7,049 research outputs found
Online Learning with Multiple Operator-valued Kernels
We consider the problem of learning a vector-valued function f in an online
learning setting. The function f is assumed to lie in a reproducing Hilbert
space of operator-valued kernels. We describe two online algorithms for
learning f while taking into account the output structure. A first contribution
is an algorithm, ONORMA, that extends the standard kernel-based online learning
algorithm NORMA from scalar-valued to operator-valued setting. We report a
cumulative error bound that holds both for classification and regression. We
then define a second algorithm, MONORMA, which addresses the limitation of
pre-defining the output structure in ONORMA by learning sequentially a linear
combination of operator-valued kernels. Our experiments show that the proposed
algorithms achieve good performance results with low computational cost
Multiclass Learning with Simplex Coding
In this paper we discuss a novel framework for multiclass learning, defined
by a suitable coding/decoding strategy, namely the simplex coding, that allows
to generalize to multiple classes a relaxation approach commonly used in binary
classification. In this framework, a relaxation error analysis can be developed
avoiding constraints on the considered hypotheses class. Moreover, we show that
in this setting it is possible to derive the first provably consistent
regularized method with training/tuning complexity which is independent to the
number of classes. Tools from convex analysis are introduced that can be used
beyond the scope of this paper
Generalization Properties of Doubly Stochastic Learning Algorithms
Doubly stochastic learning algorithms are scalable kernel methods that
perform very well in practice. However, their generalization properties are not
well understood and their analysis is challenging since the corresponding
learning sequence may not be in the hypothesis space induced by the kernel. In
this paper, we provide an in-depth theoretical analysis for different variants
of doubly stochastic learning algorithms within the setting of nonparametric
regression in a reproducing kernel Hilbert space and considering the square
loss. Particularly, we derive convergence results on the generalization error
for the studied algorithms either with or without an explicit penalty term. To
the best of our knowledge, the derived results for the unregularized variants
are the first of this kind, while the results for the regularized variants
improve those in the literature. The novelties in our proof are a sample error
bound that requires controlling the trace norm of a cumulative operator, and a
refined analysis of bounding initial error.Comment: 24 pages. To appear in Journal of Complexit
Online semi-parametric learning for inverse dynamics modeling
This paper presents a semi-parametric algorithm for online learning of a
robot inverse dynamics model. It combines the strength of the parametric and
non-parametric modeling. The former exploits the rigid body dynamics equa-
tion, while the latter exploits a suitable kernel function. We provide an
extensive comparison with other methods from the literature using real data
from the iCub humanoid robot. In doing so we also compare two different
techniques, namely cross validation and marginal likelihood optimization, for
estimating the hyperparameters of the kernel function
Harder, Better, Faster, Stronger Convergence Rates for Least-Squares Regression
We consider the optimization of a quadratic objective function whose
gradients are only accessible through a stochastic oracle that returns the
gradient at any given point plus a zero-mean finite variance random error. We
present the first algorithm that achieves jointly the optimal prediction error
rates for least-squares regression, both in terms of forgetting of initial
conditions in O(1/n 2), and in terms of dependence on the noise and dimension d
of the problem, as O(d/n). Our new algorithm is based on averaged accelerated
regularized gradient descent, and may also be analyzed through finer
assumptions on initial conditions and the Hessian matrix, leading to
dimension-free quantities that may still be small while the " optimal " terms
above are large. In order to characterize the tightness of these new bounds, we
consider an application to non-parametric regression and use the known lower
bounds on the statistical performance (without computational limits), which
happen to match our bounds obtained from a single pass on the data and thus
show optimality of our algorithm in a wide variety of particular trade-offs
between bias and variance
Matrix completion and extrapolation via kernel regression
Matrix completion and extrapolation (MCEX) are dealt with here over
reproducing kernel Hilbert spaces (RKHSs) in order to account for prior
information present in the available data. Aiming at a faster and
low-complexity solver, the task is formulated as a kernel ridge regression. The
resultant MCEX algorithm can also afford online implementation, while the class
of kernel functions also encompasses several existing approaches to MC with
prior information. Numerical tests on synthetic and real datasets show that the
novel approach performs faster than widespread methods such as alternating
least squares (ALS) or stochastic gradient descent (SGD), and that the recovery
error is reduced, especially when dealing with noisy data
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