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Random features is one of the most popular techniques to speed up kernelmethods in large-scale problems. Related works have been recognized by theNeurIPS Test-of-Time award in 2017 and the ICML Best Paper Finalist in 2019.The body of work on random features has grown rapidly, and hence it isdesirable to have a comprehensive overview on this topic explaining theconnections among various algorithms and theoretical results. In this survey,we systematically review the work on random features from the past ten years.First, the motivations, characteristics and contributions of representativerandom features based algorithms are summarized according to their samplingschemes, learning procedures, variance reduction properties and how theyexploit training data. Second, we review theoretical results that center aroundthe following key question: how many random features are needed to ensure ahigh approximation quality or no loss in the empirical/expected risks of thelearned estimator. Third, we provide a comprehensive evaluation of popularrandom features based algorithms on several large-scale benchmark datasets anddiscuss their approximation quality and prediction performance forclassification. Last, we discuss the relationship between random features andmodern over-parameterized deep neural networks (DNNs), including the use ofhigh dimensional random features in the analysis of DNNs as well as the gapsbetween current theoretical and empirical results. This survey may serve as agentle introduction to this topic, and as a users' guide for practitionersinterested in applying the representative algorithms and understandingtheoretical results under various technical assumptions. We hope that thissurvey will facilitate discussion on the open problems in this topic, and moreimportantly, shed light on future research directions.

Random Features for Kernel Approximation: A Survey on Algorithms, Theory, and Beyond