249,425 research outputs found
On optimal heuristic randomized semidecision procedures, with application to proof complexity
The existence of a (p-)optimal propositional proof system is a major open
question in (proof) complexity; many people conjecture that such systems do not
exist. Krajicek and Pudlak (1989) show that this question is equivalent to the
existence of an algorithm that is optimal on all propositional tautologies.
Monroe (2009) recently gave a conjecture implying that such algorithm does not
exist.
We show that in the presence of errors such optimal algorithms do exist. The
concept is motivated by the notion of heuristic algorithms. Namely, we allow
the algorithm to claim a small number of false "theorems" (according to any
samplable distribution on non-tautologies) and err with bounded probability on
other inputs.
Our result can also be viewed as the existence of an optimal proof system in
a class of proof systems obtained by generalizing automatizable proof systems.Comment: 11 pages, accepted to STACS 201
Information in propositional proofs and algorithmic proof search
We study from the proof complexity perspective the (informal) proof search
problem:
Is there an optimal way to search for propositional proofs?
We note that for any fixed proof system there exists a time-optimal proof
search algorithm. Using classical proof complexity results about reflection
principles we prove that a time-optimal proof search algorithm exists w.r.t.
all proof systems iff a p-optimal proof system exists.
To characterize precisely the time proof search algorithms need for
individual formulas we introduce a new proof complexity measure based on
algorithmic information concepts. In particular, to a proof system we
attach {\bf information-efficiency function} assigning to a
tautology a natural number, and we show that:
- characterizes time any -proof search algorithm has to use on
and that for a fixed there is such an information-optimal algorithm,
- a proof system is information-efficiency optimal iff it is p-optimal,
- for non-automatizable systems there are formulas with short
proofs but having large information measure .
We isolate and motivate the problem to establish {\em unconditional}
super-logarithmic lower bounds for where no super-polynomial size
lower bounds are known. We also point out connections of the new measure with
some topics in proof complexity other than proof search.Comment: Preliminary version February 202
Euclidean algorithms are Gaussian
This study provides new results about the probabilistic behaviour of a class
of Euclidean algorithms: the asymptotic distribution of a whole class of
cost-parameters associated to these algorithms is normal. For the cost
corresponding to the number of steps Hensley already has proved a Local Limit
Theorem; we give a new proof, and extend his result to other euclidean
algorithms and to a large class of digit costs, obtaining a faster, optimal,
rate of convergence. The paper is based on the dynamical systems methodology,
and the main tool is the transfer operator. In particular, we use recent
results of Dolgopyat.Comment: fourth revised version - 2 figures - the strict convexity condition
used has been clarifie
Optimal Sparse Decision Trees
Decision tree algorithms have been among the most popular algorithms for
interpretable (transparent) machine learning since the early 1980's. The
problem that has plagued decision tree algorithms since their inception is
their lack of optimality, or lack of guarantees of closeness to optimality:
decision tree algorithms are often greedy or myopic, and sometimes produce
unquestionably suboptimal models. Hardness of decision tree optimization is
both a theoretical and practical obstacle, and even careful mathematical
programming approaches have not been able to solve these problems efficiently.
This work introduces the first practical algorithm for optimal decision trees
for binary variables. The algorithm is a co-design of analytical bounds that
reduce the search space and modern systems techniques, including data
structures and a custom bit-vector library. Our experiments highlight
advantages in scalability, speed, and proof of optimality.Comment: 33rd Conference on Neural Information Processing Systems (NeurIPS
2019), Vancouver, Canad
Tensor Product Approach to Quantum Control
In this proof-of-concept paper we show that tensor product approach is
efficient for control of large quantum systems, such as Heisenberg spin wires,
which are essential for emerging quantum computing technologies. We compute
optimal control sequences using GRAPE method, applying the recently developed
tAMEn algorithm to calculate evolution of quantum states represented in the
tensor train format to reduce storage. Using tensor product algorithms we can
overcome the curse of dimensionality and compute the optimal control pulse for
a 41 spin system on a single workstation with fully controlled accuracy and
huge savings of computational time and memory. The use of tensor product
algorithms opens new approaches for development of quantum computers with 50 to
100 qubits.Comment: To appear in Proc. IMSE 201
Near-Optimal Sensor Scheduling for Batch State Estimation: Complexity, Algorithms, and Limits
In this paper, we focus on batch state estimation for linear systems. This
problem is important in applications such as environmental field estimation,
robotic navigation, and target tracking. Its difficulty lies on that limited
operational resources among the sensors, e.g., shared communication bandwidth
or battery power, constrain the number of sensors that can be active at each
measurement step. As a result, sensor scheduling algorithms must be employed.
Notwithstanding, current sensor scheduling algorithms for batch state
estimation scale poorly with the system size and the time horizon. In addition,
current sensor scheduling algorithms for Kalman filtering, although they scale
better, provide no performance guarantees or approximation bounds for the
minimization of the batch state estimation error. In this paper, one of our
main contributions is to provide an algorithm that enjoys both the estimation
accuracy of the batch state scheduling algorithms and the low time complexity
of the Kalman filtering scheduling algorithms. In particular: 1) our algorithm
is near-optimal: it achieves a solution up to a multiplicative factor 1/2 from
the optimal solution, and this factor is close to the best approximation factor
1/e one can achieve in polynomial time for this problem; 2) our algorithm has
(polynomial) time complexity that is not only lower than that of the current
algorithms for batch state estimation; it is also lower than, or similar to,
that of the current algorithms for Kalman filtering. We achieve these results
by proving two properties for our batch state estimation error metric, which
quantifies the square error of the minimum variance linear estimator of the
batch state vector: a) it is supermodular in the choice of the sensors; b) it
has a sparsity pattern (it involves matrices that are block tri-diagonal) that
facilitates its evaluation at each sensor set.Comment: Correction of typos in proof
Algorithms for LQR via Static Output Feedback for Discrete-Time LTI Systems
Randomized and deterministic algorithms for the problem of LQR optimal control via static-output-feedback (SOF) for discrete-time systems are suggested in this chapter. The randomized algorithm is based on a recently introduced randomized optimization method named the Ray-Shooting Method that efficiently solves the global minimization problem of continuous functions over compact non-convex unconnected regions. The randomized algorithm presented here has a proof of convergence in probability to the global optimum. The suggested deterministic algorithm is based on the gradient method and thus can be proved to converge to local optimum only. A comparison between the algorithms is provided as well as the performance of the hybrid algorithm
Stochastic filtering via L2 projection on mixture manifolds with computer algorithms and numerical examples
We examine some differential geometric approaches to finding approximate
solutions to the continuous time nonlinear filtering problem. Our primary focus
is a new projection method for the optimal filter infinite dimensional
Stochastic Partial Differential Equation (SPDE), based on the direct L2 metric
and on a family of normal mixtures. We compare this method to earlier
projection methods based on the Hellinger distance/Fisher metric and
exponential families, and we compare the L2 mixture projection filter with a
particle method with the same number of parameters, using the Levy metric. We
prove that for a simple choice of the mixture manifold the L2 mixture
projection filter coincides with a Galerkin method, whereas for more general
mixture manifolds the equivalence does not hold and the L2 mixture filter is
more general. We study particular systems that may illustrate the advantages of
this new filter over other algorithms when comparing outputs with the optimal
filter. We finally consider a specific software design that is suited for a
numerically efficient implementation of this filter and provide numerical
examples.Comment: Updated and expanded version published in the Journal reference
below. Preprint updates: January 2016 (v3) added projection of Zakai Equation
and difference with projection of Kushner-Stratonovich (section 4.1). August
2014 (v2) added Galerkin equivalence proof (Section 5) to the March 2013 (v1)
versio
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