In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.

Example Input: We theoretically analyze and compare the following five popular multiclass classification methods: One vs. All, All Pairs, Tree-based classifiers, Error Correcting Output Codes (ECOC) with randomly generated code matrices, and Multiclass SVM. In the first four methods, the classification is based on a reduction to binary classification. We consider the case where the binary classifier comes from a class of VC dimension d, and in particular from the class of halfspaces over R. We analyze both the estimation error and the approximation error of these methods. Our analysis reveals interesting conclusions of practical relevance, regarding the success of the different approaches under various conditions. Our proof technique employs tools from VC theory to analyze the approximation error of hypothesis classes. This is in sharp contrast to most, if not all, previous uses of VC theory, which only deal with estimation error.
Example Output: Multiclass Learning Approaches: A Theoretical Comparison with Implications

Example Input: We report on a series of experiments with convolutional neural networks (CNN) trained on top of pre-trained word vectors for sentence-level classification tasks. We first show that a simple CNN with little hyperparameter tuning and static vectors achieves excellent results on multiple benchmarks. Learning task-specific vectors through fine-tuning offers further gains in performance. We additionally propose a simple modification to the architecture to allow for the use of both taskspecific and static word vectors. The CNN models discussed herein improve upon the state-of-the-art on 4 out of 7 tasks, which include sentiment analysis and question classification.
Example Output: Convolutional Neural Networks for Sentence Classification

Example Input: We study the power index voting game design problem for weighted voting games: the problem of finding a weighted voting game in which the power of the players is as close as possible to a certain target distribution. Our goal is to find algorithms that solve this problem exactly. Thereto, we consider various subclasses of simple games, and their associated representation methods. We survey algorithms and impossibility results for the synthesis problem, i.e., converting a representation of a simple game into another representation. We contribute to the synthesis problem by showing that it is impossible to compute in polynomial time the list of ceiling coalitions (also known as shift-maximal losing coalitions) of a game from its list of roof coalitions (also known as shift-minimal winning coalitions), and vice versa. Then, we proceed by studying the problem of enumerating the set of weighted voting games. We present first a naive algorithm for this, running in doubly exponential time. Using our knowledge of the synthesis problem, we then improve on this naive algorithm, and we obtain an enumeration algorithm that runs in quadratic exponential time (that is, O(2 2 · p(n)) for a polynomial p). Moreover, we show that this algorithm runs in output-polynomial time, making it the best possible enumeration algorithm up to a polynomial factor. Finally, we propose an exact anytime algorithm for the power index voting game design problem that runs in exponential time. This algorithm is straightforward and general: it computes the error for each game enumerated, and outputs the game that minimizes this error. By the genericity of our approach, our algorithm can be used to find a weighted voting game that optimizes any exponential time computable function. We implement our algorithm for the case of the normalized Banzhaf in∗Algorithms, Combinatorics and Optimization; Centrum Wiskunde & Informatica; The Netherlands; Email: keijzer@cwi.nl. †Algorithmics; Delft University of Technology; The Netherlands; Email: T.B.Klos@tudelft.nl. ‡Department of Econometrics; Erasmus University Rotterdam; The Netherlands; Email:
Example Output:
Solving Weighted Voting Game Design Problems Optimally: Representations, Synthesis, and Enumeration