This paper proposes the Proximal Iteratively REweighted (PIRE) algorithm for solving a general problem, which involves a large body of nonconvex sparse and structured sparse related problems. Comparing with previous iterative solvers for nonconvex sparse problem, PIRE is much more general and efficient. The computational cost of PIRE in each iteration is usually as low as the state-of-the-art convex solvers. We further propose the PIRE algorithm with Parallel Splitting (PIRE-PS) and PIRE algorithm with Alternative Updating (PIRE-AU) to handle the multi-variable problems. In theory, we prove that our proposed methods converge and any limit solution is a stationary point. Extensive experiments on both synthesis and real data sets demonstrate that our methods achieve comparative learning performance, but are much more efficient, by comparing with previous nonconvex solvers.
Furthermore, previous iterative algorithms can only solve the problem with only one variable. They cannot be naively generalized to solve multi-variable problems. However, there are many problems involving two or more variables, e.g. stable robust principle component analysis @cite_0 and robust multi-task feature learning @cite_4 . So it is desirable to extend the iteratively reweighted algorithms for the multi-variable case.
@cite_0 @cite_4
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Realization of cooperative behavior in multi-agent system is important for improving problem solving ability. Reinforcement learning is one of the learning methods for such cooperative behavior of agents. In this paper, we consider pursuit problem for multi-agent reinforcement learning with communication between the agents. In our study, the agents obtain communication codes through learning. Here, the codes are rules for communicating appropriate information under various situations. We call the learning of communication codes signal learning. The signal is expressed by bit sequence, and its length is set to be variable. We carried out experiment for performance comparison with varying the signal length from 0 to 4 bits. As a result, it has been shown that, in learning precision, the case of 1 bit or more bits communication outperformed the case of no communication. It also has been shown that 4 bits communication produced the best result among the five cases, while learning with longer signals required much more iterations. In this paper, we consider the problem of multifunctional compression with side information. The problem is how we can compress a source X so that the receiver is able to compute some deterministic functions f1(X, Y1), ..., fm(X, Ym), where Yi, 1  i  m, are available at the receiver as side information. In [1], Wyner and Ziv considered this problem for the special case of m = 1 and f1(X, Y1) = X and derived a rate-distortion function. Yamamoto extended this result in [2] to the case of having one general function f1(X, Y1). Both of these results were in terms of an auxiliary random variable. For the case of zero distortion, in [3], Orlitsky and Roche gave an interpretation of this variable in terms of properties of the characteristic graph which led to a particular coding scheme. This result was extended in [4] by providing an achievable scheme based on colorings of the characteristic graph. In a recent work, reference [5] has considered this problem for a general tree network where intermediate nodes are allowed to perform some computations. These previous works only considered the case where the receiver only wants to compute one function (m=1). Here, we want to consider the case in which the receiver wants to compute several functions with different side information random variables and zero distortion. Our results do not depend on the fact that all functions are desired in one receiver and one can apply them to the case of having several receivers with different desired functions (i.e., functions are separable). We define a new concept named the multi-functional graph entropy which is an extension of the graph entropy defined by Korner in [6]. We show that the minimum achievable