arXiv subjects

Alexandra Carpentier

Publications and source records attributed to Alexandra Carpentier.

64 records · Page 4Linked to original sources

arXiv2015

Simple regret for infinitely many armed bandits

We consider a stochastic bandit problem with infinitely many arms. In this setting, the learner has no chance of trying all the arms even once and has to dedicate its limited number of samples only to a certain number of arms. All previous algorithms for this setting were designed for minimizing the cumulative regret of the learner. In this paper, we propose an algorithm aiming at minimizing the simple regret. As in the cumulative regret setting of infinitely many armed bandits, the rate of the simple regret will depend on a parameter $β$ characterizing the distribution of the near-optimal arms. We prove that depending on $β$, our algorithm is minimax optimal either up to a multiplicative constant or up to a $\log(n)$ factor. We also provide extensions to several important cases: when $β$ is unknown, in a natural setting where the near-optimal arms have a small variance, and in the case of unknown time horizon.

cs.LG↗

arXiv2015

Testing the regularity of a smooth signal

We develop a test to determine whether a function lying in a fixed $L_2$-Sobolev-type ball of smoothness $t$, and generating a noisy signal, is in fact of a given smoothness $s\geq t$ or not. While it is impossible to construct a uniformly consistent test for this problem on every function of smoothness $t$, it becomes possible if we remove a sufficiently large region of the set of functions of smoothness $t$. The functions that we remove are functions of smoothness strictly smaller than $s$, but that are very close to $s$-smooth functions. A lower bound on the size of this region has been proved to be of order $n^{-t/(2t+1/2)}$, and in this paper, we provide a test that is consistent after the removal of a region of such a size. Even though the null hypothesis is composite, the size of the region we remove does not depend on the complexity of the null hypothesis.

math.ST↗

arXiv2015

Implementable confidence sets in high dimensional regression

We consider the setting of linear regression in high dimension. We focus on the problem of constructing adaptive and honest confidence sets for the sparse parameter θ, i.e. we want to construct a confidence set for theta that contains theta with high probability, and that is as small as possible. The l_2 diameter of a such confidence set should depend on the sparsity S of θ- the larger S, the wider the confidence set. However, in practice, S is unknown. This paper focuses on constructing a confidence set for θwhich contains θwith high probability, whose diameter is adaptive to the unknown sparsity S, and which is implementable in practice.

stat.ML↗

arXiv2014

Adaptive confidence intervals for the tail coefficient in a wide second order class of Pareto models

We study the problem of constructing honest and adaptive confidence intervals for the tail coefficient in the second order Pareto model, when the second order coefficient is unknown. This problem is translated into a testing problem on the second order parameter. By constructing an appropriate model and an associated test statistic, we provide a uniform and adaptive confidence interval for the first order parameter. We also provide an almost matching lower bound, which proves that the result is minimax optimal up to a logarithmic factor.

math.ST↗

arXiv2014

Adaptive and minimax optimal estimation of the tail coefficient

We consider the problem of estimating the tail index $α$ of a distribution satisfying a $(α, β)$ second-order Pareto-type condition, where βis the second-order coefficient. When $β$ is available, it was previously proved that $α$ can be estimated with the oracle rate $n^{-β/(2β+1)}$. On the contrary, when $β$ is not available, estimating $α$ with the oracle rate is challenging; so additional assumptions that imply the estimability of $β$ are usually made. In this paper, we propose an adaptive estimator of $α$, and show that this estimator attains the rate $(n/\log\log n)^{-β/(2β+1)}$ without a priori knowledge of $β$ and any additional assumptions. Moreover, we prove that this $(\log\log n)^{β/(2β+1)}$ factor is unavoidable by obtaining the companion lower bound.

math.ST↗

arXiv2013

Honest and adaptive confidence sets in Lp

We consider the problem of constructing honest and adaptive confidence sets in Lp-loss (with p>=1 and p < infinity) over sets of Sobolev-type classes, in the setting of non-parametric Gaussian regression. The objective is to adapt the diameter of the confidence sets with respect to the smoothness degree of the underlying function, while ensuring that the true function lies in the confidence interval with high probability.When p >=2, we identify two main regimes, (i) one where adaptation is possible without any restrictions on the model, and (ii) one where critical regions have to be removed. We also prove by a matching lower bound that the size of the regions that we remove can not be chosen significantly smaller. These regimes are shown to depend in a qualitative way on the index p, and a continuous transition from p = 2 to p = infinity is exhibited.

math.ST↗

arXiv2013

Toward Optimal Stratification for Stratified Monte-Carlo Integration

We consider the problem of adaptive stratified sampling for Monte Carlo integration of a noisy function, given a finite budget n of noisy evaluations to the function. We tackle in this paper the problem of adapting to the function at the same time the number of samples into each stratum and the partition itself. More precisely, it is interesting to refine the partition of the domain in area where the noise to the function, or where the variations of the function, are very heterogeneous. On the other hand, having a (too) refined stratification is not optimal. Indeed, the more refined the stratification, the more difficult it is to adjust the allocation of the samples to the stratification, i.e. sample more points where the noise or variations of the function are larger. We provide in this paper an algorithm that selects online, among a large class of partitions, the partition that provides the optimal trade-off, and allocates the samples almost optimally on this partition.

stat.ML↗

arXiv2012

Adaptive Stratified Sampling for Monte-Carlo integration of Differentiable functions

We consider the problem of adaptive stratified sampling for Monte Carlo integration of a differentiable function given a finite number of evaluations to the function. We construct a sampling scheme that samples more often in regions where the function oscillates more, while allocating the samples such that they are well spread on the domain (this notion shares similitude with low discrepancy). We prove that the estimate returned by the algorithm is almost similarly accurate as the estimate that an optimal oracle strategy (that would know the variations of the function everywhere) would return, and provide a finite-sample analysis.

stat.ML↗

arXiv2012

Bandit Theory meets Compressed Sensing for high dimensional Stochastic Linear Bandit

We consider a linear stochastic bandit problem where the dimension $K$ of the unknown parameter $θ$ is larger than the sampling budget $n$. In such cases, it is in general impossible to derive sub-linear regret bounds since usual linear bandit algorithms have a regret in $O(K\sqrt{n})$. In this paper we assume that $θ$ is $S-$sparse, i.e. has at most $S-$non-zero components, and that the space of arms is the unit ball for the $||.||_2$ norm. We combine ideas from Compressed Sensing and Bandit Theory and derive algorithms with regret bounds in $O(S\sqrt{n})$.

math.ST↗

arXiv2012

Minimax Number of Strata for Online Stratified Sampling given Noisy Samples

We consider the problem of online stratified sampling for Monte Carlo integration of a function given a finite budget of $n$ noisy evaluations to the function. More precisely we focus on the problem of choosing the number of strata $K$ as a function of the budget $n$. We provide asymptotic and finite-time results on how an oracle that has access to the function would choose the partition optimally. In addition we prove a \textit{lower bound} on the learning rate for the problem of stratified Monte-Carlo. As a result, we are able to state, by improving the bound on its performance, that algorithm MC-UCB, defined in \citep{MC-UCB}, is minimax optimal both in terms of the number of samples n and the number of strata K, up to a $\sqrt{\log(nK)}$. This enables to deduce a minimax optimal bound on the difference between the performance of the estimate outputted by MC-UCB, and the performance of the estimate outputted by the best oracle static strategy, on the class of Hölder continuous functions, and upt to a $\sqrt{\log(n)}$.

math.ST↗