A PAC-Bayesian Margin Bound for Linear Classifiers: Why SVMs work

Neural Information Processing Systems

We present a bound on the generalisation error of linear classifiers in terms of a refined margin quantity on the training set. The result is obtained in a PAC-Bayesian framework and is based on geometrical arguments in the space of linear classifiers. The new bound constitutes an exponential improvement of the so far tightest margin bound by Shawe-Taylor et al. [8] and scales logarithmically in the inverse margin. Even in the case of less training examples than input dimensions sufficiently large margins lead to nontrivial bound values and - for maximum margins - to a vanishing complexity term.Furthermore, the classical margin is too coarse a measure for the essential quantity that controls the generalisation error: the volume ratio between the whole hypothesis space and the subset of consistent hypotheses. The practical relevance of the result lies in the fact that the well-known support vector machine is optimal w.r.t. the new bound only if the feature vectors are all of the same length. As a consequence we recommend to use SVMs on normalised feature vectors only - a recommendation that is well supported by our numerical experiments on two benchmark data sets. 1 Introduction Linear classifiers are exceedingly popular in the machine learning community due to their straightforward applicability and high flexibility which has recently been boosted by the so-called kernel methods [13]. A natural and popular framework for the theoretical analysis of classifiers is the PAC (probably approximately correct) framework[11] which is closely related to Vapnik's work on the generalisation error [12]. For binary classifiers it turned out that the growth function is an appropriate measureof "complexity" and can tightly be upper bounded by the VC (Vapnik-Chervonenkis) dimension [14].


Vector-Space Markov Random Fields via Exponential Families

arXiv.org Machine Learning

We present Vector-Space Markov Random Fields (VS-MRFs), a novel class of undirected graphical models where each variable can belong to an arbitrary vector space. VS-MRFs generalize a recent line of work on scalar-valued, uni-parameter exponential family and mixed graphical models, thereby greatly broadening the class of exponential families available (e.g., allowing multinomial and Dirichlet distributions). Specifically, VS-MRFs are the joint graphical model distributions where the node-conditional distributions belong to generic exponential families with general vector space domains. We also present a sparsistent $M$-estimator for learning our class of MRFs that recovers the correct set of edges with high probability. We validate our approach via a set of synthetic data experiments as well as a real-world case study of over four million foods from the popular diet tracking app MyFitnessPal. Our results demonstrate that our algorithm performs well empirically and that VS-MRFs are capable of capturing and highlighting interesting structure in complex, real-world data. All code for our algorithm is open source and publicly available.


Extrofitting: Enriching Word Representation and its Vector Space with Semantic Lexicons

arXiv.org Artificial Intelligence

We propose post-processing method for enriching not only word representation but also its vector space using semantic lexicons, which we call extrofitting. The method consists of 3 steps as follows: (i) Expanding 1 or more dimension(s) on all the word vectors, filling with their representative value. (ii) Transferring semantic knowledge by averaging each representative values of synonyms and filling them in the expanded dimension(s). These two steps make representations of the synonyms close together. (iii) Projecting the vector space using Linear Discriminant Analysis, which eliminates the expanded dimension(s) with semantic knowledge. When experimenting with GloVe, we find that our method outperforms Faruqui's retrofitting on some of word similarity task. We also report further analysis on our method in respect to word vector dimensions, vocabulary size as well as other well-known pretrained word vectors (e.g., Word2Vec, Fasttext).


Supervised Learning with Indefinite Topological Kernels

arXiv.org Machine Learning

Topological Data Analysis (TDA) is a recent and growing branch of statistics devoted to the study of the shape of the data. In this work we investigate the predictive power of TDA in the context of supervised learning. Since topological summaries, most noticeably the Persistence Diagram, are typically defined in complex spaces, we adopt a kernel approach to translate them into more familiar vector spaces. We define a topological exponential kernel, we characterize it, and we show that, despite not being positive semi-definite, it can be successfully used in regression and classification tasks.


Fast, smooth and adaptive regression in metric spaces

Neural Information Processing Systems

It was recently shown that certain nonparametric regressors can escape the curse of dimensionality in the sense that their convergence rates adapt to the intrinsic dimension of data (\cite{BL:65, SK:77}). We prove some stronger results in more general settings. In particular, we consider a regressor which, by combining aspects of both tree-based regression and kernel regression, operates on a general metric space, yields a smooth function, and evaluates in time $O(\log n)$. We derive a tight convergence rate of the form $n^{-2/(2+d)}$ where $d$ is the Assouad dimension of the input space.