We present and analyze an agnostic active learning algorithm that works without keeping a version space. This is unlike all previous approaches where a restricted set of candidate hypotheses is maintained throughout learning, and only hypotheses from this set are ever returned. By avoiding this version space approach, our algorithm sheds the computational burden and brittleness associated with maintaining version spaces, yet still allows for substantial improvements over supervised learning for classification.
Many of the previously mentioned algorithms are analyzed in the agnostic learning model, where no assumption is made about the noise distribution (see also @cite_21 ). In this setting, the label complexity of active learning algorithms cannot generally improve over supervised learners by more than a constant factor @cite_17 @cite_2 . However, under a parameterization of the noise distribution related to Tsybakov's low-noise condition @cite_15 , active learning algorithms have been shown to have improved label complexity bounds over what is achievable in the purely agnostic setting @cite_4 @cite_14 @cite_11 @cite_20 @cite_6 . We also consider this parameterization to obtain a tighter label complexity analysis.
@cite_14 @cite_4 @cite_11 @cite_21 @cite_6 @cite_2 @cite_15 @cite_20 @cite_17
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We present a framework for margin based active learning of linear separators. We instantiate it for a few important cases, some of which have been previously considered in the literature.We analyze the effectiveness of our framework both in the realizable case and in a specific noisy setting related to the Tsybakov small noise condition. This paper analyzes the potential advantages and theoretical challenges of "active learning" algorithms. Active learning involves sequential sampling procedures that use information gleaned from previous samples in order to focus the sampling and accelerate the learning process relative to "passive learning" algorithms, which are based on nonadaptive (usually random) samples. There are a number of empirical and theoretical results suggesting that in certain situations active learning can be significantly more effective than passive learning. However, the fact that active learning algorithms are feedback systems makes their theoretical analysis very challenging. This paper aims to shed light on achievable limits in active learning. Using minimax analysis techniques, we study the achievable rates of classification error convergence for broad classes of distributions characterized by decision boundary regularity and noise conditions. The results clearly indicate the conditions under which one can expect significant gains through active learning. Furthermore, we show that the learning rates derived are tight for "boundary fragment" classes in d-dimensional feature spaces when the feature marginal density is bounded from above and below. We study the label complexity of pool-based active learning in the agnostic PAC model. Specifically, we derive general bounds on the number of label requests made by the A2 algorithm proposed by Balcan, Beygelzimer & Langford (, 2006). This represents the first nontrivial general-purpose upper bound on label complexity in the agnostic PAC model. Sequential algorithms of active learning based on the estimation of the level sets of the empirical risk are discussed in the paper. Localized Rademacher complexities are used in the algorithms to estimate the sample sizes needed to achieve the required accuracy of learning in an adaptive way. Probabilistic bounds on the number of active examples have been proved and several applications to binary classification problems are considered. We present a practical and statistically consistent scheme for actively learning binary classifiers under general loss functions. Our algorithm uses importance weighting to correct sampling bias, and by controlling the variance, we are able to give rigorous label complexity bounds for the learning process. We study the rates of convergence in classification error achievable by active learning in the presence of label noise. Additionally, we study the more general problem of active learning with a nested hierarchy of hypothesis classes, and propose an algorithm whose error rate provably converges to the best achievable error among classifiers in the hierarchy at a rate adaptive to both the complexity of the optimal classifier and the noise conditions. In particular, we state sufficient conditions for these rates to be dramatically faster than those achievable by passive learning. Most of the existing active learning algorithms are based on the realizability assumption: The learner's hypothesis class is assumed to contain a target function that perfectly classifies all training and test examples. This assumption can hardly ever be justified in practice. In this paper, we study how relaxing the realizability assumption affects the sample complexity of active learning. First, we extend existing results on query learning to show that any active learning algorithm for the realizable case can be transformed to tolerate random bounded rate class noise. Thus, bounded rate class noise adds little extra complications to active learning, and in particular exponential label complexity savings over passive learning are still possible. However, it is questionable whether this noise model is any more realistic in practice than assuming no noise at all. Our second result shows that if we move to the truly non-realizable model of statistical learning theory, then the label complexity of active learning has the same dependence Ω(1 ∈ 2 ) on the accuracy parameter e as the passive learning label complexity. More specifically, we show that under the assumption that the best classifier in the learner's hypothesis class has generalization error at most β > 0, the label complexity of active learning is Ω(β 2 ∈ 2 log(1 δ)), where the accuracy parameter e measures how close to optimal within the hypothesis class the active learner has to get and δ is the confidence parameter. The implication of this lower bound is that exponential savings should not be expected in realistic models of active learning, and thus the label complexity goals in active learning should be refined.