arXiv subjects

Robert Kleinberg

Publications and source records attributed to Robert Kleinberg.

At least 55 records · Page 3Linked to original sources

arXiv2019

Bandits and Experts in Metric Spaces

In a multi-armed bandit problem, an online algorithm chooses from a set of strategies in a sequence of trials so as to maximize the total payoff of the chosen strategies. While the performance of bandit algorithms with a small finite strategy set is quite well understood, bandit problems with large strategy sets are still a topic of very active investigation, motivated by practical applications such as online auctions and web advertisement. The goal of such research is to identify broad and natural classes of strategy sets and payoff functions which enable the design of efficient solutions. In this work we study a very general setting for the multi-armed bandit problem in which the strategies form a metric space, and the payoff function satisfies a Lipschitz condition with respect to the metric. We refer to this problem as the "Lipschitz MAB problem". We present a solution for the multi-armed bandit problem in this setting. That is, for every metric space we define an isometry invariant which bounds from below the performance of Lipschitz MAB algorithms for this metric space, and we present an algorithm which comes arbitrarily close to meeting this bound. Furthermore, our technique gives even better results for benign payoff functions. We also address the full-feedback ("best expert") version of the problem, where after every round the payoffs from all arms are revealed.

cs.DS↗

arXiv2018

An Alternative View: When Does SGD Escape Local Minima?

Stochastic gradient descent (SGD) is widely used in machine learning. Although being commonly viewed as a fast but not accurate version of gradient descent (GD), it always finds better solutions than GD for modern neural networks. In order to understand this phenomenon, we take an alternative view that SGD is working on the convolved (thus smoothed) version of the loss function. We show that, even if the function $f$ has many bad local minima or saddle points, as long as for every point $x$, the weighted average of the gradients of its neighborhoods is one point convex with respect to the desired solution $x^*$, SGD will get close to, and then stay around $x^*$ with constant probability. More specifically, SGD will not get stuck at "sharp" local minima with small diameters, as long as the neighborhoods of these regions contain enough gradient information. The neighborhood size is controlled by step size and gradient noise. Our result identifies a set of functions that SGD provably works, which is much larger than the set of convex functions. Empirically, we observe that the loss surface of neural networks enjoys nice one point convexity properties locally, therefore our theorem helps explain why SGD works so well for neural networks.

cs.LG↗

arXiv2018

The Growth Rate of Tri-Colored Sum-Free Sets

Let $G$ be an abelian group. A tri-colored sum-free set in $G^n$ is a collection of triples $({\bf a}_i, {\bf b}_i, {\bf c}_i)$ in $G^n$ such that ${\bf a}_i+{\bf b}_j+{\bf c}_k=0$ if and only if $i=j=k$. Fix a prime $q$ and let $C_q$ be the cyclic group of order $q$. Let $θ= \min_{ρ>0} (1+ρ+\cdots + ρ^{q-1}) ρ^{-(q-1)/3}$. Blasiak, Church, Cohn, Grochow, Naslund, Sawin, and Umans (building on previous work of Croot, Lev and Pach, and of Ellenberg and Gijswijt) showed that a tri-colored sum-free set in $C_q^n$ has size at most $3 θ^n$. Between this paper and a paper of Pebody, we will show that, for any $δ> 0$, and $n$ sufficiently large, there are tri-colored sum-free sets in $C_q^n$ of size $(θ-δ)^n$. Our construction also works when $q$ is not prime.

math.CO↗

arXiv2018

Can Deep Reinforcement Learning Solve Erdos-Selfridge-Spencer Games?

Deep reinforcement learning has achieved many recent successes, but our understanding of its strengths and limitations is hampered by the lack of rich environments in which we can fully characterize optimal behavior, and correspondingly diagnose individual actions against such a characterization. Here we consider a family of combinatorial games, arising from work of Erdos, Selfridge, and Spencer, and we propose their use as environments for evaluating and comparing different approaches to reinforcement learning. These games have a number of appealing features: they are challenging for current learning approaches, but they form (i) a low-dimensional, simply parametrized environment where (ii) there is a linear closed form solution for optimal behavior from any state, and (iii) the difficulty of the game can be tuned by changing environment parameters in an interpretable way. We use these Erdos-Selfridge-Spencer games not only to compare different algorithms, but test for generalization, make comparisons to supervised learning, analyse multiagent play, and even develop a self play algorithm. Code can be found at: https://github.com/rubai5/ESS_Game

cs.AI↗

arXiv2018

Delegated Search Approximates Efficient Search

There are many settings in which a principal performs a task by delegating it to an agent, who searches over possible solutions and proposes one to the principal. This describes many aspects of the workflow within organizations, as well as many of the activities undertaken by regulatory bodies, who often obtain relevant information from the parties being regulated through a process of delegation. A fundamental tension underlying delegation is the fact that the agent's interests will typically differ -- potentially significantly -- from the interests of the principal, and as a result the agent may propose solutions based on their own incentives that are inefficient for the principal. A basic problem, therefore, is to design mechanisms by which the principal can constrain the set of proposals they are willing to accept from the agent, to ensure a certain level of quality for the principal from the proposed solution. In this work, we investigate how much the principal loses -- quantitatively, in terms of the objective they are trying to optimize -- when they delegate to an agent. We develop a methodology for bounding this loss of efficiency, and show that in a very general model of delegation, there is a family of mechanisms achieving a universal bound on the ratio between the quality of the solution obtained through delegation and the quality the principal could obtain in an idealized benchmark where they searched for a solution themself. Moreover, it is possible to achieve such bounds through mechanisms with a natural threshold structure, which are thus structurally simpler than the optimal mechanisms typically considered in the literature on delegation. At the heart of our framework is an unexpected connection between delegation and the analysis of prophet inequalities, which we leverage to provide bounds on the behavior of our delegation mechanisms.

cs.GT↗

arXiv2017

Bandits with Knapsacks

Multi-armed bandit problems are the predominant theoretical model of exploration-exploitation tradeoffs in learning, and they have countless applications ranging from medical trials, to communication networks, to Web search and advertising. In many of these application domains the learner may be constrained by one or more supply (or budget) limits, in addition to the customary limitation on the time horizon. The literature lacks a general model encompassing these sorts of problems. We introduce such a model, called "bandits with knapsacks", that combines aspects of stochastic integer programming with online learning. A distinctive feature of our problem, in comparison to the existing regret-minimization literature, is that the optimal policy for a given latent distribution may significantly outperform the policy that plays the optimal fixed arm. Consequently, achieving sublinear regret in the bandits-with-knapsacks problem is significantly more challenging than in conventional bandit problems. We present two algorithms whose reward is close to the information-theoretic optimum: one is based on a novel "balanced exploration" paradigm, while the other is a primal-dual algorithm that uses multiplicative updates. Further, we prove that the regret achieved by both algorithms is optimal up to polylogarithmic factors. We illustrate the generality of the problem by presenting applications in a number of different domains including electronic commerce, routing, and scheduling. As one example of a concrete application, we consider the problem of dynamic posted pricing with limited supply and obtain the first algorithm whose regret, with respect to the optimal dynamic policy, is sublinear in the supply.

cs.DS↗

arXiv2017

The Lovász Theta Function for Random Regular Graphs and Community Detection in the Hard Regime

We derive upper and lower bounds on the degree $d$ for which the Lovász $\vartheta$ function, or equivalently sum-of-squares proofs with degree two, can refute the existence of a $k$-coloring in random regular graphs $G_{n,d}$. We show that this type of refutation fails well above the $k$-colorability transition, and in particular everywhere below the Kesten-Stigum threshold. This is consistent with the conjecture that refuting $k$-colorability, or distinguishing $G_{n,d}$ from the planted coloring model, is hard in this region. Our results also apply to the disassortative case of the stochastic block model, adding evidence to the conjecture that there is a regime where community detection is computationally hard even though it is information-theoretically possible. Using orthogonal polynomials, we also provide explicit upper bounds on $\vartheta(\overline{G})$ for regular graphs of a given girth, which may be of independent interest.

cs.CC↗

arXiv2017

Beating 1-1/e for Ordered Prophets

Hill and Kertz studied the prophet inequality on iid distributions [The Annals of Probability 1982]. They proved a theoretical bound of $1-\frac{1}{e}$ on the approximation factor of their algorithm. They conjectured that the best approximation factor for arbitrarily large n is $\frac{1}{1+1/e} \approx 0.731$. This conjecture remained open prior to this paper for over 30 years. In this paper we present a threshold-based algorithm for the prophet inequality with n iid distributions. Using a nontrivial and novel approach we show that our algorithm is a 0.738-approximation algorithm. By beating the bound of $\frac{1}{1+1/e}$, this refutes the conjecture of Hill and Kertz. Moreover, we generalize our results to non-iid distributions and discuss its applications in mechanism design.

cs.DS↗

arXiv2017

Inferential Privacy Guarantees for Differentially Private Mechanisms

The correlations and network structure amongst individuals in datasets today---whether explicitly articulated, or deduced from biological or behavioral connections---pose new issues around privacy guarantees, because of inferences that can be made about one individual from another's data. This motivates quantifying privacy in networked contexts in terms of "inferential privacy"---which measures the change in beliefs about an individual's data from the result of a computation---as originally proposed by Dalenius in the 1970's. Inferential privacy is implied by differential privacy when data are independent, but can be much worse when data are correlated; indeed, simple examples, as well as a general impossibility theorem of Dwork and Naor, preclude the possibility of achieving non-trivial inferential privacy when the adversary can have arbitrary auxiliary information. In this paper, we ask how differential privacy guarantees translate to guarantees on inferential privacy in networked contexts: specifically, under what limitations on the adversary's information about correlations, modeled as a prior distribution over datasets, can we deduce an inferential guarantee from a differential one? We prove two main results. The first result pertains to distributions that satisfy a natural positive-affiliation condition, and gives an upper bound on the inferential privacy guarantee for any differentially private mechanism. This upper bound is matched by a simple mechanism that adds Laplace noise to the sum of the data. The second result pertains to distributions that have weak correlations, defined in terms of a suitable "influence matrix". The result provides an upper bound for inferential privacy in terms of the differential privacy parameter and the spectral norm of this matrix.

cs.DS↗

arXiv2017

Exponential Segregation in a Two-Dimensional Schelling Model with Tolerant Individuals

We prove that the two-dimensional Schelling segregation model yields monochromatic regions of size exponential in the area of individuals' neighborhoods, provided that the tolerance parameter is a constant strictly less than 1/2 but sufficiently close to it. Our analysis makes use of a connection with the first-passage percolation model from the theory of stochastic processes.

cs.GT↗

arXiv2016

Descending Price Optimally Coordinates Search

Investigating potential purchases is often a substantial investment under uncertainty. Standard market designs, such as simultaneous or English auctions, compound this with uncertainty about the price a bidder will have to pay in order to win. As a result they tend to confuse the process of search both by leading to wasteful information acquisition on goods that have already found a good purchaser and by discouraging needed investigations of objects, potentially eliminating all gains from trade. In contrast, we show that the Dutch auction preserves all of its properties from a standard setting without information costs because it guarantees, at the time of information acquisition, a price at which the good can be purchased. Calibrations to start-up acquisition and timber auctions suggest that in practice the social losses through poor search coordination in standard formats are an order of magnitude or two larger than the (negligible) inefficiencies arising from ex-ante bidder asymmetries.

cs.GT↗

arXiv2016

A nearly tight upper bound on tri-colored sum-free sets in characteristic 2

A tri-colored sum-free set in an abelian group $H$ is a collection of ordered triples in $H^3$, $\{(a_i,b_i,c_i)\}_{i=1}^m$, such that the equation $a_i+b_j+c_k=0$ holds if and only if $i=j=k$. Using a variant of the lemma introduced by Croot, Lev, and Pach in their breakthrough work on arithmetic-progression-free sets, we prove that the size of any tri-colored sum-free set in $\mathbb{F}_2^n$ is bounded above by $6 {n \choose \lfloor n/3 \rfloor}$. This upper bound is tight, up to a factor subexponential in $n$: there exist tri-colored sum-free sets in $\mathbb{F}_2^n$ of size greater than ${n \choose \lfloor n/3 \rfloor} \cdot 2^{-\sqrt{16 n / 3}}$ for all sufficiently large $n$.

math.CO↗

arXiv2016

Simultaneous Nearest Neighbor Search

Motivated by applications in computer vision and databases, we introduce and study the Simultaneous Nearest Neighbor Search (SNN) problem. Given a set of data points, the goal of SNN is to design a data structure that, given a collection of queries, finds a collection of close points that are compatible with each other. Formally, we are given $k$ query points $Q=q_1,\cdots,q_k$, and a compatibility graph $G$ with vertices in $Q$, and the goal is to return data points $p_1,\cdots,p_k$ that minimize (i) the weighted sum of the distances from $q_i$ to $p_i$ and (ii) the weighted sum, over all edges $(i,j)$ in the compatibility graph $G$, of the distances between $p_i$ and $p_j$. The problem has several applications, where one wants to return a set of consistent answers to multiple related queries. This generalizes well-studied computational problems, including NN, Aggregate NN and the 0-extension problem. In this paper we propose and analyze the following general two-step method for designing efficient data structures for SNN. In the first step, for each query point $q_i$ we find its (approximate) nearest neighbor point $\hat{p}_i$; this can be done efficiently using existing approximate nearest neighbor structures. In the second step, we solve an off-line optimization problem over sets $q_1,\cdots,q_k$ and $\hat{p}_1,\cdots,\hat{p}_k$; this can be done efficiently given that $k$ is much smaller than $n$. Even though $\hat{p}_1,\cdots,\hat{p}_k$ might not constitute the optimal answers to queries $q_1,\cdots,q_k$, we show that, for the unweighted case, the resulting algorithm is $O(\log k/\log \log k)$-approximation. Also, we show that the approximation factor can be in fact reduced to a constant for compatibility graphs frequently occurring in practice. Finally, we show that the "empirical approximation factor" provided by the above approach is very close to 1.

cs.DS↗

arXiv2016

Kulfi: Robust Traffic Engineering Using Semi-Oblivious Routing

Wide-area network traffic engineering enables network operators to reduce congestion and improve utilization by balancing load across multiple paths. Current approaches to traffic engineering can be modeled in terms of a routing component that computes forwarding paths, and a load balancing component that maps incoming flows onto those paths dynamically, adjusting sending rates to fit current conditions. Unfortunately, existing systems rely on simple strategies for one or both of these components, which leads to poor performance or requires making frequent updates to forwarding paths, significantly increasing management complexity. This paper explores a different approach based on semi-oblivious routing, a natural extension of oblivious routing in which the system computes a diverse set of paths independent of demands, but also dynamically adapts sending rates as conditions change. Semi-oblivious routing has a number of important advantages over competing approaches including low overhead, nearly optimal performance, and built-in protection against unexpected bursts of traffic and failures. Through in-depth simulations and a deployment on SDN hardware, we show that these benefits are robust, and hold across a wide range of topologies, demands, resource budgets, and failure scenarios.

cs.NI↗

arXiv2015

On the Complexity of Computing an Equilibrium in Combinatorial Auctions

We study combinatorial auctions where each item is sold separately but simultaneously via a second price auction. We ask whether it is possible to efficiently compute in this game a pure Nash equilibrium with social welfare close to the optimal one. We show that when the valuations of the bidders are submodular, in many interesting settings (e.g., constant number of bidders, budget additive bidders) computing an equilibrium with good welfare is essentially as easy as computing, completely ignoring incentives issues, an allocation with good welfare. On the other hand, for subadditive valuations, we show that computing an equilibrium requires exponential communication. Finally, for XOS (a.k.a. fractionally subadditive) valuations, we show that if there exists an efficient algorithm that finds an equilibrium, it must use techniques that are very different from our current ones.

cs.GT↗

arXiv2015

Secretary Problems with Non-Uniform Arrival Order

For many online problems, it is known that the uniform arrival order enables the design of algorithms with much better performance guarantees than under worst-case. The quintessential example is the secretary problem. If the sequence of elements is presented in uniformly random order there is an algorithm that picks the maximum value with probability 1/e, whereas no non-trivial performance guarantee is possible if the elements arrive in worst-case order. This work initiates an investigation into relaxations of the random-ordering hypothesis in online algorithms, by focusing on the secretary problems. We present two sets of properties of distributions over permutations as sufficient conditions, called the block-independence property and uniform-induced-ordering property. We show these two are asymptotically equivalent by borrowing some techniques from the approximation theory. Moreover, we show they both imply the existence of secretary algorithms with constant probability of correct selection, approaching the optimal constant 1/e in the limit. We substantiate our idea by providing several constructions of distributions that satisfy block-independence. We also show that Θ(log log n) is the minimum entropy of any permutation distribution that permits constant probability of correct selection in the secretary problem with n elements. While our block-independence condition is sufficient for constant probability of correct selection, it is not necessary; however, we present complexity-theoretic evidence that no simple necessary and sufficient criterion exists. Finally, we explore the extent to which the performance guarantees of other algorithms are preserved when one relaxes the uniform random ordering assumption, obtaining a positive result for Kleinberg's multiple-choice secretary algorithm and a negative result for the weighted bipartite matching algorithm of Korula and Pal.

cs.DS↗

arXiv2014

Merlin: A Language for Provisioning Network Resources

This paper presents Merlin, a new framework for managing resources in software-defined networks. With Merlin, administrators express high-level policies using programs in a declarative language. The language includes logical predicates to identify sets of packets, regular expressions to encode forwarding paths, and arithmetic formulas to specify bandwidth constraints. The Merlin compiler uses a combination of advanced techniques to translate these policies into code that can be executed on network elements including a constraint solver that allocates bandwidth using parameterizable heuristics. To facilitate dynamic adaptation, Merlin provides mechanisms for delegating control of sub-policies and for verifying that modifications made to sub-policies do not violate global constraints. Experiments demonstrate the expressiveness and scalability of Merlin on real-world topologies and applications. Overall, Merlin simplifies network administration by providing high-level abstractions for specifying network policies and scalable infrastructure for enforcing them.

cs.NI↗

arXiv2014

Optimal Auctions for Correlated Buyers with Sampling

Crémer and McLean [1985] showed that, when buyers' valuations are drawn from a correlated distribution, an auction with full knowledge on the distribution can extract the full social surplus. We study whether this phenomenon persists when the auctioneer has only incomplete knowledge of the distribution, represented by a finite family of candidate distributions, and has sample access to the real distribution. We show that the naive approach which uses samples to distinguish candidate distributions may fail, whereas an extended version of the Crémer-McLean auction simultaneously extracts full social surplus under each candidate distribution. With an algebraic argument, we give a tight bound on the number of samples needed by this auction, which is the difference between the number of candidate distributions and the dimension of the linear space they span.

cs.GT↗