Fast Multiplier Methods to Optimize Non-exhaustive, Overlapping Clustering

Clustering is one of the most fundamental and important tasks in data mining. Traditional clustering algorithms, such as K-means, assign every data point to exactly one cluster. However, in real-world datasets, the clusters may overlap with each other. Furthermore, often, there are outliers that should not belong to any cluster. We recently proposed the NEO-K-Means (Non-Exhaustive, Overlapping K-Means) objective as a way to address both issues in an integrated fashion. Optimizing this discrete objective is NP-hard, and even though there is a convex relaxation of the objective, straightforward convex optimization approaches are too expensive for large datasets. A practical alternative is to use a low-rank factorization of the solution matrix in the convex formulation. The resulting optimization problem is non-convex, and we can locally optimize the objective function using an augmented Lagrangian method. In this paper, we consider two fast multiplier methods to accelerate the convergence of an augmented Lagrangian scheme: a proximal method of multipliers and an alternating direction method of multipliers (ADMM). For the proximal augmented Lagrangian or proximal method of multipliers, we show a convergence result for the non-convex case with bound-constrained subproblems. These methods are up to 13 times faster---with no change in quality---compared with a standard augmented Lagrangian method on problems with over 10,000 variables and bring runtimes down from over an hour to around 5 minutes.Comment: 9 pages. 2 figure

Real-Time Reinforcement Learning of Constrained Markov Decision Processes with Weak Derivatives

We present on-line policy gradient algorithms for computing the locally optimal policy of a constrained, average cost, finite state Markov Decision Process. The stochastic approximation algorithms require estimation of the gradient of the cost function with respect to the parameter that characterizes the randomized policy. We propose a spherical coordinate parametrization and present a novel simulation based gradient estimation scheme involving weak derivatives (measure-valued differentiation). Such methods have substantially reduced variance compared to the widely used score function method. Similar to neuro-dynamic programming algorithms (e.g. Q-learning or Temporal Difference methods), the algorithms proposed in this paper are simulation based and do not require explicit knowledge of the underlying parameters such as transition probabilities. However, unlike neuro-dynamic programming methods, the algorithms proposed here can handle constraints and time varying parameters. Numerical examples are given to illustrate the performance of the algorithms. This paper was originally written in 2004. One reason we are putting this on arxiv now is that the score function gradient estimator continues to be used in the online reinforcement learning literature even though its variance grows as $O(n)$ given $n$ data points (for a Markov process). In comparison the weak derivative estimator has significantly smaller variance of $O(1)$ as reported in this paper (and elsewhere)

Some Inexact Hybrid Proximal Augmented Lagrangian Algorithms

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