17,932 research outputs found
Bit-Vector Model Counting using Statistical Estimation
Approximate model counting for bit-vector SMT formulas (generalizing \#SAT)
has many applications such as probabilistic inference and quantitative
information-flow security, but it is computationally difficult. Adding random
parity constraints (XOR streamlining) and then checking satisfiability is an
effective approximation technique, but it requires a prior hypothesis about the
model count to produce useful results. We propose an approach inspired by
statistical estimation to continually refine a probabilistic estimate of the
model count for a formula, so that each XOR-streamlined query yields as much
information as possible. We implement this approach, with an approximate
probability model, as a wrapper around an off-the-shelf SMT solver or SAT
solver. Experimental results show that the implementation is faster than the
most similar previous approaches which used simpler refinement strategies. The
technique also lets us model count formulas over floating-point constraints,
which we demonstrate with an application to a vulnerability in differential
privacy mechanisms
Some Speed-Ups and Speed Limits for Real Algebraic Geometry
We give new positive and negative results (some conditional) on speeding up
computational algebraic geometry over the reals: (1) A new and sharper upper
bound on the number of connected components of a semialgebraic set. Our bound
is novel in that it is stated in terms of the volumes of certain polytopes and,
for a large class of inputs, beats the best previous bounds by a factor
exponential in the number of variables. (2) A new algorithm for approximating
the real roots of certain sparse polynomial systems. Two features of our
algorithm are (a) arithmetic complexity polylogarithmic in the degree of the
underlying complex variety (as opposed to the super-linear dependence in
earlier algorithms) and (b) a simple and efficient generalization to certain
univariate exponential sums. (3) Detecting whether a real algebraic surface
(given as the common zero set of some input straight-line programs) is not
smooth can be done in polynomial time within the classical Turing model (resp.
BSS model over C) only if P=NP (resp. NP<=BPP). The last result follows easily
from an unpublished result of Steve Smale.Comment: This is the final journal version which will appear in Journal of
Complexity. More typos are corrected, and a new section is added where the
bounds here are compared to an earlier result of Benedetti, Loeser, and
Risler. The LaTeX source needs the ajour.cls macro file to compil
A Bayesian information criterion for singular models
We consider approximate Bayesian model choice for model selection problems
that involve models whose Fisher-information matrices may fail to be invertible
along other competing submodels. Such singular models do not obey the
regularity conditions underlying the derivation of Schwarz's Bayesian
information criterion (BIC) and the penalty structure in BIC generally does not
reflect the frequentist large-sample behavior of their marginal likelihood.
While large-sample theory for the marginal likelihood of singular models has
been developed recently, the resulting approximations depend on the true
parameter value and lead to a paradox of circular reasoning. Guided by examples
such as determining the number of components of mixture models, the number of
factors in latent factor models or the rank in reduced-rank regression, we
propose a resolution to this paradox and give a practical extension of BIC for
singular model selection problems
S2: An Efficient Graph Based Active Learning Algorithm with Application to Nonparametric Classification
This paper investigates the problem of active learning for binary label
prediction on a graph. We introduce a simple and label-efficient algorithm
called S2 for this task. At each step, S2 selects the vertex to be labeled
based on the structure of the graph and all previously gathered labels.
Specifically, S2 queries for the label of the vertex that bisects the *shortest
shortest* path between any pair of oppositely labeled vertices. We present a
theoretical estimate of the number of queries S2 needs in terms of a novel
parametrization of the complexity of binary functions on graphs. We also
present experimental results demonstrating the performance of S2 on both real
and synthetic data. While other graph-based active learning algorithms have
shown promise in practice, our algorithm is the first with both good
performance and theoretical guarantees. Finally, we demonstrate the
implications of the S2 algorithm to the theory of nonparametric active
learning. In particular, we show that S2 achieves near minimax optimal excess
risk for an important class of nonparametric classification problems.Comment: A version of this paper appears in the Conference on Learning Theory
(COLT) 201
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