We present a simple noise-robust margin-based active learning algorithm to find homogeneous (passing the origin) linear separators and analyze its error convergence when labels are corrupted by noise. We show that when the imposed noise satisfies the Tsybakov low noise condition (Mammen, Tsybakov, and others 1999; Tsybakov 2004) the algorithm is able to adapt to unknown level of noise and achieves optimal statistical rate up to poly-logarithmic factors. We also derive lower bounds for margin based active learning algorithms under Tsybakov noise conditions (TNC) for the membership query synthesis scenario (Angluin 1988). Our result implies lower bounds for the stream based selective sampling scenario (Cohn 1990) under TNC for some fairly simple data distributions. Quite surprisingly, we show that the sample complexity cannot be improved even if the underlying data distribution is as simple as the uniform distribution on the unit ball. Our proof involves the construction of a well separated hypothesis set on the d-dimensional unit ball along with carefully designed label distributions for the Tsybakov noise condition. Our analysis might provide insights for other forms of lower bounds as well.
In terms of negative results, it is well-known that the @math upper bound is tight up to polylogarithmic factors. In particular, Theorem 4.3 in @cite_9 shows that for any stream-based active learning algorithm, there exists a distribution @math satisfying TNC such that the excess risk @math is lower bounded by @math . The marginal data distribution @math is constructed in an adversarial manner and it is unclear whether the same lower bound applies when @math is some simple (e.g., uniform or Gaussian) distribution. @cite_1 proved lower bounds for stream-based active learning under each log-concave data distribution. However, their proof only applies to the separable case and shows an exponential error convergence. In contrast, we consider Tsybakov noise settings with parameter @math , for which polynomial error convergence is expected @cite_9 .
@cite_9 @cite_1
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We present an algorithm for the statistical learning setting with a bounded exp-concave loss in @math dimensions that obtains excess risk @math with probability at least @math . The core technique is to boost the confidence of recent in-expectation @math excess risk bounds for empirical risk minimization (ERM), without sacrificing the rate, by leveraging a Bernstein condition which holds due to exp-concavity. We also show that with probability @math the standard ERM method obtains excess risk @math . We further show that a regret bound for any online learner in this setting translates to a high probability excess risk bound for the corresponding online-to-batch conversion of the online learner. Lastly, we present two high probability bounds for the exp-concave model selection aggregation problem that are quantile-adaptive in a certain sense. The first bound is a purely exponential weights type algorithm, obtains a nearly optimal rate, and has no explicit dependence on the Lipschitz continuity of the loss. The second bound requires Lipschitz continuity but obtains the optimal rate. We introduce efficient margin-based algorithms for selective sampling and filtering in binary classification tasks. Experiments on real-world textual data reveal that our algorithms perform significantly better than popular and similarly efficient competitors. Using the so-called Mammen-Tsybakov low noise condition to parametrize the instance distribution, and assuming linear label noise, we show bounds on the convergence rate to the Bayes risk of a weaker adaptive variant of our selective sampler. Our analysis reveals that, excluding logarithmic factors, the average risk of this adaptive sampler converges to the Bayes risk at rate N ?(1+?)(2+?) 2(3+?) where N denotes the number of queried labels, and ?>0 is the exponent in the low noise condition. For all @math this convergence rate is asymptotically faster than the rate N ?(1+?) (2+?) achieved by the fully supervised version of the base selective sampler, which queries all labels. Moreover, for ??? (hard margin condition) the gap between the semi- and fully-supervised rates becomes exponential. We provide new results concerning label efficient, polynomial time, passive and active learning of linear separators. We prove that active learning provides an exponential improvement over PAC (passive) learning of homogeneouslinear separators under nearly log-concave distributions. Building on this, we provide a computationally efficient PAC algorithm with optimal (up to a constant factor) sample complexity for such problems. This resolves an open question of (Long, 1995, 2003; , 2009) concerning the sample complexity of efficient PAC algorithms under the uniformdistribution in the unit ball. Moreover,it providesthe first bound for a polynomial-time PAC algorithm that is tight for an interesting infinite class of hypothesis functions under a general and natural class of data-distributions, providing significant progress towards a longstanding open question of (, 1989; , 1989). We also provide new bounds for active and passive learning in the case that the data might not be linearly separable, both in the agnostic case and and underthe Tsybakovlow-noisecondition. To derive our results, we provide new structural results for (nearly) log-concave distributions, which might be of independent interest as well. We prove the first nontrivial (superlinear) lower bound in the noisy broadcast model, defined by El Gamal in [Open problems presented at the @math workshop on Specific Problems in Communication and Computation sponsored by Bell Communication Research, in Open Problems in Communication and Computation, T. M. Cover and B. Gopinath, eds., Springer-Verlag, New York, 1987, pp. 60-62]. In this model there are @math processors @math , each of which is initially given a private input bit @math . The goal is for @math to learn the value of @math , for some specified function @math , using a series of noisy broadcasts. At each step a designated processor broadcasts one bit to all of the other processors, and the bit received by each processor is flipped with fixed probability (independently for each recipient). In 1988, Gallager [IEEE Trans. Inform. Theory, 34 (1988), pp. 176-180] gave a noise-resistant protocol that allows @math to learn the entire input with constant probability in @math broadcasts. We prove that Gallager's protocol is optimal, up to a constant factor. Our lower bound follows by reduction from a lower bound for generalized noisy decision trees, a new model which may be of independent interest. For this new model we show a lower bound of @math on the depth of a tree that learns the entire input. While the above lower bound is for an @math -bit function, we also show an @math lower bound for the number of broadcasts required to compute certain explicit boolean-valued functions, when the correct output must be attained with probability at least @math for a constant parameter @math (this bound applies to all threshold functions as well as any other boolean-valued function with linear sensitivity). This bound also follows by reduction from a lower bound of @math on the depth of generalized noisy decision trees that compute the same functions with the same error. We also show a (nontrivial) @math lower bound on the depth of generalized noisy decision trees that compute such functions with small constant error. Finally, we show the first protocol in the noisy broadcast model that computes the Hamming weight of the input using a linear number of broadcasts.