Sampling-based lower bounds for counting queries

Abstract

It is well known that computing relative approximations of weighted counting queries such as the probability of evidence in a Bayesian network, the partition function of a Markov network, and the number of solutions of a constraint satisfaction problem is NP-hard. In this paper, we settle therefore on an easier problem of computing highconfidence lower bounds and propose an algorithm based on importance sampling and Markov inequality for it. However, a straight-forward application of Markov inequality often yields poor lower bounds because it uses only one sample. We therefore propose several new schemes that extend it to multiple samples. Empirically, we show that our new schemes are quite powerful, often yielding substantially higher (better) lower bounds than state-of-the-art schemes.

On the analysis of independent sets via multilevel splitting

“…The quality (tightness) of these bounds depends on the variance of the MS algorithm's output. As noted in [27], the MSB bound tends to decrease (unless the variance is very small) when the number of samples grows. The ASB and MASB schemes are based on the assumption that the sample average and maximum are likely to get larger as more samples are drawn.…”

“…Namely, to discover a good approximation to each coefficient of 's IP. Taking into consideration the computational complexity of such a procedure for large graphs, a reasonable approach is to use a probabilistic lower bound, which was first introduced by , and is provided next.…”

“…When running a Monte Carlo algorithm which provides an unbiased estimator, one can obtain a probabilistic lower bound . In particular, the following holds.Theorem (Probabilistic lower bounds).…”

“…In this article, we develop an adaptation of the general MS algorithm for the #IS problem. The proposed method is computationally efficient, enjoys a good practical performance and, similar to [27], has a probabilistic lower bound on its output. The latter allows us to reliably handle large graph instances.…”

Interleave Variational Optimization with Monte Carlo Sampling: A Tale of Two Approximate Inference Paradigms

“…The confidence parameter δ for our probabilistic bounds is set to 0.025. In the experiments, we also used Zδ, a (1 − δ) probabilistic lower bound by the Markov inequality (Gogate and Dechter 2011), and switched to our lower bound Z − ∆ when it becomes non-trivial. We also replaced Z + ∆ with the best deterministic upper bound reached so far if the latter is tighter.…”