UCB uses a padding function such as $$c_t(i)=B\sqrt{\xi \log(t) / N_t(i)},$$ where $B$ is an upper-bound on the reward.

This description comes from reference 2 (also see reference 3). But I have seen it described it as such elsewhere.

I see many people using $B=1$.

My question is, what is $B$ exactly? An upper-bound on the reward distribution? But too often, the reward distribution does not have an upper bound, it follows a Gaussian say. I am new to the literature, so I must be missing something...


  1. The original paper: Sample Mean Based Index Policies with O(log n) Regret for the Multi-Armed Bandit, by Rajeev Agrawal
  2. On Upper-Confidence Bound Policies for Non-Stationary Bandit Problems (see page 3), by Aurélien Garivier and Eric Moulines
  3. A Survey of Online Experiment Design with the Stochastic Multi-Armed Bandit, by Giuseppe Burtini, Jason Loeppky, Ramon Lawrence
  • $\begingroup$ I believe it's an upper bound on the expected reward, so if you have $k$ arms and a collection of reward distributions over time $\mu_t(k)$, $B = \max\mu_t(k)$. Could be wrong, though. $\endgroup$ – jbowman Feb 27 '18 at 18:26
  • $\begingroup$ Based on the discussion in the paper just before the Algorithm 1 block, I think it's an upper bound on the random variables. A lot of the early bandit work focused on variables in [0,1], which is consistent with the OP saying he sees "B=1" often. $\endgroup$ – combo Mar 3 '18 at 0:00

The term "UCB" stands for "Upper confidence bound", because the algorithm optimistically selects arms which might be very good. The term $B\sqrt{\xi}$ controls what precisely this means. From your reference 2 (page 3):

$B$ is an upper bound on your rewards and $\xi$ is an appropriate constant

We don't need to limit the analysis to finite rewards - in fact your reference 1 considers Gaussian and Poisson models (see section 5). Typically the constants fall out of the regret analysis and are chosen to minimize the expected regret. Sebastian Bubeck gives a very good review on UCB algorithms which describes this in the context of subgaussian rewards.

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