Put simply, the multi-armed bandit problem, and in general every exploration problem, can be solved either through random strategies, or through smarter techniques. The most notorious algorithm that belongs to the first category, is called 
-greedy; whereas optimistic exploration, such as UCB, and posterior exploration, such as Thompson sampling, belong to the second category. In this section, we'll take a look particularly at the -greedy and UCB strategies.
It's all about balancing the risk and the reward. But, how can we measure the quality of an exploration algorithm? Through regret. Regret is defined as the opportunity lost in one step that is, the regret, 
, at time, 
, is as follows:
Here, 
denotes the optimal value, and 
the action-value of 
.
Thus, the goal is to find a trade-off between exploration and exploitation, by minimizing the...
