We typically have seminars on Wednesdays at noon in Malone 228. All seminar announcements will be sent to the theory mailing list.
Speaker: Alex Slivkins
Affiliation: Microsoft Research – New York
Title: Bandits with Resource Constraints
Abstract:
Multi-armed bandits is the predominant theoretical model for exploration-exploitation tradeoff in machine learning, with countless applications ranging from medical trials, to communication networks, to Web search and advertising, to dynamic pricing. In many of these application domains the learner may be constrained by one or more supply/budget limits, in addition to the customary limitation on the time horizon. We introduce a general model that encompasses such problems, combining aspects of stochastic integer programming with online learning. A distinctive feature (and challenge) in our model, compared to the conventional bandit problems, is that the optimal policy for a given problem instance may significantly outperform the policy that always chooses the best fixed action. Our main result is an algorithm with near-optimal regret relative to the optimal policy. Also, we extend this result to contextual bandits, and detail an application to dynamic pricing.
Mohammad Hajiaghayi
University of Maryland – College Park
Title: Parameterized and Promised Streaming: Matching and Vertex Cover
Abstract:
As graphs continue to grow in size, we seek ways to effectively
process such data at scale. The model of streaming graph processing, in
which a compact summary is maintained as each edge insertion/deletion
is observed, is an attractive one. However, few results are known for
optimization (often NP-hard) problems over such dynamic graph streams.
In this talk, we introduce a new approach to handling graph streams,
by instead seeking solutions for the parameterized (and promised) versions of
these problems. Here, we are given a parameter k and the objective is to
decide whether there is a solution bounded by k. By combining
kernelization techniques with randomized sketch structures, we obtain the
first streaming algorithms for the parameterized versions of Maximal
Matching and Vertex Cover. We consider various models for a graph stream on n
nodes: the insertion-only model where the edges can only be added, and
the dynamic model where edges can be both inserted and deleted.
Gordon Wilfong
Alcatel-Lucent Bell Labs
Title: Optimal Path Encoding
Abstract: Packet networks need to maintain state in the form
of forwarding tables at each switch. The cost of this state
increases as networks support ever more sophisticated per-flow
routing, traffic engineering, and service chaining. Per-flow or per-path
state at the switches can be eliminated by encoding each
packet’s desired path in its header. A key component of such a
method is an efficient encoding of paths.
We introduce a mathematical formulation of this optimal path encoding
problem. We prove that the problem is APX-hard, by
showing that approximating it to within a factor less than 8/7
is NP-hard. We then present an algorithm
approximating the optimal path-encoding problem to within a
factor 2. Finally, we provide empirical results illustrating the
effectiveness of the proposed algorithm.
Joint work with A. Hari (Bell Labs) and U. Niesen (Qualcomm)
Speaker: Lisa Zhang
Affiliation: Alcatel-Lucent Bell Labs
Title: Analysis of k-Anonymity Algorithms for Streaming Location Data
Abstract:
We propose and analyze algorithms to achieve k-anonymity for streaming location data. We consider a framework motivated by European Union privacy requirements, in which location information arrives online into a buffer of fixed size m. Whenever the buffer fills, some data must move to permanent storage in a k-anonymized fashion. This notion of anonymity refers to recording a coarse common region containing at least k points instead of separate exact locations. One primary goal is to minimize the recorded region size so that the anonymized location data is as accurate as possible.
We observe that under competitive analysis, any online algorithm can be arbitrarily bad in terms of the recorded region size. We therefore assume a more benign model in which the location distribution is known. For a uniform distribution, we analyze a simple, natural algorithm that partitions the space into m/k identical regions to ensure k-anonymity, and picks the region with the largest occupancy whenever the buffer fills. Our detailed analysis shows
that the largest occupancy converges to 2k. This implies, perhaps somewhat unintuitively, that it is sufficient to achieve k-anonymity by partitioning space into $2m/k$ regions, which reduces and thereby improves the recorded region size by a factor of 2. We also present an almost matching lower bound of 2m/k. Finally, we discuss generalizations to nonuniform distributions by partitioning the space to match the given distribution.
Speaker: Sanjeev Khanna
Affiliation: University of Pennsylvania
Title: Tight Bounds for Linear Sketches of Approximate Matchings
Abstract:
We consider the problem of approximating a maximum matching in dynamic graph streams where the stream may include both edge insertions and deletions. Our main result is a resolution of the space complexity of linear sketches for approximating the maximum matching in this model.
Specifically, we show that for any $\eps > 0$, there exists a single-pass streaming algorithm, which only maintains a linear sketch of size roughly $n^{2-3\eps}$ bits and recovers an $n^\epsilon$-approximate maximum matching in dynamic graph streams, where $n$ is the number of vertices in the graph. We complement this result with the following lower bound result: any linear sketch for approximating the maximum matching to within a factor of $n^\eps$ has to be of size at least $n^{2-3\eps -o(1)}$ bits.
This is based on joint work with Sepehr Assadi, Yang Li, and Grigory Yaroslavtsev.
Details and abstract will be added when available.
Title:
Streaming Algorithms for Estimating the Matching Size in Planar Graphs and Beyond.
Abstract:
We consider the problem of estimating the size of a maximum matching when the edges are revealed in a streaming fashion. Consider a graph G=(V,E) with n vertices and m edges. The input stream is a permutation of edges S= (e_1,…,e_m) chosen by an adversary.
The goal is to output an estimation of the size of a maximum matching. The algorithm is only allowed to use a small amount of memory (much smaller than n).
When the underlying graph is planar, we present a simple and elegant streaming algorithm that with high probability estimates the size of a maximum matching within a constant factor using O-tilde(n^(2/3)) space. The approach generalizes to the family of graphs that have bounded arboricity. Graphs with bounded arboricity include, among other families of graphs, graphs with an excluded constant-size minor. To the best of our knowledge, this is the first result for estimating the size of a maximum matching in the adversarial-order streaming model (as opposed to the random-order streaming model). We circumvent the barriers inherent in the adversarial-order model by exploiting several structural properties of planar graphs, and more generally, graphs with bounded arboricity. We hope that this approach finds applications in estimating other properties of graphs in the adversarial-order streaming model. We further reduce the required memory size to O-tilde(sqrt(n)) for three restricted settings: (i) when the underlying graph is a forest; (ii) when we have 2-passes over the stream of edges of a graph with bounded arboricity; and (iii) when the edges arrive in random order and the underlying graph has bounded arboricity.
Finally, by introducing a communication complexity problem, we show that the approximation factor of a deterministic algorithm cannot be better than a constant using o(n) space, even if the underlying graph is a collection of paths. We can show that under a plausible conjecture for the hardness of the communication complexity problem, randomized algorithms with o(sqrt(n)) space cannot have an approximation factor better than a fixed constant.
Details and abstract will be added when available.
Speaker: Sofya Raskhodnikova
Abstract:
How quickly can we determine if an object satisfies some basic geometric property? For example, is the object a half-plane? Is it convex? Is it connected? If we need to answer such a question exactly, it requires at least as much time as it takes to read the object. In this talk, we will focus on approximate versions of these questions and will discuss how to solve them in time that depends only on the approximation parameter, but not the size of the input.
Specifically, an algorithm is given access to a discretized image represented by an n x n matrix of 0/1 pixel values. Another input to the algorithm is an approximation parameter, epsilon. The algorithm is required to accept images with the desired property and reject (with high probability) images that are far from having the desired property. An image is far if at least an epsilon fraction of its pixels has to be changed to get an image with the required property. For example, in this model, if the algorithm is allowed to read pixels of its choice, the half-plane property and convexity can be tested in time O(1/epsilon). If the algorithm receives access to pixels chosen uniformly and independently at random, then the half-plane property still takes O(1/epsilon) time, but for convexity the (optimal) bound on the running time is O(1/epsilon^(4/3)).
Based on joint work with Piotr Berman and Meiram Murzabulatov.
—
Title: Inapproximability of Truthful Mechanisms via Generalizations of the VC Dimension
Speaker: Gal Shahaf
Affiliation: The Hebrew University of Jerusalem
Abstract:
Algorithmic mechanism design (AMD) studies the delicate interplay between computational efficiency, truthfulness, and economic optimality. We focus on AMD’s paradigmatic problem: combinatorial auctions, and present new inapproximability results for truthful mechanisms in this scenario. Our main technique is a generalization of the classical VC dimension and the corresponding Sauer-Shelah Lemma.
Joint work with Amit Daniely and Michael Schapira
The talk is designed to be accessible to M.Sc. students, and includes an elementary introduction to VC dimension, combinatorial auctions and VCG mechanisms.
SPEAKER: Ilya Razenshteyn (MIT)
TITLE: Sketching and Embedding are Equivalent for Norms
ABSTRACT: Imagine the following communication task. Alice and Bob each have a point from a metric space. They want to transmit a few bits and decide whether their points are close to each other or are far apart. Of particular interest are sketching protocols: Alice and Bob both compute short summaries of their inputs and then a referee, given these summaries, makes the decision; sketches are very useful for the nearest neighbor search, streaming, randomized linear algebra etc. Indyk (FOCS 2000) showed that for the l_p spaces with 0 < p <= 2 the above problem allows a very efficient sketching protocol. Consequently, any metric that can be mapped into the l_p space with all the distances being approximately preserved has a good protocol as well.
I will show that for normed spaces (a very important class of metric spaces) embedding into l_p is the only possible technique for solving the communication problem. Slightly more formally, we show that any normed space that admits a good communication (in particular, sketching) protocol for distinguishing close and far pairs of points embeds well into l_p with p being close to 1. The proof uses tools from communication complexity and functional analysis.
As a corollary, we will show communication lower bounds for the planar Earth Mover’s Distance (minimum-cost matching metric) and for the trace norm (the sum of the singular values of a matrix) by deriving them from the (known) non-embeddability theorems and (the contrapositive of) our result.
The talk is based on a joint paper with Alexandr Andoni and Robert Krauthgamer (arXiv:1411.2577).
Title: Nonconvex Statistical Optimization: Algorithm and Model-based Optimization Theory
Abstract: Nonconvex regularized M-estimators have been widely applied to high dimensional data analysis. Existing statisticaltheory has established their statistical properties
in high dimensions only when the global optimum or certain local optimum can be obtained. Though practitioners have proposed numerous heuristic computational algorithms for
solving these nonconvex optimization problems, existing optimization theory does not necessarily guarantee these algorithms to obtain the global or local optima with
desired statistical properties in polynomial time. Therefore, there exists a significant gap between theory and practice: What is actually computed is not the same as what has
been proved. To bridge this gap, we propose a new generation of nonconvex statistical optimization algorithms and model-based theory, which incorporate the statistical thinking
into modern optimization. When developing computational algorithms, we take underlying sparse statistical models into consideration. Particularly, for nonconvex regularized
M-estimation problems, our proposed algorithms devise three different optimization schemes, under which the solutions achieved by the optimization algorithm always falls within
a restricted sparse set. Thus the nonconvex objective function mimics the behavior of a strongly convex function, which eventually allows our proposed algorithms to obtain an
estimator with the desired optimal statistical properties in polynomial time with high probability
Speaker: Aravind Srinivasan
Affiliation: University of Maryland
Title: Properties and Generalizations of the Moser-Tardos Process
Abstract: Moser and Tardos have developed an elegant and powerful algorithmic version of the Lovasz Local Lemma. Since the publication of this work, it has become apparent that this algorithm has some very interesting properties and extensions, and can be viewed as a stochastic process of independent interest. I will survey some of these, especially the ideas of “partial resampling” and the “LLL-distribution” (the properties of the output distribution of Moser-Tardos). I will draw from joint works with Haeupler and Saha, with Harris, and with Chen and Harris.
Title: Fault Resilient Graph Structures
Speaker: Merav Parter (MIT)
Abstract:
A fault-tolerant (FT) structure for a network is required to continue functioning following the failure of some of the network’s edges or vertices. Fault-resilience can be introduced into the network in several different ways. This talk will focus on a notion of fault-tolerance whereby the structure at hand is augmented (by adding to it various components) so that subsequent to the failure of some of the network’s vertices or edges, the surviving part of the structure is still operational. As this augmentation carries certain costs, it is desirable to minimize the number of added components.We will revise several constructions of sparse fault tolerant structures such as FT spanner and FT shortest-path trees. I will also present a new model for fault resilient network structures that mix two orthogonal protection mechanisms: (a) backup, namely, augmenting the structure with many (redundant) low-cost and fault-prone components, and (b) reinforcement, namely, acquiring high-cost but fault-resistant components. A trade-off between these two mechanisms will be presented in a concrete setting of shortest-path trees.
Talk Title:
Dependent Random Graphs and Multiparty Pointer Jumpin
Abstract:
We initiate a study of a relaxed version of the standard Erdos-Renyi random graph model, where each edge may depend on a few other edges. We call such graphs *dependent random graphs* and give tight bounds on the clique and chromatic numbers of such graphs. Surprisingly, some of the bounds in the standard random graph model continue to hold in this relaxed setting. For example, the size of the largest clique in a dependent random graph remains roughly log(n)/log(1/p).
As an application, we give a new upper bound on communication complexity of the Multiparty Pointer Jumping (MPJ) problem in the number-on-the-forehead (NOF) model. NOF communication lies at the current frontier of our understanding of communication complexity, and MPJ is one of the canonical problems in this setting. Furthermore, sufficiently strong bounds for MPJ would have important consequences for circuit complexity.
No prior knowledge is assumed aside from basic discrete probability. I hope to motivate both random graphs and the application and demonstrate why NOF communication is an important active research area.
This talk is based on research that is joint work with Mario Sanchez.
Bio:
Joshua Brody received a bachelor’s degree in Mathematics/Computer Science from Carnegie Mellon University, a Master’s Degree in Computer Science from New York University, and a PhD in Computer Science from Dartmouth College. Prior to graduate school, he served in the Peace Corps teaching high school mathematics in Burkina Faso, West Africa. Dr. Brody is currently an Assistant Professor in the Computer Science department at Swarthmore College. His primary research area is communication complexity. Additional research interests include several areas of theoretical computer science, including streaming algorithms, property testing, and data structures.