Asymptotic normality and efficiency analysis of the cyclic seesaw stochastic optimization algorithm

In many real-world optimization problems, the loss function to minimize is unknown and only observable in the presence of noise. In other words, it is possible to obtain a noisy measurement of the loss function at a given input parameter. Here, stochastic optimization algorithms (which are often iterative in nature) can be used for optimization. Cyclic stochastic optimization is a type of stochastic optimization where only a subset of the parameter vector is updated at a time. While cyclic algorithms are commonly used in practice, most of the results on the asymptotic behavior of the iterates are restricted to the deterministic optimization setting (where the loss function is known). In this paper, we derive asymptotic normality results for the iterates produced by cyclic stochastic optimization algorithms. We also define and investigate the relative efficiency between a stochastic optimization algorithm and its cyclic formulation.

[1]  James Demmel,et al.  Exploiting Data Sparsity in Parallel Matrix Powers Computations , 2013, PPAM.

[2]  V. Fabian On Asymptotic Normality in Stochastic Approximation , 1968 .

[3]  James C. Spall,et al.  Simulation-based examination of the limits of performance for decentralized multi-agent surveillance and tracking of undersea targets , 2014, Defense + Security Symposium.

[4]  Seok Lee,et al.  Cyclic optimization algorithms for simultaneous structure and motion recovery in computer vision , 2008 .

[5]  Damiano Varagnolo,et al.  Newton-Raphson Consensus for Distributed Convex Optimization , 2015, IEEE Transactions on Automatic Control.

[6]  R. Has’minskiĭ,et al.  Stochastic Approximation and Recursive Estimation , 1976 .

[7]  Discrete simultaneous perturbation stochastic approximation on loss function with noisy measurements , 2011, Proceedings of the 2011 American Control Conference.

[8]  Karla Hernandez Cyclic stochastic optimization via arbitrary selection procedures for updating parameters , 2016, 2016 Annual Conference on Information Science and Systems (CISS).

[9]  J. Spall Multivariate stochastic approximation using a simultaneous perturbation gradient approximation , 1992 .

[10]  Ion Necoara,et al.  Random Coordinate Descent Algorithms for Multi-Agent Convex Optimization Over Networks , 2013, IEEE Transactions on Automatic Control.

[11]  James C. Spall,et al.  Introduction to Stochastic Search and Optimization. Estimation, Simulation, and Control (Spall, J.C. , 2007 .

[12]  James C. Spall,et al.  Cyclic stochastic optimization with noisy function measurements , 2014, 2014 American Control Conference.

[13]  James C. Spall,et al.  Cyclic Seesaw Process for Optimization and Identification , 2012, J. Optim. Theory Appl..

[14]  James C. Spall,et al.  Introduction to stochastic search and optimization - estimation, simulation, and control , 2003, Wiley-Interscience series in discrete mathematics and optimization.