3 Proof of Theorem 1 using the Primal - Dual method

In this section we show how the refined upper bound on the regret of the EXP algorithm proved using the potential function approach (KL divergence) also gives us a better bound for the expert game setup with bandit feedback. Last lecture we showed how in the case of expert prediction with bandit feedback using the Exp3 algorithm, the regret is upper bounded by T 2/3n1/3 using a rough upper bound on the regret for the EXP algorithm, a special instance of the multiplicative weight update. We also proved a tighter bound on the EXP regret as in the following theorem