arXiv subjects

Sébastien Bubeck

Publications and source records attributed to Sébastien Bubeck.

At least 55 records · Page 3Linked to original sources

arXiv2018

Multi-scale Online Learning and its Applications to Online Auctions

We consider revenue maximization in online auction/pricing problems. A seller sells an identical item in each period to a new buyer, or a new set of buyers. For the online posted pricing problem, we show regret bounds that scale with the best fixed price, rather than the range of the values. We also show regret bounds that are almost scale free, and match the offline sample complexity, when comparing to a benchmark that requires a lower bound on the market share. These results are obtained by generalizing the classical learning from experts and multi-armed bandit problems to their multi-scale versions. In this version, the reward of each action is in a different range, and the regret w.r.t. a given action scales with its own range, rather than the maximum range.

cs.GT↗

arXiv2018

Entropic CLT and phase transition in high-dimensional Wishart matrices

We consider high dimensional Wishart matrices $\mathbb{X} \mathbb{X}^{\top}$ where the entries of $\mathbb{X} \in {\mathbb{R}^{n \times d}}$ are i.i.d. from a log-concave distribution. We prove an information theoretic phase transition: such matrices are close in total variation distance to the corresponding Gaussian ensemble if and only if $d$ is much larger than $n^3$. Our proof is entropy-based, making use of the chain rule for relative entropy along with the recursive structure in the definition of the Wishart ensemble. The proof crucially relies on the well known relation between Fisher information and entropy, a variational representation for Fisher information, concentration bounds for the spectral norm of a random matrix, and certain small ball probability estimates for log-concave measures.

math.PR↗

arXiv2018

An homotopy method for $\ell_p$ regression provably beyond self-concordance and in input-sparsity time

We consider the problem of linear regression where the $\ell_2^n$ norm loss (i.e., the usual least squares loss) is replaced by the $\ell_p^n$ norm. We show how to solve such problems up to machine precision in $O^*(n^{|1/2 - 1/p|})$ (dense) matrix-vector products and $O^*(1)$ matrix inversions, or alternatively in $O^*(n^{|1/2 - 1/p|})$ calls to a (sparse) linear system solver. This improves the state of the art for any $p\not\in \{1,2,+\infty\}$. Furthermore we also propose a randomized algorithm solving such problems in {\em input sparsity time}, i.e., $O^*(Z + \mathrm{poly}(d))$ where $Z$ is the size of the input and $d$ is the number of variables. Such a result was only known for $p=2$. Finally we prove that these results lie outside the scope of the Nesterov-Nemirovski's theory of interior point methods by showing that any symmetric self-concordant barrier on the $\ell_p^n$ unit ball has self-concordance parameter $\tildeΩ(n)$.

math.OC↗

arXiv2018

Optimal Algorithms for Non-Smooth Distributed Optimization in Networks

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.

math.OC↗

arXiv2018

Adversarial examples from computational constraints

Why are classifiers in high dimension vulnerable to "adversarial" perturbations? We show that it is likely not due to information theoretic limitations, but rather it could be due to computational constraints. First we prove that, for a broad set of classification tasks, the mere existence of a robust classifier implies that it can be found by a possibly exponential-time algorithm with relatively few training examples. Then we give a particular classification task where learning a robust classifier is computationally intractable. More precisely we construct a binary classification task in high dimensional space which is (i) information theoretically easy to learn robustly for large perturbations, (ii) efficiently learnable (non-robustly) by a simple linear separator, (iii) yet is not efficiently robustly learnable, even for small perturbations, by any algorithm in the statistical query (SQ) model. This example gives an exponential separation between classical learning and robust learning in the statistical query model. It suggests that adversarial examples may be an unavoidable byproduct of computational limitations of learning algorithms.

stat.ML↗

arXiv2018

Make the Minority Great Again: First-Order Regret Bound for Contextual Bandits

Regret bounds in online learning compare the player's performance to $L^*$, the optimal performance in hindsight with a fixed strategy. Typically such bounds scale with the square root of the time horizon $T$. The more refined concept of first-order regret bound replaces this with a scaling $\sqrt{L^*}$, which may be much smaller than $\sqrt{T}$. It is well known that minor variants of standard algorithms satisfy first-order regret bounds in the full information and multi-armed bandit settings. In a COLT 2017 open problem, Agarwal, Krishnamurthy, Langford, Luo, and Schapire raised the issue that existing techniques do not seem sufficient to obtain first-order regret bounds for the contextual bandit problem. In the present paper, we resolve this open problem by presenting a new strategy based on augmenting the policy space.

cs.LG↗

arXiv2017

Sparsity, variance and curvature in multi-armed bandits

In (online) learning theory the concepts of sparsity, variance and curvature are well-understood and are routinely used to obtain refined regret and generalization bounds. In this paper we further our understanding of these concepts in the more challenging limited feedback scenario. We consider the adversarial multi-armed bandit and linear bandit settings and solve several open problems pertaining to the existence of algorithms with favorable regret bounds under the following assumptions: (i) sparsity of the individual losses, (ii) small variation of the loss sequence, and (iii) curvature of the action set. Specifically we show that (i) for $s$-sparse losses one can obtain $\tilde{O}(\sqrt{s T})$-regret (solving an open problem by Kwon and Perchet), (ii) for loss sequences with variation bounded by $Q$ one can obtain $\tilde{O}(\sqrt{Q})$-regret (solving an open problem by Kale and Hazan), and (iii) for linear bandit on an $\ell_p^n$ ball one can obtain $\tilde{O}(\sqrt{n T})$-regret for $p \in [1,2]$ and one has $\tildeΩ(n \sqrt{T})$-regret for $p>2$ (solving an open problem by Bubeck, Cesa-Bianchi and Kakade). A key new insight to obtain these results is to use regularizers satisfying more refined conditions than general self-concordance

cs.LG↗

arXiv2017

Local max-cut in smoothed polynomial time

In 1988, Johnson, Papadimitriou and Yannakakis wrote that "Practically all the empirical evidence would lead us to conclude that finding locally optimal solutions is much easier than solving NP-hard problems". Since then the empirical evidence has continued to amass, but formal proofs of this phenomenon have remained elusive. A canonical (and indeed complete) example is the local max-cut problem, for which no polynomial time method is known. In a breakthrough paper, Etscheid and Röglin proved that the smoothed complexity of local max-cut is quasi-polynomial, i.e., if arbitrary bounded weights are randomly perturbed, a local maximum can be found in $n^{O(\log n)}$ steps. In this paper we prove smoothed polynomial complexity for local max-cut, thus confirming that finding local optima for max-cut is much easier than solving it.

cs.DS↗

arXiv2017

Optimal algorithms for smooth and strongly convex distributed optimization in networks

In this paper, we determine the optimal convergence rates for strongly convex and smooth distributed optimization in two settings: centralized and decentralized communications over a network. For centralized (i.e. master/slave) algorithms, we show that distributing Nesterov's accelerated gradient descent is optimal and achieves a precision $\varepsilon > 0$ in time $O(\sqrt{κ_g}(1+Δτ)\ln(1/\varepsilon))$, where $κ_g$ is the condition number of the (global) function to optimize, $Δ$ is the diameter of the network, and $τ$ (resp. $1$) is the time needed to communicate values between two neighbors (resp. perform local computations). For decentralized algorithms based on gossip, we provide the first optimal algorithm, called the multi-step dual accelerated (MSDA) method, that achieves a precision $\varepsilon > 0$ in time $O(\sqrt{κ_l}(1+\fracτ{\sqrtγ})\ln(1/\varepsilon))$, where $κ_l$ is the condition number of the local functions and $γ$ is the (normalized) eigengap of the gossip matrix used for communication between nodes. We then verify the efficiency of MSDA against state-of-the-art methods for two problems: least-squares regression and classification by logistic regression.

math.OC↗

arXiv2016

Asymptotic behavior of the Eden model with positively homogeneous edge weights

Let $d\in\mathbb N$, $α\in\mathbb R$, and let $f :\mathbb R^d\setminus \{0\} \rightarrow (0,\infty)$ be locally Lipschitz and positively homogeneous of degree $α$ (e.g. $f$ could be the $α$th power of a norm on $\mathbb R^d$). We study a generalization of the Eden model on $\mathbb Z^d$ wherein the next edge added to the cluster is chosen from the set of all edges incident to the current cluster with probability proportional to the value of $f$ at the midpoint of this edge, rather than uniformly. This model is equivalent to a variant of first passage percolation where the edge passage times are independent exponential random variables with parameters given by the value of $f$ at the midpoint of the edge. We prove that the $f$-weighted Eden model clusters have an a.s. deterministic limit shape if $α< 1$, which is an explicit functional of $f$ and the limit shape of the standard Eden model, and estimate the rate of convergence to this limit shape. We also prove that if $α>1$, then there is a norm $ν$ on $\mathbb R^d$ (depending on $α$) such that if we set $f(z) = ν(z)^{ α}$, then the $f$-weighted Eden model clusters are a.s.\ contained in a Euclidean cone with opening angle $<π$ for all time. We further show that there does \emph{not} exist a norm on $\mathbb R^d$ for which this latter statement holds for all $α>1$; and that there is no choice of function $f$ for which the above statement holds with $α=1$. Our basic approach is to compare the local behavior of the $f$-weighted first passage percolation to that of unweighted first passage percolation with iid exponential edge weights (which is equivalent to the unweighted Eden model). We include a list of open problems and several computer simulations.

math.PR↗

arXiv2016

Basic models and questions in statistical network analysis

Extracting information from large graphs has become an important statistical problem since network data is now common in various fields. In this minicourse we will investigate the most natural statistical questions for three canonical probabilistic models of networks: (i) community detection in the stochastic block model, (ii) finding the embedding of a random geometric graph, and (iii) finding the original vertex in a preferential attachment tree. Along the way we will cover many interesting topics in probability theory such as Pólya urns, large deviation theory, concentration of measure in high dimension, entropic central limit theorems, and more.

math.ST↗

arXiv2016

Kernel-based methods for bandit convex optimization

We consider the adversarial convex bandit problem and we build the first $\mathrm{poly}(T)$-time algorithm with $\mathrm{poly}(n) \sqrt{T}$-regret for this problem. To do so we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning rate). The basic version of our algorithm achieves $\tilde{O}(n^{9.5} \sqrt{T})$-regret, and we show that a simple variant of this algorithm can be run in $\mathrm{poly}(n \log(T))$-time per step at the cost of an additional $\mathrm{poly}(n) T^{o(1)}$ factor in the regret. These results improve upon the $\tilde{O}(n^{11} \sqrt{T})$-regret and $\exp(\mathrm{poly}(T))$-time result of the first two authors, and the $\log(T)^{\mathrm{poly}(n)} \sqrt{T}$-regret and $\log(T)^{\mathrm{poly}(n)}$-time result of Hazan and Li. Furthermore we conjecture that another variant of the algorithm could achieve $\tilde{O}(n^{1.5} \sqrt{T})$-regret, and moreover that this regret is unimprovable (the current best lower bound being $Ω(n \sqrt{T})$ and it is achieved with linear functions). For the simpler situation of zeroth order stochastic convex optimization this corresponds to the conjecture that the optimal query complexity is of order $n^3 / ε^2$.

cs.LG↗

arXiv2016

Black-box optimization with a politician

We propose a new framework for black-box convex optimization which is well-suited for situations where gradient computations are expensive. We derive a new method for this framework which leverages several concepts from convex optimization, from standard first-order methods (e.g. gradient descent or quasi-Newton methods) to analytical centers (i.e. minimizers of self-concordant barriers). We demonstrate empirically that our new technique compares favorably with state of the art algorithms (such as BFGS).

math.OC↗

arXiv2016

On paths, stars and wyes in trees

We further the study of local profiles of trees. Bubeck and Linial showed that the set of 5-profiles contains a certain polytope, namely the convex hull of d-millipedes, and they proved that the segment [0-millipede, 1-millipede] corresponds to a face of the set of 5-profiles. Our main result shows that the segment [1-millipede, 2-millipede] also corresponds to a face. Surprisingly we also show that for d > 3 the segment [d-millipede, (d+1)-millipede] is not a face of the set of 5-profiles. We do so by exhibiting new trees which are generalized millipedes with intriguing patterns for their degree sequence. The plot thickens, and the set of 5-profiles remains a mysterious convex set.

math.CO↗

arXiv2015

Finding Adam in random growing trees

We investigate algorithms to find the first vertex in large trees generated by either the uniform attachment or preferential attachment model. We require the algorithm to output a set of $K$ vertices, such that, with probability at least $1-ε$, the first vertex is in this set. We show that for any $ε$, there exist such algorithms with $K$ independent of the size of the input tree. Moreover, we provide almost tight bounds for the best value of $K$ as a function of $ε$. In the uniform attachment case we show that the optimal $K$ is subpolynomial in $1/ε$, and that it has to be at least superpolylogarithmic. On the other hand, the preferential attachment case is exponentially harder, as we prove that the best $K$ is polynomial in $1/ε$. We conclude the paper with several open problems.

math.PR↗

arXiv2015

Testing for high-dimensional geometry in random graphs

We study the problem of detecting the presence of an underlying high-dimensional geometric structure in a random graph. Under the null hypothesis, the observed graph is a realization of an Erdős-Rényi random graph $G(n,p)$. Under the alternative, the graph is generated from the $G(n,p,d)$ model, where each vertex corresponds to a latent independent random vector uniformly distributed on the sphere $\mathbb{S}^{d-1}$, and two vertices are connected if the corresponding latent vectors are close enough. In the dense regime (i.e., $p$ is a constant), we propose a near-optimal and computationally efficient testing procedure based on a new quantity which we call signed triangles. The proof of the detection lower bound is based on a new bound on the total variation distance between a Wishart matrix and an appropriately normalized GOE matrix. In the sparse regime, we make a conjecture for the optimal detection boundary. We conclude the paper with some preliminary steps on the problem of estimating the dimension in $G(n,p,d)$.

math.ST↗

arXiv2015

Convex Optimization: Algorithms and Complexity

This monograph presents the main complexity theorems in convex optimization and their corresponding algorithms. Starting from the fundamental theory of black-box optimization, the material progresses towards recent advances in structural optimization and stochastic optimization. Our presentation of black-box optimization, strongly influenced by Nesterov's seminal book and Nemirovski's lecture notes, includes the analysis of cutting plane methods, as well as (accelerated) gradient descent schemes. We also pay special attention to non-Euclidean settings (relevant algorithms include Frank-Wolfe, mirror descent, and dual averaging) and discuss their relevance in machine learning. We provide a gentle introduction to structural optimization with FISTA (to optimize a sum of a smooth and a simple non-smooth term), saddle-point mirror prox (Nemirovski's alternative to Nesterov's smoothing), and a concise description of interior point methods. In stochastic optimization we discuss stochastic gradient descent, mini-batches, random coordinate descent, and sublinear algorithms. We also briefly touch upon convex relaxation of combinatorial problems and the use of randomness to round solutions, as well as random walks based methods.

math.OC↗

arXiv2015

Detecting Markov Random Fields Hidden in White Noise

Motivated by change point problems in time series and the detection of textured objects in images, we consider the problem of detecting a piece of a Gaussian Markov random field hidden in white Gaussian noise. We derive minimax lower bounds and propose near-optimal tests.

math.ST↗