Efficient Algorithms for Online Decision Problems

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In an online decision problem, one makes a sequence of decisions without knowledge of the future. Tools from learning such as Weighted Majority and its many variants [4, 13, 18] demonstrate that online algorithms can perform nearly as well as the best single decision chosen in hindsight, even when there are exponentially many possible decisions. However, the naive application of these algorithms is inefficient for such large problems. For some problems with nice structure, specialized efficient solutions have been developed [3, 6, 10, 16, 17].

We show that a very simple idea, used in Hannan’s seminal 1957 paper [9], gives efficient solutions to all of these problems. Essentially, in each period, one chooses the decision that worked best in the past. To guarantee low regret, it is necessary to add randomness. Surprisingly, this simple approach gives additive ε regret per period, efficiently. We present a simple general analysis and several extensions, including a (1+ε)-competitive algorithm as well as a lazy one that rarely switches between decisions.