632 research outputs found
Geometry and convergence of natural policy gradient methods
We study the convergence of several natural policy gradient (NPG) methods in
infinite-horizon discounted Markov decision processes with regular policy
parametrizations. For a variety of NPGs and reward functions we show that the
trajectories in state-action space are solutions of gradient flows with respect
to Hessian geometries, based on which we obtain global convergence guarantees
and convergence rates. In particular, we show linear convergence for
unregularized and regularized NPG flows with the metrics proposed by Kakade and
Morimura and co-authors by observing that these arise from the Hessian
geometries of conditional entropy and entropy respectively. Further, we obtain
sublinear convergence rates for Hessian geometries arising from other convex
functions like log-barriers. Finally, we interpret the discrete-time NPG
methods with regularized rewards as inexact Newton methods if the NPG is
defined with respect to the Hessian geometry of the regularizer. This yields
local quadratic convergence rates of these methods for step size equal to the
penalization strength.Comment: 33 pages, 5 figures, under revie
Design and implementation of symbolic algorithms for the computation of generalized asymptotes
In this paper we present two algorithms for computing the g-asymptotes or generalized
asymptotes, of a plane algebraic curve, C , implicitly or parametrically defined. The asymptotes of a curve C reflect the status of C at points with sufficiently large coordinates. It
is well known that an asymptote of a curve C is a line such that the distance between C
and the line approaches zero as they tend to infinity. However, a curve C may have more
general curves than lines describing the status of C at infinity. These curves are known as
g-asymptotes or generalized asymptotes. The pseudocodes of these algorithms are presented,
as well as the corresponding implementations. For this purpose, we use the algebra software
Maple. A comparative analysis of the algorithms is carried out, based on some properties of
the input curves and their results to analyze the efficiency of the algorithms and to establish
comparative criteria. The results presented in this paper are a starting point to generalize
this study to surfaces or to curves defined by a non-rational parametrization, as well as to
improve the efficiency of the algorithms. Additionally, the methods developed can provide
a new and different approach in prediction (regression) or classification algorithms in the
machine learning field.Agencia Estatal de Investigació
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