Distributed Strategic Learning for Wireless Engineers

Proposition 9.3.5.7. The convergence time to be close to the risk-sensitive payoff with error tolerance η is at most

$\underset{j}{\max {\left(G^j\right)}^{-1}\left(\log \left[\frac{error\left(0\right)}{\eta }\right]\right)}$

where (G^j)^-1 is the inverse of the mapping $t\mapsto G^j(t):={\displaystyle {\int }_0^t{g}_j(t^{\prime })d{t}^{\prime }}$ and $erro{r}_j(0):=\left|\right|{\widehat{\mathrm{r}}}_{j,0}-\mathbb{E}\frac{1}{{\mu }_j}(e^{{\mu }_j{\tilde{U}}_j}-1)\left|\right|$, error = max_j error_j. In particular for g_j = 1 (almost active case), the convergence time is of order of $\log \left[\frac{error(0)}{\eta }\right]$.

Proof. We verify that the solution of the ODE is $\left|\right|{\widehat{\mathrm{r}}}_{j,t}^{rs}-\mathbb{E}\frac{1}{{\mu }_j}(e^{{\mu }_j{\tilde{U}}_j}-1)\left|\right|=\mathrm{erro}{r}_j{e}^{-}{{\displaystyle {\int }_0^t{g}_j(s){}^{ds}}}^{}$. From the assumptions, the primitive function G^j is a bijection and $\Vert {\widehat{\mathrm{r}}}_{j,t}^{rs}-\mathbb{E}\frac{1}{{\mu }_j}\left(e^{{\mu }_j{\tilde{U}}_j}-1\right)\Vert \le \eta ift\ge {\max }_j{\left(G^j\right)}^{-1}\left(\log \left[\frac{error\left(0\right)}{\eta }\right]\right)$. The last assertion is obtained for g_j = 1. This completes the proof.

9.3.5.8    Explicit Solutions

As promised in our introduction of this Chapter, ...