There are studies about approximation algorithms for NP complete problems in Polynomial time and exact algorithms in exponential time. Are there studies about approximation algorithms for NP complete problems in subexponential time of form $2^{n^{\delta_2}}$ where $\delta_2\in(0,1)$?I am particularly interested in what is known about hard to polynomial time approximable problems such as Independence number and Clique number in subexponential time? Note that ETH only prohibits exact computation in such a time frame. Say Independence number is $\alpha(G)=2^{r(n)n}$ on a graph with vertex count $|V|=2^{s(n)n}$ for some $0<r(n)<s(n)$. Is an $2^{(r(n)n)^{\delta_1}}$-factor approximation scheme possible for Independence number in time $2^{|V|^{\delta_2}}=2^{2^{\delta_2s(n) n}}$ where $0<\delta_1<1$ and $0<\delta_2<1$ are some fixed positive reals?That is for every $\delta_1\in(0,1)$ is there a $\delta_2\in (0,1)$ such that $\alpha(G)$ can be approximated within $2^{\log_2^{\delta_1}(\alpha(G))}=2^{(r(n)n)^{\delta_1}}$ factor in time $2^{|V|^{\delta_2}}=2^{2^{\delta_2s(n) n}}$? 
Approximation in subexponental time
approximation algorithms;approximation hardness