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| <p> $\{z^{j}_{t+1}\}_{j\in[M]}$ </p> |
| <p> $\mu(a,b)=\mathbb{E}\left[f_{\theta}(a,b)\right]$ </p> |
| <p> $\log(M\sqrt{T})/\log(1+\sigma_{w}^{-2})+2\beta_{T}=\mathcal{O}(\log\mathcal{A}% |
| T+\log T+\log\log\mathcal{A}T).$ </p> |
| <p> $(M=10,20,30)$ </p> |
| <p> $\Re_{\operatorname{adv}}(a;T,\text{Hedge},\tilde{R})=\mathcal{O}(2c\sqrt{T\log% |
| \mathcal{A}})$ </p> |
| <p> $R_{t+1,A_{t},B_{t}}=\theta_{A_{t},B_{t}}$ </p> |
| <p> $a^{+}=\max\{a,0\}$ </p> |
| <p> $o=f_{\theta}(a)+w$ </p> |
| <p> $\mathbb{P}\left(\neg{\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})}\right)$ </p> |
| <p> $B_{0:T}$ </p> |
| <p> $\displaystyle\mathbb{E}\left[\sum_{t=0}^{T-1}I_{t}(\theta;A_{t},B_{t},R_{t+1,A% |
| _{t},B_{t}})\right]$ </p> |
| <p> $\displaystyle\gamma_{T}:=\max_{A_{0:T},B_{0:T}}I(\theta;A_{0},B_{0},\ldots,A_{% |
| T-1},B_{T-1})$ </p> |
| <p> $f_{\theta}(a,b)=\phi(a,b)^{\top}\theta$ </p> |
| <p> $\Re_{t}\in\mathbb{R}^{\mathcal{A}}$ </p> |
| <p> $\displaystyle=R_{t+1,A_{t},B_{t}}\left(\frac{\tilde{C}_{t-1,A_{t}}^{+}}{X_{t,A% |
| _{t}}}-\frac{\hat{X}_{t,A_{t}}}{{X}_{t,A_{t}}}\sum_{a}\tilde{C}_{t-1,a}^{+}\right)$ </p> |
| <p> $\tilde{R}^{\operatorname{est}}=(\tilde{R}_{t+1},t=0,1,\ldots)$ </p> |
| <p> $\tilde{R}_{t}=\left[\tilde{R}_{t1},\tilde{R}_{t2}\right]$ </p> |
| <p> $L=(L_{t}\>|\>t\in\mathbb{N})$ </p> |
| <p> $\mathbb{E}_{t}\left[R_{t,a,b}^{2}\right]:=\mathbb{E}_{t}\left[(f_{\theta]}(a,b% |
| )+W_{t+1})^{2}\right]=\mathbb{E}_{t}\left[f_{\theta]}(a,b)^{2}+W_{t+1}^{2}% |
| \right]\leqslant 1+\sigma_{w}^{2}$ </p> |
| <p> $\epsilon_{t}\sim\mathcal{N}(0,0.1)$ </p> |
| <p> $\tilde{R}^{\operatorname{est}}$ </p> |
| <p> $N_{t}=\sum\limits_{k=2}^{t}m_{k}\sim\mathcal{N}\left(-\sum\limits_{k=2}^{t}% |
| \frac{k}{k+\sigma_{n}^{2}}(1-\Delta),\ \sum\limits_{k=2}^{t}(1+\frac{\sigma_{n% |
| }^{2}}{k+\sigma_{n}^{2}})\right)\triangleq\mathcal{N}(\mu_{t},\sigma_{t}^{2}).$ </p> |
| <p> $\displaystyle=\log\prod_{t=1}^{\infty}\mathbb{P}(\omega_{t}|\Omega_{t-1})=\sum% |
| _{t=1}^{\infty}\log\mathbb{P}(\omega_{t}|\Omega_{t-1})$ </p> |
| <p> $\displaystyle=\sum_{t=1}^{\infty}\log\Phi\left(\frac{\sum_{k=2}^{t}\frac{k}{k+% |
| \sigma_{n}^{2}}(1-\Delta)}{\sqrt{\sum_{k=2}^{t}(1+\frac{\sigma_{n}^{2}}{k+% |
| \sigma_{n}^{2}})}}\right)$ </p> |
| <p> $\Re_{\operatorname{full}}(a;T,\operatorname{adv},\tilde{R})$ </p> |
| <p> $\displaystyle\mathbb{E}_{0}\left[\sum_{t=1}^{T}\sum_{a}\left(\tilde{\Re}_{t,a}% |
| \right)^{2}\right]$ </p> |
| <p> $\tilde{R}_{t}(2nd)$ </p> |
| <p> $X_{1}=[0.5,0.5]$ </p> |
| <p> $\mathcal{E}(c)$ </p> |
| <p> $\displaystyle=\mathbb{E}\left[\max_{a\in\mathcal{A}}\sum_{t=0}^{T-1}\mathbb{E}% |
| \left[R_{t+1,a,B_{t}}-R_{t+1,A_{t},B_{t}}\mid\theta\right]\right]$ </p> |
| <p> $\textbf{r}_{t}=(f_{\theta}(a,B_{t}))_{a\in\mathcal{A}}$ </p> |
| <p> $m_{1}<0$ </p> |
| <p> ${N}(\mu_{p},\Sigma_{p})$ </p> |
| <p> $\mathcal{O}\big{(}\sqrt{T\mathcal{A}}+\sqrt{\gamma_{T}\beta T}\big{)}$ </p> |
| <p> $c^{\prime}=-0.62$ </p> |
| <p> $(b^{i},b^{-i})$ </p> |
| <p> $U^{\prime}_{t}\geqslant L_{t}$ </p> |
| <p> $r^{i}(a^{i},a^{-i})=-\ell^{i}(a^{i},a^{-i})$ </p> |
| <p> $H_{t+1}=(H_{t},A_{t},B_{t},R_{t+1,A_{t},B_{t}})$ </p> |
| <p> $\mathcal{O}\left(\left(\sqrt{\log\mathcal{A}}+\sqrt{\log(\mathcal{A}T)\log(T)^% |
| {d+1}}\right)\sqrt{T}\right)$ </p> |
| <p> $\displaystyle\beta_{t}I_{t}(\theta;R_{t+1,A_{t},B_{t}}\>|\>A_{t}=a,B_{t}=b)% |
| \geqslant\beta^{\prime}_{t}\sigma^{2}_{t}(a,b).$ </p> |
| <p> $\mathcal{E}^{c}_{t}(\tilde{R},U^{\prime},A_{t},B_{t}):=\{\tilde{R}_{t+1}(A_{t}% |
| )\leqslant U^{\prime}_{t}(A_{t},B_{t})\}.$ </p> |
| <p> $X_{1}=\hat{X}_{1}$ </p> |
| <p> $k_{\rm L}(\cdot,\cdot)$ </p> |
| <p> $(a+b)^{+}\leqslant(a^{+}+b)^{+}\leqslant\left|a^{+}+b\right|.$ </p> |
| <p> $\tilde{R}_{t+1}$ </p> |
| <p> $\displaystyle\Re^{*}(T,\pi,\theta)$ </p> |
| <p> $\displaystyle=\sum_{a}(\gamma_{t}\hat{X}_{t,a}-\gamma_{t}/\mathcal{A})\sum_{b}% |
| Y_{t,b}f_{\theta}(a,b)$ </p> |
| <p> $\sqrt{\gamma_{T}}$ </p> |
| <p> $\displaystyle\mathbb{P}(\left|f_{\theta}(a,b)-{\mu_{t}(a,b)}\right|\geqslant% |
| \sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b),\forall a\in\mathcal{A}\mid H_{t})% |
| \leqslant 2\mathcal{A}\exp(-\beta^{\prime}_{t}/2),$ </p> |
| <p> $\displaystyle=\sum_{a}\frac{\mathbb{E}_{t}\left[R_{t+1,a,B_{t}}^{2}\right]}{X_% |
| {t,a}}+\sum_{a}\frac{\hat{X}_{t,a}}{{X}_{t,a}}\mathbb{E}_{t}\left[R_{t+1,A_{t}% |
| ,B_{t}}^{2}\right]\left(\left|\mathcal{A}\right|\hat{X}_{t,a}-2\right)$ </p> |
| <p> $\tilde{R}^{\operatorname{est}}=(\tilde{R}_{t},t\in\mathbb{Z}_{++})$ </p> |
| <p> $\displaystyle\leqslant\mathbb{E}_{0}\left[\tilde{C}_{T,a}^{+}\right]=\mathbb{E% |
| }_{0}\left[\sqrt{(\tilde{C}_{T,a}^{+})^{2}}\right]\leqslant\mathbb{E}_{0}\left% |
| [\sqrt{\sum_{a}\left(\tilde{C}_{T,a}^{+}\right)^{2}}\right],$ </p> |
| <p> $M_{1}=M_{2}=\ldots=M_{T}=M=\mathcal{O}(\log\mathcal{A}T)$ </p> |
| <p> $\sigma(H_{t},A_{t},R_{t+1,A_{t},B_{t}})$ </p> |
| <p> $\mathcal{R}:\mathbb{R}\mapsto[0,1]$ </p> |
| <p> $\displaystyle\mathbb{E}_{0}\left[\sqrt{\sum_{a}\left(\tilde{C}^{+}_{T,i}\right% |
| )^{2}}\right]\leqslant\mathbb{E}_{0}\left[\sqrt{\sum_{t=1}^{T}\sum_{a}\left(% |
| \tilde{\Re}_{t,a}\right)^{2}}\right]$ </p> |
| <p> $H_{t},B_{t}$ </p> |
| <p> $\tilde{R}_{t2}$ </p> |
| <p> $\displaystyle\leqslant\min_{t>0}\frac{\exp(\sigma^{2}t^{2}/2)}{\exp(tc)}=\exp(% |
| -c^{2}/\sigma^{2})$ </p> |
| <p> $\displaystyle\mu_{t+1}=\Sigma_{t+1}\left(\Sigma_{t}^{-1}\mu_{t}+\frac{R_{t+1,A% |
| _{t},B_{t}}}{\sigma_{w}^{2}}\phi(A_{t},B_{t})\right)$ </p> |
| <p> $\displaystyle\mathbb{P}(X-\mu\geqslant c)$ </p> |
| <p> $\pi_{t}(H_{t})$ </p> |
| <p> $v\in\mathbb{R}^{\mathcal{A}}$ </p> |
| <p> $\sigma_{p}\leqslant 1$ </p> |
| <p> $\hat{X}_{t+1}$ </p> |
| <p> $\tilde{R}_{t+1}(a)\in[0,C]$ </p> |
| <p> $\tilde{R}_{t}=\left[z_{t},\frac{t}{t+\sigma_{n}^{2}}(1-\Delta)+\sqrt{\frac{% |
| \sigma_{n}^{2}}{\sigma_{n}^{2}+t}}z^{\prime}_{t}\right],$ </p> |
| <p> $H_{t},A_{t},B_{t}$ </p> |
| <p> $\displaystyle\leqslant(1+\sigma_{w}^{2})\left(\sum_{a}\frac{1}{{X}_{t,a}}+\sum% |
| _{a}\frac{\hat{X}_{t,a}}{{X}_{t,a}}\left(\left|\mathcal{A}\right|\hat{X}_{t,a}% |
| -2\right)\right)$ </p> |
| <p> $f(x)=\theta^{\top}x,\theta\sim N(0,\sigma_{0}I)$ </p> |
| <p> $\mathbb{E}_{t}\left[\cdot\right]=\mathbb{E}\left[\cdot\mid H_{t},\theta\right]$ </p> |
| <p> $\bm{a}=(a^{i},a^{-i})$ </p> |
| <p> $\displaystyle\geqslant\sum_{t=1}^{\infty}\left(-\frac{1}{\sqrt{2\pi}f_{t}(% |
| \Delta,\sigma_{n})e^{f_{t}^{2}(\Delta,\sigma_{n})/2}}\right)$ </p> |
| <p> $\displaystyle V^{*}=\max_{P\in\mathcal{D}(\mathcal{A})}\min_{Q\in\mathcal{D}(% |
| \mathcal{B})}\mathbb{E}_{A\sim P,B\sim Q}\left[f_{\theta}(A,B)\right],$ </p> |
| <p> $\mathbb{P}(f_{\theta}\in\mathcal{F}_{t})\geqslant 1-2\mathcal{A}\exp(-\beta^{% |
| \prime}_{t}/2).$ </p> |
| <p> $\operatorname{clip}_{[-c,c]}(x)\geqslant\min(x,c)$ </p> |
| <p> $0.01\sqrt{2\log\mathcal{A}\mathcal{B}\sqrt{T}}$ </p> |
| <p> $\sum_{t=0}^{T-1}\mathbb{E}\left[R_{t+1,A_{t},B_{t}}\>|\>\theta\right]$ </p> |
| <p> $\text{SINR}(a,b;\theta)=\phi(a,b)^{T}\theta$ </p> |
| <p> $\delta=1/\sqrt{t}$ </p> |
| <p> $\displaystyle\lim_{t\to\infty}\log\mathbb{P}(\Omega_{t})$ </p> |
| <p> $\text{Regret}^{i}(T)=\frac{1}{T}\max_{a\in\Delta^{\mathcal{D}(\mathcal{A}^{i})% |
| }}\mathbb{E}\left[\sum_{t=1}^{T}\phi\left(a,x_{t}^{-i}\right)-\phi\left(x_{t}^% |
| {i},x_{t}^{-i}\right)\right],$ </p> |
| <p> $\displaystyle\leqslant\Re_{\operatorname{full}}(T,\operatorname{adv})+\sqrt{% |
| \beta I(\theta;H_{T})T},$ </p> |
| <p> $\displaystyle\mathbb{E}_{0}\left[\sum_{t=1}^{T}\Re_{t}(a)\right]\leqslant 2^{4% |
| /3}(1+\sigma_{w}^{2})^{1/3}\left|\mathcal{A}\right|^{2/3}T^{2/3}.$ </p> |
| <p> $g_{t}(\cdot):\mathbb{R}_{+}^{\mathcal{A}}\times\mathbb{R}_{+}^{\mathcal{A}}% |
| \mapsto\mathbb{R}_{+}^{\mathcal{A}}$ </p> |
| <p> $m_{t}=z_{t}-\frac{t}{t+\sigma_{n}^{2}}(1-\Delta)-\sqrt{\frac{\sigma_{n}^{2}}{t% |
| +\sigma_{n}^{2}}}z^{\prime}_{t}=z_{t}-\sqrt{\frac{\sigma_{n}^{2}}{t+\sigma_{n}% |
| ^{2}}}z^{\prime}_{t}-\frac{t}{t+\sigma_{n}^{2}}(1-\Delta)$ </p> |
| <p> $f_{t}(\Delta,\sigma_{n})=\frac{(1-\Delta)\left(t-\sigma_{n}^{2}\ln{(t+\sigma_{% |
| n}^{2})}+2\sigma_{n}^{2}\ln{\sigma_{n}}-1/(\sigma_{n}^{2}+1)\right)}{\sqrt{t+% |
| \sigma_{n}^{2}\ln{(t+\sigma_{n}^{2})}-2\sigma_{n}^{2}\ln{\sigma_{n}}-(\sigma_{% |
| n}^{2}+2)/(\sigma_{n}^{2}+1)}},$ </p> |
| <p> $\displaystyle\leqslant\sum_{a}\left|\gamma_{t}\hat{X}_{t,a}-\gamma_{t}/% |
| \mathcal{A}\right|$ </p> |
| <p> $\tilde{R}_{t1}\geqslant\tilde{R}_{t2}$ </p> |
| <p> $(\gamma_{t})_{t\geqslant 0}$ </p> |
| <p> $\displaystyle\mathbb{P}(\mathcal{E}_{t}(\tilde{R},U,B_{t})\mid H_{t},B_{t})$ </p> |
| <p> $\phi(a,b)\in\mathbb{R}^{d}$ </p> |
| <p> $\sigma^{2}_{t}(a,b)=k((a,b),(a,b))-\mathbf{k}_{t}((a,b))^{\top}(\mathbf{K}_{t}% |
| +\sigma^{2}{\bm{I}}_{t})\mathbf{k}_{t}(a,b)$ </p> |
| <p> $\tilde{C}^{+}_{t-1,a}=0$ </p> |
| <p> $\mathcal{A}_{J}=\mathcal{F}$ </p> |
| <p> $(r_{t})_{t}$ </p> |
| <p> $\displaystyle\geqslant 1-\delta_{1}-(1-\delta_{1})\exp{\left(\frac{-Mf}{\sqrt{% |
| 2\pi}(f^{2}+1)e^{f^{2}/2}}\right)}$ </p> |
| <p> $\displaystyle(U^{\prime}_{t}(A_{t},B_{t})-L_{t}(A_{t},B_{t}))$ </p> |
| <p> $Y_{t}=\max_{y}X_{t}^{\top}\theta y$ </p> |
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