Convex Calibration Dimension for Multiclass Loss Matrices

We study consistency properties of surrogate loss functions for general multiclass learning problems, defined by a general multiclass loss matrix. We extend the notion of classification calibration, which has been studied for binary and multiclass 0-1 classification problems (and for certain other specific learning problems), to the general multiclass setting, and derive necessary and sufficient conditions for a surrogate loss to be calibrated with respect to a loss matrix in this setting. We then introduce the notion of convex calibration dimension of a multiclass loss matrix, which measures the smallest `size' of a prediction space in which it is possible to design a convex surrogate that is calibrated with respect to the loss matrix. We derive both upper and lower bounds on this quantity, and use these results to analyze various loss matrices. In particular, we apply our framework to study various subset ranking losses, and use the convex calibration dimension as a tool to show both the existence and non-existence of various types of convex calibrated surrogates for these losses. Our results strengthen recent results of Duchi et al. (2010) and Calauzenes et al. (2012) on the non-existence of certain types of convex calibrated surrogates in subset ranking. We anticipate the convex calibration dimension may prove to be a useful tool in the study and design of surrogate losses for general multiclass learning problems.Comment: Accepted to JMLR, pending editin

Learning to Rank: Online Learning, Statistical Theory and Applications.

Learning to rank is a supervised machine learning problem, where the output space is the special structured space of emph{permutations}. Learning to rank has diverse application areas, spanning information retrieval, recommendation systems, computational biology and others. In this dissertation, we make contributions to some of the exciting directions of research in learning to rank. In the first part, we extend the classic, online perceptron algorithm for classification to learning to rank, giving a loss bound which is reminiscent of Novikoff's famous convergence theorem for classification. In the second part, we give strategies for learning ranking functions in an online setting, with a novel, feedback model, where feedback is restricted to labels of top ranked items. The second part of our work is divided into two sub-parts; one without side information and one with side information. In the third part, we provide novel generalization error bounds for algorithms applied to various Lipschitz and/or smooth ranking surrogates. In the last part, we apply ranking losses to learn policies for personalized advertisement recommendations, partially overcoming the problem of click sparsity. We conduct experiments on various simulated and commercial datasets, comparing our strategies with baseline strategies for online learning to rank and personalized advertisement recommendation.PhDStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133334/1/sougata_1.pd

On Consistent Surrogate Risk Minimization and Property Elicitation

Abstract Surrogate risk minimization is a popular framework for supervised learning; property elicitation is a widely studied area in probability forecasting, machine learning, statistics and economics. In this paper, we connect these two themes by showing that calibrated surrogate losses in supervised learning can essentially be viewed as eliciting or estimating certain properties of the underlying conditional label distribution that are sufficient to construct an optimal classifier under the target loss of interest. Our study helps to shed light on the design of convex calibrated surrogates. We also give a new framework for designing convex calibrated surrogates under low-noise conditions by eliciting properties that allow one to construct 'coarse' estimates of the underlying distribution

On Consistent Surrogate Risk Minimization and Property Elicitation

Abstract Surrogate risk minimization is a popular framework for supervised learning; property elicitation is a widely studied area in probability forecasting, machine learning, statistics and economics. In this paper, we connect these two themes by showing that calibrated surrogate losses in supervised learning can essentially be viewed as eliciting or estimating certain properties of the underlying conditional label distribution that are sufficient to construct an optimal classifier under the target loss of interest. Our study helps to shed light on the design of convex calibrated surrogates. We also give a new framework for designing convex calibrated surrogates under low-noise conditions by eliciting properties that allow one to construct 'coarse' estimates of the underlying distribution

On Connections Between Machine Learning And Information Elicitation, Choice Modeling, And Theoretical Computer Science

Machine learning, which has its origins at the intersection of computer science and statistics, is now a rapidly growing area of research that is being integrated into almost every discipline in science and business such as economics, marketing and information retrieval. As a consequence of this integration, it is necessary to understand how machine learning interacts with these disciplines and to understand fundamental questions that arise at the resulting interfaces. The goal of my thesis research is to study these interdisciplinary questions at the interface of machine learning and other disciplines including mechanism design/information elicitation, preference/choice modeling, and theoretical computer science