Training Support Vector Machines Using Frank-Wolfe Optimization Methods

Training a Support Vector Machine (SVM) requires the solution of a quadratic programming problem (QP) whose computational complexity becomes prohibitively expensive for large scale datasets. Traditional optimization methods cannot be directly applied in these cases, mainly due to memory restrictions. By adopting a slightly different objective function and under mild conditions on the kernel used within the model, efficient algorithms to train SVMs have been devised under the name of Core Vector Machines (CVMs). This framework exploits the equivalence of the resulting learning problem with the task of building a Minimal Enclosing Ball (MEB) problem in a feature space, where data is implicitly embedded by a kernel function. In this paper, we improve on the CVM approach by proposing two novel methods to build SVMs based on the Frank-Wolfe algorithm, recently revisited as a fast method to approximate the solution of a MEB problem. In contrast to CVMs, our algorithms do not require to compute the solutions of a sequence of increasingly complex QPs and are defined by using only analytic optimization steps. Experiments on a large collection of datasets show that our methods scale better than CVMs in most cases, sometimes at the price of a slightly lower accuracy. As CVMs, the proposed methods can be easily extended to machine learning problems other than binary classification. However, effective classifiers are also obtained using kernels which do not satisfy the condition required by CVMs and can thus be used for a wider set of problems

Exploring the boundaries of vinylogous Mukaiyama aldol processes: stereoselective access to polyunsaturated homoallylic alcohols

Catalytic enantioselective vinylogous aldol reactions using extended enolates are of prominent value in synthetic organic chemistry. Here, we report our advances in the development of enantioselective bis-vinylogous and hyper-vinylogous Mukaiyama aldol reactions between a series of polyenylsilyloxy furans or polyenylsilyoxy indoles and aromatic aldehydes, realized by use of the enabling catalyst combination of silicon tetrachloride and Denmark’s chiral bis-phosphoramide base (R,R)-I. Several crucial issues such as the remote site-, enantio- and geometrical selectivity of the reaction will be highlighted, ultimately focusing on one main question: how far can we push the limits of the vinylogous reactivity transmittal?</i

Personalized Web Search via Query Expansion based on User’s Local Hierarchically-Organized Files

Users of Web search engines generally express information needs with short and ambiguous queries, leading to irrelevant results. Personalized search methods improve users’ experience by automatically reformulating queries before sending them to the search engine or rearranging received results, according to their specific interests. A user profile is often built from previous queries, clicked results or in general from the user’s browsing history; different topics must be distinguished in order to obtain an accurate profile. It is quite common that a set of user files, locally stored in sub-directory, are organized by the user into a coherent taxonomy corresponding to own topics of interest, but only a few methods leverage on this potentially useful source of knowledge. We propose a novel method where a user profile is built from those files, specifically considering their consistent arrangement in directories. A bag of keywords is extracted for each directory from text documents with in it. We can infer the topic of each query and expand it by adding the corresponding keywords, in order to obtain a more targeted formulation. Experiments are carried out using benchmark data through a repeatable systematic process, in order to evaluate objectively how much our method can improve relevance of query results when applied upon a third-party search engin

A Novel Frank-Wolfe Algorithm. Analysis and Applications to Large-Scale SVM Training

Recently, there has been a renewed interest in the machine learning community for variants of a sparse greedy approximation procedure for concave optimization known as {the Frank-Wolfe (FW) method}. In particular, this procedure has been successfully applied to train large-scale instances of non-linear Support Vector Machines (SVMs). Specializing FW to SVM training has allowed to obtain efficient algorithms but also important theoretical results, including convergence analysis of training algorithms and new characterizations of model sparsity. In this paper, we present and analyze a novel variant of the FW method based on a new way to perform away steps, a classic strategy used to accelerate the convergence of the basic FW procedure. Our formulation and analysis is focused on a general concave maximization problem on the simplex. However, the specialization of our algorithm to quadratic forms is strongly related to some classic methods in computational geometry, namely the Gilbert and MDM algorithms. On the theoretical side, we demonstrate that the method matches the guarantees in terms of convergence rate and number of iterations obtained by using classic away steps. In particular, the method enjoys a linear rate of convergence, a result that has been recently proved for MDM on quadratic forms. On the practical side, we provide experiments on several classification datasets, and evaluate the results using statistical tests. Experiments show that our method is faster than the FW method with classic away steps, and works well even in the cases in which classic away steps slow down the algorithm. Furthermore, these improvements are obtained without sacrificing the predictive accuracy of the obtained SVM model.Comment: REVISED VERSION (October 2013) -- Title and abstract have been revised. Section 5 was added. Some proofs have been summarized (full-length proofs available in the previous version