139 research outputs found
Solving Satisfiability Modulo Counting for Symbolic and Statistical AI Integration With Provable Guarantees
Satisfiability Modulo Counting (SMC) encompasses problems that require both
symbolic decision-making and statistical reasoning. Its general formulation
captures many real-world problems at the intersection of symbolic and
statistical Artificial Intelligence. SMC searches for policy interventions to
control probabilistic outcomes. Solving SMC is challenging because of its
highly intractable nature(-complete), incorporating
statistical inference and symbolic reasoning. Previous research on SMC solving
lacks provable guarantees and/or suffers from sub-optimal empirical
performance, especially when combinatorial constraints are present. We propose
XOR-SMC, a polynomial algorithm with access to NP-oracles, to solve highly
intractable SMC problems with constant approximation guarantees. XOR-SMC
transforms the highly intractable SMC into satisfiability problems, by
replacing the model counting in SMC with SAT formulae subject to randomized XOR
constraints. Experiments on solving important SMC problems in AI for social
good demonstrate that XOR-SMC finds solutions close to the true optimum,
outperforming several baselines which struggle to find good approximations for
the intractable model counting in SMC
XOR-Sampling for Network Design with Correlated Stochastic Events
Many network optimization problems can be formulated as stochastic network
design problems in which edges are present or absent stochastically.
Furthermore, protective actions can guarantee that edges will remain present.
We consider the problem of finding the optimal protection strategy under a
budget limit in order to maximize some connectivity measurements of the
network. Previous approaches rely on the assumption that edges are independent.
In this paper, we consider a more realistic setting where multiple edges are
not independent due to natural disasters or regional events that make the
states of multiple edges stochastically correlated. We use Markov Random Fields
to model the correlation and define a new stochastic network design framework.
We provide a novel algorithm based on Sample Average Approximation (SAA)
coupled with a Gibbs or XOR sampler. The experimental results on real road
network data show that the policies produced by SAA with the XOR sampler have
higher quality and lower variance compared to SAA with Gibbs sampler.Comment: In Proceedings of the Twenty-sixth International Joint Conference on
Artificial Intelligence (IJCAI-17). The first two authors contribute equall
On the Complexity of Random Satisfiability Problems with Planted Solutions
The problem of identifying a planted assignment given a random -SAT
formula consistent with the assignment exhibits a large algorithmic gap: while
the planted solution becomes unique and can be identified given a formula with
clauses, there are distributions over clauses for which the best
known efficient algorithms require clauses. We propose and study a
unified model for planted -SAT, which captures well-known special cases. An
instance is described by a planted assignment and a distribution on
clauses with literals. We define its distribution complexity as the largest
for which the distribution is not -wise independent ( for
any distribution with a planted assignment).
Our main result is an unconditional lower bound, tight up to logarithmic
factors, for statistical (query) algorithms [Kearns 1998, Feldman et. al 2012],
matching known upper bounds, which, as we show, can be implemented using a
statistical algorithm. Since known approaches for problems over distributions
have statistical analogues (spectral, MCMC, gradient-based, convex optimization
etc.), this lower bound provides a rigorous explanation of the observed
algorithmic gap. The proof introduces a new general technique for the analysis
of statistical query algorithms. It also points to a geometric paring
phenomenon in the space of all planted assignments.
We describe consequences of our lower bounds to Feige's refutation hypothesis
[Feige 2002] and to lower bounds on general convex programs that solve planted
-SAT. Our bounds also extend to other planted -CSP models, and, in
particular, provide concrete evidence for the security of Goldreich's one-way
function and the associated pseudorandom generator when used with a
sufficiently hard predicate [Goldreich 2000].Comment: Extended abstract appeared in STOC 201
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Complexity Theory
Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes
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