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Online regression competitive with reproducing kernel Hilbert spaces
, 2005
"... We consider the problem of online prediction of realvalued labels of new objects. The prediction algorithm’s performance is measured by the squared deviation of the predictions from the actual labels. No probabilistic assumptions are made about the way the labels and objects are generated. Instead ..."
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Cited by 12 (4 self)
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We consider the problem of online prediction of realvalued labels of new objects. The prediction algorithm’s performance is measured by the squared deviation of the predictions from the actual labels. No probabilistic assumptions are made about the way the labels and objects are generated. Instead, we are given a benchmark class of prediction rules some of which are hoped to produce good predictions. We show that for a wide range of infinitedimensional benchmark classes one can construct a prediction algorithm whose cumulative loss over the first N examples does not exceed the cumulative loss of any prediction rule in the class plus O ( √ N). Our proof technique is based on the recently developed method of defensive forecasting. 1
Competing with wild prediction rules
 Machine Learning
"... We consider the problem of online prediction competitive with a benchmark class of continuous but highly irregular prediction rules. It is known that if the benchmark class is a reproducing kernel Hilbert space, there exists a prediction algorithm whose average loss over the first N examples does n ..."
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Cited by 3 (2 self)
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We consider the problem of online prediction competitive with a benchmark class of continuous but highly irregular prediction rules. It is known that if the benchmark class is a reproducing kernel Hilbert space, there exists a prediction algorithm whose average loss over the first N examples does not exceed the average loss of any prediction rule in the class plus a “regret term ” of O(N −1/2). The elements of some natural benchmark classes, however, are so irregular that these classes are not Hilbert spaces. In this paper we develop Banachspace methods to construct a prediction algorithm with a regret term of O(N −1/p), where p ∈ [2, ∞) and p − 2 reflects the degree to which the benchmark class fails to be a Hilbert space. Only the square loss function is considered. 1
Predictions as statements and decisions (draft: comments welcome)
, 2007
"... Prediction is a complex notion, and different predictors (such as people, computer programs, and probabilistic theories) can pursue very different goals. In this paper I will review some popular kinds of prediction and argue that the theory of competitive online learning can benefit from the kinds ..."
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Prediction is a complex notion, and different predictors (such as people, computer programs, and probabilistic theories) can pursue very different goals. In this paper I will review some popular kinds of prediction and argue that the theory of competitive online learning can benefit from the kinds of prediction that are now foreign to it. The standard goal for predictor in learning theory is to incur a small loss for a given loss function measuring the discrepancy between the predictions and the actual outcomes. Competitive online learning concentrates on a “relative ” version of this goal: the predictor is to perform almost as well as the best strategies in a given benchmark class of prediction strategies. Such predictions can be interpreted as decisions made by a “small ” decision maker (i.e., one whose decisions do not affect the future outcomes). Predictions, or probability forecasts, considered in the foundations of
ABSTRACT GAMETHEORETIC PROBABILITY AND DEFENSIVE FORECASTING
"... theory can replace measure theory as a foundation for classical probability theory, discrete and continuous (Probability and Finance: Its Only a Game!, Wiley 2001). In the gametheoretic framework, classical probability theorems are proven by betting strategies that make a player rich without riskin ..."
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theory can replace measure theory as a foundation for classical probability theory, discrete and continuous (Probability and Finance: Its Only a Game!, Wiley 2001). In the gametheoretic framework, classical probability theorems are proven by betting strategies that make a player rich without risking bankruptcy if the theorem’s prediction fails. These strategies can be specified explicitly, and so the theory has a constructive flavor that lends itself to applications in economics and statistics. Defensive forecasting is one of the most interesting of these applications. It identifies a comprehensive betting strategy, which becomes rich if the probabilities fail in a relevant way (say by being uncalibrated or having poor resolution), and it chooses probabilities to defeat this comprehensive betting strategy. The fact that this is possible gives us new insight into the very meaning of probability. 1
$25 Leading strategies in competitive online prediction
, 2007
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unknown title
, 2007
"... Gametheoretic probability and some of its applications: references and addenda ..."
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Gametheoretic probability and some of its applications: references and addenda