18 research outputs found
On-line regression competitive with reproducing kernel Hilbert spaces
We consider the problem of on-line prediction of real-valued labels, assumed
bounded in absolute value by a known constant, of new objects from known
labeled objects. The prediction algorithm's performance is measured by the
squared deviation of the predictions from the actual labels. No stochastic
assumptions are made about the way the labels and objects are generated.
Instead, we are given a benchmark class of prediction rules some of which are
hoped to produce good predictions. We show that for a wide range of
infinite-dimensional benchmark classes one can construct a prediction algorithm
whose cumulative loss over the first N examples does not exceed the cumulative
loss of any prediction rule in the class plus O(sqrt(N)); the main differences
from the known results are that we do not impose any upper bound on the norm of
the considered prediction rules and that we achieve an optimal leading term in
the excess loss of our algorithm. If the benchmark class is "universal" (dense
in the class of continuous functions on each compact set), this provides an
on-line non-stochastic analogue of universally consistent prediction in
non-parametric statistics. We use two proof techniques: one is based on the
Aggregating Algorithm and the other on the recently developed method of
defensive forecasting.Comment: 37 pages, 1 figur
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
Separability of reproducing kernel Hilbert spaces
We demonstrate that a reproducing kernel Hilbert or Banach space of functions on a separable absolute Borel space or an analytic subset of a Polish space is separable if it possesses a Borel measurable feature map
Prediction with Expert Advice under Discounted Loss
We study prediction with expert advice in the setting where the losses are
accumulated with some discounting---the impact of old losses may gradually
vanish. We generalize the Aggregating Algorithm and the Aggregating Algorithm
for Regression to this case, propose a suitable new variant of exponential
weights algorithm, and prove respective loss bounds.Comment: 26 pages; expanded (2 remarks -> theorems), some misprints correcte
Online Nonparametric Regression
We establish optimal rates for online regression for arbitrary classes of
regression functions in terms of the sequential entropy introduced in (Rakhlin,
Sridharan, Tewari, 2010). The optimal rates are shown to exhibit a phase
transition analogous to the i.i.d./statistical learning case, studied in
(Rakhlin, Sridharan, Tsybakov 2013). In the frequently encountered situation
when sequential entropy and i.i.d. empirical entropy match, our results point
to the interesting phenomenon that the rates for statistical learning with
squared loss and online nonparametric regression are the same.
In addition to a non-algorithmic study of minimax regret, we exhibit a
generic forecaster that enjoys the established optimal rates. We also provide a
recipe for designing online regression algorithms that can be computationally
efficient. We illustrate the techniques by deriving existing and new
forecasters for the case of finite experts and for online linear regression