265 research outputs found
Runtime Distributions and Criteria for Restarts
Randomized algorithms sometimes employ a restart strategy. After a certain
number of steps, the current computation is aborted and restarted with a new,
independent random seed. In some cases, this results in an improved overall
expected runtime. This work introduces properties of the underlying runtime
distribution which determine whether restarts are advantageous. The most
commonly used probability distributions admit the use of a scale and a location
parameter. Location parameters shift the density function to the right, while
scale parameters affect the spread of the distribution. It is shown that for
all distributions scale parameters do not influence the usefulness of restarts
and that location parameters only have a limited influence. This result
simplifies the analysis of the usefulness of restarts. The most important
runtime probability distributions are the log-normal, the Weibull, and the
Pareto distribution. In this work, these distributions are analyzed for the
usefulness of restarts. Secondly, a condition for the optimal restart time (if
it exists) is provided. The log-normal, the Weibull, and the generalized Pareto
distribution are analyzed in this respect. Moreover, it is shown that the
optimal restart time is also not influenced by scale parameters and that the
influence of location parameters is only linear
Gaussian Bounds for Noise Correlation of Functions
In this paper we derive tight bounds on the expected value of products of
{\em low influence} functions defined on correlated probability spaces. The
proofs are based on extending Fourier theory to an arbitrary number of
correlated probability spaces, on a generalization of an invariance principle
recently obtained with O'Donnell and Oleszkiewicz for multilinear polynomials
with low influences and bounded degree and on properties of multi-dimensional
Gaussian distributions. The results derived here have a number of applications
to the theory of social choice in economics, to hardness of approximation in
computer science and to additive combinatorics problems.Comment: Typos and references correcte
The Potential of Restarts for ProbSAT
This work analyses the potential of restarts for probSAT, a quite successful
algorithm for k-SAT, by estimating its runtime distributions on random 3-SAT
instances that are close to the phase transition. We estimate an optimal
restart time from empirical data, reaching a potential speedup factor of 1.39.
Calculating restart times from fitted probability distributions reduces this
factor to a maximum of 1.30. A spin-off result is that the Weibull distribution
approximates the runtime distribution for over 93% of the used instances well.
A machine learning pipeline is presented to compute a restart time for a
fixed-cutoff strategy to exploit this potential. The main components of the
pipeline are a random forest for determining the distribution type and a neural
network for the distribution's parameters. ProbSAT performs statistically
significantly better than Luby's restart strategy and the policy without
restarts when using the presented approach. The structure is particularly
advantageous on hard problems.Comment: Eurocast 201
Neural Networks for Predicting Algorithm Runtime Distributions
Many state-of-the-art algorithms for solving hard combinatorial problems in
artificial intelligence (AI) include elements of stochasticity that lead to
high variations in runtime, even for a fixed problem instance. Knowledge about
the resulting runtime distributions (RTDs) of algorithms on given problem
instances can be exploited in various meta-algorithmic procedures, such as
algorithm selection, portfolios, and randomized restarts. Previous work has
shown that machine learning can be used to individually predict mean, median
and variance of RTDs. To establish a new state-of-the-art in predicting RTDs,
we demonstrate that the parameters of an RTD should be learned jointly and that
neural networks can do this well by directly optimizing the likelihood of an
RTD given runtime observations. In an empirical study involving five algorithms
for SAT solving and AI planning, we show that neural networks predict the true
RTDs of unseen instances better than previous methods, and can even do so when
only few runtime observations are available per training instance
Correlation Decay and Tractability of CSPs
The algebraic dichotomy conjecture of Bulatov, Krokhin and Jeavons yields an elegant characterization of the complexity of constraint satisfaction problems. Roughly speaking, the characterization asserts that a CSP L is tractable if and only if there exist certain non-trivial operations known as polymorphisms to combine solutions to L to create new ones.
In this work, we study the dynamical system associated with repeated applications of a polymorphism to a distribution over assignments. Specifically, we exhibit a correlation decay phenomenon that makes two variables or groups of variables that are not perfectly correlated become independent after repeated applications of a polymorphism.
We show that this correlation decay phenomenon can be utilized in designing algorithms for CSPs by exhibiting two applications:
1. A simple randomized algorithm to solve linear equations over a prime field, whose analysis crucially relies on correlation decay.
2. A sufficient condition for the simple linear programming relaxation for a 2-CSP to be sound (have no integrality gap) on a given instance
Effects of the lack of selective pressure on the expected run-time distribution in genetic programming
Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. D. F. Barrero, M. D. R-Moreno, B. Castano, and D. Camacho, "Effects of the lack of selective pressure on the expected run-time distribution in genetic programming", in IEEE Congress on Evolutionary Computation, CEC 2013, pp. 1748 - 1755Run-time analysis is a powerful tool to analyze algorithms. It is focused on studying the time required by an algorithm to find a solution, the expected run-time, which is one of the most relevant algorithm attributes. Previous research has associated the expected run-time in GP with the lognormal distribution. In this paper we provide additional evidence in that regard and show how the algorithm parametrization may change the resulting run-time distribution. In particular, we explore the influence of the selective pressure on the run-time distribution in tree-based GP, finding that, at least in two problem instances, the lack of selective pressure generates an expected run-time distribution well described by the Weibull probability distribution.This work has been partly supported by Spanish Ministry
of Science and Education under project ABANT (TIN2010-
19872)
Noise stability of functions with low influences: invariance and optimality
In this paper we study functions with low influences on product probability
spaces. The analysis of boolean functions with low influences has become a
central problem in discrete Fourier analysis. It is motivated by fundamental
questions arising from the construction of probabilistically checkable proofs
in theoretical computer science and from problems in the theory of social
choice in economics.
We prove an invariance principle for multilinear polynomials with low
influences and bounded degree; it shows that under mild conditions the
distribution of such polynomials is essentially invariant for all product
spaces. Ours is one of the very few known non-linear invariance principles. It
has the advantage that its proof is simple and that the error bounds are
explicit. We also show that the assumption of bounded degree can be eliminated
if the polynomials are slightly ``smoothed''; this extension is essential for
our applications to ``noise stability''-type problems.
In particular, as applications of the invariance principle we prove two
conjectures: the ``Majority Is Stablest'' conjecture from theoretical computer
science, which was the original motivation for this work, and the ``It Ain't
Over Till It's Over'' conjecture from social choice theory
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