Regret formulation in stochastic multi-armed bandit problem

Question

In this paper, there are two formulations of regret at time $n$:

1. $R_n=n\mu^*-\sum_{t=1}^{n}E[\mu_{I_{t}}]$

2. $R_n=\sum_{i=1}^{K}\Delta_iE[T_i(n+1)]$

Where

$\mu_i$ is an expected value of some unknown distribution $v_i$ of arm $i$ (distribution that generates rewards)

$\mu^*=max\{{\mu_i}\}$

$I_t\in\{{1, ..., K}\}$ is the action taken at time $t$

$\Delta_i=\mu^*-\mu_i$

$T_i(t)=\sum_{l=1}^{t-1}1_{\{I_l=i\}}$ is the number of times arm $i$ was pulled strictly before time $t\geq2$

Can someone show why are they equivalent?

Answer 1

Source: Regret Analysis of Stochastic and Nonstochastic Multi-armed Bandit Problems Sebastien Bubeck and Nicolo Cesa-Bianchi2

I feel this should be self explanatory. Do let me know it is not !