I am referring to the paper [2012] [Regret Analysis of Stochastic and NonStochastic MAB]1 to understand regret analysis of stochastic multi-armed bandits. In the paper, in Section 2.2, it says the following.

It is assumed that the distribution of rewards $X$ satisfy the following moment conditions. There exists a convex function $\psi$ on the reals such that, for all $\lambda \geq 0$,

Equation 1: \begin{equation} \ln \mathbb{E} e^{\lambda(X- \mathbb{E}[X])} \leq \psi(\lambda) \text{ and } \ln \mathbb{E} e^{\lambda(\mathbb{E}[X]-X)} \leq \psi(\lambda) \end{equation}.

They use the above to construct an upper bound estimate on the mean of each arm at some fixed confidence level, and the MAB strategy is to choose the arm that looks best under this estimate. To do so, they use the Legendre-Fenchel transform $\psi$, defined by

\begin{equation} \psi^*(\epsilon) = \sup_{\lambda \in \mathbb{R}} (\lambda \epsilon - \psi(\lambda)) \end{equation}.

Question 1: Why are the authors using this transformation? Can someone explain what does $\psi^*(\epsilon)$ represent really?

Now, let $\hat{\mu_{i,s}}$ be the sample mean of rewards obtained by pulling arm $i$ for $s$ times. Note that since the rewards are i.i.d in stochastic MAB setting, we have that distribution $\hat{\mu}_{i,s}$ is equal to $\frac{1}{s} \sum_{t=1}^{s} X_{i,t}$

Next, the authors say, using Markov's inequality, from Equation 1, one can obtain that

Equation 2: \begin{equation} P(\mu_i - \hat{\mu}_{i,s} > \epsilon) \leq e^{-s\psi^*(\epsilon)} \end{equation}

Question 2: How can I derive Equation 2 from Equation 1? Following is my attempt. Please show me how to proceed. \begin{equation} \ln \mathbb{E} e^{\lambda(X- \mathbb{E}[X])} \leq \psi(\lambda) \text{ (from Eq 1)}\\ \mathbb{E} e^{\lambda(X- \mathbb{E}[X])} \leq e^{\psi(\lambda)}\\ \frac{P(X - \mathbb{E[X]} < \epsilon)}{\epsilon} \leq e^{\psi(\lambda)} \end{equation}

Furthermore, authors say that in other words Equation 2 is, with probability at least $1-\delta$,

Equation 3: \begin{equation} \hat{\mu_{i,s}} + (\psi^*)^{-1}(\frac{1}{s} \ln \frac{1}{\delta}) > \mu_i \end{equation}

Question 3: Is the way I derive Equation 3 from Equation 2 correct? \begin{equation} P(X - \mathbb{E}[X] > \epsilon) < \delta \text{ (error should be small) } \\ e^{-s \psi^*(\epsilon)} \leq \delta \text{ (from Eq.2) } \\ -s \psi^*(\epsilon) \leq \ln \delta \\ s \psi^*(\epsilon) \geq -\ln \delta \\ \psi^*(\epsilon) \geq \frac{1}{s}\ln \frac{1}{\delta}\\ \epsilon \geq \psi^{-1}(\frac{1}{s}\ln \frac{1}{\delta})\\ \text{We need } \mu_i - \hat{\mu}_{i,a} < \epsilon. \text{So,} \\ \mu_i - \hat{\mu}_{i,a} < \psi^{-1}(\frac{1}{s}\ln \frac{1}{\delta})\\ \mu_i < \hat{\mu}_{i,a} + \psi^{-1}(\frac{1}{s}\ln \frac{1}{\delta}) \end{equation}


Let me start by answering to questions 2 and 3 first.

Question 2:

$\begin{aligned} P(\mu_i - \hat{\mu}_{i,s} > \epsilon) &= P \bigg( E[X_{i,t}]-\frac{1}{s}\sum_{t=1}^s X_{i,t} > \epsilon \bigg)\\ &= P \bigg( \sum_{t=1}^s (E[X_{i,t}] - X_{i,t}) > s \epsilon \bigg)\\ &\leq E \big[ e^{\lambda \sum_{t=1}^s (E[X_{i,t}] - X_{i,t})}\big] e^{-\lambda s \epsilon} \qquad \text{by Markov's inequality}\\ &= E \big[ e^{\lambda (E[X_{i,t}] - X_{i,t})}\big]^s e^{-\lambda s \epsilon} \qquad \text{by independence of }\{X_{i,t}\}_{t=1}^s \\ &\leq e^{\psi(\lambda)s} e^{-\lambda s \epsilon} \qquad \text{by equation 1}\\ &= e^{-s(\lambda \epsilon - \psi(\lambda))}. \end{aligned}$

Therefore it holds also for $\psi^*(\epsilon) = \sup(\lambda \epsilon - \psi(\lambda))$, where $e^{-s \psi^*(\epsilon)}$ is a tighter upper bound.

Question 3:

Your steps are correct. Let $P(\mu_i - \hat{\mu}_{i,s} > \epsilon) \leq e^{-s\psi^*(\epsilon)} = \delta$, then $\epsilon = (\psi^*)^{-1}(\frac{1}{s}\ln(1/\delta))$. Therefore,

$\begin{aligned} P(\mu_i - \hat{\mu}_{i,s} > \epsilon) &\leq \delta\\ 1-P(\mu_i - \hat{\mu}_{i,s} > \epsilon) &\geq 1-\delta\\ P(\mu_i - \hat{\mu}_{i,s} < \epsilon) &\geq 1-\delta\\ P\big(\mu_i - \hat{\mu}_{i,s} < (\psi^*)^{-1}(\frac{1}{s}\ln(1/\delta))\big) &\geq 1-\delta. \end{aligned}$

Now, regarding question 1, I do not have an intuition for the meaning of $\psi^*(\epsilon)$. However, as they explain in section 2.2 after formula (2.2), if you consider a problem in which the rewards are rescaled such that they are bounded by $[0,1]$ and let $\psi(\lambda) = \lambda^2/8$, then (2.2) becomes the Hoeffding's Lemma and $\psi^*(\epsilon) = 2\epsilon^2$. You can see that this special case makes the result in (2.3) and (2.4) more tractable.

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