We study the problem of agnostically learning halfspaces which is defined by a fixed but unknown distribution @math on @math . We define @math as the least error of a halfspace classifier for @math . A learner who can access @math has to return a hypothesis whose error is small compared to @math . Using the recently developed method of the author, Linial and Shalev-Shwartz we prove hardness of learning results under a natural assumption on the complexity of refuting random @math - @math formulas. We show that no efficient learning algorithm has non-trivial worst-case performance even under the guarantees that @math for arbitrarily small constant @math , and that @math is supported in @math . Namely, even under these favorable conditions its error must be @math for every @math . In particular, no efficient algorithm can achieve a constant approximation ratio. Under a stronger version of the assumption (where @math can be poly-logarithmic in @math ), we can take @math for arbitrarily small @math . Interestingly, this is even stronger than the best known lower bounds (Arora et. al. 1993, Feldamn et. al. 2006, Guruswami and Raghavendra 2006) for the case that the learner is restricted to return a halfspace classifier (i.e. proper learning).
Several hardness assumptions imply that it is hard to agnostically learn halfspaces. Feldman, Gopalan, Khot and Ponnuswami @cite_49 have shown that based on the security of the Ajtai-Dwork cryptosystem. Kalai, Klivans, Mansour and Servedio showed the same conclusion @cite_8 based on the hardness of learning parity with noise. Daniely and Shalev-Shwartz @cite_21 derived the same conclusion based on the hardness of refuting random @math -SAT formulas. Klivans and Kothari @cite_6 showed that assuming that learning sparse parity is hard, it is hard to learn halfspaces even when the marginal distribution is Gaussian. We note that all these results only rule out exact algorithms, but say nothing about approximation algorithms. We note however that by @cite_0 @cite_5 , an algorithm with non trivial performance of @math -realizable distributions will result with quasi-polynomial time algorithm for learning constant depth circuits.
@cite_8 @cite_21 @cite_6 @cite_0 @cite_49 @cite_5
2168002876 2106458073 2026103826 2065923584
We give the first representation-independent hardness result for agnostically learning halfspaces with respect to the Gaussian distribution. We reduce from the problem of learning sparse parities with noise with respect to the uniform distribution on the hypercube (sparse LPN), a notoriously hard problem in theoretical computer science and show that any algorithm for agnostically learning halfspaces requires n (log (1 )) time under the assumption that k-sparse LPN requires n ( k) time, ruling out a polynomial time algorithm for the problem. As far as we are aware, this is the first representation-independent hardness result for supervised learning when the underlying distribution is restricted to be a Gaussian. We also show that the problem of agnostically learning sparse polynomials with respect to the Gaussian distribution in polynomial time is as hard as PAC learning DNFs on the uniform distribution in polynomial time. This complements the surprising result of Andoni et. al. [1] who show that sparse polynomials are learnable under random Gaussian noise in polynomial time. Taken together, these results show the inherent diculty of designing supervised learning algorithms in Euclidean space even in the presence of strong distributional assumptions. Our results use a novel embedding of random labeled examples from the uniform distribution on the Boolean hypercube into random labeled examples from the Gaussian distribution that allows us to relate the hardness of learning problems on two dierent domains and distributions. 1998 ACM Subject Classification F.2.0. Analysis of Algorithms and Problem Complexity We give a computationally efficient algorithm that learns (under distributional assumptions) a halfspace in the difficult agnostic framework of Kearns, Schapire, and Sellie [Mach. Learn., 17 (1994), pp. 115-141], where a learner is given access to a distribution on labelled examples but where the labelling may be arbitrary (similar to malicious noise). It constructs a hypothesis whose error rate on future examples is within an additive @math of the optimal halfspace, in time poly @math for any constant @math , for the uniform distribution over @math or unit sphere in @math as well as any log-concave distribution in @math . It also agnostically learns Boolean disjunctions in time @math with respect to any distribution. Our algorithm, which performs @math polynomial regression, is a natural noise-tolerant arbitrary-distribution generalization of the well-known “low-degree” Fourier algorithm of Linial, Mansour, and Nisan. We observe that significant improvements on the running time of our algorithm would yield the fastest known algorithm for learning parity with noise, a challenging open problem in computational learning theory. We address well-studied problems concerning the learnability of parities and halfspaces in the presence of classification noise. Learning of parities under the uniform distribution with random classification noise, also called the noisy parity problem is a famous open problem in computational learning. We reduce a number of basic problems regarding learning under the uniform distribution to learning of noisy parities. We show that under the uniform distribution, learning parities with adversarial classification noise reduces to learning parities with random classification noise. Together with the parity learning algorithm of [5], this gives the first nontrivial algorithm for learning parities with adversarial noise. We show that learning of DNF expressions reduces to learning noisy parities of just logarithmic number of variables. We show that learning of k-juntas reduces to learning noisy parities of k variables. These reductions work even in the presence of random classification noise in the original DNF or junta. We then consider the problem of learning halfspaces over Q ^n with adversarial noise or finding a halfspace that maximizes the agreement rate with a given set of examples. We prove an essentially optimal hardness factor of 2- , improving the factor of 85 84 - due to Bshouty and Burroughs [8]. Finally, we show that majorities of halfspaces are hard to PAC-learn using any representation, based on the cryptographic assumption underlying the Ajtai-Dwork cryptosystem. Learning an unknown halfspace (also called a perceptron) from labeled examples is one of the classic problems in machine learning. In the noise-free case, when a halfspace consistent with all the training examples exists, the problem can be solved in polynomial time using linear programming. However, under the promise that a halfspace consistent with a fraction @math of the examples exists (for some small constant @math ), it was not known how to efficiently find a halfspace that is correct on even 51 of the examples. Nor was a hardness result that ruled out getting agreement on more than 99.9 of the examples known. In this work, we close this gap in our understanding and prove that even a tiny amount of worst-case noise makes the problem of learning halfspaces intractable in a strong sense. Specifically, for arbitrary @math , we prove that given a set of examples-label pairs from the hypercube, a fraction @math of which can be explained by a halfspace, it is NP-hard to find a halfspace that correctly labels a fraction @math of the examples. The hardness result is tight since it is trivial to get agreement on @math the examples. In learning theory parlance, we prove that weak proper agnostic learning of halfspaces is hard. This settles a question that was raised by , in their work on learning halfspaces in the presence of random classification noise [Algorithmica, 22 (1998), pp. 35-52], and raised by authors of some more recent works as well. Along the way, we also obtain a strong hardness result for another basic computational problem: solving a linear system over the rationals.