63,015 research outputs found
Near-optimal Linear Decision Trees for k-SUM and Related Problems
We construct near-optimal linear decision trees for a variety of decision problems in combinatorics and discrete geometry. For example, for any constant
k
, we construct linear decision trees that solve the
k
-SUM problem on
n
elements using
O
(
n
log
2
n
) linear queries. Moreover, the queries we use are comparison queries, which compare the sums of two
k
-subsets; when viewed as linear queries, comparison queries are 2
k
-sparse and have only { â1,0,1} coefficients. We give similar constructions for sorting sumsets A+B and for solving the SUBSET-SUM problem, both with optimal number of queries, up to poly-logarithmic terms.
Our constructions are based on the notion of âinference dimension,â recently introduced by the authors in the context of active classification with comparison queries. This can be viewed as another contribution to the fruitful link between machine learning and discrete geometry, which goes back to the discovery of the VC dimension
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
Cover Tree Bayesian Reinforcement Learning
This paper proposes an online tree-based Bayesian approach for reinforcement
learning. For inference, we employ a generalised context tree model. This
defines a distribution on multivariate Gaussian piecewise-linear models, which
can be updated in closed form. The tree structure itself is constructed using
the cover tree method, which remains efficient in high dimensional spaces. We
combine the model with Thompson sampling and approximate dynamic programming to
obtain effective exploration policies in unknown environments. The flexibility
and computational simplicity of the model render it suitable for many
reinforcement learning problems in continuous state spaces. We demonstrate this
in an experimental comparison with least squares policy iteration
Approximation Algorithms for Stochastic Boolean Function Evaluation and Stochastic Submodular Set Cover
Stochastic Boolean Function Evaluation is the problem of determining the
value of a given Boolean function f on an unknown input x, when each bit of x_i
of x can only be determined by paying an associated cost c_i. The assumption is
that x is drawn from a given product distribution, and the goal is to minimize
the expected cost. This problem has been studied in Operations Research, where
it is known as "sequential testing" of Boolean functions. It has also been
studied in learning theory in the context of learning with attribute costs. We
consider the general problem of developing approximation algorithms for
Stochastic Boolean Function Evaluation. We give a 3-approximation algorithm for
evaluating Boolean linear threshold formulas. We also present an approximation
algorithm for evaluating CDNF formulas (and decision trees) achieving a factor
of O(log kd), where k is the number of terms in the DNF formula, and d is the
number of clauses in the CNF formula. In addition, we present approximation
algorithms for simultaneous evaluation of linear threshold functions, and for
ranking of linear functions.
Our function evaluation algorithms are based on reductions to the Stochastic
Submodular Set Cover (SSSC) problem. This problem was introduced by Golovin and
Krause. They presented an approximation algorithm for the problem, called
Adaptive Greedy. Our main technical contribution is a new approximation
algorithm for the SSSC problem, which we call Adaptive Dual Greedy. It is an
extension of the Dual Greedy algorithm for Submodular Set Cover due to Fujito,
which is a generalization of Hochbaum's algorithm for the classical Set Cover
Problem. We also give a new bound on the approximation achieved by the Adaptive
Greedy algorithm of Golovin and Krause
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