On the Impossibility of Convex Inference in Human Computation

TL;DR

This paper proves that under natural assumptions, convex objective functions for inference in human computation are impossible, but they can be constructed if spammers are not modeled.

Contribution

It establishes the theoretical impossibility of convex inference in human computation under natural axioms and provides conditions for convexity when spammers are excluded.

Findings

Convex inference is impossible under natural axioms.

Convex objective functions can be constructed without modeling spammers.

The results clarify limitations and possibilities in designing inference algorithms for crowdsourcing.

Abstract

Human computation or crowdsourcing involves joint inference of the ground-truth-answers and the worker-abilities by optimizing an objective function, for instance, by maximizing the data likelihood based on an assumed underlying model. A variety of methods have been proposed in the literature to address this inference problem. As far as we know, none of the objective functions in existing methods is convex. In machine learning and applied statistics, a convex function such as the objective function of support vector machines (SVMs) is generally preferred, since it can leverage the high-performance algorithms and rigorous guarantees established in the extensive literature on convex optimization. One may thus wonder if there exists a meaningful convex objective function for the inference problem in human computation. In this paper, we investigate this convexity issue for human computation. We take an axiomatic approach by formulating a set of axioms that impose two mild and natural assumptions on the objective function for the inference. Under these axioms, we show that it is unfortunately impossible to ensure convexity of the inference problem. On the other hand, we show that interestingly, in the absence of a requirement to model "spammers", one can construct reasonable objective functions for crowdsourcing that guarantee convex inference.