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| | <title>MathJax Example</title> |
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| | <p> $0.01$ </p> |
| | <p> $\underset{\pm 0.10}{2.15}$ </p> |
| | <p> $(\operatorname{\bm{\theta}}_{\text{client}}^{(t)}=\operatorname{\bm{\theta}}_{% |
| | \text{client}}^{(0)}$ </p> |
| | <p> $r=64$ </p> |
| | <p> $297.78$ </p> |
| | <p> $\mathbf{0.43}$ </p> |
| | <p> $\operatorname{\mathbf{v}}_{i}\in\mathcal{V}$ </p> |
| | <p> $0.76$ </p> |
| | <p> $\underset{\pm 0.66}{45.89}$ </p> |
| | <p> $0$ </p> |
| | <p> $\rho$ </p> |
| | <p> $0.26$ </p> |
| | <p> $0.95$ </p> |
| | <p> $p\approx 8.69\times 10^{-8}$ </p> |
| | <p> $0.69$ </p> |
| | <p> $47.32$ </p> |
| | <p> $\mathbf{0.69}$ </p> |
| | <p> $2.30$ </p> |
| | <p> $\mathbf{A}\in\mathbb{R}^{d\times r}$ </p> |
| | <p> $\operatorname{\bm{\theta}}_{\text{client}}^{(t)}$ </p> |
| | <p> $500$ </p> |
| | <p> $\operatorname{\mathbf{v}}_{i}$ </p> |
| | <p> $0.78$ </p> |
| | <p> $308$ </p> |
| | <p> $\mathbf{0.36}$ </p> |
| | <p> $-\frac{1}{|\mathcal{D}^{(t)}_{\bigtriangledown}|}\sum_{\operatorname{\mathbf{d% |
| | }}^{(t)}\in\mathcal{D}^{(t)}_{\bigtriangledown}}\log p_{(\cdot)}\big{(}% |
| | \operatorname{\mathbf{d}}^{(t)}\big{|}\operatorname{\bm{\theta}}_{(\cdot)}^{(t% |
| | )}),$ </p> |
| | <p> $\operatorname{\mathbf{pr}}_{\text{client}}$ </p> |
| | <p> $p$ </p> |
| | <p> $0.66$ </p> |
| | <p> $2.65$ </p> |
| | <p> $\operatorname{\bm{\theta}}_{\text{client}}^{(0)}$ </p> |
| | <p> $25$ </p> |
| | <p> $277.25$ </p> |
| | <p> $\%$ </p> |
| | <p> $57.16$ </p> |
| | <p> $0.74$ </p> |
| | <p> $0.77$ </p> |
| | <p> $\tau=0.5$ </p> |
| | <p> $256$ </p> |
| | <p> $4.3$ </p> |
| | <p> $\operatorname{\bm{\theta}}_{\text{agent}}^{(0)}$ </p> |
| | <p> $0.90$ </p> |
| | <p> $0.80$ </p> |
| | <p> $4$ </p> |
| | <p> $0.8$ </p> |
| | <p> $9000$ </p> |
| | <p> $\operatorname{\bm{\theta}}_{\text{client}}$ </p> |
| | <p> $F_{1}$ </p> |
| | <p> $\mathbf{55.25}$ </p> |
| | <p> $0.60$ </p> |
| | <p> $\operatorname{\mathbf{v}}_{\text{next}}\in\text{Children}(\operatorname{% |
| | \mathbf{v}}_{i})$ </p> |
| | <p> $\displaystyle\geq bx_{j}^{\prime}+\sum_{i>j}x_{i}^{\prime}+(b-1)\sum_{i>j}x_{i}$ </p> |
| | <p> $x_{i}^{\prime}\geq x_{i}$ </p> |
| | <p> $A_{1}$ </p> |
| | <p> $\displaystyle(b^{k}-b^{k-1}-(b-1)^{k}+b^{k-1})x_{1}$ </p> |
| | <p> $(u,w)$ </p> |
| | <p> $C_{1},C_{2}$ </p> |
| | <p> $P:(u,v)\cup P^{\prime}$ </p> |
| | <p> $e_{>i}$ </p> |
| | <p> $(u_{1},u_{2},\dots,u_{k},v)$ </p> |
| | <p> $\displaystyle=\frac{(b-1)^{k-1}}{b^{k}-(b-1)^{k}}x.$ </p> |
| | <p> $(v,z)\in N(u)\times N(u)$ </p> |
| | <p> $1\leq i\leq k$ </p> |
| | <p> $x_{\tau+1},\dots,x_{k}\mapsto\textsf{Chunk-Shortest-Edge}(k-\tau,x-\delta)$ </p> |
| | <p> $\sum_{i}x_{i}<x$ </p> |
| | <p> $\displaystyle=bx+c(v\to t)-c(u,w)-c(w\to t)=\delta+(b-1)x$ </p> |
| | <p> $p(e_{k}^{O})<\beta$ </p> |
| | <p> $i\geq\tau$ </p> |
| | <p> $\displaystyle=\begin{cases*}c(u,y)+cost[y,y,i]&if $(y,z)\in\mathcal{P}^{\prime% |
| | }(u,y)$\\ |
| | \infty&otherwise\end{cases*}$ </p> |
| | <p> $y^{*}\leq\frac{x}{k-1}$ </p> |
| | <p> $x-\delta$ </p> |
| | <p> $p(e;b_{i})$ </p> |
| | <p> $\displaystyle=\frac{b^{k-1}}{b^{k-1}-(b-1)^{k-1}}\left(\frac{b^{k}-(b-1)^{k-1}% |
| | (b-1+1)}{b^{k}-(b-1)^{k}}\right)x+c(v\to t)$ </p> |
| | <p> $(s,s_{1})$ </p> |
| | <p> $i+j\leq k$ </p> |
| | <p> $\displaystyle=\left(\frac{b^{k-1}}{b^{k-1}-(b-1)^{k-1}}-1\right)x$ </p> |
| | <p> $\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}$ </p> |
| | <p> $(u_{3},z)$ </p> |
| | <p> $\displaystyle\frac{(b-1)(b^{k-1}-(b-1)^{k-1})+b^{k-1}}{b^{k-1}-(b-1)^{k-1}}x_{1}$ </p> |
| | <p> $\sum_{l>i}x_{l}\leq x$ </p> |
| | <p> $\displaystyle(b-1)x_{1}+x$ </p> |
| | <p> $c^{n}$ </p> |
| | <p> $\delta/k$ </p> |
| | <p> $p(e_{i})=bx_{i}+c(u,w)+c(w\to t)$ </p> |
| | <p> $\alpha=\beta$ </p> |
| | <p> $p(e_{i};b_{1})\leq\alpha_{u}^{(j)}$ </p> |
| | <p> $p(e_{i}^{C})=bx_{i}^{C}+c(u_{i}^{C}\to t)$ </p> |
| | <p> $x_{i}-x_{i}^{\prime}\geq 0$ </p> |
| | <p> $p(e_{i})=\beta^{\prime}$ </p> |
| | <p> $\displaystyle\min_{(v,z)\in\mathcal{P}(u,y)}\min(C_{1}(u,v,y),C_{2}(u,y,z),C_{% |
| | 3}(u,v,y,z)).$ </p> |
| | <p> $c(u,v)$ </p> |
| | <p> $\beta^{*}=p(e_{i})$ </p> |
| | <p> $p(e_{j};b_{1})\leq\alpha_{u}^{(1)}$ </p> |
| | <p> $\displaystyle=\frac{x}{1-\left(\frac{b-1}{b}\right)^{k-\tau+1}}+c(v\to t)-c(w% |
| | \to t)-c(u,w).$ </p> |
| | <p> $x_{1}$ </p> |
| | <p> $O(|E|^{2}k^{3}\log k+|V|)$ </p> |
| | <p> $\beta\leftarrow\frac{x-\delta}{z_{\tau}}+c(v\to t)$ </p> |
| | <p> $min\_bottleneck\leftarrow\min(\alpha,\beta)$ </p> |
| | <p> $\delta/\tau$ </p> |
| | <p> $(y,z)$ </p> |
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