453 research outputs found
Parallel coordinate descent for the Adaboost problem
We design a randomised parallel version of Adaboost based on previous studies
on parallel coordinate descent. The algorithm uses the fact that the logarithm
of the exponential loss is a function with coordinate-wise Lipschitz continuous
gradient, in order to define the step lengths. We provide the proof of
convergence for this randomised Adaboost algorithm and a theoretical
parallelisation speedup factor. We finally provide numerical examples on
learning problems of various sizes that show that the algorithm is competitive
with concurrent approaches, especially for large scale problems.Comment: 7 pages, 3 figures, extended version of the paper presented to
ICMLA'1
Distributed Optimisation with Linear Equality and Inequality Constraints using PDMM
In this paper, we consider the problem of distributed optimisation of a
separable convex cost function over a graph, where every edge and node in the
graph could carry both linear equality and/or inequality constraints. We show
how to modify the primal-dual method of multipliers (PDMM), originally designed
for linear equality constraints, such that it can handle inequality constraints
as well. In contrast to most existing algorithms for optimisation with
inequality constraints, the proposed algorithm does not need any slack
variables. Using convex analysis, monotone operator theory and fixed-point
theory, we show how to derive the update equations of the modified PDMM
algorithm by applying Peaceman-Rachford splitting to the monotonic inclusion
related to the extended dual problem. To incorporate the inequality
constraints, we impose a non-negativity constraint on the associated dual
variables. This additional constraint results in the introduction of a
reflection operator to model the data exchange in the network, instead of a
permutation operator as derived for equality constraint PDMM. Convergence for
both synchronous and stochastic update schemes of PDMM are provided. The latter
includes asynchronous update schemes and update schemes with transmission
losses.Comment: 9 page
On Distributed Nonconvex Optimisation Via Modified ADMM
This paper addresses the problem of nonconvex nonsmooth decentralised
optimisation in multi-agent networks with undirected connected communication
graphs. Our contribution lies in introducing an algorithmic framework designed
for the distributed minimisation of the sum of a smooth (possibly nonconvex and
non-separable) function and a convex (possibly nonsmooth and non-separable)
regulariser. The proposed algorithm can be seen as a modified version of the
ADMM algorithm where, at each step, an "inner loop" needs to be iterated for a
number of iterations. The role of the inner loop is to aggregate and
disseminate information across the network. We observe that a naive
decentralised approach (one iteration of the inner loop) may not converge. We
establish the asymptotic convergence of the proposed algorithm to the set of
stationary points of the nonconvex problem where the number of iterations of
the inner loop increases logarithmically with the step count of the ADMM
algorithm. We present numerical results demonstrating the proposed method's
correctness and performance.Comment: 6 pages, 1 Figur
Distributed Convex Optimisation using the Alternating Direction Method of Multipliers (ADMM) in Lossy Scenarios
The Alternating Direction Method of Multipliers (ADMM) is an extensively studied algorithm suitable for solving convex distributed optimisation problems. This Thesis presents a formulation of the ADMM that is guaranteed to converge if the communications among agents are faulty and the agents perform updates asynchronously. With strongly convex costs, the proposed algorithm is shown to converge exponentially fast. The further extension to partition-based problems is presented
Asynchronous Distributed Optimization over Lossy Networks via Relaxed ADMM: Stability and Linear Convergence
In this work we focus on the problem of minimizing the sum of convex cost
functions in a distributed fashion over a peer-to-peer network. In particular,
we are interested in the case in which communications between nodes are prone
to failures and the agents are not synchronized among themselves. We address
the problem proposing a modified version of the relaxed ADMM, which corresponds
to the Peaceman-Rachford splitting method applied to the dual. By exploiting
results from operator theory, we are able to prove the almost sure convergence
of the proposed algorithm under general assumptions on the distribution of
communication loss and node activation events. By further assuming the cost
functions to be strongly convex, we prove the linear convergence of the
algorithm in mean to a neighborhood of the optimal solution, and provide an
upper bound to the convergence rate. Finally, we present numerical results
testing the proposed method in different scenarios.Comment: To appear in IEEE Transactions on Automatic Contro
Cloud-Based Centralized/Decentralized Multi-Agent Optimization with Communication Delays
We present and analyze a computational hybrid architecture for performing
multi-agent optimization. The optimization problems under consideration have
convex objective and constraint functions with mild smoothness conditions
imposed on them. For such problems, we provide a primal-dual algorithm
implemented in the hybrid architecture, which consists of a decentralized
network of agents into which centralized information is occasionally injected,
and we establish its convergence properties. To accomplish this, a central
cloud computer aggregates global information, carries out computations of the
dual variables based on this information, and then distributes the updated dual
variables to the agents. The agents update their (primal) state variables and
also communicate among themselves with each agent sharing and receiving state
information with some number of its neighbors. Throughout, communications with
the cloud are not assumed to be synchronous or instantaneous, and communication
delays are explicitly accounted for in the modeling and analysis of the system.
Experimental results are presented to support the theoretical developments
made.Comment: 8 pages, 4 figure
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