18 research outputs found
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
Distributed Maximum Likelihood using Dynamic Average Consensus
This paper presents the formulation and analysis of a novel distributed
maximum likelihood algorithm that utilizes a first-order optimization scheme.
The proposed approach utilizes a static average consensus algorithm to reach
agreement on the initial condition to the iterative optimization scheme and a
dynamic average consensus algorithm to reach agreement on the gradient
direction. The current distributed algorithm is guaranteed to exponentially
recover the performance of the centralized algorithm. Though the current
formulation focuses on maximum likelihood algorithm built on first-order
methods, it can be easily extended to higher order methods. Numerical
simulations validate the theoretical contributions of the paper
Asymptotic convergence rates for coordinate descent in polyhedral sets
We consider a family of parallel methods for constrained optimization based
on projected gradient descents along individual coordinate directions. In the
case of polyhedral feasible sets, local convergence towards a regular solution
occurs unconstrained in a reduced space, allowing for the computation of tight
asymptotic convergence rates by sensitivity analysis, this even when global
convergence rates are unavailable or too conservative. We derive linear
asymptotic rates of convergence in polyhedra for variants of the coordinate
descent approach, including cyclic, synchronous, and random modes of
implementation. Our results find application in stochastic optimization, and
with recently proposed optimization algorithms based on Taylor approximations
of the Newton step.Comment: 20 pages. A version of this paper will be submitted for publicatio
Optimization Methods for a Class of Rosenbrock Functions
This paper gives an in-depth review of the most common iterative methods for
unconstrained optimization using two functions that belong to a class of
Rosenbrock functions as a performance test. This study covers the Steepest
Gradient Descent Method, the Newton-Raphson Method, and the Fletcher-Reeves
Conjugate Gradient method. In addition, four different step-size selecting
methods the including fixed-step-size, variable step-size, quadratic-fit, and
golden section method were considered. Due to the computational nature of
solving minimization problems, testing the algorithms is an essential part of
this paper. Therefore, an extensive set of numerical test results is also
provided to present an insightful and a comprehensive comparison of the
reviewed algorithms. This study highlights the differences and the trade-offs
involved in comparing these algorithms.Comment: Title of the paper was changed and this new version focuses on the
comparison of these methods without the application to malicious agent
Generalized gradient optimization over lossy networks for partition-based estimation
We address the problem of distributed convex unconstrained optimization over
networks characterized by asynchronous and possibly lossy communications. We
analyze the case where the global cost function is the sum of locally coupled
local strictly convex cost functions. As discussed in detail in a motivating
example, this class of optimization objectives is, for example, typical in
localization problems and in partition-based state estimation. Inspired by a
generalized gradient descent strategy, namely the block Jacobi iteration, we
propose a novel solution which is amenable for a distributed implementation and
which, under a suitable condition on the step size, is provably locally
resilient to communication failures. The theoretical analysis relies on the
separation of time scales and Lyapunov theory. In addition, to show the
flexibility of the proposed algorithm, we derive a resilient gradient descent
iteration and a resilient generalized gradient for quadratic programming as two
natural particularizations of our strategy. In this second case, global
robustness is provided. Finally, the proposed algorithm is numerically tested
on the IEEE 123 nodes distribution feeder in the context of partition-based
smart grid robust state estimation in the presence of measurements outliers.Comment: 20 pages, 5 figure
A Decentralized Second-Order Method with Exact Linear Convergence Rate for Consensus Optimization
This paper considers decentralized consensus optimization problems where
different summands of a global objective function are available at nodes of a
network that can communicate with neighbors only. The proximal method of
multipliers is considered as a powerful tool that relies on proximal primal
descent and dual ascent updates on a suitably defined augmented Lagrangian. The
structure of the augmented Lagrangian makes this problem non-decomposable,
which precludes distributed implementations. This problem is regularly
addressed by the use of the alternating direction method of multipliers. The
exact second order method (ESOM) is introduced here as an alternative that
relies on: (i) The use of a separable quadratic approximation of the augmented
Lagrangian. (ii) A truncated Taylor's series to estimate the solution of the
first order condition imposed on the minimization of the quadratic
approximation of the augmented Lagrangian. The sequences of primal and dual
variables generated by ESOM are shown to converge linearly to their optimal
arguments when the aggregate cost function is strongly convex and its gradients
are Lipschitz continuous. Numerical results demonstrate advantages of ESOM
relative to decentralized alternatives in solving least squares and logistic
regression problems
Decentralized Quasi-Newton Methods
We introduce the decentralized Broyden-Fletcher-Goldfarb-Shanno (D-BFGS)
method as a variation of the BFGS quasi-Newton method for solving decentralized
optimization problems. The D-BFGS method is of interest in problems that are
not well conditioned, making first order decentralized methods ineffective, and
in which second order information is not readily available, making second order
decentralized methods impossible. D-BFGS is a fully distributed algorithm in
which nodes approximate curvature information of themselves and their neighbors
through the satisfaction of a secant condition. We additionally provide a
formulation of the algorithm in asynchronous settings. Convergence of D-BFGS is
established formally in both the synchronous and asynchronous settings and
strong performance advantages relative to first order methods are shown
numerically
Distributed Adaptive Newton Methods with Globally Superlinear Convergence
This paper considers the distributed optimization problem over a network
where the global objective is to optimize a sum of local functions using only
local computation and communication. Since the existing algorithms either adopt
a linear consensus mechanism, which converges at best linearly, or assume that
each node starts sufficiently close to an optimal solution, they cannot achieve
globally superlinear convergence. To break through the linear consensus rate,
we propose a finite-time set-consensus method, and then incorporate it into
Polyak's adaptive Newton method, leading to our distributed adaptive Newton
algorithm (DAN). To avoid transmitting local Hessians, we adopt a low-rank
approximation idea to compress the Hessian and design a communication-efficient
DAN-LA. Then, the size of transmitted messages in DAN-LA is reduced to
per iteration, where is the dimension of decision vectors and is the same
as the first-order methods. We show that DAN and DAN-LA can globally achieve
quadratic and superlinear convergence rates, respectively. Numerical
experiments on logistic regression problems are finally conducted to show the
advantages over existing methods.Comment: Submitted to IEEE Transactions on Automatic Control. 14 pages, 4
figure
A Distributed Newton Method for Large Scale Consensus Optimization
In this paper, we propose a distributed Newton method for consensus
optimization. Our approach outperforms state-of-the-art methods, including
ADMM. The key idea is to exploit the sparsity of the dual Hessian and recast
the computation of the Newton step as one of efficiently solving symmetric
diagonally dominant linear equations. We validate our algorithm both
theoretically and empirically. On the theory side, we demonstrate that our
algorithm exhibits superlinear convergence within a neighborhood of optimality.
Empirically, we show the superiority of this new method on a variety of machine
learning problems. The proposed approach is scalable to very large problems and
has a low communication overhead
Convergence of Limited Communications Gradient Methods
Distributed optimization increasingly plays a central role in economical and
sustainable operation of cyber-physical systems. Nevertheless, the complete
potential of the technology has not yet been fully exploited in practice due to
communication limitations posed by the real-world infrastructures. This work
investigates fundamental properties of distributed optimization based on
gradient methods, where gradient information is communicated using limited
number of bits. In particular, a general class of quantized gradient methods
are studied where the gradient direction is approximated by a finite
quantization set. Sufficient and necessary conditions are provided on such a
quantization set to guarantee that the methods minimize any convex objective
function with Lipschitz continuous gradient and a nonempty and bounded set of
optimizers. A lower bound on the cardinality of the quantization set is
provided, along with specific examples of minimal quantizations. Convergence
rate results are established that connect the fineness of the quantization and
the number of iterations needed to reach a predefined solution accuracy.
Generalizations of the results to a relevant class of constrained problems
using projections are considered. Finally, the results are illustrated by
simulations of practical systems.Comment: 16 pages, 8 figure