The multiplicative weights algorithm requires advance knowledge of the time horizon $T$ to set the parameter $$\backslash$eta$$.$ Modify the algorithm so that it does not need to know $T$ a priori. Your algorithm should have expected regret at most $b$\backslash$sqrt$\{\backslash$frac$\{\backslash$ln n$\}\{$T$\}\}$$ for all sufficiently large $T$ and for every adversary, where $b $>0$$ is a constant independent of $n$ and $T$$.$

multiplicative weights algorithm is defined in the attached imageMultiplicative Weights $($MW$)$ Algorithm

initialize $w^1(a)=1$ for every ainA

for each time step $t=1,2,$dots,$T$ do

use the distribution $p^t=\frac{w^t}{{}^t}$ over actions,

where ${}^t={\displaystyle \underset{\text{a}\text{i}\text{n}\text{A}}{\overset{?}{}}}{w}^t(a)$ is the sum of the

weights

given the cost vector $c^t,$ for every action ainA

use the formula $w^{t+1}(a)=w^t(a)*(1-c^t(a))$ to

update its weight