[Bandit] 2. Explore-Then-Commit algorithm

$\newcommand{\argmin}{\mathop{\mathrm{argmin}}\limits}$ $\newcommand{\argmax}{\mathop{\mathrm{argmax}}\limits}$

Here, we continue to describe the multi-armed bandit problem in detail. The notion of regret will be introduced. Then our first bandit algorithm, explore-then-commit (ETC) will be described.


Multi-armed bandit (continued)

Suppose we have a set of possible actions $\mathcal{A}$ and a stochastic bandit (a collection of bandits) $\nu$ where

\[\nu := (P_{\theta_i})_{i \in \mathcal{A}}.\]

The algorithm interact with bandits through time $t=1,\cdots,T.$ At each time $t$ the algorithm choose an arm $a(t) \in \mathcal{A}.$ Then the reward $r_i(t) \sim P_{\theta_{a(t)}}$ is generated and revealed to the algorithm. Such interactions (action-reward pairs) between the algorithm and the environment is logged into history $\mathcal{H}$ where

\[\mathcal{H}_{t-1} := \{\left(a(t), r(t)\right): t=1,\cdots,t-1\}.\]

The algorithm can be characterized by a sequence of probabilities $(\pi_t)_{t=1,\cdots,T}.$ Arms are chosen according to conditional probability given history $\pi_t = \pi(\cdot|\mathcal{H}_{t-1}).$ Our goal is to find $(\pi_t)$ that chooses the best actions at each $t.$

Usually such problem is solved by not maximizing the cumulative reward, but by minimizing the regret function $R$. As the name implies, $R$ is the value that represents the total loss that we had under our choices. To formally define the regret, let $\mu_i$ be the (unknown) expected reward from the $i$th arm, and $\mu_{i^*}$ be the expected reward from the (also unknown) best $i^*$th arm.

\[\mu_i = E r_i(t) \tag{1}, \\ \mu_{i^*} \ge \mu_j,~ \forall j \ne i^*.\]

Let $\Delta_j$ be the difference between the mean reward from the best choice and the choice $j$.

\[\Delta_j := \mu_{i^*} - \mu_j.\]

The regret $R$ is defines as follows:

\[\begin{aligned} R(T) &:= E \sum_{t=1}^T \left( \mu_{i^*} - r_{a(t)}(t) \right) \\ &= \sum_{t=1}^T \Delta_{a(t)}. \end{aligned}\]


Exploration-exploitation trade-off

What makes the bandit problem so challenging is that only a small portion of rewards are revealed: we do not know underlying $P_{\theta_i}$’s unless we chose them to get enough samples. This problem is called exploration-exploitation trade-off. Simply put, we need to find and use the best among $\nu$ in limited amount of time $T.$

  • Exploration: Try different arms to get information about $P_{\theta_i}.$
  • Exploitation: Pick the best arm according to estimates of rewards based on current information about $P_{\theta_i}.$

One thing to note is that in exploitation, we not only need to get the estimate, but also the variabiliy of it.


Theoretical tools

The followings are some of the tools that will be used in regret analysis.

Markov and Chebeshev’s inequality

For a random variable $X \ge 0$ and $\epsilon > 0,$ $$ P(X \ge \epsilon) \le \frac{EX}{\epsilon}. $$
For a random variable $X$ and $\epsilon > 0,$ $$ P(|X - EX| \ge \epsilon) \le \frac{\text{Var}(X)}{\epsilon^2}. $$


subgaussianity


A random variable $X$ is $\sigma$-subgaussian, if there exists a $\sigma>0$ such that $$ E e^{\lambda X} \le e^{\lambda^2\sigma^2/2}. $$ That is, the MGF of $X$ is bounded above by another random variable that follows $\mathcal{N}(0,\sigma^2).$

A $\sigma$-subgaussian $X$ can be viewed as a random variable that has tail probabilities smaller than that of $\mathcal{N}(0, \sigma^2).$ To be specific, the right tail probablity can be achieved by Chernoff method. Be careful since the bound is not tight for normal random variables.

For a $\sigma$-subgaussian random variable $X,$ $$ P(X \ge \epsilon) \le e^{-\frac{\epsilon^2}{2\sigma^2}}. $$

Let $\lambda = \epsilon / \sigma^2,$ then $$ \begin{aligned} P(X \ge \epsilon) &= P(e^{\lambda X} \ge e^{\lambda \epsilon}) \\ &\le Ee^{\lambda X}/e^{\lambda \epsilon} \\ &\le e^{\lambda^2\sigma^2/2}/e^{\lambda\epsilon} \\ &= e^{-\frac{\epsilon^2}{2\sigma^2}}. \end{aligned} $$ The first inequality follows from Markov's inequality and the second follows from subgaussianity.

Similarly the same inequality holds for the lower tail, thus we get the bound

\[P(|X| \ge \epsilon) \le 2 e^{-\frac{\epsilon^2}{2\sigma^2}},\]

which is (as we set $\delta = 2e^{-\frac{\epsilon^2}{2\sigma^2}}$) equivalent to

\[P\left(|X| \ge \sqrt{2\sigma^2 \log\left(\frac{2}{\delta}\right)}\right) \le \delta.\]
$X$ is $\sigma$-subgaussian, $X_1, X_2$ are $\sigma_1$- and $\sigma_2$-subgaussian, respectively. Then
(i) $EX=0$ and $\text{Var}(X) \le \sigma^2.$
(ii) $cX$ is $|c|\sigma$-subgaussian for all $c \in \mathbb{R}.$
(iii) $X_1+X_2$ is $\sqrt{\sigma_1^2 + \sigma_2^2}$-subgaussian.

With the help of the lemma, applying Chernoff bound to the sample mean $\hat \mu=\frac{\sum_{t=1}^T X_t}{T}$ of $\sigma$-subgaussian $X$ yields

\[P(|\hat \mu - \mu| \ge \epsilon) \le 2 e^{-T\frac{\epsilon^2}{2\sigma^2}}.\]


Bernstein inequality

If $P(|X| \le c) = 1$ and $EX=0,$ $\text{Var}(X)=\sigma^2,$ then for any $t>0,$ $$ Ee^{tX} \le \exp\left( t^2\sigma^2\frac{e^{tc} - 1 - tc}{(tc)^2} \right). $$

$$ \begin{aligned} Ee^{tX} &= E\left(1 + tX + \sum_{r=2}^\infty \frac{t^rX^r}{r!} \right) \\ &= 1 + t^2\sigma^2F \\ &\le e^{t^2\sigma^2F} \end{aligned} $$ where $F=\sum_{r=2}^\infty \frac{t^{r-2}}{\sigma^2} \frac{EX^r}{r!}.$ Since $EX^r \le c^{r-2}\sigma^2$ for $r\ge2,$ we get the desired result.
Let $X_t$'s be independent random variables with $P(|X_t| \le c) = 1$ and $EX_i=0,$ $\sigma^2 = \sum_{t=1}^T \text{Var}(X_t).$ Then for any $\epsilon>0,$ $$ P(|\hat \mu| \ge \epsilon) \le 2 \exp\left( -\frac{T\epsilon^2}{2\sigma^2 + 2c\epsilon/3} \right). $$


Explore-then-Commit (ETC)

The algorithm

The first and probably the simplest form of the bandit algorithm is explore-then-commit (ETC). As the name speaks for it self, ETC divides total time $T$ into exploration step and exploitation step.

  1. Exploration step. During rounds $1,\cdots,mN,$ the algorithm plays each $N$ arms $m$ times.

  2. Exploitation step. The algorithm selects the arm with the highest empirical mean reward

    \[a = \argmax_{i=1,\cdots,N} \hat \mu_i (mN)\]

    where

    \[\hat \mu_i(t) := \frac{\sum_{\tau=1}^t r_i(\tau) \mathbf{1}(i=a(\tau))}{\sum_{\tau=1}^t \mathbf{1}(i=a(\tau))}.\]

    During the rest $mN+1,\cdots,T$ rounds, the algorithm chooses that $a$. That is, $a(t) = a$ for all $t=mN+1,\cdots,T.$

Although the algorithm is simple, it is (quite predictively) not a great performing one. Choice of the exploration parameter $m$ depends on the underlying knowledge of $\Delta_i$’s, which is not available. In addition, the algorithm is suboptimal in regret analysis.


Regret analysis

[To be appended]


References

  • Lattimore, Szepesvari. 2020. Bandit Algorithms. Cambridge University .
  • Seminar in Recent Development of Applied Statistics (Spring, 2021) @ Seoul National University, Republic of Korea (instructor: Prof. Myunghee Cho Paik).