arXiv subjects

Sébastien Bubeck

Publications and source records attributed to Sébastien Bubeck.

At least 37 records · Page 2Linked to original sources

arXiv2021

A single gradient step finds adversarial examples on random two-layers neural networks

Daniely and Schacham recently showed that gradient descent finds adversarial examples on random undercomplete two-layers ReLU neural networks. The term "undercomplete" refers to the fact that their proof only holds when the number of neurons is a vanishing fraction of the ambient dimension. We extend their result to the overcomplete case, where the number of neurons is larger than the dimension (yet also subexponential in the dimension). In fact we prove that a single step of gradient descent suffices. We also show this result for any subexponential width random neural network with smooth activation function.

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arXiv2021

Complexity of Highly Parallel Non-Smooth Convex Optimization

A landmark result of non-smooth convex optimization is that gradient descent is an optimal algorithm whenever the number of computed gradients is smaller than the dimension $d$. In this paper we study the extension of this result to the parallel optimization setting. Namely we consider optimization algorithms interacting with a highly parallel gradient oracle, that is one that can answer $\mathrm{poly}(d)$ gradient queries in parallel. We show that in this case gradient descent is optimal only up to $\tilde{O}(\sqrt{d})$ rounds of interactions with the oracle. The lower bound improves upon a decades old construction by Nemirovski which proves optimality only up to $d^{1/3}$ rounds (as recently observed by Balkanski and Singer), and the suboptimality of gradient descent after $\sqrt{d}$ rounds was already observed by Duchi, Bartlett and Wainwright. In the latter regime we propose a new method with improved complexity, which we conjecture to be optimal. The analysis of this new method is based upon a generalized version of the recent results on optimal acceleration for highly smooth convex optimization.

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arXiv2020

How to trap a gradient flow

We consider the problem of finding an $\varepsilon$-approximate stationary point of a smooth function on a compact domain of $\mathbb{R}^d$. In contrast with dimension-free approaches such as gradient descent, we focus here on the case where $d$ is finite, and potentially small. This viewpoint was explored in 1993 by Vavasis, who proposed an algorithm which, for any fixed finite dimension $d$, improves upon the $O(1/\varepsilon^2)$ oracle complexity of gradient descent. For example for $d=2$, Vavasis' approach obtains the complexity $O(1/\varepsilon)$. Moreover for $d=2$ he also proved a lower bound of $Ω(1/\sqrt{\varepsilon})$ for deterministic algorithms (we extend this result to randomized algorithms). Our main contribution is an algorithm, which we call gradient flow trapping (GFT), and the analysis of its oracle complexity. In dimension $d=2$, GFT closes the gap with Vavasis' lower bound (up to a logarithmic factor), as we show that it has complexity $O\left(\sqrt{\frac{\log(1/\varepsilon)}{\varepsilon}}\right)$. In dimension $d=3$, we show a complexity of $O\left(\frac{\log(1/\varepsilon)}{\varepsilon}\right)$, improving upon Vavasis' $O\left(1 / \varepsilon^{1.2} \right)$. In higher dimensions, GFT has the remarkable property of being a logarithmic parallel depth strategy, in stark contrast with the polynomial depth of gradient descent or Vavasis' algorithm. In this higher dimensional regime, the total work of GFT improves quadratically upon the only other known polylogarithmic depth strategy for this problem, namely naive grid search. We augment this result with another algorithm, named \emph{cut and flow} (CF), which improves upon Vavasis' algorithm in any fixed dimension.

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arXiv2020

Metrical task systems on trees via mirror descent and unfair gluing

We consider metrical task systems on tree metrics, and present an $O(\mathrm{depth} \times \log n)$-competitive randomized algorithm based on the mirror descent framework introduced in our prior work on the $k$-server problem. For the special case of hierarchically separated trees (HSTs), we use mirror descent to refine the standard approach based on gluing unfair metrical task systems. This yields an $O(\log n)$-competitive algorithm for HSTs, thus removing an extraneous $\log\log n$ in the bound of Fiat and Mendel (2003). Combined with well-known HST embedding theorems, this also gives an $O((\log n)^2)$-competitive randomized algorithm for every $n$-point metric space.

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arXiv2020

A law of robustness for two-layers neural networks

We initiate the study of the inherent tradeoffs between the size of a neural network and its robustness, as measured by its Lipschitz constant. We make a precise conjecture that, for any Lipschitz activation function and for most datasets, any two-layers neural network with $k$ neurons that perfectly fit the data must have its Lipschitz constant larger (up to a constant) than $\sqrt{n/k}$ where $n$ is the number of datapoints. In particular, this conjecture implies that overparametrization is necessary for robustness, since it means that one needs roughly one neuron per datapoint to ensure a $O(1)$-Lipschitz network, while mere data fitting of $d$-dimensional data requires only one neuron per $d$ datapoints. We prove a weaker version of this conjecture when the Lipschitz constant is replaced by an upper bound on it based on the spectral norm of the weight matrix. We also prove the conjecture in the high-dimensional regime $n \approx d$ (which we also refer to as the undercomplete case, since only $k \leq d$ is relevant here). Finally we prove the conjecture for polynomial activation functions of degree $p$ when $n \approx d^p$. We complement these findings with experimental evidence supporting the conjecture.

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arXiv2020

Cooperative and Stochastic Multi-Player Multi-Armed Bandit: Optimal Regret With Neither Communication Nor Collisions

We consider the cooperative multi-player version of the stochastic multi-armed bandit problem. We study the regime where the players cannot communicate but have access to shared randomness. In prior work by the first two authors, a strategy for this regime was constructed for two players and three arms, with regret $\tilde{O}(\sqrt{T})$, and with no collisions at all between the players (with very high probability). In this paper we show that these properties (near-optimal regret and no collisions at all) are achievable for any number of players and arms. At a high level, the previous strategy heavily relied on a $2$-dimensional geometric intuition that was difficult to generalize in higher dimensions, while here we take a more combinatorial route to build the new strategy.

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arXiv2020

Network size and weights size for memorization with two-layers neural networks

In 1988, Eric B. Baum showed that two-layers neural networks with threshold activation function can perfectly memorize the binary labels of $n$ points in general position in $\mathbb{R}^d$ using only $\ulcorner n/d \urcorner$ neurons. We observe that with ReLU networks, using four times as many neurons one can fit arbitrary real labels. Moreover, for approximate memorization up to error $ε$, the neural tangent kernel can also memorize with only $O\left(\frac{n}{d} \cdot \log(1/ε) \right)$ neurons (assuming that the data is well dispersed too). We show however that these constructions give rise to networks where the magnitude of the neurons' weights are far from optimal. In contrast we propose a new training procedure for ReLU networks, based on complex (as opposed to real) recombination of the neurons, for which we show approximate memorization with both $O\left(\frac{n}{d} \cdot \frac{\log(1/ε)}ε\right)$ neurons, as well as nearly-optimal size of the weights.

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arXiv2020

Metrical Service Systems with Transformations

We consider a generalization of the fundamental online metrical service systems (MSS) problem where the feasible region can be transformed between requests. In this problem, which we call T-MSS, an algorithm maintains a point in a metric space and has to serve a sequence of requests. Each request is a map (transformation) $f_t\colon A_t\to B_t$ between subsets $A_t$ and $B_t$ of the metric space. To serve it, the algorithm has to go to a point $a_t\in A_t$, paying the distance from its previous position. Then, the transformation is applied, modifying the algorithm's state to $f_t(a_t)$. Such transformations can model, e.g., changes to the environment that are outside of an algorithm's control, and we therefore do not charge any additional cost to the algorithm when the transformation is applied. The transformations also allow to model requests occurring in the $k$-taxi problem. We show that for $α$-Lipschitz transformations, the competitive ratio is $Θ(α)^{n-2}$ on $n$-point metrics. Here, the upper bound is achieved by a deterministic algorithm and the lower bound holds even for randomized algorithms. For the $k$-taxi problem, we prove a competitive ratio of $\tilde O((n\log k)^2)$. For chasing convex bodies, we show that even with contracting transformations no competitive algorithm exists. The problem T-MSS has a striking connection to the following deep mathematical question: Given a finite metric space $M$, what is the required cardinality of an extension $\hat M\supseteq M$ where each partial isometry on $M$ extends to an automorphism? We give partial answers for special cases.

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arXiv2020

Online Learning for Active Cache Synchronization

Existing multi-armed bandit (MAB) models make two implicit assumptions: an arm generates a payoff only when it is played, and the agent observes every payoff that is generated. This paper introduces synchronization bandits, a MAB variant where all arms generate costs at all times, but the agent observes an arm's instantaneous cost only when the arm is played. Synchronization MABs are inspired by online caching scenarios such as Web crawling, where an arm corresponds to a cached item and playing the arm means downloading its fresh copy from a server. We present MirrorSync, an online learning algorithm for synchronization bandits, establish an adversarial regret of $O(T^{2/3})$ for it, and show how to make it practical.

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arXiv2020

Coordination without communication: optimal regret in two players multi-armed bandits

We consider two agents playing simultaneously the same stochastic three-armed bandit problem. The two agents are cooperating but they cannot communicate. We propose a strategy with no collisions at all between the players (with very high probability), and with near-optimal regret $O(\sqrt{T \log(T)})$. We also argue that the extra logarithmic term $\sqrt{\log(T)}$ should be necessary by proving a lower bound for a full information variant of the problem.

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arXiv2020

Online Multiserver Convex Chasing and Optimization

We introduce the problem of $k$-chasing of convex functions, a simultaneous generalization of both the famous k-server problem in $R^d$, and of the problem of chasing convex bodies and functions. Aside from fundamental interest in this general form, it has natural applications to online $k$-clustering problems with objectives such as $k$-median or $k$-means. We show that this problem exhibits a rich landscape of behavior. In general, if both $k > 1$ and $d > 1$ there does not exist any online algorithm with bounded competitiveness. By contrast, we exhibit a class of nicely behaved functions (which include in particular the above-mentioned clustering problems), for which we show that competitive online algorithms exist, and moreover with dimension-free competitive ratio. We also introduce a parallel question of top-$k$ action regret minimization in the realm of online convex optimization. There, too, a much rougher landscape emerges for $k > 1$. While it is possible to achieve vanishing regret, unlike the top-one action case the rate of vanishing does not speed up for strongly convex functions. Moreover, vanishing regret necessitates both intractable computations and randomness. Finally we leave open whether almost dimension-free regret is achievable for $k > 1$ and general convex losses. As evidence that it might be possible, we prove dimension-free regret for linear losses via an information-theoretic argument.

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arXiv2019

Near-optimal method for highly smooth convex optimization

We propose a near-optimal method for highly smooth convex optimization. More precisely, in the oracle model where one obtains the $p^{th}$ order Taylor expansion of a function at the query point, we propose a method with rate of convergence $\tilde{O}(1/k^{\frac{ 3p +1}{2}})$ after $k$ queries to the oracle for any convex function whose $p^{th}$ order derivative is Lipschitz.

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arXiv2019

Improved Path-length Regret Bounds for Bandits

We study adaptive regret bounds in terms of the variation of the losses (the so-called path-length bounds) for both multi-armed bandit and more generally linear bandit. We first show that the seemingly suboptimal path-length bound of (Wei and Luo, 2018) is in fact not improvable for adaptive adversary. Despite this negative result, we then develop two new algorithms, one that strictly improves over (Wei and Luo, 2018) with a smaller path-length measure, and the other which improves over (Wei and Luo, 2018) for oblivious adversary when the path-length is large. Our algorithms are based on the well-studied optimistic mirror descent framework, but importantly with several novel techniques, including new optimistic predictions, a slight bias towards recently selected arms, and the use of a hybrid regularizer similar to that of (Bubeck et al., 2018). Furthermore, we extend our results to linear bandit by showing a reduction to obtaining dynamic regret for a full-information problem, followed by a further reduction to convex body chasing. We propose a simple greedy chasing algorithm for squared 2-norm, leading to new dynamic regret results and as a consequence the first path-length regret for general linear bandit as well.

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arXiv2019

Non-Stochastic Multi-Player Multi-Armed Bandits: Optimal Rate With Collision Information, Sublinear Without

We consider the non-stochastic version of the (cooperative) multi-player multi-armed bandit problem. The model assumes no communication at all between the players, and furthermore when two (or more) players select the same action this results in a maximal loss. We prove the first $\sqrt{T}$-type regret guarantee for this problem, under the feedback model where collisions are announced to the colliding players. Such a bound was not known even for the simpler stochastic version. We also prove the first sublinear guarantee for the feedback model where collision information is not available, namely $T^{1-\frac{1}{2m}}$ where $m$ is the number of players.

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arXiv2019

Parametrized Metrical Task Systems

We consider parametrized versions of metrical task systems and metrical service systems, two fundamental models of online computing, where the constrained parameter is the number of possible distinct requests $m$. Such parametrization occurs naturally in a wide range of applications. Striking examples are certain power management problems, which are modeled as metrical task systems with $m=2$. We characterize the competitive ratio in terms of the parameter $m$ for both deterministic and randomized algorithms on hierarchically separated trees. Our findings uncover a rich and unexpected picture that differs substantially from what is known or conjectured about the unparametrized versions of these problems. For metrical task systems, we show that deterministic algorithms do not exhibit any asymptotic gain beyond one-level trees (namely, uniform metric spaces), whereas randomized algorithms do not exhibit any asymptotic gain even for one-level trees. In contrast, the special case of metrical service systems (subset chasing) behaves very differently. Both deterministic and randomized algorithms exhibit gain, for $m$ sufficiently small compared to $n$, for any number of levels. Most significantly, they exhibit a large gain for uniform metric spaces and a smaller gain for two-level trees. Moreover, it turns out that in these cases (as well as in the case of metrical task systems for uniform metric spaces with $m$ being an absolute constant), deterministic algorithms are essentially as powerful as randomized algorithms. This is surprising and runs counter to the ubiquitous intuition/conjecture that, for most problems that can be modeled as metrical task systems, the randomized competitive ratio is polylogarithmic in the deterministic competitive ratio.

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arXiv2018

A Nearly-Linear Bound for Chasing Nested Convex Bodies

Friedman and Linial introduced the convex body chasing problem to explore the interplay between geometry and competitive ratio in metrical task systems. In convex body chasing, at each time step $t \in \mathbb{N}$, the online algorithm receives a request in the form of a convex body $K_t \subseteq \mathbb{R}^d$ and must output a point $x_t \in K_t$. The goal is to minimize the total movement between consecutive output points, where the distance is measured in some given norm. This problem is still far from being understood, and recently Bansal et al. gave an algorithm for the nested version, where each convex body is contained within the previous one. We propose a different strategy which is $O(d \log d)$-competitive algorithm for this nested convex body chasing problem, improving substantially over previous work. Our algorithm works for any norm. This result is almost tight, given an $Ω(d)$ lower bound for the $\ell_{\infty}$.

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arXiv2018

Adversarial Examples from Cryptographic Pseudo-Random Generators

In our recent work (Bubeck, Price, Razenshteyn, arXiv:1805.10204) we argued that adversarial examples in machine learning might be due to an inherent computational hardness of the problem. More precisely, we constructed a binary classification task for which (i) a robust classifier exists; yet no non-trivial accuracy can be obtained with an efficient algorithm in (ii) the statistical query model. In the present paper we significantly strengthen both (i) and (ii): we now construct a task which admits (i') a maximally robust classifier (that is it can tolerate perturbations of size comparable to the size of the examples themselves); and moreover we prove computational hardness of learning this task under (ii') a standard cryptographic assumption.

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arXiv2018

Competitively Chasing Convex Bodies

Let $\mathcal{F}$ be a family of sets in some metric space. In the $\mathcal{F}$-chasing problem, an online algorithm observes a request sequence of sets in $\mathcal{F}$ and responds (online) by giving a sequence of points in these sets. The movement cost is the distance between consecutive such points. The competitive ratio is the worst case ratio (over request sequences) between the total movement of the online algorithm and the smallest movement one could have achieved by knowing in advance the request sequence. The family $\mathcal{F}$ is said to be chaseable if there exists an online algorithm with finite competitive ratio. In 1991, Linial and Friedman conjectured that the family of convex sets in Euclidean space is chaseable. We prove this conjecture.

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