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| <p> $X_{t,a}$ </p> |
| <p> $U=(\mu_{t}(a,b)+\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b):t\in\mathbb{N}),\quad |
| U% |
| ^{\prime}=(\mu_{t}(a,b)+\sqrt{2\log(M\sqrt{t})}\sigma_{t}(a,b):t\in\mathbb{N}).$ </p> |
| <p> $(f_{\theta}(a,b):(a,b)\in\mathcal{A}\times\mathcal{B})$ </p> |
| <p> $\mathcal{O}\big{(}\sqrt{T\mathcal{A}}\big{)}$ </p> |
| <p> $a^{i},b^{i}\in\mathcal{A}^{i}$ </p> |
| <p> $\displaystyle=\left(\frac{\sqrt{2\log(M\sqrt{t})}}{\sqrt{\beta^{\prime}_{t}}}+% |
| 1\right)\sqrt{\beta_{t}I_{t}(\theta;A_{t},B_{t},R_{t+1,A_{t},B_{t}})}$ </p> |
| <p> $\displaystyle\geqslant\sum_{t=1}^{\infty}\log\left(1-\frac{1}{\sqrt{2\pi}f_{t}% |
| (\Delta,\sigma_{n})e^{f_{t}^{2}(\Delta,\sigma_{n})/2}}\right)$ </p> |
| <p> $\Delta,\sigma_{n}$ </p> |
| <p> $\displaystyle\leqslant\left(\frac{\sqrt{2\log(M\sqrt{t})}}{\sqrt{\beta^{\prime% |
| }_{t}}}+1\right)\sqrt{\beta_{t}I_{t}(\theta;R_{t+1,A_{t},B_{t}}\>|\>A_{t},B_{t% |
| })}$ </p> |
| <p> $\left|\mathcal{A}\right|$ </p> |
| <p> $k(x,x^{\prime})=\exp(-(2l^{2})^{-1}\left\|x-x^{\prime}\right\|^{2}s)$ </p> |
| <p> $\Phi(\beta^{\prime}_{t})^{M}=1/\sqrt{t}$ </p> |
| <p> $\tilde{R}_{t}\in[0,1]^{\mathcal{A}}$ </p> |
| <p> $m_{t}=\tilde{R}_{t1}-\tilde{R}_{t2}$ </p> |
| <p> $\overline{x}_{T}=\frac{1}{T}\sum_{t=1}^{T}x_{t},\quad\overline{y}_{T}=\frac{1}% |
| {T}\sum_{t=1}^{T}y_{t}.$ </p> |
| <p> $k((a,b),(a^{\prime},b^{\prime}))$ </p> |
| <p> $\left\langle\tilde{C}_{t-1}^{+},\tilde{\Re}_{t}\right\rangle\leqslant 0$ </p> |
| <p> $N(0,\sigma_{w}^{2})$ </p> |
| <p> $\displaystyle=\mathbb{E}_{0}\left[\sum_{t=1}^{T}R_{t+1,A_{t},B_{t}}^{2}\left(% |
| \frac{1}{X_{t,A_{t}}^{2}}-\frac{2\hat{X}_{t,A_{t}}}{X_{t,A_{t}}^{2}}+\left|% |
| \mathcal{A}\right|\frac{\hat{X}_{t,A_{t}}^{2}}{X_{t,A_{t}}^{2}}\right)\right]$ </p> |
| <p> $\displaystyle\tilde{\Re}_{t,a}=\frac{\mathbb{I}_{A_{t}=a}R_{t+1,A_{t},B_{t}}}{% |
| X_{t,a}}-R_{t+1,A_{t},B_{t}}\frac{\hat{X}_{t,A_{t}}}{X_{t,A_{t}}}$ </p> |
| <p> $(X_{a})_{a\in\mathcal{A}}$ </p> |
| <p> $Y_{t}=\min_{y}X_{t}^{\top}\theta y$ </p> |
| <p> $\log(\text{average regret})\propto(1/2)\log(M+N)$ </p> |
| <p> $H_{t}=\left(A_{0},B_{0},Y_{1,A_{0},B_{0}},\ldots,A_{t-1},B_{t-1},Y_{t,A_{t-1},% |
| B_{t-1}}\right)$ </p> |
| <p> $U^{\prime}\geqslant U\geqslant L$ </p> |
| <p> $\mathcal{O}(T^{d(d+1)/(2\nu+d(d+1))}(\log T))$ </p> |
| <p> $\mathcal{O}\big{(}\sqrt{T\mathcal{A}\log\mathcal{A}}\big{)}$ </p> |
| <p> $\displaystyle:=\operatorname{clip}_{[0,1]}\left(\tilde{f}_{t+1}^{\operatorname% |
| {OTS}}(a,B_{t})\right).$ </p> |
| <p> $n_{t}(a,b)$ </p> |
| <p> $\tilde{\Re}_{t}$ </p> |
| <p> $\Phi(x)\leqslant 1-\frac{x}{\sqrt{2\pi}(x^{2}+1)e^{x^{2}/2}}$ </p> |
| <p> $\displaystyle=\sum_{t=1}^{\infty}\log\Phi(-\frac{\mu_{t}}{\sigma_{t}})$ </p> |
| <p> $g_{t}:\Delta^{\mathcal{A}}\times[0,1]^{\mathcal{A}}\mapsto\Delta^{\mathcal{A}}$ </p> |
| <p> $c=0.54$ </p> |
| <p> $\tilde{\mathcal{O}}(\sqrt{MN/T})$ </p> |
| <p> $X_{t+1}=g_{t}(X_{t},(f_{\theta}(a,B_{t}))_{a\in\mathcal{A}})$ </p> |
| <p> $\displaystyle=\sum_{a}(\hat{X}_{t,a}-X_{t,a})\sum_{b}Y_{t,b}f_{\theta}(a,b)$ </p> |
| <p> $\mathbb{P}(\neg\mathcal{E}^{c}_{t}(f_{\theta},B_{t}))$ </p> |
| <p> $Z_{t+1}$ </p> |
| <p> $\displaystyle\geqslant(1-\delta_{1})\mathbb{P}(\max_{i}x_{i}\geqslant\max_{i}y% |
| _{i}\mid\epsilon)$ </p> |
| <p> $X_{2}=[1,0]$ </p> |
| <p> $\mathcal{O}\big{(}\sqrt{T\log\mathcal{A}}\big{)}$ </p> |
| <p> $\displaystyle\mathcal{F}_{t}:=\left\{f_{\theta}:\left|f_{\theta}(a,B_{t})-\mu_% |
| {t}(a,B_{t})\right|\leqslant\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,B_{t}),% |
| \forall a\in\mathcal{A}\right\}$ </p> |
| <p> $w\sim N\left(0,\sigma_{w}^{2}\right)$ </p> |
| <p> $\frac{\sqrt{2\log M\sqrt{t}}}{\sqrt{\beta^{\prime}_{t}}}=\sqrt{\frac{2\log M% |
| \sqrt{t}}{\log\mathcal{A}\sqrt{t}}}$ </p> |
| <p> $G_{J}$ </p> |
| <p> $\displaystyle\tilde{R}_{t+1}(A_{t})-f_{\theta}(A_{t},B_{t})\leqslant(\mathds{1% |
| }_{\mathcal{E}^{c}_{t}(\tilde{R},U^{\prime},A_{t},B_{t})})(U^{\prime}_{t}(A_{t% |
| },B_{t})-f_{\theta}(A_{t},B_{t}))+C(1-\mathds{1}_{\mathcal{E}^{c}_{t}(\tilde{R% |
| },U^{\prime},A_{t},B_{t})}).$ </p> |
| <p> $\displaystyle=\sum_{t=1}^{\infty}\log\mathbb{P}(\mu_{t}+\sigma_{t}Z\leqslant 0% |
| ),\quad Z\sim\mathcal{N}(0,1)$ </p> |
| <p> $\text{regret-matching}^{+}(\text{RM}^{+})$ </p> |
| <p> $\displaystyle f_{\theta}(a,B_{t})-f_{\theta}(A_{t},B_{t})=\underbrace{\tilde{R% |
| }_{t+1}(a)-\tilde{R}_{t+1}(A_{t})}_{(I)\leavevmode\nobreak\ \operatorname{adv}% |
| _{t+1}(a)}+\underbrace{f_{\theta}(a,B_{t})-\tilde{R}_{t+1}(a)}_{(II)% |
| \leavevmode\nobreak\ \operatorname{pess}_{t+1}(a)}+\underbrace{\tilde{R}_{t+1}% |
| (A_{t})-f_{\theta}(A_{t},B_{t})}_{(III)\operatorname{est}_{t+1}}$ </p> |
| <p> $\sum_{a}\tilde{C}_{t-1,a}^{+}\leqslant 0$ </p> |
| <p> $\displaystyle f_{\theta}(a,B_{t})-\tilde{R}_{t+1}(a)\leqslant C(1-\mathds{1}_{% |
| \mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})\cap\mathcal{E}^{c}_{t}(f_{\theta},B_{t}% |
| )}),\quad\forall a\in\mathcal{A}.$ </p> |
| <p> $\displaystyle\leqslant\sqrt{T\left(8\log(M\sqrt{T})/\log(1+\sigma_{w}^{-2})+2% |
| \beta_{T}\right)I(\theta;H_{T})}$ </p> |
| <p> $\mathbb{P}(\tilde{R}_{t1}\geqslant\tilde{R}_{t2})$ </p> |
| <p> $\displaystyle\Re(T,\pi^{\operatorname{alg}},\pi^{B})$ </p> |
| <p> $\displaystyle=\mathbb{P}(0.5m_{1}+\sum_{k=2}^{t}m_{k}\leqslant 0|\Omega_{t-1})$ </p> |
| <p> $(H_{t},A_{t},B_{t},\theta)$ </p> |
| <p> $\beta=\mathcal{O}(\log\mathcal{A}T)$ </p> |
| <p> $\sigma_{n}=0.1$ </p> |
| <p> $[-c,c]^{\mathcal{A}}$ </p> |
| <p> $\displaystyle\geqslant\mathbb{P}(\sum_{k=2}^{t}m_{k}\leqslant 0|\Omega_{t-1})$ </p> |
| <p> $\lim_{t\to\infty}\mathbb{P}(\Omega_{t})\geqslant c>0,$ </p> |
| <p> $\displaystyle=\mathbb{P}\left(\max_{j\in[M]}z^{j}_{t+1}\geqslant\sqrt{\beta^{% |
| \prime}_{t}}\right)$ </p> |
| <p> $\sigma_{t}(a,b)=\left\|\phi(a,b)\right\|_{\Sigma_{t}}$ </p> |
| <p> $\mathbb{P}(\Omega_{t})\geqslant\lim\limits_{t\to\infty}\mathbb{P}(\Omega_{t})\geqslant |
| c$ </p> |
| <p> $\displaystyle\operatorname{NashRegret}_{t}=\mathbb{E}\left[V^{*}-R_{t+1,A_{t},% |
| B_{t}}\right]$ </p> |
| <p> $\tilde{R}_{t}\in\mathbb{R}^{\mathcal{A}}$ </p> |
| <p> $\displaystyle\geqslant 1-\delta_{1}-(1-\delta_{1})\Phi^{M}\left(\frac{t}{t+% |
| \sigma_{n}^{2}}(1-\Delta)+\sqrt{\frac{2\sigma_{n}^{2}\log(M/\delta_{1})}{t+% |
| \sigma_{n}^{2}}}\right)$ </p> |
| <p> $I(\theta;H_{T})=I(\theta;A_{0},B_{0},\ldots,A_{T-1},B_{T-1})\leqslant\gamma_{T}.$ </p> |
| <p> $\displaystyle\mathbb{P}(\left|f_{\theta}(a,b)-{\mu_{t}(a,b)}\right|\geqslant% |
| \sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,b)\mid H_{t})$ </p> |
| <p> $r=(\sqrt{2\nu}/l)\left\|x-x^{\prime}\right\|$ </p> |
| <p> $(a,b),(a^{\prime},b^{\prime})\in\mathcal{A}\times\mathcal{B}$ </p> |
| <p> $\mu_{t}(a,b)$ </p> |
| <p> $\sum_{a}\left({\tilde{C}_{T,a}^{+}}\right)^{2}\leqslant\sum_{t=1}^{T}\sum_{a}% |
| \left(\tilde{\Re}_{t,a}\right)^{2}$ </p> |
| <p> $\tilde{R}_{t}$ </p> |
| <p> $\displaystyle\tilde{f}^{\operatorname{OTS}}_{t+1}(a,B_{t}):=(\max_{j\in[M]}z_{% |
| t+1}^{j})\cdot\sqrt{\beta^{\prime}_{t}}\sigma_{t}(a,B_{t})+\mu_{t}(a,B_{t})% |
| \leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \tilde{R}_{t+1}(a)=% |
| \operatorname{clip}_{[-c,c]}\left(\tilde{f}_{t+1}^{\operatorname{OTS}}(a,B_{t}% |
| )\right),\forall a\in\mathcal{A}.$ </p> |
| <p> $\displaystyle\tilde{R}^{\operatorname{OTS}}_{t+1}(a)$ </p> |
| <p> $\displaystyle=R_{t+1,A_{t},B_{t}}\left(\frac{\tilde{C}_{t-1,A_{t}}^{+}}{X_{t,A% |
| _{t}}}-\frac{\tilde{C}_{t-1,A_{t}}^{+}/\sum_{a}\tilde{C}_{t-1,a}^{+}}{{X}_{t,A% |
| _{t}}}\sum_{a}\tilde{C}_{t-1,a}^{+}\right)=0$ </p> |
| <p> $\gamma=\sqrt[3]{((1+\sigma_{w}^{2})\left|\mathcal{A}\right|^{2})/{2T}}$ </p> |
| <p> $\tilde{R}_{t+1}=E(H_{t+1},Z_{t+1})\in\mathbb{R}^{\mathcal{A}}.$ </p> |
| <p> $\displaystyle\leqslant\Re_{\operatorname{full}}(T,\operatorname{adv},\tilde{R}% |
| )+\sum_{t=0}^{T-1}\mathbb{E}\left[(U^{\prime}_{t}(A_{t},B_{t})-L_{t}(A_{t},B_{% |
| t}))\right]$ </p> |
| <p> $\displaystyle\leqslant\sum_{t=0}^{T-1}\mathbb{E}\left[(U^{\prime}_{t}(A_{t},B_% |
| {t})-L_{t}(A_{t},B_{t}))\right]$ </p> |
| <p> $\displaystyle=\mathbb{E}\left[f_{\theta}({a,B_{t}})-\tilde{R}_{t+1}(a)\mid% |
| \theta\right],$ </p> |
| <p> $(III)$ </p> |
| <p> $\displaystyle f_{\theta}(a,B_{t})-\tilde{R}_{t+1}(a)\leqslant\mathds{1}_{% |
| \mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})}(f_{\theta}(a,B_{t})-U_{t}(a,B_{t}))+C(% |
| 1-\mathds{1}_{{\mathcal{E}}^{o}_{t}(\tilde{R},U,B_{t})}).$ </p> |
| <p> $\Re(T)\geqslant 2\mathbb{P}(\Omega_{T})\Delta\cdot T.$ </p> |
| <p> $\displaystyle\mathbb{E}\left[R_{t+1,a,B_{t}}-R_{t+1,A_{t},B_{t}}\>|\>\theta% |
| \right]=(I)+(II)+(III)$ </p> |
| <p> $reg_{t}$ </p> |
| <p> $\displaystyle=I(\theta;Z_{0},\ldots,Z_{T-1})=I(\theta;H_{T})$ </p> |
| <p> $w\in\mathbb{R}_{+}$ </p> |
| <p> $\mathbb{P}(\Omega_{t})$ </p> |
| <p> $\displaystyle\leqslant\left(\frac{\sqrt{2\log(M\sqrt{t})}}{\sqrt{\beta^{\prime% |
| }_{t}}}+1\right)\sqrt{\beta^{\prime}_{t}}\sigma_{t}(A_{t},B_{t})$ </p> |
| <p> $\Re_{\operatorname{full}}(a;T,\text{RM},\tilde{R})=\mathcal{O}(2c\sqrt{T% |
| \mathcal{A}})$ </p> |
| <p> $\sigma^{2}_{t}(a,b)\leqslant k((a,b),(a,b))\leqslant 1$ </p> |
| <p> $\displaystyle\mathbb{P}(\neg\mathcal{E}^{c}_{t}(\tilde{R},U^{\prime},A_{t},B_{% |
| t}))\leqslant\frac{1}{\sqrt{t}}.$ </p> |
| <p> $\displaystyle\leqslant 2\exp\left(-\frac{\beta^{\prime}_{t}}{2}\right)$ </p> |
| <p> $\mathbb{P}(f_{\theta}(a,B_{t})\geqslant\tilde{R}_{t+1}(a)\mid\theta)\leqslant% |
| \mathcal{O}(1/\sqrt{T}).$ </p> |
| <p> $\displaystyle=\mathbb{P}(\max_{i}x_{i}\geqslant\max_{i}y_{i})$ </p> |
| <p> $\sqrt{2\log\mathcal{A}\sqrt{T}},0.2\sqrt{2\log\mathcal{A}\mathcal{B}\sqrt{T}},% |
| 0.05\sqrt{2\log\mathcal{A}\mathcal{B}\sqrt{T}}$ </p> |
| <p> $\mathcal{E}^{o}_{t}(\tilde{R},U,B_{t})$ </p> |
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