arXiv subjects

Lev Reyzin

Publications and source records attributed to Lev Reyzin.

27 records · Page 2Linked to original sources

arXiv2017

Network Construction with Ordered Constraints

In this paper, we study the problem of constructing a network by observing ordered connectivity constraints, which we define herein. These ordered constraints are made to capture realistic properties of real-world problems that are not reflected in previous, more general models. We give hardness of approximation results and nearly-matching upper bounds for the offline problem, and we study the online problem in both general graphs and restricted sub-classes. In the online problem, for general graphs, we give exponentially better upper bounds than exist for algorithms for general connectivity problems. For the restricted classes of stars and paths we are able to find algorithms with optimal competitive ratios, the latter of which involve analysis using a potential function defined over pq-trees.

cs.DS↗

arXiv2016

Statistical Algorithms and a Lower Bound for Detecting Planted Clique

We introduce a framework for proving lower bounds on computational problems over distributions against algorithms that can be implemented using access to a statistical query oracle. For such algorithms, access to the input distribution is limited to obtaining an estimate of the expectation of any given function on a sample drawn randomly from the input distribution, rather than directly accessing samples. Most natural algorithms of interest in theory and in practice, e.g., moments-based methods, local search, standard iterative methods for convex optimization, MCMC and simulated annealing can be implemented in this framework. Our framework is based on, and generalizes, the statistical query model in learning theory (Kearns, 1998). Our main application is a nearly optimal lower bound on the complexity of any statistical query algorithm for detecting planted bipartite clique distributions (or planted dense subgraph distributions) when the planted clique has size $O(n^{1/2-δ})$ for any constant $δ> 0$. The assumed hardness of variants of these problems has been used to prove hardness of several other problems and as a guarantee for security in cryptographic applications. Our lower bounds provide concrete evidence of hardness, thus supporting these assumptions.

cs.CC↗

arXiv2015

On the Computational Complexity of MapReduce

In this paper we study MapReduce computations from a complexity-theoretic perspective. First, we formulate a uniform version of the MRC model of Karloff et al. (2010). We then show that the class of regular languages, and moreover all of sublogarithmic space, lies in constant round MRC. This result also applies to the MPC model of Andoni et al. (2014). In addition, we prove that, conditioned on a variant of the Exponential Time Hypothesis, there are strict hierarchies within MRC so that increasing the number of rounds or the amount of time per processor increases the power of MRC. To the best of our knowledge we are the first to approach the MapReduce model with complexity-theoretic techniques, and our work lays the foundation for further analysis relating MapReduce to established complexity classes.

cs.CC↗

arXiv2014

Network installation and recovery: approximation lower bounds and faster exact formulations

We study the Neighbor Aided Network Installation Problem (NANIP) introduced previously which asks for a minimal cost ordering of the vertices of a graph, where the cost of visiting a node is a function of the number of neighbors that have already been visited. This problem has applications in resource management and disaster recovery. In this paper we analyze the computational hardness of NANIP. In particular we show that this problem is NP-hard even when restricted to convex decreasing cost functions, give a linear approximation lower bound for the greedy algorithm, and prove a general sub-constant approximation lower bound. Then we give a new integer programming formulation of NANIP and empirically observe its speedup over the original integer program.

cs.DM↗

arXiv2014

Data Stability in Clustering: A Closer Look

We consider the model introduced by Bilu and Linial (2010), who study problems for which the optimal clustering does not change when distances are perturbed. They show that even when a problem is NP-hard, it is sometimes possible to obtain efficient algorithms for instances resilient to certain multiplicative perturbations, e.g. on the order of $O(\sqrt{n})$ for max-cut clustering. Awasthi et al. (2010) consider center-based objectives, and Balcan and Liang (2011) analyze the $k$-median and min-sum objectives, giving efficient algorithms for instances resilient to certain constant multiplicative perturbations. Here, we are motivated by the question of to what extent these assumptions can be relaxed while allowing for efficient algorithms. We show there is little room to improve these results by giving NP-hardness lower bounds for both the $k$-median and min-sum objectives. On the other hand, we show that constant multiplicative resilience parameters can be so strong as to make the clustering problem trivial, leaving only a narrow range of resilience parameters for which clustering is interesting. We also consider a model of additive perturbations and give a correspondence between additive and multiplicative notions of stability. Our results provide a close examination of the consequences of assuming stability in data.

cs.LG↗

arXiv2014

On Coloring Resilient Graphs

We introduce a new notion of resilience for constraint satisfaction problems, with the goal of more precisely determining the boundary between NP-hardness and the existence of efficient algorithms for resilient instances. In particular, we study $r$-resiliently $k$-colorable graphs, which are those $k$-colorable graphs that remain $k$-colorable even after the addition of any $r$ new edges. We prove lower bounds on the NP-hardness of coloring resiliently colorable graphs, and provide an algorithm that colors sufficiently resilient graphs. We also analyze the corresponding notion of resilience for $k$-SAT. This notion of resilience suggests an array of open questions for graph coloring and other combinatorial problems.

cs.CC↗

arXiv2013

Anti-Coordination Games and Stable Graph Colorings

Motivated by understanding non-strict and strict pure strategy equilibria in network anti-coordination games, we define notions of stable and, respectively, strictly stable colorings in graphs. We characterize the cases when such colorings exist and when the decision problem is NP-hard. These correspond to finding pure strategy equilibria in the anti-coordination games, whose price of anarchy we also analyze. We further consider the directed case, a generalization that captures both coordination and anti-coordination. We prove the decision problem for non-strict equilibria in directed graphs is NP-hard. Our notions also have multiple connections to other combinatorial questions, and our results resolve some open problems in these areas, most notably the complexity of the strictly unfriendly partition problem.

cs.GT↗

arXiv2011

Contextual Bandit Algorithms with Supervised Learning Guarantees

We address the problem of learning in an online, bandit setting where the learner must repeatedly select among $K$ actions, but only receives partial feedback based on its choices. We establish two new facts: First, using a new algorithm called Exp4.P, we show that it is possible to compete with the best in a set of $N$ experts with probability $1-δ$ while incurring regret at most $O(\sqrt{KT\ln(N/δ)})$ over $T$ time steps. The new algorithm is tested empirically in a large-scale, real-world dataset. Second, we give a new algorithm called VE that competes with a possibly infinite set of policies of VC-dimension $d$ while incurring regret at most $O(\sqrt{T(d\ln(T) + \ln (1/δ))})$ with probability $1-δ$. These guarantees improve on those of all previous algorithms, whether in a stochastic or adversarial environment, and bring us closer to providing supervised learning type guarantees for the contextual bandit setting.

cs.LG↗

arXiv2011

Efficient Optimal Learning for Contextual Bandits

We address the problem of learning in an online setting where the learner repeatedly observes features, selects among a set of actions, and receives reward for the action taken. We provide the first efficient algorithm with an optimal regret. Our algorithm uses a cost sensitive classification learner as an oracle and has a running time $\mathrm{polylog}(N)$, where $N$ is the number of classification rules among which the oracle might choose. This is exponentially faster than all previous algorithms that achieve optimal regret in this setting. Our formulation also enables us to create an algorithm with regret that is additive rather than multiplicative in feedback delay as in all previous work.

cs.LG↗