4,119 research outputs found
Unbiased Black-Box Complexities of Jump Functions
We analyze the unbiased black-box complexity of jump functions with small,
medium, and large sizes of the fitness plateau surrounding the optimal
solution.
Among other results, we show that when the jump size is , that is, only a small constant fraction of the fitness values
is visible, then the unbiased black-box complexities for arities and higher
are of the same order as those for the simple \textsc{OneMax} function. Even
for the extreme jump function, in which all but the two fitness values
and are blanked out, polynomial-time mutation-based (i.e., unary unbiased)
black-box optimization algorithms exist. This is quite surprising given that
for the extreme jump function almost the whole search space (all but a
fraction) is a plateau of constant fitness.
To prove these results, we introduce new tools for the analysis of unbiased
black-box complexities, for example, selecting the new parent individual not by
comparing the fitnesses of the competing search points, but also by taking into
account the (empirical) expected fitnesses of their offspring.Comment: This paper is based on results presented in the conference versions
[GECCO 2011] and [GECCO 2014
Complexity Theory for Discrete Black-Box Optimization Heuristics
A predominant topic in the theory of evolutionary algorithms and, more
generally, theory of randomized black-box optimization techniques is running
time analysis. Running time analysis aims at understanding the performance of a
given heuristic on a given problem by bounding the number of function
evaluations that are needed by the heuristic to identify a solution of a
desired quality. As in general algorithms theory, this running time perspective
is most useful when it is complemented by a meaningful complexity theory that
studies the limits of algorithmic solutions.
In the context of discrete black-box optimization, several black-box
complexity models have been developed to analyze the best possible performance
that a black-box optimization algorithm can achieve on a given problem. The
models differ in the classes of algorithms to which these lower bounds apply.
This way, black-box complexity contributes to a better understanding of how
certain algorithmic choices (such as the amount of memory used by a heuristic,
its selective pressure, or properties of the strategies that it uses to create
new solution candidates) influences performance.
In this chapter we review the different black-box complexity models that have
been proposed in the literature, survey the bounds that have been obtained for
these models, and discuss how the interplay of running time analysis and
black-box complexity can inspire new algorithmic solutions to well-researched
problems in evolutionary computation. We also discuss in this chapter several
interesting open questions for future work.Comment: This survey article is to appear (in a slightly modified form) in the
book "Theory of Randomized Search Heuristics in Discrete Search Spaces",
which will be published by Springer in 2018. The book is edited by Benjamin
Doerr and Frank Neumann. Missing numbers of pointers to other chapters of
this book will be added as soon as possibl
An Active Learning Algorithm for Ranking from Pairwise Preferences with an Almost Optimal Query Complexity
We study the problem of learning to rank from pairwise preferences, and solve
a long-standing open problem that has led to development of many heuristics but
no provable results for our particular problem. Given a set of
elements, we wish to linearly order them given pairwise preference labels. A
pairwise preference label is obtained as a response, typically from a human, to
the question "which if preferred, u or v?u,v\in V{n\choose 2}$ possibilities only. We present an active learning algorithm for
this problem, with query bounds significantly beating general (non active)
bounds for the same error guarantee, while almost achieving the information
theoretical lower bound. Our main construct is a decomposition of the input
s.t. (i) each block incurs high loss at optimum, and (ii) the optimal solution
respecting the decomposition is not much worse than the true opt. The
decomposition is done by adapting a recent result by Kenyon and Schudy for a
related combinatorial optimization problem to the query efficient setting. We
thus settle an open problem posed by learning-to-rank theoreticians and
practitioners: What is a provably correct way to sample preference labels? To
further show the power and practicality of our solution, we show how to use it
in concert with an SVM relaxation.Comment: Fixed a tiny error in theorem 3.1 statemen
Black-Box Complexity of the Binary Value Function
The binary value function, or BinVal, has appeared in several studies in
theory of evolutionary computation as one of the extreme examples of linear
pseudo-Boolean functions. Its unbiased black-box complexity was previously
shown to be at most , where is the problem
size. We augment it with an upper bound of ,
which is more precise for many values of . We also present a lower bound of
. Additionally, we prove that BinVal is an easiest
function among all unimodal pseudo-Boolean functions at least for unbiased
algorithms.Comment: 24 pages, one figure. An extended two-page abstract of this work will
appear in proceedings of the Genetic and Evolutionary Computation Conference,
GECCO'1
Two Structural Results for Low Degree Polynomials and Applications
In this paper, two structural results concerning low degree polynomials over
finite fields are given. The first states that over any finite field
, for any polynomial on variables with degree , there exists a subspace of with dimension on which is constant. This result is shown to be tight.
Stated differently, a degree polynomial cannot compute an affine disperser
for dimension smaller than . Using a recursive
argument, we obtain our second structural result, showing that any degree
polynomial induces a partition of to affine subspaces of dimension
, such that is constant on each part.
We extend both structural results to more than one polynomial. We further
prove an analog of the first structural result to sparse polynomials (with no
restriction on the degree) and to functions that are close to low degree
polynomials. We also consider the algorithmic aspect of the two structural
results.
Our structural results have various applications, two of which are:
* Dvir [CC 2012] introduced the notion of extractors for varieties, and gave
explicit constructions of such extractors over large fields. We show that over
any finite field, any affine extractor is also an extractor for varieties with
related parameters. Our reduction also holds for dispersers, and we conclude
that Shaltiel's affine disperser [FOCS 2011] is a disperser for varieties over
.
* Ben-Sasson and Kopparty [SIAM J. C 2012] proved that any degree 3 affine
disperser over a prime field is also an affine extractor with related
parameters. Using our structural results, and based on the work of Kaufman and
Lovett [FOCS 2008] and Haramaty and Shpilka [STOC 2010], we generalize this
result to any constant degree
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