17,184 research outputs found
On the complexity of probabilistic trials for hidden satisfiability problems
What is the minimum amount of information and time needed to solve 2SAT? When
the instance is known, it can be solved in polynomial time, but is this also
possible without knowing the instance? Bei, Chen and Zhang (STOC '13)
considered a model where the input is accessed by proposing possible
assignments to a special oracle. This oracle, on encountering some constraint
unsatisfied by the proposal, returns only the constraint index. It turns out
that, in this model, even 1SAT cannot be solved in polynomial time unless P=NP.
Hence, we consider a model in which the input is accessed by proposing
probability distributions over assignments to the variables. The oracle then
returns the index of the constraint that is most likely to be violated by this
distribution. We show that the information obtained this way is sufficient to
solve 1SAT in polynomial time, even when the clauses can be repeated. For 2SAT,
as long as there are no repeated clauses, in polynomial time we can even learn
an equivalent formula for the hidden instance and hence also solve it.
Furthermore, we extend these results to the quantum regime. We show that in
this setting 1QSAT can be solved in polynomial time up to constant precision,
and 2QSAT can be learnt in polynomial time up to inverse polynomial precision.Comment: 24 pages, 2 figures. To appear in the 41st International Symposium on
Mathematical Foundations of Computer Scienc
Optimal Testing for Planted Satisfiability Problems
We study the problem of detecting planted solutions in a random
satisfiability formula. Adopting the formalism of hypothesis testing in
statistical analysis, we describe the minimax optimal rates of detection. Our
analysis relies on the study of the number of satisfying assignments, for which
we prove new results. We also address algorithmic issues, and give a
computationally efficient test with optimal statistical performance. This
result is compared to an average-case hypothesis on the hardness of refuting
satisfiability of random formulas
Learning Task Specifications from Demonstrations
Real world applications often naturally decompose into several sub-tasks. In
many settings (e.g., robotics) demonstrations provide a natural way to specify
the sub-tasks. However, most methods for learning from demonstrations either do
not provide guarantees that the artifacts learned for the sub-tasks can be
safely recombined or limit the types of composition available. Motivated by
this deficit, we consider the problem of inferring Boolean non-Markovian
rewards (also known as logical trace properties or specifications) from
demonstrations provided by an agent operating in an uncertain, stochastic
environment. Crucially, specifications admit well-defined composition rules
that are typically easy to interpret. In this paper, we formulate the
specification inference task as a maximum a posteriori (MAP) probability
inference problem, apply the principle of maximum entropy to derive an analytic
demonstration likelihood model and give an efficient approach to search for the
most likely specification in a large candidate pool of specifications. In our
experiments, we demonstrate how learning specifications can help avoid common
problems that often arise due to ad-hoc reward composition.Comment: NIPS 201
A variational description of the ground state structure in random satisfiability problems
A variational approach to finite connectivity spin-glass-like models is
developed and applied to describe the structure of optimal solutions in random
satisfiability problems. Our variational scheme accurately reproduces the known
replica symmetric results and also allows for the inclusion of replica symmetry
breaking effects. For the 3-SAT problem, we find two transitions as the ratio
of logical clauses per Boolean variables increases. At the first one
, a non-trivial organization of the solution space in
geometrically separated clusters emerges. The multiplicity of these clusters as
well as the typical distances between different solutions are calculated. At
the second threshold , satisfying assignments disappear
and a finite fraction of variables are overconstrained and
take the same values in all optimal (though unsatisfying) assignments. These
values have to be compared to obtained
from numerical experiments on small instances. Within the present variational
approach, the SAT-UNSAT transition naturally appears as a mixture of a first
and a second order transition. For the mixed -SAT with , the
behavior is as expected much simpler: a unique smooth transition from SAT to
UNSAT takes place at .Comment: 24 pages, 6 eps figures, to be published in Europ. Phys. J.
Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs
Laplacian mixture models identify overlapping regions of influence in
unlabeled graph and network data in a scalable and computationally efficient
way, yielding useful low-dimensional representations. By combining Laplacian
eigenspace and finite mixture modeling methods, they provide probabilistic or
fuzzy dimensionality reductions or domain decompositions for a variety of input
data types, including mixture distributions, feature vectors, and graphs or
networks. Provable optimal recovery using the algorithm is analytically shown
for a nontrivial class of cluster graphs. Heuristic approximations for scalable
high-performance implementations are described and empirically tested.
Connections to PageRank and community detection in network analysis demonstrate
the wide applicability of this approach. The origins of fuzzy spectral methods,
beginning with generalized heat or diffusion equations in physics, are reviewed
and summarized. Comparisons to other dimensionality reduction and clustering
methods for challenging unsupervised machine learning problems are also
discussed.Comment: 13 figures, 35 reference
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