Exponential Error Convergence in Data Classification with Optimized Random Features: Acceleration by Quantum Machine Learning

Random features are a central technique for scalable learning algorithms based on kernel methods. A recent work has shown that an algorithm for machine learning by quantum computer, quantum machine learning (QML), can exponentially speed up sampling of optimized random features, even without imposing restrictive assumptions on sparsity and low-rankness of matrices that had limited applicability of conventional QML algorithms; this QML algorithm makes it possible to significantly reduce and provably minimize the required number of features for regression tasks. However, a major interest in the field of QML is how widely the advantages of quantum computation can be exploited, not only in the regression tasks. We here construct a QML algorithm for a classification task accelerated by the optimized random features. We prove that the QML algorithm for sampling optimized random features, combined with stochastic gradient descent (SGD), can achieve state-of-the-art exponential convergence speed of reducing classification error in a classification task under a low-noise condition; at the same time, our algorithm with optimized random features can take advantage of the significant reduction of the required number of features so as to accelerate each iteration in the SGD and evaluation of the classifier obtained from our algorithm. These results discover a promising application of QML to significant acceleration of the leading classification algorithm based on kernel methods, without ruining its applicability to a practical class of data sets and the exponential error-convergence speed.