Probable convexity and its application to Correlated Topic Models
This addresses the problem of inefficient non-convex optimization for researchers and practitioners in machine learning, offering a new theoretical framework and algorithm, though it is incremental in applying convexity analysis to specific models.
The paper tackles the intractability of non-convex optimization in probabilistic modeling by introducing the concept of probable convexity, showing that inference in Correlated Topic Models can be concave under certain conditions, leading to a novel algorithm that is significantly more scalable and qualitative than existing methods.
Non-convex optimization problems often arise from probabilistic modeling, such as estimation of posterior distributions. Non-convexity makes the problems intractable, and poses various obstacles for us to design efficient algorithms. In this work, we attack non-convexity by first introducing the concept of \emph{probable convexity} for analyzing convexity of real functions in practice. We then use the new concept to analyze an inference problem in the \emph{Correlated Topic Model} (CTM) and related nonconjugate models. Contrary to the existing belief of intractability, we show that this inference problem is concave under certain conditions. One consequence of our analyses is a novel algorithm for learning CTM which is significantly more scalable and qualitative than existing methods. Finally, we highlight that stochastic gradient algorithms might be a practical choice to resolve efficiently non-convex problems. This finding might find beneficial in many contexts which are beyond probabilistic modeling.