{act}}_{D})^{c}}^{T}=0$ $\mathcal{S}^{\prime\prime}\subset\mathcal{P}^{\prime}$ $D^{+}\in(D_{M},\infty)$ $\widetilde{\Gamma}_{D}\cap\mathcal{F}_{1}\neq\emptyset$ $f^{T}\geq 0$ $\displaystyle\quad\mymathbf{1}^{\top}f=D.$ $\displaystyle\widetilde{\lambda}^{\operatorname{vec}}(D)$ $D\in\{\frac{2}{3}\}\cup[2,\infty)$ $\widetilde{\mathcal{P}}=\mathcal{P}$ $f^{D_{M}}\in\mathcal{F}_{D_{M}}$ $b_{\mathcal{Q}}$ ${\mathcal{D}:=\{D_{0},D_{1},\cdots,D_{M},D_{M+1}\}\subset{\mathbb{R}}_{\geq 0}%
+\cup\{+\infty\}}$ $\lambda^{\operatorname{vec}}(D^{-})=\lambda^{\operatorname{vec}}(D_{i})+(D^{-}%
+-D_{i})\delta C^{i}$ $D^{\prime}_{j}=0$ $\Gamma_{D}\subseteq\operatorname{SOL}(\mathcal{M}_{D},A)$ $\widetilde{C}(f)=Af$ ${\mathcal{J}^{\operatorname{act}}_{i-1}=\mathcal{R}^{\operatorname{act}}_{D_{i%
+}}}$ ${g:\real\rightarrow{}^{n}}$ $D $D^{\prime\prime}_{k}>0$ $f^{\delta}\in\operatorname{SOL}(\mathcal{F}_{1},A)$ $\widetilde{\mathcal{F}}_{D}\subseteq\mathcal{F}_{D}$ $\displaystyle C_{e_{7}}(f_{e_{7}})=f_{e_{7}}.$ $f^{D_{M}}\geq 0$ $f^{D+\epsilon}\in\mathcal{W}_{D+\epsilon}$ $f^{\delta,i}\in\Gamma_{D}$ $\mathcal{J}^{\operatorname{act}}_{M}$ $f^{\delta}\in\mathcal{M}_{D}$ $u_{\widetilde{\mathcal{P}},\widetilde{M}}$ $\vec{D}_{i}\in\vec{\mathcal{D}}$ $\lambda^{\operatorname{vec}}_{p_{4}}$ $T^{+}>T$ ${\widetilde{\lambda}^{\operatorname{WE}}(T)\geq\lambda^{\operatorname{WE}}(T)}$ $D^{-} $\mathcal{P}^{\prime\prime}:=\mathcal{P}\setminus\mathcal{S}^{\prime\prime}$ $u_{\mathcal{P},i}$ ${\lambda^{\operatorname{WE}}(T)=\lambda^{\operatorname{WE}}(D_{i})+(T-D_{i})%
+\delta\lambda^{i}}$ $\displaystyle+(T-D_{M})\delta\lambda^{M}-(T-\widetilde{D}_{\widetilde{M}})%
+\delta\widetilde{\lambda}^{\widetilde{M}}$ $D\in{\mathbb{R}}_{\geq 0}$ $(\frac{1}{2},\infty)$ ${f^{\delta}\in\operatorname{SOL}(\mathcal{M},A)}$ $D\geq D^{\operatorname{BP}}$ ${A_{pr}=\sum_{e_{k}\in(p\cap r)}\alpha_{e_{k}}}$ $f^{\delta}\in\Gamma^{i}\cap\Gamma^{i+1}$ $(Af^{\delta})^{\top}(f-f^{\delta})\leq 0$ $\displaystyle\widetilde{\mathcal{W}}_{D}$ $C_{p}(f^{D^{+}})\leq C_{r}(f^{D^{+}})$ ${\lambda^{\operatorname{vec}}(D_{M})=Af^{D_{M}}+\beta}$ $(f^{\delta})^{\top}\lambda^{\operatorname{vec}}(D_{M})=\lambda^{\operatorname{%
+WE}}(D_{M})$ $e_{k}\in p_{i}$ $\widetilde{f}_{p}>0$ $\lambda^{\operatorname{vec}}_{p}(D_{M})=\lambda^{\operatorname{WE}}(D_{M})%
+\quad\text{ for all }p\in\mathcal{P}\text{ satisfying }f^{\delta}_{p}>0.$ $f^{\delta,i}\in\mathcal{H}_{1}$ $C_{e_{k}}(f^{D})=C_{e_{k}}(\widehat{f}^{D})$ $\tilde{\mathcal{M}}\subseteq\mathcal{M}$ $\min_{r\in\widetilde{\mathcal{Q}}}A_{r}\widetilde{f}^{\delta}<\min_{r\in%
+\mathcal{Q}}A_{r}f^{\delta}.$ $\Gamma_{D}=\operatorname{SOL}(\mathcal{M}_{D},A).$ $\vec{\mathcal{P}}=\mathcal{P}\setminus\{(e_{1},e_{4})\}$ $\alpha_{e_{k}},\beta_{e_{k}}\in{\mathbb{R}}_{\geq 0}$ $\widetilde{\mathcal{R}}^{\operatorname{use}}_{D}=\widetilde{\mathcal{P}}$ $V(D)<\widetilde{V}(D)$ $\widetilde{\Gamma}_{D}$ $f^{T}\in\mathcal{F}_{T}$ $f^{\delta}\in\Gamma_{D}\cap\mathcal{F}_{1}$ $u_{\vec{\mathcal{P}},0}$ $T\in[D_{i},D_{i+1}]$ $D^{+}=0$ $\delta\widetilde{C}^{i}$ $\displaystyle\mathcal{M}^{-}$ $T\in(\widetilde{D}_{j},\widetilde{D}_{j+1})$ $\delta\lambda^{i}=\min_{r\in\mathcal{R}^{\operatorname{act}}_{D_{i}}}A_{r}f^{\delta}$ $\widecheck{\mathcal{P}}$ $\delta C^{i}_{p}=\delta C^{i}_{r}$ $\mymathbf{1}$ $\mathcal{Q}:=\mathcal{R}^{\operatorname{act}}_{D}$ $\displaystyle\widetilde{V}(D):=$ $\mathcal{Q}=\mathcal{R}^{\operatorname{act}}_{D}$ $\displaystyle C_{e_{4}}(f_{e_{4}})=f_{e_{4}},\hskip 8.0pt$ $f^{\delta_{0}}:=(D_{i+1}-D_{i})^{-1}(f^{D_{i+1}}-f^{D}).$ ${f^{\delta}\in\Gamma_{D}\cap\operatorname{SOL}(\mathcal{F}_{1},A)}$ $\displaystyle C(f):=\big{(}C_{1}(f),$ $\lambda^{\prime\operatorname{WE}}(\cdot)$ $0 $\displaystyle:=\{p\in\widetilde{\mathcal{P}}\;|\;\widetilde{\lambda}^{%
+\operatorname{vec}}_{p}(D)\leq\widetilde{\lambda}^{\operatorname{vec}}_{r}(D)%
+\text{ for all }r\in\widetilde{\mathcal{P}}\},$ $\mathcal{M}_{T}=\mathcal{M}$ $\mathcal{C}\subset\mathcal{K}$ $(\vec{\mathcal{P}},\mathcal{C})$ ${T\in(D_{M},\infty)}$ $\widetilde{f}^{\delta}\in\widetilde{\Gamma}_{D}$ ${\ker(A)=\{f\in\real\;|\;f_{1}=f_{2}=-f_{3}\}}$ $\mathcal{M}:=\{f^{\delta}\in\mathcal{H}_{T}\;|\;f^{\delta}_{\mathcal{Q}%
+\setminus\mathcal{R}}\geq 0,\quad f^{\delta}_{\mathcal{Q}^{c}}=0\}.$ $f^{\delta}\in\Gamma^{M}$ $\displaystyle=\min_{r\in\mathcal{R}^{\operatorname{act}}_{D}}A_{r}f^{\delta}%
+\quad\hskip 10.0pt\text{for all }f^{\delta}\in\operatorname{SOL}(\mathcal{M}_{%
+D},A),$ $\displaystyle f_{r}=0\quad$ $\mathcal{R}_{i}:=\{r\in\mathcal{P}\;|\;f^{\delta,i}_{r}<0\}$ $\displaystyle:=\operatorname{SOL}(\widetilde{\mathcal{M}}_{D},A).$ $\widehat{f}^{D}\in\mathcal{F}_{D}$ $\vec{\mathcal{P}}\subseteq\mathcal{P}$ $\mathcal{P}^{\prime\prime}\subset\mathcal{P}^{\prime}$ $A\in{}^{n\times n}$ $f^{\delta,i}_{r}\leq nt(T_{i}-D)^{-1}+1$ $\mathcal{R}=\mathcal{R}^{\operatorname{use}}_{D}$ $\displaystyle C_{e_{2}}(f_{e_{2}})=f_{e_{2}}+1,$ $D^{+}\in(D^{-},D_{i+1}]$ $r^{\prime}\in\mathcal{J}^{\operatorname{act}}_{i+1}$ $\displaystyle\forall r\in\mathcal{I}_{2},$ $(D_{i},D_{i+1})$ $\mathcal{R}^{\operatorname{act}}_{T}\neq\mathcal{R}^{\operatorname{act}}_{D}$ $Af^{\delta}=\delta C^{M}$ $\frac{\partial^{+}}{\partial x}g(x):=\lim_{h\rightarrow 0^{+}}\frac{g(x+h)-g(h%
+)}{h}$ $D\in(D^{\prime}_{j-1},D^{\prime}_{j})$ $(\cdot)^{\prime\prime\prime}$ ${\frac{\partial}{\partial T}V(T)=\lambda^{\operatorname{WE}}(T)}$ $V(D)\leq\widetilde{V}(D)$ ${f^{D}\in{\mathbb{R}}_{\geq 0}^{n}}$ $\widetilde{f}^{\delta}_{p}>0$ $\widetilde{f}\in\widetilde{\mathcal{F}}_{D}$ $\displaystyle=\{r\in\mathcal{P}\;|\;f^{\delta}_{r}=0,\hskip 2.0pt\delta C^{M}_%
+{r}=\delta\lambda^{M}\}.$ $(f^{\delta})^{\top}Af^{D_{M}}=D_{M}\delta\lambda^{M}$ $f^{\delta}\in\operatorname{SOL}(\mathcal{M}_{T},A)$ $\lambda^{\operatorname{vec}}_{p_{4}}(D)=\begin{cases}2&\text{for }D\in[0,2],\\
+\frac{1}{3}D+\frac{4}{3}&\text{for }D\in[2,\infty).\end{cases}$ $\displaystyle=\{r\in\mathcal{P}\;|\;f^{\delta}_{r}>0\},$ $T\in\real$ $\lambda^{\operatorname{vec}}_{r}(T)<\lambda^{\operatorname{WE}}(T)$ $\mathcal{J}^{\operatorname{act}}_{M}=\mathcal{R}^{\operatorname{act}}_{D^{+}}$ $\delta\lambda^{+}(D)$ $\widetilde{f}^{D}\in\widetilde{\mathcal{F}}_{D}$ $T\in[D_{i},D_{i}+\epsilon)$ $\mathcal{M}_{D}\neq\mathcal{M}_{D_{i}}$ $C_{p}(f^{*})=\delta\lambda^{M}D+\bar{\beta}\quad\text{for all }p\text{ such %
+that }f^{*}_{p}\neq 0.$ $f^{\prime D}_{p}=0$ $D\in[D^{\prime}_{j},D^{\prime}_{j+1})$ $V(T)=\widetilde{V}(T)$ $\mathcal{R}^{\operatorname{use}}_{D^{-}}\neq\mathcal{R}^{\operatorname{use}}_{%
+D^{+}}$ $f^{\delta}\in\operatorname{SOL}(\mathcal{M},A)$ ${\mu\in[0,1]}$ $\displaystyle=\lambda^{\operatorname{WE}}(D_{M})-\widetilde{\lambda}^{%
+\operatorname{WE}}(\widetilde{D}_{\widetilde{M}})$ ${(f^{\delta})^{\top}\lambda^{\operatorname{vec}}(D)=\lambda^{\operatorname{WE}%
+}(D)}$ $p_{3}=(e_{1},e_{5},e_{4})$ $Q_{k,k}=\alpha_{e_{k}}\geq 0$ $v^{\operatorname{in}}_{k},v^{\operatorname{out}}_{k}\in\mathcal{V}$ $\displaystyle:=\{f^{\delta}\in\mathcal{H}_{1}\;|\;f^{\delta}_{\mathcal{Q}%
+\setminus\mathcal{R}}\geq 0,\quad f^{\delta}_{\mathcal{Q}^{c}}=0\},$ $\displaystyle\lambda^{\operatorname{vec}}(D^{+})=\lambda^{\operatorname{vec}}(%
+D^{-})+(D^{+}-D^{-})Af^{\delta}.$ $\lambda^{\operatorname{vec}}(D)$ ${f^{\delta}_{T}>0}$ $\mathcal{J}^{\prime\operatorname{use}}_{j-1}\subset\mathcal{P}^{\prime}$ $D\in(D_{i},D_{i+2})$ $\frac{\partial}{\partial T}\widetilde{V}(T)=\widetilde{\lambda}^{\operatorname%
+{WE}}(T)$ $f^{T}_{p}\geq 0$ $D\in(2.2,\infty)$ $\enskip e_{6}$ $f^{\delta}\in\mathcal{M}$ $A_{r}f^{\delta} $f_{\mathcal{R}}$ $\Gamma_{D}\neq\Gamma_{D_{i}}$ $\widetilde{f}^{D}=\left(\begin{array}[]{cc}\frac{D}{2},&\frac{D}{2}\end{array}%
+\right)^{\top},$ $\displaystyle u_{\mathcal{P}^{\prime\prime},k-1}$ $\displaystyle(\widetilde{f}^{\delta})^{\top}A\widetilde{f}^{\delta}=\min_{r\in%
+\widetilde{\mathcal{Q}}}A_{r}\widetilde{f}^{\delta}<\min_{r\in\mathcal{Q}}A_{r%
+}f^{\delta}$ $e_{k}\in\mathcal{E}$ $f^{\delta}\in\mathcal{F}_{1}\cap\Gamma_{D}$ $[D^{-},D^{+}]$ $u_{\widetilde{\mathcal{P}},\widetilde{M}}(D)<\lambda^{\operatorname{WE}}(D)$ $\lambda^{\operatorname{WE}}(D_{M})=\min_{r\in\mathcal{P}}\lambda^{%
+\operatorname{vec}}_{r}(D_{M})$ $\displaystyle C(f^{\prime})$ $\widetilde{\mathcal{W}}_{D}$ $\displaystyle\leq C_{r}(f^{T}),\quad\forall r\in\mathcal{P}.$ $\mathcal{R}\subseteq\mathcal{Q}$ $\{p_{1},p_{2}\}$ $\mathcal{J}^{\operatorname{act}}_{M}\cap\mathcal{I}_{2}=\emptyset$ $\delta C^{i}_{p^{\prime}}>\delta C^{i}_{r^{\prime}}$ ${\delta\widetilde{\lambda}^{\widetilde{M}}=\delta\lambda^{M}}$ $\lambda^{\prime\operatorname{WE}}$ $\lambda^{\operatorname{WE}}(T)<\widetilde{\lambda}^{\operatorname{WE}}(T)\quad%
+\text{for all }T\in(D^{-},D^{+}).$ $Af^{D_{M}}+\beta=\lambda^{\operatorname{vec}}(D_{M})$ $\widehat{\Gamma}^{\prime}_{D}\cap\mathcal{F}_{1}=\emptyset$ $f^{\delta}(D):=\frac{\partial^{+}}{\partial D}f^{D}=\begin{cases}\left(\begin{%
+array}[]{ccc}0,&0,&1\end{array}\right)^{\top}&\text{for }D\in[0,1),\\
+\left(\begin{array}[]{ccc}1,&1,&-1\end{array}\right)^{\top}&\text{for }D\in[1,%
+2),\\
+\left(\begin{array}[]{ccc}\frac{1}{2},&\frac{1}{2},&0\end{array}\right)^{\top}%
+&\text{for }D\in[2,\infty).\end{cases}$ $\mathcal{J}^{\operatorname{act}}_{i}\subseteq\mathcal{P}$ $f_{(\mathcal{R}^{\operatorname{act}}_{D})^{c}}^{\delta}=0$ $p\in\mathcal{J}^{\operatorname{use}}_{i}$ $\beta:=(\beta_{p})_{p\in\mathcal{P}}$ ${f^{D}+\epsilon f^{\delta}}$ $D^{-}\in(0,D^{+})$ $f^{T}\in\mathcal{W}_{T}$ $\displaystyle\widetilde{\Gamma}_{D}$ $D<\widetilde{D}_{\widetilde{M}}$ $\delta\widetilde{\lambda}^{-}(D^{+})\geq\delta\lambda^{-}(D^{+})$ $\{f^{\delta,i_{k}}\}_{k\in\mathbb{N}}$ $A_{p}\widetilde{f}^{\delta}=A_{r}\widetilde{f}^{\delta}$ $\widehat{\lambda}^{\prime\prime\operatorname{WE}}(D)<\widehat{\lambda}^{\prime%
+\operatorname{WE}}(D)$ $p\in\mathcal{I}_{3}$ $\mathcal{R}^{\operatorname{act}}_{D}=\mathcal{J}^{\operatorname{act}}_{i},%
+\quad\mathcal{R}^{\operatorname{use}}_{D}=\mathcal{J}^{\operatorname{use}}_{i}.$ $\displaystyle=C(f^{T})+(D-T)Af^{\delta}.$ $\displaystyle C_{e_{1}}(f_{e_{1}})=f_{e_{1}},$ $T\in[D^{-},D^{+}]^{c}$ $\mathcal{J}^{\operatorname{act}}_{i-1}=\mathcal{R}^{\operatorname{act}}_{D_{i}}$ $\displaystyle=-\min_{r\in\mathcal{R}^{\operatorname{act}}_{D}}A_{r}f^{\delta}%
+\quad\text{for all }f^{\delta}\in\operatorname{SOL}(\mathcal{M}_{D}^{-},A).$ $\mathcal{J}^{\prime\operatorname{use}}_{j-1}=\emptyset$ ${0\leq D<\frac{2}{3}}$ $(\widecheck{\mathcal{P}},\mathcal{C})$ $\mathcal{D}:=(D_{0},D_{1},\cdots,D_{M},D_{M+1})$ $r\in\mathcal{R}^{\operatorname{use}}_{D}$ ${\widetilde{\mathcal{P}}:=\mathcal{P}\setminus\mathcal{S}^{\texttt{rem}}=%
+\mathcal{J}^{\operatorname{use}}_{i}}$ $\mathcal{J}^{\operatorname{use}}_{i}\neq\mathcal{J}^{\operatorname{use}}_{j}$ $p\in\mathcal{R}^{\operatorname{use}}_{T}=\mathcal{J}^{\operatorname{use}}_{M}$ $\Gamma_{D_{i}}\subseteq\Gamma^{i}$ $\displaystyle Af^{\delta}=\delta C^{M},$ $\displaystyle=z\big{(}\widetilde{V}(z)-V(z)\big{)}|_{0}^{D}-\int_{0}^{D}%
+\widetilde{V}(z)-V(z)dz$ $\widehat{f}^{\delta}\in\Gamma_{D_{M}}$ $\mathcal{R}^{\operatorname{use}}_{T}\subseteq\mathcal{R}^{\operatorname{act}}_%
+{D}$ ${j\in[M+1]}$ $\displaystyle=\lambda^{\operatorname{vec}}(D^{-})+(D^{+}-D^{-})\delta C^{i},$ $\Gamma_{D}=\operatorname{SOL}(\mathcal{M}_{D},A)$ $\widehat{\lambda}^{\operatorname{WE}}(D)\leq u_{\mathcal{P},i}(D)$ $p\in\mathcal{R}^{\operatorname{act}}_{D_{i}}$ $u_{\widetilde{\mathcal{P}},\widetilde{M}}(D)<\lambda^{\operatorname{WE}}(D).$ $\widecheck{\lambda}^{\operatorname{WE}}_{i,D}(D)\leq u_{\widetilde{\mathcal{P}%
+},i}(D).$ $\lambda^{\operatorname{WE}}(D)>\widetilde{\lambda}^{\operatorname{WE}}(D)$ $D\leq\widetilde{D}_{i+1}$ $A_{p}f^{\delta}=\min_{r\in\mathcal{P}}A_{r}f^{\delta}$ $\delta C^{M}_{p}>\delta\lambda^{M}$ $\delta\lambda^{i-1}=\min_{r\in\mathcal{J}^{\operatorname{act}}_{i-1}}\delta C^%
+{i-1}_{r}$ $\widetilde{(\cdot)}$ $\operatorname{SOL}(\mathcal{M},A)$ $\displaystyle(Af^{\delta_{2}})^{\top}(f^{\delta_{1}}-f^{\delta_{2}})$ $D\geq D_{M}$ $D_{1}=1$ $f:=f^{\delta}-\epsilon(e_{p}-e_{r})\in\mathcal{M}$ $f^{D+\epsilon}\geq 0$ $p\in\mathcal{R}^{\operatorname{use}}_{T^{+}}$ $\displaystyle\lambda^{\operatorname{vec}}(D^{+})$ $\displaystyle C_{e_{2}}(f_{e_{2}})=1,$ $\lambda^{\prime\operatorname{WE}}(D^{\prime}_{j})=u_{\mathcal{P},i}(D^{\prime}%
+_{j})$ $\displaystyle C_{e_{5}}(f_{e_{5}})=0.$ $\mathcal{S}^{\prime}=(\mathcal{J}^{\operatorname{use}}_{i})^{c}$ $\enskip e_{4}$ $f^{D}=\sum_{r\in\mathcal{R}^{\operatorname{use}}_{D}}\mu_{r}f^{r}$ ${D,T\in(D_{i},D_{i+1})}$ ${\widetilde{\lambda}^{\operatorname{WE}}(D)<\lambda^{\operatorname{WE}}(D)}$ $\displaystyle f\geq 0.$ $\displaystyle\lambda^{\operatorname{WE}}(T)$ $D_{i},D_{i+1},D_{j},D_{j+1}\in\mathcal{D}$ $p\in\mathcal{R}$ ${-t(T_{1}-D)^{-1}\leq f^{\delta,i}_{r}\leq-nt(T_{1}-D)^{-1}+1}$ $C_{p}(f^{D^{+}})=\delta\lambda^{M}D^{+}+\bar{\beta}.$ $\mathcal{R}^{\operatorname{act}}_{D}=\mathcal{R}^{\operatorname{act}}_{T}$ $\Gamma_{D_{M}}\subseteq\Gamma^{M}$ $\Gamma^{M}$ $\displaystyle=\int_{0}^{D}V(z)-\widetilde{V}(z)dz.$ $f\in{}^{n}$ ${T=D_{i+1}}$ $f^{D_{i}}$ ${\widetilde{f}^{\delta-}\in\operatorname{SOL}(\widetilde{\mathcal{M}}^{-},A)}$ $\displaystyle\leq C_{r}(f^{D}),\quad\forall r\in\mathcal{P},$ $(\mathcal{P}^{\prime},\mathcal{C})$ $\delta\lambda^{-}(D)$ $\delta\lambda^{i}=\min_{r\in\mathcal{J}^{\operatorname{act}}_{i}}\delta C^{i}_%
+{r}$ $\widetilde{\mathcal{R}}\subseteq\widetilde{\mathcal{Q}}$ $T>D$ $[D_{M},\infty)$ $\displaystyle\widetilde{\mathcal{R}}^{\operatorname{use}}_{D}$ $\widetilde{f}^{D}$ $B_{k,i}=1$ $f^{-},f^{+}\in{}^{p}$ $\displaystyle $u_{\mathcal{P},2}$ $f^{D}+(T-D)f^{\delta}$ $\widetilde{f}^{\delta}\in\operatorname{SOL}(\widetilde{\mathcal{M}}_{D},A)$ $\lambda^{\prime\prime\operatorname{WE}}$ $f^{D}+\epsilon f^{\delta,i}$ $\displaystyle C_{e}(f)$ $p_{2}=(e_{3},e_{4})$ $\lim_{k\rightarrow\infty}T_{i_{k}}=\infty$ $\displaystyle=u_{\emptyset,i}(D_{i})+(D-D_{i})\delta\lambda^{i}.$ ${\epsilon\in[0,\epsilon^{*})}$ $\displaystyle\quad f\in\mathcal{M}.$ ${C_{p}(f^{D})=\min_{r\in\mathcal{P}}C_{r}(f^{D})}$ ${\mathcal{H}_{D}:=\{f\in{}^{n}\;|\;\sum_{i\in[n]}f_{i}=D\}}$ $\mymathbf{0}_{n}$ $D\in[1,\frac{1}{2}]$ $\displaystyle\widetilde{\mathcal{M}}^{-}$ $\widetilde{f}^{\delta}\in\widetilde{\Gamma}_{D}\cap\mathcal{F}_{1}$ $f^{\delta,i}:=(T_{i}-D)^{-1}(f^{T_{i}}-f^{D}).$ ${D\mapsto\lambda^{\operatorname{vec}}(D):=C(f^{D})}$ $\mathcal{H}_{-1}$ $(\widetilde{\mathcal{P}},\mathcal{C})$ $\mathcal{S}^{\texttt{rem}}\notin\mathcal{N}_{D}$ $\displaystyle\widetilde{\mathcal{H}}_{D}$ $C(f)=Af+b=\left(\begin{array}[]{ccccc}1&0&1&0\\
+0&1&1&0\\
+1&1&3&0\\
+0&0&0&1\\
+\end{array}\right)f+\left(\begin{array}[]{c}1\\
+1\\
+0\\
+2\\
+\end{array}\right)$ $u_{\widetilde{\mathcal{P}},i}(D)$ $C(f)=Af+b=\left(\begin{array}[]{ccccc}1&0&1&0\\
+0&1&1&0\\
+1&1&2&0\\
+0&0&0&0\\
+\end{array}\right)f+\left(\begin{array}[]{c}1\\
+1\\
+0\\
+2.1\\
+\end{array}\right)$ $\displaystyle=f^{D_{i}}+(1-\mu)(f^{D_{i+1}}-f^{D_{i}})$ $f^{\prime}_{p}=0$ ${p\in\mathcal{R}^{\operatorname{use}}_{D}}$ ${C_{p}(f^{D^{+}})\geq\delta\lambda^{M}D^{+}+\bar{\beta}}$ $\widehat{\mathcal{S}}^{\prime}=(\widehat{\mathcal{R}}^{\operatorname{use}}_{D}%
+)^{c}$ $\mathcal{J}^{\operatorname{act}}_{i}\subseteq\mathcal{R}^{\operatorname{act}}_%
+{D_{i}}$ $\mathcal{X}\subset{}^{n}$ $p\in PP^{\prime}$ $\displaystyle\frac{\partial}{\partial D}\big{(}\widetilde{V}(T)-V(T)\big{)}$ $\delta\widetilde{\lambda}^{-}(D^{+})>\delta\lambda^{-}(D^{+})$ $f^{T}_{p}=0$ $u_{\widetilde{\mathcal{P}},i}$ ${\widetilde{\mathcal{R}}^{\operatorname{use}}_{D}\subseteq\mathcal{R}^{%
+\operatorname{use}}_{D}\cap(\mathcal{S}^{\texttt{rem}})^{c}}$ $p\in\mathcal{R}^{\operatorname{use}}_{D^{-}}$ $\displaystyle:=\operatorname{SOL}(\widetilde{\mathcal{F}}_{D},C),$ ${\widetilde{\lambda}^{\operatorname{WE}}(D)=\widetilde{V}^{\prime}(D)}$ $f^{D}=\begin{cases}\left(\begin{array}[]{cccc}0,&0,&D,&0\end{array}\right)^{%
+\top}&\text{for }D\in[0,1],\\
+\left(\begin{array}[]{cccc}D-1,&D-1,&2-D,&0\end{array}\right)^{\top}&\text{for%
+ }D\in[1,2],\\
+\left(\begin{array}[]{cccc}\frac{D}{2},&\frac{D}{2},&0,&0\end{array}\right)^{%
+\top}&\text{for }D\in[2,2.2],\\
+\left(\begin{array}[]{cccc}1.1,&1.1,&0,&D-2.2\end{array}\right)^{\top}&\text{%
+for }D\in[2.2,\infty).\end{cases}$ $\mathcal{R}^{\operatorname{use}}_{D_{i}}\subseteq\mathcal{R}^{\operatorname{%
+use}}_{T}$ ${C_{p}(f^{T})=\lambda^{\operatorname{WE}}(T)=\delta\lambda^{M}T+\bar{\beta}}$ $\widetilde{f}^{\delta}\in\operatorname{SOL}(\mathcal{M}_{D},A)$ $\lambda^{\operatorname{vec}}_{p_{3}}$ $\delta\lambda^{i}\geq 0$ ${(D-D_{M})(f^{\delta})^{\top}Af^{\delta}=(D-D_{M})\delta\lambda^{M}}$ $D\in[D_{i},D_{i+1})$ $\mathcal{S}^{\texttt{rem}}_{i,D}\subset\mathcal{P}$ $\delta\lambda^{M}=\min_{r\in\mathcal{P}}\delta C^{M}$ $\mathcal{S}^{\texttt{rem}}=\{p_{3}\}$ $\widetilde{\mathcal{M}}_{D}=\{\widetilde{f}^{\delta}\in\widetilde{\mathcal{H}}%
+_{1}\;|\;\widetilde{f}_{\widetilde{\mathcal{R}}^{\operatorname{act}}_{D}%
+\setminus\widetilde{\mathcal{R}}^{\operatorname{use}}_{D}}^{\delta}\geq 0,%
+\enskip\widetilde{f}_{(\widetilde{\mathcal{R}}^{\operatorname{act}}_{D})^{c}}^%
+{\delta}=0\}.$ $B\in{}^{q\times n}$ $\Gamma^{M}\cap\operatorname{SOL}(\mathcal{F}_{1},A)$ ${f^{\delta}\in\mathcal{H}_{1}}$ $\operatorname{coco}_{\mu}(f^{-},f^{+}):=\mu f^{-}+(1-\mu)f^{+}$ $\lim_{k\to\infty}f^{\delta,i_{k}}_{r}\geq 0$ $f^{D^{+}}:=f^{*}+(D^{+}-D)f^{\delta}\in\mathcal{W}_{D^{+}}.$ ${\widetilde{V}(T)=V(T)}$ $\displaystyle\lambda^{\operatorname{vec}}(T_{\mu})$ $(\mathcal{P},\mathcal{C})$ $T\in(D^{+}-\epsilon,D^{+}]$ ${C_{p}(f^{\prime})\geq\delta\lambda^{M}D+\bar{\beta}}$ $\widetilde{\lambda}^{\operatorname{WE}}(T)$ $\lambda^{\prime\prime\operatorname{WE}}(D)=u_{\mathcal{P}^{\prime},j-1}\quad%
+\text{for all }D\in[D^{\prime\prime}_{k},D^{\prime\prime}_{k+1}),$ ${\lambda^{\operatorname{WE}}(D)=\widetilde{\lambda}^{\operatorname{WE}}(D)}$ $C(f)=Af+b=\left(\begin{array}[]{ccccc}3&0&2&0\\
+0&4&2&1\\
+2&2&4&0\\
+0&1&0&2\\
+\end{array}\right)f+\left(\begin{array}[]{c}1\\
+1\\
+0\\
+6\\
+\end{array}\right).$ $D\in[0,2)$ $\mathcal{J}^{\operatorname{act}}_{i}\subseteq\mathcal{J}^{\operatorname{act}}_%
+{i+1}$ $D\in(\frac{2}{3},1)$ ${T\in[0,D]}$ $f^{T}_{p}>0$ $Q\in{}^{q\times q}$ $f^{\delta}\in\mathcal{F}_{1}$ $\widetilde{f}^{\delta}\in\Gamma_{D}$ $\displaystyle\delta\widetilde{\lambda}^{-}(D)$ ${f^{\mu}:=\operatorname{coco}_{\mu}(f^{D},f^{T})}$ $\displaystyle=(Af^{\delta_{1}})^{\top}(f^{\delta_{1}}-f^{\delta_{2}})-(Af^{%
+\delta_{2}})^{\top}(f^{\delta_{1}}-f^{\delta_{2}})\leq 0,$ $f^{T}=f^{\mu}+(T-T_{\mu})f^{\delta}$ $[D,D_{i+1}]$ $\Gamma^{\prime}_{D}\cap\mathcal{F}_{1}=\emptyset$ $\displaystyle\widetilde{\mathcal{R}}^{\operatorname{act}}_{D}$ $\begin{split}C_{1}(f)&=f_{1}+f_{3}+1,\\
+C_{2}(f)&=f_{2}+f_{3}+1,\\
+C_{3}(f)&=f_{1}+f_{2}+2f_{3},\end{split}$ $\displaystyle=Af^{T}+\beta+(D-T)Af^{\delta},$ $\displaystyle\Gamma_{D}:=\{f^{\delta}$ $\lambda^{\operatorname{vec}}_{p^{\prime}}(D)>\lambda^{\operatorname{vec}}_{r^{%
+\prime}}(D)$ $\mathcal{R}^{\operatorname{use}}_{D_{i}}\subseteq\mathcal{J}^{\operatorname{%
+use}}_{i}$ $\mathcal{S}^{\prime\prime}=(\mathcal{J}^{\prime\operatorname{use}}_{j-1})^{c}$ $V(2)=\widetilde{V}(2)$ $\widetilde{\lambda}^{\operatorname{WE}}(T)\geq\lambda^{\operatorname{WE}}(T)$ $\displaystyle\mathcal{R}^{\operatorname{use}}_{D_{i+1}}$ $D>6$ $\displaystyle\quad C_{r}(f)\geq\delta\lambda^{M}D+\bar{\beta}$ $f^{D^{\prime}}\in\mathcal{W}_{D^{\prime}}$ ${f^{\delta}\in\mathcal{F}_{1}}$ $\mathcal{R}^{\operatorname{use}}_{D_{i+1}}\subseteq\mathcal{J}^{\operatorname{%
+use}}_{i+1}\subseteq\mathcal{J}^{\operatorname{act}}_{i+1}$ $f^{\delta}\in\mathcal{H}_{1}$ $f^{D},f^{T}\geq 0$ $A\widetilde{f}^{\delta}=\delta C^{M}$ $\displaystyle:=\{\widetilde{f}^{\delta}\in\widetilde{\mathcal{H}}_{1}\;|\;%
+\widetilde{f}_{\widetilde{\mathcal{R}}^{\operatorname{act}}_{D}\setminus%
+\widetilde{\mathcal{R}}^{\operatorname{use}}_{D}}^{\delta}\geq 0,\enskip%
+\widetilde{f}_{(\widetilde{\mathcal{R}}^{\operatorname{act}}_{D})^{c}}^{\delta%
+}=0\},$ $\delta\widetilde{\lambda}^{-}(D^{+})=\delta\lambda^{-}(D^{+})$ $J(D)=\int_{0}^{D}\widetilde{\lambda}^{\operatorname{WE}}(z)-\lambda^{%
+\operatorname{WE}}(z)dz$ $n=|\mathcal{P}|$ $\displaystyle\widetilde{\mathcal{M}}$ $\mathcal{M}^{-}_{D}$ $\displaystyle\min_{r\in\widetilde{\mathcal{Q}}}A_{r}\widetilde{f}^{\delta}$ ${T\in(T_{\mu},D_{i+1})}$ $f^{D_{M}}_{p}>0$ $\displaystyle=\widetilde{\lambda}^{\operatorname{WE}}(T)-\lambda^{%
+\operatorname{WE}}(T)$ ${p_{3}=(e_{1},e_{5},e_{4})}$ $T\notin\mathcal{D}$ $\lambda^{\operatorname{WE}}(T)>\widetilde{\lambda}^{\operatorname{WE}}(T)\quad%
+\text{for all }T\in(D^{-},D^{+}).$ $\displaystyle:=\{\widetilde{f}\in{\mathbb{R}}_{\geq 0}^{n}\;|\;\sum_{p\in%
+\widetilde{\mathcal{P}}}\widetilde{f}_{p}=D,\enskip\widetilde{f}_{\mathcal{S}^%
+{\texttt{rem}}}=0\},$ $D,T\in(D_{i},D_{i+1})$ $\epsilon\in[0,T_{i}-D)$ $(f^{\delta})^{\top}Af^{D_{M}}=D_{M}\delta\lambda^{M}\quad\text{ for all }f^{%
+\delta}\in\operatorname{SOL}(\mathcal{F}_{1},A).$ $D\in(\frac{1}{2},2)$ $A(f^{\delta_{1}}-f^{\delta_{2}})=0$ $J(D)\geq 0\text{ for all }D\in[0,\infty),$ $\displaystyle\text{(respectively},\quad\min_{r\in\widetilde{\mathcal{Q}}}A_{r}%
+\widetilde{f}^{\delta-}$ $f^{*}\in f^{D_{M}}$ ${0\leq D^{-}\leq D^{+}}$ $\displaystyle:=\{f^{\delta}\in\mathcal{H}_{-1}\;|\;f^{\delta}_{\mathcal{Q}%
+\setminus\mathcal{R}}\geq 0,\quad f^{\delta}_{\mathcal{Q}^{c}}=0\},$ $p\in\mathcal{R}^{\operatorname{act}}_{D_{M}}$ $\lambda^{\operatorname{WE}}(1)=2$ $v^{\operatorname{in}}_{p_{1}}=v_{o}$ ${\mathcal{R}^{\operatorname{use}}_{D}=\{p_{3}\}}$ ${D\in(D_{i},D_{i+1})}$ $D^{+}>0$ $\displaystyle=D\delta\lambda^{M}+\beta^{\top}f^{\delta}.$ $\displaystyle\quad f_{r}\geq 0$ ${\Gamma_{D}\cap\mathcal{F}_{1}=\emptyset}$ $f^{\delta}\in\operatorname{SOL}(\mathcal{M}_{D},A)$ $D\geq D_{M}=1$ $\mathcal{M}_{D}:=\{f^{\delta}\in\mathcal{H}_{1}\;|\;f_{\mathcal{R}^{%
+\operatorname{act}}_{D}\setminus\mathcal{R}^{\operatorname{use}}_{D}}^{\delta}%
+\geq 0,\enskip f_{(\mathcal{R}^{\operatorname{act}}_{D})^{c}}^{\delta}=0\},$ $f^{D}\in\mathcal{F}_{0}$ $\frac{\partial}{\partial D}V(D)=\lambda^{\operatorname{WE}}(D).$ $\lambda^{\operatorname{WE}}(D)=\begin{cases}3D\quad&\text{for }D\in[0,\frac{1}%
+{2}],\\
+\frac{1}{3}D+\frac{4}{3}\quad&\text{for }D\in[\frac{1}{2},\infty).\end{cases}$ $r\in(\mathcal{R}^{\operatorname{act}}_{D_{i}})^{c}$ $\widetilde{\lambda}^{\operatorname{WE}}(T)<\lambda^{\operatorname{WE}}(T)$ $p\in\mathcal{I}_{2}$ $\delta\lambda^{M}=\min_{r\in\mathcal{P}}\delta C^{M}_{p}$ $\mathcal{M}_{D}^{-}$ $p_{3}:=(e_{1},e_{5},e_{4})$ $\widetilde{\Gamma}_{D}\cap\operatorname{SOL}(\mathcal{F}_{1},A)$ $(f^{\delta})^{\top}Af^{D_{M}}$ $f^{D}+\epsilon f^{\delta}\in\mathcal{F}_{D+\epsilon}$ $\displaystyle\lambda^{\prime\prime\operatorname{WE}}(D)$ $\mathcal{R}^{\operatorname{act}}_{D}=\mathcal{R}^{\operatorname{act}}_{D^{-}}=%
+\mathcal{R}^{\operatorname{act}}_{D^{+}}$ $\displaystyle\min_{f\in\mathcal{F}_{D}}\sum_{e_{k}\in\mathcal{E}}\int_{0}^{f_{%
+e_{k}}}C_{e_{k}}(z)dz,$ $\delta\widetilde{\lambda}^{\widetilde{M}}\geq\delta\lambda^{M}$ $\displaystyle=\epsilon(A_{p}f^{\delta}-A_{r}f^{\delta})<0.$ $f_{(\mathcal{R}^{\operatorname{act}}_{D})^{c}}^{D}=0$ ${T\in(D,D_{i+1}]}$ $D\in[D^{-},D^{+}]$ ${T\in[D_{i-1},D_{i})}$ ${C_{p}(f^{D^{-}})\leq C_{r}(f^{D^{-}})}$ $\mathcal{J}^{\prime\operatorname{use}}_{j-1}\neq\mathcal{J}^{\prime%
+\operatorname{use}}_{j}$ $T\mapsto u_{\vec{\mathcal{P}},i}(T):=\vec{\lambda}^{\operatorname{WE}}(\vec{D}%
+_{i})+(T-\vec{D}_{i})\delta\vec{\lambda}^{i},$ ${f^{D}\in\mathcal{W}_{D}}$ $\mathcal{M}_{D}=\{f^{\delta}\in\mathcal{H}_{1}\;|\;f^{\delta}_{1},f^{\delta}_{%
+2},f^{\delta}_{4}\geq 0\}$ $\displaystyle C_{p}(f)\leq C_{r}(f)\quad$ $\widetilde{\lambda}^{\operatorname{WE}}_{p}(D)=\widetilde{\lambda}^{%
+\operatorname{WE}}_{r}(D)$ $\Gamma_{D}=\Gamma^{i}$ $\mymathbf{1}^{\top}f=D$ $D^{\prime}\in(D_{i},D_{i+1})$ $f\in\mathcal{F}_{D}$ $p\in\mathcal{R}^{\operatorname{act}}_{D_{i}}\cap\mathcal{R}^{\operatorname{act%
+}}_{D_{i+1}}$ $D^{\operatorname{BP}}\geq 0$ $\mathcal{M}_{D_{i}}$ $\operatorname{coco}_{\mu}(f^{D},f^{T})\in\mathcal{W}_{\operatorname{coco}_{\mu%
+}(D,T)}$ $(\mathcal{P}^{\prime\prime},\mathcal{C})$ $\displaystyle\mathcal{R}^{\operatorname{use}}_{D_{i}}$ $\mathcal{R}^{\operatorname{use}}_{D}$ $D\mapsto\mathcal{W}_{D}$ $C(f)=Af+\beta,$ $\operatorname{SOL}(\mathcal{M}_{D},A)$ $\delta\lambda^{i-1}<\delta\lambda^{i}$ $\left\lvert{\mathcal{P}}\right\rvert$ $\displaystyle=(f^{\delta})^{\top}\Big{(}A\big{(}f^{D_{M}}+(D-D_{M})f^{\delta}%
+\big{)}+\beta\Big{)}$ $\operatorname{SOL}(\mathcal{M}_{D},A)=\operatorname{SOL}(\mathcal{F}_{1},A)$ $\operatorname{SOL}(\mathcal{F},A):=\operatorname{SOL}(\mathcal{F},C)$ $\displaystyle C_{p}(f^{\prime})$ $\displaystyle=C(f^{D_{i}})+(T_{\mu}-D_{i})Af^{\delta_{0}}$ $T\in(D^{-},D^{+})$ $\left\lvert{T-D}\right\rvert<\delta$ $\widetilde{\mathcal{M}}_{D}=\{f^{\delta}\in\mathcal{H}_{1}\;|\;f^{\delta}_{%
+\widetilde{\mathcal{P}}^{c}}=0\},$ $\displaystyle C_{e_{5}}(f_{e_{5}})=0,$ $r\in(\mathcal{R}^{\operatorname{act}}_{T})^{c}$ $\lambda^{\prime\operatorname{WE}}(D) $\displaystyle=C(f^{\mu})$ ${T_{\mu}=\operatorname{coco}_{\mu}(D_{i},{D_{i+1}})}$ $f^{D^{+}}_{p}=0$ $G(x^{*})^{\top}(x-x^{*})\geq 0,\enskip\forall x\in\mathcal{X}.$ $\widetilde{V}(T) $\displaystyle=\mathcal{R}^{\operatorname{act}}_{D_{i}}\Rightarrow\delta\lambda%
+^{i-1}<\delta\lambda^{i},\qquad\mathcal{J}^{\operatorname{act}}_{i}=\mathcal{R%
+}^{\operatorname{act}}_{D_{i}}\Rightarrow\delta\lambda^{i-1}>\delta\lambda^{i}.$ $p_{1}=(e_{1},e_{2})$ $\lambda^{\operatorname{vec}}(T)$ $u_{\mathcal{P},M}$ ${A_{p}f^{\delta}=\min_{r\in\mathcal{R}^{\operatorname{act}}_{D}}A_{r}f^{\delta}}$ $\widetilde{\mathcal{P}}\subset\mathcal{P}$ $\lambda^{\operatorname{WE}}(D)=V^{\prime}(D)$ $\lambda^{\operatorname{WE}}(D)=\lambda^{\operatorname{WE}}(D_{M})+(D-D_{M})%
+\delta\lambda^{M}$ $i\in[\widetilde{M}]$ $f_{e_{k}}:=\sum_{\{p\in\mathcal{P}\;|\;e_{k}\in p\}}f_{p}.$ $\displaystyle=(f^{\delta})^{\top}\lambda^{\operatorname{vec}}(D)$ ${\mathcal{P}^{\prime}:=\mathcal{P}\setminus\mathcal{S}^{\prime}}$ $\lambda^{\prime\operatorname{WE}}(D)<\lambda^{\operatorname{WE}}(D)$ $D^{\operatorname{BP}}$ $f^{\delta}_{p}\neq 0$ $\alpha_{e_{k}}=0$ $\mathcal{P}^{\prime\prime}=\emptyset$ $f^{D}\in\operatorname{SOL}(\mathcal{F}_{D},C)$ $A_{p}f^{\delta}>A_{r}f^{\delta}$ $\displaystyle:=\{f^{\delta}\in\mathcal{H}_{-1}\;|\;f^{\delta}_{\widetilde{%
+\mathcal{Q}}\setminus\widetilde{\mathcal{R}}}\geq 0,\quad f^{\delta}_{%
+\widetilde{\mathcal{Q}}^{c}}=0\}.$ $[D^{+},D]$ $p,r\in\mathcal{J}^{\operatorname{act}}_{i}$ $f^{D^{+}}\in\mathcal{F}_{D^{+}}$ $\displaystyle\widetilde{\mathcal{F}}_{D}$ $f\in{\mathbb{R}}_{\geq 0}^{n}$ $\displaystyle C_{p}(f^{T})$ $\displaystyle=\epsilon(f^{\delta})^{\top}A(e_{p}-e_{r})$ $p_{2}:=(e_{3},e_{4})$ $C_{p}(f^{*})\geq\delta\lambda^{M}D+\bar{\beta}$ $f^{D^{-}}\in\mathcal{W}_{D^{-}}$ ${f^{\delta}\in\operatorname{SOL}(\mathcal{M}_{D},A)}$ $f^{*}\in\mathcal{W}_{D_{M}}$ $\mathcal{J}^{\operatorname{act}}_{i}\cap\mathcal{J}^{\operatorname{act}}_{i+1}=\emptyset$ $\delta C^{M}=Af^{\delta}$ ${\lambda^{\operatorname{vec}}(D_{M})\geq\lambda^{\operatorname{WE}}(D_{M})%
+\mymathbf{1}}$ $D\in[D^{\prime}_{j-1},D^{\prime}_{j})$ ${\widetilde{f}^{\delta}\in\Gamma_{D}\cap\operatorname{SOL}(\mathcal{F}_{1},A)}$ $D\in(D_{i-1},D_{i})$ $f_{p_{4}}=0$ $f_{\mathcal{R}^{\operatorname{act}}_{D}\setminus\mathcal{R}^{\operatorname{use%
+}}_{D}}^{\delta}\geq 0$ $C_{p}(\widetilde{f}^{D})\leq C_{r}(\widetilde{f}^{D})\quad\text{for all }r\in%
+\widetilde{\mathcal{P}}.$ $p\in\mathcal{I}_{1}\cup\mathcal{I}_{3}$ $\displaystyle\subseteq\mathcal{J}^{\operatorname{use}}_{i}\subseteq\mathcal{J}%
+^{\operatorname{act}}_{i}\subseteq\mathcal{R}^{\operatorname{act}}_{D_{i+1}}.$ $D\in(0,2)$ $f^{\delta}\in\Gamma_{T}$ $\beta_{p}=\sum_{e_{k}\in p}\beta_{e_{k}}$ $C_{p}(f)=\delta\lambda^{M}D+\bar{\beta}$ $\widetilde{\mathcal{F}}_{D}$ $\widetilde{\mathcal{P}}:=\mathcal{P}\setminus\mathcal{S}^{\texttt{rem}}$ $(\widetilde{f}^{\delta}-f^{\delta})^{\top}A(f^{\delta}-\widetilde{f}^{\delta})%
+\leq 0.$ $\mathcal{D}=\{0,1,\infty\}$ $\widetilde{\lambda}^{\operatorname{WE}}(D)=u_{\widetilde{\mathcal{P}},%
+\widetilde{M}}(D)$ ${\mathcal{S}^{\texttt{rem}}:=(\mathcal{J}^{\operatorname{use}}_{i})^{c}}$ $C_{e_{k}}(f_{e_{k}}):=\alpha_{e_{k}}f_{e_{k}}+\beta_{e_{k}},$ $\widetilde{\lambda}^{\operatorname{WE}}(D)$ $(\mathcal{R}^{\operatorname{act}}_{D})^{c}=\mathcal{P}\setminus\mathcal{R}^{%
+\operatorname{act}}_{D}$ $A_{p}f^{\delta}=\min_{r\in\mathcal{Q}}A_{r}f^{\delta}$ ${[D^{\prime}_{j-1},D^{\prime}_{j})\subseteq[D^{\prime\prime}_{k},D^{\prime%
+\prime}_{k+1})}$ ${\widetilde{f}^{D}\in\mathcal{W}_{D}}$ $f^{D}_{e_{k}}=\widehat{f}^{D}_{e_{k}}$ $\displaystyle\text{ for all }p\in\mathcal{J}^{\operatorname{act}}_{M}\text{ %
+and }r\in\mathcal{P},$ $f^{\delta_{0}}\in\mathcal{M}$ $\lambda^{\operatorname{vec}}_{p^{\prime}}(D)<\lambda^{\operatorname{vec}}_{r^{%
+\prime}}(D)$ $(\widetilde{f}^{\delta})^{\top}\lambda^{\operatorname{vec}}(D_{M})=\lambda^{%
+\operatorname{WE}}(D_{M}),$ $\widetilde{f}^{\delta}$ $\widetilde{f}^{\delta}\in\widetilde{\Gamma}_{T}$ ${\widetilde{\mathcal{M}}_{D}\subseteq\mathcal{M}_{D}}$ $\{f^{\delta,i}\}_{i\in\mathbb{N}}\subset\Gamma_{D}$ $\displaystyle:=\{p\in\widetilde{\mathcal{P}}\;|\;\exists\widetilde{f}^{D}\in%
+\widetilde{\mathcal{W}}_{D}\text{ such that }\widetilde{f}^{D}_{p}>0\},$ $\lambda^{\operatorname{vec}}_{p}(D_{M})=\lambda^{\operatorname{WE}}(D_{M})$ $T\in[0,D]$ $\displaystyle\mathcal{J}^{\operatorname{use}}_{i-1}$ $\operatorname{SOL}(\mathcal{M}_{D},A)=\ker(A)\cap\mathcal{M}_{D}$ $T\in(D,\infty)$ $\displaystyle\mathcal{W}_{D}:=\enskip\begin{cases}\left\{\left(\begin{array}[]%
+{cccc}0&0&D&0\end{array}\right)^{\top}\right\}&\text{for }D\in[0,1],\\
+\{f\in\mathcal{F}_{D}\;|\;f_{1}+f_{3}=1,f_{2}+f_{3}=1\}&\text{for }D\geq 1.\\
+\end{cases}$ $\mathcal{M}_{D}=\{f^{\delta}\in\mathcal{H}_{1}\;|\;f^{\delta}_{\mathcal{J}^{%
+\operatorname{act}}_{i-1}\setminus\mathcal{J}^{\operatorname{use}}_{i-1}}\geq 0%
+,\enskip f^{\delta}_{(\mathcal{J}^{\operatorname{act}}_{i-1})^{c}}=0\}.$ $\mathcal{P}^{\prime}=\mathcal{R}^{\prime\operatorname{use}}_{D}=\mathcal{R}^{%
+\prime\operatorname{use}}_{T}$ $\Gamma_{D}\cap\mathcal{F}_{1}=\emptyset$ $\widetilde{f}_{\mathcal{S}^{\texttt{rem}}}=0$ $\displaystyle C_{e_{3}}(f_{e_{3}})=1,$ $\Gamma^{i+1}\cap\Gamma^{i}=\emptyset$ $e_{3}\enskip$ $\mathcal{R}^{\operatorname{use}}_{D}\subseteq\mathcal{R}^{\operatorname{act}}_%
+{D}$ $v^{\operatorname{out}}_{pk}=v^{\operatorname{in}}_{pk+1}$ $f^{D}\in\mathcal{F}_{D}$ $\mathcal{W}_{D_{M}}$ $u_{\mathcal{P}^{\prime},j-1}$ ${\mathcal{R}^{\operatorname{act}}_{D}=\{p_{1},p_{2},p_{3},p_{4}\}}$ $\delta C_{M}$ $\displaystyle V(D)=$ $f^{\delta}\in\mathcal{H}_{-1}$ $f^{D_{i}}\in\mathcal{W}_{D_{i}}$ $\displaystyle\quad\frac{1}{2}f^{\top}Af$ $(e_{p_{1}},\cdots,e_{p_{l}})$ $T\in(D_{M},\infty)$ $\widetilde{f}^{D}+\epsilon\widetilde{f}^{\delta}\geq 0$ $k\in[l-1]$ $\widecheck{\mathcal{P}}\subseteq\widetilde{\mathcal{P}}$ $\widehat{\lambda}^{\operatorname{WE}}(D)<\widetilde{\lambda}^{\operatorname{WE%
+}}(D)$ ${(\widetilde{f}^{\delta})^{\top}A(f^{\delta}-\widetilde{f}^{\delta})\leq 0}$ $r^{\prime}\in(\mathcal{J}^{\operatorname{act}}_{i})^{c}\cap\mathcal{J}^{%
+\operatorname{act}}_{i+1}$ $\displaystyle:=C(\widetilde{f}^{D})\text{ for any }\widetilde{f}^{D}\in%
+\widetilde{\mathcal{W}}_{D},$ $\displaystyle:=\big{(}C_{p_{1}}(f),C_{p_{2}}(f),\cdots,C_{p_{m}}(f)\big{)}^{%
+\top}.$ $\displaystyle C_{p}(f^{D+\epsilon})$ $f^{D^{+}}_{p}>0$ $(\widehat{\mathcal{P}}\setminus\widehat{\mathcal{S}}^{\prime},\mathcal{C})$ $T\in(D_{i},D_{i+1})$ $T\in[D_{i},D_{i+1})$ ${A=B^{\top}QB}$ $\displaystyle\widetilde{\mathcal{M}}_{D}$ $(f^{\delta})^{\top}A(\widetilde{f}^{\delta}-f^{\delta})\geq 0,$ $T\geq D$ $\lambda^{\operatorname{WE}}(T)$ $Af^{\delta_{0}}=Af^{\delta}$ ${f^{\prime}:=f^{T}+(D-T)f^{\delta}}.$ $\lambda^{\operatorname{WE}}(D):=\min_{p\in\mathcal{P}}C_{p}(f^{D})=C_{r}(f^{D}%
+),\quad r\in\mathcal{R}^{\operatorname{act}}_{D}.$ $f^{*}+\epsilon f^{\delta}$ ${D^{+}\in(D_{j},D_{j+1})}$ $(\mathcal{R}^{\operatorname{act}}_{D})^{c}\subseteq(\mathcal{R}^{\operatorname%
+{use}}_{T})^{c}$ $\delta\lambda^{+}(T)=\delta\lambda^{-}(\bar{D})$ $Af^{0}=\mymathbf{0}$ $f^{D_{i+1}}\in\mathcal{W}_{D_{i+1}}$ $(e_{i})_{i}=1$ $\lambda^{\operatorname{WE}}(T)=\lambda^{\operatorname{WE}}(D_{i})+(T-D_{i})%
+\delta\lambda^{i}.$ $\mathcal{Q}^{c}$ $\displaystyle=(Af^{\delta})^{\top}(f-f^{\delta})\geq 0.$ $\mathcal{M}_{D}\subset\mathcal{M}_{D_{i}}$ $\displaystyle\geq\delta\lambda^{+}(D),$ $\delta\lambda^{+}(D)=\min_{r\in\mathcal{R}^{\operatorname{act}}_{D}}A_{r}f^{%
+\delta},\quad\delta\widetilde{\lambda}^{+}(D)=\min_{r\in\widetilde{\mathcal{R}%
+}^{\operatorname{act}}_{D}}A_{r}\widetilde{f}^{\delta},$ $\displaystyle\widetilde{\lambda}^{\operatorname{WE}}(D)$ $\widetilde{D}_{i+1}$ ${\delta\lambda^{M}=\min_{r\in\mathcal{J}^{\operatorname{act}}_{M}}A_{r}f^{%
+\delta}}$ $(f^{\prime})^{\top}Af^{\prime}+(f^{\prime})^{\top}$ $\displaystyle\geq\min_{r\in\mathcal{Q}}A_{r}f^{\delta},$ $\displaystyle\forall r\in\mathcal{I}_{3},$ $\widetilde{\lambda}^{\operatorname{WE}}(D)<\lambda^{\operatorname{WE}}(D)$ $T_{1}\leq T_{i}$ $D^{\prime\prime}_{k-1}\in\mathcal{D}^{\prime\prime}$ $\widetilde{V}(D)=V(D)$ ${(f^{\delta})^{\top}Af^{\delta}=\min_{r\in\mathcal{Q}}A_{r}f^{\delta}}$ $\displaystyle=(Af^{\delta})^{\top}(f-f^{\delta})+(Af^{\delta})^{\top}(f^{%
+\delta}-f^{\delta_{0}})$ $u_{\vec{\mathcal{P}},i}$ $f^{\delta}:=\lim_{k\to\infty}f^{\delta,i_{k}}\geq 0.$ $p_{3}=(e_{1},e_{4})$ $f^{\delta}_{\mathcal{Q}^{c}}=0$ $p\in\mathcal{J}^{\operatorname{act}}_{i}$ $p\in\mathcal{R}^{\operatorname{use}}_{T}\subseteq\mathcal{R}^{\operatorname{%
+act}}_{T}$ $f^{D_{M}}$ $\widetilde{\mathcal{R}}^{\operatorname{act}}_{T}=\widetilde{\mathcal{J}}^{%
+\operatorname{act}}_{j}=\widetilde{\mathcal{P}}$ $\lambda^{\operatorname{WE}}(D)=\lambda^{\operatorname{vec}}_{p}(D)$ $\mathcal{M}_{D_{i}}=\{f^{\delta}\in\mathcal{H}_{1}\;|\;f^{\delta}_{\mathcal{R}%
+^{\operatorname{act}}_{D_{i}}\setminus\mathcal{R}^{\operatorname{use}}_{D_{i}}%
+}\geq 0,\enskip f^{\delta}_{(\mathcal{R}^{\operatorname{act}}_{D_{i}})^{c}}=0\}.$ $\{\mathcal{J}^{\operatorname{act}}_{0},\mathcal{J}^{\operatorname{act}}_{1},%
+\cdots,\mathcal{J}^{\operatorname{act}}_{M}\}$ $p\in\mathcal{I}_{1}$ $D\geq\widetilde{D}_{\widetilde{M}}$ $T\in(0,D)$ $A\widetilde{f}^{\delta}=\delta C^{i-1}$ $\frac{2}{3} ${(f^{\prime})^{\top}Af^{\prime}+(f^{\prime})^{\top}\beta=(f^{\prime})^{\top}C(%
+f^{\prime})=D(\delta\lambda^{M}D+\bar{\beta})}$ ${\lambda^{\prime\prime\operatorname{WE}}(D)<\lambda^{\prime\operatorname{WE}}(%
+D)}$ $\mathcal{N}_{D}:=\{\mathcal{S}^{\texttt{nec}}\subseteq\mathcal{P}\;|\;f^{D}_{%
+\mathcal{S}^{\texttt{nec}}}\neq 0\text{ for all }f^{D}\in\mathcal{W}_{D}\}.$ $A_{p}f^{\delta}=\min_{r\in\mathcal{J}^{\operatorname{act}}_{M}}A_{r}f^{\delta}$ $\mathcal{J}^{\operatorname{use}}_{i}$ $\delta C^{i}=\delta C^{i+1}$ $\displaystyle=A\big{(}f^{T}+(D-T)f^{\delta}\big{)}+\beta,$ ${u_{\mathcal{P},M}(D)<\lambda^{\operatorname{WE}}(D)}$ $\mathcal{J}^{\operatorname{act}}_{i}$ $(\mathcal{J}^{\operatorname{use}}_{i})^{c}$ $\{f^{T_{i}}\}_{i\in\mathbb{N}}$ $\mathcal{J}^{\operatorname{use}}_{M}$ $A_{p}f^{\delta}=\min_{r\in\mathcal{R}^{\operatorname{act}}_{T}}A_{r}f^{\delta}$ $f^{D^{\prime}}_{p}>0$ $D\in(D_{i},D_{i+1})$ ${\delta C^{i}=\delta C^{i+1}}$ $\displaystyle:=\{\widetilde{f}\in{}^{n}\;|\;\sum_{p\in\widetilde{\mathcal{P}}}%
+\widetilde{f}_{p}=D,\enskip\widetilde{f}_{\mathcal{S}^{\texttt{rem}}}=0\}.$ ${p,r\in\mathcal{J}^{\operatorname{act}}_{i}}$ $Af^{\delta}=\delta C$ $f^{D},\widehat{f}^{D}\in\mathcal{W}_{D}$ $D,T\in[D_{i},D_{i+1}]$ $\displaystyle\quad f^{\top}Af+f^{\top}\beta$ $p\in\mathcal{R}^{\operatorname{use}}_{D^{\prime}}$ $\delta\lambda^{\prime j-1}>\delta\lambda^{\prime j}$ $\delta C^{M}_{p}=\delta\lambda^{M}$ $\lambda^{\operatorname{WE}}$ $\delta\lambda^{i-1}=\min_{r\in\mathcal{J}^{\operatorname{act}}_{i-1}}A_{r}f^{\delta}$ $\widetilde{\lambda}^{\operatorname{WE}}$ $\displaystyle:=(f_{p_{1}},f_{p_{2}},\cdots,f_{p_{m}})^{\top},$ $(f^{\delta})^{\top}\Big{(}A\big{(}f^{D_{M}}+(D-D_{M})f^{\delta}\big{)}+\beta%
+\Big{)},$ $\displaystyle=\{r\in\mathcal{P}\;|\;f^{\delta}_{r}=0,\hskip 2.0pt\delta C^{M}_%
+{r}>\delta\lambda^{M}\},$ $\displaystyle\quad C_{r}(f)=\delta\lambda^{M}D+\bar{\beta}$ $\displaystyle=(Af^{\delta})^{\top}(f-f^{\delta_{0}})$ $\widetilde{f}^{D}+\epsilon\widetilde{f}^{\delta}\in\mathcal{W}_{D+\epsilon}$ $\delta C^{M}_{p}>\min_{r\in\mathcal{P}}\delta C^{M}_{r}$ $\{\mathcal{J}^{\operatorname{use}}_{0},\mathcal{J}^{\operatorname{use}}_{1},%
+\cdots,\mathcal{J}^{\operatorname{use}}_{M}\}$ $\widetilde{f}^{D}\in\mathcal{W}_{D}\cap\widetilde{\mathcal{F}}_{D}$ $\displaystyle\leq\min_{r\in\mathcal{P}}\bigl{(}C_{r}(f^{D})+\epsilon A_{r}f^{%
+\delta}\bigr{)}$ ${\operatorname{coco}_{\mu}(f^{-},f^{+})=\mu f^{-}+(1-\mu)f^{+}}$ $\widetilde{f}^{D}=(\frac{1}{2},\enskip\frac{1}{2})^{\top}$ $W(D)=\int_{0}^{D}z\big{(}\widetilde{\lambda}^{\operatorname{WE}}(z)-\lambda^{%
+\operatorname{WE}}(z)\big{)}dz.$ $\mathcal{R}^{\operatorname{act}}_{D}:=\{p\in\mathcal{P}\;|\;C_{p}(f^{D})\leq C%
+_{r}(f^{D}),\,\,\forall f^{D}\in\mathcal{W}_{D},\,\,\forall r\in\mathcal{P}\}.$ $Af^{\delta}=A\widetilde{f}^{\delta}$ $Af^{\delta}=\delta C^{i},$ $\delta C^{i-1}_{r}=Af^{\delta}$ ${p_{4}:=(e_{6})}$ $C_{p}(f^{\mu})\leq C_{r}(f^{\mu})$ $\mathcal{S}^{\texttt{rem}}$ $D^{\prime}_{j-1}>0$ $\widetilde{\mathcal{M}}\subseteq\mathcal{M}$ ${[n]_{0}:=\{0,1,\cdots,n\}}$ $p\in\mathcal{R}^{\operatorname{use}}_{D}$ $B_{k,i}=0$ $\displaystyle(f^{\delta})^{\top}(Af^{D_{M}}+\beta)$ $\displaystyle(f^{*})^{\top}Af^{*}+(f^{*})^{\top}\beta=D(\delta\lambda^{M}D+%
+\bar{\beta})$ $\Gamma^{-}_{D}$ $\displaystyle\mathcal{I}_{3}$ $\|f^{D}-f^{T}\|<\epsilon$ $v^{\operatorname{out}}_{p_{l}}=v_{d}$ $f^{\mu}\in\mathcal{F}_{T_{\mu}}$ $\vec{\lambda}^{\operatorname{WE}}$ $(\mathcal{J}^{\operatorname{use}}_{i})^{c}\notin\mathcal{N}_{D+\epsilon}$ $\delta C^{i}=\delta C^{i-1}$ $\delta C_{p}^{M}>\delta\lambda^{M}$ $\displaystyle:=\{f^{\delta}\in\mathcal{H}_{1}\;|\;f^{\delta}_{\widetilde{%
+\mathcal{Q}}\setminus\widetilde{\mathcal{R}}}\geq 0,\quad f^{\delta}_{%
+\widetilde{\mathcal{Q}}^{c}}=0\},$ $\displaystyle=\lambda^{\operatorname{WE}}(D_{M})+(T-D_{M})\delta\lambda^{M},$ $\displaystyle:=\big{(}C_{e_{1}}(f_{e_{1}}),\cdots,C_{e_{q}}(f_{e_{q}})\big{)}^%
+{\top},$ $f^{\delta}\in\Gamma_{D_{M}}\cap\mathcal{F}_{1}$ $\Gamma_{D}\subseteq\Gamma^{M}$ $\widetilde{\mathcal{J}}^{\operatorname{act}}_{j}=\widetilde{\mathcal{R}}^{%
+\operatorname{act}}_{\widetilde{D}_{j+1}}$ $\mathcal{R}^{\operatorname{act}}_{D}=\mathcal{J}^{\operatorname{act}}_{i}$ $\displaystyle\lambda^{\operatorname{WE}}(D)$ $\mathcal{J}^{\operatorname{act}}_{i-1}\subset\mathcal{R}^{\operatorname{act}}_%
+{D_{i}}$ $D^{\prime\prime}_{k-1} ${\lambda^{\operatorname{vec}}(T)=\lambda^{\operatorname{WE}}(D_{i})+(T_{\mu}-D%
+_{i})Af^{\delta_{0}}}$ $\displaystyle=\int_{0}^{D}z\big{(}\widetilde{\lambda}^{\operatorname{WE}}(z)-%
+\lambda^{\operatorname{WE}}(z)\big{)}dz$ ${\mathcal{R}^{\operatorname{act}}_{D}=\{p_{1},p_{2},p_{4}\}}$ $\displaystyle=(f^{\delta})^{\top}Af^{\delta}$ $C_{e_{5}}(f_{e_{5}}):=f_{e_{5}},\quad C_{e_{6}}(f_{e_{6}}):=2+f_{e_{6}}.$ $(f^{*})^{\top}Af^{*}+(f^{*})^{\top}\beta=D(\delta\lambda^{M}D+\bar{\beta})$ $C(f)=Af+b=\left(\begin{array}[]{cccc}1&0&1&0\\
+0&1&1&0\\
+1&1&2&0\\
+0&0&0&0\\
+\end{array}\right)f+\left(\begin{array}[]{c}1\\
+1\\
+0\\
+2\\
+\end{array}\right).$ $r\notin\mathcal{R}^{\operatorname{act}}_{D^{+}}$ $\delta C^{M}$ $\mathcal{S}^{\texttt{rem}}_{T}\subset\mathcal{P}$ $\mathcal{R}^{\prime\operatorname{act}}_{D^{\prime}_{j}}=\mathcal{J}^{\prime%
+\operatorname{act}}_{j}$ $D_{M}=1$ $\displaystyle\text{ for all }r\in(\mathcal{J}^{\operatorname{act}}_{M})^{c},$ $(\mathcal{P}\setminus\widehat{\mathcal{S}})$ $f^{D}=\begin{cases}\left(\begin{array}[]{cccc}0&0&D&0\end{array}\right)^{\top}%
+&\text{for }D\in[0,\frac{1}{2}],\\
+\left(\begin{array}[]{cccc}\frac{8D-4}{12}&\frac{6D-3}{12}&\frac{7-2D}{12}&0%
+\end{array}\right)^{\top}&\text{for }D\in[\frac{1}{2},\frac{7}{2}],\\
+\left(\begin{array}[]{cccc}\frac{4D}{7}&\frac{3D}{7}&0&0\end{array}\right)^{%
+\top}&\text{for }D\in[\frac{7}{2},\frac{35}{9}],\\
+\left(\begin{array}[]{cccc}\frac{7D+15}{19}&\frac{3D+20}{19}&0&\frac{9D-35}{19%
+}\end{array}\right)^{\top}&\text{for }D\in[\frac{35}{9},6],\\
+\left(\begin{array}[]{cccc}\frac{10D+27}{29}&\frac{4D+34}{29}&\frac{D-6}{29}&%
+\frac{14D-55}{29}\end{array}\right)^{\top}&\text{for }D\in[6,\infty).\end{cases}$ ${p_{1}:=(e_{1},e_{2})}$ $p\in\mathcal{R}^{\operatorname{act}}_{T}$ $G:{}^{n}\rightarrow{}^{n}$ $\displaystyle(f^{\prime})^{\top}Af^{\prime}+(f^{\prime})^{\top}\beta$ $\alpha_{e_{k}}>0$ $f^{D^{+}}\in\mathcal{W}_{D^{+}}$ $(\mathcal{P},\mathcal{C},D)$ $f^{-},f^{+}\in{}^{n}$ $\displaystyle\mathcal{J}^{\operatorname{act}}_{i-1}$ $T\geq D^{\operatorname{BP}}$ $\widetilde{\lambda}^{\operatorname{vec}}$ $f^{\delta_{0}}\in\operatorname{SOL}(\mathcal{M},A)$ $D_{i}>0$ $\displaystyle=(Af^{\delta})^{\top}(f-f^{\delta}+f^{\delta}-f^{\delta}_{0})$ $[D_{i},D_{i+1})$ $\lambda^{\operatorname{vec}}(T)=\lambda^{\operatorname{vec}}(D)+(T-D)Af^{\delta}$ ${Af^{\delta}=Af^{\delta_{0}}}$ $\mathcal{R}:=\mathcal{R}^{\operatorname{use}}_{D}$ $D^{-}\in[D_{i},D_{i+1})$ $f^{*}_{p}<0$ $f^{\delta}\in\Gamma_{T_{\mu}}=\Gamma^{i}$ $\displaystyle\quad f_{r}=0$ $C_{p}(f)\geq\delta\lambda^{M}D+\bar{\beta}$ ${D\in{\mathbb{R}}_{\geq 0}}$ $\mathcal{R}^{\operatorname{use}}_{D_{i+1}}$ $\widehat{f}^{\delta}\in\mathcal{M}_{D_{M}}$ $\displaystyle=\min_{r\in\mathcal{P}}C_{r}(f^{D+\epsilon}).$ $\widetilde{f}\in{\mathbb{R}}_{\geq 0}^{\left\lvert{\widetilde{\mathcal{P}}}%
+\right\rvert}$ $D^{\prime\prime}_{k},D^{\prime\prime}_{k+1}\in\mathcal{D}^{\prime\prime}$ $\widetilde{\lambda}(D)<\lambda^{\operatorname{WE}}(D)$ $\displaystyle:=\mathcal{P}\setminus\mathcal{S}^{\texttt{rem}},$ $f^{\prime}_{p}\neq 0$ $\lambda^{\operatorname{vec}}_{p}(D)$ $\widetilde{f}^{D}+\epsilon\widetilde{f}^{\delta}$ $\lambda^{\operatorname{vec}}_{p}(D)=\lambda^{\operatorname{vec}}_{r}(D)$ $V(0)=\widetilde{V}(0)=0$ $\delta C^{M}_{r}<\delta\lambda^{M}$ $\displaystyle\leq\min_{r\in\mathcal{Q}}A_{r}f^{\delta-}).$ $\displaystyle u_{\mathcal{P},i}(D)$ $p\in\mathcal{R}^{\operatorname{use}}_{D_{i+1}}$ $p,r\in\widetilde{\mathcal{P}}$ $n=\left\lvert{\mathcal{P}}\right\rvert$ $\widetilde{f}^{\delta}\in\widetilde{\mathcal{M}}\subseteq\mathcal{M}$ $(D^{-},D^{+})$ $D_{i},D_{i+1}\in\mathcal{D}$ $(e_{3},e_{2})$ $i\in[M]_{0}$ $\displaystyle $(2,\infty)$ $f^{D_{M}}+(D-D_{M})f^{\delta}\geq 0$ $f\in\ker(A)\cap\mathcal{M}_{D}$ $\displaystyle\beta^{\top}f^{\delta}$ $\displaystyle\min_{\widetilde{f}\in\widetilde{\mathcal{F}}_{D}}\sum_{e_{k}\in%
+\mathcal{E}}\int_{0}^{\widetilde{f}_{e_{k}}}C_{e_{k}}(z)dz.$ $\mathcal{M}_{D_{i}}\subseteq\mathcal{M}_{D}$ $\displaystyle(Af^{\delta})^{\top}(f-f^{\delta})$ $\mathcal{J}^{\operatorname{act}}_{i}\cap\mathcal{J}^{\operatorname{act}}_{i+1}\neq\emptyset$ ${r^{\prime}\in\mathcal{R}^{\operatorname{act}}_{D_{i+1}}}$ $\widetilde{f}^{D}\in\widetilde{\mathcal{W}}_{D}$ $C_{p}(f^{*})=\delta\lambda^{M}D+\bar{\beta}$ $T\in(D_{i},D_{i+1}]$ $f^{\delta}\in\Gamma_{D_{i}}$ $\displaystyle=\lambda^{\operatorname{vec}}(D_{i})+(D^{-}-D_{i})\delta C^{i}+(D%
+^{+}-D^{-})\delta C^{i}$ ${D_{i},D_{i+1}\in\mathcal{D}}$ $1\leq i\leq j\leq l$ $\widetilde{D}_{\widetilde{M}}$ $r\in(\mathcal{J}^{\operatorname{act}}_{i})^{c}$ $T\in[D^{+},D]$ $\displaystyle=D(\delta\lambda^{M}D+\bar{\beta}).$ ${\lambda^{\prime\operatorname{WE}}(D)=u_{\mathcal{P},i}(D)}$ $D^{\prime}=\mu D+(1-\mu)T$ $\lambda^{\operatorname{WE}}(D)=D\delta\lambda^{M}+\bar{\beta}.$ $\delta C^{i+1}=\delta C^{i}$ $\operatorname{VI}(\mathcal{X},G)$ $T\in{\mathbb{R}}_{\geq 0}$ $\delta\lambda^{i}=\min_{r\in\mathcal{R}^{\operatorname{act}}_{D_{i}}}\delta C^%
+{i}$ $\mathcal{J}^{\prime\operatorname{use}}_{j-1}\subseteq\mathcal{P}^{\prime}$ ${\mathcal{S}^{\texttt{rem}}\notin\mathcal{N}_{D}}$ $r\in\mathcal{I}_{2}$ ${f^{\delta}\in\operatorname{SOL}(\mathcal{M}_{D_{i}},A)}$ ${\ker(A)\cap\mathcal{M}_{D}}$ $\delta C\in{}^{n}$ $\displaystyle=f^{D_{i}}+(T_{\mu}-D_{i})f^{\delta_{0}}.$ $(\widetilde{f}^{\delta})^{\top}A\widetilde{f}^{\delta}=\min_{r\in\widetilde{%
+\mathcal{Q}}}A_{r}\widetilde{f}^{\delta}$ $f^{D_{i+1}}$ $C_{p}(f^{D^{+}})\geq\delta\lambda^{M}D^{+}+\bar{\beta}.$ $\min_{r\in\widetilde{\mathcal{R}}^{\operatorname{act}}_{D}}A_{r}\widetilde{f}^%
+{\delta}\geq\min_{r\in\mathcal{R}^{\operatorname{act}}_{D}}A_{r}f^{\delta}.$ $\sum_{r\in\mathcal{R}_{i}}f^{\delta}_{r}\geq-nt(T_{i}-D)^{-1}.$ $f^{\delta}\in\mathcal{M}_{D_{M}}$ $\widetilde{\mathcal{P}}:=\mathcal{J}^{\operatorname{use}}_{i}$ $t:=\max_{r\in\mathcal{P}}f^{D}_{r}$ $\mymathbf{1}^{\top}f^{*}=D_{M}$ $\mathcal{V}=[N]$ $V(D):=\min_{f\in\mathcal{F}_{D}}\sum_{e_{k}\in\mathcal{E}}\int_{0}^{f_{e_{k}}}%
+C_{e_{k}}(z)dz.$ $C(f+\epsilon f^{0})=C(f)$ $\mathcal{J}^{\operatorname{use}}_{i}\subseteq\mathcal{P}$ $\lambda^{\operatorname{WE}}(D^{+})=\widetilde{\lambda}^{\operatorname{WE}}(D^{%
++})$ $(\widetilde{f}^{\delta})^{\top}A(f^{\delta}-\widetilde{f}^{\delta})>0.$ $v_{o}\in\mathcal{V}$ $\widecheck{\lambda}^{\operatorname{WE}}_{i,D}$ $\lambda^{\operatorname{WE}}(D_{i})=u_{\emptyset,i}(D_{i})$ $(D_{i+1}-\epsilon,D_{i+1}]$ $\displaystyle=(f^{\delta})^{\top}Af^{D_{M}}+(D-D_{M})(f^{\delta})^{\top}Af^{%
+\delta}+\beta^{\top}f^{\delta}.$ $f^{D}=\begin{cases}\left(\begin{matrix}0,&0,&D,&0\end{matrix}\right)^{\top}&%
+\text{for }D\in[0,\frac{1}{2}],\\
+\frac{1}{3}\left(\begin{matrix}2D-1,&2D-1,&2-D,&0\end{matrix}\right)^{\top}&%
+\text{for }D\in[\frac{1}{2},2],\\
+\frac{1}{3}\left(\begin{matrix}D+1,&D+1,&0,&D-2\end{matrix}\right)^{\top}&%
+\text{for }D\in[2,\infty).\end{cases}$ ${p_{1}=(e_{1},e_{2})}$ ${p_{4}=(e_{3},e_{6})}$ $\displaystyle=\widetilde{V}(z)|_{0}^{D}-V(z)|_{0}^{D}$ $f^{D^{+}}$ $D\in(\frac{2}{3},2)$ $p\notin\mathcal{R}^{\operatorname{act}}_{T}$ $\lambda^{\operatorname{vec}}$ $\displaystyle(Af^{\delta_{0}})^{\top}(f-f^{\delta_{0}})$ $\widehat{\mathcal{S}}^{\prime\prime}\subset\widehat{\mathcal{P}}^{\prime}$ ${D\geq D_{M}=\max\big{(}\mathcal{D}\setminus\{\infty\}\big{)}}$ $\displaystyle=\lambda^{\operatorname{vec}}(D_{i})+(T_{\mu}-D_{i})Af^{\delta_{0%
+}}.$ $C(f^{D^{+}})=C(f^{*})+(D^{+}-D)Af^{\delta}.$ $p\in\mathcal{R}^{\operatorname{act}}_{D}$ $q:=\left\lvert{\mathcal{E}}\right\rvert$ $\widetilde{\lambda}(D)=u_{\widetilde{\mathcal{P}},\widetilde{M}}(D)$ $D(\delta\lambda^{M}D+\bar{\beta})$ $\mathcal{S}^{\texttt{rem}}\subset\mathcal{P}$ $f^{D_{M}}+(D-D_{M})f^{\delta}\in\mathcal{W}_{D}$ ${\frac{\partial^{-}}{\partial x}g(x):=\lim_{h\rightarrow 0^{-}}\frac{g(x+h)-g(%
+h)}{h}}$ $f^{\delta}\in\Gamma_{D}$ $D\in[\widetilde{D}_{j},\widetilde{D}_{j+1})$ $f^{\delta,i}_{r}\geq-t(T_{i}-D)^{-1}$ $f^{*}_{p}>0$ $\delta\lambda^{+}(T):=\frac{\partial^{+}}{\partial D}\lambda^{\operatorname{WE%
+}}(D)\Big{|}_{D=T},\quad\delta\lambda^{-}(T):=\frac{\partial^{-}}{\partial D}%
+\lambda^{\operatorname{WE}}(D)\Big{|}_{D=T}.$ $\displaystyle\Gamma^{-}_{D}:=\{f^{\delta}\in\mathcal{H}_{-1}\;|\;\exists f^{D}%
+\in\mathcal{W}_{D},\enskip\bar{\epsilon}>0\text{ such that }f^{D}+\epsilon f^{%
+\delta}\in\mathcal{W}_{D-\epsilon}\enskip\forall\epsilon\in[0,\bar{\epsilon}]\}.$ $\displaystyle=\lambda^{\operatorname{WE}}(D_{i})+(D-D_{i})\delta\lambda^{i-1},$ $D=D_{i}$ $D_{M+1}=\infty$ $f^{\mu}=\operatorname{coco}_{\mu}(f^{D_{i}},f^{T})$ $\mathcal{J}^{\operatorname{act}}_{i-1}\subseteq\mathcal{R}^{\operatorname{act}%
+}_{D_{i}}$ $\lambda^{\operatorname{vec}}_{r^{\prime}}(D_{i+1})\leq\lambda^{\operatorname{%
+vec}}_{p^{\prime}}(D_{i+1})$ $p\notin\mathcal{J}^{\operatorname{act}}_{M}$ $f^{\mu}_{p}>0$ $D+\epsilon\in(D,D_{i+1})$ $\lim_{i\rightarrow\infty}T_{i}=\infty$ $\displaystyle=\delta\lambda^{M}D+\bar{\beta}.$ $D^{\prime\prime}_{k-1}=0$ $\delta C^{i}:=Af^{\delta_{0}}$ $\displaystyle:=\widetilde{\lambda}^{\operatorname{vec}}_{p}(D)\text{ for any }%
+p\in\widetilde{\mathcal{R}}^{\operatorname{act}}_{D},$ $i\in[\vec{\mathcal{M}}]_{0}$ $\displaystyle W(D)$ $\mathcal{R}^{\operatorname{act}}_{T}=\mathcal{R}^{\operatorname{act}}_{D}$ $\sum_{r\in\mathcal{R}^{\operatorname{use}}_{D}}\mu_{r}=1$ $\mathcal{J}^{\operatorname{use}}_{i-1}=\mathcal{R}^{\operatorname{use}}_{D_{i}}$ $\mathcal{J}^{\operatorname{act}}_{i}\neq\mathcal{J}^{\operatorname{act}}_{i+1}$ $\{f^{\delta,i}\}_{i\in\mathbb{N}}$ $(e_{1},e_{4})$ $\delta\widetilde{\lambda}^{+}(D)\geq\delta\lambda^{+}(D).$ $f_{(\mathcal{R}^{\operatorname{use}}_{T})^{c}}^{T}=0$ $\displaystyle C_{p}(f^{D})$ $\operatorname{SOL}(\mathcal{F}_{1},A)$ $T=\operatorname{coco}_{\mu}(D^{-},D^{+})$ $f^{\mu}$ $T_{i} $f^{D+\epsilon}:=f^{D}+\epsilon f^{\delta}.$ ${u_{\mathcal{P}^{\prime},j-1}(D) $f^{T_{i}}\geq 0$ $C(f)=Af$ $\lambda^{\prime\operatorname{WE}}(D)=\lambda^{\operatorname{WE}}(D),\quad\text%
+{for all }D\in[D_{i},D_{i+1}].$ $f^{\delta}_{p}=0$ $\widehat{\mathcal{P}}\subset\widetilde{\mathcal{P}}$ $A\big{(}f^{D_{M}}+(D-D_{M})f^{\delta}\big{)}+\beta=\lambda^{\operatorname{vec}%
+}(D)$ ${\mathcal{R}^{\operatorname{act}}_{D}=\{p_{1},p_{2},p_{3}\}}$ $(\widehat{\mathcal{P}}^{\prime}\setminus\widehat{\mathcal{S}}^{\prime\prime},%
+\mathcal{C})$ ${\mathcal{P}^{c}=\{p\in[n]\;|\;p\notin\mathcal{P}\}}$ $D^{\prime}_{j}>0$ $\lambda^{\operatorname{WE}}(D)>\widehat{\lambda}^{\operatorname{WE}}(D)$ $\mathcal{S}^{\prime}\cup\mathcal{S}^{\prime\prime}$ $\displaystyle\Gamma_{D}:=\begin{cases}\{f^{\delta}\in\mathcal{H}_{1}\;|\;f^{%
+\delta}_{1},f^{\delta}_{2},f^{\delta}_{4}=0,\hskip 2.0ptf^{\delta}_{3}=1\},&%
+\text{for }D\in[0,1),\\
+\{f^{\delta}\in\mathcal{H}_{1}\;|\;f^{\delta}_{1},f^{\delta}_{2},f^{\delta}_{4%
+}\geq 0,\hskip 2.0ptf^{\delta}_{1}=f^{\delta}_{2}=-f^{\delta}_{3}\},&\text{for%
+ }D=1,\\
+\{f^{\delta}\in\mathcal{H}_{1}\;|\;f^{\delta}_{1}+f^{\delta}_{3}=0,\hskip 2.0%
+ptf^{\delta}_{2}+f^{\delta}_{3}=0\},&\text{for }D\geq 1.\end{cases}$ $f^{\delta}\in\Gamma^{M}\cap\operatorname{SOL}(\mathcal{F}_{1},A)$ $\displaystyle\widetilde{\lambda}^{\operatorname{WE}}(T)$ $C_{p}(f^{T})=\lambda^{\operatorname{WE}}(T)$ $\displaystyle(\mathcal{R}^{\operatorname{act}}_{D},\mathcal{R}^{\operatorname{%
+use}}_{D})=\begin{cases}(\{p_{3}\},\emptyset)&\text{for }D=0,\\
+(\{p_{3}\},\{p_{3}\})&\text{for }D\in(0,1),\\
+(\{p_{1},p_{2},p_{3}\},\{p_{3}\})&\text{for }D=1,\\
+(\{p_{1},p_{2},p_{3}\},\{p_{1},p_{2},p_{3}\})&\text{for }D\in(1,2),\\
+(\{p_{1},p_{2},p_{3}\},\{p_{1},p_{2}\})&\text{for }D=2,\\
+(\{p_{1},p_{2}\},\{p_{1},p_{2}\})&\text{for }D\in(2,\infty).\\
+\end{cases}$ $\mathcal{R}^{\operatorname{use}}_{D^{-}}=\mathcal{R}^{\operatorname{use}}_{D^{%
++}}$ $\Gamma_{D_{i}}=\operatorname{SOL}(\mathcal{M}_{D_{i}},A)$ $\widetilde{\lambda}^{\operatorname{WE}}(1)=1.5$ ${T>D}$ $\widetilde{f}^{D}_{\mathcal{S}^{\texttt{rem}}}=0$ $f^{\delta}\in\Gamma_{D^{-}}$ $\lambda^{\operatorname{WE}}(T)<\widetilde{\lambda}^{\operatorname{WE}}(T)$ $\Gamma_{D_{M}}=\operatorname{SOL}(\mathcal{M}_{D_{M}},A)$ $f^{T_{i}}$ $\lambda^{\operatorname{WE}}(T)=\widetilde{\lambda}^{\operatorname{WE}}(T)$ $f^{\delta}\in\Gamma_{D}\cap\operatorname{SOL}(\mathcal{F}_{1},A)$ $D^{\prime}_{j-1}=0$ $f^{D^{+}}=f^{D^{-}}+(D^{+}-D^{-})f^{\delta}.$ $\delta\lambda^{M}$ $\widetilde{\mathcal{M}}^{-}\subseteq\mathcal{M}^{-}$ $\displaystyle f^{\mu}$ $\displaystyle=\lambda^{\operatorname{WE}}(D)$ $\mathcal{W}_{D}$ $\delta C^{i}\neq\delta C^{i+1}$ $T\geq\max(D_{M},\widetilde{D}_{\widetilde{M}})$ $\Gamma_{D}\cap\mathcal{F}_{1}$ $\mathcal{M}=\mathcal{M}_{D_{i}}$ $\mymathbf{0}$ $\displaystyle C_{e_{6}}(f_{e_{6}})=f_{e_{6}}+5,$ $f^{D^{+}}\geq 0$ $\operatorname{coco}_{\mu}(f^{D^{-}},f^{D^{+}})\in\mathcal{W}_{T}$ ${T_{\mu}\in(D_{i},D_{i+1})}$ $(v^{\operatorname{in}}_{k},v^{\operatorname{out}}_{k})$ $[\vec{D}_{i},\vec{D}_{i+1})$ $v^{\operatorname{out}}_{pj}\neq v^{\operatorname{in}}_{p_{i}}$ $\displaystyle C_{\mathcal{R}}(f)$ $\delta\lambda^{\prime j}=\delta\lambda^{i}$ $D\in(2,\infty)$ $C_{p}(f^{\prime})=\delta\lambda^{M}D+\bar{\beta}$ $f^{\prime}_{p}\geq 0$ $\mathcal{J}^{\operatorname{act}}_{i}\neq\mathcal{J}^{\operatorname{act}}_{j}$ $p\in\mathcal{R}^{\operatorname{use}}_{D_{M}}$ $\delta C_{p}^{M}<\delta\lambda^{M}$ $\mathcal{S}\notin\mathcal{N}_{D}$ $D^{-},D^{+}$ ${\mathcal{R}=\{p_{1},p_{2},\cdots,p_{m}\}\subseteq\mathcal{P}}$ $\displaystyle(Af^{\delta_{1}})^{\top}(f^{\delta_{2}}-f^{\delta_{1}})$ $A\widehat{f}^{\delta}=\delta C^{M}$ $\mathcal{R}^{\operatorname{use}}_{D}=\mathcal{J}^{\operatorname{use}}_{i}$ $f^{\delta}\in\Gamma_{D_{M}}$ $\mathcal{S}^{\texttt{rem}}_{T}$ $\displaystyle=Af^{\prime}+\beta,$ $\lambda^{\operatorname{vec}}:{\mathbb{R}}_{\geq 0}\rightarrow{\mathbb{R}}_{%
+\geq 0}^{n}$ ${T\geq\max(D_{M},\widetilde{D}_{\widetilde{M}})}$ $\widetilde{f}^{D}+\epsilon\widetilde{f}^{\delta}\in\widetilde{\mathcal{W}}_{D+\epsilon}$ $(b_{1},\dots b_{|V^{lca}|})\leftarrow\textsc{Sort}(V^{lca})$ $V^{olca}\leftarrow V^{olca}\cup\{\textsc{LCA}(v_{i_{j}},v_{i_{j+1}})\}$ $O\left(k\log k\right)$ $l=LCA(v,q)$ $d_{c}=\textsc{DistToClosest}(u)+w^{virt}(u,v)$ $x_{1},\dots,x_{i}$ $v\in V^{s}$ $t_{out}(v)\leftarrow timer$ $O_{opt}(I)=\operatorname{arg\,min}_{O\in\widetilde{O}(I)}cost(I,O)$ $O\left(k^{2}+k\log n\right)$ $\operatorname{dist}\left(1,\textsc{Parent}(v)\right)+1=\operatorname{dist}(1,v)$ $v_{i_{1}},\dots,v_{i_{k+1}}$ $V^{lca}\leftarrow V^{s}\cup V^{olca}$ $\textsc{Closest}(v)=v$ $v^{\prime}_{1},\dots,v^{\prime}_{k}$ $\textsc{Move}(\textsc{Closest}(r),\textsc{DistToClosest}(r))$ $\operatorname{dist}(l,v)\geq z$ $z\leq\operatorname{dist}(l,v)$ $|V^{lca}|\leq|V^{s}|+|V^{olca}|\leq|V^{s}|+|V^{s}|-1\leq 2|V^{s}|$ $\textsc{ComputeDistanceBase}(u)$ $u,v\in V^{s}$ $dist(x,u)+dist(x,v)=dist(u,v)$ $\textsc{UpdatePositions}(r,\textsc{DistToClosest}(r))$ $\operatorname{dist}(1,v)$ $\textsc{Move}(\textsc{Closest}(u),b)$ $t_{in}(v)\leftarrow timer$ $|V^{lca}|\leq 2|V^{s}|$ $z>\operatorname{dist}(l,v)$ $\textsc{UpdatePositions}(u,b)$ $O\left(k^{2}+k\cdot\log n\right)$ $V^{s}\subseteq V^{lca}$ $\textsc{Closest}(v)\leftarrow v_{c}$ $V^{lca}$ $\textsc{DistToClosest}(v)=0$ $v_{c}\leftarrow\textsc{Closest}(u)$ $\textsc{Children}(v)=\{u:\textsc{Parent}(u)=v\}$ $.08{\scriptstyle\ \pm.01}$ $.61{\scriptstyle\ \pm.01}$ $\mathbf{\%}$ $.39{\scriptstyle\ \pm.02}$ $.03{\scriptstyle\ \pm.00}$ $.13{\scriptstyle\ \pm.01}$ $3.37\times 10^{-5}$ $\log\ \mathcal{U}[1\times 10^{-5},0.01]$ $.22{\scriptstyle\ \pm.01}$ $.04{\scriptstyle\ \pm.00}$ $.07{\scriptstyle\ \pm.00}$ $.60{\scriptstyle\ \pm.02}$ $\underline{\mathbf{.00}}{\scriptstyle\ \pm.00}$ $.48{\scriptstyle\ \pm.00}$ $\mathcal{C}_{m}$ $.49{\scriptstyle\ \pm.01}$ $.53{\scriptstyle\ \pm.01}$ $.62{\scriptstyle\ \pm.03}$ $9.58\times 10^{-5}$ $\mathbf{1}(\hat{y}_{i}=y_{i})$ $.27{\scriptstyle\ \pm.00}$ $0.005793$ $\mathbf{.12}{\scriptstyle\ \pm.01}$ $\mathbb{E}\Big{[}\big{|}\mathbb{P}\big{(}Y=\hat{y}\ |\ P=\hat{p}\big{)}-\hat{p%
+}\big{|}\Big{]},$ $.16{\scriptstyle\ \pm.01}$ $.11{\scriptstyle\ \pm.00}$ $\mathbf{.18}{\scriptstyle\ \pm.01}$ $\tau=0.35$ $.05{\scriptstyle\ \pm.01}$ $\mathbf{.82}{\scriptstyle\ \pm.01}$ $.05{\scriptstyle\ \pm.00}$ $\hat{y}\in\mathcal{Y}$ $\mathbf{.01}{\scriptstyle\ \pm.00}$ $\mathcal{U}[1\times 10^{-4},0.05]$ $.02{\scriptstyle\ \pm.01}$ $0.01932$ $0,\ 0.33,\ 0.66$ $.70{\scriptstyle\ \pm.17}$ $8.84\times 10^{-5}$ $\hat{a}_{j}$ $.74{\scriptstyle\ \pm.01}$ $.81{\scriptstyle\ \pm.01}$ $.38{\scriptstyle\ \pm.03}$ $.11{\scriptstyle\ \pm.08}$ $.30{\scriptstyle\ \pm.00}$ $\underline{\mathbf{.03}}{\scriptstyle\ \pm.01}$ $.32{\scriptstyle\ \pm.00}$ $\hat{q}=\sigma(a\hat{p}+b)$ $.01{\scriptstyle\ \pm.00}$ $\hat{p}\in[0,1]$ $.22{\scriptstyle\ \pm.00}$ $0.008936$ $5.59\times 10^{-5}$ $.64{\scriptstyle\ \pm.02}$ $.32{\scriptstyle\ \pm.01}$ $.70{\scriptstyle\ \pm.01}$ $.55{\scriptstyle\ \pm.01}$ $\mathbf{.02}{\scriptstyle\ \pm.01}$ $\underline{\mathbf{.72}}{\scriptstyle\ \pm.02}$ $.15{\scriptstyle\ \pm.00}$ $0.73$ $.08{\scriptstyle\ \pm.00}$ $\mathbf{.03}{\scriptstyle\ \pm.01}$ $.06{\scriptstyle\ \pm.01}$ $.00{\scriptstyle\ \pm.12}$ $.26{\scriptstyle\ \pm.00}$ $.72{\scriptstyle\ \pm.02}$ $.07{\scriptstyle\ \pm.01}$ $\mathcal{B}_{m}$ $.48{\scriptstyle\ \pm.01}$ $.82{\scriptstyle\ \pm.01}$ $.19{\scriptstyle\ \pm.01}$ $.38{\scriptstyle\ \pm.00}$ $.09{\scriptstyle\ \pm.01}$ $0.01362$ $\sum_{m=1}^{M}\frac{|\mathcal{B}_{m}|}{N}\Big{|}\underbrace{\frac{1}{|\mathcal%
+{B}_{m}|}\sum_{i\in\mathcal{B}_{m}}\mathbf{1}(\hat{y}_{i}=y_{i})}_{\text{Bin %
+accuracy (target)}}-\underbrace{\frac{1}{|\mathcal{B}_{m}|}\sum_{i\in\mathcal{%
+B}_{m}}\hat{p}_{i}}_{\text{Avg. bin confidence}}\Big{|},$ $.06{\scriptstyle\ \pm.00}$ $\mathbf{.02}{\scriptstyle\ \pm.00}$ $.25{\scriptstyle\ \pm.00}$ $.19{\scriptstyle\ \pm.00}$ $\mathbf{.07}{\scriptstyle\ \pm.01}$ $\mathbf{.03}{\scriptstyle\ \pm.00}$ $\underline{\mathbf{.06}}{\scriptstyle\ \pm.01}$ $\underline{\mathbf{.12}}{\scriptstyle\ \pm.01}$ $\mathcal{C}(i)$ $.39{\scriptstyle\ \pm.01}$ $.52{\scriptstyle\ \pm.02}$ $.47{\scriptstyle\ \pm.25}$ $\underline{\mathbf{.18}}{\scriptstyle\ \pm.01}$ $.65{\scriptstyle\ \pm.02}$ $1.62\times 10^{-5}$ $.45{\scriptstyle\ \pm.01}$ $.21{\scriptstyle\ \pm.00}$ $.02{\scriptstyle\ \pm.00}$ $.60{\scriptstyle\ \pm.14}$ $.75{\scriptstyle\ \pm.01}$ $0.03184$ $.24{\scriptstyle\ \pm.00}$ $.03{\scriptstyle\ \pm.01}$ $.72{\scriptstyle\ \pm.01}$ $O(k\log k)+O(k)=O(k\log k)$ $q\in\{v^{\prime}_{1},\dots,v^{\prime}_{k}\}$ $x\in V^{lca}$ $i_{1},\dots,i_{k+1}$ $v\in V^{lca}$ $t_{in}(v)$ $V^{lca}=\{LCA(v,w):v,w\in V^{s}\}$ $|V^{s}|\leq|V^{lca}|$ $\sum_{i=1}^{k}\operatorname{dist}(v_{i},v^{\prime}_{i})$ $V^{s}\cup V^{olca}\subseteq V^{lca}$ $x\neq u,x\neq v$ $\textsc{Closest}(v)=\textsc{Closest}(u)$ $\textsc{DistToClosest}(v)=min\{\textsc{DistToClosest}(u)+w^{virt}(u,v):u\in%
+\textsc{Children}(v)\}$ $i\in\{1,\dots,h-1\}$ $b\leftarrow\min\{\textsc{Closest}(r),\textsc{Closest}(u)\}$ $\textsc{ProcessingAQuery}(q,T,v_{1},\dots,v_{k})$ $V^{olca}=\{LCA(v_{i_{j}},v_{i_{j+1}}):1\leq j\leq k\}$ $d=dist(1,v)-z$ $V^{s}\cup V^{olca}$ $timer\leftarrow timer+1$ $Result\leftarrow\textsc{LA}(v,dist(1,v)-z)$ $M=|I|$ $v\in\{v_{1},\dots,v_{k}\}$ $V^{s}=\{v_{1},\dots,v_{k},q\}$ $O\left(k(\log n)^{2}\right)$ $\textsc{InitStack}({\cal S})$ $\textsc{Closest}(u)$ $\textsc{IsEmpty}({\cal S})=False$ $\textsc{LA}(v,d)$ $v\in T_{v}$ $\textsc{ConstructVirtualTree}(T,v_{1},\dots,v_{k},q)$ $\textsc{Move}(v,z)$ $v\in\{\textsc{Closest}(u):u\in\textsc{Children}(r)\}\backslash\{\textsc{%
+Closest}(r)\}$ $\textsc{ClosestComputing}(v)$ $\operatorname{dist}(1,1)\leftarrow 0$ $\textsc{Parent}(v)$ $d_{c}\leftarrow\textsc{DistToClosest}(u)+w^{virt}(u,v)$ $\textsc{Id}(v_{i})$ $d_{c}=NULL$ $\textsc{ClosestComputing}(r)$ $\textsc{Closest}(u)=\textsc{Closest}(v)$ $\textsc{Push}({\cal S},v)$ $A(I)=(y_{1},\dots,y_{M})$ $\textsc{LCA\_Preprocessing}()$ $z-\operatorname{dist}(l,v)$ $\operatorname{dist}(v,u)$ $LCA(v,v)=v$ $par=LCA(u,v)$ $\textsc{DistToClosest}(v)\leftarrow d_{c}$ $V^{olca}$ $T^{virt}$ $\textsc{ComputeDistance}()$ $cost(I,O)$ $\textsc{DistToClosest}(v)$ $dist(1,u)\leftarrow dist(1,v)+1$ $\textsc{ComputeDistance}(v)$ $O(\log^{3}n\log^{2}k)$ $O\in\widetilde{O}(I)$ $r\leftarrow\textsc{Root}(T^{virt})$ $\textsc{ComputeDistanceBase}(v)$ $d_{c}>\textsc{DistToClosest}(u)+w^{virt}(u,v)$ $(v_{i_{1}},\dots,v_{i_{k+1}})\leftarrow\textsc{Sort}(v_{1},\dots,v_{k},q)$ $\textsc{AddEdge}(T^{virt},\textsc{Pick}({\cal S}),v_{b_{j}})$ $par\in V^{olca}$ $\textsc{ComputeDistanceBase}(1)$ $v_{i_{j}}\in T_{v}$ $V^{s}=\{3,5,6,13,14\}$ $|V^{s}|,|V^{lca}|,|V^{olca}|=O(k)$ $\textsc{LCA}(u,v)$ $O=(y_{1},\dots,y_{M})$ $v_{c}\leftarrow NULL$ $par=LCA(v_{i_{j-1}},v_{i_{j}})$ $\textsc{DistToClosest}(r)$ $\textsc{UpdatePositions}(v,b)$ $d_{c}\leftarrow NULL$ $par\not\in V^{s}\cup V^{olca}$ $\textsc{Pick}({\cal S})$ $\textsc{LA}(v,dist(1,l)+z-\operatorname{dist}(l,v))$ $V^{olca}\subseteq V^{lca}$ $I\in{\cal I}$ $cost(I,A(I))\leq c\cdot cost(I,O_{Opt}(I))+\alpha$ $\textsc{Root}(T^{virt})$ $\textsc{DistToClosest}(v)\leftarrow 0$ $\textsc{LA}(v,dist(1,v)-z)$ $\textsc{Closest}(r)$ $u\in T_{u}$ $\textsc{ComputeDistanceBase}()$ $par\in V^{lca}$ $\textsc{Move}(u,z)$ $\textsc{IsAncestor}(\textsc{Pick}({\cal S}),v_{b_{j}})=False$ $t_{out}(v)$ $\cal{I}$ $t_{in}(u)\leq t_{in}(v)$ $\textsc{Closest}(v)\neq\textsc{Closest}(u)\ \textbf{then}$ $v_{i_{j-1}}\in T_{u}\cup T_{m}$ $\textsc{Closest}(v)\leftarrow v$ $\textsc{Pop}({\cal S})$ $w^{virt}(u,v)=dist(u,v)$ $|V^{olca}|\leq|V^{s}|-1$ $u\in\textsc{Children}(v)$ $\textsc{Closest}(v)$ $\textsc{ClosestComputing}(u)$ $V^{lca}=V^{s}\cup V^{olca}$ $d=dist(1,l)+z-\operatorname{dist}(l,v)$ $\textsc{Push}({\cal S},v_{b_{j}})$ $t_{out}$ $LCA(u,v)\not\in V^{s}\cup V^{olca}$ $\textsc{Id}(v_{c})>\textsc{Id}(\textsc{Closest}(u))$ $b\leftarrow\min\{b,\textsc{DistToClosest}(v)\}$ $l=\textsc{LCA}(v,q)$ $\textsc{LA\_Preprocessing}()$ $V^{lca}\subseteq V^{s}\cup V^{olca}$ $t_{out}(v)\leq t_{out}(u)$ $|V^{s}|\leq|V^{lca}|\leq 2|V^{s}|$ $v_{1},\dots,v_{h}$ $\textsc{IsAncestor}(u,v)$ $Result\leftarrow\textsc{LA}(q,dist(1,l)+z-\operatorname{dist}(l,v))$ $\textsc{IsEmpty}({\cal S})$ $(v_{1},\dots,v_{h})$ $j\in\{2,\dots,|V^{lca}|\}$ $\textsc{Push}({\cal S},v_{b_{1}})$ $t_{in}(v_{i_{j}})\leq t_{in}(v_{i_{j+1}})$ $\textsc{AddEdge}(T^{virt},u,v)$ $\widetilde{O}(I)$ $V^{olca}\leftarrow\{\}$ $u\in\textsc{Children}(r)$ $dist(1,1)\leftarrow 0$ $I=(x_{1},\dots,x_{M})$ $\displaystyle p(\mathcal{A}_{i}|f$ $p_{k_{i}NN}$ $\mathcal{L}=\frac{\mathcal{L}_{1}}{N}+\frac{\mathcal{L}_{2}}{N}$ $k_{1},\cdot,k_{K}$ $\displaystyle\mathrm{softmax}(W_{2}^{T}[\mathrm{ReLU}(W_{1}^{T}\cdot f(\textbf%
+{x},\textbf{y}_{1:i-1}))])$ $\mathcal{A}_{i}=1$ $z=f_{\mathcal{M}}(\textbf{x},\hat{\textbf{y}}_{1:i-1})$ $\{(f(\textbf{x},\textbf{y}_{1:i-1}),y_{i})|i=1,\cdots,m\}$ $W_{2}\in\mathcal{R}^{d^{\prime}\times 2}$ $\textbf{y}=\{y_{1},\cdots,y_{m}\}$ $\displaystyle=\lambda p_{KNN}(\hat{y_{i}}|\textbf{x},\hat{\textbf{y}}_{i-1})$ $q=f(\textbf{x},\hat{\textbf{y}}_{i-1})$ $\displaystyle+\sum_{j=1}^{K}p_{\text{Meta}}(k_{j})\cdot p_{k_{i}NN}(\hat{y_{i}%
+}|\textbf{x},\hat{\textbf{y}}_{1:i-1})$ $W_{1}\in\mathcal{R}^{d\times d^{\prime}}$ $\mathcal{L}_{2}=\sum_{i}[-\sum_{\hat{y}_{i}\in V}[y_{i}==\hat{y}_{i}]\log p(%
+\hat{y}_{i}|\textbf{x},\hat{\textbf{y}}_{1:i-1})]$ $p_{kNN}(\hat{y_{i}}|\textbf{x},\hat{\textbf{y}}_{1:i-1})$ $p_{KNN}(\hat{y_{i}}|\textbf{x},\hat{\textbf{y}}_{i-1})=\mathrm{sofmtax}(\frac{%
+-d(q,k_{j})}{T}),j=1,\cdots,K$ $\displaystyle G\approx$ $p(\hat{y}_{i}|\textbf{x},\hat{\textbf{y}}_{1:i-1})=\left\{\begin{matrix}p_{MT}%
+(\hat{y}_{i}|\textbf{x},\hat{\textbf{y}}_{1:i-1})&\mathcal{A}_{i}=0\\
+p_{\text{{combined}}}(\hat{y}_{i}|\textbf{x},\hat{\textbf{y}}_{1:i-1})&%
+\mathcal{A}_{i}=1\end{matrix}\right.$ $p_{combined}(\hat{y_{i}}|\textbf{x},\hat{\textbf{y}}_{1:i-1})$ $\hat{\textbf{y}}_{1:i-1}$ $\mathcal{S}(z)==0$ $\displaystyle\hat{\textbf{y}}_{i-1})$ $g_{m}=-\log(-\log(u))$ $p_{MT}(\hat{y_{i}}|\textbf{x},\hat{\textbf{y}}_{1:i-1})$ $\textbf{x}=\{x_{1},\cdots,x_{n}\}$ $(\textbf{x},\hat{\textbf{y}}_{i-1})$ $p(\hat{y}_{i}|\textbf{x},\hat{\textbf{y}}_{1:i-1})$ $\displaystyle\mathcal{L}_{1}$ $\displaystyle p_{\text{{combined}}}(\hat{y_{i}}|\textbf{x},$ $\displaystyle p_{\text{{combined}}}(\hat{y_{i}}|\textbf{x},\hat{\textbf{y}}_{i%
+-1})$ $\displaystyle=p_{\text{Meta}}(f(\textbf{x},\textbf{y}_{1:i-1}))\cdot p_{MT}(%
+\hat{y_{i}}|\textbf{x},\hat{\textbf{y}}_{1:i-1})$ $\mathcal{A}_{i}=\mathrm{argmax}_{\mathcal{A}}p(\mathcal{A}|f(\textbf{x},%
+\textbf{y}_{1:i-1})),\mathcal{A}\in\{0,1\}$ $\displaystyle+(1-\lambda)p_{MT}(\hat{y_{i}}|\textbf{x},\hat{\textbf{y}}_{1:i-1})$ $f(\textbf{x},\textbf{y}_{1:i-1})$ $\hat{y}_{i}=\mathcal{M}(f(\textbf{x},\textbf{y}_{1:i-1}))$ $p_{MT}(\hat{y_{i}}|\textbf{x},\hat{\textbf{y}}_{1:i-1})=\mathcal{M}(z)$ $\mathcal{A}_{i}\in\{0,1\}$ $\displaystyle(\textbf{x},\textbf{y}_{1:i-1}))$ $p_{\text{Meta}}$ $\displaystyle=\sum_{i}[-\frac{N}{B}[l_{i}=0]\log p(\mathcal{A}_{i}=0|f(\textbf%
+{x},\textbf{y}_{1:i-1}))$ $\displaystyle\nabla_{W}\frac{\exp((\log p(\mathcal{A}|f(\textbf{x},\textbf{y}_%
+{1:i-1});W))+g_{m}(\mathcal{A}^{\prime}))/\tau)}{\sum\limits_{\mathcal{A}^{%
+\prime}\in\{0,1\}}\exp((\log p(\mathcal{A}^{\prime}|f(\textbf{x},\textbf{y}_{1%
+:i-1});W))+g_{m}(\mathcal{A}^{\prime}))/\tau)}$ $\displaystyle-(1-\frac{N}{B})[l_{i}=1]\log p(\mathcal{A}_{i}=1|f(\textbf{x},%
+\textbf{y}_{1:i-1}))]$ $next\_state_{\_}0(1,s)=3s+1$ $v\in V(\tau(\mathcal{A}))$ $\mathrm{f}_{\_}k(Int\ \tau(\Delta^{n}))$ $(v_{\_}p^{\prime},w_{\_}p^{\prime})\in G_{\_}p(\mathcal{A})$ $\mathrm{Ch}\ \mathcal{A}$ $next\_state$ $P(k,1)\in\Theta(f(k,r))$ $\textit{next\_state}_{\_}i:view,state,round\mapsto view$ $\mathrm{f}_{\_}1(\mathrm{St}^{\circ}(\mathrm{Ch}^{r}\ \mathcal{A},v))\in\Theta%
+\big{(}(2^{n-1}n)^{r}\big{)}$ $\sigma\in Skel^{i}\mathcal{A}$ $\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v))=\sum_{\_}{i=k%
+}^{n}{\binom{n}{i}\sum_{\_}{j=1}^{k}{\binom{i}{j}\mathrm{f}_{\_}{k-j}(\mathrm{%
+St}^{\circ}(\mathrm{Ch}\ \Delta^{i-j},v))}}$ $\mathrm{f}_{\_}k(K)$ $R(k,n)\in\Theta(f(k,n))\overset{\text{Lemma~{}\ref{lemma:ThetaBoundFkdelta}}}{%
+\iff}\sum_{\_}{j=1}^{k}{\binom{n}{j}\frac{(k-j+1)^{i-k}(i-j)_{\_}{k-j}}{\ln(2)%
+^{k-j-1}}}\\
+=\frac{(n)_{\_}k}{\ln(2)^{k-1}}\sum_{\_}{j=1}^{k}{\frac{(k-j+1)^{n-k}\ln(2)^{j%
+}}{j!}}\in\Theta(f(k,n)\overset{\forall k\leq n}{\iff}\lim_{\_}{n\to\infty}{%
+\sum_{\_}{j=1}^{k}{(\frac{k-j+1}{k+1})^{n-k}\frac{\ln(2)^{j}}{j!}}}=C>0\$ $|\operatorname{Im}\delta_{\_}p|\leq\Delta(G_{\_}p(\mathcal{A}))+1$ $\Xi_{\_}b$ $|\operatorname{Im}\textit{encode}|\leq\Delta(G)+1$ $\log_{\_}3(\lceil{\frac{1}{\epsilon}}\rceil)$ $(c,\sigma)$ $\mathrm{Ch}\ \Delta^{n}$ $\mathrm{f}_{\_}{k-i}(\Delta^{n-i})$ $(\mathcal{I},\mathcal{O},\Delta)$ $\Xi(\mathcal{I})$ $v^{\prime}\in V(\Delta^{i})$ $\textit{encode}:V(\mathcal{A})\rightarrow E$ $\mathrm{f}_{\_}k(\tau(Skel^{i}\mathcal{A}))=0$ $w_{\_}q$ $\mathrm{f}_{\_}k(K)+\mathrm{f}_{\_}k(\tau(Skel^{i-1}\mathcal{A}))=\mathrm{f}_{%
+\_}k(\tau(Skel^{i}\mathcal{A}))$ $\frac{k-j+1}{k+1}<1$ $dim(\delta*v)=1$ $O(rn\log n)$ $\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v))\in\Theta(f(k,%
+n))\overset{\text{IH}}{\iff}\sum_{\_}{i=k}^{n}{\binom{n}{i}\sum_{\_}{j=1}^{k}{%
+\binom{i}{j}\frac{(k-j+1)^{i-k}(i-j)_{\_}{k-j}}{\ln(2)^{k-j-1}}}}\\
+\overset{\text{Lemma~{}\ref{lemma:A1}}}{=}\sum_{\_}{j=1}^{k}{\frac{\ln(2)^{j}}%
+{\ln(2)^{k-1}j!}(k-j+2)^{n-k}(n)_{\_}k}=\frac{(n)_{\_}k}{\ln(2)^{k-1}}\sum_{\_%
+}{j=1}^{k}{\frac{(k-j+2)^{n-k}\ln(2)^{j}}{j!}}\in\Theta(f(k,n))\\
+\overset{\forall k\leq n}{\iff}\lim_{\_}{n\to\infty}{\frac{\frac{(n)_{\_}k}{%
+\ln(2)^{k-1}}\sum_{\_}{j=1}^{k}{\frac{(k-j+2)^{n-k}\ln(2)^{j}}{j!}}}{(n)_{\_}k%
+(k+1)^{n-k}\ln(2)^{-k+1}}}=C>0\overset{\forall k\leq n}{\iff}\lim_{\_}{n\to%
+\infty}{\sum_{\_}{j=1}^{k}{(\frac{k-j+2}{k+1})^{n-k}\frac{\ln(2)^{j}}{j!}}}=C>0$ $\tau(Skel^{i}\mathcal{A})$ $\mathrm{f}_{\_}k(\tau(Skel^{i-1}\mathcal{A}))=S(i-1)$ $f(k,n)$ $\delta_{\_}i(v)$ $\forall k\leq n$ $Int\ \tau(\sigma)\cap\tau(\sigma^{\prime})=\emptyset$ $\mathcal{A}\text{ is distinguishable under }\textit{encode}\iff\forall p\in\Pi%
+,\textit{encode}\text{ is a proper vertex coloring of }G_{\_}p(\mathcal{A})$ $S(i)=\mathrm{f}_{\_}k(\tau(Skel^{i}\mathcal{A}))$ $(s,p_{\_}{\{0,1\}})$ $Int\ \mathrm{St}^{\circ}(\tau(\Delta^{i}),r)$ $|\operatorname{Im}\textit{encode}|\geq\max_{\_}{v\in V(\mathrm{Ch}\ \mathcal{A%
+})}{\mathrm{f}_{\_}1(\mathrm{St}^{\circ}(\mathrm{Ch}^{r}\ \mathcal{A},v))}$ $w_{\_}p\in\mathrm{Lk}(\mathcal{A},\mathrm{St}(\mathcal{A},v_{\_}p))\implies(v_%
+{\_}p,w_{\_}p)\in G_{\_}p(\mathcal{A})$ $f(k,r):=\bigg{(}\frac{(k+1)^{n-k}(n)_{\_}k}{\ln(2)^{k-1}}\bigg{)}^{r}$ $next\_state:\mathcal{S}\rightarrow\mathcal{S}$ $\mathrm{f}_{\_}0(\mathrm{Lk}(\mathrm{Ch}^{r}\ \mathcal{A},\mathrm{St}(\mathrm{%
+Ch}^{r}\ \mathcal{A},v)))$ $\mathrm{Lk}(\mathcal{A},v_{\_}p)\raisebox{-2.15277pt}{$|$}_{q}$ $3s+i$ $\Xi^{r}_{\_}b$ $\mathcal{H}_{\_}{ij}=\{\{(p_{\_}0,v_{\_}i),(p_{\_}1,v_{\_}{ij})\},$ $view$ $\mathcal{S}\subset\mathbb{N}$ $dim(Skel^{i}\mathcal{A})\leq i$ $r\leq r\leq n$ $V(\mathcal{A})\cup V(\mathcal{B})$ $(c,\Delta^{n})$ $s_{\_}p,t_{\_}q$ $\Delta((p_{\_}1,1))={(p_{\_}1,1)}$ $\Xi_{\_}\epsilon^{r}$ $\sum_{\_}{i=k}^{n}{\binom{n}{i}\binom{i}{r}b^{i-\alpha}(i-r)_{\_}{k-r}}=\sum_{%
+\_}{i=k}^{n}{\frac{n!}{(n-i)!r!(i-k)!}b^{i-\alpha}}=\frac{n!}{r!}\frac{(n-k)!}%
+{(n-k)!}\sum_{\_}{i=0}^{n-k}{\frac{b^{i+k-\alpha}}{(n-k-i)!i!}}\\
+=\frac{(n)_{\_}k}{r!}b^{k-\alpha}\sum_{\_}{i=0}^{n-k}{\binom{n-k}{i}b^{i}}=%
+\frac{b^{k-\alpha}}{r!}(b+1)^{n-k}(n)_{\_}k\squareforqed$ $\frac{T(k,n)}{f(k,n)}$ $T(k,n)$ $\mathrm{Ch}^{2}\ {\Delta^{2}}$ $\mathrm{f}_{\_}i(\mathrm{St}^{\circ}(\mathcal{A},v))$ $next\_state_{\_}i(v,s,k)$ $\Xi_{\_}b(\mathcal{I})\cong\mathrm{Ch}\ \mathcal{I}$ $(w_{\_}p,v_{\_}p)$ $\mathrm{St}^{\circ}(\tau(\Delta^{i})$ $\leq\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\tau^{r+1}(\mathcal{A}),v))$ $v_{\_}p\in V(\mathcal{A})\raisebox{-2.15277pt}{$|$}_{p}$ $\lim_{\_}{n\to\infty}{\sum_{\_}{j=1}^{k}{\big{(}\frac{k-j+2}{k+1}\big{)}^{n-k}%
+\frac{\ln(2)^{j}}{j!}}}>\lim_{\_}{n\to\infty}(\frac{k+1}{k+1})^{n-k}\frac{\ln 2%
+}{1!}=\ln 2>0\ \ \forall{k\leq n}$ $S(i-1)$ $P(k,r)\in\Theta(f(k,r))\overset{\text{HI}}{\iff}\sum_{\_}{i=k}^{n}{f(k,r-1)%
+\sum_{\_}{j=1}^{k}{\binom{i}{j}T(k-j,i-k)}}\in\Theta(f(k,r))\\
+\overset{\text{Lemma~{}\ref{lemma:bound_intch}}}{=}f(k,r-1)\sum_{\_}{i=k}^{n}{%
+\frac{(k+1)^{i-k}(i)_{\_}k}{\ln(2)^{k-1}}}=f(k,r-1)\frac{1}{\ln(2)^{k-1}}\sum_%
+{\_}{i=k}^{n}{(k+1)^{i-k}(i)_{\_}k}\in\Theta(f(k,r))\\
+\iff f(k,r-1)\cdot f(k,1)\in\Theta(f(k,r))$ $n^{\prime}\in[0,n-1]$ $\mathrm{Ch}^{r}\ \mathcal{A}$ $\mathrm{St}^{\circ}(\tau(\mathcal{A}),v)$ $\displaystyle\overset{\text{Lemma~{}\ref{lemma:bound_intch}}}{\iff}\sum_{\_}{i%
+=k}^{n}{P(i,0)\frac{(k+1)^{i-k}(i)_{\_}k}{\ln(2)^{k-1}}}\in\Theta(f(k,r))$ $V(\tau^{r+1}(\mathcal{A}))\setminus V(\tau^{r}(\mathcal{A}))$ $\mathcal{A}*\mathcal{B}$ $\mathrm{f}_{\_}n(\mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v))\sim\frac{n!}{%
+2\ln(2)^{n+1}}$ $\mathrm{f}_{\_}k(\tau(\mathcal{A}))=\sum_{\_}{i=k}^{n}{\mathrm{f}_{\_}i(%
+\mathcal{A})\ \mathrm{f}_{\_}k(Int\ \tau(\Delta^{i}))}$ $view,state\mapsto decode_{\_}p(view,state)$ $\emptyset\in\mathcal{A}$ $v\in V(\mathcal{A})$ $\delta_{\_}1(0)=1$ $\raisebox{-2.15277pt}{$|$}_{p}$ $snapshot(M[k])$ $\textit{encode}:V(\mathcal{I})\rightarrow E$ $\delta*v$ $f(k,n):=\frac{(k+1)^{n-k}(n)_{\_}k}{ln(2)^{k+1}}$ $\mathrm{f}_{\_}0(\mathrm{Lk}(\mathrm{Ch}^{r}\ \mathcal{A},\mathrm{St}(\mathrm{%
+Ch}^{r}\ \mathcal{A},v)))\in\Theta\bigg{(}\bigg{(}\frac{n!n^{n}}{\ln(2)^{n-1}}%
+\bigg{)}^{r}\bigg{)}$ $\tau(\partial(\sigma))\subseteq\tau(Skel^{i-1}\mathcal{A})$ $V(\Delta)-1$ $\delta_{\_}i(s)$ $(s^{\prime},p_{\_}1)$ $\Xi_{\_}\epsilon^{0}(\mathcal{I})=\mathcal{I}$ $next\_state_{\_}1(\bot,s)=3s+1$ $s,t\in V(\mathcal{A})$ $\partial\ \mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v)$ $\mathcal{A}\text{ is distinguishable under }\textit{encode}\implies|%
+\operatorname{Im}\textit{encode}|\geq\max_{\_}{p\in\Pi}{\omega(G_{\_}p(%
+\mathcal{A}))}$ $\Theta(\log_{\_}2((2^{n-1}n)^{r}))$ $v=input(i)$ $w_{\_}p,v_{\_}p\in\mathrm{Lk}(\mathcal{A},t_{\_}q)$ $v,w\in V(\mathcal{A})\raisebox{-2.15277pt}{$|$}_{p}:v\neq w$ $\mathrm{f}_{\_}k(Int\ \mathrm{St}^{\circ}(\tau(\Delta^{i}),v))$ $(i-1)(3s-1+2(s\mod 2))+i(3s+2(1-s\mod 2))$ $w_{\_}q\in\mathrm{Lk}(\mathcal{A},s_{\_}p)\arrowvert_{\_}q$ $\Xi\cong\mathrm{Ch}$ $\mathrm{f}_{\_}i(\mathcal{A})$ $\Xi_{\_}b(\mathcal{I})\ncong\mathrm{Ch}\ \mathcal{I}$ $G_{\_}\Pi(\mathcal{A})=\{G_{\_}p(\mathcal{A})\}_{\_}{p\in\Pi}$ $\Xi_{\_}b(\mathcal{I})\cong\Xi(\mathcal{I})\implies\Xi_{\_}b(\mathcal{I})\cong%
+\mathrm{Ch}\ \mathcal{I}$ $h(k,n)^{r}$ $\mathrm{St}^{\circ}(\tau^{r}(\mathcal{A}),v)$ $\mathrm{f}_{\_}i(\mathcal{A})\cdot\mathrm{f}_{\_}k(Int\ \tau(\Delta^{i}))$ $(0,p_{\_}0)$ $(v,w)\in G_{\_}p(\mathcal{A})\iff\exists t\in V(\mathcal{A}):v,w\in V(\mathrm{%
+Lk}(\mathcal{A},t))\raisebox{-2.15277pt}{$|$}_{p}$ $\alpha\cup\beta,\alpha\in\mathcal{A}$ $\lim_{\_}{n\to\infty}{\sum_{\_}{j=1}^{k}{\big{(}\frac{k-j+2}{k+1}\big{)}^{n-k}%
+\frac{\ln(2)^{j}}{j!}}}<\lim_{\_}{n\to\infty}{\sum_{\_}{j=1}^{\infty}{\big{(}%
+\frac{k-j+2}{k+1}\big{)}^{n-k}\frac{\ln(2)^{j}}{j!}}}<1\ \ \forall{k\leq n}$ $\lceil\log_{\_}3(\epsilon)\rceil$ $(n)_{\_}k$ $\mathrm{St}^{\circ}(\mathcal{A},v_{\_}p)$ $H=G_{\_}{p_{\_}0}(\mathcal{H})$ $\mathcal{I}=\{v_{\_}p,w_{\_}q,t_{\_}q,\{v_{\_}p,w_{\_}q\},\{v_{\_}p,t_{\_}q\},%
+\{\}\}$ $\mathrm{f}_{\_}k(\tau(\sigma))=\mathrm{f}_{\_}k(\tau(\partial(\sigma)))+%
+\mathrm{f}_{\_}k(\tau(Int\ \sigma))$ $\textit{encode}(s)=1$ $G_{\_}\Pi(\mathcal{A})$ $\log_{\_}2(|\operatorname{Im}\textit{encode}|)$ $\delta:\Xi(\mathcal{I})\rightarrow\mathcal{O}$ $O(log(\epsilon))$ $v_{\_}p$ $|\operatorname{Im}\textit{encode}|$ $r\geq log_{\_}3(\epsilon)$ $\mathcal{H}=\bigcup_{\_}{(v_{\_}i,v_{\_}j)\in H}\mathcal{H}_{\_}{ij}$ $next\_state_{\_}i$ $p_{\_}0$ $(c^{\prime},\sigma^{\prime})$ $r\in V(\Delta^{i})$ $\mathcal{S}\subseteq\mathcal{A}$ $G_{\_}{p_{\_}0}(\mathcal{H})=H$ $\mathrm{St}^{\circ}(\mathcal{A},\mathcal{S})=\{\sigma\in\mathcal{A}:\mathcal{S%
+}\subseteq\sigma\}$ $n,k,r\in\mathbb{N}$ $M[k]$ $t\in\mathrm{St}(\mathcal{A},v_{\_}p)$ $|\Pi|^{r}$ $\max_{\_}{p\in\Pi}{\Delta(G_{\_}p(\mathcal{A}))}=\max_{\_}{p\in\Pi}{\max_{\_}{%
+v\in V(\mathcal{A})}{\mathrm{f}_{\_}0(\mathrm{Lk}(\mathcal{A},\mathrm{St}(%
+\mathcal{A},v))\raisebox{-2.15277pt}{$|$}_{p})}}$ $\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v)\in\Theta\bigg{%
+(}\frac{(k+1)^{n-k}(n)_{\_}k}{\ln(2)^{k-1}}\bigg{)}$ $\sum^{\infty}_{\_}{j=1}{\frac{\ln(2)^{j}}{j!}}=1$ $\delta_{\_}i:\mathcal{S}\rightarrow\{\frac{k}{\epsilon}:k\in[0,\epsilon]\},\ %
+\delta_{\_}i(s)=\frac{2s+i}{\epsilon}$ $\Xi_{\_}\epsilon$ $\delta_{\_}1(\frac{3^{r}-1}{2})=\frac{3^{r}}{\epsilon}=1$ $\Xi_{\_}\epsilon(\mathcal{I})$ $(v_{\_}i,v_{\_}j)\in H$ $M[k,i]$ $\mathrm{St}(\mathcal{A},\mathcal{S})$ $\textit{encode}:\mathcal{S}\rightarrow\{1,2\},\ \textit{encode}(s)=2-(s\mod 2)$ $|\operatorname{Im}\textit{encode}|\leq\max_{\_}{p\in\Pi}{\Delta(G_{\_}p(%
+\mathcal{A}))}+1$ $next\_state_{\_}0(\bot,s)=3s$ $\Xi_{\_}b^{r}(\mathcal{I})\cong\mathrm{Ch}^{r}\ \mathcal{I}\iff\forall r^{%
+\prime}\in[0,r-1],\mathrm{Ch}^{r^{\prime}}\ \mathcal{I}\text{ is %
+distinguishable under }\textit{encode}$ $K\cap\tau(Skel^{i-1}\mathcal{A})=\emptyset$ $|\operatorname{Im}\textit{encode}|\leq\max_{\_}{v\in V(\mathrm{Ch}\ \mathcal{A%
+})}{\mathrm{f}_{\_}0(\mathrm{Lk}(\mathrm{Ch}^{r}\ \mathcal{A},\mathrm{St}(%
+\mathrm{Ch}^{r}\ \mathcal{A},v)))}+1$ $\Delta((p_{\_}0,0))={(p_{\_}0,0)}$ $\{\mathrm{f}_{\_}n(\mathrm{St}^{\circ}(\mathrm{Ch}\ (\Delta^{n},v))\}_{\_}{n%
+\geq 0}$ $\tau:\mathcal{A}\rightarrow B$ $\Delta:\mathcal{I}\rightarrow 2^{\mathcal{O}}$ $\sigma^{\prime}\in Skel^{i}\mathcal{A}:\sigma^{\prime}\neq\sigma$ $\mathrm{f}_{\_}k(\mathcal{A})=|\{\sigma\in\mathcal{A}:dim(\sigma)=k\}|$ $\mathrm{f}_{\_}0(\mathrm{Lk}(\mathrm{Ch}^{r}\ \mathcal{A},\mathrm{St}(\mathrm{%
+Ch}^{r}\ \mathcal{A},v)))\in\Theta(f_{\_}n(k,r))\overset{\text{Lemma~{}\ref{%
+lemma:f_linkstar}}}{\iff}\sum_{\_}{i=1}^{n}{\mathrm{f}_{\_}i(\mathrm{St}^{%
+\circ}(\mathrm{Ch}^{r}\mathcal{A},v))}\in\Theta(f_{\_}n(k,r))\\
+\overset{\text{Theorem~{}\ref{theorem:finalBound}}}{\iff}\sum_{\_}{i=1}^{n}{%
+\bigg{(}\frac{(i+1)^{n-i}(n)_{\_}i}{\ln(2)^{i-1}}\bigg{)}^{r}}\in\Theta(f(k,r)%
+)\iff\lim_{\_}{n\to\infty}{\frac{\sum_{\_}{i=1}^{n}{\bigg{(}\frac{(i+1)^{n-i}(%
+n)_{\_}i}{\ln(2)^{i+1}}\bigg{)}^{r}}}{\bigg{(}\frac{n!n^{n}}{\ln(2)^{n-1}}%
+\bigg{)}^{r}}}\\
+=\lim_{\_}{n\to\infty}{\sum_{\_}{i=1}^{n}{\bigg{(}\frac{\ln(2)^{n-i}}{(n-i)!}%
+\cdot\bigg{(}\frac{i+1}{n}\bigg{)}^{n}\cdot\frac{1}{(i+1)^{i}}\bigg{)}^{r}}}=C>0$ $\displaystyle P(k,1)\in\Theta(f(k,r))$ $\max_{\_}{v\in V(\mathrm{Ch}\ \mathcal{I})}{\mathrm{f}_{\_}1(\mathrm{St}^{%
+\circ}(\mathrm{Ch}^{r}\ \mathcal{I},v))}\leq|\operatorname{Im}\textit{encode}|%
+\leq\max_{\_}{v\in V(\mathrm{Ch}\ \mathcal{I})}{\mathrm{f}_{\_}0(\mathrm{Lk}(%
+\mathrm{Ch}^{r}\ \mathcal{I},\mathrm{St}(\mathrm{Ch}^{r}\ \mathcal{I},v)))}+1$ $M[1],\ldots,M[r]$ $\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\mathrm{Ch}^{r}\ \mathcal{A},v))\in\Theta%
+\bigg{(}\bigg{(}\frac{(k+1)^{n-k}(n)_{\_}k}{\ln(2)^{k-1}}\bigg{)}^{r}\bigg{)}$ $\mathrm{f}_{\_}1(St^{\circ}(\mathrm{Ch}^{r}\ \Delta^{1},v))\in\Theta(1)$ $\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\mathrm{Ch}^{r}\ \mathcal{A},v))=\sum_{\_%
+}{i=k}^{n}{\mathrm{f}_{\_}i(\mathrm{St}^{\circ}(\mathrm{Ch}^{r-1}\ \mathcal{A}%
+,v))\sum_{\_}{j=1}^{k}{\binom{i}{j}\mathrm{f}_{\_}{k-j}(\mathrm{St}^{\circ}(%
+\mathrm{Ch}\ \Delta^{i-j},v^{\prime}))}}$ $V(\mathcal{O})=\{(p_{\_}0,0),(p_{\_}1,\frac{1}{\epsilon}),(p_{\_}0,\frac{2}{%
+\epsilon}),\dots,(p_{\_}0,\frac{\epsilon-1}{\epsilon}),(p_{\_}1,1)\}$ $\mathrm{f}(\mathcal{A})=(\mathrm{f}_{\_}{-1}(\mathcal{A}),\mathrm{f}_{\_}0(%
+\mathcal{A}),\dots,\mathrm{f}_{\_}n(\mathcal{A}))$ $\mathcal{I}=\{0,1,\{0,1\}\}$ $\operatorname{Im}\textit{encode}_{\_}i$ ${T(k^{\prime},n^{\prime}):k^{\prime}\in[0,k-1]\land n^{\prime}\in[0,n-1]}$ $T(k,n)\squareforqed$ $\Xi^{r}$ $\mathrm{f}_{\_}n(\mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v))$ $v\in V(\Delta^{n})$ $\mathrm{Ch}\ {\Delta^{2}}$ $w_{\_}p$ $next\_state_{\_}p$ $\sum_{\_}{i=k}^{n}{\binom{n}{i}\binom{i}{r}b^{i-\alpha}(i-r)_{\_}{k-r}}=\frac{%
+b^{k-\alpha}}{r!}(b+1)^{n-k}(n)_{\_}k$ $\Pi=\{p_{\_}1,\dots,p_{\_}{n+1}\}$ $G_{\_}p$ $\{(s,p_{\_}0),(s,p_{\_}1)\}$ $(s,p_{\_}0)$ $next\_state_{\_}i(v,s)$ $(3s,p_{\_}0)-(3s,p_{\_}1)-(3s+1,p_{\_}0)-(3s+1,p_{\_}1)$ $\mathrm{Ch}\ \mathcal{I}$ $\textit{encode}_{\_}i$ $\{G_{\_}p(\mathcal{A})\}_{\_}{p\in\Pi}$ $G_{\_}p(\mathcal{A})$ $\displaystyle\iff\frac{1}{\ln(2)^{k-1}}\sum_{\_}{i=k}^{n}{P(i,0)(k+1)^{i-k}(i)%
+_{\_}k}\in\Theta(f(k,r))$ $Int\ \mathcal{A}=\mathcal{A}\setminus\partial(\mathcal{A})$ $\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\mathrm{Ch}\ (\Delta^{n}),v))$ $\mathrm{f}_{\_}k(Int\ \mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v))=\sum_{\_%
+}{i=1}^{k}{\binom{n}{i}\mathrm{f}_{\_}{k-i}(\mathrm{St}^{\circ}(\mathrm{Ch}\ %
+\Delta^{n-i},v))}$ $\sum_{\_}{i=k}^{n}{(k+1)^{i-k}(i)_{\_}k}\in\Theta((k+1)^{n-k}(n)_{\_}k)$ $\Xi_{\_}b(\mathcal{I})$ $\mathrm{f}_{\_}k(Int\ \mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v))$ $f(k,r-1)\cdot f(k,1)=f(k,r)$ $s=next\_state_{\_}i(v,\bot,0)$ $p_{\_}1$ $\textit{encode}_{\_}i:state,round\mapsto state$ $R(k,n):=\sum_{\_}{j=1}^{k}{\binom{n}{j}T(k-j,n-j)}$ $\partial(\mathcal{A})=\{\sigma\in\mathcal{A}:\sigma\subset G\text{ for a %
+unique facet }G\in\mathcal{A}\}\cup\{\emptyset\}$ $(\frac{3^{r}-1}{2},p_{\_}1)$ $\forall\tilde{v}\in V(\tau^{r}(\mathcal{A})),\mathrm{f}_{\_}k(\mathrm{St}^{%
+\circ}(\tau^{r}(\mathcal{A}),\tilde{v}))\leq\mathrm{f}_{\_}k(\mathrm{St}^{%
+\circ}(\tau^{r}(\mathcal{A}),v))$ $\frac{1}{\ln(2)^{k-1}}\sum_{\_}{i=k}^{n}{P(i,0)(k+1)^{i-k}(i)_{\_}k}<\frac{1}{%
+\ln(2)^{k-1}}\sum_{\_}{i=k}^{n}{P(i,0)(k+1)^{n-k}(n)_{\_}k}\\
+=\frac{(k+1)^{n-k}(n)_{\_}k}{\ln(2)^{k-1}}\sum_{\_}{i=k}^{n}{P(i,0)}\implies P%
+(k,1)\in O(f(k,1))\\
+\frac{1}{\ln(2)^{k-1}}\sum_{\_}{i=k}^{n}{P(i,0)(k+1)^{i-k}(i)_{\_}k}>\frac{(k+%
+1)^{n-k}(n)_{\_}k}{\ln(2)^{k-1}}P(n,0)\implies P(k,1)\in\Omega(f(k,1))$ $Int\ \tau(\sigma^{i})$ $\mathrm{f}_{\_}k(\tau(\partial(\sigma)))$ $\Xi_{\_}b^{r}\cong\Xi^{r}$ $(p_{\_}0,v_{\_}i),(p_{\_}0,v_{\_}j)\in V(\mathrm{Lk}(\mathcal{H},(p_{\_}1,v_{%
+\_}{ij})))\ \forall i,j$ $dim(\Delta)$ $|\operatorname{Im}{\textit{encode}}|$ $f_{\_}{-1}(\mathcal{A})=1$ $r-th$ $\sum_{\_}{j=1}^{k}{\binom{n}{j}\mathrm{f}_{\_}{k-j}(\mathrm{St}^{\circ}(%
+\mathrm{Ch}\ \Delta^{n-j},v))}\in\Theta\bigg{(}\frac{(k+1)^{n-k}(n)_{\_}k}{\ln%
+(2)^{k-1}}\bigg{)}$ $\{(s+1,p_{\_}0),(s,p_{\_}1)\}$ $k^{\prime}\in[0,k-1]$ $\mathrm{f}_{\_}{k^{\prime}}(\mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n^{\prime%
+}},v))$ $v=\operatorname*{arg\,max}_{\_}{v\in V(\tau^{r}(\mathcal{A}))}{\mathrm{f}_{\_}%
+k(\mathrm{St}^{\circ}(\tau^{r}(\mathcal{A}),v))}$ $v\in\Delta^{n}$ $S(i)=S(i-1)+\mathrm{f}_{\_}i(\mathcal{A})\cdot\mathrm{f}_{\_}k(Int\ \tau(%
+\Delta^{i}))$ $\Delta(G_{\_}p(\mathcal{A}))+1$ $\Xi_{\_}b^{r}$ $q\in\Pi$ $\mathrm{Lk}(\mathcal{A},v_{\_}p)$ $v[(1-i)]=1$ $\mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v)$ $\tau(\mathcal{A})$ $skel^{0}(\mathcal{A})=V(\mathcal{A})$ $\max_{\_}{p\in\Pi}{\omega(G_{\_}p(\mathcal{A}))}\geq\max_{\_}{p\in\Pi}{\max_{%
+\_}{v\in V(\mathcal{A})}{\mathrm{f}_{\_}1(\mathrm{St}^{\circ}(\mathcal{A},v)%
+\raisebox{-2.15277pt}{$|$}_{p})}}$ $\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\tau(\mathcal{A}),v))=\sum_{\_}{i=k}^{n}{%
+\mathrm{f}_{\_}i(\mathrm{St}^{\circ}(\mathcal{A},v))\ \mathrm{f}_{\_}k(Int\ %
+\mathrm{St}^{\circ}(\tau(\Delta^{i}),v))}$ $\Theta(\log_{\_}2((\frac{n!n^{n}}{\ln(2)^{n-1}})^{r}))$ $\textit{encode}(v_{\_}p)=\textit{encode}(w_{\_}p)$ $\mathrm{Lk}(\mathcal{I},v_{\_}p)$ $Int\ \tau(\Delta^{i})$ $\Delta(\{(p_{\_}0,0),(p_{\_}1,1)\})=\mathcal{O}$ $\delta\circ\Xi(\mathcal{I})\subseteq\Delta(\mathcal{I})$ $|\delta_{\_}0(s)-\delta_{\_}1(s^{\prime})|=|\frac{2(s-s^{\prime})-1}{\epsilon}%
+|\leq\frac{1}{\epsilon}$ $v_{\_}p\in V(\mathcal{A})$ $\delta_{\_}0(0)=0$ $-1\leq k\leq n$ $\mathrm{St}(\mathcal{A},v_{\_}p)$ $t_{\_}q\in V(\mathcal{A})$ $v[(1-i)]=\bot$ $\Xi_{\_}b(\mathcal{I})\cong\Xi(\mathcal{I})$ $G_{\_}p:\mathcal{A}\times\Pi\rightarrow(V(\mathcal{A}),V(\mathcal{A})\times V(%
+\mathcal{A}))$ $t_{\_}q$ $next\_state_{\_}i(s,\bot)=3s+i$ $P(k,r):=\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\mathrm{Ch}^{r}\ \mathcal{A},v))$ $decode$ $v_{\_}p,w_{\_}q\in V(\sigma)$ $\mathrm{St}^{\circ}(\mathcal{A},\mathcal{S})$ $|\operatorname{Im}\textit{encode}_{\_}i| $\delta\in\sigma:dim(\delta)=i-1$ $\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\tau^{r+1}(\mathcal{A}),v))$ $\mathrm{f}_{\_}1(\mathrm{St}^{\circ}(\mathrm{Ch}^{r}\ \mathcal{A},v))$ $\forall\tilde{v}\in V(\tau^{r}(\mathcal{A})),\mathrm{f}_{\_}k(\mathrm{St}^{%
+\circ}(\tau^{r+1}(\mathcal{A}),\tilde{v}))$ $\mathrm{f}_{\_}0$ $\Xi_{\_}b(\mathcal{I})\cong\mathrm{Ch}\ \mathcal{I}\iff\mathcal{I}\text{ is %
+distinguishable under }\textit{encode}$ $p_{\_}i\in\Pi$ $M[r]$ $\Xi^{r}(\mathcal{I})\cong\Xi_{\_}b^{r}(\mathcal{I})$ $f_{\_}n(k,r):=\bigg{(}\bigg{(}\frac{n!n^{n}}{\ln(2)^{n-1}}\bigg{)}^{r}\bigg{)}$ $v[(1-i)]=2$ $K\cup\tau(Skel^{i-1}\mathcal{A})=\tau(Skel^{i}\mathcal{A})$ $P(k,0)$ $encode_{\_}i(s,k)$ $\mathrm{Lk}(\mathcal{A},\mathcal{S})$ $\beta\in\mathcal{B}$ $\textit{encode}(t_{\_}q)=\textit{encode}(w_{\_}q)$ $\frac{k-j+2}{k+1}\leq 1\ \forall k\leq n$ $dim(\delta*v)=k$ $\delta_{\_}p$ $\sigma\subseteq\sigma^{\prime}$ $\operatorname*{arg\,max}_{\_}{v\in V(\tau^{r}(\mathcal{A}))}{\mathrm{f}_{\_}k(%
+\mathrm{St}^{\circ}(\tau^{r}(\mathcal{A}),v))}\in V(\tau(\mathcal{A})),\ \ \ %
+\ \forall k\leq n,\forall r\geq 1$ $next\_state_{\_}1(1,s)=3s$ $(\ln 2,1)$ $K:=\tau(Skel^{i}\mathcal{A})-\tau(Skel^{i-1}\mathcal{A})$ $s-s^{\prime}\leq 1$ $T(0,n)\overset{\mathrm{def}}{=}1\in\Theta(1)=\Theta(f(0,n)),\forall n\in%
+\mathbb{N}$ $skel^{l}(\mathcal{A})$ $s_{\_}p$ $\log_{\_}3(\epsilon)$ $\{(p_{\_}1,v_{\_}{ij}),(p_{\_}0,v_{\_}j)\},\{(p_{\_}0,v_{\_}i)\},\{(p_{\_}1,v_%
+{\_}{ij})\},\{(p_{\_}0,v_{\_}j)\},\emptyset\}$ $T(k,n):=\mathrm{f}_{\_}k(\mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v))$ ${\max_{\_}{v\in V(\mathrm{Ch}^{r}\ \mathcal{A})}{\mathrm{f}_{\_}0(\mathrm{Lk}(%
+\mathrm{Ch}^{r}\ \mathcal{A},\mathrm{St}(\mathrm{Ch}^{r}\ \mathcal{A},v)))%
+\raisebox{-2.15277pt}{$|$}_{p}}}\in V(\mathrm{Ch}\ (\mathcal{A}))$ $decode_{\_}p(\textit{encode}_{\_}q(w_{\_}q),v_{\_}p)=w_{\_}q$ $Skel^{n}\mathcal{A}=\mathcal{A}$ $(v_{\_}p,w_{\_}p)\in G_{\_}p(\mathcal{A})$ $Int\ \mathrm{Ch}\ \Delta^{n}$ $\mathrm{St}(\mathrm{Ch}\ \mathcal{A},v_{\_}p)$ $\textit{encode}_{\_}q(w_{\_}q)=\textit{encode}_{\_}q(t_{\_}q)=e$ $\mathrm{f}_{\_}0(\mathrm{Ch}\ \mathcal{A},\mathrm{Lk}(\mathrm{St}(\mathrm{Ch}%
+\ \mathcal{A},v_{\_}p))\raisebox{-2.15277pt}{$|$}_{p})=\sum_{\_}{i=1}^{n}{%
+\mathrm{f}_{\_}i(\mathrm{St}^{\circ}(\mathcal{A},v_{\_}p)))}$ $f(k,n):=\frac{(k+1)^{n-k}(n)_{\_}k}{\ln(2)^{k+1}}$ $\mathrm{f}_{\_}0(Int\ \mathrm{St}^{\circ}(\mathrm{Ch}\ \Delta^{n},v))=0$ $g:\mathbb{R}^{d}\rightarrow\mathbb{R}^{k}$ $c\rightarrow y$ $\{x^{(i)},y^{(i)},c^{(i)}\}_{i=1}^{n}$ $f:\mathbb{R}^{k}\rightarrow\mathcal{Y}$ $\{y^{(i)},c^{(i)}\}_{i=1}^{n}$ $\{x^{(i)},c^{(i)}\}_{i=1}^{n}$ $x\rightarrow c$ $\displaystyle t_{i}^{*}=(B-1)\left(t_{i}-t_{1}\right)/\left(t_{N}-t_{1}\right),$ $\hat{f_{j}^{i}}$ $\mathbf{V}_{0\xrightarrow{}4}$ $bins{=}13$ $\mathbf{V}_{0\xrightarrow{}5}$ $\mathbf{V}_{0\xrightarrow{}6}$ $\mathbf{V}_{0\xrightarrow{}8}$ $bins=21$ $\mathbf{V}_{0\xrightarrow{}7}$ $V_{0,i}$ $100{\times}\frac{i}{14}ms$ $dt{=}1$ ${>}120dB$ $288{\times}384$ $\displaystyle k_{b}(a)=\max(0,1-|a|),$ $\mathrm{FWL}:=\frac{\sigma^{2}(I(E,\mathbf{V}))}{\sigma^{2}(I(E,0))}$ $640{\times}480$ $7.14ms$ $\displaystyle VG(x,y,t)=\sum_{i}p_{i}k_{b}\left(x-x_{i}\right)k_{b}\left(y-y_{%
+i}\right)k_{b}\left(t-t_{i}^{*}\right),$ $60dB$ $1{\times}1e^{-3}$ $f_{i}^{1}{\to}f_{4}^{N}$ $140Hz$ $dt{=}4$ $\mathbf{V}_{0\xrightarrow{}3}$ $B{\times}H{\times}W$ $20Hz$ $\mathbf{V}_{0,1}\to\mathbf{V}_{0,B-1}$ $I(E,\phi)=\left(\begin{array}[]{l}x_{i}^{\prime}\\
+y_{i}^{\prime}\end{array}\right)=\left(\begin{array}[]{l}x_{i}\\
+y_{i}\end{array}\right)+\left(t^{\prime}-t_{i}\right)\mathbf{V}(x_{i},y_{i}).$ $t_{b}-\tau $\mathbf{V}_{0,j}^{i}$ $\left\{\left(x_{i},y_{i},t_{i},p_{i}\right)\right\}_{i\in[1,N]}$ $E=(\left\{\left(x_{i},y_{i},t_{i},p_{i}\right)\right\}_{i\in[1,N]})$ $5{\times}1e^{-4}$ $4{:}1$ $\mathbf{V}_{0\xrightarrow{}13}$ $N{\times}B,H,W$ $\mathbf{V}_{0\xrightarrow{}14}$ $k_{b}(a)$ $\mathbf{V}_{0\xrightarrow{}12}$ $\mathbf{V}_{0\xrightarrow{}2}$ $50Hz$ $\mathbf{V}_{0\xrightarrow{}11}$ $bins{=}15$ $Loss=||\mathbf{F}_{gt}-\mathbf{F}_{pre}||_{1}.$ $bins{=}3$ $\Delta\mathbf{V}_{0,j}^{i}$ $\mathrm{RFWL}:=\sigma^{2}(\frac{I(E,\mathbf{V})}{\sum I(E,\mathbf{V})})/\sigma%
+^{2}(\frac{I(E,0)}{\sum I(E,0)}).$ $93ms$ $(B-1)\times\tau$ $\displaystyle UVG=concat([Bin_{0},Bin_{1},...,Bin_{B-1}]).$ $\mathbf{V}_{0,B-1}$ $\mathbf{V}_{0,i}$ $10FPS$ $3.3ms$ $\mathbf{V}_{0,j}^{i-1}$ $346{\times}260$ $\mathbf{V}_{0\xrightarrow{}10}$ $bins{=}4$ $300Hz$ $\mathbf{V}_{0\xrightarrow{}1}$ $N,B,H,W$ $7.1ms$ $5ms$ $j^{i}$ $bins{=}9$ $Ht_{j}^{i-1}$ $\mathbf{V}_{0\xrightarrow{}9}$ $9.2ms$ $\displaystyle Bin_{b}(x,y)=\sum_{i}p_{i}k_{b}\left(x-x_{i}\right)k_{b}\left(y-%
+y_{i}\right)k_{b}\left(\frac{t-t_{b}}{\tau}\right)$ $200Hz$ $Experts$ $Feedback$ $Probed\ system$ $Gating\ model$ $\bm{\lambda}^{\sf DMD}_{i,k}=G^{T}_{i}\bm{\lambda}_{k}=G^{T}_{i}A_{\lambda}%
+\mathbf{y}_{k-1}\,.$ $h=1/N$ $\delta_{i}\approx 2h$ $A_{\lambda}(\bm{\mu}_{j})$ $\mathcal{F}_{i}(\bm{\lambda}_{k-1};\mathbf{u}_{1,k},\mathbf{u}_{2,k}):=G^{T}_{%
+i}\bm{\lambda}_{k}=G^{T}_{i}{A}_{\lambda}\mathbf{y}_{k-1}\,,$ $\kappa_{i}>0$ $\bm{\Lambda}_{i,\beta_{i,k-1}}$ $\Delta_{128}t=1.67\times 10^{-3}$ $\mathbf{u}_{i,k+1}=\mathbf{u}_{i,k}+\Delta tM_{i}^{-1}\left(\mathbf{b}_{i,k}+(%
+-1)^{i}\bm{\lambda}_{i,k}\right)\,,\quad i=1,2\,.$ $q_{128}=3761$ $\bm{\mu}_{j}=\{\kappa_{1,j},\kappa_{2,j}\}$ $n_{l,\gamma}\times n_{{\delta_{1}},D}$ $u^{h}_{i,M}$ $\mathbf{u}_{i,k+1}=\mathbf{u}_{i,k}+\Delta tM_{i}^{-1}\left(\mathbf{f}_{i,k}-K%
+_{i}\mathbf{u}_{i,k}+(-1)^{i}\bm{\lambda}_{i,k}\right)\,.$ $\dot{u}_{1}(\bm{x},t)=\dot{u}_{2}(\bm{x},t)\quad\mbox{on}\quad\gamma\times[0,T%
+]\,.$ $n_{1,D}$ $A_{\lambda,u_{1}}$ $\bm{\mu}_{j}\in\mathcal{M}_{m}$ $2n^{2}_{1,\gamma}$ $\displaystyle(\dot{u}_{2},v_{2})_{0,\Omega_{2}}+(F_{2}(u_{2}),\nabla v_{2})_{0%
+,\Omega_{2}}-\left<\lambda,v_{2}\right>_{\gamma}$ $\left $\bm{\lambda}_{i,k-1}$ $N_{\sf FS}=n_{l,\gamma}+n_{1,D}+n_{2,D}$ $\bm{\lambda}_{i,k}:=G^{T}_{i}\bm{\lambda}_{k}$ $\widetilde{y}_{i}=U^{T}_{k}y_{i}$ $\mathbf{y}_{k-1}=\left(\bm{\lambda}_{k-1},\mathbf{u}_{1,k}(\delta_{1}),\mathbf%
+{u}_{2,k}(\delta_{2})\right)^{T}$ $\mathbf{U}_{i,\alpha_{i,k}}$ $u_{i,0}$ $u^{h}_{i,D}\in S^{h}_{i,D}$ $n_{\delta_{i},D}=O(n_{i,\gamma})$ $q=1866$ $\widetilde{A}_{k}:={U}_{k}^{T}A_{k}U_{k}\in\mathbb{R}^{k\times k}$ $E_{k}(Y)$ $\bm{\mu}=\{\kappa_{1},\kappa_{2}\}$ ${H}^{1}(\Omega_{i})$ $u^{h}_{i}=u^{h}_{i,D}+g^{h}_{i}$ $B(\bm{\mu},R)$ ${H}^{1}_{D}(\Omega_{i})$ $n_{1,D}=n_{2,D}$ $\{A_{\lambda}(\bm{\mu}_{j})\}_{j=1}^{m}$ $A=\mathbf{Y}^{\prime}\mathbf{Y}^{+}$ $\mathbf{u}_{1,k}$ $\displaystyle=(f_{i},v^{h}_{i})_{0,\Omega_{i}}+(-1)^{i}\left<\lambda,v^{h}_{i}%
+\right>_{\gamma}\quad\forall v^{h}_{i}\in S^{h}_{i,D}\,.$ $G_{i}M_{i}^{-1}$ $u^{h}_{i,X}$ $\displaystyle=(f_{1},v_{1})_{0,\Omega_{1}}\qquad\forall v_{1}\in H_{D}^{1}(%
+\Omega_{1})$ $\bm{\mu}\notin\mathcal{M}_{m}$ $O(n_{i,D}n_{l,\gamma})$ $H_{i,\gamma}=G_{i}{M}_{i,\gamma}^{-1}$ $\gamma^{h}_{i}$ $\mathbf{y}_{k+1}=A\mathbf{y}_{k}\,,$ $\mathbf{u}_{i,\gamma}\in\mathbb{R}^{n_{i,\gamma}}$ $\Omega^{h}_{1}$ $\Delta_{64}t=3.37\times 10^{-3}$ $\times 3.41$ $\displaystyle=\mathbf{f}_{2}$ $[\mathbf{Y}^{\prime}(\bm{\mu}_{i})]_{i=1}^{m}$ $\bm{\mu}_{i}\in\mathcal{M}_{m}$ $\lambda=F_{1}(u_{1})\cdot\bm{n}_{\gamma}=F_{2}(u_{2})\cdot\bm{n}_{\gamma}$ $\displaystyle(\dot{u}_{i},v_{i})_{0,\Omega_{i}}+(F_{i}(u_{i}),\nabla v_{i})_{0%
+,\Omega_{i}}$ $\mathcal{M}=([1,2]\times[3,4])\times 10^{-3}$ $\mathcal{M}:=[\kappa_{1,\min},\kappa_{1,\max}]\times[\kappa_{2,\min},\kappa_{2%
+,\max}]\subset\mathbb{R}^{2}\,.$ $\displaystyle=0\hskip 64.58313pt\forall\mu\in H^{-1/2}(\gamma)$ $\displaystyle M_{2}\dot{\mathbf{u}}_{2}+K_{2}\mathbf{u}_{2}-G_{2}^{T}\bm{\lambda}$ $\mathbf{u}_{i,m}$ $\displaystyle=(f_{i},v_{i})_{0,\Omega_{i}}+(-1)^{i}\left<\lambda,v_{i}\right>_%
+{\gamma}\quad\forall v_{i}\in H_{D}^{1}(\Omega_{i})\,.$ $\mathbf{u}_{i,0}$ $\ell_{\bm{\mu}_{j}}$ $u_{i,0}(\bm{x})$ $n_{i,\Gamma}$ $\bm{v}=\left(0.5-y,x-0.5\right)$ $\left\{\begin{aligned} \dot{u}_{i}-\nabla\cdot F_{i}(u_{i})&=f_{i}\quad\mbox{%
+in}\quad\Omega_{i}\times[0,T]\\[2.15277pt]
+{u}_{i}&=g_{i}\quad\mbox{in}\quad\Gamma_{i}\times[0,T]\\[2.15277pt]
+u_{i}(\bm{x},0)&=u_{i,0}(\bm{x})\quad\mbox{in}\quad\Omega_{i}\end{aligned}%
+\right.\quad i=1,2,$ $q_{32}=918$ $\bm{x}_{i,r}\in\Gamma^{h}_{i}$ $\bm{\mu}=(1.5,2.5)\times 10^{-3}$ $\displaystyle E_{k}(Y):=\frac{\sum_{i=1}^{k}\sigma_{i}^{2}}{\sum_{i=1}^{n}%
+\sigma_{i}^{2}}.$ $Y=U\Sigma V^{T}$ $\bm{\lambda}_{i,m}$ $\mathbf{u}_{i}=(\mathbf{u}_{i,\gamma},\mathbf{u}_{i,0},\mathbf{u}_{i,\Gamma})$ $S=\{(x,y)\in\Omega_{1}\,|\,0\leq x\leq 0.5\ \mbox{and}\ y=0.5\}$ $\displaystyle(\dot{u}_{1},v_{1})_{0,\Omega_{1}}+(F_{1}(u_{1}),\nabla v_{1})_{0%
+,\Omega_{1}}+\left<\lambda,v_{1}\right>_{\gamma}$ $\mathcal{M}\subset\mathbb{R}^{M}$ $\mathcal{F}_{i}(\mathbf{u}_{1},\mathbf{u}_{2})=R_{j\mapsto i}K_{j,\gamma}%
+\mathbf{u}_{j}$ $\bm{\mu}=(1.5,3.5)\times 10^{-3}$ ${A}_{\lambda}(\bm{\mu}):=\sum_{\bm{\mu}_{j}\in\mathcal{M}_{m}(\bm{\mu},r)}\ell%
+_{\bm{\mu}_{j}}(\bm{\mu}){A}_{\lambda}(\bm{\mu}_{j})$ $\lambda^{h}\in S^{h}_{l,\gamma}$ $\displaystyle=(f_{2},v_{2})_{0,\Omega_{2}}\qquad\forall v_{2}\in H_{D}^{1}(%
+\Omega_{2})$ $\mathbf{Y}_{k}^{+}$ $(\cdot)_{\gamma}$ ${V}_{k}$ $\langle\cdot,\cdot\rangle_{\gamma}$ $\mathbf{y}_{k-1}=\left(\bm{\lambda}_{k-1},\mathbf{u}_{1,k-1},\mathbf{u}_{2,k-1%
+}\right)^{T}$ $F_{i}(u^{h}_{i})$ $\mathbf{b}_{i,k}=\mathbf{f}_{i,k}-K_{i}\mathbf{u}_{i,k}$ ${M}_{i}=\mbox{diag}({M}_{i,\gamma},{M}_{0,\gamma})$ $B(\bm{\mu},r)\subset\mathcal{M}$ $\displaystyle=\mathbf{f}_{1}$ $F_{i}(u_{i})=\kappa_{i}\nabla u_{i}-\bm{v}u_{i},\quad i=1,2;$ $\Omega_{2}=[0.5,1]\times[0,1]$ $\mathbf{u}_{i,\Gamma}\in\mathbb{R}^{n_{i,\Gamma}}$ ${M}_{i,\gamma}$ $q_{64}=1866$ $u^{h}_{M}$ $t_{k+1}=t_{k}+\Delta t$ $A_{k_{j}}(\bm{\mu}_{j})$ $\mathbf{u}_{i,k}$ $u^{h}_{i,L}$ $\bm{\lambda}^{\sf DMD}_{i,k}$ $\displaystyle\left $(\cdot,\cdot)_{0,\Omega_{i}}$ $\mathbf{u}_{i,0}\in\mathbb{R}^{n_{i,0}}$ $\Omega^{h}_{i}$ $n_{1,\gamma}=n_{2,\gamma}$ $\Omega_{1}=[0,0.5]\times[0,1]$ $\mathcal{F}_{i,\bm{\mu}}(\bm{\lambda}_{k-1};\mathbf{u}_{1,k},\mathbf{u}_{2,k})%
+:=G^{T}_{i}\bm{\lambda}_{k}=G^{T}_{i}{A}_{\lambda}(\bm{\mu})\mathbf{y}_{k-1}\,.$ $u_{i}(\cdot,t)\in H^{1}(\Omega_{i})$ $m=1,\ldots,4$ $\{A(\bm{\mu}_{j})\}_{j=1}^{m}$ $\ell_{i,r}(\bm{x}_{i,s})=\delta_{rs}$ $\bm{\mu}\in\mathcal{M}$ $\mathbf{y}_{k-1}:=\begin{bmatrix}\bm{\lambda}_{k-1}\\
+\mathbf{u}_{1,k}(\delta_{1})\\
+\mathbf{u}_{2,k}(\delta_{2})\end{bmatrix}\,,\quad k=1,2,\ldots\,.$ $A_{\lambda}(\bm{\mu})$ $2n_{1,\gamma}(2n_{1,\gamma}-1)\approx O(n^{2}_{1,\gamma})$ $\delta_{i}=5/2h$ $\text{dim}\,S_{i}^{h}=n_{i}$ $L^{2}(\Omega_{i})$ $\mathbf{y}_{i}:=\mathbf{y}(t_{i})$ $\mathbf{v}=\left(0.5-y,x-0.5\right)$ $\mathbf{u}_{i,k}(\delta_{i})=\left\{(\mathbf{u}_{i,k})_{j}\,|\,\exists\bm{x}_{%
+j}\in\Omega^{h}_{i}\ \mbox{s.t.}\ d(\bm{x}_{j},\gamma)<\delta_{i}\right\}\in%
+\mathbb{R}^{n_{\delta_{i},D}}$ $1-E_{k}(Y)\leq\epsilon\,,$ $\mathcal{M}_{m}:=\{\bm{\mu}_{j}\}_{j=1}^{m}$ $\Gamma^{h}_{i}$ $t_{1},t_{2},\ldots t_{n}$ $\mathbf{P}_{q}$ $S^{h}_{i,\Gamma}$ $R_{2\mapsto 1}$ $\epsilon=10^{-13}$ ${M}_{0,\gamma}$ $\gamma^{h}_{1}$ $\mathbf{Y}(\bm{\mu}_{i})$ $\mathbf{Y}^{\prime}(\bm{\mu}_{i})$ $\mathcal{M}_{m}=\left\{(1,3),(1,4),(2,3),(2,4)\right\}\times 10^{-3}\,.$ $u^{h}_{i}\in S_{i}^{h}$ $n_{i,\gamma}$ $n_{i,D}\times n_{i,D}$ $S=G_{1}M_{1}^{-1}G_{1}^{T}+G_{2}M_{2}^{-1}G_{2}^{T}$ $\mathbf{c}_{k}=H_{1,\gamma}(\mathbf{b}_{1,k})_{\gamma}-H_{2,\gamma}(\mathbf{b}%
+_{2,k})_{\gamma}\,,$ $\partial\Omega_{i}$ $N\in\{16,32,64,128\}$ $g^{h}_{i}\in S^{h}_{i,\Gamma}$ $\mathcal{F}_{i}(\mathbf{u}_{1},\mathbf{u}_{2}):=G^{T}_{i}\bm{\lambda}=G^{T}_{i%
+}\left(G_{1}M_{1}^{-1}G_{1}^{T}+G_{2}M_{2}^{-1}G_{2}^{T}\right)^{-1}\left(G_{1%
+}M_{1}^{-1}\mathbf{b}_{1}-G_{2}M_{2}^{-1}\mathbf{b}_{2}\right)\,.$ $\beta_{i,k-1}$ $\Omega^{h}_{2}$ $O(n_{1,\gamma}n_{1,D})$ $\mathcal{F}_{1}(\mathbf{u}_{1},\mathbf{u}_{2}):=K_{1,\gamma}\widehat{\mathbf{u%
+}}_{1}=K_{1,\gamma}R_{2\mapsto 1}\mathbf{u}_{2}\quad\mbox{and}\quad\mathcal{F}%
+_{2}(\mathbf{u}_{1},\mathbf{u}_{2}):=K_{2,\gamma}\widehat{\mathbf{u}}_{2}=K_{2%
+,\gamma}R_{1\mapsto 2}\mathbf{u}_{1}\,,$ $\mathcal{M}_{m}(\bm{\mu},r)$ $K_{i,s}$ $\lambda(\cdot,t)\in H^{-1/2}(\gamma)$ $u^{h}_{i,D}$ $S^{h}_{i,D}$ $\mathbf{Y}^{\prime}:=[\mathbf{y}_{i}]_{i=1}^{s}$ $\displaystyle M_{1}\dot{\mathbf{u}}_{1}+K_{1}\mathbf{u}_{1}+G_{1}^{T}\bm{\lambda}$ $k=0,1,\ldots,N-1$ $u^{h}(\cdot,t)\in S^{h}_{i}$ $\mathcal{M}_{m}=\left\{(1,2),(1,3),(2,2),(2,3)\right\}\times 10^{-3}\,.$ $\mathcal{M}_{m}:=\{\bm{\mu}_{i}\}_{i=1}^{m}$ $\mathbf{Y}=U\Sigma V^{T}$ $S^{h}_{l,\gamma}$ $\mathcal{E}^{0}_{X}$ $\displaystyle=\mathbf{f}_{i}+(-1)^{i}\bm{\lambda}_{i}\,,\quad t\in(0,T]\quad%
+\mbox{and}\quad\mathbf{u}_{i}(0)=\mathbf{u}_{i,0}\,,$ $n_{1,\gamma}\times n_{i,\gamma}$ $q_{16}=444$ $\left\{\begin{aligned} \dot{u}_{i}-\nabla\cdot F_{i}(u_{i})&=f_{i}&\text{in}\ %
+&\Omega_{i}\times(0,T]\\[2.15277pt]
+{u}_{i}&=g_{i}&\text{on}\ &\Gamma_{i}\times(0,T]\\[2.15277pt]
+F_{i}(u_{i})\cdot\bm{n}_{i}&=(-1)^{i}\lambda&\text{on}\ &\gamma\times(0,T]\\[2%
+.15277pt]
+u_{i}(\bm{x},0)&=u_{i,0}(\bm{x})&\text{in}\ &\quad\Omega_{i}\end{aligned}%
+\right.\,;\quad i=1,2.$ $\|\cdot\|_{1,\Omega_{i}}$ $\bm{\lambda}^{\sf DMD}_{i,k}=G^{T}_{i}\bm{\lambda}_{k}=G^{T}_{i}A_{\lambda}(%
+\bm{\mu})\mathbf{y}_{k-1}\,.$ $\bm{\mu}_{j}=(\kappa_{1,j},\kappa_{2,j})$ $\kappa_{1}\neq\kappa_{2}$ $\mathbf{c}_{k}:=H_{1}\mathbf{b}_{1,k}-H_{2}\mathbf{b}_{2,k}$ $\mathbf{b}_{i}=\mathbf{f}_{i}-K_{i}\mathbf{u}_{i}$ $u^{h}_{i}$ $X\in\{C,L,D\}$ $\mathbf{Y}^{+}$ $u^{h}_{i,C}$ $\times 16.66$ $\mathcal{E}^{r}_{X}=\frac{1}{2}\sum_{i=1}^{2}\frac{\|u^{h}_{i,X}-u^{h}_{i,M}\|%
+_{r,\Omega_{i}}}{\|u^{h}_{i,M}\|_{r,\Omega_{i}}}\,,r=0,1\,.$ $\{A_{k_{j}}(\bm{\mu}_{j})\,|\,\bm{\mu}_{j}\in\mathcal{M}_{m}(\bm{\mu},r)\}$ $O(n^{2}_{l,\gamma})$ $H^{-1/2}(\gamma)$ $K_{i,\gamma}$ $({u}^{h}_{i,0},v^{h}_{i})_{0,\Omega_{i}}=(u_{i,0},v^{h}_{i})_{0,\Omega_{i}}%
+\quad\forall v^{h}_{i}\in S^{h}_{i}\,.$ $\mathbf{y}_{k}=\bm{\lambda}_{k}$ $\mathbf{y}_{k}=(\bm{\lambda}_{k},\mathbf{u}_{1,k+1}(\delta_{1}),\mathbf{u}_{1,%
+k+1}(\delta_{2}))^{T}$ $R_{1\mapsto 2}$ $N_{\sf FS}=n_{l,\gamma}+n_{\delta_{1},D}+n_{\delta_{2},D}$ $g^{h}_{i}(\bm{x}_{i,r})=g_{i}(\bm{x}_{i,r})$ $[\mathbf{Y}(\bm{\mu}_{i})]_{i=1}^{m}$ $\Gamma_{i}:=\partial\Omega_{i}\backslash\gamma$ $\bm{\mu}=\{\kappa_{1},\kappa_{2}\}\in\mathcal{M}$ $\bm{\mu}\in\mathbb{R}^{m}$ $\mathbf{b}_{i,k}$ $\bm{\lambda}_{i,k}=\mathcal{F}_{i}(\bm{\Lambda}_{1,\beta_{1,k-1}},\bm{\Lambda}%
+_{2,\beta_{2,k-1}};\mathbf{U}_{1,\alpha_{1,k}},\mathbf{U}_{2,\alpha_{2,k}})\,,%
+\quad i=1,2\,.$ $\mathbf{Y}^{\prime}\approx{A}\mathbf{Y}.$ $u_{1}(\bm{x},t)=u_{2}(\bm{x},t)\quad\text{and}\quad F_{1}(u_{1})\cdot\bm{n}_{%
+\gamma}=F_{2}(u_{2})\cdot\bm{n}_{\gamma}\quad\mbox{on}\quad\gamma\times[0,T]\,,$ $\bm{\lambda}_{i,k}$ $S^{h}_{i,\gamma}$ $S^{h}_{i}\subset H^{1}(\Omega_{i})$ $\mathcal{M}=([1,2]\times[2,3])\times 10^{-3}$ $\mathbf{y}_{k}\in\mathbb{R}^{N_{\sf FS}}$ $\bm{\mu}_{j}\in\mathcal{M}_{m}(\bm{\mu},r):=\mathcal{M}_{m}\cap B(\bm{\mu},r)\,.$ ${U}_{k}$ $A_{\lambda}=\begin{bmatrix}A_{\lambda,\lambda}&A_{\lambda,u_{1}}&A_{\lambda,u_%
+{2}}\end{bmatrix}\,.$ $n_{i,\gamma}\times n_{i,\gamma}$ $\displaystyle M_{i}\dot{\mathbf{u}}_{i}+K_{i}\mathbf{u}_{i}$ $\Omega\in\mathbb{R}^{\nu}$ $\Delta_{32}t=6.84\times 10^{-3}$ $\mathbf{u}_{i}(t)$ $S\bm{\lambda}=G_{1}M_{1}^{-1}\mathbf{b}_{1}-G_{2}M_{2}^{-1}\mathbf{b}_{2}\,,$ $\psi(x,y;x_{0},y_{0})=e^{-\frac{(x-x_{0})^{2}+(y-y_{0})^{2}}{2\sigma^{2}}}.$ $n_{1,\gamma}$ $N_{\sf FS}\times N_{\sf FS}$ $H_{i}:=G_{i}M^{-1}_{i}$ $O(n^{2}_{1,\gamma})$ $\bm{\lambda}_{1,k}(\bm{\mu}_{j})$ $\mathbf{u}_{2,k}$ $\mathbf{y}_{k-1}=(\bm{\lambda}_{k-1},\mathbf{u}_{1,k}(\delta_{1}),\mathbf{u}_{%
+1,k}(\delta_{2}))^{T}$ $\mathbf{y}_{k+1}$ $\{u_{1}(\cdot,t),u_{2}(\cdot,t),\lambda(\cdot,t)\}\in H^{1}(\Omega_{1})\times H%
+^{1}(\Omega_{2})\times H^{-1/2}(\gamma)$ $u_{i,0}\in H^{1}(\Omega_{i})$ $\text{dim}\,S_{i,\Gamma}^{h}=n_{i,\Gamma}$ $T=2\pi$ $u(x,y)=\left\{\begin{array}[]{ll}\displaystyle t(x+2y+3)&\mbox{if $(x,y)\in%
+\bar{\Omega}_{1}$}\\[8.61108pt]
+\displaystyle t\left(\frac{\kappa_{1}}{\kappa_{2}}x+2y+\frac{\kappa_{2}-\kappa%
+_{1}}{2\kappa_{2}}+3\right)&\mbox{if $(x,y)\in{\Omega}_{2}$}\end{array}\right.\,,$ $\displaystyle(\dot{u}^{h}_{i},v^{h}_{i})_{0,\Omega_{i}}+(F_{i}(u^{h}_{i}),%
+\nabla v^{h}_{i})_{0,\Omega_{i}}$ $\gamma^{h}_{2}$ $0<\delta_{i}<\mbox{diam}(\Omega_{i})$ $\mathbf{y}_{k-1}$ $\|\cdot\|_{0,\Omega_{i}}$ $\mathbf{y}_{k-1}:=\begin{bmatrix}\bm{\lambda}_{k-1}\\
+\mathbf{u}_{1,k}\\
+\mathbf{u}_{2,k}\end{bmatrix}\,,\quad k=1,2,\ldots\,.$ $\delta_{rs}$ $A\approx{A}_{k}:=\mathbf{Y}^{\prime}\mathbf{Y}_{k}^{+}=\mathbf{Y}^{\prime}{V}_%
+{k}{\Sigma}_{k}^{+}{U}_{k}^{T}\,.$ $\mathbf{u}_{2,k}(\bm{\mu}_{j})$ $\mathbf{u}_{i,k+1}$ $f_{i}(\cdot,t)\in H^{-1}(\Omega_{i})$ ${A}({\bm{\mu}})$ $S=LL^{T}$ $0=t_{0} $\displaystyle G_{1}\dot{\mathbf{u}}_{1}-G_{2}\dot{\mathbf{u}}_{2}$ $\bm{n}_{2}$ ${\mathbf{y}}_{k-1}$ $\mathbf{u}_{i,D}=(\mathbf{u}_{i,\gamma},\mathbf{u}_{i,0},\mathbf{0})$ $\bm{n}_{\gamma}$ $\Delta_{16}t=1.42\times 10^{-2}$ $\nu=2,3$ $\kappa_{1}=1\times 10^{-3}$ $\lambda(\bm{x},t_{m})$ $\mathbf{Y}^{+}=V\Sigma^{+}U^{T}\approx{V}_{k}{\Sigma}_{k}^{+}{U}_{k}^{T}=:%
+\mathbf{Y}_{k}^{+}\,,$ ${A}_{\lambda}(\bm{\mu})$ $\widehat{\mathbf{u}}_{1}=R_{2\mapsto 1}\mathbf{u}_{2}\quad\mbox{and}\quad%
+\widehat{\mathbf{u}}_{2}=R_{1\mapsto 2}\mathbf{u}_{1}\,,$ $\kappa_{2}=3\times 10^{-3}$ $\bm{x}_{i,r}$ $\text{dim}\,S^{h}_{i,\gamma}=n_{i,\gamma}$ $\mathbf{Y}:=[\mathbf{y}_{i}]_{i=0}^{s-1}$ $N_{\sf FS}(2N_{\sf FS}-1)$ $0\leq m\leq N$ $\mathbf{y}_{i+1}\approx A\mathbf{y}_{i}\,.$ $\mathbf{u}_{1,k}(\bm{\mu}_{j})$ $S_{i,D}^{h}$ $S^{h}_{i,D}\subset H^{1}_{D}(\Omega_{i})$ $\Delta t=3.37E-3$ $\Sigma_{k}^{+}$ $(N_{\sf FS}-n_{l,\gamma})(2N_{\sf FS}-1)$ $\kappa_{1}=\kappa_{2}=1\times 10^{-3}$ $A=\begin{bmatrix}A_{\lambda,\lambda}&A_{\lambda,u_{1}}&A_{\lambda,u_{2}}\\
+A_{u_{1},\lambda}&A_{u_{1},u_{1}}&A_{u_{1},u_{2}}\\
+A_{u_{2},\lambda}&A_{u_{2},u_{1}}&A_{u_{2},u_{2}}\\
+\end{bmatrix}$ $S_{i}^{h}$ $2n_{1,\gamma}(2n_{1,D}-1)$ $\mathbf{u}_{i,k}(\delta_{i})$ $q=2\pi/\Delta t$ $H^{1/2}(\gamma)$ $\mathcal{E}^{1}_{X}$ $n_{l,\gamma}$ $\bm{\lambda}_{k-1}$ $\mathbf{u}_{i}\in\mathbb{R}^{n_{i}}$ $u^{h}_{i}(\bm{x},t_{m})$ $\{{u}^{j}_{i,0}\}_{j=1}^{P}$ $g_{i}(\cdot,t)\in H^{1/2}_{00}(\Gamma_{i})$ $\mathbf{u}_{i,k+1}=\mathbf{u}_{i,k}+\Delta tM_{i}^{-1}\left(\mathbf{f}_{i,k}-K%
+_{i}\mathbf{u}_{i,k}+(-1)^{i}\bm{\lambda}^{DMD}_{i,k}\right)\,.$ $\{\ell_{i,r}\}$ $\mathbf{y}_{k-1}(\bm{\mu}_{j})=\left(\bm{\lambda}_{1,k-1}(\bm{\mu}_{j}),%
+\mathbf{u}_{1,k}(\delta_{1};\bm{\mu}_{j}),\mathbf{u}_{2,k}(\delta_{2};\bm{\mu}%
+_{j})\right)^{T}$ $\mathbf{u}_{i,k}(\delta_{i})\subset\mathbf{u}_{i,k}$ $\text{dim}\,S_{i,D}^{h}=n_{i,D}:=n_{i}-n_{i,\Gamma}$ $(X_{*},\mathbf{y}_{*})$ $N_{*}\times 1$ $\phi_{i}(\mathbf{x})$ $\bm{\Phi}_{(\cdot)}$ $\bm{\mu}_{*}$ $\phi_{\mathbf{n}}(\mathbf{x})$ $\mathbf{y}_{*}=\{y_{*i}\in\mathbb{R}\,|\,i=1,\ldots,N_{*}\}$ $\nu\sim\mathcal{N}(0,\,\sigma_{n}^{2})$ $\displaystyle\phi_{\mathbf{n}}(\mathbf{x})$ $\displaystyle i=1,\ldots,n$ $\tilde{K}=(K+\sigma^{2}I)$ $\mathbf{f}_{*}|\mathbf{f},\mathbf{y},\bm{\theta}\sim\mathcal{N}(\bm{\mu}_{*},%
+\Sigma_{*})$ $\displaystyle\approx m(X_{*})+W(\mathbf{y}-m(X))$ $p(\mathbf{f})=\mathcal{N}(\mathbf{0},\,k(X,X))$ $y=\sum_{i=1}^{p}\cos{x_{i}}+\nu$ $X=\{\mathbf{x}_{*i}\in\mathbb{R}^{p}\,|\,i=1,\ldots,N_{*}\}$ $y=f(\mathbf{x})+\varepsilon$ $\displaystyle\Sigma_{*}$ $n^{p}\times n^{p}$ $\displaystyle=\mathbf{\Phi}_{(X_{*})}\Lambda\mathbf{\Phi}_{(X)}^{T}\left(%
+\Sigma_{N}^{-1}-\Sigma_{N}^{-1}\mathbf{\Phi}_{(X)}\bar{\Lambda}^{-1}\mathbf{%
+\Phi}_{(X)}^{T}\Sigma_{N}^{-1}\right)$ $\Sigma_{n}^{-1}$ $\displaystyle\hskip 15.0pt\bm{\Phi}_{(X_{*})}\Lambda\bm{\Phi}^{T}_{(X)}\left(%
+\bm{\Phi}_{(X)}\Lambda\bm{\Phi}^{T}_{(X)}+\sigma_{n}^{2}I\right)^{-1}\bm{\Phi}%
+_{(X)}\Lambda\bm{\Phi}^{T}_{(X_{*})}$ $\Sigma_{*}$ $\displaystyle=\prod_{j=1}^{p}\phi_{n_{j}}(x_{j};\varepsilon_{j},\rho_{j})$ $\displaystyle=K_{**}-K_{*}(K+\sigma^{2}I)^{-1}K_{*}^{T}$ $9^{2}\times 9^{2}$ $k_{SE}^{ARD}(\mathbf{x},\mathbf{x^{\prime}})=\exp\left(-\varepsilon_{1}^{2}(x_%
+{1}-x_{1}^{\prime})^{2}-\ldots-\varepsilon_{p}^{2}(x_{p}-x_{p}^{\prime})^{2}\right)$ $(X,\mathbf{y})$ $\displaystyle=\Lambda^{-1}+\mathbf{\Phi}_{(X)}^{T}\Sigma_{N}^{-1}\mathbf{\Phi}%
+_{(X)}$ $\lambda_{n_{j}}(\varepsilon_{j},\rho_{j})$ $\Sigma_{n}=\sigma_{n}^{2}I$ $\displaystyle=\left(1+\left(\frac{2\varepsilon}{\rho}\right)^{2}\right)^{\frac%
+{1}{4}},$ $k(\mathbf{x},\mathbf{x^{\prime}})\approx\sum_{i=1}^{n}\lambda_{i}\phi_{i}(%
+\mathbf{x})\phi_{i}(\mathbf{x^{\prime}})$ $\displaystyle\lambda_{i}=\sqrt{\frac{\rho^{2}}{\rho^{2}+\delta^{2}+\varepsilon%
+^{2}}}\left(\frac{\varepsilon^{2}}{\rho^{2}+\delta^{2}+\varepsilon^{2}}\right)%
+^{i-1}$ $X=\{\mathbf{x}_{i}\in\mathbb{R}^{p}\,|\,i=1,\ldots,N\}$ $N_{*}\times N_{*}$ $\displaystyle=m(X_{*})+K_{*}(K+\sigma^{2}I)^{-1}(\mathbf{y}-m(X))$ $\varepsilon\sim\mathcal{N}(0,\,\sigma^{2})$ $K_{**}=k(X_{*},X_{*})$ $N_{*}\times N$ $K=k(X,X)$ $p(\mathbf{f})$ $\displaystyle\delta^{2}=\frac{\rho}{2}(\beta^{2}-1),$ $\displaystyle\approx\mathbf{\Phi}_{(X_{*})}\Lambda\mathbf{\Phi}_{(X_{*})}^{T}-%
+W\mathbf{\Phi}_{(X)}\Lambda\mathbf{\Phi}_{(X_{*})}^{T}$ $N\times 9^{2}$ $m(X)=m(X_{*})=\mathbf{0}$ $\bar{\Lambda}=\left(\Lambda^{-1}+\bm{\Phi}^{T}_{(X)}\Sigma_{n}^{-1}\bm{\Phi}_{%
+(X)}\right)$ $K_{*}=k(X_{*},X)$ $\displaystyle\approx m(X_{*})+\bm{\Phi}_{(X_{*})}\Lambda\bm{\Phi}^{T}_{(X)}%
+\left(\bm{\Phi}_{(X)}\Lambda\bm{\Phi}^{T}_{(X)}+\Sigma_{n}\right)^{-1}(\mathbf%
+{y}-m(X))$ $\Lambda=\begin{bmatrix}\lambda_{1}&&\\
+&\ddots&\\
+&&\lambda_{n}\end{bmatrix}$ $\mathbf{y}=\{y_{i}\in\mathbb{R}\,|\,i=1,\ldots,N\}$ $\geq 3.18$ $\bm{\Phi}_{(X)}=\begin{bmatrix}|&&|\\
+\phi_{i}(X)&\ldots&\phi_{n}(X)\\
+|&&|\end{bmatrix}$ $\approx 10.5$ $\displaystyle\gamma_{i}=\sqrt{\frac{\beta}{2^{i-1}\Gamma_{(i)}}},$ $\displaystyle\lambda_{\mathbf{n}}$ $\left(\bm{\Phi}_{(X)}\Lambda\bm{\Phi}^{T}_{(X)}+\Sigma_{n}\right)^{-1}=\Sigma_%
+{n}^{-1}-\Sigma_{n}^{-1}\bm{\Phi}_{(X)}\left(\Lambda^{-1}+\bm{\Phi}^{T}_{(X)}%
+\Sigma_{n}^{-1}\bm{\Phi}_{(X)}\right)^{-1}\bm{\Phi}^{T}_{(X)}\Sigma_{n}^{-1}$ $k(X,X^{\prime})\approx\bm{\Phi}_{(X)}\Lambda\bm{\Phi}^{T}_{(X^{\prime})}$ $\mathbf{f}=f(\mathbf{x})$ $\displaystyle\bm{\mu}_{*}$ $\displaystyle=\prod_{j=1}^{p}\lambda_{n_{j}}(\varepsilon_{j},\rho_{j})$ $k_{SE}^{ARD}{(\mathbf{x},\mathbf{x^{\prime}})}\approx\sum_{\mathbf{n}\in%
+\mathbb{N}^{p}}\lambda_{\mathbf{n}}\phi_{\mathbf{n}}(\mathbf{x})\phi_{\mathbf{%
+n}}(\mathbf{x^{\prime}})$ $\phi_{n_{j}}(x_{j};\varepsilon_{j},\rho_{j})$ $\displaystyle\bar{\Lambda}$ $\mathbf{x}=\begin{bmatrix}x_{1}&x_{2}&\ldots&x_{p}\end{bmatrix}\in\mathbb{R}^{p}$ $N\times n^{p}$ $\mathbf{\Phi}_{(X)}$ $\displaystyle\approx\bm{\Phi}_{(X_{*})}\Lambda\bm{\Phi}^{T}_{(X_{*})}-$ $k_{SE}(x,x^{\prime})=\exp\left(-\varepsilon^{2}(x-x^{\prime})^{2}\right)$ $k(\mathbf{x},\mathbf{x^{\prime}})=\sum_{i=1}^{\infty}\lambda_{i}\phi_{i}(%
+\mathbf{x})\phi_{i}(\mathbf{x^{\prime}})$ $\lambda_{\mathbf{n}}$ $\displaystyle\phi_{i}(x)=\gamma_{i}\exp^{-\delta^{2}x^{2}}H_{i-1}(\rho\beta x)$ $\phi(a,b)=\frac{a^{T}b}{\left\lVert a\right\rVert\left\lVert b\right\rVert}$ $\mathcal{L}=-\sum_{i=1}^{T}\log\frac{e^{\phi(z_{i},z^{\prime}_{i})/\tau}}{\sum%
+_{j\in D_{i}}e^{\phi(z_{i},z^{\prime}_{j})/\tau}}$ $WER=\frac{1}{|S^{\prime}|}\sum_{(x,z)\in S^{\prime}}\left\{\begin{array}[]{l l%
+}0\text{ if }h(x)=z\\
+1\text{ otherwise }\end{array}\right.$ $a\in A,b\in B,x\neq a\in A$ $\tilde{X}=r(X,M)$ $z_{t}=h(x_{t})$ $D_{SSIMI}$ $ABX$ $d(a,x) $\mathcal{L}=\sum_{t\in M}\log p_{f}(z_{t}|\tilde{X},t)$ $(a,b,x)$ $PVQ$ $X=\left[x_{1},\cdots x_{T}\right]$ $t_{\operatorname*{DMN}}$ $|N_{v}\setminus U^{d}_{[v]_{\Pi_{d^{\prime}}}}|+|[v]_{\Pi_{d^{\prime}}}%
+\setminus N_{v}|>d^{\prime}$ $t_{\text{$\operatorname*{DMN}$}}$ $6.7\cdot 10^{2}$ $\displaystyle=|N_{v}\setminus[v]_{\Pi}|+|[v]_{\Pi}\setminus N_{v}|$ $made\_join=True$ $\displaystyle\geq|N_{v}|-|N_{v}|/2=|N_{v}|/2>2d\enspace,$ $u^{\prime}\in C^{\prime}$ $\{u,v\}\in\binom{V}{2}$ $v\in N_{w}$ $\{u,v\}\in N_{w}$ $\varphi(\Pi)=d$ $\Pi=\{\{0,1\},\{2\},\{3\},\{4,5,6\}\}$ $U^{d}_{[v]_{\Pi_{d}}}$ $|N_{u}\triangle N_{u^{\prime}}|\leq 2d$ $U^{d}_{C}=\{v\in V\mid|N_{w}\triangle N_{u}|\leq 2d\;\forall w\in[v]_{\Pi_{d}}%
+\;\forall u\in C\}$ $\displaystyle[v]_{\Pi_{d}}=U^{d}_{[v]_{\Pi_{d}}}=\{u\in V\mid|N_{u}\cap N_{v}|%
+>|N_{v}|/2\}\enspace.$ $I_{\{u,v\}}$ $[v]_{\Pi_{d}}\subseteq[v]_{\Pi}\subseteq U^{d}_{[v]_{\Pi_{d}}}$ $1.996+\epsilon$ $C\in\Pi_{d}$ $\varphi(\Pi)\leq 2$ $6.4\cdot 10^{4}$ $4.1\cdot 10^{4}$ $1.7\cdot 10^{5}$ $U^{2}_{\{4\}}=\{0,4,5,6\}$ $|N_{w}\triangle N_{v}|$ $d=\max_{u\in C}|N_{u}\triangle C|$ $N_{v}:=\{v\}\cup\{w\in V\mid\{v,w\}\in E\}$ $|N_{u}\cap N_{v}|>2\varphi(\Pi)$ $C\subseteq U^{d}_{C}$ $x_{\{0,1\}}=x_{\{0,2\}}=\tfrac{1}{2}$ $E_{d}=\{\{u,v\}\in\binom{V}{2}\mid|N_{u}\cap N_{v}|>2d\}$ $x_{\{1,3\}}=x_{\{2,4\}}=\tfrac{3}{4}$ $1.2\cdot 10^{5}$ $x_{\{3,4\}}=x_{\{3,5\}}=x_{\{4,5\}}=\tfrac{1}{4}$ $U^{d}_{C}$ $\Pi_{2}=\{\{v\}\mid v\in V\}$ $t_{\operatorname*{DMN}}^{++}$ $\displaystyle=|(N_{u}\setminus N_{v})\cap C|+|(N_{u}\setminus N_{v})\setminus C|$ $\displaystyle|N_{4}\triangle[4]_{\Pi}|$ $\tfrac{1}{2}\sum_{w\in N_{u}\cap N_{v}}\min(\theta_{uw},\theta_{vw})$ $u\notin U^{d}_{[v]_{\Pi_{d}}}$ $\tfrac{7}{4}$ $|N_{u}\cap N_{v}|\leq 2\varphi(\Pi)$ $1.5\cdot 10^{6}$ $[u]_{\Pi}$ $E^{+}=E$ $U^{1}_{\{0\}}=\{0,1,2\}$ $-|N_{v}\triangle N_{w}|$ $|N_{v}|/4$ $[v]_{\Pi_{d}}$ $\mathcal{O}(n^{2}\log_{2}(\delta)+n\delta^{2})$ $|N_{w}|\leq\delta$ $\mathcal{O}(n^{2}\delta^{2})$ $N_{u}\cap N_{v}$ $I_{\{u,v\}}>2d$ $C=[u]_{\Pi}=[v]_{\Pi}$ $d=\varphi(\Pi)$ $x_{\{4,5\}}=0$ $\Pi_{1}=\{\{0\},\{1\},\{2\},\{3,4,5\}\}$ $\sum_{w\in N_{u}\cap N_{v}}\min(\theta_{uw},\theta_{vw})>2\varphi(\Pi)$ $|[v]_{\Pi_{d}}\setminus N_{v}|$ $\displaystyle=|N_{u}\triangle C|+|N_{v}\triangle C|$ $\displaystyle\min_{\Pi\in P_{V}}\underbrace{\max_{v\in V}\quad\left|[v]_{\Pi}%
+\triangle N_{v}\right|}_{=:\ \varphi(\Pi)}$ $made\_join=False$ $x_{\{0,1\}}=x_{\{0,2\}}=x_{\{0,3\}}=x_{\{1,4\}}=x_{\{1,5\}}=x_{\{2,4\}}=x_{\{2%
+,5\}}=x_{\{3,4\}}=x_{\{3,5\}}=\tfrac{1}{2}$ $\displaystyle|N_{u}\triangle N_{v}|$ $\mathcal{O}(n\delta^{2})$ $t_{\mathcal{A}^{*}}$ $C=\{u\in V\mid|N_{u}\cap N_{v}|>|N_{v}|/2\}$ $.08{\scriptstyle\ \pm.12}$ $0.03495$ $.80{\scriptstyle\ \pm.01}$ $\mathbf{.18}{\scriptstyle\ \pm.00}$ $.12{\scriptstyle\ \pm.00}$ $7.42\times 10^{-4}$ $2.96\times 10^{-5}$ $.15{\scriptstyle\ \pm.01}$ $\underline{\mathbf{.01}}{\scriptstyle\ \pm.00}$ $55\%$ $\underline{\mathbf{.83}}{\scriptstyle\ \pm.01}$ $.72{\scriptstyle\ \pm.03}$ $.52{\scriptstyle\ \pm.01}$ $.04{\scriptstyle\ \pm.02}$ $\mathbf{.81}{\scriptstyle\ \pm.01}$ $\mathbb{P}\big{(}Y=\hat{y}\ |\ P=\hat{p}\big{)}=\hat{p},$ $q\left(x_{t}\mid x_{t-1}\right)=N\left(x_{t};\sqrt{1-\beta_{t}}x_{t-1},\beta_{%
+t}I\right)$ $T_{o}\leq T$ $I^{k}_{s}$ $x_{o}$ $q\left(x_{1:T}\mid x_{0}\right)=\prod_{t=1}^{T}q\left(x_{t}\mid x_{t-1}\right)$ $v_{o}\in\mathbb{R}^{1\times 6}$ $T\rightarrow\infty$ $v_{c}=\Phi(I_{s})$ $x_{0},x_{1},\ldots,x_{T}$ $\sigma^{k}_{o}$ $t\sim\mathcal{U}(\{1,\ldots,T\})$ $k\in\{R,G,B\}$ $x_{t-1}=\frac{1}{\sqrt{\alpha_{t}}}\left(x_{t}-\frac{1-\alpha_{t}}{\sqrt{1-%
+\bar{\alpha}_{t}}}\epsilon_{\theta}(x_{t},I_{c},v_{c},t)\right)+\sigma_{t}z$ $L_{err}=\mathbb{E}_{x_{0},t,\epsilon\sim\mathcal{N}(0,I),I_{c},v_{c}}\left[%
+\left\|\epsilon-\epsilon_{\theta}\left(x_{t},t,I_{c},v_{c}\right)\right\|\right]$ $2^{-16}$ $\forall t\sim\mathcal{U}(\{1,\ldots,T\})$ $x_{T_{o}}=I_{c}$ $v_{c}\in\mathbb{R}^{1\times 6}$ $\begin{split}I^{k}_{c}=\frac{I^{k}_{s}-\mu^{k}_{s}}{\sigma^{k}_{s}+\varepsilon%
+}\sigma^{k}_{c}+\mu^{k}_{c},\end{split}$ $t=T_{o},T_{o-1},\dots,1$ ${L}_{2}(\Theta)=\|v_{o}-v_{c}\|_{2},$ $p_{\theta}\left(x_{t-1}\mid x_{t}\right)=\mathcal{N}\left(x_{t-1};\mu_{\theta}%
+\left(x_{t},t\right),{\sigma_{\theta}\left(x_{t},t\right)}^{2}I\right)$ $(I_{s},I_{o})\sim P$ $v_{c}$ $x_{t}=\sqrt{\bar{\alpha}_{t}}x_{0}+\epsilon\sqrt{1-\bar{\alpha}_{t}}\\$ $T_{o}$ $I^{k}_{c}$ $\sigma_{\theta}\left(x_{t},t\right)^{2}$ $\nabla_{\theta}\|\epsilon-\epsilon_{\theta}(x_{t},I_{c},v_{c},t)\|$ $x_{t}=\sqrt{\bar{\alpha}_{t}}I_{o}+\epsilon\sqrt{1-\bar{\alpha}_{t}}$ $\sigma^{k}_{c}$ $\mu^{k}_{o}$ $I_{c}=\text{ReNormalize}(v_{c},I_{s})$ $t\in\{T,\ldots,1\}$ $\mu_{\theta}\left(x_{t},t\right)$ $z\sim\mathcal{N}(0,I)$ $v_{o}$ $\mu^{k}_{c}$ $P=\{(I_{s}^{n},I_{o}^{n})\}_{n=1}^{N}$ $L_{err}=\mathbb{E}_{x_{o},t,\epsilon\sim\mathcal{N}(0,I)}\left[\left\|\epsilon%
+-\epsilon_{\theta}\left(x_{t},t\right)\right\|\right]$ $\mu^{k}_{s}$ $\sigma^{k}_{s}$ $I_{o}$ $q(x_{t-1}\mid x_{t})$ $x_{T}\sim\mathcal{N}(0,I)$ $s(A,B,X,Y)=\sum_{x\in X}s(x,A,B)-\sum_{y\in Y}s(y,A,B)$ $d=\frac{\mu(\{s(x,A,B)\}_{x\in X})-\mu(\{s(y,A,B)\}_{y\in Y})}{\sigma(\{s(t,X,%
+Y)\}_{t\in A\cup B})}$ $s(w,A,B)\!=\!\frac{1}{|A|}\!\sum_{a\in A}\!\cos(w,a)\!-\!\frac{1}{|B|}\!\sum_{%
+b\in B}\!\cos(w,b).$ $G=2$ $\mathbf{W}^{(x)}\in\mathbb{R}^{C^{\prime}\times C}$ $1\times{}1$ $\hat{\mathbf{q}}$ $\mathbf{g}_{:hw}=\textrm{softmax}\left(\mathbf{W}^{(g)}\cdot\textrm{ReLU}\left%
+(\mathbf{W}^{(x)}\hat{\mathbf{x}}_{:hw}\odot\mathbf{W}^{(q)}\hat{\mathbf{q}}%
+\right)\right),$ $C\cdot{}G+Q$ $\mathbf{g}\in\mathbb{R}^{G\times{}H\times{}W}$ $\hat{\mathbf{q}}\in\mathbb{R}^{Q}$ $\hat{\mathbf{x}}^{\prime}_{cghw}=\mathbf{g}_{ghw}\cdot\hat{\mathbf{x}}_{chw}%
+\cdot\left(\mathbf{m}\downarrow_{H\times{}W}\right)_{hw},$ $\mathbf{W}^{(g)}$ $(\mathbf{q},\mathbf{m})$ $\hat{\mathbf{x}}$ $\hat{\mathbf{x}}^{\prime}$ $\mathbf{W}^{(g)}\in\mathbb{R}^{G\times C^{\prime}}$ $\hat{a}=\operatorname*{arg\,max}_{a\in\mathcal{A}}p_{\theta}(a\mid\mathbf{q},%
+\mathbf{x},\mathbf{m}).$ $\hat{\mathbf{x}}^{\prime}=\textrm{att}(\hat{\mathbf{q}},\hat{\mathbf{x}},%
+\mathbf{m})$ $\mathbf{W}^{(q)}\in\mathbb{R}^{C^{\prime}\times Q}$ $\mathbf{m}\downarrow_{H\times{}W}$ $\hat{\mathbf{x}}\in\mathbb{R}^{C\times{}H\times{}W}$ ${:}hw$ $\mathbf{W}^{(x)}$ $Q_{g}=diag(q_{g})$ $q_{g}\in R^{K}$ $(M)_{i*}$ $p_{g}\in R^{D}$ $\in[0.2,0.4,0.6]$ $1\leq g\leq G$ $n_{c}=\lfloor a/((c-1)^{-\gamma}+b)\rceil$ $p\in[1,5,10,20,50]$ $\lfloor{.}\rceil$ $R_{D}$ $q_{g}$ $\begin{split}\Omega(W,P,Q)&=\lambda_{1}R_{W}(W,P,Q)+\lambda_{2}R_{D}(P,Q)+%
+\lambda_{3}R_{E}(P,Q)\end{split}$ $\gamma\in\{2.0,1.0,0.6,0.20\}$ $l_{2,1}$ $p_{g}$ $\displaystyle\leq|C\setminus N_{v}|+|N_{u}\setminus C|+|C\setminus N_{u}|+|N_{%
+v}\setminus C|$ $d=\operatorname*{CLB}(G)$ $[v]_{\Pi_{d}}\subseteq U^{d}_{[v]_{\Pi_{d}}}$ $\displaystyle=|N_{u}\cup N_{v}|-|N_{u}\cap N_{v}|$ $C=[v]_{\Pi}\cup[w]_{\Pi}$ $2\varphi(\Pi)$ $\displaystyle|N_{v}\setminus U^{d}_{[v]_{\Pi_{d}}}|=|\{u\in N_{v}\mid\{[u]_{%
+\Pi_{d}},[v]_{\Pi_{d}}\}\notin E^{\prime}\}|\enspace,$ $|N_{u}\cap N_{v}|>|N_{v}|/2$ $t_{\operatorname*{CLB}}$ $\Pi\in P_{V}$ $x_{\{1,6\}}=x_{\{2,6\}}=x_{\{3,6\}}=\tfrac{3}{4}$ $|N_{v}|>4\varphi(\Pi)$ $U^{d}$ $U^{1}_{\{1\}}=\{0,1\}$ $V\setminus U^{d}_{C}$ $-|N_{w}\triangle N_{v}|$ $\varphi(\Pi)=0$ $I_{\{u,v\}}=0$ $U^{d}_{[v]_{\Pi_{d}}}\subseteq\{u\in V\mid|N_{u}\cap N_{v}|>|N_{v}|/2\}%
+\subseteq[v]_{\Pi_{d}}$ $\mathcal{O}(n^{2}+n\delta^{2})$ $\mathcal{O}(n\delta)$ $\operatorname*{CLB}(G)$ $\operatorname*{DMN}$ $|N_{u}\cap N_{v}|\leq|N_{v}|/2$ $\varphi(\Pi)\geq|(N_{u}\cap N_{v})\setminus[v]_{\Pi}|$ $\displaystyle|N_{v}\triangle[v]_{\Pi}|$ $\displaystyle=|N_{u}\triangle[u]_{\Pi}|+|N_{v}\triangle[v]_{\Pi}|\leq 2\varphi%
+(\Pi)\enspace.\qquad\square$ $|N_{v}|/2>2d$ $I_{\{u,v\}}=|N_{u}\cap N_{v}|$ $\tfrac{5}{4}$ $\mathcal{O}(\log_{2}(\delta))$ $\Pi:=\Pi\setminus\{[v]_{\Pi},[w]_{\Pi}\}\cup\{C\}$ $\varphi(\Pi)=1$ $4.1\cdot 10^{5}$ $\varphi(\Pi)\geq|N_{v}\triangle[v]_{\Pi}|>d^{\prime}$ $2.0\cdot 10^{5}$ $\operatorname*{CLB}$ $\Pi_{0}=\{V\}$ $\varphi(\Pi)\geq|N_{u}\triangle[u]_{\Pi}|\geq|N_{u}\setminus[u]_{\Pi}|\geq|(N_%
+{u}\cap N_{v})\setminus[u]_{\Pi}|$ $\displaystyle=|N_{4}\setminus[4]_{\Pi}|+|[4]_{\Pi}\setminus N_{4}|$ $|N_{u}\triangle N_{v}|\leq 2\varphi(\Pi)$ $|N_{u}\triangle N_{v}|>2\varphi(\Pi)$ $d>|N_{v}\triangle[v]_{\Pi}|$ $0\leq\operatorname*{CLB}(G)\leq\delta$ $\Pi_{d}\in P_{V}$ $|N_{w}\cap N_{v}|$ $[v]_{\Pi}=\{u\in V\mid|N_{u}\cap N_{v}|>|N_{v}|/2\}$ $1.6\cdot 10^{3}$ $|N_{u}\cap N_{v}|\leq|(N_{u}\cap N_{w})\setminus[u]_{\Pi}|+|(N_{u}\cap N_{w})%
+\setminus[v]_{\Pi}|\leq 2\varphi(\Pi)$ $[u]_{\Pi}\neq[v]_{\Pi}$ $\displaystyle\geq|N_{v}\setminus U^{d}_{[v]_{\Pi_{d}}}|+|[v]_{\Pi_{d}}%
+\setminus N_{v}|\enspace.$ $\displaystyle\quad+|(N_{v}\setminus N_{u})\cap C|+|(N_{v}\setminus N_{u})%
+\setminus C|$ $w\in\operatorname*{argmax}_{v\in V}|N_{v}\triangle[v]_{\Pi}|$ $\varphi(\Pi)=d^{\prime} $[v]_{\Pi}=[w]_{\Pi}$ $\min(\theta_{uw},\theta_{vw})$ $|N_{w}\cap N_{v}|-|N_{w}\triangle N_{v}|$ $U^{1}_{\{2\}}=\{0,2\}$ $|N_{v}\setminus U^{d}_{[v]_{\Pi_{d}}}|$ $(V,E^{+}\cup E^{-})$ $t_{\text{LP}}$ $U^{1}_{\{3,4,5\}}=\{3,4,5\}$ $made\_join$ $\left|[v]_{\Pi}\triangle N_{v}\right|$ $\varphi(\Pi)$ $C,C^{\prime}\in\Pi_{d}$ $f\in\{0,50,100,\dots,1000\}$ $\displaystyle\geq|N_{4}\setminus\{0,4,5,6\}|+|\{4\}\setminus N_{4}|=3$ $[u]_{\Pi}=[v]_{\Pi}$ $G_{d}=(V,E_{d})$ $\displaystyle\max_{v\in V}\;\;\Bigl{|}N_{v}\setminus U^{d}_{[v]_{\Pi_{d}}}%
+\Bigr{|}+\Bigl{|}[v]_{\Pi_{d}}\setminus N_{v}\Bigr{|}\quad\leq\quad d\enspace,$ $|N_{v}|>4d$ $\displaystyle=|N_{u}\setminus N_{v}|+|N_{v}\setminus N_{u}|$ $7.06\times 10^{54})$ $\mu=1149$ $1\text{\times}{10}^{-25}$ $\sigma=334$ $\bm{\epsilon}\sim\mathcal{N}(\bm{0},\mathbf{Q})$ ${\tilde{\mathcal{X}}}_{t}^{j}$ $\mathrm{DecSTER}$ $\mathcal{Z}^{j}_{t}$ $y_{i^{\prime}}\in\mathcal{Y}$ $\textbf{x}_{t,1}^{j},\dots,\textbf{x}_{t,\rho}^{j}$ $g(\mathbf{z}|\textbf{x},\mathbf{q})$ $\{\hat{\textbf{x}}_{1},\dots,\hat{\textbf{x}}_{\hat{n}_{\mathcal{G}}}\}$ $\textbf{x}_{t+1}=\mathbf{F}\textbf{x}_{t}+\bm{\epsilon}$ $\tilde{\rho}=100$ $(t,\mathbf{a}^{j}_{t},\mathcal{Z}^{j}_{t})$ $\hat{\mathcal{X}}_{t+1}^{j}$ $\displaystyle\psi_{\mathbf{z},\mathbf{q}}(\textbf{x})=g(\mathbf{z}|\textbf{x},%
+\mathbf{q})p_{d}(\textbf{x}|\mathbf{q})$ ${\hat{\mathcal{X}^{\prime}}}_{t+1}^{j}$ $\nu_{t}^{j}=\{(w_{1}^{j},\textbf{x}_{1}^{j}),\dots,(w_{\rho}^{j},\textbf{x}_{%
+\rho}^{j})\}$ $p_{d}=0.9$ $\hat{n}=\sum_{i}w_{i},\forall\textbf{x}_{i}\in E$ $\bar{\nu}_{t+1}^{j}$ $\int_{E}\nu(x)dx$ $\{w_{1},\dots,w_{\rho}\}$ $n_{l}\times n_{w}=16\times 16$ $\bm{\omega}\sim\mathcal{N}(\bm{0},\sigma^{2}\mathbf{I})$ ${w^{\prime}}_{i}$ $\mathbb{D}_{t}^{j}=\{(t^{\prime},\mathbf{a}^{\mathbf{q}_{j}^{\prime}}_{t^{%
+\prime}},\mathcal{Z}^{j^{\prime}}_{t^{\prime}})\}_{t^{\prime} $v_{x},v_{y}\in[-v_{\text{max}},v_{\text{max}}]$ $\nu_{t+1}^{j}$ $\tilde{n}\sim\text{Poisson}(\hat{n}_{\mathcal{G}})$ $\mathbf{a}_{t}^{j}$ $\lambda\in\{0.005,0.04,1,5\}$ $\psi_{\mathbf{z},\mathbf{q}}(\textbf{x})$ $\text{OSPA}(\mathcal{X},\mathcal{Y})$ $\mathbf{q}=\begin{bmatrix}q_{x},q_{y}\end{bmatrix}^{\text{T}}$ $\tilde{\mathcal{X}}=\{\tilde{\textbf{x}}_{1},\dots,\tilde{\textbf{x}}_{\tilde{%
+n}}\}$ ${\hat{\mathcal{X}}}_{t}^{j}$ $|\mathcal{X}|=m\leq|\mathcal{Y}|=n$ $p_{s}(\textbf{x})$ $\hat{\mathcal{X}}_{t}^{j}$ $\mathcal{Z}=\{\mathbf{z}_{1},\dots,\mathbf{z}_{m}\}$ $\tilde{\mathcal{X}}_{t}^{j}$ $\textbf{x}=\begin{bmatrix}{l_{x},l_{y},v_{x},v_{y}}\end{bmatrix}^{\text{T}}$ $\mathbf{H}=\begin{bmatrix}1&0&0&0\\
+0&1&0&0\end{bmatrix}$ $t,\mathbf{a}^{\mathbf{q}_{j}}_{t},\mathcal{Z}^{j}_{t}$ ${\nu}_{t}^{j}$ $\tilde{n}=\sum_{i=1}^{\tilde{\rho}}w_{i}$ $p_{d}(\textbf{x}|\mathbf{q})$ $\mathbb{D}_{t}^{j}$ $\mathbf{z}=h(\textbf{x})+\bm{\omega}$ $\mathcal{X}=\{\textbf{x}_{1},\dots,\textbf{x}_{|\mathcal{X}|}\}$ $\mathbf{z}=\mathbf{H}\textbf{x}+\bm{\omega}$ $\tilde{\mathcal{X}}_{R}$ $\tilde{n}\sim\text{Poisson}(\hat{n})$ $\mathcal{Z}_{t}^{j}$ $\bar{\mathcal{Z}}_{t}^{j}$ $\mathcal{Y}=\{\mathbf{y}_{1},\dots,\mathbf{y}_{|\mathcal{Y}|}\}$ $(l_{x},l_{y})\in[0,n_{w}]\times[0,n_{l}]$ $\{(w_{1},\textbf{x}_{1}),\dots,(w_{\rho},\textbf{x}_{\rho})\}$ $v_{\max}=0.1$ $\displaystyle\bar{\nu}_{t}(\textbf{x})=b(\textbf{x})+\int_{E}f(\textbf{x}|\bm{%
+\xi})p_{s}(\bm{\xi})\nu_{t-1}(\bm{\xi})d\bm{\xi}$ $\displaystyle\sum_{i=1}^{\rho}\bar{w}_{i}+\frac{\alpha}{1-\alpha}\sum_{i=1}^{%
+\rho}{w^{\prime}}_{i}-\frac{1}{1-\alpha}\sum_{i=1}^{\rho}{w^{\prime}}_{i}^{%
+\alpha}\bar{w}_{i}^{1-\alpha}$ $\mathcal{Y}_{t}^{j}$ $\nu=\{(w_{1},\textbf{x}_{1}),\dots,(w_{\rho},\textbf{x}_{\rho})\}$ $f(\mathbf{x}|\bm{\xi})$ $\hat{n}_{\mathcal{G}}$ $\lambda_{\mathbf{q}}$ $\displaystyle\eta_{\mathbf{z}}(\nu)=\kappa(\mathbf{z}|\mathbf{q})+\int_{E}\psi%
+_{\mathbf{z},\mathbf{q}}(\textbf{x})\nu(\textbf{x})d\textbf{x}$ $\mathbf{a}^{\mathbf{q}_{j}}_{t}$ $|\mathcal{A}|=340$ $\text{OSPA}(\mathcal{X},\mathcal{Y})=\big{(}\frac{1}{n}\min_{\pi\in\Pi_{n}}%
+\sum_{i=1}^{m}d_{c}(x_{i},y_{\pi(i)})^{p}+c^{p}(n-m)\big{)}^{\frac{1}{p}}$ $\hat{n}=\sum_{i}w_{i}$ $E\subseteq\mathcal{G}$ $\displaystyle\mathbf{a}^{j}_{t}=\operatorname*{arg\;min}_{\mathbf{a}}\mathbb{E%
+}_{\mathcal{Y}_{t}^{j}|\tilde{\mathcal{X}}_{t}^{j},\mathbf{a}}[\text{OSPA}(%
+\tilde{\mathcal{X}}_{t}^{j},\mathcal{Y}_{t}^{j})]$ $\displaystyle\nu_{t}(\textbf{x})=(1-p_{d}(\textbf{x}|\mathbf{q}))\bar{\nu}_{t}%
+(\textbf{x})+\sum_{\mathbf{z}\in\mathcal{Z}_{t}}\frac{\psi_{\mathbf{z},\mathbf%
+{q}}(\textbf{x})\bar{\nu}_{t}(\textbf{x})}{\eta_{\mathbf{z}}(\bar{\nu}_{t})}$ $p_{s}=1$ $d_{c}(x,y)=min(c,||x-y||)$ $w_{t,1}^{j},\dots,w_{t,\rho}^{j}$ $\sum_{i=1}^{\rho}w_{i}$ $h(\textbf{x})=\begin{bmatrix}l_{x},l_{y}\end{bmatrix}^{\text{T}}$ $\nu_{t}^{j}=\{(w_{t,1}^{j},\textbf{x}_{t,1}^{j}),\dots,(w_{t,\rho}^{j},\textbf%
+{x}_{t,\rho}^{j})\}$ ${\nu^{\prime}}_{t+1}^{j}$ $\nu_{t}^{j}$ $\kappa(z)$ $\mathbf{a}_{\mathbf{q}}$ $\mathbf{F}=\begin{bmatrix}1&0&\Delta T&0\\
+0&1&0&\Delta T\\
+0&0&1&0\\
+0&0&0&1\end{bmatrix}$ $\{\textbf{x}_{i}\}_{i=1}^{\tilde{\rho}}$ $\tilde{\mathcal{X}}_{t}^{j}\sim\bar{\nu}_{t+1}^{j}$ $\{1,\dots,J\}$ $\bm{\omega}\sim\mathcal{N}(\mathbf{0},\sigma_{h}^{2}\mathbf{I})$ $\hat{n}=\int_{E}\nu(\textbf{x})d\textbf{x}$ $\hat{\mathcal{X}}\cup\tilde{\mathcal{X}}_{R}$ $p\in\{0.05,0.25,0.50,0.75,1\}$ $\Delta T=1$ $\mathbf{Q}=\begin{bmatrix}0.03&0&0.05&0\\
+0&0.03&0&0.05\\
+0.05&0&0.1&0\\
+0&0.05&0&0.1\end{bmatrix}$ $b(\textbf{x})$ $T_{b}\leftarrow$ $L(D)=\min_{h\in\mathcal{H}}(L(D\mid h)+L(h))$ $Z_{b}\in\mathbb{R}^{b\times max(|T|)\times d}$ $P\left(q_{3}\mid q_{3}\right)=P\left(q_{5}\mid q_{3}\right)=0<\frac{1}{5}$ $i,q\in enumerate(Q_{n})$ ${ord}_{th}$ $clf$ $P\left(q_{3}\mid q_{4}\right)=\frac{3}{4}>\theta$ $\{a,X,b\}$ $X,Y,X,Y\cdots$ $q_{1}\ldots q_{x}q_{y}$ $Z_{n}\leftarrow$ $F=[Z||X]$ $q_{1},q_{2},\cdots,q_{m}$ $Z\leftarrow Z_{b}.\text{mean(axis=1,keepdim=False)}$ $\textit{result}\text{.append(}Z\text{)}$ $q_{1}q_{2}q_{3}$ $P\left(q_{3}\mid q_{3}\right)$ $\tau=1/|\operatorname{set}(S)|=1/5$ $P\left(q_{5}\mid q_{3}\right)$ $Z_{o}[\text{idx\_new}]\leftarrow Z_{n}$ $Z_{o}\leftarrow 0^{|Q|\times d}$ $q_{1},q_{2},\cdots,q_{n}$ $q_{2}q_{3}$ $P\left(q_{2}\mid q_{1}\right)=\frac{1}{2}<\theta$ $P\left(q_{4}\mid q_{3}\right)=\frac{2}{3}<\theta$ $P=\{p_{i}|i=1,2,\cdots,n\}$ $\begin{split}\mathbb{C}\left(S,M^{{ord}}\right)=2\times(\log{ord}+\log m+1)\\
++m(({ord}+1)\log|\operatorname{set}(S)|+2\log|S|)\\
+-\log\mathrm{P}\left(S\mid M^{{ord}}\right)\end{split}$ $x_{th}$ $Z_{o}\in\mathbb{R}^{b\times d}$ $P\left(q_{4}\mid q_{3}\right)=\frac{\operatorname{value}\left(n_{12}\right)}{%
+\operatorname{value}\left(n_{3}\right)}=\frac{2}{3}>\frac{1}{5}$ $S_{T}\leftarrow len(T\;\text{for}\;T\;\text{in}T_{b})$ ${max\_{ord}}=1$ $\mathrm{P}\left(S\mid M^{{ord}}\right)$ $\mathrm{P}\left(q_{x}\mid\mathrm{s}\right)=\frac{\operatorname{value}\left(n_{%
+c}\right)}{\operatorname{value}\left(n_{p}\right)}$ $Z\leftarrow Z_{b}.\text{max(axis=1,keepdim=False)}$ $\operatorname{set}(S)$ $\begin{split}\mathrm{P}\left(S\mid M^{{ord}}\right)=P\left(q_{1}\right)\times P%
+\left(q_{2}\mid q_{1}\right)\times P\left(q_{3}\mid q_{2}\right)\times P\left(%
+q_{4}\mid q_{3}\right)\\
+\times P\left(q_{3}\mid q_{4}\right)\times\cdots\times P\left(q_{5}\mid q_{2}%
+\right)\end{split}$ $Z_{o}\in\mathbb{R}^{|Q|\times d}$ $M^{ord}$ $\mathrm{P}\left(q_{y}\mid q_{1}\ldots q_{x}\right)\geq\theta$ $\begin{split}&P\left(X_{t}=x_{t}\mid X_{t-1}=x_{t-1},\ldots,X_{1}=x_{1}\right)%
+\\
+&=P\left(X_{t}=x_{t}\mid X_{t-1}=x_{t-1},\ldots,X_{t-m}=x_{t-m}\right)\\
+\end{split}$ $\{c,Y\}$ $q\not\in D$ $n_{8}=2$ $query\_batch$ $max\_ord+1$ $Z_{o}$ $max(|T|)$ $S=q_{1}q_{2}q_{3}q_{4}q_{3}q_{4}q_{2}q_{3}q_{4}q_{3}q_{1}q_{3}\\
+q_{4}q_{3}q_{2}q_{5}$ $Z_{b}\leftarrow$ $q_{1}q_{2}...q_{1}q_{4}$ $P\left(q_{1}\mid q_{3}\right)$ $max(|T|)\times d$ $\textit{batch\_idx}\leftarrow\textit{batch\_idx}+b$ $A,B,C,\cdots$ $Z\in\mathbb{R}^{|T|\times d}$ $max\_ord$ $Z_{o}[i]\leftarrow D[q]$ $\textit{batch\_idx}<|Q|$ $max\_ord+2$ $\textit{batch\_end}=min(\textit{batch\_idx}+b,|Q|)$ $\mathrm{P}\left(q_{x}\mid\mathrm{s}\right)<\tau$ $P\left(q_{4}\mid q_{3}\right)$ $\textit{batch\_idx}\leftarrow$ $i,q\in\textsf{enumerate}(Q)$ $\tau=1/|\operatorname{set}(S)|$ $query\_batch=Q[\textit{batch\_idx}:\textit{batch\_end}]$ $clf(F)$ $P\left(q_{2}\mid q_{3}\right)=P\left(q_{1}\mid q_{3}\right)=\frac{%
+\operatorname{value}\left(n_{10}\right)}{\text{ value }\left(n_{3}\right)}=%
+\frac{1}{6}<\frac{1}{5}$ $\left(1-P\left(q_{4}\mid q_{3}\right)\right)/4=\frac{1}{12}$ $q_{4}q_{3}$ $D[q]\leftarrow Z_{n}[i]$ $\{Y,X\}$ $P\left(q_{2}\mid q_{3}\right)$ $\mathrm{P}\left(q_{x}\mid\mathrm{s}\right)\geq\tau$ $15-n$ $\displaystyle P_{\text{ana}}^{i}$ $\displaystyle=\bm{F_{agg}}^{i}\;(\bm{F_{ml}}^{i}(C_{i},E_{i}),\;\bm{F_{res}}^{%
+i}(C_{i}))$ $P=CV^{2}$ $\text{MAPE}=1/n*\sum_{k=1}^{n}{|y_{k}-\hat{y_{k}}|/y_{k}}$ $1.1/0.8$ $\displaystyle=\bm{F_{agg}}^{i}\;(\textit{\#Hit, \#Miss, Energy per hit/miss})$ $\bm{F_{res}}^{i}\;(C_{i})$ $\hat{y_{k}}$ $P_{\text{ana}}^{i}$ $\displaystyle P_{\text{ana}}^{i}=\bm{F_{agg}}^{i}\;$ $\displaystyle=\textit{ICacheWay}*\textit{ICacheFetchBytes}$ $P_{\text{ml}}$ $\displaystyle=\bm{F_{ml}}^{i}(C_{i},E_{i})*\bm{F_{res}}^{i}\;(C_{i})$ $F_{res}=4,8,16$ $\bm{F_{res}^{i}}$ $P_{\text{PANDA}}^{i}=\bm{F_{ml}}^{i}(C_{i},E_{i})*\textit{ICacheWay}*\textit{ICacheFetchBytes}$ $/\bm{F_{res}}^{i}$ $P_{\text{ml}}=\bm{F_{ml}}\;(\{C,E\})$ $C=\{C_{i}\,|\,i\in[1,N]\}$ $F_{res}$ $40/28$ $f_{\text{ReserveStationNum}}$ $\bm{F_{res}}^{i}(C_{i})$ $\bm{F_{ml}}^{i}(C_{i},E_{i})$ $n\in[1,14]$ $\displaystyle\bm{F_{res}}^{i}(C_{i})$ $P_{\text{ml}}=\bm{F_{ml}}\;(\{C_{i}\,|\,i\in[1,N]\},\;\{E_{i}\,|\,i\in[1,N]\})$ $\displaystyle P_{\text{PANDA}}^{i}$ $P_{\text{ana}}=\sum_{i\in[1,N]}P_{\text{ana}}^{i}$ $E=\{E_{i}\,|\,i\in[1,N]\}$ $\bm{F_{res}}^{i}$ $40/28*(1.1/0.8)^{2}$ $\bm{F_{ml}^{i}}$ $\bm{F_{res}}^{i}\;(\textit{ICacheWay},\;\textit{ICacheFetchBytes})=\textit{%
+Energy per hit/miss}$ $\bm{F_{agg}}^{i}$ $\bm{F_{ml}}^{i}$ $\displaystyle(E_{i},\;\bm{F_{res}}^{i}(C_{i}))$ $P^{i}_{\text{PANDA}}=\bm{F_{agg}}^{i}\;(\bm{F_{ml}}^{i}(C_{i},E_{i}),\;\bm{F_{%
+res}}^{i}(C_{i}))$ $F_{res}=2$ $\displaystyle=\frac{\textit{\#Hit}*\textit{Energy per hit}+\textit{\#Miss}*%
+\textit{Energy per miss}}{\textit{Total benchmark execution time}}$ $CV^{2}$ $\bm{F_{ml}}$ $j+\frac{D}{2}\ \mathrm{mod}\ D\in S_{J}$ $\displaystyle=e^{\frac{\imath 2\pi\sum_{i,k\in\left[d\right]}2^{i+k}a_{i}b_{i}%
+}{D}}\left\lvert a\right\rangle_{h}=e^{\frac{\imath 2\pi ab}{D}}\left\lvert a%
+\right\rangle_{h}$ $p=\Pr\left(ab\equiv c(\ \mathrm{mod}\ D)\right)$ $r_{4}=c_{1}-r_{2}r_{3}$ $2^{d-(d-d_{1})}$ $\forall x\in S_{X},\forall C\in S_{C},\Pr(X=x)=p_{x}=\frac{1}{\left\lvert S_{X%
+}\right\lvert},\Pr(C=c)=p_{c}=\frac{1}{\left\lvert S_{C}\right\lvert}$ $\displaystyle\left\lvert a\right\rangle_{h}\left\lvert 0\right\rangle_{t}%
+\overset{\mathcal{BMUL}(b)_{(h,t)}}{\longrightarrow}\left\lvert a\right\rangle%
+_{h}\left\lvert ab\right\rangle_{t}\overset{\mathcal{BMUL}(-b^{-1})_{(t,h)}}{\longrightarrow}$ $-b\equiv D-b\ (\ \mathrm{mod}\ D)$ $S_{\mathbf{c}}=\left[D\right]^{3}\times\left[D\right]^{o}$ $h,t_{1},t_{2},g$ $ja_{2}$ $\rho_{s_{i}}=\left\lvert\psi_{s_{i}}\right\rangle\left\langle\psi_{s_{i}}\right\lvert$ $\displaystyle=\sum_{i=0}^{d-l-1}a_{i}^{(l)}2^{i}+2^{d-l}\left(\sum_{i=0}^{l-1}%
+a_{i+d-l}^{(l)}2^{l}+wa_{d-1-l}-k2^{l}\right)$ $\mathcal{CP}(i):\left\lvert a\right\rangle\left\lvert b\right\rangle%
+\rightarrow e^{\frac{\imath 2\pi 2^{i}}{D}ab}\left\lvert a\right\rangle\left%
+\lvert b\right\rangle$ $d_{1}+d_{2}\geq d$ $\displaystyle\rho=\sum_{s_{i}\in\left[D\right]^{o}}\frac{2}{D}\rho_{s_{i}}$ $H(X_{B}|\hat{M})=H(X_{B}|M)$ $s_{i}=2a+1$ $\mathcal{BROT}$ $tr_{P}(\cdot)$ $\displaystyle=\frac{4v+4\sum^{n}_{i=1}x_{i}y_{i}\ \mathrm{mod}\ D}{4}=\frac{4(%
+v+\sum^{n}_{i=1}x_{i}y_{i})-a2^{d}}{4}$ $\displaystyle\overset{\mathcal{SUM}(r_{4})_{g}}{\longrightarrow}\frac{1}{\sqrt%
+{D}}\sum_{j\in\left[D\right]}\omega^{jM_{i}}\left\lvert j\right\rangle_{h}%
+\left\lvert j+c_{1}\right\rangle_{t_{1}}$ $\frac{d^{2}+d-2}{2\cdot 2^{d}}=\frac{5}{8}$ $\displaystyle=\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{jM_{i}}\left%
+\lvert j\right\rangle_{h}\left\lvert j+c_{1}\right\rangle_{t_{1}}\left\lvert jp%
+_{i}+c_{2}\right\rangle_{t_{2}}$ $\left\lvert r_{3}\right\rangle_{e_{1}}\left\lvert r_{4}\right\rangle_{e_{2}}$ $p_{s_{i}}=\frac{2}{D}$ $\displaystyle=H(X_{B})-H(X_{B}|M)-H(u).$ $\displaystyle=v+\sum^{n}_{i=1}x_{i}y_{i}\ \mathrm{mod}\ D=u.$ $\displaystyle\overset{\mathcal{SUM}(-c_{1})_{t_{1}}\mathcal{SUM}(-c_{2})_{t_{2%
+}}}{\longrightarrow}$ $\left[2^{d-d_{3}+d_{1}}\right]$ $H(u|X_{B})=0$ $\rho\left\lvert\phi_{j+}\right\rangle=0\left\lvert\phi_{j+}\right\rangle,\rho%
+\left\lvert\phi_{j-}\right\rangle=\frac{2}{\left\lvert S_{J}\right\lvert}\left%
+\lvert\phi_{j+}\right\rangle.$ $\displaystyle\equiv a+\sum_{i=1}^{d-1}2^{i}wa_{i-1}(\ \mathrm{mod}\ 2^{d}).$ $4v_{i}\equiv\hat{M_{i}}-4x_{i}y_{i}(\ \mathrm{mod}\ D)$ $\rho=\frac{1}{\left\lvert S_{J}\right\lvert}\sum_{j\in S_{J}}\left(\left\lvert
+j%
+\right\rangle-\left\lvert j+\frac{D}{2}\right\rangle\right)\left\langle j%
+\right\lvert.$ $\mu(m_{X})=\Omega(1/poly(m_{X}))$ $j\in S_{J}$ $\displaystyle I_{B}=H(M:X_{B}|u)=H(X_{B}:M,u)-H(X_{B}:u).$ $D|(a-b)$ $\hat{M}_{i}\equiv M_{i}-2x_{i}\equiv 4v_{i}+4x_{i}y_{i}(\ \mathrm{mod}\ D)$ $\displaystyle\overset{\mathcal{BSUM}_{(t_{1},g)}\mathcal{BSUM}_{(t_{2},g)}}{\longrightarrow}$ $\mathcal{ROT}(b)$ $w_{1}^{-1}\in\left[2^{d-d_{3}}\right]^{o}$ $h(x,c),f(j,x,c)$ $\displaystyle\left\lvert a\right\rangle_{h}\left\lvert 0\right\rangle_{t}%
+\overset{\mathcal{SUM}(2^{0}b)^{h_{0}}_{t}}{\longrightarrow}\left\lvert a%
+\right\rangle_{h}\left\lvert 2^{0}a_{0}b\right\rangle_{t}\overset{\mathcal{SUM%
+}(2^{1}b)^{h_{1}}_{t}}{\longrightarrow}$ $\mathcal{U}_{h}$ $\displaystyle Output=\frac{\sum^{n}_{i=1}\left(M_{i}-2x_{i}\right)\ \mathrm{%
+mod}\ D}{4}$ $\left\lvert S_{J}\right\lvert=0$ $\mathcal{MUL}(b)_{h}:\left\lvert a\right\rangle_{h}\rightarrow\left\lvert ab%
+\right\rangle_{h}.$ $k_{1},k_{2},k_{3}\in\left[D\right]^{o}$ $\forall j\in S_{J}$ $\left\lvert\psi\right\rangle$ $\displaystyle=\frac{\sum^{n}_{i=1}\left(s_{i}+p_{i}q_{i}-2x_{i}\right)\ %
+\mathrm{mod}\ D}{4}$ $r_{2}=c_{1}k_{1}+c_{2}k_{2}+c_{4}k_{3}$ $\mathcal{BSUM}_{(h,t)}=\mathcal{QFT}^{\dagger}_{t}\mathcal{BROT}_{(h,t)}%
+\mathcal{QFT}_{t}$ $\mathcal{SUM}(b)$ $\sum_{s_{i}\in\left[D\right]^{o}}\omega^{(j^{\prime}-j)s_{i}}=\frac{D}{2}$ $\left\lvert\psi\right\rangle_{h}$ $\displaystyle=\frac{1}{D\left\lvert S_{C}\right\lvert}\sum_{b\in\left[L\right]%
+}\alpha_{bx}\left\lvert b\right\rangle\left\langle b\right\lvert_{Q}.$ $\displaystyle=\begin{pmatrix}\sum_{i=1}^{n}{a_{1i}b_{i1}}&\sum_{i=1}^{n}{a_{1i%
+}b_{i2}}&\cdots&\sum_{i=1}^{n}{a_{1i}b_{in}}\\
+\sum_{i=1}^{n}{a_{2i}b_{i1}}&\sum_{i=1}^{n}{a_{2i}b_{i2}}&\cdots&\vdots\\
+\vdots&\vdots&\ddots&\vdots\\
+\sum_{i=1}^{n}{a_{ki}b_{i1}}&\sum_{i=1}^{n}{a_{ki}b_{i2}}&\cdots&\sum_{i=1}^{n%
+}{a_{ki}b_{in}}\end{pmatrix}+\mathbf{V}$ $\left\lvert S_{J}\right\lvert=\frac{2^{d}}{2^{d-d_{l}}}=2^{d_{l}}$ $d_{l} $\displaystyle=\frac{2^{d-d_{l}-1}}{2^{d-1}}=\frac{1}{2^{d_{l}}};$ $u=\mathbf{x}\cdot\mathbf{y}+v\ \mathrm{mod}\ N=\sum_{i=1}^{n}{x_{i}y_{i}}+v\ %
+\mathrm{mod}\ N$ $\left(k_{1}-r_{3}^{-1}\right)b_{1}+k_{2}b_{2}+k_{3}b_{3}+r_{4}r_{3}^{-1}=0.$ $\displaystyle=-{N}^{n+1}{N}^{n-1}\frac{1}{N^{n-1}}\frac{1}{N^{n+1}}\log_{2}{%
+\frac{1}{N^{n}}}=nm.$ $\left[N\right]^{o}$ $\displaystyle\overset{\mathcal{MUL}(p_{i}^{-1})_{t_{2}}}{\longrightarrow}\frac%
+{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{jM_{i}}\left\lvert j\right%
+\rangle_{h}\left\lvert j\right\rangle_{t_{1}}\left\lvert j\right\rangle_{t_{2}}$ $\displaystyle=\log_{2}{N}-\left\lvert Im(g_{B})\right\lvert\frac{DN\log_{2}%
+\left(DN\right)-ND\log_{2}D}{DN\frac{D^{4}}{2}}$ $\left\lvert\psi_{s_{i}}\right\rangle=\frac{1}{\sqrt{\left\lvert S_{J}\right%
+\lvert}}\sum_{j\in S_{J}}\omega^{js_{i}}\left\lvert j\right\rangle.$ $\mathbf{y}=(y_{1},y_{2},\cdots,y_{n})$ $\mathcal{T}:\left\lvert a\right\rangle\left\lvert b\right\rangle\left\lvert c%
+\right\rangle\rightarrow\left\lvert a\right\rangle\left\lvert b\right\rangle%
+\left\lvert c\oplus a\cdot b\right\rangle$ $\left[D\right]^{o}$ $\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{js_{i}}\left%
+\lvert j\right\rangle_{t_{1}}\frac{1}{\sqrt{D}}\sum_{j^{\prime}\in\left[D%
+\right]}\omega^{j^{\prime}q_{i}}$ $\displaystyle\rho_{s_{i}}=\left\lvert\psi_{s_{i}}\right\rangle\left\langle\psi%
+_{s_{i}}\right\lvert=\frac{1}{\left\lvert S_{J}\right\lvert}\sum_{j^{\prime},j%
+\in S_{J}}\omega^{(j^{\prime}-j)s_{i}}\left\lvert j^{\prime}\right\rangle\left%
+\langle j\right\lvert$ $D=2^{d}$ $\left\langle f(j^{\prime},x,c)\lvert f(j,x,c)\right\rangle=\delta_{j^{\prime}j}$ $d_{r}=d$ $I_{B}=(n+1)m-nm-m=0$ $\displaystyle\overset{\mathcal{ROT}(s_{i})_{t_{1}}\mathcal{ROT}(q_{i})_{t_{2}}%
+}{\longrightarrow}\frac{1}{\sqrt{D}}\omega^{(j+c_{1})s_{i}}\omega^{(jp_{i}+c_{%
+2})q_{i}}$ $\mathbf{V}=\left(v_{ij}\right)_{k\times n},v_{ij}\in\left[N\right]$ $X,C$ $v\in\left[N\right]$ $\displaystyle\ \left\lvert S_{X}\right\lvert\frac{1}{\left\lvert S_{X}\right%
+\lvert}\left(\log_{2}\left(D\left\lvert S_{C}\right\lvert\right)-\frac{1}{D%
+\left\lvert S_{C}\right\lvert}\sum_{b\in\left[L\right]}\alpha_{bx}\log_{2}%
+\alpha_{bx}\right)$ $p_{\left\lvert S_{J}\right\lvert}=\left\{\begin{matrix}\frac{1}{\left\lvert S_%
+{J}\right\lvert},&\ \mathrm{if}\ \left\lvert S_{J}\right\lvert $I_{B}=O\left(\frac{d^{2}}{2^{d}}\right)$ $\displaystyle\left\lvert jp_{i}+c_{2}\right\rangle_{t_{2}}\left\lvert j+c_{1}%
+\right\rangle_{g}$ $\mathcal{CP}(i+k)_{(h_{i},t_{i})}$ $a_{d-l-1}=1$ $2^{d-(d_{3}-d_{1})}$ $\displaystyle\left\lvert g(j^{\prime},x,c)\right\rangle\left\langle g(j,x,c)%
+\right\lvert_{Q}$ $\left\lvert Im(g_{B})\right\lvert=D^{3}\cdot\frac{D}{2}=\frac{D^{4}}{2}$ $p(X_{B})=\frac{1}{N^{n+1}}$ $\displaystyle=\frac{\sum_{b\in\left[L\right]}\alpha_{bx}}{D\left\lvert S_{C}%
+\right\lvert}\log_{2}\left(D\left\lvert S_{C}\right\lvert\right)-\frac{1}{D%
+\left\lvert S_{C}\right\lvert}\sum_{b\in\left[L\right]}\alpha_{bx}\log_{2}%
+\alpha_{bx}$ $t_{1},t_{2},g,e$ $2^{d_{1}+d_{2}}w_{1}w_{2}=2^{d_{3}}\left(w_{3}+k2^{d-d_{3}}\right)$ $\displaystyle=H(M,u)-H(M,u|X_{B})-H(u)$ $S\left(\rho_{s_{i}}\right)=0$ $\mathcal{MUL},\mathcal{BSUM}$ $\displaystyle=m-(d+m)+d=0.$ $\displaystyle\sum_{s_{i}\in\left[D\right]^{o}}\omega^{(j^{\prime}-j)s_{i}}=%
+\sum_{a\in\left[\frac{D}{2}\right]}e^{\frac{\imath 2\pi}{D}\cdot(j^{\prime}-j)%
+(2a+1)}$ $\displaystyle+\mathbf{V}=\mathbf{A}\cdot\mathbf{B}+\mathbf{V}.$ $\displaystyle\overset{\mathcal{QFT}_{h}}{\longrightarrow}\frac{1}{\sqrt{D}}%
+\sum_{j\in\left[D\right]}\left\lvert j\right\rangle_{h}\left\lvert 0\right%
+\rangle_{t_{1}}\left\lvert 0\right\rangle_{t_{2}}\left\lvert 0\right\rangle_{g}$ $d_{3}=d$ $X_{B}=(\mathbf{y},v)$ $\displaystyle=\Pr\left(jk_{1}r_{3}+j^{\prime}k_{2}r_{3}+r_{4}=j\right)$ $a,b,c\in\left[D\right]$ $\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{js_{i}}\left%
+\lvert j\right\rangle_{t_{1}}\left\lvert(jk_{1}r_{3}+r_{4})\oplus j\right%
+\rangle_{g}.$ $2^{d_{1}+d_{2}}w_{1}w_{2}=(k+1)2^{d}\equiv c(\ \mathrm{mod}\ D)$ $w=\frac{b-1}{2}$ $u=\sum_{i=1}^{n}{x_{i}y_{i}}+v\ \mathrm{mod}\ N$ $\displaystyle a^{(l+1)}=\sum_{i=0}^{d-l-1}a_{i}^{(l)}2^{i}$ $\mathcal{H}:\left\lvert a\right\rangle\rightarrow\frac{\left\lvert 0\right%
+\rangle+(-1)^{a}\left\lvert 1\right\rangle}{\sqrt{2}}$ $1-k_{1}k_{3}=2^{d_{l}}w_{l}$ $\displaystyle\sum_{j\in\left[D\right]}\left\lvert j\right\rangle_{h}\left%
+\lvert j+c_{1}\right\rangle_{t_{1}}\left\lvert jp_{i}+c_{2}\right\rangle_{t_{2%
+}}\left\lvert jr_{1}+r_{2}\right\rangle_{g}$ $O(2\epsilon^{-2}+n^{2})$ $\displaystyle=\omega^{j^{\prime}-j}\sum_{a\in\left[\frac{D}{2}\right]}e^{\frac%
+{\imath 2\pi}{D}\cdot 2(j^{\prime}-j)a}=\omega^{j^{\prime}-j}\frac{1-e^{\imath
+2%
+\pi(j^{\prime}-j)}}{1-e^{\frac{\imath 2\pi}{D}\cdot 2(j^{\prime}-j)}}$ $h=\left(h_{d-1},h_{d-2},\cdots,h_{0}\right)$ $\alpha_{bx_{i}}=D$ $\rho^{Q}_{xc}=\frac{1}{D}\sum_{j\in\left[D\right]}\left\lvert g(j,x,c)\right%
+\rangle\left\langle g(j,x,c)\right\lvert_{Q}.$ $\displaystyle\left\lvert jp_{i}+c_{2}\right\rangle_{t_{2}}\left\lvert jr_{1}+r%
+_{2}\right\rangle_{g}$ $jk_{1}r_{3}+r_{4}=j$ $g^{-1}(b)$ $S\left(\rho\right)=-\frac{\left\lvert S_{J}\right\lvert}{2}\cdot\frac{2}{\left%
+\lvert S_{J}\right\lvert}\log_{2}\frac{2}{\left\lvert S_{J}\right\lvert}=\log_%
+{2}\left\lvert S_{J}\right\lvert-1.$ $j-j^{\prime}\equiv\mathrm{odd}(\ \mathrm{mod}\ D)$ $c_{1},c_{2},c_{4}\in\left[D\right]$ $\displaystyle=\Pr\left(2^{d-d_{1}}\lvert b\right)=\frac{2^{d-(d-d_{1})}}{2^{d}%
+}=\frac{1}{2^{d-d_{1}}}.$ $h(x,c)$ $\displaystyle S\left(\rho^{B}\right)=-\sum_{b\in\left[L\right]}\frac{\beta_{b}%
+}{D\left\lvert S_{X}\right\lvert\left\lvert S_{C}\right\lvert}\log_{2}\frac{%
+\beta_{b}}{D\left\lvert S_{X}\right\lvert\left\lvert S_{C}\right\lvert}$ $\displaystyle p_{2}=\Pr\left(ab\equiv c(\ \mathrm{mod}\ D)\lvert d>d_{3}\geq d%
+_{1}\right)=p_{2a}\cdot p_{2b}$ $\left\lvert\mathbf{x}\right\lvert$ $\displaystyle-\frac{\sum_{b\in\left[L\right]}\beta_{b}\log_{2}{\beta_{b}}-%
+\left\lvert S_{x_{i}}\right\lvert\sum_{b\in\left[L\right]}\alpha_{b{x_{i}}}%
+\log_{2}{\alpha_{b{x_{i}}}}}{D\left\lvert S_{x_{i}}\right\lvert\left\lvert S_{%
+\mathbf{c}}\right\lvert}$ $\mathcal{SWAP}_{(h,t)}=\mathcal{XOR}_{(h,t)}\mathcal{XOR}_{(t,h)}\mathcal{XOR}%
+_{(h,t)}:\left\lvert a\right\rangle_{h}\left\lvert b\right\rangle_{t}%
+\rightarrow\left\lvert b\right\rangle_{h}\left\lvert a\right\rangle_{t}$ $\beta_{b}=DN$ $b=b_{1}\parallel b_{2}\parallel b_{3}\parallel r_{3}\parallel r_{4}$ $\mathcal{MUL}(b)$ $H(X)=m_{X}$ $\displaystyle=\frac{2^{d_{1}}}{2^{d-d_{3}+d_{1}-1}}=\frac{1}{2^{d-d_{3}-1}}.$ $w_{1}w_{2}=w_{3}+k2^{d-d_{3}}$ $\displaystyle\overset{\mathcal{XOR}_{(h,t_{1})}\mathcal{XOR}_{(h,t_{2})}%
+\mathcal{XOR}_{(h,g)}}{\longrightarrow}$ $w_{3}=1$ $\displaystyle\ \ \ \ +2^{d-l}\left(\sum_{i=0}^{l-1}a_{i+d-l}^{(l)}2^{l}+wa_{d-%
+1-l}\ \mathrm{mod}\ 2^{l}\right),$ $\left\lvert 0\right\rangle_{t_{1}}\left\lvert 0\right\rangle_{t_{2}}$ $\frac{1}{2^{d-d_{l}}}$ $\hat{M}=(\hat{M}_{1},\cdots,\hat{M}_{n})$ $\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right\rangle_{t_{1}}%
+\left\lvert ja_{1}\right\rangle_{t_{2}}\left\lvert ja_{2}\right\rangle_{g}$ $\displaystyle Output=(Output_{ij})_{k\times n}.$ $\mathcal{P}(i):\left\lvert a\right\rangle\rightarrow e^{\frac{\imath 2\pi 2^{i%
+}}{D}a}\left\lvert a\right\rangle$ $Output_{ij}$ $\mathcal{SUM}^{h_{i}}$ $2^{d_{3}-d_{1}+1}\nmid b$ $\displaystyle\cdots=\sum_{i=0}^{d-l-1}a_{i}^{(0)}2^{i}=\sum_{i=0}^{d-l-1}a_{i}%
+2^{i}.$ $\left\lvert 0\right\rangle_{t}$ $\mathcal{U}_{\varepsilon}:\left\lvert j\right\rangle_{t}\left\lvert 0\right%
+\rangle_{e}\rightarrow\sqrt{\eta_{j}}\left\lvert j\right\rangle_{t}\left\lvert%
+\varepsilon(j)\right\rangle_{e}+\sqrt{1-\eta_{j}}\left\lvert V(j)\right\rangle%
+_{(t,e)},$ $\displaystyle(1-k_{1}r_{3})\left(j+\frac{D}{2}\right)\equiv(1-k_{1}r_{3})j%
+\equiv r_{4}(\ \mathrm{mod}\ D).$ $Output=\frac{\sum^{n}_{i=1}\left(M_{i}-2x_{i}\right)\ \mathrm{mod}\ D}{4}$ $2^{d_{3}-d_{1}}\lvert b$ $\displaystyle\left\lvert 0\right\rangle_{h}\left\lvert 0\right\rangle_{t_{1}}%
+\left\lvert 0\right\rangle_{t_{2}}\left\lvert 0\right\rangle_{g}$ $H(Z_{B}:X_{A})=0$ $\omega^{j(a_{1}b_{1}+a_{2}b_{2})}$ $c_{3}=c_{4}=0$ $\mathcal{ROT}(b_{2})$ $\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right\rangle_{P}\left%
+\lvert ja_{1}\right\rangle_{Q_{1}}\left\lvert ja_{2}\right\rangle_{Q_{2}}$ $\left\langle\mathbf{x}\lvert\mathbf{y}\right\rangle$ $i\in\left[d\right]$ $\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{jM_{i}}\left%
+\lvert j\right\rangle_{h}\left\lvert j+c_{1}\right\rangle_{t_{1}}\left\lvert jp%
+_{i}+c_{2}\right\rangle_{t_{2}}$ $r_{4}=2^{d_{r}}w_{r}$ $\displaystyle=\log_{2}\left(D\left\lvert S_{C}\right\lvert\right)-\frac{1}{D%
+\left\lvert S_{C}\right\lvert}\sum_{b\in\left[L\right]}\alpha_{bx}\log_{2}%
+\alpha_{bx},$ $\displaystyle\ \ \ \frac{\sum_{b\in\left[L\right]}\beta_{b}\log_{2}{\beta_{b}}%
+-\left\lvert S_{X}\right\lvert\sum_{b\in\left[L\right]}\alpha_{bx}\log_{2}{%
+\alpha_{bx}}}{D\left\lvert S_{X}\right\lvert\left\lvert S_{C}\right\lvert}.$ $\displaystyle=\frac{1}{D}\frac{(d-1)(d+2)}{2}=\frac{d^{2}+d-2}{2\cdot 2^{d}}=O%
+\left(\frac{d^{2}}{2^{d}}\right).$ $\left\lfloor a\right\rfloor,\left\lceil a\right\rceil$ $H(M,u|X_{B})=H(u|M,X_{B})+H(M|X_{B})=H(M,X_{B})-H(X_{B})$ $1-\frac{\left\lvert S_{J}\right\lvert}{D}$ $x_{i},y_{i}\in\left[N\right]$ $O(n\epsilon^{-2})$ $j\in\left[D\right]$ $d_{1}+d_{2}\neq d_{3}$ $p_{\left\lvert S_{J}\right\lvert}=\frac{2}{2^{d-1}}$ $r_{3}=r_{1}^{-1}$ $\displaystyle\sum_{i=0}^{d-1}a_{i}^{(l+1)}2^{i}=a^{(l)}+wa_{d-1-l}2^{d-l}\ %
+\mathrm{mod}\ D$ $\displaystyle\left\lvert j+c_{1}\right\rangle_{t_{1}}\left\lvert jp_{i}+c_{2}%
+\right\rangle_{t_{2}}\left\lvert jc_{3}+c_{4}\right\rangle_{g}.$ $\displaystyle Output=v+\sum^{n}_{i=1}x_{i}y_{i}-a2^{m}$ $\displaystyle\left\lvert j^{\prime}\right\rangle_{t_{2}}\left\lvert(jk_{1}r_{3%
+}+j^{\prime}k_{2}r_{3}+r_{4})\oplus j\right\rangle_{g}.$ $4v\in\left[D\right]$ $\displaystyle=\Pr\left(d_{1}+d_{2}\geq d_{3}\right)=\Pr\left(d_{2}\geq d-d_{1}\right)$ $H(Z:X)$ $2^{d_{3}-d_{1}}$ $1-k_{1}r_{3}=0$ $\displaystyle=2^{d-d_{3}}(2^{d_{1}}-l)-2,$ $\frac{2}{\left\lvert S_{J}\right\lvert}$ $\omega^{ja_{1}b_{1}}$ $p(\hat{M}|X_{B})=\frac{1}{N^{n-1}}$ $ab\equiv c(\ \mathrm{mod}\ D)$ $\displaystyle\frac{\sum_{b\in\left[L\right]}\beta_{b}\log_{2}{\beta_{b}}-\left%
+\lvert S_{X}\right\lvert\sum_{b\in\left[L\right]}\alpha_{bx}\log_{2}{\alpha_{%
+bx}}}{D\left\lvert S_{X}\right\lvert\left\lvert S_{C}\right\lvert},$ $\mathcal{SUM}(b\ \mathrm{mod}\ 2^{2})$ $0=D=2^{d}\times 1$ $0\leq i\leq d-l-1$ $2^{d_{1}+d_{2}}w_{1}w_{2}=(k+1)2^{d}$ $\displaystyle\mathcal{QFT}_{h}:\left\lvert a\right\rangle_{h}\rightarrow\frac{%
+1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{aj}\left\lvert j\right\rangle_{h},$ $\displaystyle=\sum_{i=0}^{d-l-1}a_{i}^{(l)}2^{i}+\sum_{i=d-l}^{d-1}a_{i}^{(l)}%
+2^{i}+wa_{d-1-l}2^{d-l}-k2^{d}$ $X_{A}=x_{i}$ $\displaystyle=\frac{2}{D\left\lvert S_{J}\right\lvert}\sum_{j\in S_{J}}\sum_{j%
+^{\prime}\in S_{J}}\sum_{s_{i}\in\left[D\right]^{o}}\omega^{(j^{\prime}-j)s_{i%
+}}\left\lvert j^{\prime}\right\rangle\left\langle j\right\lvert.$ $\mathcal{MUL}(r)$ $w_{2}=w_{1}^{-1}w_{3}+k2^{d-d_{3}}\in\left[2^{d-d_{3}+d_{1}}\right]^{o},$ $O\left(n2^{4m}\right)$ $\displaystyle I_{B}=H(Z_{A}:s_{i})\leq\sum_{\left\lvert S_{J}\right\lvert}%
+\frac{\left\lvert S_{J}\right\lvert}{D}p_{\left\lvert S_{J}\right\lvert}\log_{%
+2}\left\lvert S_{J}\right\lvert-1.$ $S_{x_{i}}=\left[N\right]$ $\displaystyle\left\lvert jp_{i}+c_{2}\right\rangle_{t_{2}}\left\lvert jr_{1}+r%
+_{2}\right\rangle_{g},$ $g:\left[D\right]\times S_{X}\times S_{C}\to\left[L\right]$ $\left[\begin{array}[]{c|c}\begin{matrix}1&0&0&0\\
+0&1&0&0\\
+0&0&j&1\\
+0&0&k_{3}&0\\
+k_{1}-r_{3}^{-1}&k_{2}&0&k_{3}\end{matrix}&\begin{matrix}b_{1}-j\\
+b_{2}-jp_{i}\\
+b_{3}\\
+r_{3}^{-1}-k_{1}-p_{i}k_{2}\\
+-r_{4}r_{3}^{-1}\end{matrix}\end{array}\right].$ $\displaystyle\ \ \frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{jh(x,c)}%
+\left\lvert f(j,x,c)\right\rangle_{P}\left\lvert g(j,x,c)\right\rangle_{Q},$ $O\left(d^{2}\right)$ $M_{i}=s_{i}+p_{i}q_{i}$ $2^{d_{1}+d_{2}}w_{1}w_{2}=2^{d_{3}}w_{3}+k2^{d}$ $\displaystyle=\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right%
+\rangle_{h}\left\lvert jk_{1}+c_{1}k_{1}\right\rangle_{t_{1}}\left\lvert jp_{i%
+}k_{2}+c_{2}k_{2}\right\rangle_{t_{2}}$ $\left\lvert j\right\rangle_{g}\to\left\lvert 0\right\rangle_{g}$ $\left\lvert\mathbf{y}\right\lvert=\Theta\left(2^{m}\right)$ $I_{B}<\frac{d^{2}+d-2}{2\cdot 2^{d}}=O\left(\frac{d^{2}}{2^{d}}\right)$ $\displaystyle=\frac{1}{D}\sum_{j\in\left[D\right]}\left\lvert g(j,x,c)\right%
+\rangle\left\langle g(j,x,c)\right\lvert_{Q}.$ $\mathcal{QFT}$ $v_{i},c_{1},c_{2},c_{3},c_{4},k_{1},k_{2},k_{3}$ $\left[D\right]$ $\displaystyle p_{1}=\Pr\left(ab\equiv c(\ \mathrm{mod}\ D)\lvert d=d_{3}\right)$ $1-\frac{1}{D}$ $d>d_{3}\geq d_{1}$ $b\in Im(g_{B})$ $\mathcal{BMUL}(b)_{(h,t)}$ $\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right%
+\rangle_{h}\left\lvert jk_{1}+c_{1}k_{1}\right\rangle_{t_{1}}\left\lvert jp_{i%
+}k_{2}+c_{2}k_{2}\right\rangle_{t_{2}}$ $4\sum_{i=1}^{n}{v_{i}}\equiv 4v(\ \mathrm{mod}\ D)$ $\displaystyle\left\lvert jp_{i}+c_{2}\right\rangle_{t_{2}}\left\lvert 0\right%
+\rangle_{g},$ $\displaystyle\overset{\mathcal{MUL}(k_{1}^{-1})_{t_{1}}\mathcal{MUL}(k_{2}^{-1%
+})_{t_{2}}}{\longrightarrow}\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left%
+\lvert j\right\rangle_{h}\left\lvert j+c_{1}\right\rangle_{t_{1}}$ $\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right\rangle_{t_{1}},%
+\left\lvert 0\right\rangle_{t_{2}},\left\lvert 0\right\rangle_{g}$ $\displaystyle\mathbf{x}_{i}=\left(a_{i1},a_{i2},\cdots,a_{in}\right),$ $q_{i}=2y_{i}+1$ $\left\{b\lvert g(a)=b,a\in A\right\}$ $\displaystyle\left\lvert jr_{1}+r_{2}\right\rangle_{g}$ $\displaystyle\left\lvert 0\right\rangle_{t_{1}}\left\lvert 0\right\rangle_{t_{%
+2}},$ $\displaystyle ja_{2}\to jk_{1}+jk_{2}a_{1}+jk_{3}a_{2}=j(k_{1}+k_{2}a_{1}+k_{3%
+}a_{2}),$ $X\in S_{X}$ $\displaystyle=\frac{4\sum^{n}_{i=1}v_{i}+4\sum^{n}_{i=1}x_{i}y_{i}\ \mathrm{%
+mod}\ D}{4}$ $p=\frac{1}{2^{d-d_{1}}}$ $\displaystyle Output_{ij}=\mathbf{x}_{i}\cdot\mathbf{y}_{j}+v_{ij}.$ $\displaystyle\left[\begin{array}[]{c|c}\begin{matrix}1&0&0&0\\
+0&1&0&0\\
+0&0&1&0\\
+0&0&0&1\\
+0&0&0&0\end{matrix}&\begin{matrix}b_{1}-j\\
+b_{2}-jp_{i}\\
+k_{3}^{-1}\left(r_{3}^{-1}-k_{1}-p_{i}k_{2}\right)\\
+b_{3}-jk_{3}^{-1}\left(r_{3}^{-1}-k_{1}-p_{i}k_{2}\right)\\
+\left(k_{1}-r_{3}^{-1}\right)b_{1}+k_{2}b_{2}+k_{3}b_{3}+r_{4}r_{3}^{-1}\end{%
+matrix}\end{array}\right].$ $1-k_{1}r_{3}$ $\displaystyle\frac{1}{D}\sum_{j,j^{\prime}\in\left[D\right]}\left\lvert j%
+\right\rangle_{t_{1}}\left\lvert j^{\prime}\right\rangle_{t_{2}}\left\lvert 0%
+\right\rangle_{g}\rightarrow$ $\mathbf{B}=\left(b_{ij}\right)_{k\times n}$ $(h_{d-1},h_{d-2},\cdots,h_{d-l})$ $\ \mathrm{mod}\ N$ $\left\lvert\sum_{i\in\left[2\right]}2^{i}a_{i+d-2}+b\ \mathrm{mod}\ 2^{2}\right\rangle$ $s_{i}=4v_{i}-2y_{i}-1\ \mathrm{mod}\ D$ $\forall b_{1},b_{2},b_{3}\in\left[D\right],r_{3}\in\left[D\right]^{o}$ $\displaystyle\Pr\left((jk_{1}r_{3}+j^{\prime}k_{2}r_{3}+r_{4})\oplus j=0\right)$ $I_{B}=0$ $\frac{2}{D}$ $\displaystyle\mathbf{A}=\left(a_{ij}\right)_{k\times n}=\begin{pmatrix}a_{11}&%
+a_{12}&\cdots&a_{1n}\\
+a_{21}&a_{22}&\cdots&\vdots\\
+\vdots&\vdots&\ddots&\vdots\\
+a_{k1}&a_{k2}&\cdots&a_{kn}\end{pmatrix}$ $a_{i}^{(l+1)}=a_{i}^{(l)}$ $M_{i}\equiv p_{i}q_{i}+s_{i}\equiv 4x_{i}y_{i}+4v_{i}(\ \mathrm{mod}\ D)$ $H(X_{B}:u)=H(u)$ $O(kn\cdot nm^{2})=O(kn^{2}m^{2})$ $\displaystyle=\frac{2^{d-(d_{3}-d_{1})}2^{-1}}{2^{d}}=\frac{1}{2^{d_{3}-d_{1}+%
+1}},$ $\omega^{c_{1}s_{i}+c_{2}q_{i}}$ $\left\lvert 0\right\rangle_{g}$ $\displaystyle=\frac{\sum^{n}_{i=1}\left(4v_{i}+4x_{i}y_{i}\right)\ \mathrm{mod%
+}\ D}{4}$ $\displaystyle\frac{1}{D}\sum_{j,j^{\prime}\in\left[D\right]}\omega^{js_{i}+j^{%
+\prime}q_{i}}\left\lvert j\right\rangle_{t_{1}}\left\lvert j^{\prime}\right%
+\rangle_{t_{2}}\left\lvert jk_{1}+j^{\prime}k_{2}\right\rangle_{g}\rightarrow$ $2^{d-d_{1}}$ $O\left(4^{m}n\log^{2}n\right)$ $\mathcal{P}(i+k)_{h_{i}}$ $v=u-\mathbf{x}\cdot\mathbf{y}$ $\alpha_{bx}=\left\lvert g_{x}^{-1}(b)\right\lvert=\left\lvert\left\{(j,c)%
+\lvert g(j,x,c)=b\right\}\right\lvert$ $H(Z:X)=0$ $\frac{\left\lvert S_{J}\right\lvert}{2}$ $a^{(0)}=a$ $4(v+\sum^{n}_{i=1}x_{i}y_{i})-a2^{d}\in\left[D\right]$ $\left\lvert a_{d-1}\right\rangle\left\lvert a_{d-2}\right\rangle$ $\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right\rangle_{t_{1}},%
+\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right\rangle_{t_{2}}$ $\displaystyle\overset{\mathcal{XOR}_{(t_{1},g)}}{\longrightarrow}\frac{1}{%
+\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{jM_{i}}\left\lvert j\right\rangle_{h%
+}\left\lvert j+c_{1}\right\rangle_{t_{1}}$ $O(kn\cdot nm)=O(kmn^{2})$ $\mathcal{CNOT}:\left\lvert a\right\rangle\left\lvert b\right\rangle\rightarrow%
+\left\lvert a\right\rangle\left\lvert b\oplus a\right\rangle$ $\displaystyle\rho^{Q}_{x}=\sum_{c\in S_{C}}\frac{1}{\left\lvert S_{C}\right%
+\lvert}\rho^{Q}_{xc}$ $j^{\prime}-j\equiv\frac{D}{2}(\ \mathrm{mod}\ D)$ $\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{jM_{i}}\left%
+\lvert j\right\rangle_{h}\left\lvert j\right\rangle_{t_{1}}\left\lvert jp_{i}%
+\right\rangle_{t_{2}}$ $\mathcal{SUM}(b)_{h}:\left\lvert a\right\rangle_{h}\rightarrow\left\lvert a+b%
+\right\rangle_{h}.$ $a^{(l+1)}=a^{(l)}+wa_{d-1-l}2^{d-l}\ \mathrm{mod}\ D$ $e^{\frac{\imath 2\pi}{D}}=\cos(\frac{2\pi}{D})+\imath\sin(\frac{2\pi}{D})$ $\left\lvert j\right\rangle_{t}$ $H(u)=m$ $H(X_{B})=(n+1)m$ $c_{3}\in\left[D\right]^{o}$ $p(X_{B}|\hat{M})=\frac{1}{N^{n}}$ $l2^{d-d_{3}}+1\leq w_{1}^{-1}w_{3}\leq(l+1)2^{d-d_{3}}-1,$ $1\leq i\leq k,1\leq j\leq n$ $\ \mathrm{mod}\ 2^{l}$ $\displaystyle Output=\begin{pmatrix}\mathbf{x}_{1}\cdot\mathbf{y}_{1}+v_{11}&%
+\cdots&\mathbf{x}_{1}\cdot\mathbf{y}_{n}+v_{1n}\\
+\mathbf{x}_{2}\cdot\mathbf{y}_{1}+v_{21}&\cdots&\vdots\\
+\vdots&\ddots&\vdots\\
+\mathbf{x}_{k}\cdot\mathbf{y}_{1}+v_{k1}&\cdots&\mathbf{x}_{k}\cdot\mathbf{y}_%
+{n}+v_{kn}\end{pmatrix}$ $1-k_{2}r_{3}\equiv 0(\ \mathrm{mod}\ D)$ $\displaystyle H(Z_{B}:X_{A})\leq\log_{2}{\left\lvert S_{x_{i}}\right\lvert}$ $Q=(t_{1},t_{2},g,e_{1},e_{2})$ $a^{(d)}=a^{(d-1)}+wa_{0}2^{0}\ \mathrm{mod}\ D=ab$ $g_{B}(j,x_{i},\mathbf{c})=j+c_{1}\parallel jp_{i}+c_{2}\parallel jc_{3}+c_{4}%
+\parallel r_{1}^{-1}\parallel c_{1}-r_{2}r_{3},$ $\displaystyle=-\sum_{X_{B}\in\left[N\right]^{n+1}}\sum_{v_{1},v_{2},\cdots,v_{%
+n-1}\in\left[N\right]}p(\hat{M}|X_{B})p(X_{B})\log_{2}{p(X_{B}|\hat{M})}$ $b\in\left[D\right]^{o}$ $\mathcal{ROT}(b_{1})$ $\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{jM_{i}}\left%
+\lvert j\right\rangle_{h}\left\lvert j+c_{1}\right\rangle_{t_{1}}$ $2^{d_{1}}$ $\displaystyle I_{B}=H(M:X_{B}|u)=H(X_{B}:M,u)-H(X_{B}:u)$ $\displaystyle H\left(Z_{A}:s_{i}|\rm{Alice\ passes\ the\ test}\right)$ $\omega^{bc}$ $O\left(2^{4m}\right)$ $\mathcal{MUL}$ $\displaystyle\leq 2^{d-d_{3}+d_{1}}-1-w_{1}^{-1}w_{3}\leq 2^{d-d_{3}+d_{1}}-1-%
+l2^{d-d_{3}}-1$ $w_{2}\equiv w_{1}^{-1}w_{3}(\ \mathrm{mod}\ 2^{d-d_{3}})$ $\mathcal{ROT}_{h}=\prod_{i,k\in\left[d\right]}\mathcal{P}(i+k)^{b_{i}}_{h_{i}}$ $v_{n}=\frac{4v-4\sum_{i=1}^{n-1}{v_{i}}\ \mathrm{mod}\ D}{4}$ $d_{3} $\displaystyle p_{2b}=\Pr\left(w_{2}=w_{1}^{-1}w_{3}+k2^{d-d_{3}}\lvert d_{2}=d%
+_{3}-d_{1}\right)$ $\displaystyle S\left(\rho^{Q}_{x}\right)=-\sum_{b\in\left[L\right]}\frac{%
+\alpha_{bx}}{D\left\lvert S_{C}\right\lvert}\log_{2}\frac{\alpha_{bx}}{D\left%
+\lvert S_{C}\right\lvert}$ $\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right%
+\rangle_{h}\left\lvert j\right\rangle_{t_{1}}\left\lvert j\right\rangle_{t_{2}%
+}\left\lvert j\right\rangle_{g}$ $t=\left(t_{d-1},t_{d-2},\cdots,t_{0}\right)$ $\left\lvert A\right\lvert$ $\displaystyle\left\lvert jp_{i}+c_{2}\right\rangle_{t_{2}}\left\lvert j+r_{2}r%
+_{3}\right\rangle_{g}$ $2^{d_{3}-d_{1}+1}$ $d=1+2=3$ $H(Z:X|F)$ $\sum_{b\in\left[L\right]}\alpha_{bx}=\left\lvert\left[D\right]\times S_{C}%
+\right\lvert=D\left\lvert S_{C}\right\lvert$ $I_{A}=0$ $\displaystyle=\frac{1}{D}\sum_{j^{\prime},j\in\left[D\right]}\omega^{(j^{%
+\prime}-j)h(x,c)}\left\lvert f(j^{\prime},x,c)\right\rangle\left\langle f(j,x,%
+c)\right\lvert_{A}$ $I=H(Z:X)$ $p_{\left\lvert S_{J}\right\lvert}$ $\epsilon=\Theta(2^{-2m})$ $\displaystyle=\frac{4(v+\sum^{n}_{i=1}x_{i}y_{i}-a2^{m})}{4}=v+\sum^{n}_{i=1}x%
+_{i}y_{i}-a2^{m},$ $\mathcal{BROT}_{(h,t)}:\left\lvert a\right\rangle_{h}\left\lvert b\right%
+\rangle_{t}\rightarrow\omega^{ab}\left\lvert a\right\rangle_{h}\left\lvert b%
+\right\rangle_{t}$ $l=0,1,\cdots,d-1$ $j=2^{d_{j}}w_{j}$ $f:\left[D\right]\times S_{X}\times S_{C}\to\left[2^{l_{1}}\right]$ $\mu(m_{X}):\mathbb{N}\to[0,1]$ $\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right%
+\rangle_{h}\left\lvert j\right\rangle_{t_{1}}\left\lvert jp_{i}\right\rangle_{%
+t_{2}}\left\lvert jc_{3}\right\rangle_{g}$ $\displaystyle=\sum_{d_{l}=1}^{d-1}\frac{2^{d_{l}}}{D}\cdot\frac{1}{2^{d_{l}}}%
+\left(\log_{2}2^{d_{l}}-1\right)+\frac{2^{d}}{D}\cdot\frac{2}{2^{d}}\left(\log%
+_{2}2^{d}-1\right)$ $\mathcal{SUM}(b\ \mathrm{mod}\ 2^{l})$ $\mathcal{XOR}$ $\alpha_{b{x_{i}}}=\beta_{b}=0$ $w_{1}^{-1}w_{3}$ $\displaystyle=\frac{1}{D}\sum_{d_{l}=1}^{d-1}(d_{l}-1)+\frac{2}{D}(d-1)=\frac{%
+1}{D}\frac{(d-1)(d-2)+4(d-1)}{2}$ $\forall y_{i},v\in\left[N\right]$ $Output_{ij}=\mathbf{x}_{i}\cdot\mathbf{y}_{j}+v_{ij}$ $\mathcal{BSUM}$ $\displaystyle\overset{\mathcal{MUL}(k_{1})_{t_{1}}\mathcal{MUL}(k_{2})_{t_{2}}%
+\mathcal{MUL}(k_{3})_{g}}{\longrightarrow}$ $\mathcal{SUM}(b)=\mathcal{QFT}^{\dagger}\mathcal{ROT}(b)\mathcal{QFT}$ $\displaystyle\left\lvert g(j^{\prime},x,c)\right\rangle\left\langle g(j,x,c)%
+\right\lvert_{Q}.$ $\displaystyle=\sum_{i=0}^{d-1}a_{i}^{(l)}2^{i}+wa_{d-1-l}2^{d-l}\ \mathrm{mod}\ D$ $a_{ij},b_{ij}\in\left[N\right]$ $D|(1-k_{1}r_{3})\frac{D}{2}$ $\displaystyle=\frac{1}{D\left\lvert S_{C}\right\lvert}\sum_{b\in\left[L\right]%
+}\sum_{g(j,x,c)=b}\left\lvert b\right\rangle\left\langle b\right\lvert_{Q}$ $k_{1}r_{3}$ $\displaystyle\left\lvert jc_{3}k_{3}+c_{4}k_{3}\right\rangle_{g}$ $\displaystyle\mathcal{QFT}^{\dagger}_{h}:\left\lvert a\right\rangle_{h}%
+\rightarrow\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{-aj}\left\lvert j%
+\right\rangle_{h}.$ $\displaystyle\sum_{s_{i}\in\left[D\right]^{o}}\omega^{(j^{\prime}-j)s_{i}}=%
+\sum_{s_{i}\in\left[D\right]^{o}}e^{\imath\pi s_{i}}=\sum_{s_{i}\in\left[D%
+\right]^{o}}(-1)=-\frac{D}{2};$ $w_{1}\ \mathrm{mod}\ 2^{d-d_{3}}$ $\displaystyle I_{B}\leq\left(1-\frac{1}{2^{d-1}}\right)O\left(\frac{d^{2}}{2^{%
+d}}\right)+\frac{1}{2^{d-1}}d\leq O\left(\frac{d^{2}}{2^{d}}\right).$ $\displaystyle\rho^{(P,Q)}_{xc}=\left\lvert\psi(x,c)\right\rangle\left\langle%
+\psi(x,c)\right\lvert_{(P,Q)}$ $\displaystyle=H(M)-H(M,X_{B})+H(X_{B})-H(u)$ $d_{l}=d$ $b^{-1}\ \mathrm{mod}\ D$ $-l\leq k\leq 2^{d_{1}}-l-1$ $a^{(l)}=\sum_{i=0}^{d-1}a_{i}^{(l)}2^{i}$ $\displaystyle=\frac{1}{\left\lvert S_{X}\right\lvert}\sum_{x\in S_{X}}\frac{1}%
+{\left\lvert S_{C}\right\lvert}\sum_{c\in S_{C}}\frac{1}{D}\sum_{j\in\left[D%
+\right]}\left\lvert g(j,x,c)\right\rangle\left\langle g(j,x,c)\right\lvert_{Q}$ $\displaystyle\left\lvert j\left(k_{1}+p_{i}k_{2}+c_{3}k_{3}\right)+\left(c_{1}%
+k_{1}+c_{2}k_{2}+c_{4}k_{3}\right)\right\rangle_{g}$ $p_{c}=\frac{1}{\left\lvert S_{C}\right\lvert}$ $d_{3}\geq d_{1}$ $\displaystyle p_{2a}=\Pr\left(d_{2}=d_{3}-d_{1}\right)$ $\displaystyle\cdots\mathcal{SUM}(w\ \mathrm{mod}\ 2^{2})^{h_{d-3}}\mathcal{SUM%
+}(w\ \mathrm{mod}\ 2^{1})^{h_{d-2}},$ $\displaystyle\rho^{Q}=\sum_{x\in S_{X}}\frac{1}{\left\lvert S_{X}\right\lvert}%
+\rho^{Q}_{x}$ $d_{2}=d_{3}-d_{1}$ $M=(M_{1},M_{2},\cdots,M_{n})$ $b=\sum_{k\in\left[d\right]}b_{k}2^{k}$ $\displaystyle\prod_{i,k\in\left[d\right]}\mathcal{P}(i+k)^{b_{i}}_{h_{i}}\left%
+\lvert a\right\rangle_{h}=\prod_{i,k\in\left[d\right]}e^{\frac{\imath 2\pi 2^{%
+i+k}a_{i}b_{i}}{D}}\left\lvert a\right\rangle_{h}$ $\displaystyle:X)\leq\log_{2}{\left\lvert S_{X}\right\lvert}-$ $\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right%
+\rangle_{h}\left\lvert j+c_{1}\right\rangle_{t_{1}}\left\lvert jp_{i}+c_{2}%
+\right\rangle_{t_{2}}\left\lvert jc_{3}+c_{4}\right\rangle_{g}$ $m,N=2^{m}$ $\mathcal{SUM}(c)$ $\displaystyle=\begin{pmatrix}\mathbf{x}_{1}\cdot\mathbf{y}_{1}&\mathbf{x}_{1}%
+\cdot\mathbf{y}_{2}&\cdots&\mathbf{x}_{1}\cdot\mathbf{y}_{n}\\
+\mathbf{x}_{2}\cdot\mathbf{y}_{1}&\mathbf{x}_{2}\cdot\mathbf{y}_{2}&\cdots&%
+\vdots\\
+\vdots&\vdots&\ddots&\vdots\\
+\mathbf{x}_{k}\cdot\mathbf{y}_{1}&\mathbf{x}_{k}\cdot\mathbf{y}_{2}&\cdots&%
+\mathbf{x}_{k}\cdot\mathbf{y}_{n}\end{pmatrix}+\mathbf{V}$ $\left\lvert ja_{1}+c\right\rangle_{Q_{1}}$ $\displaystyle H(X_{B}|\hat{M})=-\sum_{X_{B}\in\left[N\right]^{n+1}}\sum_{\hat{%
+M}\in 4\left[N\right]^{n}}p(X_{B},\hat{M})\log_{2}{p(X_{B}|\hat{M})}$ $H(M,u)=H(M)$ $C\in S_{C}$ $\displaystyle\cdots\overset{\mathcal{SUM}(2^{d-1}b)^{h_{d-1}}_{t}}{%
+\longrightarrow}=\left\lvert a\right\rangle_{h}\left\lvert ab\right\rangle_{t},$ $\mathcal{QFT}^{\dagger}$ $a\ \mathrm{mod}\ D$ $\displaystyle=\begin{pmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\
+a_{21}&a_{22}&\cdots&\vdots\\
+\vdots&\vdots&\ddots&\vdots\\
+a_{k1}&a_{k2}&\cdots&a_{kn}\end{pmatrix}\cdot\begin{pmatrix}b_{11}&b_{12}&%
+\cdots&b_{1n}\\
+b_{21}&b_{22}&\cdots&\vdots\\
+\vdots&\vdots&\ddots&\vdots\\
+b_{k1}&b_{k2}&\cdots&b_{kn}\end{pmatrix}$ $I_{A}=H(Z_{B}:X_{A})=0$ $\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right%
+\rangle_{t_{1}}\left\lvert 0\right\rangle_{t_{2}}\left\lvert 0\right\rangle_{g%
+}\rightarrow\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right%
+\rangle_{t_{1}}\left\lvert 0\right\rangle_{t_{2}}\left\lvert jk_{1}\right%
+\rangle_{g},$ $\left\{a\lvert g(a)=b,a\in A\right\}$ $I_{B}=H(Z_{A}:s_{i})=O\left(\frac{d^{2}}{2^{d}}\right)$ $\mathcal{ROT}(b)_{h}:\left\lvert a\right\rangle_{h}\rightarrow\omega^{ab}\left%
+\lvert a\right\rangle_{h}.$ $\mathcal{ROT}_{h}$ $j(1-k_{2}r_{3})\equiv r_{4}(\ \mathrm{mod}\ D)$ $r_{1}=k_{1}+p_{i}k_{2}+c_{3}k_{3}$ $\left\{\begin{matrix}c_{1}=b_{1}-j\\
+c_{2}=b_{2}-jp_{i}\\
+jc_{3}+c_{4}=b_{3}\\
+k_{3}c_{3}=r_{3}^{-1}-k_{1}-p_{i}k_{2}\\
+\left(k_{1}-r_{3}^{-1}\right)c_{1}+k_{2}c_{2}+k_{3}c_{4}=-r_{4}r_{3}^{-1}\end{%
+matrix}\right..$ $\displaystyle\left\lvert 0\right\rangle_{h}\left\lvert ab\right\rangle_{t}%
+\overset{\mathcal{SWAP}_{(h,t)}}{\longrightarrow}\left\lvert ab\right\rangle_{%
+h}\left\lvert 0\right\rangle_{t},$ $\mathbf{y}\in\left[N\right]^{n}$ $\displaystyle p_{\left\lvert S_{J}\right\lvert}=\Pr\left(2^{d_{l}}\lvert(1-k_{%
+1}r_{3}),\ 2^{d_{l}+1}\nmid(1-k_{1}r_{3})\right)$ $a^{(l+1)}=\sum_{i=0}^{d-1}a_{i}^{(l+1)}2^{i}$ $a=2^{d_{1}}w_{1},b=2^{d_{2}}w_{2},c=2^{d_{3}}w_{3}$ $d_{r} $S_{J}=\{j|j(1-k_{1}r_{3})=r_{4},j\in\left[D\right]\}$ $X_{B}=(y_{i},v_{i})$ $\displaystyle\overset{\mathcal{MUL}(r_{3})_{g}}{\longrightarrow}\frac{1}{\sqrt%
+{D}}\sum_{j\in\left[D\right]}\omega^{jM_{i}}\left\lvert j\right\rangle_{h}%
+\left\lvert j+c_{1}\right\rangle_{t_{1}}$ $\displaystyle=\log_{2}{\left\lvert S_{X}\right\lvert}-$ $\left\lvert\phi_{j\pm}\right\rangle=\left\lvert j\right\rangle\pm\left\lvert j%
++\frac{D}{2}\right\rangle$ $\displaystyle=\sum_{i=0}^{d-l-1}a_{i}^{(l)}2^{i}$ $p=p_{1}=p_{2}=\frac{1}{2^{d-d_{1}}}$ $\displaystyle\sum_{i=0}^{d-l-1}a_{i}^{(l+1)}2^{i}=\sum_{i=0}^{d-l-1}a_{i}^{(l)%
+}2^{i}=$ $a_{1},a_{2}\in\left[D\right]^{o}$ $\left\{0,1,\cdots,N-1\right\}$ $\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{js_{i}}\left%
+\lvert j\right\rangle_{t_{1}}\left\lvert jk_{1}\right\rangle_{g}\rightarrow$ $\displaystyle ab\equiv a(2w+1)\equiv a+2w\sum_{i=0}^{d-1}2^{i}a_{i}\equiv a+%
+\sum_{i=1}^{d}2^{i}wa_{i-1}(\ \mathrm{mod}\ 2^{d})$ $bra{j(k_{1}+k_{2}a_{1}+k_{3}a_{2})}_{g}\to\left\lvert j\right\rangle_{g}$ $\forall\hat{M_{i}}\in 4\left[N\right],y_{i}\in\left[N\right]$ $\mathbf{y},v$ $d=O(m)$ $\displaystyle\leq S\left(\rho\right)-\sum_{s_{i}\in\left[D\right]^{o}}p_{s_{i}%
+}S\left(\rho_{s_{i}}\right)=\log_{2}\left\lvert S_{J}\right\lvert-1$ $d,D=2^{d}$ $\left\{\begin{matrix}j+c_{1}=b_{1}\\
+jp_{i}+c_{2}=b_{2}\\
+jc_{3}+c_{4}=b_{3}\\
+\left(k_{1}+p_{i}k_{2}+c_{3}k_{3}\right)^{-1}=r_{3}\\
+c_{1}-\left(k_{1}c_{1}+k_{2}c_{2}+k_{3}c_{4}\right)r_{3}=r_{4}\end{matrix}%
+\right.,$ $\mathbf{U}=\mathbf{A}\cdot\mathbf{B}+\mathbf{V}$ $\left\{\begin{matrix}c_{1}=b_{1}-j\\
+c_{2}=b_{2}-jp_{i}\\
+c_{3}=k_{3}^{-1}\left(r_{3}^{-1}-k_{1}-p_{i}k_{2}\right)\\
+c_{4}=b_{3}-jk_{3}^{-1}\left(r_{3}^{-1}-k_{1}-p_{i}k_{2}\right)\end{matrix}%
+\right..$ $\mathcal{XOR}_{(h,t)}:\left\lvert a\right\rangle_{h}\left\lvert b\right\rangle%
+_{t}\rightarrow\left\lvert a\right\rangle_{h}\left\lvert b\oplus a\right%
+\rangle_{t}.$ $\mathcal{P}(i+k)^{b_{i}}_{h_{i}}$ $\displaystyle=\log_{2}\left(D\left\lvert S_{X}\right\lvert\left\lvert S_{C}%
+\right\lvert\right)-\frac{1}{D\left\lvert S_{X}\right\lvert\left\lvert S_{C}%
+\right\lvert}\sum_{b\in\left[L\right]}\beta_{b}\log_{2}\alpha_{b}-$ $Im(g)$ $b\in\left[D\right]$ $H(A,B),H(A:B),$ $\displaystyle=\Pr\left(j^{\prime}=(k_{2}r_{3})^{-1}(j(1-k_{1}r_{3})-r_{4})%
+\right)=\frac{1}{D}.$ $I_{A},I_{B}$ $\mathcal{BSUM}_{(h,t)}:\left\lvert a\right\rangle_{h}\left\lvert b\right%
+\rangle_{t}\rightarrow\left\lvert a\right\rangle_{h}\left\lvert b+a\right%
+\rangle_{t}.$ $L=2^{l_{2}}$ $\displaystyle\rho^{Q}_{xc}=tr_{P}\left(\rho^{(P,Q)}_{xc}\right)$ $j,ja_{1}$ $I_{A}=H(Z_{B}:X_{A})$ $v_{1},v_{2},\cdots,v_{n-1}\in\left[N\right]$ $r_{4}=-r_{3}\left[\left(k_{1}-r_{3}^{-1}\right)b_{1}+k_{2}b_{2}+k_{3}b_{3}%
+\right],$ $1\leq w_{1}^{-1}w_{3}+k2^{d-d_{3}}\leq 2^{d-d_{3}+d_{1}}-1$ $d_{1}+d_{2}=d_{3}$ $p_{i}=2x_{i}+1$ $v+\sum^{n}_{i=1}x_{i}y_{i}-a2^{m}\in\left[\frac{D}{4}\right]=\left[N\right]$ $Z_{A},Z_{B}$ $d=m+2$ $4v_{n}\equiv 4v-4\sum_{i=1}^{n-1}v_{i}(\ \mathrm{mod}\ D)$ $w_{1}w_{2}\equiv w_{3}(\ \mathrm{mod}\ 2^{d-d_{3}})$ $a\equiv b(\ \mathrm{mod}\ D)$ $r=(k_{1}+k_{2}a_{1}+k_{3}a_{2})^{-1}$ $\mathcal{U}_{\varepsilon}$ $t_{1},t_{2},g$ $k_{1}+p_{i}k_{2}=r_{3}^{-1}$ $\omega^{ja_{2}b_{2}}$ $j(1-k_{1}r_{3})=r_{4}$ $poly(m_{X})$ $\displaystyle\mathbf{y}_{j}=\left(b_{1j},b_{2j},\cdots,b_{nj}\right).$ $\displaystyle=\frac{1}{D\left\lvert S_{X}\right\lvert\left\lvert S_{C}\right%
+\lvert}\sum_{b\in\left[L\right]}\beta_{b}\left\lvert b\right\rangle\left%
+\langle b\right\lvert_{Q}.$ $\displaystyle=\log_{2}\left(D\left\lvert S_{X}\right\lvert\left\lvert S_{C}%
+\right\lvert\right)-\frac{1}{D\left\lvert S_{X}\right\lvert\left\lvert S_{C}%
+\right\lvert}\sum_{b\in\left[L\right]}\beta_{b}\log_{2}\alpha_{b},$ $d_{1},d_{2},d_{3}\in\left[d+1\right],w_{i}\in\left[2^{d-d_{i}}\right]^{o}$ $\forall(j,x_{i})\in\left[D\right]\times\left[N\right]$ $d_{r}\geq d_{l}$ $\displaystyle=\frac{1}{D}\sum_{j^{\prime},j\in\left[D\right]}\omega^{(j^{%
+\prime}-j)h(x,c)}\delta_{j^{\prime}j}\left\lvert g(j^{\prime},x,c)\right%
+\rangle\left\langle g(j,x,c)\right\lvert_{Q}$ $f(j,x,c),g(j,x,c)$ $t_{1},g$ $1-k_{2}r_{3}\not\equiv 0(\ \mathrm{mod}\ D)$ $d_{1}+d_{2}\geq d_{3}$ $\mathcal{U}_{\varepsilon}:\left\lvert j\right\rangle_{t}\left\lvert 0\right%
+\rangle_{e}\rightarrow\left\lvert j\right\rangle_{t}\left\lvert\varepsilon(j)%
+\right\rangle_{e}$ $\rho=\sum_{y_{i}\in\left[N\right]}\frac{1}{N}\rho_{y}$ $\displaystyle H(Z$ $\frac{\left\lvert S_{J}\right\lvert}{D}$ $4v\equiv 4\sum_{i=1}^{n}v_{i}(\ \mathrm{mod}\ D)$ $\displaystyle=\frac{1}{\left\lvert S_{C}\right\lvert}\sum_{c\in S_{C}}\frac{1}%
+{D}\sum_{j\in\left[D\right]}\left\lvert g(j,x,c)\right\rangle\left\langle g(j,%
+x,c)\right\lvert_{Q}$ $\displaystyle-(l+1)2^{d-d_{3}}+1\leq-w_{1}^{-1}w_{3}\leq k2^{d-d_{3}}$ $\displaystyle I_{B}\leq\sum_{\left\lvert S_{J}\right\lvert}\frac{\left\lvert S%
+_{J}\right\lvert}{D}p_{\left\lvert S_{J}\right\lvert}\log_{2}\left\lvert S_{J}%
+\right\lvert-1$ $\displaystyle\overset{\mathcal{XOR}_{(h,t_{1})}\mathcal{XOR}_{(h,t_{2})}}{%
+\longrightarrow}\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{jM_{i}}%
+\left\lvert j\right\rangle_{h}$ $\displaystyle\overset{\mathcal{SUM}(c_{1})_{t_{1}}\mathcal{SUM}(c_{2})_{t_{2}}%
+\mathcal{SUM}(c_{4})_{g}}{\longrightarrow}\frac{1}{\sqrt{D}}\sum_{j\in\left[D%
+\right]}\left\lvert j\right\rangle_{h}$ $(t_{1},t_{2},g)$ $O\left(nm^{2}\right)$ $w_{3}+k2^{d-d_{3}}$ $\displaystyle\mathcal{MUL}(b)=\mathcal{SUM}(w\ \mathrm{mod}\ 2^{d-1})^{h_{0}}%
+\mathcal{SUM}(w\ \mathrm{mod}\ 2^{d-2})^{h_{1}}$ $\left\lvert a_{d-1}\right\rangle\cdots\left\lvert a_{d-l}\right\rangle\ %
+\mathrm{mod}\ 2^{l}$ $\forall j^{\prime},j\in\left[D\right],\left\langle f(j^{\prime},x,c)\lvert f(j%
+,x,c)\right\rangle=\delta_{j^{\prime}j}$ $\sum_{b\in\left[L\right]}\beta_{b}=\left\lvert\left[D\right]\times S_{X}\times
+S%
+_{C}\right\lvert=D\left\lvert S_{X}\right\lvert\left\lvert S_{C}\right\lvert$ $\mathcal{ROT}$ $\beta_{b}=\left\lvert g^{-1}(b)\right\lvert=\left\lvert\left\{(j,x,c)\lvert g(%
+j,x,c)=b\right\}\right\lvert$ $\frac{1}{2^{d-1}}$ $I_{A}=H(Z_{A}:X_{B})=0$ $\displaystyle=\sum_{b\in\left[L\right]}\frac{\alpha_{bx}}{D\left\lvert S_{C}%
+\right\lvert}\left(\log_{2}\left(D\left\lvert S_{C}\right\lvert\right)-\log_{2%
+}\alpha_{bx}\right)$ $\forall u\in\left[N\right]$ $q_{i},s_{i}$ $\displaystyle\left\lvert\psi(x,c)\right\rangle_{(P,Q)}=$ $\displaystyle\frac{1}{D}\sum_{j,j^{\prime}\in\left[D\right]}\left\lvert jk%
+\right\rangle_{t_{1}}\left\lvert j^{\prime}\right\rangle_{t_{2}}\left\lvert jk%
+_{1}+j^{\prime}k_{2}\right\rangle_{g}.$ $a=\sum_{i\in\left[d\right]}a_{i}2^{i}$ $p_{i}=k_{2}^{-1}\left(r_{3}^{-1}-k_{1}\right)$ $O\left(\sqrt{2^{m}}\right)$ $p_{x}=\frac{1}{\left\lvert S_{X}\right\lvert}$ $\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\omega^{jM_{i}}\left\lvert j\right%
+\rangle_{h}\overset{\mathcal{QFT}^{\dagger}_{h}}{\longrightarrow}\left\lvert M%
+_{i}\right\rangle_{h}\ \mathrm{mod}\ D,$ $\displaystyle\frac{1}{\sqrt{D}}\sum_{j\in\left[D\right]}\left\lvert j\right%
+\rangle_{h}\left\lvert j+c_{1}\right\rangle_{t_{1}}\left\lvert jp_{i}+c_{2}%
+\right\rangle_{t_{2}}\left\lvert jr_{1}+r_{2}\right\rangle_{g}$ $\mathbf{c}=(c_{1},c_{2},c_{3},c_{4})$ $\displaystyle H(Z:X)\leq S\left(\rho^{B}\right)-\sum_{x\in S_{X}}p_{x}S\left(%
+\rho^{B}_{x}\right)$ $\displaystyle\ \ \ \ +2^{d-l}\left(\sum_{i=0}^{l-1}a_{i+d-l}^{(l)}2^{l}+wa_{d-%
+1-l}^{(l)}\ \mathrm{mod}\ 2^{l}\right),$ $\displaystyle\left\lvert jr_{1}+r_{2}\right\rangle_{g},$ $\displaystyle\overset{\mathcal{MUL}(p_{i})_{t_{2}}\mathcal{MUL}(c_{3})_{g}}{\longrightarrow}$ $\mathcal{CNOT}$ $\displaystyle=\frac{1}{D}\sum_{j^{\prime},j\in\left[D\right]}\omega^{(j^{%
+\prime}-j)h(x,c)}tr\left(\left\lvert f(j^{\prime},x,c)\right\rangle\left%
+\langle f(j,x,c)\right\lvert_{P}\right)$ $\displaystyle=\mathcal{ROT}_{h}\left\lvert a\right\rangle_{h}.$ $u=\mathbf{x}\cdot\mathbf{y}+v\ \mathrm{mod}\ N$ $\displaystyle=\frac{1}{2^{d_{3}-d_{1}+1}}\cdot\frac{1}{2^{d-d_{3}-1}}=\frac{1}%
+{2^{d-d_{1}}}=p_{1}.$ $p(u)=\frac{1}{N}$ $|\psi(\theta_{h},n)\rangle$ $\displaystyle\langle\omega(\theta_{h},5)|Z_{P(size)}|\omega(\theta_{h},5)\rangle,$ $\displaystyle W^{(127)}_{10}$