Predicting (0, 1)-functions on randomly drawn points

Editors: Ming Li and Leslie ValiantA b s t r a c t . In this paper we consider the problem of tracking a subset of a domain (called the target) which changes gradually over time. A single (unknown) probability distribution over the domain is used to generate random examples for the learning algorithm and measure the speed at which the target changes. Clearly, the more rapidly the target moves, the harder it is for the algorithm to maintain a good approximation of the target. Therefore we evaluate algorithms based on how much movement of the target can be tolerated between examples while predicting with accuracy e. Furthermore, the complexity of the class 7-/of possible targets, as measured by d, its VC-dimension, also effects the difficulty of tracking the target concept. We show that if the problem of minimizing the number of disagreements with a sample from among concepts in a class 7{ can be approximated to within a factor k, then there is a simple tracking algorithm for 7-t which can achieve a probability e of making a mistake if the target movement rate is at most a constant times e2/(k(d + k) In 1), where d is the Vapnik-Chervonenkis dimension of 7-t. Also, we show that if 7-/ is properly PAC-learnable, then there is an efficient (randomized) algorithm that with high probability approximately minimizes disagreements to within a factor of 7d + 1, yielding an efficient tracking algorithm for 7-I which tolerates drift rates up to a constant times e2/(d 2 In ¼). In addition, we prove complementary results for the classes of halfspaces and axisaligned hyperrectangles showing that the maximum rate of drift that any algorithm (even with unlimited computational power) can tolerate is a constant times e2/d.

Section: Upper Bounds On the Tolerable Amount Of Driftmentioning

confidence: 99%

Tracking drifting concepts by minimizing disagreements

Helmbold

Long

1994

“…It is easy to see that the closure algorithm corresponds to one particular orientation of the 1-inclusion graph algorithm (Haussler, Littlestone and Warmuth, 1994). Thus, besides our bound of O( 1 ε (d log d + log 1 δ )) and the bound O( 1 ε (log 1 δ + d log 1 ε )) of Blumer et al (1989), the bound of O( d ε log 1 δ ) for the 1-inclusion graph algorithm of Haussler, Littlestone and Warmuth (1994) holds as well for learning intersection-closed classes with the closure algorithm.…”

Section: Proofmentioning

confidence: 99%

“…The learning algorithm A then PAC learns a concept class, if for ε, δ > 0 there is an m = m (ε, δ), such that with probability at least 1 − δ the algorithm outputs a hypothesis with error smaller than ε, when m random examples are given to A. Bounds on m often depend on the VC-dimension, a combinatorial parameter of the concept class. For finite d the well-known bound of Blumer et al (1989) states that for any consistent learning algorithm O( 1 ε (log 1 δ + d log 1 ε )) examples suffice for PAC learning concept classes of VC-dimension d. On the other hand, for the 1-inclusion graph algorithm a bound of O( d ε log 1 δ ) was established by Haussler, Littlestone and Warmuth (1994).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A new PAC bound for intersection-closed concept classes

Auer

Ortner

2006

et al. (1989) and Haussler, Littlestone and Warmuth (1994). Our bound is established using the closure algorithm, that generates as its hypothesis the intersection of all concepts that are consistent with the positive training examples. On the other hand, we show that many intersection-closed concept classes including e.g. maximum intersection-closed classes satisfy an additional combinatorial property that allows a proof of the optimal bound of O( 1 ε (d + log 1 δ )). For such improved bounds the choice of the learning algorithm is crucial, as there are consistent learning algorithms that need ( 1 ε (d log 1 ε + log 1 δ )) examples to learn some particular maximum intersection-closed concept classes.

“…We are now ready to define the absolute mistake-bound variant of the on-line learning model (Haussler, Littlestone, & Warmuth, 1988;Littlestone, 1988). An on-line algorithm (or incremental algorithm) for C is an algorithm that runs under the following scenario.…”

Section: Preliminary Definitionsmentioning

confidence: 99%

The power of self-directed learning

Goldman

Sloan

1994

Abstract. This article studies self-directed learning, a variant of the on-line (or incremental) learning model in which the learner selects the presentation order for the instances. Alternatively, one can view this model as a variation of learning with membership queries in which the learner is only "charged" for membership queries for which it could not predict the outcome. We give tight bounds on the complexity of self-directed learning for the concept classes of monomials, monotone DNF formulas, and axis-parallel rectangles in {0, 1, . .., n -1} 6. These results demonstrate that the number of mistakes under self-directed learning can be surprisingly small. We then show that learning complexity in the model of self-directed learning is less than that of all other commonly studied on-line and query learning models. Next we explore the relationship between the complexity of self-directed learning and the Vapnik-Chervonenkis (VC-)dimension. We show that, in general, the VC-dimension and the self-directed learning complexity are incomparable. However, for some special cases, we show that the VC-dimension gives a lower bound for the self-directed learning complexity. Finally, we explore a relationship between Mitchell's version space algorithm and the existence of self-directed learning algorithms that make few mistakes.