On the Impossibility of Convex Inference in Human Computation
This addresses a foundational issue in crowdsourcing and machine learning by clarifying theoretical limits on convex optimization for inference problems, which is incremental but important for algorithm design.
The paper tackles the problem of whether a convex objective function exists for joint inference of ground-truth answers and worker abilities in human computation, showing that under mild axioms, convexity is impossible, but it can be achieved if modeling spammers is not required.
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.