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# 2.1 An -Armed Bandit Problem

Consider the following learning problem. You are faced repeatedly with a choice among different options, or actions. After each choice you receive a numerical reward chosen from a stationary probability distribution that depends on the action you selected. Your objective is to maximize the expected total reward over some time period, for example, over 1000 action selections. Each action selection is called a play.

This is the original form of the -armed bandit problem, so named by analogy to a slot machine, or "one-armed bandit," except that it has levers instead of one. Each action selection is like a play of one of the slot machine's levers, and the rewards are the payoffs for hitting the jackpot. Through repeated plays you are to maximize your winnings by concentrating your plays on the best levers. Another analogy is that of a doctor choosing between experimental treatments for a series of seriously ill patients. Each play is a treatment selection, and each reward is the survival or well-being of the patient. Today the term "-armed bandit problem" is often used for a generalization of the problem described above, but in this book we use it to refer just to this simple case.

In our -armed bandit problem, each action has an expected or mean reward given that that action is selected; let us call this the value of that action. If you knew the value of each action, then it would be trivial to solve the -armed bandit problem: you would always select the action with highest value. We assume that you do not know the action values with certainty, although you may have estimates.

If you maintain estimates of the action values, then at any time there is at least one action whose estimated value is greatest. We call this a greedy action. If you select a greedy action, we say that you are exploiting your current knowledge of the values of the actions. If instead you select one of the nongreedy actions, then we say you are exploring because this enables you to improve your estimate of the nongreedy action's value. Exploitation is the right thing to do to maximize the expected reward on the one play, but exploration may produce the greater total reward in the long run. For example, suppose the greedy action's value is known with certainty, while several other actions are estimated to be nearly as good but with substantial uncertainty. The uncertainty is such that at least one of these other actions probably is actually better than the greedy action, but you don't know which one. If you have many plays yet to make, then it may be better to explore the nongreedy actions and discover which of them are better than the greedy action. Reward is lower in the short run, during exploration, but higher in the long run because after you have discovered the better actions, you can exploit them. Because it is not possible both to explore and to exploit with any single action selection, one often refers to the "conflict" between exploration and exploitation.

In any specific case, whether it is better to explore or exploit depends in a complex way on the precise values of the estimates, uncertainties, and the number of remaining plays. There are many sophisticated methods for balancing exploration and exploitation for particular mathematical formulations of the -armed bandit and related problems. However, most of these methods make strong assumptions about stationarity and prior knowledge that are either violated or impossible to verify in applications and in the full reinforcement learning problem that we consider in subsequent chapters. The guarantees of optimality or bounded loss for these methods are of little comfort when the assumptions of their theory do not apply.

In this book we do not worry about balancing exploration and exploitation in a sophisticated way; we worry only about balancing them at all. In this chapter we present several simple balancing methods for the -armed bandit problem and show that they work much better than methods that always exploit. In addition, we point out that supervised learning methods (or rather the methods closest to supervised learning methods when adapted to this problem) perform poorly on this problem because they do not balance exploration and exploitation at all. The need to balance exploration and exploitation is a distinctive challenge that arises in reinforcement learning; the simplicity of the -armed bandit problem enables us to show this in a particularly clear form.

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Mark Lee 2005-01-04