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NONPARAMETRIC STOCHASTIC APPROXIMATION WITH LARGE STEP-SIZES
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DOI：10.1214/15-AOS1391.png)
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摘要
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We consider the random-design least-squares regression problem within the reproducing kernel Hilbert space (RKHS) framework. Given a stream of independent and identically distributed input/output data, we aim to learn a regression function within an RKHS H, even if the optimal predictor (i.e., the conditional expectation) is not in H. In a stochastic approximation framework where the estimator is updated after each observation, we show that the averaged unregularized least-mean-square algorithm (a form of stochastic gradient descent), given a sufficient large step-size, attains optimal rates of convergence for a variety of regimes for the smoothnesses of the optimal prediction function and the functions in H. Our results apply as well in the usual finite-dimensional setting of parametric least-squares regression, showing adaptivity of our estimator to the spectral decay of the covariance matrix of the covariates.
Keyword:
Reproducing kernel Hilbert space
stochastic approximation
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NONPARAMETRIC STOCHASTIC APPROXIMATION WITH LARGE STEP-SIZES具有大步长的非参数随机逼近.png)
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