969 research outputs found
Distributed Subgradient Projection Algorithm over Directed Graphs
We propose a distributed algorithm, termed the Directed-Distributed Projected
Subgradient (D-DPS), to solve a constrained optimization problem over a
multi-agent network, where the goal of agents is to collectively minimize the
sum of locally known convex functions. Each agent in the network owns only its
local objective function, constrained to a commonly known convex set. We focus
on the circumstance when communications between agents are described by a
directed network. The D-DPS augments an additional variable for each agent, to
overcome the asymmetry caused by the directed communication network. The
convergence analysis shows that D-DPS converges at a rate of , where k is the number of iterations
Distributed Subgradient Projection Algorithm over Directed Graphs: Alternate Proof
We propose Directed-Distributed Projected Subgradient (D-DPS) to solve a
constrained optimization problem over a multi-agent network, where the goal of
agents is to collectively minimize the sum of locally known convex functions.
Each agent in the network owns only its local objective function, constrained
to a commonly known convex set. We focus on the circumstance when
communications between agents are described by a \emph{directed} network. The
D-DPS combines surplus consensus to overcome the asymmetry caused by the
directed communication network. The analysis shows the convergence rate to be
.Comment: Disclaimer: This manuscript provides an alternate approach to prove
the results in \textit{C. Xi and U. A. Khan, Distributed Subgradient
Projection Algorithm over Directed Graphs, in IEEE Transactions on Automatic
Control}. The changes, colored in blue, result into a tighter result in
Theorem~1". arXiv admin note: text overlap with arXiv:1602.0065
Distributed Optimization over Directed Graphs with Row Stochasticity and Constraint Regularity
This paper deals with an optimization problem over a network of agents, where
the cost function is the sum of the individual objectives of the agents and the
constraint set is the intersection of local constraints. Most existing methods
employing subgradient and consensus steps for solving this problem require the
weight matrix associated with the network to be column stochastic or even
doubly stochastic, conditions that can be hard to arrange in directed networks.
Moreover, known convergence analyses for distributed subgradient methods vary
depending on whether the problem is unconstrained or constrained, and whether
the local constraint sets are identical or nonidentical and compact. The main
goals of this paper are: (i) removing the common column stochasticity
requirement; (ii) relaxing the compactness assumption, and (iii) providing a
unified convergence analysis. Specifically, assuming the communication graph to
be fixed and strongly connected and the weight matrix to (only) be row
stochastic, a distributed projected subgradient algorithm and its variation are
presented to solve the problem for cost functions that are convex and Lipschitz
continuous. Based on a regularity assumption on the local constraint sets, a
unified convergence analysis is given that can be applied to both unconstrained
and constrained problems and without assuming compactness of the constraint
sets or an interior point in their intersection. Further, we also establish an
upper bound on the absolute objective error evaluated at each agent's available
local estimate under a nonincreasing step size sequence. This bound allows us
to analyze the convergence rate of both algorithms.Comment: 14 pages, 3 figure
Distributed Autonomous Online Learning: Regrets and Intrinsic Privacy-Preserving Properties
Online learning has become increasingly popular on handling massive data. The
sequential nature of online learning, however, requires a centralized learner
to store data and update parameters. In this paper, we consider online learning
with {\em distributed} data sources. The autonomous learners update local
parameters based on local data sources and periodically exchange information
with a small subset of neighbors in a communication network. We derive the
regret bound for strongly convex functions that generalizes the work by Ram et
al. (2010) for convex functions. Most importantly, we show that our algorithm
has \emph{intrinsic} privacy-preserving properties, and we prove the sufficient
and necessary conditions for privacy preservation in the network. These
conditions imply that for networks with greater-than-one connectivity, a
malicious learner cannot reconstruct the subgradients (and sensitive raw data)
of other learners, which makes our algorithm appealing in privacy sensitive
applications.Comment: 25 pages, 2 figure
Approximate Projection Methods for Decentralized Optimization with Functional Constraints
We consider distributed convex optimization problems that involve a separable
objective function and nontrivial functional constraints, such as Linear Matrix
Inequalities (LMIs). We propose a decentralized and computationally inexpensive
algorithm which is based on the concept of approximate projections. Our
algorithm is one of the consensus based methods in that, at every iteration,
each agent performs a consensus update of its decision variables followed by an
optimization step of its local objective function and local constraints. Unlike
other methods, the last step of our method is not an Euclidean projection onto
the feasible set, but instead a subgradient step in the direction that
minimizes the local constraint violation. We propose two different averaging
schemes to mitigate the disagreements among the agents' local estimates. We
show that the algorithms converge almost surely, i.e., every agent agrees on
the same optimal solution, under the assumption that the objective functions
and constraint functions are nondifferentiable and their subgradients are
bounded. We provide simulation results on a decentralized optimal gossip
averaging problem, which involves SDP constraints, to complement our
theoretical results
Distributed Subgradient-based Multi-agent Optimization with More General Step Sizes
A wider selection of step sizes is explored for the distributed subgradient
algorithm for multi-agent optimization problems, for both time-invariant and
time-varying communication topologies. The square summable requirement of the
step sizes commonly adopted in the literature is removed. The step sizes are
only required to be positive, vanishing and non-summable. It is proved that in
both unconstrained and constrained optimization problems, the agents' estimates
reach consensus and converge to the optimal solution with the more general
choice of step sizes. The idea is to show that a weighted average of the
agents' estimates approaches the optimal solution, but with different
approaches. In the unconstrained case, the optimal convergence of the weighted
average of the agents' estimates is proved by analyzing the distance change
from the weighted average to the optimal solution and showing that the weighted
average is arbitrarily close to the optimal solution. In the constrained case,
this is achieved by analyzing the distance change from the agents' estimates to
the optimal solution and utilizing the boundedness of the constraints. Then the
optimal convergence of the agents' estimates follows because consensus is
reached in both cases. These results are valid for both a strongly connected
time-invariant graph and time-varying balanced graphs that are jointly strongly
connected
Fenchel Dual Gradient Methods for Distributed Convex Optimization over Time-varying Networks
In the large collection of existing distributed algorithms for convex
multi-agent optimization, only a handful of them provide convergence rate
guarantees on agent networks with time-varying topologies, which, however,
restrict the problem to be unconstrained. Motivated by this, we develop a
family of distributed Fenchel dual gradient methods for solving constrained,
strongly convex but not necessarily smooth multi-agent optimization problems
over time-varying undirected networks. The proposed algorithms are constructed
based on the application of weighted gradient methods to the Fenchel dual of
the multi-agent optimization problem, and can be implemented in a fully
decentralized fashion. We show that the proposed algorithms drive all the
agents to both primal and dual optimality asymptotically under a minimal
connectivity condition and at sublinear rates under a standard connectivity
condition. Finally, the competent convergence performance of the distributed
Fenchel dual gradient methods is demonstrated via simulations
Privacy Preservation in Distributed Subgradient Optimization Algorithms
Privacy preservation is becoming an increasingly important issue in data
mining and machine learning. In this paper, we consider the privacy preserving
features of distributed subgradient optimization algorithms. We first show that
a well-known distributed subgradient synchronous optimization algorithm, in
which all agents make their optimization updates simultaneously at all times,
is not privacy preserving in the sense that the malicious agent can learn other
agents' subgradients asymptotically. Then we propose a distributed subgradient
projection asynchronous optimization algorithm without relying on any existing
privacy preservation technique, where agents can exchange data between
neighbors directly. In contrast to synchronous algorithms, in the new
asynchronous algorithm agents make their optimization updates asynchronously.
The introduced projection operation and asynchronous optimization mechanism can
guarantee that the proposed asynchronous optimization algorithm is privacy
preserving. Moreover, we also establish the optimal convergence of the newly
proposed algorithm. The proposed privacy preservation techniques shed light on
developing other privacy preserving distributed optimization algorithms
Online Distributed Optimization on Dynamic Networks
This paper presents a distributed optimization scheme over a network of
agents in the presence of cost uncertainties and over switching communication
topologies. Inspired by recent advances in distributed convex optimization, we
propose a distributed algorithm based on a dual sub-gradient averaging. The
objective of this algorithm is to minimize a cost function cooperatively.
Furthermore, the algorithm changes the weights on the communication links in
the network to adapt to varying reliability of neighboring agents. A
convergence rate analysis as a function of the underlying network topology is
then presented, followed by simulation results for representative classes of
sensor networks.Comment: Submitted to The IEEE Transactions on Automatic Control, 201
Graph Balancing for Distributed Subgradient Methods over Directed Graphs
We consider a multi agent optimization problem where a set of agents
collectively solves a global optimization problem with the objective function
given by the sum of locally known convex functions. We focus on the case when
information exchange among agents takes place over a directed network and
propose a distributed subgradient algorithm in which each agent performs local
processing based on information obtained from his incoming neighbors. Our
algorithm uses weight balancing to overcome the asymmetries caused by the
directed communication network, i.e., agents scale their outgoing information
with dynamically updated weights that converge to balancing weights of the
graph. We show that both the objective function values and the consensus
violation, at the ergodic average of the estimates generated by the algorithm,
converge with rate , where is the number of
iterations. A special case of our algorithm provides a new distributed method
to compute average consensus over directed graphs
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