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| <p> $\tilde{\mathcal{O}}(\sqrt{(M+N)/T})$ </p> |
| <p> $\text{SINR}(a,b;\theta)$ </p> |
| <p> $\sqrt{2\log\mathcal{A}\sqrt{T}}$ </p> |
| <p> $\operatorname{NashRegret}(T)$ </p> |
| <p> $\displaystyle\geqslant\sum_{t=1}^{\infty}\log\Phi(f_{t}(\Delta,\sigma_{n}))$ </p> |
| <p> $(\Delta,\sigma_{n}^{2})$ </p> |
| <p> $\displaystyle\tilde{R}_{t+1}(a)=1-\frac{\mathbb{I}_{A_{t}=a}(1-R_{t+1,A_{t},B_% |
| {t}})}{X_{t,a}}$ </p> |
| <p> $w_{t}(a,b)=\sqrt{\beta_{t}I_{t}\left(\theta;R_{t+1,A_{t},B_{t}}\mid A_{t}=a,B_% |
| {t}=b\right)}\quad\text{ with }\quad\beta_{t}=\frac{2\beta^{\prime}_{t}}{\log(% |
| 1+\sigma_{w}^{-2})}.$ </p> |
| <p> $\mu_{t}(a,b)=\mathbf{k}_{t}((a,b))^{\top}(\mathbf{K}_{t}+\sigma^{2}{\bm{I}}_{t% |
| })^{-1}\mathbf{R}_{t}$ </p> |
| <p> $\eta_{1},\eta_{2},\ldots,\eta_{M}$ </p> |
| <p> $z_{t},z^{\prime}_{t}\sim\mathcal{N}(0,1)$ </p> |
| <p> $\theta\in\mathbb{R}^{\mathcal{A}\times\mathcal{B}}$ </p> |
| <p> $\beta_{t}=2\beta^{\prime}_{t}/\log(1+\sigma_{w}^{-2})$ </p> |
| <p> $\Re_{\operatorname{full}}(T,{\operatorname{Hedge}})=\mathcal{O}(\sqrt{T\log% |
| \mathcal{A}})$ </p> |
| <p> $\displaystyle:=\max_{j\in[M_{t+1}]}\tilde{f}^{\operatorname{TS},j}_{t+1}(a,B_{% |
| t}),$ </p> |
| <p> $\displaystyle\leqslant\max_{a\in\mathcal{A}}\Re_{\operatorname{full}}(a;T,% |
| \operatorname{adv},\tilde{R})+\mathbb{E}\left[\max_{a\in\mathcal{A}}\sum_{t=0}% |
| ^{T-1}\mathbb{E}\left[\operatorname{pess}_{t+1}(a)\mid\theta\right]\right]+% |
| \sum_{t=0}^{T-1}\mathbb{E}\left[\operatorname{est}_{t+1}\right]$ </p> |
| <p> $\tilde{r}_{t}=A_{ij}+\epsilon_{t}$ </p> |
| <p> $N(\mu_{p},\Sigma_{p})$ </p> |
| <p> $M=\frac{\log(\sqrt{t})}{\log\frac{1}{\Phi(\sqrt{\beta^{\prime}_{t}})}}$ </p> |
| <p> $\displaystyle=\mathbb{P}(\sum_{k=2}^{t}m_{k}\leqslant 0)$ </p> |
| <p> $\displaystyle\mathbb{E}\left[\sum_{t=0}^{T-1}\operatorname{est}_{t+1}\right]$ </p> |
| <p> $\displaystyle\mathbb{E}_{t}\left[\tilde{R}_{t+1}(a)\right]=1-\mathbb{E}_{t}% |
| \left[\mathbb{I}_{A_{t}=a}\frac{1-R_{t+1,a,B_{t}}}{X_{t,a}}\right]=1-\mathbb{E% |
| }_{t}\left[\mathbb{I}_{A_{t}=a}\right]\frac{1-\mathbb{E}_{t}\left[f_{\theta}(a% |
| ,B_{t})\right]}{X_{t,a}}=\mathbb{E}_{t}\left[f_{\theta}(a,B_{t})\right].$ </p> |
| <p> $\sigma_{t}(a,b)$ </p> |
| <p> $\displaystyle=k((A_{i},B_{i}),(A_{j},B_{j}))$ </p> |
| <p> $\Re(T,\pi^{\operatorname{alg}},B_{0:T},\theta)$ </p> |
| <p> $\displaystyle\mathbb{E}_{0}\left[\sum_{t=1}^{T}\Re_{t}(a)\right]=\mathbb{E}_{0% |
| }\left[\sum_{t=1}^{T}\mathbb{E}_{t}\left[\tilde{\Re}_{t,a}\right]+\sum_{t=1}^{% |
| T}2\gamma_{t}\right]=\mathbb{E}_{0}\left[\tilde{C}_{T,a}+\sum_{t=1}^{T}2\gamma% |
| _{t}\right]\leqslant\sqrt{\sum_{t=1}^{T}\frac{2(1+\sigma_{w}^{2})\left|% |
| \mathcal{A}\right|^{2}}{\gamma_{t}}}+2\gamma_{t}$ </p> |
| <p> $\displaystyle=\mathbb{P}(\tilde{R}_{t+1}(a)\geqslant U_{t}(a,B_{t}),\forall a% |
| \in\mathcal{A}\mid H_{t},B_{t})$ </p> |
| <p> $\Re_{t}(a):=\mathbb{E}\left[R_{t+1,a,B_{t}}-R_{t+1,A_{t},B_{t}}\mid\theta,H_{t% |
| }\right]=\mathbb{E}\left[f_{\theta}(a,B_{t})\mid\theta,H_{t}\right]-\sum_{a}{X% |
| }_{t,a}\mathbb{E}\left[f_{\theta}(a,B_{t})\mid\theta,H_{t}\right]$ </p> |
| <p> $\displaystyle\sum_{t=0}^{T-1}\mathbb{E}\left[R_{t+1,A^{*},B_{t}}-R_{t+1,A_{t},% |
| B_{t}}\>|\>\theta\right],$ </p> |
| <p> $\displaystyle\leqslant\Re_{\operatorname{full}}(a;T,\operatorname{adv},\tilde{% |
| R})+\sum_{t=0}^{T-1}\mathbb{E}\left[\operatorname{pess}_{t+1}(a)\mid\theta% |
| \right]+\sum_{t=0}^{T-1}\mathbb{E}\left[\operatorname{est}_{t+1}\>|\>\theta\right]$ </p> |
| <p> $\mathbb{E}\left[\tilde{R}_{t+1}(a)\>|\>H_{t},\theta\right]=\mathbb{E}\left[f_{% |
| \theta}(a,B_{t})\>|\>H_{t},\theta\right]=\mathbb{E}\left[g_{\theta}(e_{a},Y_{t% |
| })\>|\>H_{t},\theta\right]$ </p> |
| <p> $\displaystyle\leqslant\mathbb{E}\left[\sum_{t=0}^{T-1}\sqrt{\left(\sqrt{2\log(% |
| M\sqrt{t})/\beta^{\prime}_{t}}+1\right)^{2}\beta_{t}I_{t}(\theta;A_{t},B_{t},R% |
| _{t+1,A_{t},B_{t}})}\right]$ </p> |
| <p> $\Re_{\operatorname{full}}(T,\text{RM},\tilde{R}^{\operatorname{est}})=\mathcal% |
| {O}(\sqrt{T\mathcal{A}})$ </p> |
| <p> $(a=b)$ </p> |
| <p> $\mathbb{P}(\Omega_{t})\geqslant c$ </p> |
| <p> $\beta^{\prime}_{t}=2\log\mathcal{A}\sqrt{t}.$ </p> |
| <p> $\displaystyle\Re^{*}(T,\pi^{\operatorname{alg}})=\sup_{B_{0:T}\in\mathcal{B}^{% |
| T}}\Re(T,\pi^{\operatorname{alg}},B_{0:T})=o(T),$ </p> |
| <p> $\mathbf{K}_{t}$ </p> |
| <p> $\tilde{R}^{OTS}_{t+1}(1st)>\tilde{R}^{OTS}_{t+1}(2nd)$ </p> |
| <p> $\displaystyle\leqslant\mathbb{E}\left[\max_{a\in\mathcal{A}}\sum_{t=0}^{T-1}% |
| \mathbb{E}\left[C(1-\mathds{1}_{\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})\cap% |
| \mathcal{E}^{c}_{t}(f_{\theta},B_{t})})\mid\theta\right]\right]$ </p> |
| <p> $\nu\rightarrow\infty$ </p> |
| <p> $\displaystyle\quad+C\left(1-\mathds{1}_{\mathcal{E}^{c}_{t}(\tilde{R},U,A_{t},% |
| B_{t})}\mathds{1}_{\mathcal{E}^{c}_{t}(f_{\theta},B_{t})}\right)$ </p> |
| <p> $\displaystyle X_{t,a}=(1-\gamma_{t})\hat{X}_{t,a}+\gamma_{t}(1/\mathcal{A}),% |
| \forall a\in\mathcal{A}$ </p> |
| <p> $\textrm{Hedge:}\ g_{t,a}(X_{t},r_{t})=X_{t,a}\exp(\eta_{t}r_{t}(a)),\quad% |
| \textrm{RM:}\ g_{t,a}(X_{t},r_{t})=\max\left(0,\sum_{s=0}^{t}r_{t}(a)-r_{t}(A_% |
| {s})\right).$ </p> |
| <p> $\displaystyle\mathcal{E}_{t}(f_{\theta},B_{t}):=\{\forall a\in\mathcal{A},f_{% |
| \theta}(a,B_{t})\in[L_{t}(a,B_{t}),U_{t}(a,B_{t})]\}.$ </p> |
| <p> $reg_{1}$ </p> |
| <p> $f_{\theta}(a,b)\mid H_{t}$ </p> |
| <p> $[a^{i}]_{e}=0$ </p> |
| <p> $\displaystyle=\mathbb{E}_{0}\left[\sum_{t=1}^{T}\sum_{a}R_{t+1,A_{t},B_{t}}^{2% |
| }\left(\left(\frac{\mathbb{I}_{A_{t}=a}}{X_{t,A_{t}}}\right)^{2}-\frac{2\hat{X% |
| }_{t,A_{t}}}{X_{t,A_{t}}^{2}}\mathbb{I}_{A_{t}=a}+\frac{\hat{X}_{t,A_{t}}^{2}}% |
| {X_{t,A_{t}}^{2}}\right)\right]$ </p> |
| <p> $\tilde{R}_{t1}=\max\limits_{i}x_{ti}$ </p> |
| <p> $\displaystyle\mathbb{P}(\neg\mathcal{E}_{t}(\tilde{R},U,B_{t}))\leqslant\frac{% |
| 1}{\sqrt{t}}.$ </p> |
| <p> $\displaystyle\mathbf{K}_{t}(i,j)$ </p> |
| <p> $\nu>1$ </p> |
| <p> $\tilde{R}_{t+1}(a)$ </p> |
| <p> $\tilde{R}:=\{\tilde{R}_{t}:t\in\mathbb{Z}_{++}\}$ </p> |
| <p> $\mathbb{P}(A_{t}\in\cdot\>|\>\pi_{t})=\mathbb{P}(A_{t}\in\cdot\>|\>H_{t})=\pi_% |
| {t}(\cdot)$ </p> |
| <p> $\displaystyle\begin{cases}\textrm{UCB:}&\tilde{f}_{t+1}(a,B_{t})\>|\>H_{t+1}=% |
| \mu_{t}(a,B_{t})+\beta_{t}\sigma_{t}(a,B_{t}),\\ |
| &\tilde{R}_{t+1}(a)=\tilde{f}_{t+1}(a,B_{t})\wedge 1,\forall a\in\mathcal{A}.% |
| \\ |
| \textrm{TS:}&\tilde{f}_{t+1}(a,B_{t})\>|\>H_{t+1}\sim N(\mu_{t}(a,B_{t}),% |
| \sigma_{t}(a,B_{t})),\\ |
| &\tilde{R}_{t+1}(a)=\tilde{f}_{t+1}(a,B_{t})\wedge 1,\forall a\in\mathcal{A}.% |
| \end{cases}$ </p> |
| <p> $\Sigma_{0}=\Sigma_{p}$ </p> |
| <p> $\displaystyle\leqslant(1+\sigma_{w}^{2})\left(\sum_{a}\frac{1}{X_{t,a}}+\sum_{% |
| a}\frac{\hat{X}_{t,a}}{{X}_{t,a}}\left(\left|\mathcal{A}\right|-2\right)\right)$ </p> |
| <p> $\mathds{1}(a=b)$ </p> |
| <p> $\operatorname{est}$ </p> |
| <p> $\mathbf{k}_{t}((a,b))$ </p> |
| <p> $\mathcal{E}_{t}(\tilde{R},U^{\prime},A_{t},B_{t})$ </p> |
| <p> $\displaystyle\leqslant\sum_{a}\left(\left|\gamma_{t}\hat{X}_{t,a}\right|+\left% |
| |\gamma_{t}/\mathcal{A}\right|\right)=2\gamma_{t}$ </p> |
| <p> $\mathcal{B}={1,\ldots,\left|\mathcal{B}\right|}$ </p> |
| <p> $M=\frac{\log(\sqrt{t})}{\log\frac{1}{\Phi(\sqrt{\beta^{\prime}_{t}})}}.$ </p> |
| <p> $\tilde{R}_{t+1}(a)=\min(\tilde{f}_{t+1}(a,B_{t}),1)$ </p> |
| <p> $\displaystyle\mathbb{E}_{0}\left[\sum_{t=1}^{T}\Re_{t}(a)\right]\leqslant\sqrt% |
| {T}\sqrt{\frac{2(1+\sigma_{w}^{2})\left|\mathcal{A}\right|^{2}}{\gamma}}+2% |
| \gamma T,$ </p> |
| <p> $\displaystyle\mathbb{P}\left(\max_{j\in[M]}\eta_{j}\geqslant w\right)=1-\left[% |
| \Phi\left(\frac{w}{\sigma}\right)\right]^{M}$ </p> |
| <p> $Z_{t}=(A_{t},B_{t},R_{t+1,A_{t},B_{t}}))$ </p> |
| <p> $(r_{t})_{t\in[T]}\in[0,1]^{\mathcal{A}\times T}$ </p> |
| <p> $\displaystyle\stackrel{{\scriptstyle(iii)}}{{=}}1-\Phi\left(\sqrt{\beta_{t}^{% |
| \prime}}\right)^{M}.$ </p> |
| <p> $\displaystyle>-\infty,$ </p> |
| <p> $KL(\overline{y}_{T},y^{)}$ </p> |
| <p> $\displaystyle\tilde{R}_{t+1}(a)-f_{\theta}(a,B_{t})=\mathds{1}_{\mathcal{E}^{o% |
| }_{t}(\tilde{R},U,B_{t})}(\tilde{R}_{t+1}(a)-f_{\theta}(a,B_{t}))+(1-\mathds{1% |
| }_{{\mathcal{E}}^{o}_{t}(\tilde{R},U,B_{t})})(\tilde{R}_{t+1}(a)-f_{\theta}(a,% |
| B_{t}))$ </p> |
| <p> $\mathcal{A}={1,\ldots,\left|\mathcal{A}\right|}$ </p> |
| <p> $\Re^{*}(T,\text{UCB-Hedge})=\mathcal{O}(\sqrt{T\log\mathcal{A}}+\sqrt{\gamma_{% |
| T}\beta T}),\ \Re^{*}(T,\text{UCB-RM})=\mathcal{O}(\sqrt{T\mathcal{A}}+\sqrt{% |
| \gamma_{T}\beta T}).$ </p> |
| <p> $\Re_{\operatorname{full}}(T,\text{Hedge},\tilde{R}^{\operatorname{est}})=% |
| \mathcal{O}(\sqrt{T\log\mathcal{A}})$ </p> |
| <p> $U=(\mu_{t}(a,b)+\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b):t\in\mathbb{N}),\quad |
| L% |
| =(\mu_{t}(a,b)-\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b):t\in\mathbb{N}).$ </p> |
| <p> $c(\Delta,\sigma_{n})\approx 0.54$ </p> |
| <p> $k_{\rm P}(\cdot,\cdot)$ </p> |
| <p> $\sigma_{n}>0$ </p> |
| <p> $k((a,b),(a^{\prime},b^{\prime}))=\mathbb{E}\left[(f_{\theta}(a,b)-\mu(a,b))(f_% |
| {\theta}(a^{\prime},b^{\prime})-\mu(a^{\prime},b^{\prime}))\right]$ </p> |
| <p> $\displaystyle=[k((A_{0},B_{0}),(a,b)),\ldots,k((A_{t-1},B_{t-1}),(a,b))]^{\top}$ </p> |
| <p> $\sigma_{t}^{2}(a,b)$ </p> |
| <p> $\displaystyle\leqslant\sum_{a}\left(\left({\tilde{C}_{T-1,a}^{+}}\right)^{2}+2% |
| {\tilde{C}_{T-1,a}^{+}}{\tilde{\Re}_{T,a}}+\left(\tilde{\Re}_{T,a}\right)^{2}\right)$ </p> |
| <p> $A\in\mathbb{R}^{10\times 5}$ </p> |
| <p> $\displaystyle\tilde{R}_{t+1}(A_{t})-f_{\theta}(A_{t},B_{t})$ </p> |
| <p> $g(a^{-i})=\sum_{j\neq i}a^{j}$ </p> |
| <p> $\gamma_{t}=\gamma$ </p> |
| <p> $P_{a}(\theta)$ </p> |
| <p> $\displaystyle\sum_{a}\left({\tilde{C}_{T,a}^{+}}\right)^{2}$ </p> |
| <p> $\sum\limits_{k=1}^{t}\frac{1}{k+a}\leqslant\int_{0}^{t}\frac{1}{k+a}dk$ </p> |
| <p> $y^{*}=\operatorname*{arg\,min}\limits_{y\in\Delta}y^{T}(Ax),$ </p> |
| <p> $\displaystyle\mathbb{P}(\max_{j\in[M]}\eta_{j}\geqslant w)=1-\mathbb{P}(\max_{% |
| j\in[M]}\eta_{j}\leqslant w)=1-\mathbb{P}(\forall j\in[M],\eta_{j}\leqslant w)% |
| =1-\left[\Phi\left(\frac{w}{\sigma}\right)\right]^{M}.$ </p> |
| <p> $m_{1}\leqslant 0$ </p> |
| <p> $\tilde{R}_{t+1}\in[0,C]$ </p> |
| <p> $\mathcal{D}(\mathcal{A}^{i})$ </p> |
| <p> $\displaystyle\sum_{t=1}^{T}\frac{1}{\sqrt{t}}=2\sum_{t=1}^{T}\frac{t-(t-1)}{% |
| \sqrt{t}+\sqrt{t}}\leqslant 2\sum_{t=1}^{T}\frac{t-(t-1)}{\sqrt{t}+\sqrt{t-1}}% |
| =2\sum_{t=1}^{T}(\sqrt{t}-\sqrt{t-1})=2\sqrt{T}$ </p> |
| <p> $50)$ </p> |
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