1,181 research outputs found
Dual Averaging for Distributed Optimization: Convergence Analysis and Network Scaling
The goal of decentralized optimization over a network is to optimize a global
objective formed by a sum of local (possibly nonsmooth) convex functions using
only local computation and communication. It arises in various application
domains, including distributed tracking and localization, multi-agent
co-ordination, estimation in sensor networks, and large-scale optimization in
machine learning. We develop and analyze distributed algorithms based on dual
averaging of subgradients, and we provide sharp bounds on their convergence
rates as a function of the network size and topology. Our method of analysis
allows for a clear separation between the convergence of the optimization
algorithm itself and the effects of communication constraints arising from the
network structure. In particular, we show that the number of iterations
required by our algorithm scales inversely in the spectral gap of the network.
The sharpness of this prediction is confirmed both by theoretical lower bounds
and simulations for various networks. Our approach includes both the cases of
deterministic optimization and communication, as well as problems with
stochastic optimization and/or communication.Comment: 40 pages, 4 figure
Approximate Computation and Implicit Regularization for Very Large-scale Data Analysis
Database theory and database practice are typically the domain of computer
scientists who adopt what may be termed an algorithmic perspective on their
data. This perspective is very different than the more statistical perspective
adopted by statisticians, scientific computers, machine learners, and other who
work on what may be broadly termed statistical data analysis. In this article,
I will address fundamental aspects of this algorithmic-statistical disconnect,
with an eye to bridging the gap between these two very different approaches. A
concept that lies at the heart of this disconnect is that of statistical
regularization, a notion that has to do with how robust is the output of an
algorithm to the noise properties of the input data. Although it is nearly
completely absent from computer science, which historically has taken the input
data as given and modeled algorithms discretely, regularization in one form or
another is central to nearly every application domain that applies algorithms
to noisy data. By using several case studies, I will illustrate, both
theoretically and empirically, the nonobvious fact that approximate
computation, in and of itself, can implicitly lead to statistical
regularization. This and other recent work suggests that, by exploiting in a
more principled way the statistical properties implicit in worst-case
algorithms, one can in many cases satisfy the bicriteria of having algorithms
that are scalable to very large-scale databases and that also have good
inferential or predictive properties.Comment: To appear in the Proceedings of the 2012 ACM Symposium on Principles
of Database Systems (PODS 2012
Efficient and Robust Compressed Sensing Using Optimized Expander Graphs
Expander graphs have been recently proposed to construct efficient compressed sensing algorithms. In particular, it has been shown that any n-dimensional vector that is k-sparse can be fully recovered using O(klog n) measurements and only O(klog n) simple recovery iterations. In this paper, we improve upon this result by considering expander graphs with expansion coefficient beyond 3/4 and show that, with the same number of measurements, only O(k) recovery iterations are required, which is a significant improvement when n is large. In fact, full recovery can be accomplished by at most 2k very simple iterations. The number of iterations can be reduced arbitrarily close to k, and the recovery algorithm can be implemented very efficiently using a simple priority queue with total recovery time O(nlog(n/k))). We also show that by tolerating a small penal- ty on the number of measurements, and not on the number of recovery iterations, one can use the efficient construction of a family of expander graphs to come up with explicit measurement matrices for this method. We compare our result with other recently developed expander-graph-based methods and argue that it compares favorably both in terms of the number of required measurements and in terms of the time complexity and the simplicity of recovery. Finally, we will show how our analysis extends to give a robust algorithm that finds the position and sign of the k significant elements of an almost k-sparse signal and then, using very simple optimization techniques, finds a k-sparse signal which is close to the best k-term approximation of the original signal
Complexity Theory
Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes
Optimal Lower Bounds for Universal and Differentially Private Steiner Tree and TSP
Given a metric space on n points, an {\alpha}-approximate universal algorithm
for the Steiner tree problem outputs a distribution over rooted spanning trees
such that for any subset X of vertices containing the root, the expected cost
of the induced subtree is within an {\alpha} factor of the optimal Steiner tree
cost for X. An {\alpha}-approximate differentially private algorithm for the
Steiner tree problem takes as input a subset X of vertices, and outputs a tree
distribution that induces a solution within an {\alpha} factor of the optimal
as before, and satisfies the additional property that for any set X' that
differs in a single vertex from X, the tree distributions for X and X' are
"close" to each other. Universal and differentially private algorithms for TSP
are defined similarly. An {\alpha}-approximate universal algorithm for the
Steiner tree problem or TSP is also an {\alpha}-approximate differentially
private algorithm. It is known that both problems admit O(logn)-approximate
universal algorithms, and hence O(log n)-approximate differentially private
algorithms as well. We prove an {\Omega}(logn) lower bound on the approximation
ratio achievable for the universal Steiner tree problem and the universal TSP,
matching the known upper bounds. Our lower bound for the Steiner tree problem
holds even when the algorithm is allowed to output a more general solution of a
distribution on paths to the root.Comment: 14 page
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