The cross entropy method for classification

Abstract

We consider support vector machines for binary classification. As opposed to most approaches we use the number of support vectors (the "L 0 norm") as a regularizing term instead of the L 1 or L 2 norms. In order to solve the optimization problem we use the cross entropy method to search over the possible sets of support vectors. The algorithm consists of solving a sequence of efficient linear programs. We report experiments where our method produces generalization errors that are similar to support vector machines, while using a considerably smaller number of support vectors.

A Tutorial on the Cross-Entropy Method

“…The CE method has been successfully applied to a diverse range of estimation and optimization problems, including buffer allocation [1], queueing models of telecommunication systems [14,16], optimal control of HIV/AIDS spread [48,49], signal detection [30], combinatorial auctions [9], DNA sequence alignment [24,38], scheduling and vehicle routing [3,8,11,20,23,53], neural and reinforcement learning [31,32,34,52,54], project management [12], rare-event simulation with light-and heavy-tail distributions [2,10,21,28], clustering analysis [4,5,29]. Applications to classical combinatorial optimization problems including the max-cut, traveling salesman, and Hamiltonian cycle 1…”

Critical factors in the empirical performance of temporal difference and evolutionary methods for reinforcement learning

“…Finally, Kalyanakrishnan and Stone [28] compare Sarsa and the cross-entropy method [41,71], another approach to policy search, in a simple navigation task. They study how the relative performance of these methods changes with respect to several domain characteristics, including sensor and effector noise.…”

“…Other evolutionary methods such as CoSyNE [21], EANT [29], and HyperNEAT [17], an extension to NEAT based on indirect encodings, also deserve closer empirical study. Beyond evolutionary methods, other policy search approaches such as the cross-entropy method [41,71] or policy gradient approaches [3,7,34,70] could be usefully compared with TD methods. Similarly, recent developments in making value function approximation more robust, e.g., least-squares policy iteration [36], fitted Q-iteration [54] and evolutionary function approximation [79], need to be thoroughly compared to the traditional function approximation approach used in this paper.…”

Deriving and improving CMA-ES with information geometric trust regions

“…This search distribution is used to generate a population of individuals which are evaluated by their corresponding fitness values. Subsequently, a new search distribution is computed by either computing gradient based updates [19], expectation-maximisationbased updates [7,12], evolutionary strategies [10], the cross-entropy method [14] or information-theoretic policy updates [1], such that the individuals with higher fitness will have better selection probability. The Covariance Matrix Adaptation -Evolutionary Strategy (CMA-ES) is one of the most popular stochastic search algorithms [10].…”

“…However, a unique mathematical framework that explains all these update rules is so far still missing. In contrast, expectation maximisation-based algorithms [7,12,14] (Section 2.2) optimize a clearly defined objective, i.e., the maximization of a lower-bound. The maximisation of lower bound in each iteration is equivalent to weighted maximum likelihood estimation (MLE) of the distribution.…”