415 research outputs found
Stochastic Modified Equations for Continuous Limit of Stochastic ADMM
Stochastic version of alternating direction method of multiplier (ADMM) and
its variants (linearized ADMM, gradient-based ADMM) plays a key role for modern
large scale machine learning problems. One example is the regularized empirical
risk minimization problem. In this work, we put different variants of
stochastic ADMM into a unified form, which includes standard, linearized and
gradient-based ADMM with relaxation, and study their dynamics via a
continuous-time model approach. We adapt the mathematical framework of
stochastic modified equation (SME), and show that the dynamics of stochastic
ADMM is approximated by a class of stochastic differential equations with small
noise parameters in the sense of weak approximation. The continuous-time
analysis would uncover important analytical insights into the behaviors of the
discrete-time algorithm, which are non-trivial to gain otherwise. For example,
we could characterize the fluctuation of the solution paths precisely, and
decide optimal stopping time to minimize the variance of solution paths
Connections between Mean-Field Game and Social Welfare Optimization
This paper studies the connection between a class of mean-field games and a
social welfare optimization problem. We consider a mean-field game in function
spaces with a large population of agents, and each agent seeks to minimize an
individual cost function. The cost functions of different agents are coupled
through a mean-field term that depends on the mean of the population states. We
show that although the mean-field game is not a potential game, under some mild
condition the -Nash equilibrium of the mean-field game coincides with
the optimal solution to a social welfare optimization problem, and this is true
even when the individual cost functions are non-convex. The connection enables
us to evaluate and promote the efficiency of the mean-field equilibrium. In
addition, it also leads to several important implications on the existence,
uniqueness, and computation of the mean-field equilibrium. Numerical results
are presented to validate the solution, and examples are provided to show the
applicability of the proposed approach
Parallel ADMM for robust quadratic optimal resource allocation problems
An alternating direction method of multipliers (ADMM) solver is described for
optimal resource allocation problems with separable convex quadratic costs and
constraints and linear coupling constraints. We describe a parallel
implementation of the solver on a graphics processing unit (GPU) using a
bespoke quartic function minimizer. An application to robust optimal energy
management in hybrid electric vehicles is described, and the results of
numerical simulations comparing the computation times of the parallel GPU
implementation with those of an equivalent serial implementation are presented
A Proximal Zeroth-Order Algorithm for Nonconvex Nonsmooth Problems
In this paper, we focus on solving an important class of nonconvex
optimization problems which includes many problems for example signal
processing over a networked multi-agent system and distributed learning over
networks. Motivated by many applications in which the local objective function
is the sum of smooth but possibly nonconvex part, and non-smooth but convex
part subject to a linear equality constraint, this paper proposes a proximal
zeroth-order primal dual algorithm (PZO-PDA) that accounts for the information
structure of the problem. This algorithm only utilize the zeroth-order
information (i.e., the functional values) of smooth functions, yet the
flexibility is achieved for applications that only noisy information of the
objective function is accessible, where classical methods cannot be applied. We
prove convergence and rate of convergence for PZO-PDA. Numerical experiments
are provided to validate the theoretical results
Survey: Sixty Years of Douglas--Rachford
The Douglas--Rachford method is a splitting method frequently employed for
finding zeroes of sums of maximally monotone operators. When the operators in
question are normal cones operators, the iterated process may be used to solve
feasibility problems of the form: Find The success
of the method in the context of closed, convex, nonempty sets
is well-known and understood from a theoretical standpoint. However, its
performance in the nonconvex context is less understood yet surprisingly
impressive. This was particularly compelling to Jonathan M. Borwein who,
intrigued by Elser, Rankenburg, and Thibault's success in applying the method
for solving Sudoku Puzzles, began an investigation of his own. We survey the
current body of literature on the subject, and we summarize its history. We
especially commemorate Professor Borwein's celebrated contributions to the
area
Dynamical convergence analysis for nonconvex linearized proximal ADMM algorithms
The convergence analysis of optimization algorithms using continuous-time
dynamical systems has received much attention in recent years. In this paper,
we investigate applications of these systems to analyze the convergence of
linearized proximal ADMM algorithms for nonconvex composite optimization, whose
objective function is the sum of a continuously differentiable function and a
composition of a possibly nonconvex function with a linear operator. We first
derive a first-order differential inclusion for the linearized proximal ADMM
algorithm, LP-ADMM. Both the global convergence and the convergence rates of
the generated trajectory are established with the use of Kurdyka-\L{}ojasiewicz
(KL) property. Then, a stochastic variant, LP-SADMM, is delved into an
investigation for finite-sum nonconvex composite problems. Under mild
conditions, we obtain the stochastic differential equation corresponding to
LP-SADMM, and demonstrate the almost sure global convergence of the generated
trajectory by leveraging the KL property. Based on the almost sure convergence
of trajectory, we construct a stochastic process that converges almost surely
to an approximate critical point of objective function, and derive the expected
convergence rates associated with this stochastic process. Moreover, we propose
an accelerated LP-SADMM that incorporates Nesterov's acceleration technique.
The continuous-time dynamical system of this algorithm is modeled as a
second-order stochastic differential equation. Within the context of KL
property, we explore the related almost sure convergence and expected
convergence rates
Newton-Raphson Consensus under asynchronous and lossy communications for peer-to-peer networks
In this work we study the problem of unconstrained convex-optimization in a
fully distributed multi-agent setting which includes asynchronous computation
and lossy communication. In particular, we extend a recently proposed algorithm
named Newton-Raphson Consensus by integrating it with a broadcast-based average
consensus algorithm which is robust to packet losses. We show via the
separation of time scales principle that under mild conditions (i.e.,
persistency of the agents activation and bounded consecutive communication
failures) the proposed algorithm is proved to be locally exponentially stable
with respect to the optimal global solution. Finally, we complement the
theoretical analysis with numerical simulations that are based on real
datasets
Communication-Efficient Distributed Optimization of Self-Concordant Empirical Loss
We consider distributed convex optimization problems originated from sample
average approximation of stochastic optimization, or empirical risk
minimization in machine learning. We assume that each machine in the
distributed computing system has access to a local empirical loss function,
constructed with i.i.d. data sampled from a common distribution. We propose a
communication-efficient distributed algorithm to minimize the overall empirical
loss, which is the average of the local empirical losses. The algorithm is
based on an inexact damped Newton method, where the inexact Newton steps are
computed by a distributed preconditioned conjugate gradient method. We analyze
its iteration complexity and communication efficiency for minimizing
self-concordant empirical loss functions, and discuss the results for
distributed ridge regression, logistic regression and binary classification
with a smoothed hinge loss. In a standard setting for supervised learning, the
required number of communication rounds of the algorithm does not increase with
the sample size, and only grows slowly with the number of machines
An FFT-based method for computing the effective crack energy of a heterogeneous material on a combinatorially consistent grid
We introduce an FFT-based solver for the combinatorial continuous maximum
flow discretization applied to computing the minimum cut through heterogeneous
microstructures. Recently, computational methods were introduced for computing
the effective crack energy of periodic and random media. These were based on
the continuous minimum cut-maximum flow duality of G. Strang, and made use of
discretizations based on trigonometric polynomials and finite elements. For
maximum flow problems on graphs, node-based discretization methods avoid
metrication artifacts associated to edge-based discretizations. We discretize
the minimum cut problem on heterogeneous microstructures by the combinatorial
continuous maximum flow discretization introduced by Couprie et al.
Furthermore, we introduce an associated FFT-based ADMM solver and provide
several adaptive strategies for choosing numerical parameters. We demonstrate
the salient features of the proposed approach on problems of industrial scale
Distributed Control Methods for Integrating Renewable Generations and ICT Systems
With increased energy demand and decreased fossil fuels usages, the penetration of distributed generators (DGs) attracts more and more attention. Currently centralized control approaches can no longer meet real-time requirements for future power system. A proper decentralized control strategy needs to be proposed in order to enhance system voltage stability, reduce system power loss and increase operational security. This thesis has three key contributions:
Firstly, a decentralized coordinated reactive power control strategy is proposed to tackle voltage fluctuation issues due to the uncertainty of output of DG. Case study shows results of coordinated control methods which can regulate the voltage level effectively whilst also enlarging the total reactive power capability to reduce the possibility of active power curtailment. Subsequently, the communication system time-delay is considered when analyzing the impact of voltage regulation.
Secondly, a consensus distributed alternating direction multiplier method (ADMM) algorithm is improved to solve the optimal power ow (OPF) problem. Both synchronous and asynchronous algorithms are proposed to study the performance of convergence rate. Four different strategies are proposed to mitigate the impact of time-delay. Simulation results show that the optimization of reactive power allocation can minimize system power loss effectively and the proposed weighted autoregressive (AR) strategies can achieve an effective convergence result.
Thirdly, a neighboring monitoring scheme based on the reputation rating is proposed to detect and mitigate the potential false data injection attack. The simulation results show that the predictive value can effectively replace the manipulated data. The convergence results based on the predictive value can be very close to the results of normal case without cyber attack
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