73,433 research outputs found
Optimal Rates for Random Fourier Features
Kernel methods represent one of the most powerful tools in machine learning
to tackle problems expressed in terms of function values and derivatives due to
their capability to represent and model complex relations. While these methods
show good versatility, they are computationally intensive and have poor
scalability to large data as they require operations on Gram matrices. In order
to mitigate this serious computational limitation, recently randomized
constructions have been proposed in the literature, which allow the application
of fast linear algorithms. Random Fourier features (RFF) are among the most
popular and widely applied constructions: they provide an easily computable,
low-dimensional feature representation for shift-invariant kernels. Despite the
popularity of RFFs, very little is understood theoretically about their
approximation quality. In this paper, we provide a detailed finite-sample
theoretical analysis about the approximation quality of RFFs by (i)
establishing optimal (in terms of the RFF dimension, and growing set size)
performance guarantees in uniform norm, and (ii) presenting guarantees in
() norms. We also propose an RFF approximation to derivatives of
a kernel with a theoretical study on its approximation quality.Comment: To appear at NIPS-201
Encoding and processing of sensory information in neuronal spike trains
Recently, a statistical signal-processing technique has allowed the information carried by single spike trains of sensory neurons on time-varying stimuli to be characterized quantitatively in a variety of preparations. In weakly electric fish, its application to first-order sensory neurons encoding electric field amplitude (P-receptor afferents) showed that they convey accurate information on temporal modulations in a behaviorally relevant frequency range (<80 Hz). At the next stage of the electrosensory pathway (the electrosensory lateral line lobe, ELL), the information sampled by first-order neurons is used to extract upstrokes and downstrokes in the amplitude modulation waveform. By using signal-detection techniques, we determined that these temporal features are explicitly represented by short spike bursts of second-order neurons (ELL pyramidal cells). Our results suggest that the biophysical mechanism underlying this computation is of dendritic origin. We also investigated the accuracy with which upstrokes and downstrokes are encoded across two of the three somatotopic body maps of the ELL (centromedial and lateral). Pyramidal cells of the centromedial map, in particular I-cells, encode up- and downstrokes more reliably than those of the lateral map. This result correlates well with the significance of these temporal features for a particular behavior (the jamming avoidance response) as assessed by lesion experiments of the centromedial map
Learning with SGD and Random Features
Sketching and stochastic gradient methods are arguably the most common
techniques to derive efficient large scale learning algorithms. In this paper,
we investigate their application in the context of nonparametric statistical
learning. More precisely, we study the estimator defined by stochastic gradient
with mini batches and random features. The latter can be seen as form of
nonlinear sketching and used to define approximate kernel methods. The
considered estimator is not explicitly penalized/constrained and regularization
is implicit. Indeed, our study highlights how different parameters, such as
number of features, iterations, step-size and mini-batch size control the
learning properties of the solutions. We do this by deriving optimal finite
sample bounds, under standard assumptions. The obtained results are
corroborated and illustrated by numerical experiments
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