# Multi-armed bandit

A row of slot machines in Las Vegas.

In probability theory, the multi-armed bandit problem (sometimes called the K-[1] or N-armed bandit problem[2]) is the problem a gambler faces at a row of slot machines, sometimes known as "one-armed bandits", when deciding which machines to play, how many times to play each machine and in which order to play them.[3] When played, each machine provides a random reward from a distribution specific to that machine. The objective of the gambler is to maximize the sum of rewards earned through a sequence of lever pulls.[4][5]

Robbins in 1952, realizing the importance of the problem, constructed convergent population selection strategies in "Some aspects of the sequential design of experiments".[6]

A theorem, the Gittins index published first by John C. Gittins gives an optimal policy in the Markov setting for maximizing the expected discounted reward.[7]

In practice, multi-armed bandits have been used to model the problem of managing research projects in a large organization, like a science foundation or a pharmaceutical company. Given a fixed budget, the problem is to allocate resources among the competing projects, whose properties are only partially known at the time of allocation, but which may become better understood as time passes.[4][5]

In early versions of the multi-armed bandit problem, the gambler has no initial knowledge about the machines. The crucial tradeoff the gambler faces at each trial is between "exploitation" of the machine that has the highest expected payoff and "exploration" to get more information about the expected payoffs of the other machines.

## Empirical motivation

The multi-armed bandit problem models an agent that simultaneously attempts to acquire new knowledge and optimize his or her decisions based on existing knowledge. There are many practical applications:

In these practical examples, the problem requires balancing reward maximization based on the knowledge already acquired with attempting new actions to further increase knowledge. This is known as the exploitation vs. exploration tradeoff in reinforcement learning.

The model can also be used to control dynamic allocation of resources to different projects, answering the question "which project should I work on" given uncertainty about the difficulty and payoff of each possibility.

Originally considered by Allied scientists in World War II, it proved so intractable that, according to Peter Whittle, it was proposed the problem be dropped over Germany so that German scientists could also waste their time on it.[10]

The version of the problem now commonly analyzed was formulated by Herbert Robbins in 1952.

## The multi-armed bandit model

The multi-armed bandit (or just bandit for short) can be seen as a set of real distributions $B = \{R_1, \dots ,R_K\}$, each distribution being associated with the rewards delivered by one of the K levers. Let $\mu_1, \dots , \mu_K$ be the mean values associated with these reward distributions. The gambler iteratively plays one lever per round and observes the associated reward. The objective is to maximize the sum of the collected rewards. The horizon H is the number of rounds that remain to be played. The bandit problem is formally equivalent to a one-state Markov decision process. The regret $\rho$ after T rounds is defined as the expected difference between the reward sum associated with an optimal strategy and the sum of the collected rewards: $\rho = T \mu^* - \sum_{t=1}^T \widehat{r}_t$, where $\mu^*$ is the maximal reward mean, $\mu^* = \max_k \{ \mu_k \}$, and $\widehat{r}_t$ is the reward at time t. A strategy whose average regret per round $\rho / T$ tends to zero with probability 1 when the number of played rounds tends to infinity is a zero-regret strategy. Intuitively, zero-regret strategies are guaranteed to converge to an optimal strategy, not necessarily unique, if enough rounds are played.

## Variations

A common formulation is the Binary multi-armed bandit or Bernoulli multi-armed bandit, which issues a reward of one with probability p, and otherwise a reward of zero.

Another formulation of the multi-armed bandit has each arm representing an independent Markov machine. Each time a particular arm is played, the state of that machine advances to a new one, chosen according to the Markov state evolution probabilities. There is a reward depending on the current state of the machine. In a generalisation called the "restless bandit problem", the states of non-played arms can also evolve over time.[11] There has also been discussion of systems where the number of choices (about which arm to play) increases over time.[12]

Computer science researchers have studied multi-armed bandits under worst-case assumptions, obtaining positive[clarification needed] results for finite numbers of trials with both stochastic [1] and non-stochastic[13] arm payoffs.

## Bandit strategies

A major breakthrough was the construction of optimal population selection strategies, or policies (that possess uniformly maximum convergence rate to the population with highest mean) in the work described below.

### Optimal solutions

In the paper "Asymptotically efficient adaptive allocation rules", Lai and Robbins[14] (following many papers of Robbins and his co-workers going back to Robbins (1952)) constructed convergent population selection policies that possess the fastest rate of convergence (to the population with highest mean) for the case that the population reward distributions are the one-parameter exponential family. Then, in Katehakis and Robbins [15] simplifications of the policy and the main proof were given for the case of Normal populations with known variances. The next notable progress was obtained by Burnetas and Katehakis in the "Optimal adaptive policies for sequential allocation problems",[16] where index based policies with uniformly maximum convergence rate were constructed, under more general conditions that include the case in which the distributions of outcomes from each population depend on a vector of unknown parameters. Burnetas AN and Katehakis MN (1996) also provided an explicit solution for the important case in which the distributions of outcomes follow arbitrary (i.e., nonparametric) discrete, univariate distributions.

Later in "Optimal adaptive policies for Markov decision processes"[17] Burnetas and Katehakis studied the much larger model of Markov Decision Processes under partial information, where the transition law and/or the expected one period rewards may depend on unknown parameter. In this work the explicit form for a class of adaptive policies that possess uniformly maximum convergence rate properties for the total expected finite horizon reward, were constructed under sufficient assumptions of finite state-action spaces and irreducibility of the transition law. A main feature of these policies is that the choice of actions, at each state and time period, is based on indices that are inflations of the right-hand side of the estimated average reward optimality equations. These inflations have recently been called the optimistic approach in the work of Tewari and Bartlett,[18] Ortner[19] Filippi, Cappé, and Garivier,[20] and Honda and Takemura.[21]

### Approximate solutions

Many strategies exist which provide an approximate solution to the bandit problem, and can be put into the four broad categories detailed below.

#### Semi-uniform strategies

Semi-uniform strategies were the earliest (and simplest) strategies discovered to approximately solve the bandit problem. All those strategies have in common a greedy behavior where the best lever (based on previous observations) is always pulled except when a (uniformly) random action is taken.

• Epsilon-greedy strategy: The best lever is selected for a proportion $1 - \epsilon$ of the trials, and a lever is selected at random (with uniform probability) for a proportion $\epsilon$. A typical parameter value might be $\epsilon = 0.1$, but this can vary widely depending on circumstances and predilections.
• Epsilon-first strategy: A pure exploration phase is followed by a pure exploitation phase. For $N$ trials in total, the exploration phase occupies $\epsilon N$ trials and the exploitation phase $(1 - \epsilon) N$ trials. During the exploration phase, a lever is randomly selected (with uniform probability); during the exploitation phase, the best lever is always selected.
• Epsilon-decreasing strategy: Similar to the epsilon-greedy strategy, except that the value of $\epsilon$ decreases as the experiment progresses, resulting in highly explorative behaviour at the start and highly exploitative behaviour at the finish.
• Adaptive epsilon-greedy strategy based on value differences (VDBE): Similar to the epsilon-decreasing strategy, except that epsilon is reduced on basis of the learning progress instead of manual tuning (Tokic, 2010).[22] High fluctuations in the value estimates lead to a high epsilon (exploration); low fluctuations to a low epsilon (exploitation). Further improvements can be achieved by a softmax weighted action selection in case of exploratory actions (Tokic & Palm, 2011).[23]
• Contextual-Epsilon-greedy strategy: Similar to the epsilon-greedy strategy, except that the value of $\epsilon$ is computed regarding the situation in experiment processes, which let the algorithm be Context-Aware. It is based on dynamic exploration/exploitation and can adaptively balance the two aspects by deciding which situation is most relevant for exploration or exploitation, resulting in highly explorative behavior when the situation is not critical and highly exploitative behavior at critical situation (Bouneffouf et al., 2012).[24]

#### Probability matching strategies

Probability matching strategies reflect the idea that the number of pulls for a given lever should match its actual probability of being the optimal lever. Probability matching strategies are also known as Thompson sampling or Bayesian Bandits[25] are surprisingly easy to implement if you can sample from the posterior for the mean value of each alternative.

Probability matching strategies also admit solutions to so-called contextual bandit problems.

#### Pricing strategies

Pricing strategies establish a price for each lever. For example as illustrated with the POKER algorithm,[26] the price can be the sum of the expected reward plus an estimation of extra future rewards that will gain through the additional knowledge. The lever of highest price is always pulled.

#### Strategies with ethical constraints

These strategies minimize the assignment of any patient to an inferior arm ("physician's duty"). In a typical case, they minimize expected successes lost (ESL), that is, the expected number of favorable outcomes that were missed because of assignment to an arm later proved to be inferior. Another version minimizes resources wasted on any inferior, more expensive, treatment.[8]

## References

1. ^ a b Auer, P.; Cesa-Bianchi, N.; Fischer, P. (2002). Machine Learning 47 (2/3): 235. doi:10.1023/A:1013689704352. edit
2. ^ Katehakis, M. N.; Veinott, A. F. (1987). "The Multi-Armed Bandit Problem: Decomposition and Computation". Mathematics of Operations Research 12 (2): 262. doi:10.1287/moor.12.2.262. edit
3. ^ Richard R. Weber (1992). "On the Gittins Index for Multiarmed Bandits". The Annals of Applied Probability 2 (4): 1024–1033. doi:10.2307.2F2959678. JSTOR 2959678. edit
4. ^ a b c Gittins, J. C. (1989), Multi-armed bandit allocation indices, Wiley-Interscience Series in Systems and Optimization., Chichester: John Wiley & Sons, Ltd., ISBN 0-471-92059-2
5. ^ a b c Berry, Donald A.; Fristedt, Bert (1985), Bandit problems: Sequential allocation of experiments, Monographs on Statistics and Applied Probability, London: Chapman & Hall, ISBN 0-412-24810-7
6. ^ Robbins, H. (1952). "Some aspects of the sequential design of experiments". Bulletin of the American Mathematical Society 58 (5): 527–535. doi:10.1090/S0002-9904-1952-09620-8. edit
7. ^ J. C. Gittins (1979). "Bandit Processes and Dynamic Allocation Indices". Journal of the Royal Statistical Society. Series B (Methodological) 41 (2): 148–177. JSTOR 2985029. edit
8. ^ a b Press, William H. (2009), "Bandit solutions provide unified ethical models for randomized clinical trials and comparative effectiveness research", Proceedings of the National Academy of Sciences 106 (52): 22387–22392, doi:10.1073/pnas.0912378106, PMC 2793317, PMID 20018711.
9. ^ Press (1986)
10. ^ Whittle, Peter (1979), "Discussion of Dr Gittins' paper", Journal of the Royal Statistical Society, Series B 41 (2): 165, JSTOR 2985029
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