Sampling and integration of near log-concave functions

Applegate, David; Kannan, Ravi

doi:10.1145/103418.103439

Cited by 110 publications

(182 citation statements)

References 6 publications

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“…Our analysis of computational complexity builds on several fundamental papers studying the computational complexity of Metropolis procedures, especially Applegate and Kannan [2], Frieze, Kannan and Polson [16], Polson [38], Kannan, Lovász and Simonovits [27], Kannan and Li [26], Lovász and Simonovits [34], and Lovász and Vempala [35,36,37]. Many of our results and proofs rely upon and extend the mathematical tools previously developed in these works.…”

Section: Introductionmentioning

confidence: 96%

On the computational complexity of MCMC-based estimators in large samples

Belloni¹,

Chernozhukov²

2007

Working Paper Series

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In this paper we examine the implications of the statistical large sample theory for the computational complexity of Bayesian and quasiBayesian estimation carried out using Metropolis random walks. Our analysis is motivated by the Laplace-Bernstein-Von Mises central limit theorem, which states that in large samples the posterior or quasi-posterior approaches a normal density. Using this observation, we establish polynomial bounds on the computational complexity of general Metropolis random walks methods in large samples. Our analysis covers cases, where the underlying log-likelihood or extremum criterion function is possibly nonconcave, discontinuous, and of increasing dimension. However, the central limit theorem restricts the deviations from continuity and log-concavity of the log-likelihood or extremum criterion function in a very specific manner.Under minimal assumptions for the central limit theorem framework to hold, we show that the Metropolis algorithm is theoretically efficient even for the canonical Gaussian walk which is studied in detail. Specifically, we show that the running time of the algorithm in large samples is bounded in probability by a polynomial in the parameter dimension d, and, in particular, is of stochastic order d 2 in the leading cases after the burn-in period. We then give an application to exponential and curved exponential families of increasing dimension.

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Section: Introductionmentioning

confidence: 96%

On the computational complexity of MCMC-based estimators in large samples

Belloni¹,

Chernozhukov²

2007

Working Paper Series

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“…This result relies essentially on the Prékopa-Leindler inequality which is a generalization of the Brunn-Minkowski inequality (we refer the reader to the excellent survey [4]). We note that the isoperimetric inequality we prove (Theorem 2) was stated in [5] for volumes, and in [6] for continuous -log-concave distributions, in the context of efficient sampling from convex bodies. However, the proof sketched in [6] relies in an essential way on having a continuous density ( [7]).…”

Section: Introductionmentioning

confidence: 93%

“…We note that the isoperimetric inequality we prove (Theorem 2) was stated in [5] for volumes, and in [6] for continuous -log-concave distributions, in the context of efficient sampling from convex bodies. However, the proof sketched in [6] relies in an essential way on having a continuous density ( [7]). We provide a complete proof of the more general result using the Ham-Sandwich Theorem (as in [5], but using a different method) and a different reduction argument.…”

Section: Introductionmentioning

confidence: 93%

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An Inequality for Nearly Log-Concave Distributions with Applications to Learning

Caramanis

Mannor

2004

Learning Theory

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Abstract-We prove that given a nearly log-concave distribution, in any partition of the space to two well separated sets, the measure of the points that do not belong to these sets is large. We apply this isoperimetric inequality to derive lower bounds on the generalization error in learning. We further consider regression problems and show that if the inputs and outputs are sampled from a nearly log-concave distribution, the measure of points for which the prediction is wrong by more than 0 and less than 1 is (roughly) linear in 1 0 0 , as long as 0 is not too small, and 1 not too large. We also show that when the data are sampled from a nearly log-concave distribution, the margin cannot be large in a strong probabilistic sense.

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“…√ n centered at the origin inside this circle and hence C must be contained in K. For more details on how a contained cube can be determined in polynomial time, the interested reader is referred to Applegate and Kannan [1] who proved that one can find an affine mapping in polynomial time which maps K to K such that the unit cube is contained in K .…”

Section: Theorem 4 Let K Denote An Arbitrary N Dimensional Polytope Wmentioning

confidence: 99%

Using Histograms to Better Answer Queries to Probabilistic Logic Programs

2009

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Probabilistic logic programs (PLPs) define a set of probability distribution functions (PDFs) over the set of all Herbrand interpretations of the underlying logical language. When answering a query Q, a lower and upper bound on Q is obtained by optimizing (min and max) an objective function subject to a set of linear constraints whose solutions are the PDFs mentioned above. A common critique not only of PLPs but many probabilistic logics is that the difference between the upper bound and lower bound is large, thus often providing very little useful information in the query answer. In this paper, we provide a new method to answer probabilistic queries that tries to come up with a histogram that "maps" the probability that the objective function will have a value in a given interval, subject to the above linear constraints. This allows the system to return to the user a histogram where he can directly "see" what the most likely probability range for his query will be. We prove that computing these histograms is #P -hard, and show that computing these histograms is closely related to polyhedral volume computation. We show how existing randomized algorithms for volume computation can be adapted to the computation of such histograms. A prototype experimental implementation is discussed.

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Sampling and integration of near log-concave functions

Cited by 110 publications

References 6 publications

On the computational complexity of MCMC-based estimators in large samples

On the computational complexity of MCMC-based estimators in large samples

An Inequality for Nearly Log-Concave Distributions with Applications to Learning

Using Histograms to Better Answer Queries to Probabilistic Logic Programs

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