9,745 research outputs found
A local decision test for sparse polynomials
An ℓ-sparse (multivariate) polynomial is a polynomial containing at most ℓ-monomials in its explicit description. We assume that a polynomial is implicitly represented as a black-box: on an input query from the domain, the black-box replies with the evaluation of the polynomial at that input. We provide an efficient, randomized algorithm, that can decide whether a polynomial [MathML] given as a black-box is ℓ-sparse or not, provided that q is large compared to the polynomial's total degree. The algorithm makes only queries, which is independent of the domain size. The running time of our algorithm (in the bit-complexity model) is , where d is an upper bound on the degree of each variable. Existing interpolation algorithms for polynomials in the same model run in time . We provide a similar test for polynomials with integer coefficients
Learning using Local Membership Queries
We introduce a new model of membership query (MQ) learning, where the
learning algorithm is restricted to query points that are \emph{close} to
random examples drawn from the underlying distribution. The learning model is
intermediate between the PAC model (Valiant, 1984) and the PAC+MQ model (where
the queries are allowed to be arbitrary points).
Membership query algorithms are not popular among machine learning
practitioners. Apart from the obvious difficulty of adaptively querying
labelers, it has also been observed that querying \emph{unnatural} points leads
to increased noise from human labelers (Lang and Baum, 1992). This motivates
our study of learning algorithms that make queries that are close to examples
generated from the data distribution.
We restrict our attention to functions defined on the -dimensional Boolean
hypercube and say that a membership query is local if its Hamming distance from
some example in the (random) training data is at most . We show the
following results in this model:
(i) The class of sparse polynomials (with coefficients in R) over
is polynomial time learnable under a large class of \emph{locally smooth}
distributions using -local queries. This class also includes the
class of -depth decision trees.
(ii) The class of polynomial-sized decision trees is polynomial time
learnable under product distributions using -local queries.
(iii) The class of polynomial size DNF formulas is learnable under the
uniform distribution using -local queries in time
.
(iv) In addition we prove a number of results relating the proposed model to
the traditional PAC model and the PAC+MQ model
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
Sparse sum-of-squares (SOS) optimization: A bridge between DSOS/SDSOS and SOS optimization for sparse polynomials
Optimization over non-negative polynomials is fundamental for nonlinear
systems analysis and control. We investigate the relation between three
tractable relaxations for optimizing over sparse non-negative polynomials:
sparse sum-of-squares (SSOS) optimization, diagonally dominant sum-of-squares
(DSOS) optimization, and scaled diagonally dominant sum-of-squares (SDSOS)
optimization. We prove that the set of SSOS polynomials, an inner approximation
of the cone of SOS polynomials, strictly contains the spaces of sparse
DSOS/SDSOS polynomials. When applicable, therefore, SSOS optimization is less
conservative than its DSOS/SDSOS counterparts. Numerical results for
large-scale sparse polynomial optimization problems demonstrate this fact, and
also that SSOS optimization can be faster than DSOS/SDSOS methods despite
requiring the solution of semidefinite programs instead of less expensive
linear/second-order cone programs.Comment: 9 pages, 3 figure
Domain Decomposition for Stochastic Optimal Control
This work proposes a method for solving linear stochastic optimal control
(SOC) problems using sum of squares and semidefinite programming. Previous work
had used polynomial optimization to approximate the value function, requiring a
high polynomial degree to capture local phenomena. To improve the scalability
of the method to problems of interest, a domain decomposition scheme is
presented. By using local approximations, lower degree polynomials become
sufficient, and both local and global properties of the value function are
captured. The domain of the problem is split into a non-overlapping partition,
with added constraints ensuring continuity. The Alternating Direction
Method of Multipliers (ADMM) is used to optimize over each domain in parallel
and ensure convergence on the boundaries of the partitions. This results in
improved conditioning of the problem and allows for much larger and more
complex problems to be addressed with improved performance.Comment: 8 pages. Accepted to CDC 201
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