54 research outputs found
Convergence Thresholds of Newton's Method for Monotone Polynomial Equations
Monotone systems of polynomial equations (MSPEs) are systems of fixed-point
equations where
each is a polynomial with positive real coefficients. The question of
computing the least non-negative solution of a given MSPE arises naturally in the analysis of stochastic models such as stochastic
context-free grammars, probabilistic pushdown automata, and back-button
processes. Etessami and Yannakakis have recently adapted Newton's iterative
method to MSPEs. In a previous paper we have proved the existence of a
threshold for strongly connected MSPEs, such that after iterations of Newton's method each new iteration computes at least 1 new
bit of the solution. However, the proof was purely existential. In this paper
we give an upper bound for as a function of the minimal component
of the least fixed-point of . Using this result we
show that is at most single exponential resp. linear for strongly
connected MSPEs derived from probabilistic pushdown automata resp. from
back-button processes. Further, we prove the existence of a threshold for
arbitrary MSPEs after which each new iteration computes at least new
bits of the solution, where and are the width and height of the DAG of
strongly connected components.Comment: version 2 deposited February 29, after the end of the STACS
conference. Two minor mistakes correcte
Computing the Least Fixed Point of Positive Polynomial Systems
We consider equation systems of the form X_1 = f_1(X_1, ..., X_n), ..., X_n =
f_n(X_1, ..., X_n) where f_1, ..., f_n are polynomials with positive real
coefficients. In vector form we denote such an equation system by X = f(X) and
call f a system of positive polynomials, short SPP. Equation systems of this
kind appear naturally in the analysis of stochastic models like stochastic
context-free grammars (with numerous applications to natural language
processing and computational biology), probabilistic programs with procedures,
web-surfing models with back buttons, and branching processes. The least
nonnegative solution mu f of an SPP equation X = f(X) is of central interest
for these models. Etessami and Yannakakis have suggested a particular version
of Newton's method to approximate mu f.
We extend a result of Etessami and Yannakakis and show that Newton's method
starting at 0 always converges to mu f. We obtain lower bounds on the
convergence speed of the method. For so-called strongly connected SPPs we prove
the existence of a threshold k_f such that for every i >= 0 the (k_f+i)-th
iteration of Newton's method has at least i valid bits of mu f. The proof
yields an explicit bound for k_f depending only on syntactic parameters of f.
We further show that for arbitrary SPP equations Newton's method still
converges linearly: there are k_f>=0 and alpha_f>0 such that for every i>=0 the
(k_f+alpha_f i)-th iteration of Newton's method has at least i valid bits of mu
f. The proof yields an explicit bound for alpha_f; the bound is exponential in
the number of equations, but we also show that it is essentially optimal.
Constructing a bound for k_f is still an open problem. Finally, we also provide
a geometric interpretation of Newton's method for SPPs.Comment: This is a technical report that goes along with an article to appear
in SIAM Journal on Computing
RePBubLik: Reducing the Polarized Bubble Radius with Link Insertions
The topology of the hyperlink graph among pages expressing different opinions
may influence the exposure of readers to diverse content. Structural bias may
trap a reader in a polarized bubble with no access to other opinions. We model
readers' behavior as random walks. A node is in a polarized bubble if the
expected length of a random walk from it to a page of different opinion is
large. The structural bias of a graph is the sum of the radii of
highly-polarized bubbles. We study the problem of decreasing the structural
bias through edge insertions. Healing all nodes with high polarized bubble
radius is hard to approximate within a logarithmic factor, so we focus on
finding the best edges to insert to maximally reduce the structural bias.
We present RePBubLik, an algorithm that leverages a variant of the random walk
closeness centrality to select the edges to insert. RePBubLik obtains, under
mild conditions, a constant-factor approximation. It reduces the structural
bias faster than existing edge-recommendation methods, including some designed
to reduce the polarization of a graph
The effect of the back button in a random walk: application for pagerank
International audienceTheoretical analysis of the Web graph is often used to improve the efficiency of search engines. The PageRank algorithm, proposed by Page, Brin et al., is used by the Google search engine to improve the results of the queries. The purpose of this article is to describe an enhanced version of the algorithm using a realistic model for the back button. We introduce a limited history stack model (you cannot click more than m times in a row), and show that when m = 1, the computation of this Back PageRank can be as fast as that of a standard PageRank
Polynomial Time Algorithms for Multi-Type Branching Processes and Stochastic Context-Free Grammars
We show that one can approximate the least fixed point solution for a
multivariate system of monotone probabilistic polynomial equations in time
polynomial in both the encoding size of the system of equations and in
log(1/\epsilon), where \epsilon > 0 is the desired additive error bound of the
solution. (The model of computation is the standard Turing machine model.)
We use this result to resolve several open problems regarding the
computational complexity of computing key quantities associated with some
classic and heavily studied stochastic processes, including multi-type
branching processes and stochastic context-free grammars
Recursive Stochastic Games with Positive Rewards
Abstract. We study the complexity of a class of Markov decision processes and, more generally, stochastic games, called 1-exit Recursive Markov Decision Processes (1-RMDPs) and Simple Stochastic Games (1-RSSGs) with strictly positive rewards. These are a class of finitely presented countable-state zero-sum stochastic games, with total expected reward objective. They subsume standard finite-state MDPs and Condon’s simple stochastic games and correspond to optimization and game versions of several classic stochastic models, with rewards. Such stochastic models arise naturally as models of probabilistic procedural programs with recursion, and the problems we address are motivated by the goal of analyzing the optimal/pessimal expected running time in such a setting. We give polynomial time algorithms for 1-exit Recursive Markov decision processes (1-RMDPs) with positive rewards. Specifically, we show that the exact optimal value of both maximizing and minimizing 1-RMDPs with positive rewards can be computed in polynomial time (this value may be ∞). For two-player 1-RSSGs with positive rewards, we prove a “stackless and memoryless ” determinacy result, and show that deciding whether the game value is at least a given value r is in NP ∩ coNP. We also prove that a simultaneous strategy improvement algorithm converges to the value and optimal strategies for these stochastic games. We observe that 1-RSSG positive reward games are “harder ” than finite-state SSGs in several senses.
Recursive Probabilistic Models: efficient analysis and implementation
This thesis examines Recursive Markov Chains (RMCs), their natural extensions and
connection to other models. RMCs can model in a natural way probabilistic procedural
programs and other systems that involve recursion and probability. An RMC
is a set of ordinary finite state Markov Chains that are allowed to call each other recursively
and it describes a potentially infinite, but countable, state ordinary Markov
Chain. RMCs generalize in a precise sense several well studied probabilistic models
in other domains such as natural language processing (Stochastic Context-Free Grammars),
population dynamics (Multi-Type Branching Processes) and in queueing theory
(Quasi-Birth-Death processes (QBDs)). In addition, RMCs can be extended to a
controlled version called Recursive Markov Decision Processes (RMDPs) and also a
game version referred to as Recursive (Simple) Stochastic Games (RSSGs). For analyzing
RMCs, RMDPs, RSSGs we devised highly optimized numerical algorithms and
implemented them in a tool called PReMo (Probabilistic Recursive Models analyzer).
PReMo allows computation of the termination probability and expected termination
time of RMCs and QBDs, and a restricted subset of RMDPs and RSSGs. The input
models are described by the user in specifically designed simple input languages. Furthermore,
in order to analyze the worst and best expected running time of probabilistic
recursive programs we study models of RMDPs and RSSGs with positive rewards
assigned to each of their transitions and provide new complexity upper and lower
bounds of their analysis. We also establish some new connections between our models
and models studied in queueing theory. Specifically, we show that (discrete time)
QBDs can be described as a special subclass of RMCs and Tree-like QBDs, which are a
generalization of QBDs, are equivalent to RMCs in a precise sense. We also prove that
for a given QBD we can compute (in the unit cost RAM model) an approximation of
its termination probabilities within i bits of precision in time polynomial in the size of
the QBD and linear in i. Specifically, we show that we can do this using a decomposed
Newton’s method
Recursive Stochastic Games with Positive Rewards
We first show that in such games both players have optimal deterministic “stackless and memoryless” optimal strategies. We then provide polynomial-time algorithms for computing the exact optimal expected reward (which may be infinite, but is otherwise rational), and optimal strategies, for both the maximizing and minimizing single-player versions of the game, i.e., for (1-exit) Recursive Markov Decision Processes (1-RMDPs). It follows that the quantitative decision problem for positive reward 1-RSSGs is in NP ∩ coNP. We show that Condon's well-known quantitative termination problem for finite-state simple stochastic games (SSGs) which she showed to be in NP ∩ coNP reduces to a special case of the reward problem for 1-RSSGs, namely, deciding whether the value is ∞. By contrast, for finite-state SSGs with strictly positive rewards, deciding if this expected reward value is ∞ is solvable in P-time. We also show that there is a simultaneous strategy improvement algorithm that converges in a finite number of steps to the value and optimal strategies of a 1-RSSG with positive rewards
Networked Occupancy Sensor System
Energy is often wasted on systems that are used to provide services such as light, heating, air conditioning and ventilation. If these services were intelligently controlled, there is potential for significant improvements in energy conservation. A system including room sensors, database, and webserver was designed, constructed, and implemented over the course of this project. Sensors report occupancy and light status and temperature. Real-time room data is available via the webserver and is archived in the database. The system is networked via Ethernet and powered using the power over Ethernet (802.3af) standard
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