133 research outputs found
No fast exponential deviation inequalities for the progressive mixture rule
We consider the learning task consisting in predicting as well as the best
function in a finite reference set G up to the smallest possible additive term.
If R(g) denotes the generalization error of a prediction function g, under
reasonable assumptions on the loss function (typically satisfied by the least
square loss when the output is bounded), it is known that the progressive
mixture rule g_n satisfies E R(g_n) < min_{g in G} R(g) + C (log|G|)/n where n
denotes the size of the training set, E denotes the expectation w.r.t. the
training set distribution and C denotes a positive constant. This work mainly
shows that for any training set size n, there exist a>0, a reference set G and
a probability distribution generating the data such that with probability at
least a R(g_n) > min_{g in G} R(g) + c sqrt{[log(|G|/a)]/n}, where c is a
positive constant. In other words, surprisingly, for appropriate reference set
G, the deviation convergence rate of the progressive mixture rule is only of
order 1/sqrt{n} while its expectation convergence rate is of order 1/n. The
same conclusion holds for the progressive indirect mixture rule. This work also
emphasizes on the suboptimality of algorithms based on penalized empirical risk
minimization on G
Robustness of Anytime Bandit Policies
This paper studies the deviations of the regret in a stochastic multi-armed
bandit problem. When the total number of plays n is known beforehand by the
agent, Audibert et al. (2009) exhibit a policy such that with probability at
least 1-1/n, the regret of the policy is of order log(n). They have also shown
that such a property is not shared by the popular ucb1 policy of Auer et al.
(2002). This work first answers an open question: it extends this negative
result to any anytime policy. The second contribution of this paper is to
design anytime robust policies for specific multi-armed bandit problems in
which some restrictions are put on the set of possible distributions of the
different arms
Robust linear least squares regression
We consider the problem of robustly predicting as well as the best linear
combination of given functions in least squares regression, and variants of
this problem including constraints on the parameters of the linear combination.
For the ridge estimator and the ordinary least squares estimator, and their
variants, we provide new risk bounds of order without logarithmic factor
unlike some standard results, where is the size of the training data. We
also provide a new estimator with better deviations in the presence of
heavy-tailed noise. It is based on truncating differences of losses in a
min--max framework and satisfies a risk bound both in expectation and in
deviations. The key common surprising factor of these results is the absence of
exponential moment condition on the output distribution while achieving
exponential deviations. All risk bounds are obtained through a PAC-Bayesian
analysis on truncated differences of losses. Experimental results strongly back
up our truncated min--max estimator.Comment: Published in at http://dx.doi.org/10.1214/11-AOS918 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org). arXiv admin note: significant text
overlap with arXiv:0902.173
Fast learning rates for plug-in classifiers under the margin condition
It has been recently shown that, under the margin (or low noise) assumption,
there exist classifiers attaining fast rates of convergence of the excess Bayes
risk, i.e., the rates faster than . The works on this subject
suggested the following two conjectures: (i) the best achievable fast rate is
of the order , and (ii) the plug-in classifiers generally converge
slower than the classifiers based on empirical risk minimization. We show that
both conjectures are not correct. In particular, we construct plug-in
classifiers that can achieve not only the fast, but also the {\it super-fast}
rates, i.e., the rates faster than . We establish minimax lower bounds
showing that the obtained rates cannot be improved.Comment: 36 page
Fast learning rates for plug-in classifiers
It has been recently shown that, under the margin (or low noise) assumption,
there exist classifiers attaining fast rates of convergence of the excess Bayes
risk, that is, rates faster than . The work on this subject has
suggested the following two conjectures: (i) the best achievable fast rate is
of the order , and (ii) the plug-in classifiers generally converge more
slowly than the classifiers based on empirical risk minimization. We show that
both conjectures are not correct. In particular, we construct plug-in
classifiers that can achieve not only fast, but also super-fast rates, that is,
rates faster than . We establish minimax lower bounds showing that the
obtained rates cannot be improved.Comment: Published at http://dx.doi.org/10.1214/009053606000001217 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Minimax Policies for Combinatorial Prediction Games
We address the online linear optimization problem when the actions of the
forecaster are represented by binary vectors. Our goal is to understand the
magnitude of the minimax regret for the worst possible set of actions. We study
the problem under three different assumptions for the feedback: full
information, and the partial information models of the so-called "semi-bandit",
and "bandit" problems. We consider both -, and -type of
restrictions for the losses assigned by the adversary.
We formulate a general strategy using Bregman projections on top of a
potential-based gradient descent, which generalizes the ones studied in the
series of papers Gyorgy et al. (2007), Dani et al. (2008), Abernethy et al.
(2008), Cesa-Bianchi and Lugosi (2009), Helmbold and Warmuth (2009), Koolen et
al. (2010), Uchiya et al. (2010), Kale et al. (2010) and Audibert and Bubeck
(2010). We provide simple proofs that recover most of the previous results. We
propose new upper bounds for the semi-bandit game. Moreover we derive lower
bounds for all three feedback assumptions. With the only exception of the
bandit game, the upper and lower bounds are tight, up to a constant factor.
Finally, we answer a question asked by Koolen et al. (2010) by showing that the
exponentially weighted average forecaster is suboptimal against
adversaries
Regret lower bounds and extended Upper Confidence Bounds policies in stochastic multi-armed bandit problem
This paper is devoted to regret lower bounds in the classical model of
stochastic multi-armed bandit. A well-known result of Lai and Robbins, which
has then been extended by Burnetas and Katehakis, has established the presence
of a logarithmic bound for all consistent policies. We relax the notion of
consistence, and exhibit a generalisation of the logarithmic bound. We also
show the non existence of logarithmic bound in the general case of Hannan
consistency. To get these results, we study variants of popular Upper
Confidence Bounds (ucb) policies. As a by-product, we prove that it is
impossible to design an adaptive policy that would select the best of two
algorithms by taking advantage of the properties of the environment
Graph Laplacians and their convergence on random neighborhood graphs
Given a sample from a probability measure with support on a submanifold in
Euclidean space one can construct a neighborhood graph which can be seen as an
approximation of the submanifold. The graph Laplacian of such a graph is used
in several machine learning methods like semi-supervised learning,
dimensionality reduction and clustering. In this paper we determine the
pointwise limit of three different graph Laplacians used in the literature as
the sample size increases and the neighborhood size approaches zero. We show
that for a uniform measure on the submanifold all graph Laplacians have the
same limit up to constants. However in the case of a non-uniform measure on the
submanifold only the so called random walk graph Laplacian converges to the
weighted Laplace-Beltrami operator.Comment: Improved presentation, typos corrected, to appear in JML
Risk bounds in linear regression through PAC-Bayesian truncation
We consider the problem of predicting as well as the best linear combination
of d given functions in least squares regression, and variants of this problem
including constraints on the parameters of the linear combination. When the
input distribution is known, there already exists an algorithm having an
expected excess risk of order d/n, where n is the size of the training data.
Without this strong assumption, standard results often contain a multiplicative
log n factor, and require some additional assumptions like uniform boundedness
of the d-dimensional input representation and exponential moments of the
output. This work provides new risk bounds for the ridge estimator and the
ordinary least squares estimator, and their variants. It also provides
shrinkage procedures with convergence rate d/n (i.e., without the logarithmic
factor) in expectation and in deviations, under various assumptions. The key
common surprising factor of these results is the absence of exponential moment
condition on the output distribution while achieving exponential deviations.
All risk bounds are obtained through a PAC-Bayesian analysis on truncated
differences of losses. Finally, we show that some of these results are not
particular to the least squares loss, but can be generalized to similar
strongly convex loss functions.Comment: 78 page
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