8,237 research outputs found
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
Algorithms for Approximate Minimization of the Difference Between Submodular Functions, with Applications
We extend the work of Narasimhan and Bilmes [30] for minimizing set functions
representable as a difference between submodular functions. Similar to [30],
our new algorithms are guaranteed to monotonically reduce the objective
function at every step. We empirically and theoretically show that the
per-iteration cost of our algorithms is much less than [30], and our algorithms
can be used to efficiently minimize a difference between submodular functions
under various combinatorial constraints, a problem not previously addressed. We
provide computational bounds and a hardness result on the mul- tiplicative
inapproximability of minimizing the difference between submodular functions. We
show, however, that it is possible to give worst-case additive bounds by
providing a polynomial time computable lower-bound on the minima. Finally we
show how a number of machine learning problems can be modeled as minimizing the
difference between submodular functions. We experimentally show the validity of
our algorithms by testing them on the problem of feature selection with
submodular cost features.Comment: 17 pages, 8 figures. A shorter version of this appeared in Proc.
Uncertainty in Artificial Intelligence (UAI), Catalina Islands, 201
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
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