3,200 research outputs found
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
Online Isotonic Regression
We consider the online version of the isotonic regression problem. Given a
set of linearly ordered points (e.g., on the real line), the learner must
predict labels sequentially at adversarially chosen positions and is evaluated
by her total squared loss compared against the best isotonic (non-decreasing)
function in hindsight. We survey several standard online learning algorithms
and show that none of them achieve the optimal regret exponent; in fact, most
of them (including Online Gradient Descent, Follow the Leader and Exponential
Weights) incur linear regret. We then prove that the Exponential Weights
algorithm played over a covering net of isotonic functions has a regret bounded
by and present a matching
lower bound on regret. We provide a computationally efficient version of this
algorithm. We also analyze the noise-free case, in which the revealed labels
are isotonic, and show that the bound can be improved to or even to
(when the labels are revealed in isotonic order). Finally, we extend the
analysis beyond squared loss and give bounds for entropic loss and absolute
loss.Comment: 25 page
Oracle Inequalities and Optimal Inference under Group Sparsity
We consider the problem of estimating a sparse linear regression vector
under a gaussian noise model, for the purpose of both prediction and
model selection. We assume that prior knowledge is available on the sparsity
pattern, namely the set of variables is partitioned into prescribed groups,
only few of which are relevant in the estimation process. This group sparsity
assumption suggests us to consider the Group Lasso method as a means to
estimate . We establish oracle inequalities for the prediction and
estimation errors of this estimator. These bounds hold under a
restricted eigenvalue condition on the design matrix. Under a stronger
coherence condition, we derive bounds for the estimation error for mixed
-norms with . When , this result implies
that a threshold version of the Group Lasso estimator selects the sparsity
pattern of with high probability. Next, we prove that the rate of
convergence of our upper bounds is optimal in a minimax sense, up to a
logarithmic factor, for all estimators over a class of group sparse vectors.
Furthermore, we establish lower bounds for the prediction and
estimation errors of the usual Lasso estimator. Using this result, we
demonstrate that the Group Lasso can achieve an improvement in the prediction
and estimation properties as compared to the Lasso.Comment: 37 page
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