Periodic Reporting for period 2 - NACS (New Approaches to Counting and Sampling)
Berichtszeitraum: 2022-07-01 bis 2023-12-31
The project aims to study computational counting and sampling problems. Typical problems of this kind include the evaluation or estimation of expectations and probabilities from complicated, usually high-dimensional, distributions. These distributions are the backbone of many probabilistic systems nowadays, and being able to evaluate and reason about them is of significant theoretical and practical importance. The focus of the project is mainly on the theoretical side, and the goal is to find algorithms to problems that have no efficient algorithm known yet. This objective is twofold: 1) to expand the design and analysis of the existing methods; and 2) to find alternative methods that are more favourable to the existing methods, such as Markov chain Monte Carlo. Moreover, in case a problem remain elusive even with significant effort, we also seek potential explanations from the theory of computational complexity for such apparent intractability.
Since the start of the project, we have made progress in many front and designed a number of new counting and sampling algorithms. One main achievement is a new way to improve the connectivity of solution spaces by considering projected distributions. This leads to new Markov chains that are both rapidly mixing and efficiently implementable. This method has found a number of applications, such as sampling hypergraph colourings and approximately counting solutions to random k-Sat instances. Another achievement is the first efficient approximation algorithm to the total variation distance between two product distributions, a fundamental computational problem in statistics with very little known before. In addition, we have found new techniques to analyze Markov chains, new perfect samplers for spin systems, and new results on the computational complexity of counting vertices of polytopes.
We have a number of ongoing research projects. We have been working on derandomising Markov chains, which is an important yet seldomly explored topic. We have made progress by designing extremely efficient local / marginal samplers, which, in turn, can be easily derandomised. These samplers are achieved by deducing the execution history of Markov chains. With this new technique, we pushed forward the threshold of state-of-the-art deterministic approximate counting algorithms for problems related to the Lovasz local lemma. Another ongoing research is to design faster approximate counting algorithms than the traditional simulated annealing algorithms. The main insight is to design new estimators that have less variance and yet can be computed more efficiently. In addition to these directions, we expect to find new results on analyzing Markov chains on geometry related problems, as well as on the fine-grained aspects of approximate counting.