Approximating a Target Distribution using Weight Queries

TL;DR

This paper introduces a novel method for approximating a target distribution using weight queries instead of sampling, combining bandit-inspired algorithms with discrepancy estimation to achieve effective reweighting.

Contribution

The paper presents an interactive algorithm that approximates a target distribution via weight queries, with theoretical guarantees and experimental validation.

Findings

Algorithm effectively approximates target distribution with limited queries

Theoretical bounds on total variation distance achieved

Experimental results show advantages over baselines

Abstract

We consider a novel challenge: approximating a distribution without the ability to randomly sample from that distribution. We study how such an approximation can be obtained using *weight queries*. Given some data set of examples, a weight query presents one of the examples to an oracle, which returns the probability, according to the target distribution, of observing examples similar to the presented example. This oracle can represent, for instance, counting queries to a database of the target population, or an interface to a search engine which returns the number of results that match a given search. We propose an interactive algorithm that iteratively selects data set examples and performs corresponding weight queries. The algorithm finds a reweighting of the data set that approximates the weights according to the target distribution, using a limited number of weight queries. We derive an approximation bound on the total variation distance between the reweighting found by the algorithm and the best achievable reweighting. Our algorithm takes inspiration from the UCB approach common in multi-armed bandits problems, and combines it with a new discrepancy estimator and a greedy iterative procedure. In addition to our theoretical guarantees, we demonstrate in experiments the advantages of the proposed algorithm over several baselines. A python implementation of the proposed algorithm and of all the experiments can be found at https://github.com/Nadav-Barak/AWP.