$$\mathsf {WAPS}$$ : Weighted and Projected Sampling

Abstract

Given a set of constraints F and a user-defined weight function W on the assignment space, the problem of constrained sampling is to sample satisfying assignments of F conditioned on W. Constrained sampling is a fundamental problem with applications in probabilistic reasoning, synthesis, software and hardware testing. Consequently, the problem of sampling has been subject to intense theoretical and practical investigations over the years. Despite such intense investigations, there still remains a gap between theory and practice. In particular, there has been significant progress in the development of sampling techniques when W is a uniform distribution, but such techniques fail to handle general weight functions W. Furthermore, we are, often, interested in Σ 1 1 formulas, i.e., G(X) := ∃Y F (X, Y) for some F ; typically the set of variables Y are introduced as auxiliary variables during encoding of constraints to F. In this context, one wonders whether it is possible to design sampling techniques whose runtime performance is agnostic to the underlying weight distribution and can handle Σ 1 1 formulas? The primary contribution of this work is a novel technique, called WAPS, for sampling over Σ 1 1 whose runtime is agnostic to W. WAPS is based on our recently discovered connection between knowledge compilation and uniform sampling. WAPS proceeds by compiling F into a well studied compiled form, d-DNNF, which allows sampling operations to be conducted in linear time in the size of the compiled form. We demonstrate that WAPS can significantly outperform existing state-of-the-art weighted and projected sampler WeightGen, by up to 3 orders of magnitude in runtime while achieving a geometric speedup of 296× and solving 564 more instances out of 773. The distribution generated by WAPS is statistically indistinguishable from that generated by an ideal weighted and projected sampler. Furthermore, WAPS is almost oblivious to the number of samples requested.

Baital: an adaptive weighted sampling approach for improved t-wise coverage

“…To analyse the efficiency of our approach we have implemented a prototype of Baital in Python 1 . The sampling process is done by the literal-weighted sampler WAPS [20]. To evaluate our approach we designed a set of experiments helping to answer the following research questions.…”

“…To accomplish an efficient procedure, we need to tackle three challenges: Figure 1: Feature model of graph library Challenge 1: Representation of weights over assignment space To represent weights over assignment space, we turn to a literal-weighted function that assigns a non-negative weight to every literal such that the weight of an assignment is the product of the weight of its literal. The choice of literalweighted function is primarily motivated due to the observation that a wide variety of distributions arising from diverse disciplines can be represented as literal-weight function [13,20].…”

“…In [53] it was observed that advances in knowledge compilation can be used to design efficient uniform samplers and designed an efficient uniform sampler KUS. In [20] KUS was extended to design a weighted sampler WAPS. Given an input formula F and the desired number of samples s, WAPS uses a three staged process: Compilation, Annotation, and Sampling.…”

“…We make the following key observation about Annotation and Sampling in WAPS that strongly supports our choice of knowledge compilation-based approach for Baital. [20,53]. Essentially, the key observation is that Annotation is performed in a bottom-up process while visiting each node exactly once, while Sampling is performed in a top-down manner while visiting each node exactly once.…”

“…The update of weight function typically requires the underlying sampling algorithm to redo the entire work from scratch. To allow the reuse of the computation, we adapt the recently introduced knowledge compilation-based approach for weighted sampling [20].…”