arXiv subjects

Sébastien Bubeck

Publications and source records attributed to Sébastien Bubeck.

At least 73 records · Page 4Linked to original sources

arXiv2015

Multi-scale exploration of convex functions and bandit convex optimization

We construct a new map from a convex function to a distribution on its domain, with the property that this distribution is a multi-scale exploration of the function. We use this map to solve a decade-old open problem in adversarial bandit convex optimization by showing that the minimax regret for this problem is $\tilde{O}(\mathrm{poly}(n) \sqrt{T})$, where $n$ is the dimension and $T$ the number of rounds. This bound is obtained by studying the dual Bayesian maximin regret via the information ratio analysis of Russo and Van Roy, and then using the multi-scale exploration to solve the Bayesian problem.

math.MG↗

arXiv2015

Sampling from a log-concave distribution with Projected Langevin Monte Carlo

We extend the Langevin Monte Carlo (LMC) algorithm to compactly supported measures via a projection step, akin to projected Stochastic Gradient Descent (SGD). We show that (projected) LMC allows to sample in polynomial time from a log-concave distribution with smooth potential. This gives a new Markov chain to sample from a log-concave distribution. Our main result shows in particular that when the target distribution is uniform, LMC mixes in $\tilde{O}(n^7)$ steps (where $n$ is the dimension). We also provide preliminary experimental evidence that LMC performs at least as well as hit-and-run, for which a better mixing time of $\tilde{O}(n^4)$ was proved by Lov{á}sz and Vempala.

math.PR↗

arXiv2015

A geometric alternative to Nesterov's accelerated gradient descent

We propose a new method for unconstrained optimization of a smooth and strongly convex function, which attains the optimal rate of convergence of Nesterov's accelerated gradient descent. The new algorithm has a simple geometric interpretation, loosely inspired by the ellipsoid method. We provide some numerical evidence that the new method can be superior to Nesterov's accelerated gradient descent.

math.OC↗

arXiv2015

Exceptional rotations of random graphs: a VC theory

In this paper we explore maximal deviations of large random structures from their typical behavior. We introduce a model for a high-dimensional random graph process and ask analogous questions to those of Vapnik and Chervonenkis for deviations of averages: how "rich" does the process have to be so that one sees atypical behavior. In particular, we study a natural process of Erdős-Rényi random graphs indexed by unit vectors in $\mathbb{R}^d$. We investigate the deviations of the process with respect to three fundamental properties: clique number, chromatic number, and connectivity. In all cases we establish upper and lower bounds for the minimal dimension $d$ that guarantees the existence of "exceptional directions" in which the random graph behaves atypically with respect to the property. For each of the three properties, four theorems are established, to describe upper and lower bounds for the threshold dimension in the subcritical and supercritical regimes.

math.PR↗

arXiv2015

Detecting positive correlations in a multivariate sample

We consider the problem of testing whether a correlation matrix of a multivariate normal population is the identity matrix. We focus on sparse classes of alternatives where only a few entries are nonzero and, in fact, positive. We derive a general lower bound applicable to various classes and study the performance of some near-optimal tests. We pay special attention to computational feasibility and construct near-optimal tests that can be computed efficiently. Finally, we apply our results to prove new lower bounds for the clique number of high-dimensional random geometric graphs.

math.ST↗

arXiv2015

The entropic barrier: a simple and optimal universal self-concordant barrier

We prove that the Cramér transform of the uniform measure on a convex body in $\mathbb{R}^n$ is a $(1+o(1)) n$-self-concordant barrier, improving a seminal result of Nesterov and Nemirovski. This gives the first explicit construction of a universal barrier for convex bodies with optimal self-concordance parameter. The proof is based on basic geometry of log-concave distributions, and elementary duality in exponential families.

math.OC↗

arXiv2015

Bandit Convex Optimization: sqrt{T} Regret in One Dimension

We analyze the minimax regret of the adversarial bandit convex optimization problem. Focusing on the one-dimensional case, we prove that the minimax regret is $\widetildeΘ(\sqrt{T})$ and partially resolve a decade-old open problem. Our analysis is non-constructive, as we do not present a concrete algorithm that attains this regret rate. Instead, we use minimax duality to reduce the problem to a Bayesian setting, where the convex loss functions are drawn from a worst-case distribution, and then we solve the Bayesian version of the problem with a variant of Thompson Sampling. Our analysis features a novel use of convexity, formalized as a "local-to-global" property of convex functions, that may be of independent interest.

cs.LG↗

arXiv2014

On the local profiles of trees

We study the local profiles of trees. We show that, in contrast with the situation for general graphs, the limit set of k-profiles of trees is convex. We initiate a study of the defining inequalities of this convex set. Many challenging problems remain open.

math.CO↗

arXiv2014

From trees to seeds: on the inference of the seed from large trees in the uniform attachment model

We study the influence of the seed in random trees grown according to the uniform attachment model, also known as uniform random recursive trees. We show that different seeds lead to different distributions of limiting trees from a total variation point of view. To do this, we construct statistics that measure, in a certain well-defined sense, global "balancedness" properties of such trees. Our paper follows recent results on the same question for the preferential attachment model.

math.PR↗

arXiv2014

Most Correlated Arms Identification

We study the problem of finding the most mutually correlated arms among many arms. We show that adaptive arms sampling strategies can have significant advantages over the non-adaptive uniform sampling strategy. Our proposed algorithms rely on a novel correlation estimator. The use of this accurate estimator allows us to get improved results for a wide range of problem instances.

stat.ML↗

arXiv2014

On the influence of the seed graph in the preferential attachment model

We study the influence of the seed graph in the preferential attachment model, focusing on the case of trees. We first show that the seed has no effect from a weak local limit point of view. On the other hand, we conjecture that different seeds lead to different distributions of limiting trees from a total variation point of view. We take a first step in proving this conjecture by showing that seeds with different degree profiles lead to different limiting distributions for the (appropriately normalized) maximum degree, implying that such seeds lead to different (in total variation) limiting trees.

math.PR↗

arXiv2013

lil' UCB : An Optimal Exploration Algorithm for Multi-Armed Bandits

The paper proposes a novel upper confidence bound (UCB) procedure for identifying the arm with the largest mean in a multi-armed bandit game in the fixed confidence setting using a small number of total samples. The procedure cannot be improved in the sense that the number of samples required to identify the best arm is within a constant factor of a lower bound based on the law of the iterated logarithm (LIL). Inspired by the LIL, we construct our confidence bounds to explicitly account for the infinite time horizon of the algorithm. In addition, by using a novel stopping time for the algorithm we avoid a union bound over the arms that has been observed in other UCB-type algorithms. We prove that the algorithm is optimal up to constants and also show through simulations that it provides superior performance with respect to the state-of-the-art.

stat.ML↗

arXiv2013

Prior-free and prior-dependent regret bounds for Thompson Sampling

We consider the stochastic multi-armed bandit problem with a prior distribution on the reward distributions. We are interested in studying prior-free and prior-dependent regret bounds, very much in the same spirit as the usual distribution-free and distribution-dependent bounds for the non-Bayesian stochastic bandit. Building on the techniques of Audibert and Bubeck [2009] and Russo and Roy [2013] we first show that Thompson Sampling attains an optimal prior-free bound in the sense that for any prior distribution its Bayesian regret is bounded from above by $14 \sqrt{n K}$. This result is unimprovable in the sense that there exists a prior distribution such that any algorithm has a Bayesian regret bounded from below by $\frac{1}{20} \sqrt{n K}$. We also study the case of priors for the setting of Bubeck et al. [2013] (where the optimal mean is known as well as a lower bound on the smallest gap) and we show that in this case the regret of Thompson Sampling is in fact uniformly bounded over time, thus showing that Thompson Sampling can greatly take advantage of the nice properties of these priors.

stat.ML↗

arXiv2013

Regret in Online Combinatorial Optimization

We address online linear optimization problems when the possible actions of the decision maker are represented by binary vectors. The regret of the decision maker is the difference between her realized loss and the best loss she would have achieved by picking, in hindsight, the best possible action. Our goal is to understand the magnitude of the best possible (minimax) regret. We study the problem under three different assumptions for the feedback the decision maker receives: full information, and the partial information models of the so-called "semi-bandit" and "bandit" problems. Combining the Mirror Descent algorithm and the INF (Implicitely Normalized Forecaster) strategy, we are able to prove optimal bounds for the semi-bandit case. We also recover the optimal bounds for the full information setting. In the bandit case we discuss existing results in light of a new lower bound, and suggest a conjecture on the optimal regret in that case. Finally we also prove that the standard exponentially weighted average forecaster is provably suboptimal in the setting of online combinatorial optimization.

cs.LG↗

arXiv2013

Bounded regret in stochastic multi-armed bandits

We study the stochastic multi-armed bandit problem when one knows the value $μ^{(\star)}$ of an optimal arm, as a well as a positive lower bound on the smallest positive gap $Δ$. We propose a new randomized policy that attains a regret {\em uniformly bounded over time} in this setting. We also prove several lower bounds, which show in particular that bounded regret is not possible if one only knows $Δ$, and bounded regret of order $1/Δ$ is not possible if one only knows $μ^{(\star)}$

math.ST↗

arXiv2012

Regret Analysis of Stochastic and Nonstochastic Multi-armed Bandit Problems

Multi-armed bandit problems are the most basic examples of sequential decision problems with an exploration-exploitation trade-off. This is the balance between staying with the option that gave highest payoffs in the past and exploring new options that might give higher payoffs in the future. Although the study of bandit problems dates back to the Thirties, exploration-exploitation trade-offs arise in several modern applications, such as ad placement, website optimization, and packet routing. Mathematically, a multi-armed bandit is defined by the payoff process associated with each option. In this survey, we focus on two extreme cases in which the analysis of regret is particularly simple and elegant: i.i.d. payoffs and adversarial payoffs. Besides the basic setting of finitely many actions, we also analyze some of the most important variants and extensions, such as the contextual bandit model.

cs.LG↗

arXiv2012

Bandits with heavy tail

The stochastic multi-armed bandit problem is well understood when the reward distributions are sub-Gaussian. In this paper we examine the bandit problem under the weaker assumption that the distributions have moments of order 1+ε, for some $ε\in (0,1]$. Surprisingly, moments of order 2 (i.e., finite variance) are sufficient to obtain regret bounds of the same order as under sub-Gaussian reward distributions. In order to achieve such regret, we define sampling strategies based on refined estimators of the mean such as the truncated empirical mean, Catoni's M-estimator, and the median-of-means estimator. We also derive matching lower bounds that also show that the best achievable regret deteriorates when ε<1.

stat.ML↗

arXiv2012

Detection of correlations

We consider the hypothesis testing problem of deciding whether an observed high-dimensional vector has independent normal components or, alternatively, if it has a small subset of correlated components. The correlated components may have a certain combinatorial structure known to the statistician. We establish upper and lower bounds for the worst-case (minimax) risk in terms of the size of the correlated subset, the level of correlation, and the structure of the class of possibly correlated sets. We show that some simple tests have near-optimal performance in many cases, while the generalized likelihood ratio test is suboptimal in some important cases.

math.ST↗