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<p>Non-convex optimization has an important role in machine learning. However, the theoretical understanding of non-convex optimization remained rather limited. Studying efficient algorithms for non-convex optimization has attracted a great deal of attention from many researchers around the world but these problems are usually NP-hard to solve. In this paper, we have proposed a new algorithm namely GS-OPT (General Stochastic OPTimization) which is effective for solving the non-convex problems. Our idea is to combine two stochastic bounds of the objective function where they are made by a commonly discrete probability distribution namely Bernoulli. We consider GS-OPT carefully on both the theoretical and experimental aspects. We also apply GS-OPT for solving the posterior inference problem in the latent Dirichlet allocation. Empirical results show that our approach is often more efficient than previous ones.</p>

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The estimation of the posterior distribution is the core problem in topic models, unfortunately it is intractable. There are approximation and sampling methods proposed to solve it. However, most of them do not have any clear theoretical guarantee of neither quality nor rate of convergence. Online Maximum a Posteriori Estimation (OPE) is another approach with concise guarantee on quality and convergence rate, in which we cast the estimation of the posterior distribution into a non-convex optimization problem. In this paper, we propose a more general and flexible version of OPE, namely Generalized Online Maximum a Posteriori Estimation (G-OPE), which not only enhances the flexibility of OPE in different real-world situations but also preserves key advantage theoretical characteristics of OPE when comparing to the state-of-the-art methods. We employ G-OPE as inference a document within large text corpora. The experimental and theoretical results show that our new approach performs better than OPE and other state-of-the-art methods.