133,453 research outputs found
Performance of a Distributed Stochastic Approximation Algorithm
In this paper, a distributed stochastic approximation algorithm is studied.
Applications of such algorithms include decentralized estimation, optimization,
control or computing. The algorithm consists in two steps: a local step, where
each node in a network updates a local estimate using a stochastic
approximation algorithm with decreasing step size, and a gossip step, where a
node computes a local weighted average between its estimates and those of its
neighbors. Convergence of the estimates toward a consensus is established under
weak assumptions. The approach relies on two main ingredients: the existence of
a Lyapunov function for the mean field in the agreement subspace, and a
contraction property of the random matrices of weights in the subspace
orthogonal to the agreement subspace. A second order analysis of the algorithm
is also performed under the form of a Central Limit Theorem. The
Polyak-averaged version of the algorithm is also considered.Comment: IEEE Transactions on Information Theory 201
Distributed black-box optimization of nonconvex functions
We combine model-based methods and distributed stochastic approximation to propose a fully distributed algorithm for nonconvex optimization, with good empirical performance and convergence guarantees. Neither the expression of the objective nor its gradient are known. Instead, the objective is like a “black-box”, in which the agents input candidate solutions and evaluate the output. Without central coordination, the distributed algorithm naturally balances the computational load among the agents. This is especially relevant when many samples are needed (e.g., for high-dimensional objectives) or when evaluating each sample is costly. Numerical experiments over a difficult benchmark show that the networked agents match the performance of a centralized architecture, being able to approach the global optimum, while none of the individual noncooperative agents could by itself
A Stochastic Method for Solving Time-Fractional Differential Equations
We present a stochastic method for efficiently computing the solution of
time-fractional partial differential equations (fPDEs) that model anomalous
diffusion problems of the subdiffusive type. After discretizing the fPDE in
space, the ensuing system of fractional linear equations is solved resorting to
a Monte Carlo evaluation of the corresponding Mittag-Leffler matrix function.
This is accomplished through the approximation of the expected value of a
suitable multiplicative functional of a stochastic process, which consists of a
Markov chain whose sojourn times in every state are Mittag-Leffler distributed.
The resulting algorithm is able to calculate the solution at conveniently
chosen points in the domain with high efficiency. In addition, we present how
to generalize this algorithm in order to compute the complete solution. For
several large-scale numerical problems, our method showed remarkable
performance in both shared-memory and distributed-memory systems, achieving
nearly perfect scalability up to 16,384 CPU cores.Comment: Submitted to the Journal of Computational Physic
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