6,369 research outputs found
Asymptotic properties and optimization of some non-Markovian stochastic processes
summary:We study the limit behavior of certain classes of dependent random sequences (processes) which do not possess the Markov property. Assuming these processes depend on a control parameter we show that the optimization of the control can be reduced to a problem of nonlinear optimization. Under certain hypotheses we establish the stability of such optimization problems
Rocking Subdiffusive Ratchets: Origin, Optimization and Efficiency
We study origin, parameter optimization, and thermodynamic efficiency of
isothermal rocking ratchets based on fractional subdiffusion within a
generalized non-Markovian Langevin equation approach. A corresponding
multi-dimensional Markovian embedding dynamics is realized using a set of
auxiliary Brownian particles elastically coupled to the central Brownian
particle (see video on the journal web site). We show that anomalous
subdiffusive transport emerges due to an interplay of nonlinear response and
viscoelastic effects for fractional Brownian motion in periodic potentials with
broken space-inversion symmetry and driven by a time-periodic field. The
anomalous transport becomes optimal for a subthreshold driving when the driving
period matches a characteristic time scale of interwell transitions. It can
also be optimized by varying temperature, amplitude of periodic potential and
driving strength. The useful work done against a load shows a parabolic
dependence on the load strength. It grows sublinearly with time and the
corresponding thermodynamic efficiency decays algebraically in time because the
energy supplied by the driving field scales with time linearly. However, it
compares well with the efficiency of normal diffusion rocking ratchets on an
appreciably long time scale
Convergence and Convergence Rate of Stochastic Gradient Search in the Case of Multiple and Non-Isolated Extrema
The asymptotic behavior of stochastic gradient algorithms is studied. Relying
on results from differential geometry (Lojasiewicz gradient inequality), the
single limit-point convergence of the algorithm iterates is demonstrated and
relatively tight bounds on the convergence rate are derived. In sharp contrast
to the existing asymptotic results, the new results presented here allow the
objective function to have multiple and non-isolated minima. The new results
also offer new insights into the asymptotic properties of several classes of
recursive algorithms which are routinely used in engineering, statistics,
machine learning and operations research
Asymptotic Bias of Stochastic Gradient Search
The asymptotic behavior of the stochastic gradient algorithm with a biased
gradient estimator is analyzed. Relying on arguments based on the dynamic
system theory (chain-recurrence) and the differential geometry (Yomdin theorem
and Lojasiewicz inequality), tight bounds on the asymptotic bias of the
iterates generated by such an algorithm are derived. The obtained results hold
under mild conditions and cover a broad class of high-dimensional nonlinear
algorithms. Using these results, the asymptotic properties of the
policy-gradient (reinforcement) learning and adaptive population Monte Carlo
sampling are studied. Relying on the same results, the asymptotic behavior of
the recursive maximum split-likelihood estimation in hidden Markov models is
analyzed, too.Comment: arXiv admin note: text overlap with arXiv:0907.102
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