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Error Bounds and Applications for Stochastic Approximation with Non-Decaying Gain
This work analyzes the stochastic approximation algorithm with non-decaying
gains as applied in time-varying problems. The setting is to minimize a
sequence of scalar-valued loss functions at sampling times
or to locate the root of a sequence of vector-valued functions
at with respect to a parameter . The
available information is the noise-corrupted observation(s) of either
or evaluated at one or two design points only. Given
the time-varying stochastic approximation setup, we apply stochastic
approximation algorithms with non-decaying gains, so that the recursive
estimate denoted as can maintain its momentum in tracking the
time-varying optimum denoted as .
Chapter 3 provides a bound for the root-mean-squared error . Overall, the bounds are applicable
under a mild assumption on the time-varying drift and a modest restriction on
the observation noise and the bias term. After establishing the tracking
capability in Chapter 3, we also discuss the concentration behavior of
in Chapter 4. The weak convergence limit of the continuous
interpolation of is shown to follow the trajectory of a
non-autonomous ordinary differential equation. Both Chapter 3 and Chapter 4 are
probabilistic arguments and may not provide much guidance on the gain-tuning
strategies useful for one single experiment run. Therefore, Chapter 5 discusses
a data-dependent gain-tuning strategy based on estimating the Hessian
information and the noise level. Overall, this work answers the questions "what
is the estimate for the dynamical system " and "how much we can
trust as an estimate for ."Comment: arXiv admin note: text overlap with arXiv:1906.0953