Probable convexity and its application to Correlated Topic Models
Abstract
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.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2013
- DOI:
- 10.48550/arXiv.1312.4527
- arXiv:
- arXiv:1312.4527
- Bibcode:
- 2013arXiv1312.4527T
- Keywords:
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- Computer Science - Machine Learning;
- Statistics - Machine Learning
- E-Print:
- 22 pages