Multi-armed bandit

From HandWiki
Short description: Resource problem in machine learning
A row of slot machines in Las Vegas

In probability theory and machine learning, the multi-armed bandit problem (sometimes called the K-[1] or N-armed bandit problem[2]) is a problem in which a decision maker iteratively selects one of multiple fixed choices (i.e. arms or actions) when the properties of each choice are only partially known at the time of allocation, and may become better understood as time passes. A fundamental aspect of bandit problems is that choosing an arm does not affect the properties of the arm or other arms.[3]

Instances of the multi-armed bandit problem include the task of iteratively allocating a fixed, limited set of resources between competing (alternative) choices in a way that minimizes the regret.[4][5] Alternative setups for the multi-armed bandit problem include the "best arm identification" problem where the goal is instead to identify the best choice by the end of a finite number of rounds.[6]

The multi-armed bandit problem is a classic reinforcement learning problem that exemplifies the exploration–exploitation tradeoff dilemma. In contrast to general RL, the selected actions in bandit problems do not affect the reward distribution of the arms. The name comes from imagining a gambler at a row of slot machines (sometimes known as "one-armed bandits"), who has to decide which machines to play, how many times to play each machine and in which order to play them, and whether to continue with the current machine or try a different machine.[7] The multi-armed bandit problem also falls into the broad category of stochastic scheduling.

In the problem, each machine provides a random reward from a probability distribution specific to that machine, that is not known a-priori. The objective of the gambler is to maximize the sum of rewards earned through a sequence of lever pulls.[4][5] 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. The trade-off between exploration and exploitation is also faced in machine learning. In practice, multi-armed bandits have been used to model problems such as managing research projects in a large organization, like a science foundation or a pharmaceutical company.[4][5] In early versions of the problem, the gambler begins with no initial knowledge about the machines.

Herbert Robbins in 1952, realizing the importance of the problem, constructed convergent population selection strategies in "some aspects of the sequential design of experiments".[8] A theorem, the Gittins index, first published by John C. Gittins, gives an optimal policy for maximizing the expected discounted reward.[9]

Empirical motivation

How must a given budget be distributed among these research departments to maximize results?

The multi-armed bandit problem models an agent that simultaneously attempts to acquire new knowledge (called "exploration") and optimize their decisions based on existing knowledge (called "exploitation"). The agent attempts to balance these competing tasks in order to maximize their total value over the period of time considered. There are many practical applications of the bandit model, for example:

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 machine learning.

The model has also been used to control dynamic allocation of resources to different projects, answering the question of which project to work on, given uncertainty about the difficulty and payoff of each possibility.[14]

Originally considered by Allied scientists in World War II, it proved so intractable that, according to Peter Whittle, the problem was proposed to be dropped over Germany so that German scientists could also waste their time on it.[15]

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

The multi-armed bandit model

The multi-armed bandit (short: bandit or MAB) can be seen as a set of real distributions [math]\displaystyle{ B = \{R_1, \dots ,R_K\} }[/math], each distribution being associated with the rewards delivered by one of the [math]\displaystyle{ K \in \mathbb{N}^+ }[/math] levers. Let [math]\displaystyle{ \mu_1, \dots , \mu_K }[/math] 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 [math]\displaystyle{ H }[/math] 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 [math]\displaystyle{ \rho }[/math] after [math]\displaystyle{ T }[/math] rounds is defined as the expected difference between the reward sum associated with an optimal strategy and the sum of the collected rewards:

[math]\displaystyle{ \rho = T \mu^* - \sum_{t=1}^T \widehat{r}_t }[/math],

where [math]\displaystyle{ \mu^* }[/math] is the maximal reward mean, [math]\displaystyle{ \mu^* = \max_k \{ \mu_k \} }[/math], and [math]\displaystyle{ \widehat{r}_t }[/math] is the reward in round t.

A zero-regret strategy is a strategy whose average regret per round [math]\displaystyle{ \rho / T }[/math] tends to zero with probability 1 when the number of played rounds tends to infinity.[16] Intuitively, zero-regret strategies are guaranteed to converge to a (not necessarily unique) optimal strategy 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 [math]\displaystyle{ p }[/math], 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 generalization called the "restless bandit problem", the states of non-played arms can also evolve over time.[17] There has also been discussion of systems where the number of choices (about which arm to play) increases over time.[18]

Computer science researchers have studied multi-armed bandits under worst-case assumptions, obtaining algorithms to minimize regret in both finite and infinite (asymptotic) time horizons for both stochastic[1] and non-stochastic[19] 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[20] (following papers of Robbins and his co-workers going back to Robbins in the year 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[21] 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 paper "Optimal adaptive policies for sequential allocation problems",[22] 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 and Katehakis (1996) also provided an explicit solution for the important case in which the distributions of outcomes follow arbitrary (i.e., non-parametric) discrete, univariate distributions.

Later in "Optimal adaptive policies for Markov decision processes"[23] 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 parameters. In this work, the authors constructed an explicit form for a class of adaptive policies with uniformly maximum convergence rate properties for the total expected finite horizon reward 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,[24] Ortner[25] Filippi, Cappé, and Garivier,[26] and Honda and Takemura.[27]

For Bernoulli multi-armed bandits, Pilarski et al.[28] studied computation methods of deriving fully optimal solutions (not just asymptotically) using dynamic programming in the paper "Optimal Policy for Bernoulli Bandits: Computation and Algorithm Gauge."[28] Via indexing schemes, lookup tables, and other techniques, this work provided practically applicable optimal solutions for Bernoulli bandits provided that time horizons and numbers of arms did not become excessively large. Pilarski et al.[29] later extended this work in "Delayed Reward Bernoulli Bandits: Optimal Policy and Predictive Meta-Algorithm PARDI"[29] to create a method of determining the optimal policy for Bernoulli bandits when rewards may not be immediately revealed following a decision and may be delayed. This method relies upon calculating expected values of reward outcomes which have not yet been revealed and updating posterior probabilities when rewards are revealed.

When optimal solutions to multi-arm bandit tasks[30] are used to derive the value of animals' choices, the activity of neurons in the amygdala and ventral striatum encodes the values derived from these policies, and can be used to decode when the animals make exploratory versus exploitative choices. Moreover, optimal policies better predict animals' choice behavior than alternative strategies (described below). This suggests that the optimal solutions to multi-arm bandit problems are biologically plausible, despite being computationally demanding.[31]

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:[32] The best lever is selected for a proportion [math]\displaystyle{ 1 - \epsilon }[/math] of the trials, and a lever is selected at random (with uniform probability) for a proportion [math]\displaystyle{ \epsilon }[/math]. A typical parameter value might be [math]\displaystyle{ \epsilon = 0.1 }[/math], but this can vary widely depending on circumstances and predilections.
  • Epsilon-first strategy[citation needed]: A pure exploration phase is followed by a pure exploitation phase. For [math]\displaystyle{ N }[/math] trials in total, the exploration phase occupies [math]\displaystyle{ \epsilon N }[/math] trials and the exploitation phase [math]\displaystyle{ (1 - \epsilon) N }[/math] 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[citation needed]: Similar to the epsilon-greedy strategy, except that the value of [math]\displaystyle{ \epsilon }[/math] 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).[33] High fluctuations in the value estimates lead to a high epsilon (high exploration, low exploitation); low fluctuations to a low epsilon (low exploration, high exploitation). Further improvements can be achieved by a softmax-weighted action selection in case of exploratory actions (Tokic & Palm, 2011).[34]
  • Adaptive epsilon-greedy strategy based on Bayesian ensembles (Epsilon-BMC): An adaptive epsilon adaptation strategy for reinforcement learning similar to VBDE, with monotone convergence guarantees. In this framework, the epsilon parameter is viewed as the expectation of a posterior distribution weighting a greedy agent (that fully trusts the learned reward) and uniform learning agent (that distrusts the learned reward). This posterior is approximated using a suitable Beta distribution under the assumption of normality of observed rewards. In order to address the possible risk of decreasing epsilon too quickly, uncertainty in the variance of the learned reward is also modeled and updated using a normal-gamma model. (Gimelfarb et al., 2019).[35]

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,[36][37] and 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.[36]

Pricing strategies

Pricing strategies establish a price for each lever. For example, as illustrated with the POKER algorithm,[16] 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.

Contextual bandit

A useful generalization of the multi-armed bandit is the contextual multi-armed bandit. At each iteration an agent still has to choose between arms, but they also see a d-dimensional feature vector, the context vector they can use together with the rewards of the arms played in the past to make the choice of the arm to play. Over time, the learner's aim is to collect enough information about how the context vectors and rewards relate to each other, so that it can predict the next best arm to play by looking at the feature vectors.[38]

Approximate solutions for contextual bandit

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

Online linear bandits

  • LinUCB (Upper Confidence Bound) algorithm: the authors assume a linear dependency between the expected reward of an action and its context and model the representation space using a set of linear predictors.[39][40]
  • LinRel (Linear Associative Reinforcement Learning) algorithm: Similar to LinUCB, but utilizes Singular-value decomposition rather than Ridge regression to obtain an estimate of confidence.[41][42]

Online non-linear bandits

  • UCBogram algorithm: The nonlinear reward functions are estimated using a piecewise constant estimator called a regressogram in nonparametric regression. Then, UCB is employed on each constant piece. Successive refinements of the partition of the context space are scheduled or chosen adaptively.[43][44][45]
  • Generalized linear algorithms: The reward distribution follows a generalized linear model, an extension to linear bandits.[46][47][48][49]
  • KernelUCB algorithm: a kernelized non-linear version of linearUCB, with efficient implementation and finite-time analysis.[50]
  • Bandit Forest algorithm: a random forest is built and analyzed w.r.t the random forest built knowing the joint distribution of contexts and rewards.[51]
  • Oracle-based algorithm: The algorithm reduces the contextual bandit problem into a series of supervised learning problem, and does not rely on typical realizability assumption on the reward function.[52]

Constrained contextual bandit

In practice, there is usually a cost associated with the resource consumed by each action and the total cost is limited by a budget in many applications such as crowdsourcing and clinical trials. Constrained contextual bandit (CCB) is such a model that considers both the time and budget constraints in a multi-armed bandit setting. A. Badanidiyuru et al.[53] first studied contextual bandits with budget constraints, also referred to as Resourceful Contextual Bandits, and show that a [math]\displaystyle{ O(\sqrt{T}) }[/math] regret is achievable. However, their work focuses on a finite set of policies, and the algorithm is computationally inefficient.

Framework of UCB-ALP for constrained contextual bandits

A simple algorithm with logarithmic regret is proposed in:[54]

  • UCB-ALP algorithm: The framework of UCB-ALP is shown in the right figure. UCB-ALP is a simple algorithm that combines the UCB method with an Adaptive Linear Programming (ALP) algorithm, and can be easily deployed in practical systems. It is the first work that show how to achieve logarithmic regret in constrained contextual bandits. Although[54] is devoted to a special case with single budget constraint and fixed cost, the results shed light on the design and analysis of algorithms for more general CCB problems.

Adversarial bandit

Another variant of the multi-armed bandit problem is called the adversarial bandit, first introduced by Auer and Cesa-Bianchi (1998). In this variant, at each iteration, an agent chooses an arm and an adversary simultaneously chooses the payoff structure for each arm. This is one of the strongest generalizations of the bandit problem[55] as it removes all assumptions of the distribution and a solution to the adversarial bandit problem is a generalized solution to the more specific bandit problems.

Example: Iterated prisoner's dilemma

An example often considered for adversarial bandits is the iterated prisoner's dilemma. In this example, each adversary has two arms to pull. They can either Deny or Confess. Standard stochastic bandit algorithms don't work very well with these iterations. For example, if the opponent cooperates in the first 100 rounds, defects for the next 200, then cooperate in the following 300, etc. then algorithms such as UCB won't be able to react very quickly to these changes. This is because after a certain point sub-optimal arms are rarely pulled to limit exploration and focus on exploitation. When the environment changes the algorithm is unable to adapt or may not even detect the change.

Approximate solutions

Exp3[56]

EXP3 is a popular algorithm for adversarial multiarmed bandits, suggested and analyzed in this setting by Auer et al. [2002b]. Recently there was an increased interest in the performance of this algorithm in the stochastic setting, due to its new applications to stochastic multi-armed bandits with side information [Seldin et al., 2011] and to multi-armed bandits in the mixed stochastic-adversarial setting [Bubeck and Slivkins, 2012]. The paper presented an empirical evaluation and improved analysis of the performance of the EXP3 algorithm in the stochastic setting, as well as a modification of the EXP3 algorithm capable of achieving “logarithmic” regret in stochastic environment

Algorithm
 Parameters: Real [math]\displaystyle{ \gamma \in (0,1]  }[/math]
 
 Initialisation: [math]\displaystyle{ \omega_i(1) = 1 }[/math] for [math]\displaystyle{ i = 1,...,K }[/math]
 
 For each t = 1, 2, ..., T
  1. Set [math]\displaystyle{ p_i(t) = (1 - \gamma)\frac{\omega_i(t)}{\sum_{j=1}^{K}\omega_j(t)} + \frac{\gamma}{K} }[/math]       [math]\displaystyle{ i = 1,...,K }[/math]
  2. Draw [math]\displaystyle{ i_t }[/math] randomly according to the probabilities [math]\displaystyle{ p_1(t),...,p_K(t) }[/math]
  3. Receive reward [math]\displaystyle{ x_{i_t}(t) \in [0,1] }[/math]
  4. For [math]\displaystyle{ j = 1,...,K }[/math] set:
      [math]\displaystyle{ \hat{x}_j(t) = \begin{cases}x_j(t)/p_j(t) & \text{if }j = i_t \\0, & \text{otherwise}\end{cases} }[/math]
 
      [math]\displaystyle{ \omega_j(t+1) = \omega_j(t) \exp(\gamma\hat{x}_j(t)/K) }[/math]
Explanation

Exp3 chooses an arm at random with probability [math]\displaystyle{ (1 - \gamma) }[/math] it prefers arms with higher weights (exploit), it chooses with probability [math]\displaystyle{ \gamma }[/math] to uniformly randomly explore. After receiving the rewards the weights are updated. The exponential growth significantly increases the weight of good arms.

Regret analysis

The (external) regret of the Exp3 algorithm is at most [math]\displaystyle{ O(\sqrt{KTlog(K)}) }[/math]

Follow the perturbed leader (FPL) algorithm

Algorithm
 Parameters: Real [math]\displaystyle{ \eta }[/math]
 
 Initialisation: [math]\displaystyle{ \forall i: R_i(1) = 0 }[/math]
 
 For each t = 1,2,...,T
  1. For each arm generate a random noise from an exponential distribution [math]\displaystyle{ \forall i : Z_i(t) \sim Exp(\eta) }[/math]
  2. Pull arm [math]\displaystyle{ I(t) }[/math]: [math]\displaystyle{ I(t)=arg\max_i\{R_i(t) + Z_i(t)\} }[/math]
     Add noise to each arm and pull the one with the highest value
  3. Update value: [math]\displaystyle{ R_{I(t)}(t+1) = R_{I(t)}(t) + x_{I(t)}(t) }[/math]
     The rest remains the same
Explanation

We follow the arm that we think has the best performance so far adding exponential noise to it to provide exploration.[57]

Exp3 vs FPL

Exp3 FPL
Maintains weights for each arm to calculate pulling probability Doesn't need to know the pulling probability per arm
Has efficient theoretical guarantees The standard FPL does not have good theoretical guarantees
Might be computationally expensive (calculating the exponential terms) Computationally quite efficient

Infinite-armed bandit

In the original specification and in the above variants, the bandit problem is specified with a discrete and finite number of arms, often indicated by the variable [math]\displaystyle{ K }[/math]. In the infinite armed case, introduced by Agrawal (1995),[58] the "arms" are a continuous variable in [math]\displaystyle{ K }[/math] dimensions.

Non-stationary bandit

This framework refers to the multi-armed bandit problem in a non-stationary setting (i.e., in presence of concept drift). In the non-stationary setting, it is assumed that the expected reward for an arm [math]\displaystyle{ k }[/math] can change at every time step [math]\displaystyle{ t \in \mathcal{T} }[/math]: [math]\displaystyle{ \mu_{t-1}^k \neq \mu_t^k }[/math]. Thus, [math]\displaystyle{ \mu_t^k }[/math] no longer represents the whole sequence of expected (stationary) rewards for arm [math]\displaystyle{ k }[/math]. Instead, [math]\displaystyle{ \mu^k }[/math] denotes the sequence of expected rewards for arm [math]\displaystyle{ k }[/math], defined as [math]\displaystyle{ \mu^k = \{\mu_t^k\}_{t=1}^T }[/math].[59]

A dynamic oracle represents the optimal policy to be compared with other policies in the non-stationary setting. The dynamic oracle optimises the expected reward at each step [math]\displaystyle{ t \in \mathcal{T} }[/math] by always selecting the best arm, with expected reward of [math]\displaystyle{ \mu_t^* }[/math]. Thus, the cumulative expected reward [math]\displaystyle{ \mathcal{D}(T) }[/math] for the dynamic oracle at final time step [math]\displaystyle{ T }[/math] is defined as:

[math]\displaystyle{ \mathcal{D}(T) = \sum_{t=1}^T{\mu_t^*} }[/math]

Hence, the regret [math]\displaystyle{ \rho^\pi(T) }[/math] for policy [math]\displaystyle{ \pi }[/math] is computed as the difference between [math]\displaystyle{ \mathcal{D}(T) }[/math] and the cumulative expected reward at step [math]\displaystyle{ T }[/math] for policy [math]\displaystyle{ \pi }[/math]:

[math]\displaystyle{ \rho^\pi(T) = \sum_{t=1}^T{\mu_t^*} - \mathbb{E}_\pi^\mu \left[ \sum_{t=1}^T{r_t} \right] = \mathcal{D}(T) - \mathbb{E}_\pi^\mu \left[ \sum_{t=1}^T{r_t} \right] }[/math]

Garivier and Moulines derive some of the first results with respect to bandit problems where the underlying model can change during play. A number of algorithms were presented to deal with this case, including Discounted UCB[60] and Sliding-Window UCB.[61] A similar approach based on Thompson Sampling algorithm is the f-Discounted-Sliding-Window Thompson Sampling (f-dsw TS)[62] proposed by Cavenaghi et al. The f-dsw TS algorithm exploits a discount factor on the reward history and an arm-related sliding window to contrast concept drift in non-stationary environments. Another work by Burtini et al. introduces a weighted least squares Thompson sampling approach (WLS-TS), which proves beneficial in both the known and unknown non-stationary cases.[63]

Other variants

Many variants of the problem have been proposed in recent years.

Dueling bandit

The dueling bandit variant was introduced by Yue et al. (2012)[64] to model the exploration-versus-exploitation tradeoff for relative feedback. In this variant the gambler is allowed to pull two levers at the same time, but they only get a binary feedback telling which lever provided the best reward. The difficulty of this problem stems from the fact that the gambler has no way of directly observing the reward of their actions. The earliest algorithms for this problem are InterleaveFiltering,[64] Beat-The-Mean.[65] The relative feedback of dueling bandits can also lead to voting paradoxes. A solution is to take the Condorcet winner as a reference.[66]

More recently, researchers have generalized algorithms from traditional MAB to dueling bandits: Relative Upper Confidence Bounds (RUCB),[67] Relative EXponential weighing (REX3),[68] Copeland Confidence Bounds (CCB),[69] Relative Minimum Empirical Divergence (RMED),[70] and Double Thompson Sampling (DTS).[71]

Collaborative bandit

Approaches using multiple bandits that cooperate sharing knowledge in order to better optimize their performance started in 2013 with "A Gang of Bandits",[72] an algorithm relying on a similarity graph between the different bandit problems to share knowledge. The need of a similarity graph was removed in 2014 by the work on the CLUB algorithm.[73] Following this work, several other researchers created algorithms to learn multiple models at the same time under bandit feedback. For example, COFIBA was introduced by Li and Karatzoglou and Gentile (SIGIR 2016),[74] where the classical collaborative filtering, and content-based filtering methods try to learn a static recommendation model given training data.

Combinatorial bandit

The Combinatorial Multiarmed Bandit (CMAB) problem[75][76][77] arises when instead of a single discrete variable to choose from, an agent needs to choose values for a set of variables. Assuming each variable is discrete, the number of possible choices per iteration is exponential in the number of variables. Several CMAB settings have been studied in the literature, from settings where the variables are binary[76] to more general setting where each variable can take an arbitrary set of values.[77]

See also

References

  1. 1.0 1.1 Auer, P.; Cesa-Bianchi, N.; Fischer, P. (2002). "Finite-time Analysis of the Multiarmed Bandit Problem". Machine Learning 47 (2/3): 235–256. doi:10.1023/A:1013689704352. 
  2. Katehakis, Michael N.; Veinott, Jr., Arthur F. (1987). "The Multi-Armed Bandit Problem: Decomposition and Computation". Mathematics of Operations Research 12 (2): 262–268. doi:10.1287/moor.12.2.262. 
  3. Sébastien Bubeck and Nicolò Cesa-Bianchi (2012), "Regret Analysis of Stochastic and Nonstochastic Multi-armed Bandit Problems", Foundations and Trends® in Machine Learning: Vol. 5: No. 1, pp 1-122. http://dx.doi.org/10.1561/2200000024
  4. 4.0 4.1 4.2 4.3 Gittins, J. C. (1989), Multi-armed bandit allocation indices, Wiley-Interscience Series in Systems and Optimization., Chichester: John Wiley & Sons, Ltd., ISBN 978-0-471-92059-5 
  5. 5.0 5.1 5.2 5.3 Berry, Donald A.; Fristedt, Bert (1985), Bandit problems: Sequential allocation of experiments, Monographs on Statistics and Applied Probability, London: Chapman & Hall, ISBN 978-0-412-24810-8 
  6. Soare, M., Lazaric, A., & Munos, R. (2014). Best-Arm Identification in Linear Bandits. ArXiv, abs/1409.6110.
  7. Weber, Richard (1992), "On the Gittins index for multiarmed bandits", Annals of Applied Probability 2 (4): 1024–1033, doi:10.1214/aoap/1177005588 
  8. 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. 
  9. J. C. Gittins (1979). "Bandit Processes and Dynamic Allocation Indices". Journal of the Royal Statistical Society. Series B (Methodological) 41 (2): 148–177. doi:10.1111/j.2517-6161.1979.tb01068.x. 
  10. 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, PMID 20018711, Bibcode2009PNAS..10622387P. 
  11. Press (1986)
  12. Brochu, Eric; Hoffman, Matthew W.; de Freitas, Nando (September 2010), Portfolio Allocation for Bayesian Optimization, Bibcode2010arXiv1009.5419B 
  13. Shen, Weiwei; Wang, Jun; Jiang, Yu-Gang; Zha, Hongyuan (2015), "Portfolio Choices with Orthogonal Bandit Learning", Proceedings of International Joint Conferences on Artificial Intelligence (IJCAI2015), http://www.aaai.org/ocs/index.php/IJCAI/IJCAI15/paper/viewPDFInterstitial/10972/10798, retrieved 2016-03-20 
  14. Farias, Vivek F; Ritesh, Madan (2011), "The irrevocable multiarmed bandit problem", Operations Research 59 (2): 383–399, doi:10.1287/opre.1100.0891 
  15. Whittle, Peter (1979), "Discussion of Dr Gittins' paper", Journal of the Royal Statistical Society, Series B 41 (2): 148–177, doi:10.1111/j.2517-6161.1979.tb01069.x 
  16. 16.0 16.1 Vermorel, Joannes; Mohri, Mehryar (2005), Multi-armed bandit algorithms and empirical evaluation, In European Conference on Machine Learning, Springer, pp. 437–448, http://bandit.sourceforge.net/Vermorel2005poker.pdf 
  17. Whittle, Peter (1988), "Restless bandits: Activity allocation in a changing world", Journal of Applied Probability 25A: 287–298, doi:10.2307/3214163 
  18. Whittle, Peter (1981), "Arm-acquiring bandits", Annals of Probability 9 (2): 284–292, doi:10.1214/aop/1176994469 
  19. Auer, P.; Cesa-Bianchi, N.; Freund, Y.; Schapire, R. E. (2002). "The Nonstochastic Multiarmed Bandit Problem". SIAM J. Comput. 32 (1): 48–77. doi:10.1137/S0097539701398375. 
  20. Lai, T.L.; Robbins, H. (1985). "Asymptotically efficient adaptive allocation rules". Advances in Applied Mathematics 6 (1): 4–22. doi:10.1016/0196-8858(85)90002-8. 
  21. Katehakis, M.N.; Robbins, H. (1995). "Sequential choice from several populations". Proceedings of the National Academy of Sciences of the United States of America 92 (19): 8584–5. doi:10.1073/pnas.92.19.8584. PMID 11607577. Bibcode1995PNAS...92.8584K. 
  22. Burnetas, A.N.; Katehakis, M.N. (1996). "Optimal adaptive policies for sequential allocation problems". Advances in Applied Mathematics 17 (2): 122–142. doi:10.1006/aama.1996.0007. 
  23. Burnetas, Apostolos N. (1997). "Optimal adaptive policies for Markov decision processes". Mathematics of Operations Research 22 (1): 222–255. doi:10.1287/moor.22.1.222. 
  24. Tewari, A.; Bartlett, P.L. (2008). "Optimistic linear programming gives logarithmic regret for irreducible MDPs". Advances in Neural Information Processing Systems 20. http://books.nips.cc/papers/files/nips20/NIPS2007_0673.pdf. Retrieved 2012-10-12. 
  25. Ortner, R. (2010). "Online regret bounds for Markov decision processes with deterministic transitions". Theoretical Computer Science 411 (29): 2684–2695. doi:10.1016/j.tcs.2010.04.005. 
  26. Filippi, S. and Cappé, O. and Garivier, A. (2010). "Online regret bounds for Markov decision processes with deterministic transitions", Communication, Control, and Computing (Allerton), 2010 48th Annual Allerton Conference on, pp. 115–122
  27. Honda, J.; Takemura, A. (2011). "An asymptotically optimal policy for finite support models in the multi-armed bandit problem". Machine Learning 85 (3): 361–391. doi:10.1007/s10994-011-5257-4. 
  28. 28.0 28.1 Pilarski, Sebastian; Pilarski, Slawomir; Varró, Dániel (February 2021). "Optimal Policy for Bernoulli Bandits: Computation and Algorithm Gauge". IEEE Transactions on Artificial Intelligence 2 (1): 2–17. doi:10.1109/TAI.2021.3074122. ISSN 2691-4581. 
  29. 29.0 29.1 Pilarski, Sebastian; Pilarski, Slawomir; Varro, Daniel (2021). "Delayed Reward Bernoulli Bandits: Optimal Policy and Predictive Meta-Algorithm PARDI". IEEE Transactions on Artificial Intelligence 3 (2): 152–163. doi:10.1109/TAI.2021.3117743. ISSN 2691-4581. 
  30. Averbeck, B.B. (2015). "Theory of choice in bandit, information sampling, and foraging tasks". PLOS Computational Biology 11 (3): e1004164. doi:10.1371/journal.pcbi.1004164. PMID 25815510. Bibcode2015PLSCB..11E4164A. 
  31. Costa, V.D.; Averbeck, B.B. (2019). "Subcortical Substrates of Explore-Exploit Decisions in Primates". Neuron 103 (3): 533–535. doi:10.1016/j.neuron.2019.05.017. PMID 31196672. 
  32. Sutton, R. S. & Barto, A. G. 1998 Reinforcement learning: an introduction. Cambridge, MA: MIT Press.
  33. Tokic, Michel (2010), "Adaptive ε-greedy exploration in reinforcement learning based on value differences", KI 2010: Advances in Artificial Intelligence, Lecture Notes in Computer Science, 6359, Springer-Verlag, pp. 203–210, doi:10.1007/978-3-642-16111-7_23, ISBN 978-3-642-16110-0, http://www.tokic.com/www/tokicm/publikationen/papers/AdaptiveEpsilonGreedyExploration.pdf .
  34. Tokic, Michel; Palm, Günther (2011), "Value-Difference Based Exploration: Adaptive Control Between Epsilon-Greedy and Softmax", KI 2011: Advances in Artificial Intelligence, Lecture Notes in Computer Science, 7006, Springer-Verlag, pp. 335–346, ISBN 978-3-642-24455-1, http://www.tokic.com/www/tokicm/publikationen/papers/KI2011.pdf .
  35. Gimelfarb, Michel; Sanner, Scott; Lee, Chi-Guhn (2019), "ε-BMC: A Bayesian Ensemble Approach to Epsilon-Greedy Exploration in Model-Free Reinforcement Learning", Proceedings of the Thirty-Fifth Conference on Uncertainty in Artificial Intelligence, AUAI Press, pp. 162, http://auai.org/uai2019/proceedings/papers/162.pdf .
  36. 36.0 36.1 Scott, S.L. (2010), "A modern Bayesian look at the multi-armed bandit", Applied Stochastic Models in Business and Industry 26 (2): 639–658, doi:10.1002/asmb.874 
  37. Olivier Chapelle; Lihong Li (2011), "An empirical evaluation of Thompson sampling", Advances in Neural Information Processing Systems (Curran Associates) 24: 2249–2257, http://papers.nips.cc/paper/4321-an-empirical-evaluation-of-thompson-sampling 
  38. Langford, John; Zhang, Tong (2008), "The Epoch-Greedy Algorithm for Contextual Multi-armed Bandits", Advances in Neural Information Processing Systems, 20, Curran Associates, Inc., pp. 817–824, http://papers.nips.cc/paper/3178-the-epoch-greedy-algorithm-for-multi-armed-bandits-with-side-information 
  39. Lihong Li; Wei Chu; John Langford; Robert E. Schapire (2010), "A contextual-bandit approach to personalized news article recommendation", Proceedings of the 19th international conference on World wide web, pp. 661–670, doi:10.1145/1772690.1772758, ISBN 9781605587998, Bibcode2010arXiv1003.0146L 
  40. Wei Chu; Lihong Li; Lev Reyzin; Robert E. Schapire (2011), "Contextual bandits with linear payoff functions", Proceedings of the 14th International Conference on Artificial Intelligence and Statistics (AISTATS): 208–214, http://proceedings.mlr.press/v15/chu11a/chu11a.pdf 
  41. Auer, P. (2000). "Using upper confidence bounds for online learning". IEEE Comput. Soc. pp. 270–279. doi:10.1109/sfcs.2000.892116. ISBN 978-0769508504. 
  42. Hong, Tzung-Pei; Song, Wei-Ping; Chiu, Chu-Tien (November 2011). "Evolutionary Composite Attribute Clustering". 2011 International Conference on Technologies and Applications of Artificial Intelligence. IEEE. pp. 305–308. doi:10.1109/taai.2011.59. ISBN 9781457721748. 
  43. Rigollet, Philippe; Zeevi, Assaf (2010), Nonparametric Bandits with Covariates, Conference on Learning Theory, COLT 2010, Bibcode2010arXiv1003.1630R 
  44. Slivkins, Aleksandrs (2011), Contextual bandits with similarity information., Conference on Learning Theory, COLT 2011, http://www.jmlr.org/papers/volume15/slivkins14a/slivkins14a.pdf 
  45. Perchet, Vianney; Rigollet, Philippe (2013), "The multi-armed bandit problem with covariates", Annals of Statistics 41 (2): 693–721, doi:10.1214/13-aos1101 
  46. Sarah Filippi; Olivier Cappé; Aurélien Garivier; Csaba Szepesvári (2010), "Parametric Bandits: The Generalized Linear Case", Advances in Neural Information Processing Systems (Curran Associates) 23: 586–594, http://papers.nips.cc/paper/4166-parametric-bandits-the-generalized-linear-case 
  47. Lihong Li; Yu Lu; Dengyong Zhou (2017), "Provably optimal algorithms for generalized linear contextual bandits", Proceedings of the 34th International Conference on Machine Learning (ICML): 2071–2080, Bibcode2017arXiv170300048L, http://proceedings.mlr.press/v70/li17c.html 
  48. Kwang-Sung Jun; Aniruddha Bhargava; Robert D. Nowak; Rebecca Willett (2017), "Scalable generalized linear bandits: Online computation and hashing", Advances in Neural Information Processing Systems (Curran Associates) 30: 99–109, Bibcode2017arXiv170600136J, http://papers.nips.cc/paper/6615-scalable-generalized-linear-bandits-online-computation-and-hashing 
  49. Branislav Kveton; Manzil Zaheer; Csaba Szepesvári; Lihong Li; Mohammad Ghavamzadeh; Craig Boutilier (2020), "Randomized exploration in generalized linear bandits", Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS), Bibcode2019arXiv190608947K 
  50. Michal Valko; Nathan Korda; Rémi Munos; Ilias Flaounas; Nello Cristianini (2013), Finite-Time Analysis of Kernelised Contextual Bandits, 29th Conference on Uncertainty in Artificial Intelligence (UAI 2013) and (JFPDA 2013)., Bibcode2013arXiv1309.6869V 
  51. Féraud, Raphaël; Allesiardo, Robin; Urvoy, Tanguy; Clérot, Fabrice (2016). "Random Forest for the Contextual Bandit Problem". Aistats: 93–101. http://jmlr.org/proceedings/papers/v51/feraud16.html. Retrieved 2016-06-10. 
  52. Alekh Agarwal; Daniel J. Hsu; Satyen Kale; John Langford; Lihong Li; Robert E. Schapire (2014), "Taming the monster: A fast and simple algorithm for contextual bandits", Proceedings of the 31st International Conference on Machine Learning (ICML): 1638–1646, Bibcode2014arXiv1402.0555A, http://proceedings.mlr.press/v32/agarwalb14.html 
  53. Badanidiyuru, A.; Langford, J.; Slivkins, A. (2014), "Resourceful contextual bandits", Proceeding of Conference on Learning Theory (COLT), http://www.jmlr.org/proceedings/papers/v35/badanidiyuru14.pdf 
  54. 54.0 54.1 Wu, Huasen; Srikant, R.; Liu, Xin; Jiang, Chong (2015), "Algorithms with Logarithmic or Sublinear Regret for Constrained Contextual Bandits", The 29th Annual Conference on Neural Information Processing Systems (NIPS) (Curran Associates) 28: 433–441, Bibcode2015arXiv150406937W, https://papers.nips.cc/paper/6008-algorithms-with-logarithmic-or-sublinear-regret-for-constrained-contextual-bandits 
  55. Burtini, Giuseppe, Jason Loeppky, and Ramon Lawrence. "A survey of online experiment design with the stochastic multi-armed bandit." arXiv preprint arXiv:1510.00757 (2015).
  56. Seldin, Y., Szepesvári, C., Auer, P. and Abbasi-Yadkori, Y., 2012, December. Evaluation and Analysis of the Performance of the EXP3 Algorithm in Stochastic Environments. In EWRL (pp. 103–116).
  57. Hutter, M. and Poland, J., 2005. Adaptive online prediction by following the perturbed leader. Journal of Machine Learning Research, 6(Apr), pp.639–660.
  58. Agrawal, Rajeev. The Continuum-Armed Bandit Problem. SIAM J. of Control and Optimization. 1995.
  59. Besbes, O.; Gur, Y.; Zeevi, A. Stochastic multi-armed-bandit problem with non-stationary rewards. In Proceedings of the Advances in Neural Information Processing Systems, Montreal, QC, Canada, 8–13 December 2014; pp. 199–207<https://proceedings.neurips.cc/paper/2014/file/903ce9225fca3e988c2af215d4e544d3-Paper.pdf>
  60. Discounted UCB, Levente Kocsis, Csaba Szepesvári, 2006
  61. On Upper-Confidence Bound Policies for Non-Stationary Bandit Problems, Garivier and Moulines, 2008 <https://arxiv.org/abs/0805.3415>
  62. Cavenaghi, E.; Sottocornola, G.; Stella, F.; Zanker, M. Non Stationary Multi-Armed Bandit: Empirical Evaluation of a New Concept Drift-Aware Algorithm. Entropy 2021, 23, 380. <https://doi.org/10.3390/e23030380>
  63. Improving Online Marketing Experiments with Drifting Multi-armed Bandits, Giuseppe Burtini, Jason Loeppky, Ramon Lawrence, 2015 <http://www.scitepress.org/DigitalLibrary/PublicationsDetail.aspx?ID=Dx2xXEB0PJE=&t=1>
  64. 64.0 64.1 Yue, Yisong; Broder, Josef; Kleinberg, Robert; Joachims, Thorsten (2012), "The K-armed dueling bandits problem", Journal of Computer and System Sciences 78 (5): 1538–1556, doi:10.1016/j.jcss.2011.12.028 
  65. Yue, Yisong; Joachims, Thorsten (2011), "Beat the Mean Bandit", Proceedings of ICML'11 
  66. Urvoy, Tanguy; Clérot, Fabrice; Féraud, Raphaël; Naamane, Sami (2013), "Generic Exploration and K-armed Voting Bandits", Proceedings of the 30th International Conference on Machine Learning (ICML-13), http://www.jmlr.org/proceedings/papers/v28/urvoy13.pdf, retrieved 2016-04-29 
  67. Zoghi, Masrour; Whiteson, Shimon; Munos, Remi; Rijke, Maarten D (2014), "Relative Upper Confidence Bound for the $K$-Armed Dueling Bandit Problem", Proceedings of the 31st International Conference on Machine Learning (ICML-14), http://www.jmlr.org/proceedings/papers/v32/zoghi14.pdf, retrieved 2016-04-27 
  68. Gajane, Pratik; Urvoy, Tanguy; Clérot, Fabrice (2015), "A Relative Exponential Weighing Algorithm for Adversarial Utility-based Dueling Bandits", Proceedings of the 32nd International Conference on Machine Learning (ICML-15), http://jmlr.org/proceedings/papers/v37/gajane15.pdf, retrieved 2016-04-29 
  69. Zoghi, Masrour; Karnin, Zohar S; Whiteson, Shimon; Rijke, Maarten D (2015), "Copeland Dueling Bandits", Advances in Neural Information Processing Systems, NIPS'15, Bibcode2015arXiv150600312Z 
  70. Komiyama, Junpei; Honda, Junya; Kashima, Hisashi; Nakagawa, Hiroshi (2015), "Regret Lower Bound and Optimal Algorithm in Dueling Bandit Problem", Proceedings of the 28th Conference on Learning Theory, http://jmlr.org/proceedings/papers/v40/Komiyama15.pdf, retrieved 2016-04-27 
  71. Wu, Huasen; Liu, Xin (2016), "Double Thompson Sampling for Dueling Bandits", The 30th Annual Conference on Neural Information Processing Systems (NIPS), Bibcode2016arXiv160407101W 
  72. Cesa-Bianchi, Nicolo; Gentile, Claudio; Zappella, Giovanni (2013), A Gang of Bandits, Advances in Neural Information Processing Systems 26, NIPS 2013 
  73. Gentile, Claudio; Li, Shuai; Zappella, Giovanni (2014), "Online Clustering of Bandits", The 31st International Conference on Machine Learning, Journal of Machine Learning Research (ICML 2014), Bibcode2014arXiv1401.8257G 
  74. Li, Shuai; Alexandros, Karatzoglou; Gentile, Claudio (2016), "Collaborative Filtering Bandits", The 39th International ACM SIGIR Conference on Information Retrieval (SIGIR 2016), Bibcode2015arXiv150203473L 
  75. {{citation |author1=Gai, Y. |author2=Krishnamachari, B. |author3=Jain, R. |contribution=Learning multiuser channel allocations in cognitive radio networks: A combinatorial multi-armed bandit formulation |title=2010 IEEE Symposium on New Frontiers in Dynamic Spectrum |pages=1–9 |year=2010 |url=http://www.academia.edu/download/30758682/DySPAN2010.pdf
  76. 76.0 76.1 Chen, Wei; Wang, Yajun; Yuan, Yang (2013), "Combinatorial multi-armed bandit: General framework and applications", Proceedings of the 30th International Conference on Machine Learning (ICML 2013), pp. 151–159, http://www.jmlr.org/proceedings/papers/v28/chen13a.pdf, retrieved 2019-06-14 
  77. 77.0 77.1 Santiago Ontañón (2017), "Combinatorial Multi-armed Bandits for Real-Time Strategy Games", Journal of Artificial Intelligence Research 58: 665–702, doi:10.1613/jair.5398, Bibcode2017arXiv171004805O, https://www.jair.org/index.php/jair/article/download/11053/26230 

Further reading

  • Allesiardo, Robin (2014), "A Neural Networks Committee for the Contextual Bandit Problem", Neural Information Processing – 21st International Conference, ICONIP 2014, Malaisia, November 03-06,2014, Proceedings, Lecture Notes in Computer Science, 8834, Springer, pp. 374–381, doi:10.1007/978-3-319-12637-1_47, ISBN 978-3-319-12636-4 .

External links