We consider the problems of robust PAC learning from distributed and streaming data, which may contain malicious errors and outliers, and analyze their fundamental complexity questions. In particular, we establish lower bounds on the communication complexity for distributed robust learning performed on multiple machines, and on the space complexity for robust learning from streaming data on a single machine. These results demonstrate that gaining robustness of learning algorithms is usually at the expense of increased complexities. As far as we know, this work gives the first complexity results for distributed and online robust PAC learning.
Several recent works on communication complexity for distributed PAC learning include @cite_21 @cite_9 @cite_1 and there are also some works about distributed statistical estimation @cite_5 @cite_10 . In particular, Duchi @cite_7 demonstrated an analysis tool based on the information Fano inequalities. Garg @cite_16 investigated how the communication cost scales in the parameter dimensionality for distributed statistical estimation. Kremer @cite_24 provide a connection between communication complexity and VC-dimension. However, those works generally focus on the case no outliers or malicious errors are presented in the learning process.
@cite_7 @cite_9 @cite_21 @cite_1 @cite_24 @cite_5 @cite_16 @cite_10
2017753243 2127180992 2953101345 2963422939
Learnability in Valiant's PAC learning model has been shown to be strongly related to the existence of uniform laws of large numbers. These laws define a distribution-free convergence property of means to expectations uniformly over classes of random variables. Classes of real-valued functions enjoying such a property are also known as uniform Glivenko-Cantelli classes. In this paper, we prove, through a generalization of Sauer's lemma that may be interesting in its own right, a new characterization of uniform Glivenko-Cantelli classes. Our characterization yields Dudley, Gine´, and Zinn's previous characterization as a corollary. Furthermore, it is the first based on a Gine´, and Zinn's previous characterization as a corollary. Furthermore, it is the first based on a simple combinatorial quantity generalizing the Vapnik-Chervonenkis dimension. We apply this result to obtain the weakest combinatorial condition known to imply PAC learnability in the statistical regression (or “agnostic”) framework. Furthermore, we find a characterization of learnability in the probabilistic concept model, solving an open problem posed by Kearns and Schapire. These results show that the accuracy parameter plays a crucial role in determining the effective complexity of the learner's hypothesis class. We consider the problem of PAC-learning from distributed data and analyze fundamental communication complexity questions involved. We provide general upper and lower bounds on the amount of communication needed to learn well, showing that in addition to VC-dimension and covering number, quantities such as the teaching-dimension and mistake-bound of a class play an important role. We also present tight results for a number of common concept classes including conjunctions, parity functions, and decision lists. For linear separators, we show that for non-concentrated distributions, we can use a version of the Perceptron algorithm to learn with much less communication than the number of updates given by the usual margin bound. We also show how boosting can be performed in a generic manner in the distributed setting to achieve communication with only logarithmic dependence on 1 epsilon for any concept class, and demonstrate how recent work on agnostic learning from class-conditional queries can be used to achieve low communication in agnostic settings as well. We additionally present an analysis of privacy, considering both differential privacy and a notion of distributional privacy that is especially appealing in this context. We explore the connection between dimensionality and communication cost in distributed learning problems. Specifically we study the problem of estimating the mean @math of an unknown @math dimensional gaussian distribution in the distributed setting. In this problem, the samples from the unknown distribution are distributed among @math different machines. The goal is to estimate the mean @math at the optimal minimax rate while communicating as few bits as possible. We show that in this setting, the communication cost scales linearly in the number of dimensions i.e. one needs to deal with different dimensions individually. Applying this result to previous lower bounds for one dimension in the interactive setting ZDJW13 and to our improved bounds for the simultaneous setting, we prove new lower bounds of @math and @math for the bits of communication needed to achieve the minimax squared loss, in the interactive and simultaneous settings respectively. To complement, we also demonstrate an interactive protocol achieving the minimax squared loss with @math bits of communication, which improves upon the simple simultaneous protocol by a logarithmic factor. Given the strong lower bounds in the general setting, we initiate the study of the distributed parameter estimation problems with structured parameters. Specifically, when the parameter is promised to be @math -sparse, we show a simple thresholding based protocol that achieves the same squared loss while saving a @math factor of communication. We conjecture that the tradeoff between communication and squared loss demonstrated by this protocol is essentially optimal up to logarithmic factor. We study the tradeoff between the statistical error and communication cost of distributed statistical estimation problems in high dimensions. In the distributed sparse Gaussian mean estimation problem, each of the m machines receives n data points from a d-dimensional Gaussian distribution with unknown mean θ which is promised to be k-sparse. The machines communicate by message passing and aim to estimate the mean θ. We provide a tight (up to logarithmic factors) tradeoff between the estimation error and the number of bits communicated between the machines. This directly leads to a lower bound for the distributed sparse linear regression problem: to achieve the statistical minimax error, the total communication is at least Ω(min n,d m), where n is the number of observations that each machine receives and d is the ambient dimension. These lower results improve upon Shamir (NIPS'14) and Steinhardt-Duchi (COLT'15) by allowing multi-round iterative communication model. We also give the first optimal simultaneous protocol in the dense case for mean estimation. As our main technique, we prove a distributed data processing inequality, as a generalization of usual data processing inequalities, which might be of independent interest and useful for other problems.